Theory of Knowledge: Structures and Processes 9814522678, 9789814522670, 9789814522687, 9814522686, 263-307-309-3
This book aims to synthesize different directions in knowledge studies into a unified theory of knowledge and knowledge
524 99 6MB
English Pages 400 [950] Year 2017
Table of contents :
Content: Introduction: Knowledge as an Epistemic Structure, as a Vital Resource of Society and as an Object of Theoretical Studies
Knowledge Characteristics and Typology
Knowledge Evaluation and Justification
Knowledge Structure and Functioning: Microlevel or Quantum Theory of Knowledge
Knowledge Structure and Functioning: Macrolevel or Common Theory of Knowledge
Knowledge Structure and Functioning: Megalevel or Global Theory of Knowledge
Knowledge Acquisition, Representation and Application
Knowledge, Data and Information
Conclusion.
Citation preview
World Scientific Series in Information Studies —
Vol. 5
Theory of Knowledge Structures and Processes
Mark Burgin University of California, Los Angeles, USA
World Scientific
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Names: Burgin, M. S. (Mark Semenovich), author. Title: Theory of knowledge : structures and processes / Mark Burgin. Description: New Jersey : World Scientific, 2016. | Series: World Scientific series in information studies ; Volume 5 | Includes bibliographical references and index. Identifiers: LCCN 2015049963 | ISBN 9789814522670 (hc : alk. paper) Subjects: LCSH: Knowledge, Theory of. Classification: LCC BD161 .B865 2216 | DDC 121--dc23 LC record available at http://lccn.loc.gov/2015049963
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
Printed in Singapore
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Contents Preface
ix
Acknowledgments
xiii
About the Author
xv
1. Introduction
1
1.1. The role of knowledge in the contemporary society . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. A brief history of knowledge studies . . . . . . . . . . 9 1.3. Structure of the book . . . . . . . . . . . . . . . . . . 39 2. Knowledge Characteristics and Typology 2.1. The differentiation and classification of knowledge 2.2. Existential characteristics of knowledge . . . . . . 2.3. Descriptive properties of knowledge and corresponding typology . . . . . . . . . . . . . 2.3.1. Dimensions and other characteristics of knowledge . . . . . . . . . . . . . . . . . 2.3.2. Correctness, relevance, and consistency of knowledge . . . . . . . . . . . . . . . . . 2.3.3. Confidence in and certainty of knowledge . 2.3.4. Complexity and clarity of knowledge . . . . 2.3.5. Significance of knowledge . . . . . . . . . . 2.3.6. Efficiency of knowledge . . . . . . . . . . . 2.3.7. Reliability of knowledge . . . . . . . . . . .
45 . . 45 . . 77 . . 91 . . 94 . . . . . .
. . . . . .
96 119 122 131 134 136
2.3.8. Abstractness and generality of knowledge . 2.3.9. Completeness of knowledge versus precision of knowledge . . . . . . . . . . . . . . . . . 2.3.10. Meaning of knowledge . . . . . . . . . . . . 2.3.11. Other descriptive properties of knowledge . 2.4. Metaknowledge and metadata . . . . . . . . . . . .
. . 137 . . . .
3. Knowledge Evaluation and Validation in the Context of Epistemic Structures 3.1. Knowledge in the context of epistemic structures and knowledge scales . . . . . . . . . . . . . . . . 3.2. Knowledge evaluation, justification, and testing . 3.2.1. Knowledge evaluation . . . . . . . . . . . 3.2.2. Knowledge validation, justification, and testing . . . . . . . . . . . . . . . . . 3.3. Local consistency versus global consistency in knowledge representation . . . . . . . . . . . .
. . . .
139 140 149 151
169 . . . 170 . . . 215 . . . 215 . . . 240 . . . 263
4. Knowledge Structure and Functioning: Microlevel or Quantum Theory of Knowledge 4.1. Basic structures of knowledge units on the quantum level — knowledge quanta and semantic links . . . . 4.1.1. Quantum theory of knowledge (QTK) . . . . 4.1.2. Semantic link network theory (SLNT) and Semantic link theory of knowledge (SLTK) . 4.1.3. QTK–SLTK connection . . . . . . . . . . . . 4.2. Signs and symbols as quantum units of knowledge . 4.3. Operations with and relations between quantum knowledge units . . . . . . . . . . . . . . . . . . . . . 4.3.1. Properties of and relations between nodes and links in SLN and knowledge quanta in QTK . . . . . . . . . . . . . . . . . . . . . 4.3.2. Operations with extended knowledge quanta . . . . . . . . . . . . . . . . . . . . . 4.3.3. Operations with symbolic knowledge quanta and complete semantic links . . . . . . . . .
307 . 309 . 310 . 329 . 340 . 343 . 358
. 360 . 369 . 380
5. Knowledge Structure and Functioning: Macrolevel or Theory of Average Knowledge 5.1. Language as a universal tool for knowledge representation . . . . . . . . . . . . . . . . . . . . 5.1.1. Natural languages . . . . . . . . . . . . . 5.1.2. Languages of science and mathematics . . 5.1.3. Algorithmic and programming languages 5.2. Logic as a tool for knowledge representation and production . . . . . . . . . . . . . . . . . . . 5.2.1. Concepts, names, terms, and objects . . . 5.2.2. Statements, queries, and instructions . . 5.2.3. Logical systems of inference . . . . . . . . 5.3. Theory of abstract properties . . . . . . . . . . . 5.4. Semantic networks and ontology . . . . . . . . . 5.5. Scripts and productions . . . . . . . . . . . . . . 5.6. Frames and Schemas . . . . . . . . . . . . . . . . 6. Knowledge Structure and Functioning: Megalevel or Global Theory of Knowledge 6.1. A typology of structures and scientific knowledge 6.2. Nuclear and comprehensive knowledge systems . 6.3. Logic-linguistic knowledge system and descriptive knowledge . . . . . . . . . . . . . . . . . . . . . . 6.4. Model-representation knowledge system and representational knowledge . . . . . . . . . . . . 6.5. Procedural, axiological and instrumental knowledge systems, and operational knowledge . 6.6. Relations between and operations with global knowledge systems . . . . . . . . . . . . . . . . . 6.7. Hierarchies of knowledge systems . . . . . . . . . 7. Knowledge Production, Acquisition, Engineering, and Application
395 . . . .
. . . .
. . . .
402 403 411 423
. . . . . . . .
. . . . . . . .
. . . . . . . .
428 446 481 491 500 518 527 536
593 . . . 595 . . . 603 . . . 612 . . . 617 . . . 622 . . . 631 . . . 636
643
7.1. Knowledge production, learning, and acquisition as basic cognitive processes . . . . . . . . . . . . . . . 644
7.1.1. Scientific cognition . . . . . . . . . . . . . . 7.1.2. Intuition as a cognitive instrument . . . . . 7.1.3. Computers and networks as cognitive tools 7.1.4. Learning . . . . . . . . . . . . . . . . . . . 7.1.5. Knowledge creation in organizations . . . . 7.2. Knowledge organization and engineering . . . . . . 7.3. Knowledge management and application . . . . . .
. . . . . . .
. . . . . . .
8. Knowledge, Data, and Information
658 669 688 696 705 711 714 721
8.1. Epistemic structures and cognitive information 8.2. Structural aspects of knowledge–information duality . . . . . . . . . . . . . . . . . . . . . . . 8.3. Information as a source of knowledge . . . . . . 8.4. Dynamic aspects of knowledge, data, and information interaction . . . . . . . . . . . . . . 8.5. Knowledge as a measure of information . . . .
. . . . 722 . . . . 727 . . . . 760 . . . . 766 . . . . 791
9. Conclusion
803
Appendix
809
A. B. C. D. E.
Set theoretical foundations . . . . . . . Elements of the theory of algorithms . . Elements of algebra and category theory Numbers and numerical functions . . . . Topological, metric and normed spaces .
. . . . .
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809 819 825 831 833
Bibliography
837
Subject Index
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Preface If the extent of . . . knowledge is the hallmark of our civilization, the use to be made of it may be its crisis. S. Dilon Ripley An investment in knowledge pays the best interest. Benjamin Franklin
Knowledge has always been important in society and all educated people have always understood importance of knowledge. That is why Western philosophers have studied knowledge as an important phenomenon from the time of Plato and Aristotle. Thinkers from other countries, such as China and India, also tried to understand the essence of knowledge from ancient times. In contemporary society, importance of knowledge is much higher and continues to grow very fast. Researchers concluded that knowledge had become the key strategic asset for the 21st century and for every organization. Consequently, the necessity in developing the best strategy for identifying, developing, and applying the knowledge assets has become critical. Every organization needs to invest in creating and implementing the best knowledge networks, processes, methods, tools, and technologies. The growing needs in knowledge and efficient knowledge organization intensified studies of knowledge. There are three main directions
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in these studies: — The philosophical and methodological direction, which comprises epistemology and the methodology of science and mathematics. — The area of artificial intelligence (AI), in which knowledge is perceived as the base of intelligence. — The field of knowledge management where knowledge is treated as the main asset of companies and organizations. AI is typically directed at knowledge representation and processing. Epistemology is largely interested in knowledge definition and acquisition (cognition). Knowledge management is mostly concerned with knowledge organization and utilization. In addition, knowledge is also explored in psychology, sociology, and linguistics. Intensification of studies in area of knowledge brought forth a quantity of books on a variety of issues and problems of knowledge. So, why is this book different? It is different because its main goal is to present, organize and synthesize the basic ideas, results, and concepts from these three directions, which are loosely related now, into a unified theory of knowledge and knowledge processes. It is called the synthetic theory of knowledge. It is multidisciplinary and transdisciplinary at the same time. The approach presented in this book provides a new explanation of important relations between knowledge and information demonstrating new kinds of possibilities for knowledge management, information technology, data mining, information sciences, computer science, knowledge engineering, psychology, social sciences, genetics, and education that are made available by the synthetic theory of knowledge. Explanation of knowledge essence, structure and functioning is given in this book, as well as answers to the following questions: — How knowledge is related to information and data? — How knowledge is modeled by mathematical and logical structures?
— How these models are used to better understand and utilize computers and Internet, cognition and education, communication and computation? Knowledge is inseparable from information. People acquire knowledge receiving cognitive information. At the same time, knowledge, by its essence, contains information and this is the main feature of knowledge. This intrinsic unity of knowledge and information forms the base of the synthetic theory of knowledge.
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Chapter 1
Introduction All men by nature desire knowledge. Aristotle
There is an abundance of different books and papers treating various problems and studying different issues of knowledge (cf., for example, (Aune, 1967; Polanyi, 1974; Cleveland, 1985; Chisholm, 1989; Bloor, 1991; Burgin, 1997; Boisot, 1998; Choo, 1998; Rao, 1998; Pollock and Cruz, 1999; Bernecker and Dretske, 2000; Bean and Green, 2001; Popper, 2002; Goldman, 2004; Dalkir, 2005; Leydesdorff, 2006; Magnani, 2007; Nguen, 2008; Fantl and McGrath, 2009; Zhuge, 2012)). A lot of ideas, models, and several theories have been suggested in this area. The whole area of knowledge related activities consists of three parts: 1. Knowledge studies (theoretical and experimental). 2. Knowledge engineering. 3. Knowledge utilization and management. The two latter parts belong to knowledge technology — knowledge engineering deals with technology of knowledge production, organization, transformation, management, preservation, capture and acquisition, while knowledge utilization studies how people and organizations use knowledge, developing new techniques and approaches for this purpose.
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There are three types of knowledge theories: 1. Philosophical theories comprised by the philosophical discipline called epistemology are interested in three fundamental problems: (1) knowledge definition, i.e., trying to find what knowledge is and how to separate knowledge from beliefs; (2) limits of knowledge acquisition, i.e., what it is possible to know; and (3) ways of knowledge creation and acquisition, i.e., how knowledge is obtained. 2. Mathematical theories include mathematical logic, which provides means for formal knowledge representation and formation; theory of algorithms, which provides means for knowledge transformation and preservation dealing mostly with procedural or operational knowledge (cf., Chapter 6); and mathematical linguistics, which studies informal knowledge representation and formation. 3. Empirical theories are oriented at the practice of knowledge functioning, including theories of many disciplines, such as artificial intelligence, knowledge management, knowledge bases, cognitology, knowledge acquisition, cognitive psychology, cognitive neuroscience, cognitive anthropology, cognitive sociology, education, and the sociology of knowledge. Experimental exploration of knowledge emerged in ancient times. A brilliant example of such an experimentation is presented in the Plato dialogue Theætetus describing how Sokrates and Theaetetus discuss and investigate the essence and nature of knowledge. For a long time, people used mental experiments for knowledge studies. With the advance of computers, computer experiment has become crucial in AI and knowledge management. Besides, various experiments have been conducted with physical carriers of knowledge. For instance, psychologists, educators and sociologists organized various experiments examining how people acquire, store and disseminate knowledge. All research in the area of knowledge can be divided into three directions: • Structural analysis of knowledge strives to understand how knowledge is built and what properties it has.
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Introduction
• Axiological analysis of knowledge aims at explanation of those features that are primary for knowledge as a social and technological phenomenon. • Functional analysis of knowledge tries to find how knowledge functions, how it is produced and acquired. Structural analysis of knowledge is the main tool for the system theory of knowledge, knowledge bases, and artificial intelligence (AI). Axiological analysis of knowledge is the core instrument for the philosophy of knowledge, psychology, and social sciences, including the sociology of knowledge, which is the study of the relationship between human creativity and the social context within which it arises, of the effects knowledge has on individuals, organizations and societies dealing with broad fundamental questions, of the extent and limits of social influences on cognition, and of the social and cultural foundations of knowledge about the world. Functional analysis of knowledge is the key device for epistemology, knowledge engineering, and cognitology. 1.1. The role of knowledge in the contemporary society Knowledge is power. Francis Bacon
To survive and to prosper, people have always needed knowledge. Through the ages, philosophers contemplated problems of knowledge and cognition. The importance of knowledge has grown all the time and now active knowledge assets become crucial. This is true for all levels of society. Simply to function in the contemporary society, any individual needs some basic knowledge. Many organizations feel obliged to run their business based on efficient knowledge management just to keep up. More and more people and organizations are coming to the understanding that the optimal generation, acquisition, and application of knowledge is the key to success. Although the role of knowledge in the economy is not new, in recent years, knowledge has gained increased importance, both
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quantitatively and qualitatively, due to the development and utilization of information processing and communication technologies (Foray, 2004). The main roles of knowledge are (Tuomi, 1999): a resource, a product, and a restriction. Indeed, knowledge is clearly the primary resource in the technologically advanced industries, such as the computer, communication and software industries, and other knowledge-intensive industries, such as pharmaceuticals, but it is fast becoming the primary source of wealth in more traditional sectors of the economy as well (Stata, 1989). It is also estimated that knowledge now accounts for approximately three-fourths of the value increase in the manufacturing sector (Stewart, 1997). At the same time, in contrast to many other resources, people can produce knowledge, which now plays the role of a product. As a result, importance of knowledge production and creation grows very fast. Governments and other organizations invest more and more into knowledge production. Knowledge has become an intellectual property, attached to a name or group of names and certified by copyright, or some other form of social recognition, e.g., publication or awarding prizes (Granstrand, 1999). As an economical commodity, knowledge and knowledge production are paid for in the research, communication, and educational areas. As the result, knowledge has moved to the social overhead investment of society in the form presented in books, articles, patents or computer programs, written down, printed or recorded at some point for transmission and utilization (Bell, 1973). Our civilization is based on knowledge and information processing. In contemporary knowledge-driven economy, organizations ultimately gain their value from intellectual and knowledge-based assets rather than material commodities. That is why it is so important to know properties of knowledge and how to work with it. For instance, the principal problem for computer science as well as for computer technology is to process not only data but also knowledge. Knowledge processing and management make problem solving much more efficient and are crucial (if not vital) for big companies and institutions (Ueno, 1987; Osuga, 1989; Dalkir, 2005). To achieve this goal, it is necessary to make a distinction between knowledge and
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knowledge representation to know regularities of knowledge structure, functioning and representation, and to develop software (and in some cases, hardware) that is based on theses regularities. Many intelligent systems search knowledge spaces, which are explicitly or implicitly predefined by the choice of knowledge representation. In effect, the knowledge representation serves as a strong bias. People increasingly rely on AI processing systems, which in turn, depend on their software, while information is processed in the search of knowledge. Sophisticated safety-critical software is embedded in a diversity of systems across most industry sectors, ranging from automotive and aerospace to energy and maritime (Kandel and Dick, 2005). This situation once more demonstrates importance of knowledge because software is a form of operational knowledge representation. At the same time, the National Institute of Standards and Technology (NIST) reported that low quality software costs the U.S. economy almost $60 billion per year (Tassey, 2002; Thibodeau, 2002). Besides, only one quarter of software projects are judged a success (Standish Group). Software defects are accepted as inevitable by both the software industry and the long-suffering user community. In any other engineering discipline, this defect rate would be unacceptable. Moreover, when safety and security are at stake, the extent of current software vulnerability also becomes unsustainable (Croxford and Chapman, 2005). Therefore, validation of operational knowledge in the form of software has become an urgent task for contemporary society. In our time, importance of knowledge has grown very fast with the advancement of society. Thus, in the 20th century, with the advent of computers, knowledge has become a concern of science. As a result, now knowledge is studied in such areas as AI, computer science, dataand knowledge bases, global networks (e.g., the Internet), information science, knowledge engineering, and knowledge management. Philosophers also continue their studies of knowledge (Chisholm, 1989). However, knowledge is not an easy concept to understand. As Land et al. (2007) write, knowledge is understood to be a slippery
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concept, which has many definitions. This is apparent in the many questions philosophers and other thinkers ask themselves about the essence, distinctive characteristics, functions and roles of knowledge in society. These questions can vary from theoretical considerations to practical applications. For instance, relations between knowledge and information are blurred in contemporary society. Some comprehend knowledge as a kind of information (cf., for example, (Osuga and Saeki, 1990; Davenport, 1997; Probst et al., 1999; Gundry, 2001; Stenmark, 2002; Dalkir, 2005)), while others claim that information is a kind of knowledge (cf., for example, (Kogut and Zander, 1992; Tuomi, 1999)). In addition, there are opinions that information and knowledge are essentially different essences (cf., for example, (Davenport and Prusak, 1998; Lenski, 2004; Burgin, 2010)). All basic questions about knowledge are related to the way in which we organize and direct the development and application of knowledge on different levels — from individuals through companies and organizations through the whole society. For instance, in many organizations, knowledge management has come to occupy a central place in their functioning. It is a role that makes great demands on an organization’s strategic insight, problem solving ability, and successful development. As Kalfoglou et al. (2004) write, managing knowledge is a difficult and tricky enterprise. A wide variety of technologies have to be invoked in providing support for knowledge requirements, ranging from the acquisition, modeling, maintenance, transmission, dissemination, retrieval, reuse, and publishing of knowledge. Knowledge is a valuable asset and resource. So, any toolset capable of providing support for operating with knowledge would be valuable as its effects can percolate down to all the application domains structured around the domain representation. To reflect importance of knowledge, the term knowledge society was coined as a description of the contemporary society by its pivotal characteristic. Some researchers suggest that knowledge society is the next stage of the information society. In essence, every society has its own knowledge assets. However, in our times, knowledge
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together with information is becoming the key tool not only for further development but also for present survival in conditions of the knowledge economy. To describe the role of knowledge in contemporary society, Fritz Machlup (1902–1983) introduced the concept knowledge economy in the book (Machlup, 1962). The knowledge economy is a particular knowledge-driven stage of economical development, based on knowledge, succeeding a phase based on physical assets such as workforce, energy, and matter. Knowledge is in the process of taking the place of the workforce and other resources making possible getting better results with less workforce and other resources. Knowledge is substance and money substitutable, meaning that knowledge can replace, to some extent, capital, labor, or physical materials. Namely, knowledge allows one to use less money, labor, or physical materials than it is possible to do without this knowledge. As a result, the created wealth is measured less by the output of work itself but more and more by the general level of scientific and technological development (Jaffe and Trajtenberg, 2002). Amidon explained that knowledge about how to produce different products and provide services as well as their embedded knowledge is often more valuable than the products and services themselves or the materials they contain (Amidon, 1997). That is why Machlup (1962) defined knowledge as a commodity, developing techniques for measuring the magnitude of its production and distribution within a modern economy. He correctly assumed that all devices involved in knowledge production, dissemination, and utilization have to be taken into account in these measurements. A diversity of activities linked to research, education, and services, tend to assume increasing importance in the knowledge economy. Besides, the importance of knowledge in economic activity is not confined to the high-tech sectors but also pervades modes of organization of production and commerce in apparently low-tech sectors, which have also been essentially transformed. Toffler explains that knowledge is a wealth and force multiplier, in that it augments what is available or reduces the amount of resources needed to achieve a
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given purpose (Toffler, 1990). Stewart calls knowledge the intellectual capital (Stewart, 2002). Many researchers, economists, authors, governments, policymakers, international organizations, and think tanks declare that people now live in a knowledge-based economy as knowledge is the basis for various decisions in different areas, as well as a priceless asset to individuals and organizations. Moreover, few concepts introduced by economists have been more successful than that of a knowledgebased economy reflecting a qualitative transition in economic conditions (Foray and Lundvall, 1996; Leydesdorff, 2006a). To represent and study this new situation, the economical triple helix of university–industry–government relations was introduced (Etzkowitz and Leydesdorff, 1995; 1997; 1998; Leydesdorff, 2006; 2006a). Governance is treated as the force that instantiates and organizes systems in the socio-geographical dimension of the model. Industry is the main mover of material production and exchange, while academe plays the leading role in the organization of the knowledge production function. As a result, knowledge production and exchange becomes an economy in itself (Foray, 2004) and the development of a knowledge base turns out to be essentially dependent on the condition that knowledge production is socially organized and regulated. Naturally, the global economy now places much greater value on knowledge production and dissemination activities such as design with an emphasis on Research and Development including patenting, on education and on information effort such as marketing, networking, computation, and communication. Information is a source for knowledge, while knowledge is a base for producing and retrieving information. Naturally, importance of knowledge grows very rapidly as society becomes more and more advanced. As a result, in the 20th century, with the advent of computers, knowledge has become a concern of science and now knowledge is studied in such areas as AI, computer science, data and knowledge bases, global networks (e.g., Internet), information science, knowledge engineering, and knowledge
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management. Philosophers also continue their studies of knowledge (Chisholm, 1989).
1.2. A brief history of knowledge studies Some people drink deeply from the fountain of knowledge. Others just gargle. Grant M. Bright
Knowledge has been always important in society. That is why the best minds have been concerned with the problem of knowledge from ancient times. Studies of knowledge formed one of the pivotal philosophical disciplines, which is called epistemology from Greek words episteme, which means knowledge, and logos, which means cognition, study or reason. In other words, epistemology is the philosophical theory of knowledge and cognition. In this section, we give a very brief exposition of the epistemological research presenting approaches of some leading philosophers in the history of the human civilization and starting with the most ancient explorations and ideas. In Upanishad, which is one of the principal classical texts in Indian culture written from the end of the second millennium B.C.E. to the middle of the second millennium C.E., two kinds of knowledge, higher knowledge and lower knowledge, were discerned. Later Nyaya school of Hindu philosophy considered four types of knowledge acquisition: perception when senses make contact with an object, inference, analogy, and verbal testimony of reliable persons. Inference was used in three forms: a priory inference, a posteriory inference, and inference by common sense. In general, theory of knowledge has a long-standing tradition in Indian philosophy with many achievements and interesting insights. Let us get some glimpses on this big knowledge field developed in ancient India. In his book “Theories of knowledge”, Rao presents eight directions in the philosophical and methodological studies of knowledge in India
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(Rao, 1998): — — — — — — — —
Samkhya (Yoga) theory of knowledge Vedantins’ theories of knowledge Visistadvaita theory of knowledge Madhva theory of knowledge Mimansaka theories of knowledge Jaina theory of knowledge Buddhist theories of knowledge Logician’s (Nyaya) theory of knowledge
The Samkhya (Yoga) theory of knowledge Samkhya, also Sankhya, S¯ am ankhya, ˙ is one of the most . khya, or S¯ prominent and one of the oldest directions in Indian philosophy. It belongs to the six basic schools of the classical Indian philosophy. Bhagavad Gita identifies Samkhya with understanding of knowledge. The word Samkhya is based upon the Sanskrit word samkhya which means ‘number’ or ‘perfect knowledge’. An eminent, great sage Kapila (between 8th and 6th B.C.E.) was the founder of the Samkhya philosophy. Samkhya may be characterized as a dualistic realism. It is dualistic because it advocates two ultimate realities: Prakriti, matter and Purusha, self, spirit or consciousness. At the same time, Samkhya is a kind of realism as it considers that both matter and spirit are equally real. In addition, Samkhya is pluralistic because it is teaching that Purusha is not one but many. Samkhya has a developed theory of knowledge discerning three sources of valid knowledge: perception, inference based on Sankhya syllogism and valid testimony. The procedure of knowledge acquisition starts when the sense-organs come in contact with an object causing sensations and impressions to come to the manas (mind). The manas processes these impressions into proper forms and converts them into definite percepts. These percepts are carried to the Mahat (intellect) inducing changes in Mahat, and Mahat takes the form of the object, from which these sensations come. This transformation
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of Mahat is known as vritti or modification of buddhi. As Mahat is a physical entity, the process of knowledge formation is not complete. Thus, the consciousness of the Purusha (self) transforms Mahat producing in it consciousness of the form of the object, from which these sensations come. To better explain this, the following analogy is used. A mirror cannot produce an image by itself. It needs light to reflect and produce the image and thereby reveal the object. In a similar way, Mahat needs the “light” of the consciousness of the Purusha to produce knowledge. Besides, Samkhya discerns two types of perceptions: indeterminate (nirvikalpa) perceptions and determinate (savikalpa) perceptions. Indeterminate perceptions are like pure sensations or crude impressions containing no knowledge of the form or the name of the object. There is only vague awareness about an object. Determinate perceptions are the mature form of perceptions obtained from sensations, which have been processed, categorized and interpreted properly. In turn, determinate perceptions generate knowledge by inference based on analogy. Samkhya is related to Yoga, which is a specific religious system within Hinduism emerging from the older Samkhya system. The theoretical part of Yoga, i.e., its philosophy, was derived almost entirely from Samkhya. The Vedanta Theory of knowledge Vedanta is one the most prominent and philosophically advanced six basic schools of the classical Indian philosophy. According to Balasubramanian (2000), the Vedantic philosophy is as old as the Vedas, since the basic ideas of the Vedanta systems are derived from the Vedas during the Vedic period (1500–600 B.C.E.). The term veda means “knowledge” and the term anta means “end”. Thus, Vedanta means complete knowledge of the Veda. Originally, Vedanta denoted the Upanishads, a collection of foundational texts in Hinduism considered as the final layer of the Vedic canon. By the 8th century, the meaning of Vedanta changed for standing for all philosophical
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traditions concerned with interpreting the three basic texts of Hinduist philosophy, namely, the Upanishads, the Brahma Sutras, and the Bhagavad Gita. There are at least 10 schools of Vedanta as the system of philosophy that further develops the implications in the Upanishads that all reality is a single principle, Brahman, teaching that the believer’s goal is to transcend the limitations of self-identity and achieve unity with Brahman. According to the Vedanta Theory of knowledge, Brahman is selfindulgent and knowledge is not different from Brahman. Therefore, knowledge is eternal and without beginning. However, ignorance also exists until it is destroyed by knowledge. Although knowledge is without beginning, the state of knowing is produced by mental modification (Vrtti) of the internal organ (Abhivyanjaka). The Vrtti is four-fold consisting of doubt, definite knowledge, egoism, and recollection. Knowledge is produced with the help of two causes, the material cause (Upadana) and the efficient cause (Nimitta). The Vedanta Theory discerned two types of knowledge: the mediate knowledge (Paroksa) and the immediate knowledge (Aparoksa). An example of mediate knowledge is the statement “Brahman is”, while an example of immediate knowledge is the statement “I am Brahman is” (cf., (Rao, 1998)). Here is another example. The statement “I see fire” is immediate knowledge, while “I see smoke, so there is fire” is mediate knowledge. It might be interesting to compare this knowledge classification with a similar classification of Kant who considered knowledge of two kinds: intuitions as immediate knowledge and concepts as mediate knowledge. The Visistadvaita theory of knowledge Visistadvaita is a philosophy of religion, in which the central idea is integration and harmonization of all knowledge, while knowledge, jnaana, is obtained through sense perception, inference, and revelation. According to the Upanisads, knowledge comes from Brahman as “he who knows the Brahman attains the highest”. This asserts unity
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of the threefold system of Vedantic wisdom known as tattva, hita, and purusartha. Answering the basic question of epistemology about the origin and possibility of knowledge, Visistadvaita affirms possibility of getting knowledge about reality stating that people can know things as they are. Knowledge essentially presupposes a knowing self and an object of thought and is obtained in the process of ascent from the corresponding sensation to the self. Namely, this process starts with sensations, which form the raw material of knowledge and become percepts by action of the “a priori” form prescribed by the mind. The perceived objects are conceived and arranged by the synthetic mind or understanding which brings together the perceived objects producing judgments. Then reason unifies these judgments and forming conception in the self as the synthetic unity of knowledge. This shows that knowledge is not a plain synthetic construction, but originates in a process by which things are revealed. The objects in nature exist by themselves and are not created by thought, which only reveals them. Thus, knowledge is the self-revelation of a real object as a holistic system, while the object is not the copy of the idea, nor is the idea the archetype of the object, neither is deduced from the other. The Visistadvaita theory of knowledge assumes the integrity of experience on all its levels and forms, which constitute pratyaksa (perception), anumana (inference), and sastra (scripture). As a result, Visistadvaita is a dualistic philosophy assuming independent existence of the perceiving self, and of the external world that is perceived. The Madhva theory of knowledge The Dvaita or “dualist” school of Hindu Vedanta philosophy originated by Sri Madhvacarya, or Madhva (ca. 1238–1317), who considered himself an avatara of the wind-god Vayu and taught the fundamental difference between the individual self or Atman and the ultimate reality, Brahman. Thus, according to Madhva, there are three orders of reality: (1) the independent ultimate reality, Brahman;
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and the dependent reality, paratantra, which consists of (2) souls (jivas), and (3) lifeless objects (jada). Madhva’s pluralistic ontology is founded on his realistic epistemology. He argues that God and the human soul are separate because our daily experience of separateness from God and of plurality in general is given to people as an undeniable fact, fundamental to our knowledge of all things. Madhva considered two means of valid knowledge (Pramana): valid knowledge itself (Kevala Pramana), and the instrument of knowledge (Anupramana). In turn, Anupramana consists of three sources of knowledge: sense perception (Pratyaksha), inference (Anum¯ ana), and testimony of Vedic literature (Aagama) (Sharma, 1994). Further, existence of invalid knowledge acquired by sense perception demands permanent questioning of the knowledge content. The Mimansaka theories of knowledge M¯ım¯ am a is a Sanskrit word meaning “revered thought”. It is also the . s¯ name of one of the six astika (orthodox) schools of Hindu philosophy based on the Vedas. Its core tenets are ritualism, anti-asceticism, and anti-mysticism. The central aim of the school is explanation of the nature of dharma to maintain the harmony of the universe and provide the personal well-being of the person who follows ritual obligations and prerogatives. The Mimamsa school traces the source of the knowledge of dharma neither to sense-experience nor inference, but to verbal cognition (knowledge of words and meanings). In order to understand the correct dharma for specific situations, it is necessary to rely on examples of explicit or implicit commands in the Vedic texts. An implicit command must be understood by studying parallels in other, similar passages. If one text does not provide details for how a priest should proceed with a particular action, the details must be sought in other, related Vedic texts. This preoccupation with precision and accuracy required meticulous examination of the structures of sentences conveying commands, and led to an extensive exegesis of the Vedas and a detailed analysis of semantics.
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The Mimamsa made notable contributions to Indian thought in the fields of logic and epistemology. The Mimamsa doctrine of knowledge affirms that the world is real. Mimamsa introduced two additional means of valid knowledge in addition to the four traditional means of perception, inference, comparison and testimony, recognized by other schools of Hinduism. They are arthapatti (pre-conception or postulation) and abhava (absence, negation, nonexistence). Mimamsa advanced the unique epistemological theory that all cognition is valid. All knowledge is true, until it is superseded by further cognition. What is to be proved is not the truth of a cognition, but its falsity. Mimamsakas drew on this theory of validity to establish the unchallengeable validity of the Vedas.
The Jaina theory of knowledge The concept of soul is central in philosophy. Knowledge (Jnana) according to Jainas, is the soul’s intrinsic, inherent, inseparable, and inalienable attribute, without which no soul can exist. Knowledge plays an important part in the conception of soul and its emancipation. As a result, Jain epistemology or Jain theory of knowledge thus becomes vital in Jaina philosophy including the theory of knowledge along with various topics such as psychology, teaching about feelings, emotions, and passions, theory of causation, logic, philosophy of non-absolutism, and the conditional mode of predication (Shah, 1990). Consciousness (Cetana), according to Jainas, is the power of the soul knowledge and operates through understanding (Upyoga). It gets experience in three ways: (1) some experience is the fruit of karma; (2) other experience comes from activity of the soul; and (3) one more kind of experience is knowledge itself (Shah, 1990). According to Jaina thinkers, Cetana (consciousness) culminates in pure and perfect knowledge and knowledge itself has grades and modes. In turn, understanding (Upyoga) is divided into two: sensation (Darsana) and Cognition (Jnana). Uma Svati says: “Understanding is the distinguishing characteristic of the soul. It is of two sets — Jnana and
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Darsana. The first is of eight kinds and the second, of four” (Shah, 1990). Namely, sensation (Darsana) is of four kinds: • • • •
Visual (Cakshusa) Non-visual (Acakshusa) Clairvoyant (Avadhi Dersana) Pure (Kevala)
Each piece of knowledge is experienced with reference to its characteristic (Dharma) and its substratum (Dharmin). In addition, Jainas discerned two kinds of knowledge: direct knowledge and indirect knowledge. Direct knowledge does not demand the medium of another knowledge in contrast to indirect knowledge. According to Jainas, it is possible to obtain indirect knowledge by five techniques: recollection, recognition, Reductio ad Absurdum (Tarka), inference, and syllogism. The Buddhist theories of knowledge Being a strict empiricist, Siddhartha Gautama Sakyamuni (the “Buddha” or “awakened one”) believed that people can have knowledge of only those things that can be directly experienced. It is impossible to achieve ultimate knowledge until the follies and weakness of human life bring one to despair. That is why Buddha famously refused to answer ultimate questions such as “Does the world have a beginning or not?”, “Does God exist?”, and “Does the soul perish after death or not?”. Later, Buddhists developed a technique of denying all the logically alternative answers to such questions. For instance, the answer to the first question has to be: “No, the world does not have a beginning, it does not fail to have a beginning, it does not have and not have a beginning, nor does it neither have nor not have a beginning”. Knowledge in the Buddhist understanding is of prime importance to people. One of the principles of Buddhist philosophy instructs that the pleasure of advancing knowledge becomes a duty. Theory of knowledge in Buddhism is not treated as relative but is presumed to be perfectly true and absolute.
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With respect to their ontological assumptions, Buddhist religious directions are separated into four classes (Rao, 1998): — Madhyamika presupposes that the entire world is void, everything is fleeting and all activity goes in the dream state. — Yogasaras hold that there are no external objects in the world asserting that the object cognized and the cognizing person are the same. — Sautrantikas admit existence of the objective world, which cannot be perceived by senses but it is only inferred. — Vaibhastika admits existence of the objective world but rejects existence of objects of inference claiming that only indeterminate knowledge is valid. As reasoning is an important procedure in knowledge acquisition, three features of reason are explicated and utilized: — Existence only in the subject (Paksa). — Existence in the homologue (Sapaksa). — Non-existence only in the heterologue (Vipaksa). In addition, reason in the Buddhist theory of knowledge has three types: — Non-cognition (Anupalabdhi). — Cause in itself (Svabhava). — Effect (Karya). Besides, the Buddhist theory of knowledge uses four forms of predication: 1. S is P , e.g., “a square is a rectangle” or “there is a world of ideas”. 2. S is not P , e.g., “a square is not a circle” or “there is no world of ideas”. 3. S is and is not P , e.g., “a ball that is partially green and partially yellow is green and is not green” or “there is and is no world of ideas”.
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4. S neither is nor is not P , e.g., “a ball neither is green nor it is not green” or “the world of ideas neither is real nor it is not real”. The Buddhists assume that at least one of these alternatives is always true in any meaningful situation and use this assumption for logical classification. However, when the question is considered meaningless, all four alternatives are rejected. At the same time when the answer is ‘yes’ to each of the alternatives, it was treated as misleading and all four alternatives are also excluded. The Logician’s (Nyaya) theory of knowledge In the Logician’s theory of knowledge, knowledge (Buddhi or J˜ n¯ ana) is a special property of the soul, while mind (Manas) is a separate substance (Rao, 1998). Knowledge is obtained by experience (Anubhava) and recollection (Smrti). In turn, experience gives (is) twofold — valid knowledge (Yatharthanubhava or Prama) and invalid knowledge (Ayatharthanubhava or Bhrama). There are four ways for getting valid knowledge (Yatharthanubhava or Prama): — — — —
Perception gives perceptual knowledge (Pratyksa). Inference (Anumana) gives inferential knowledge (Anumiti). Analogy (Upamana) gives analogical knowledge (Upamiti). Utilization of language (verbal testimony) gives verbal knowledge (Sabda).
According to Gautama, there are four factors involved in direct perception (Pratyksa): — — — —
the the the the
senses (indriyas). sensual objects (artha). contact of the senses and the objects (sannikarsa). cognition produced by this contact (jnana).
In addition, the Nyaya believed that the five sense organs — eye, ear, nose, tongue, and skin — have the five elements — light, ether, earth, water, and air — as their field, with corresponding qualities of color, sound, smell, taste, and touch.
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According to logicians, there are also three ways for getting invalid knowledge (Ayatharthanubhava or Bhrama): — Doubt gives (is) uncertain knowledge (Samsaya). — Wrong reasoning gives invalid knowledge (Viparyaya). — Reductio ad absurdum gives (is) invalid knowledge (Tarka). Tarka includes: — — — — —
Faults of self-dependence (Atmasraya). Faults of mutual dependence (Anyonyasraya). Faults of dependence on a cycle (Cakrakasraya). Faults of infinite regress (Anavastha). Statements of undesirable effects (Anistaprasanga).
Inference (Anumana) is knowledge from the perceived about the unperceived and this relation may be of three sorts: — the inferred constituent may be the cause of the element perceived. — the inferred constituent may be the effect of the element perceived. — both may be the joint effects of something else. In addition, inference (and the results of inference) has two types: — Inference for one’s own sake (Svartha). — Inference for another’s own sake (Parartha). Verbal testimony (and its results) has two types: — Scriptural testimony (Vaidika). — Non-scriptural testimony (Laukika). Perception (and the results of perception) has two types: — Determinate perception (Nirvikalpaka). — Indeterminate perception (Savikalpaka).
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In addition, there three kinds of transcendental perception (Alaukika): — Perception in the Samanyalak¸sana supernormal contact. — Perception in the Jnanalak¸sana supernormal contact. — Perception in the Yogaja (Lak¸sana) supernormal contact. In the process of cognition, mind (Manas) mediates between the self and the senses. When the mind is in contact with one sense organ, it cannot be so with another. It is therefore said to be atomic in dimension. The Nyaya assumed that due to the nature of the mind that experiences of people are discrete and linear, although quick succession of impressions may give the appearance of simultaneity. It is possible to read more about Indian theories of knowledge in the book (Rao, 1998). In other countries, philosophers also paid considerable attention to the problems of knowledge and cognition. In China, Confucius (551–479 B.C.E.) thoroughly considered knowledge and its sources. He discerned two kinds of knowledge: one was innate, while the other came from learning. According to him, knowledge consisted of two components: knowledge of facts (statics) and skills of reasoning (dynamics). The contemporary methodology of science classifies the first type as a part of the logic-linguistic subsystem, which contains declarative knowledge, while the second type is a part of the procedural subsystem of a developed knowledge system, which contains procedural knowledge (Burgin and Kuznetsov, 1994). For Confucius, to know was to know people. He was not interested in knowledge about nature, studied by modern science. The philosophy of Confucius had the main impact on Chinese society for many centuries. Besides, Chinese philosophers paid much attention to names as carriers (bearers) of knowledge reflecting intrinsic aspects of reality. In this respect, Confucius writing about names and their rectification, asserted (Confucius, 1979): “If names be not correct, language is not in accordance with the truth of things. If language be not in accordance with the truth of things, affairs cannot be carried on to success.
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When affairs cannot be carried on to success, proprieties and music do not flourish. When proprieties and music do not flourish, punishments will not be properly awarded. When punishments are not properly awarded, the people do not know how to move hand or foot”.
One of the basic aims of name rectification was to create a consistent knowledge representation in language that would allow each word to have a consistent and universal meaning, providing accurate knowledge of things and actions, while avoiding confusion of multiple Ways (Dao). Later Xun Zi, also called Hs¨ un Tzu, (ca. 312–230 B.C.E.) continued exploration of names as knowledge representations. Xun Zi wrote a tract on the rectification of names, arguing for the rectification of names so, that a ruler could adequately control his people in accordance with Dao (the Way), without being misunderstood. Indeed, when misapprehension became easy, then Dao would not effectively be put into action. Xun Zi explained (cf., (Watson, 2003)): “When the ruler’s accomplishments are long lasting and his undertakings are brought to completion, this is the height of a good government. All of this is the result of being careful to see that men stick to the names which have been agreed upon”. Necessity for rectifying names is both political and epistemological. On one hand, there is a need to distinguish the higher from the lower in terms of the social rank, while on the other hand, it is necessary to discriminate the different states and qualities of things. “When the distinctions between the noble and the humble are clear and similarities and differences [of things] are discriminated, there will be no danger of ideas being misunderstood and work encountering difficulties or being neglected” (cf., (Ding, 2008)). Besides, explaining that understanding right and wrong causes morality to be more unbiased, Xun Zi argued that without universally accepted interpretations of names, knowledge of right and wrong would become hazy. According to Xun Zi, the ancient knowledgeable kings chose names that gave correct knowledge of actualities, but later generations confused terminology, coined new names, and thus could no longer differentiate right from wrong.
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Xun Zi assumed that utilization of senses through seeing, hearing, smelling, tasting, and touching is the key source for getting knowledge of distinctions between things and thus, allowing people to give names based on the sameness or difference between various things. Consequently, this was the way of producing true knowledge of the world, i.e., true knowledge was achieved through naming. Xun Zi also wrote about “things which share the same form but occupy different places, and things which have different forms but occupy the same place”. The former, e.g., two identical flutes, should be distinguished as two separate things, although they have the same form and other properties, because they occupy different places. At the same time, as one of these identical things, e.g., flutes, is used and becomes damaged or broken over time, it appears to change into something else. But even though it seems to become something different, it is still the same things, e.g., flute, and should be regarded as such. Another representative of the School of Names Gongsun Long (ca. 325–250 B.C.E.) asserted in his work “On Names and Actualities” that because all things in the world come into sight in particular shapes and substances, they are given different names. To know if the meaning of a word correctly corresponds to the essence of the thing named by it or not, it is necessary to know the conditions which give rise to it. Gongsun Long writes (cf., (Ding, 2008)): “A name is to designate an actuality. If we know that this is not this and know that this is not here, we shall not call it [‘this’]. If we know that is not that and know that is not there, we shall not call it [‘there’]”. In ancient Greece, Plato (427–347 B.C.E.) performed even more profound analysis of the problem of knowledge. For instance, in one of Plato dialogues, Theætetus, Socrates and Theaetetus discuss the nature of knowledge and Socrates asks the question that permanently puzzles him: “What is knowledge?”. To answer this question, three approaches are suggested. At first, the conjecture “knowledge and perception are the same” is proposed. Socrates refutes this idea by explaining that it is possible to perceive without knowing and it is possible to know without perceiving. For
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instance, it is possible to see a text in a foreign language without us knowing it. The second hypothesis is that true belief is knowledge. Socrates invalidates this idea by giving the following example. When a jury believes a defendant is guilty by listening to the prosecutor instead of looking at solid evidence, it cannot be said that jurors know that the accused is guilty even if, in fact, he is. The third proposition is that true belief with a rational validation is knowledge. However, Socrates also challenges this approach because all interpretations of this definition look inadequate. Thus, Socrates demonstrates that all three definitions of knowledge: knowledge as nothing but perception, knowledge as true judgment, and, finally, knowledge as a true judgment with justification, are unsatisfactory. In spite of this, according to Cornford (2003), in many of his works, e.g., Meno, Phaedo, Phaedo, Symposium, Republic, and Timaeus, Plato treated knowledge as a justified true belief, and this approach prevailed becoming a stable tradition in philosophy. Much later Bertrand Russell in (Russell, 1912; 1948), Edmund Gettier in (Gettier, 1963), Elliot Sober in (Sober, 1991) and some other thinkers gave persuasive examples demonstrating that the definition of knowledge as a justified true belief is not adequate. Let us consider an example demonstrating deficiencies of this definition (Russell, 1912; 1948; Scheffler, 1965). A woman looks at a clock at 3 p.m. The clock shows 3 p.m. So, the woman thinks that it is 3 p.m. Thus, she has a belief, which is true and justified by observation of the clock. Now suppose that the clock is not going though the woman thinks it is. Thus, it seems wrong to hold that she knows that it is 3 p.m. Plato was also interested in the problem of knowledge acquisition. His idea was that people learn in this life by remembering knowledge originally acquired in a previous life. In essence, the soul has all knowledge and knowledge acquisition is recollection of what the soul already knows. Plato conceived it is possible to achieve correct knowledge only through the knowledge of the forms, or ideas (eidos), because what
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came through our senses is not knowledge of the thing itself but only knowledge of the imperfect changing copy of the form. Thus, the only possible way to acquire correct knowledge of the forms was through reasoning as senses could provide only opinion. For a long time, philosophers were not able to clearly and consistently explain what Plato forms, or ideas (eidos), are. Only at the end of the 20th century, it was discovered that the concept structure provides the scientific representation of Plato forms, while the existence of the world of structures was postulated and proved (Burgin, 1997; 2010; 2012). Another great philosopher Aristotle (384–322 B.C.E.) studied problems of knowledge categorizing knowledge with respect to knowledge domains (objects) and the relative certainty with which one could know those domains (objects). He assumed that certain domains (such as in mathematics or logic) permit one to have absolute knowledge that is true all the time. However, his examples of absolute knowledge, such as two plus two is always equal to four or all swans are white, failed when new discoveries were made. For instance, the statement two plus two always equals four was disproved when non-Diophantine arithmetics were discovered (Burgin, 1977; 1997c; 2007; 2010c). The statement “all swans are white” was invalidated when Europeans came to Australia and found black swans. According to Aristotle, absolute knowledge, e.g., mathematical knowledge, is characterized by certainty and precise explanations. However, unlike Plato and Socrates, Aristotle did not demand certainty in everything. Some domains, such as human behavior, do not permit precise knowledge. The corresponding vague knowledge involves expectations, chances, and imprecise explanations. Knowledge that falls into this category is related to ethics, psychology, or politics. One cannot expect the same level of certainty in politics or ethics that one can demand in geometry or logic. In his work Ethics, Aristotle defines the difference between knowledge in different areas in the following way: “we must be satisfied to indicate the truth with a rough and general sketch: when the subject and the basis of a discussion consist of matters which hold good only as a general rule, but not always, the conclusions
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Introduction reached must be of the same order ... For a well-schooled man is one who searches for that degree of precision in each kind of study which the nature of the subject at hand admits: it is obviously just as foolish to accept arguments of probability from a mathematician as to demand strict demonstrations from an orator”. (Aristotle, 1984)
Aristotle was deeply interested in how people got knowledge. He identified three sources of knowledge: sensation as the passive capacity for the soul to be changed through the contact of the associated body with external objects, thought as the more active process of engaging in the manipulation of forms without any contact with external objects at all, and desire as the origin of movement towards some goal. Developing logic as a tool for knowledge acquisition, Aristotle constructed rules of logical inference. The basic rule is called syllogism. It has the following form All A are B. C is A. Therefore, C is B. Here is the famous example of a syllogism: All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Treating syllogism as the main tool of knowledge acquisition, Aristotle conceives of knowledge as hierarchically structured by inference. He puts this claim forward in the Posterior Analytics (Aristotle, 1984). To have knowledge of a fact, it is not enough simply to be able to repeat the fact, while in many cases, for example, in history, it is impossible to repeat the fact. Thus, to have knowledge, it is also necessary to be able to give the reasons why that fact is true. Aristotle calls this process demonstration, which is essentially a matter of showing that the fact in question is the conclusion to a valid syllogism. Thus, knowledge that is premises for obtaining other knowledge is logically prior to the knowledge that follows from it. Eventually,
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there must be one or several “first principles”, from which all other knowledge follows and which themselves do not follow from anything. However, if these first principles do not follow from anything, then by Aristotle, they cannot count as knowledge because there are no reasons or premises we can give to prove that they are true. Aristotle suggests that these first principles are a kind of intuition of the facts and ideas we recognize in experience. Aristotle believes that knowledge domains or objects are structured hierarchically. Consequently, he treats definition as a process of division and specification. For instance, defining whale, we observe that whales are animals, which is the genus to which they belong. Then we search for various conditions, which distinguish whales from other animals such as: whales live in water, unlike tigers, and they are very big, unlike mice. While true knowledge is derived from knowledge of first principles, actual argument and debate is much less immaculate. When two people argue, they do not go back to first principles to ground every claim but simply suggest premises they both acquiesce. The essence of the debates is to find premises your opponent can agree with and then show that conclusions different from your opponent’s position to follow necessarily from these premises. In the Topics, Aristotle classifies the kinds of conclusions that can be drawn from different kinds of premises, while in the Sophistical Refutations, he explores various logical ploys used to trick people into accepting a faulty line of reasoning. Thus, we can see that Aristotle strives to organize knowledge in the manner of a well-structured, architectural construction with a firm foundation of unshakable first principles and an upper structure of propositions firmly attached to the foundation by steadfast inference. In such a way, the Euclid’s geometry and virtually any axiomatic mathematical system is built. It has a foundation of definitions, postulates, and axioms or common notions as first principles and an upper structure of deduced propositions — theorems and lemmas. In the first millennium, the distinguished philosopher Ab¨ u Na¸sr al-Farabi (870–950) also studied knowledge and its sources. He
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defined the highest level of knowledge as theoretical or genuine knowledge, which is the excellence of the theoretical part of the soul and comes from (or is) science (‘ilm). As al-Farabi wrote, for genuine knowledge, certainty is achieved within the soul. For the entities that do not depend on human production, their existence and determination of what each one of them is and how it is can be accomplished by demonstration of true, necessary, universal and primary premises, securely grasped and naturally known by reasons (Fusul, p. 51). As a result, genuine knowledge is indispensable, unchangeable, and universal. The highest type of this theoretical knowledge, for al-Farabi, is wisdom (hikmah), which is the knowledge of the ultimate causes of all existing entities (metaphysics) as well as the proximate causes of everything caused (physics). According to al-Farabi, certain knowledge is threefold: — certain knowledge that the thing exists, which is called the knowledge of existence; — certain knowledge of the cause of the thing, which is called knowledge why; — the certain knowledge of the both together. The syllogisms used in attaining this threefold epistemic certainty are also three: — syllogisms used to prove only existence of the thing; — syllogisms used to prove only its cause; — syllogisms used to prove the two together. Later many outstanding philosophers, such as Thomas Aquinas (1224–1274), Ren´e Descartes (1596–1650), Baruch Spinoza (1632 – 1677), John Locke (1634–1704), George Berkeley (1685–1753), David Hume (1711–1776), Immanuel Kant (1724–1804), Georg Wilhelm Friedrich Hegel (1770–1831), Bertrand Russell (1872–1970), Ludwig Josef Johann Wittgenstein (1889–1951), Michael Polanyi (1891– 1976) and Karl Raimund Popper (1902–1993) studied problems of knowledge.
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The great medieval philosopher Thomas Aquinas assumed that all knowledge of people comes from sense perception writing: “. . . it is natural to man to attain to intellectual truths through sensible objects, because all of our knowledge originates from the sense.”
In its turn, sense perception comes from the actual things themselves, while the human mind does not have inborn ideas. At the same time, people possess a natural ability to abstract knowledge. When people see an object such as a tree, the actual tree is what the person observes and perceives its reflection by senses. The mind knows that what it is seeing corresponds to reality and as a result, an individual attains knowledge about the tree. The form of the real object, e.g., a tree, is not generated by the senses, or the mind of the perceiver, but is impressed by the object itself. All external knowledge obtained through sense is combined by the common sense, which causes the unifying process of the senses into a single perception, which is then presented to the mind. The mind forms a representation sent to the intellect, which generates the universal idea from it by abstraction and names it by a word. The great French philosopher Rene Descartes evaluates knowledge in terms of doubt and certainty, distinguishing certain rigorous knowledge (scientia) and knowledge with lesser grades of certainty (persuasio). Descartes posits that doubt and certainty are complementary feelings — when certainty increases, doubt decreases, and vice versa. Consequently, according to Descartes, knowledge is conviction based on a reason so strong that it could never be shaken by any stronger reason. As a result, knowledge becomes absolute and utterly indefeasible. Descartes writes: “. . . we reject all such merely probable knowledge and make it a rule to trust only what is completely known and incapable of being doubted”. (Descartes, 1984)
That is why Cartesian methodology of cognition starts with assessing convictions or beliefs by doubting and reasoning in the process of discovering innate truths and obtaining knowledge. This
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method demands not merely to apply doubt to all candidates for knowledge, but to apply doubt collectively to these candidates and describes this process in the following way (Descartes, 1984): “. . . those who have never philosophized correctly have various opinions in their minds which they have begun to store up since childhood, and which they therefore have reason to believe may in many cases be false. They then attempt to separate the false beliefs from the others, so as to prevent their contaminating the rest and making the whole lot uncertain. Now the best way they can accomplish this is to reject all their beliefs together in one go, as if they were all uncertain and false. They can then go over each belief in turn and re-adopt only those which they recognize to be true and indubitable”.
Descartes promotes skeptic arguments precisely in acknowledgement that there is a definite reason for the overall doubt, while it is necessary to have valid arguments for truth recognition. Note that although Descartes suggests applying doubt universally to all candidates for knowledge, he does not recommend to do this with tools for founding knowledge. Besides, understanding of cognitive processes by Descartes is similar to Plato’s doctrine of recollection as Descartes writes that cognition seems not so much learning something new as remembering what was known before. Descartes also evokes that there are three possible options for the kind of external essences causing sensations: (1) God (2) Material/corporeal substance (3) Other created substance. However, Descartes discards options (a) and (c) leaving only the second possibility for sensations. The basic Descartes’ principle of doubting any knowledge claim, as well as every attempt at justification of knowledge claims gained much support in traditional epistemology. It has been assumed that it is vital to find a bedrock of certain knowledge immune to all possible doubt. However, this search did not bring conclusive solutions.
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The great European philosopher Baruch Spinoza called by the name Benedito de Espinosa when he was born and later by the name Benedict de Spinoza also studied problems of knowledge elaborating the triadic typology of knowledge:
✧ The first kind of knowledge is obtained in two ways — from opinion or random experience and from imagination. ✧ The second kind of knowledge arises from the intellect, which employs common notions and elaborates adequate ideas of the properties of things. ✧ The third kind of knowledge comes from intuition allowing people to have adequate knowledge, and therefore, to get absolute truth about things. Treating problems of knowledge, the outstanding British philosopher John Locke, at first, explains the origin of ideas that people have and the use of words to signify them. He assumes, being in good agreement with Chinese philosophers, that making of the names of substances is a kind of discovery through an abstract general idea, which is named and then introduced into language. By Locke, names of substances are supposed to copy the properties of the substances they refer to. After this, Locke gives a simple definition of knowledge writing: “Knowledge then seems to me to be nothing but the perception of the connection and agreement, or disagreement and repugnancy of any of our Ideas. In this alone it consists” (Locke, 1975).
Thus, genuine knowledge occurs only when people actually are perceiving. Locke also considered habitual knowledge, which is related to what was known in the past but is not perceived now. Besides, he rejected innate knowledge arguing that otherwise children (and mental defectives) would be the most pure and reliable guides to logical truth. Observing the development of knowledge in individual cases, it is possible to see gradual acquisition of the requisite ideas, perception of agreement or disagreement of which forms knowing, although there is self-evident knowledge.
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Locke’s definition of knowledge as perception of agreement or disagreement of ideas involves two criteria for knowledge acquisition: first, it is necessary to have the requisite ideas and then to perceive the connection, i.e., agreement or disagreement, between them. As a result, knowledge has to be relational in structure and propositional in form. Locke recognizes four types of knowledge: — Knowledge of identity and diversity, which rests upon recognition of the difference of each idea from any other. — Knowledge of relation, which reflects positive non-identical connections among ideas. — Knowledge of co-existence, which is based on coincident appearance of qualities. — Knowledge of real existence, which presumes some connection between an idea and the real thing it represents. In addition, Locke supposes that these types of knowledge can occur in any of three forms: — Intuitive knowledge is a certain and unquestionable perception of identity and relation of any two ideas without the mediation of any other. It is the clearest and the most certain of all degrees of human knowledge. It accounts for self-evident truths serving as the foundation upon which all other genuine knowledge is built. — Demonstrative knowledge is obtained through a series of connections between intermediate ideas by means of reasoning. The standard area of demonstrative human knowledge is mathematics, where our possession of distinct ideas of particular quantities yield the requisite clarity, while disciplined reasoning helps to uncover the intermediate links that establish knowledge of identity and relation. However, Locke thinks that it is possible to have demonstrative knowledge of moral relations. — Sensitive knowledge provides some evidence of the existence of particular objects outside ourselves, although it is not always true that there must exist an external object corresponding
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to each idea of sensation. Locke makes serious reservations about the reliability of our sensitive knowledge of the natural world. Another outstanding British philosopher George Berkeley studied problems of knowledge in his Treatise Concerning the Principles of Human Knowledge (1710). The goal was to make an inquiry into the first principles of human knowledge for discovering what had led to doubt, uncertainty, absurdity, and contradiction in philosophy. Berkeley claimed that the mind cannot conceive abstract ideas and declared that words, such as names, do not signify abstract ideas. To the contrary, he stressed that people could only think of particular things that had perceived. Thus, names denoted general ideas, not abstract ideas. General ideas represent any one of several particular ideas, while existence of an idea of a thing was actually the state of perception of a perceiver. Based on this approach, Berkeley came to a conclusion that all motion is relative, which perfectly correlates with contemporary physics. Human minds know ideas, not objects. Ideas, which constitute knowledge, are brought forth by sensation, thought and imagination. When several ideas are associated together, they are comprehended as ideas of one distinct thing, which is then signified by one name. Even more, according to Berkeley, the outside world is composed only of ideas because “ideas can only resemble ideas”. However, the world possesses logic and regularity given by God. Berkeley challenged that even if some things exist outside the human mind, we cannot know this. Indeed, knowledge through our senses only gives us knowledge of our senses but not of any of the unperceived things. Knowledge through reason does not guarantee that there are, necessarily, unperceived objects while imagination, the third source of knowledge, has proved to produce mostly nonexisting (imaginary) entities. For instance, in dreams, people have ideas that do not correspond to external objects. Another outstanding British philosopher David Hume also tried to solve the enigma of knowledge. He claimed that all knowledge
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stemmed from sense experience being justified in terms of what was in peoples’ minds. Thus, all knowledge consists of impressions and ideas. The former are vivid and clear perceptions, while the latter are less vivid and clear copies of impressions. Hume contended that it is possible to have knowledge of two kinds — the relations between ideas and matters of fact. “Relations between ideas can be known with absolute certainty, and can be known by the “mere operations of thought”. In his treatise, Hume cites only mathematics as an example of relations of ideas. At the same time, “matters of fact” can never be known with the same degree of certainty, and cannot be known by the mere operations of thought. Knowledge of matters of fact is always a posteriori and synthetic as people obtain it by using observation and employing induction and reasoning about what is probable. The foundation of this knowledge is what people experience in the present or can remember from the past. Knowledge that goes beyond testimony of the senses or the records of our memory rests on causal inference. Discussing inference, Hume questions validity of induction arguing that our belief that the future will resemble the past is not based on reason at all. Hume was especially interested in different ways used to justify that some belief we had in essence was knowledge maintaining that all knowledge comes from and must be justified by experience. For instance, matters of fact are justified by probable arguments and not by deductive reasoning. Immanuel Kant is one of the most influential philosophers in the history of Western thought. His ideas in metaphysics, epistemology, ethics, and aesthetics have made a profound impact on almost every philosophical movement that followed his work. A substantial part of Kant’s philosophy addresses the question “What can we know?” In answering this question, he discerned three parts of theoretical knowledge: — logic, which, according to Kant, gives absolute knowledge and have not changed from the time of Aristotle. — arithmetic and geometry, which give the most reliable knowledge.
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— the fundamental principles of natural sciences, which are changing with time giving relative knowledge. Besides, Kant defined items of human knowledge as representations, dividing representations into two classes: — Intuitions, which are “immediate” representations. — Concepts, which are “mediate” representations. These representations could be pure without any relation to experience or empirical, coming from experience. Pure intuitions gave perceptions of basic forms, e.g., intuitions of space and time, which turn unorganized sensations into perceptions. Pure concepts give their basic forms of conceptual knowledge facilitating understanding and comprising the 12 categories described by Kant in accordance with the Aristotelian logic. Being necessary for experience of physical objects, their causal behavior and structural properties, the conceptual categories cannot be circumvented to achieve a mindindependent world. Reason, according to Kant, is structured by forms of experience and categories, giving practical and logical arrangement to people’s everyday experience. In addition, Kant brought in two kinds of knowledge: — analytic knowledge (analytic representations), which is expressed by self-justifying judgments about properties of objects that exist in these objects by definition, e.g., propositions the predicate concept of which is contained in its subject concept. — synthetic knowledge (synthetic representations), which is expressed by judgments about properties of objects that are added to these objects, e.g., propositions, the predicate concept of which is not contained in its subject concept These two kinds were related to two classes of knowledge: — a priori knowledge (representations) known before and independent of experience. — a posteriori knowledge (representations) is obtained from experience.
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As a result, Kant discriminated three kinds of knowledge: • analytical a priori knowledge, which is exact and certain but mostly uninformative as it expounds only what is contained in definitions; • synthetic a posteriori knowledge, which conveys information about what is learned from experience, but it is subject to the errors of the senses; • synthetic a priori knowledge, which is uncovered by pure intuition and is both exact and certain, for it expresses the necessary conditions that the mind imposes on all objects of experience. According to Kant, mathematics and philosophy give synthetic a priori knowledge. Kant explains that analytic knowledge is a priori knowledge, while synthetic knowledge is sometimes a posteriori knowledge and sometimes a priori knowledge. For instance, the statement “Any natural number is larger than or equal to one” is analytic because this property is contained in the definition of natural numbers, which start with one and are built by consecutive addition of one. At the same time, the statement “Any natural number is either prime or compound” is synthetic. As the majority of philosophers, Kant assumed that knowledge was characterized by propositions or statements. Analytic propositions are true by nature of the meaning of the words involved in the sentence, while synthetic statements only tell people something about the world. Thus, the truth or falsehood of synthetic statements comes from something outside of their linguistic content. However, Kant does not demand coincidence of analytic and a priori knowledge explaining that elementary mathematics, e.g., arithmetic, is synthetic and a priori because its statements provide new knowledge, but knowledge that is not derived from experience. This becomes part of his main argument for transcendental idealism, in which the possibility of experience depends on certain necessary conditions called a priori forms, which organize comprehension of the world of experience.
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It is possible to suggest that knowledge of basic arithmetic does not demand any empirical experience to know that 2 + 2 = 4, which is essentially analytic. However, Kant disproves this explaining that if the number 2 in this calculation is examined, there is nothing to be found in it by which the number 4 can be inferred. Thus, it is selfevident, and undeniably a priori, but at the same time, it is synthetic. It is interesting that this might be true for arithmetic as an empirical science. When axioms of arithmetic, e.g., Peano axioms (cf., for example, (Shoenfield, 2001)), were constructed, arithmetic propositions that were not axioms became analytic and a posteriori. This became even more transparent with the discovery of non-Diophantine arithmetics (Burgin, 1977; 1997c; 2007; 2010c). Moreover the whole mathematics is, in a definite sense, an empirical science where experiments, mostly mental experiments, play the leading role (Burgin, 1998). Consequently, the majority of mathematical propositions, even many axioms, are a posteriori by their nature. In the 20th century, Bertrand Russell also studied problems of knowledge. His views are exposed in the article (Russell, 1926). He writes, “the question how knowledge should be defined is perhaps the most important and difficult” of all problems related to knowledge. “This may seem surprising”: he continues, — at first sight, it might be thought that knowledge might be defined as belief which is in agreement with the facts. The trouble is that no one knows what a belief is, no one knows what a fact is, and no one knows what sort of agreement would make a belief true”. According to Russell, theory of knowledge is partly logical, partly psychological and we can add, partly algorithmic. Connection between these parts is not very pronounced. Taking precision and certainty as the basic characteristics of knowledge, Russell assumes that they have different degrees, or in modern terms, precision and certainty are fuzzy properties. In essence, there is no absolutely precise knowledge or knowledge with absolute certainty — all knowledge is more or less uncertain and more or less vague. Often vague knowledge seems more reliable than precise knowledge, but is less useful. Russell believes that one of the aims of science is to increase
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precision without diminishing certainty although it is incorrect to restrict understanding of knowledge to what has the highest degree of both precision and certainty. It is interesting that Russell treats data as a kind of knowledge, namely, immediate knowledge. Indeed, he writes: “the separation into data and inferences belongs to a well-developed stage of knowledge, and is absent in its beginnings” (Russell, 1926).
At the same time, he assumes that all knowledge is represented by propositions and is obtained by observations (data) and inference (inferred knowledge). Traditionally, two sorts of data are considered: one physical, derived from the senses, the other mental, derived from introspection. Russell suggests that the difference between the physical and the mental belongs to inferences and constructions and not to data. Russell also distinguishes two kinds of inference, deduction and induction. Deduction, as he maintains, is obviously of great practical importance, since it embraces the whole of mathematics. Inductive inferences are essential to the conduct of life. Russell implies that “we have to accept merely probable knowledge in daily life, and theory of knowledge must help us to decide when it really is probable, and not mere animal prejudice”. In addition, Russell also writes about analogy as a way of inference. Interestingly, that parallel to conventional inference, Russell acknowledges animal inference explaining why there are grounds for doing this. Studying knowledge, the outstanding Austrian philosopher Ludwig Wittgenstein asserts that some statements, such as “here is a hand” or “the world has existed for more than five minutes”, look like empirical propositions saying something factual about the world and open to doubt. However, in essence, they are similar to logical propositions because they function in language, which makes possible for empirical propositions to make sense. This compels us to take such propositions for granted to allow us to speak about things in the external world. According to Wittgenstein, a proposition has no meaning unless it is placed within a particular context.
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It is not the goal of Wittgenstein to refute skeptical doubts about the existence of an external world. Instead, he tries to circumvent them by explaining that the doubts, as they are understood in philosophy, do not do what they are meant to do. By ascribing logical nature to certain fundamental propositions, Wittgenstein explicates their structural role in communication and behavior of people. For instance, the statement “Here is a hand” is an implicit definition of the word hand by showing an example. In addition, this statement indicates how the word hand is used rather than making an empirical claim about the presence of a hand. Doubts aimed at such propositions destroy language and its utilization. Communication and rational thought are only possible, provided there is some sort of common ground. Although skeptical doubts are sensible in rational debate, doubting too much undermines rationality and as a consequence, very foundation for doubt. In addition, similar to many Chinese philosophers mentioned above, Wittgenstein explores how words acquire meaning. However, in contrast to them, Wittgenstein derives meaning from usage and not vice versa. In doing this, he asserts that one should look to real language to answer questions about the meaning of words. Wittgenstein demonstrates that many philosophical problems arose from philosophers’ redefining words and then applying their own definitions to promote their ideas and to defeat their opponents. Wittgenstein does not try to define knowledge but suggests looking at the way the word knowledge is used in natural languages. He apprehends knowledge as an instance of a family resemblance reconstructing the concept of knowledge as a cluster conception that comprises relevant features but cannot be adequately captured by any precise definition. Wittgenstein also discusses the distinction between sense-data and reality indicating that people learn what a tree is by being shown trees and not by being given tree sense-data. According to him, the tree sense-data are irrelevant in this case and do not bear on anything useful for people. The possibility of illusion is there, of course, but there are criteria for deciding what constitutes an illusion. These
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criteria work for most people. Those for whom they fail are called “mad” and are widely disregarded. However, according to contemporary psychological and neurophysiological theories, an individual sees a tree only if she receives tree sense-data through her senses. However, the reception of sensedata is not enough. To see a tree as a tree, the brain has to correctly process sense-data, building a relevant image and assigning the correct name “tree” to this image. Besides, it is possible to know what a tree is by observing not trees but their images, e.g., pictures or movies with trees. After images of trees are stored in the memory, an individual can see a tree in her dreams. In this case, the brain simulates acceptance of sense-data from a physical tree or from its picture. 1.3. Structure of the book There’s only one solution: look at the map. Umberto Eco, Foucalt’s Pendulum The map is not the territory, and the name is not the thing named. Alfred Korzybski
The main goal of this book is to achieve a synthesized understanding of the complex, multifaceted phenomenon called knowledge by building a synthetic theory of knowledge, which allows systematizing and binding together existing approaches to knowledge in one unified theoretical system. However, we do not try to represent all approaches and directions of knowledge studies in a complete form or even to give all important results of this area. Our goal is to give an introduction to the main approaches and directions, explaining their basics and demonstrating how they can be comprehended in the context of the general theory of knowledge. Besides, references are given to sources where an interested reader can find more information about these approaches and
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directions of knowledge studies. The goal is to present a broad picture of contemporary knowledge studies, provide a unifying theory of knowledge and synthesize all existing approaches in an amalgamated structure of ideas, constructions, methods, and applications. That is why in Chapter 1, we explain the leading role of knowledge in the contemporary society and describe a brief history of knowledge studies in different countries and cultures exhibiting not only the development of knowledge studies in Western countries but also achievements of the Eastern civilizations in the fields of logic and epistemology. Chapter 2 studies properties of knowledge and its classifications. Usually, knowledge is studied in the context of beliefs (cf., for example, (Gettier, 1963; Pollock, 1974; Pollock and Cruz, 1999; Dretske, 2000)). In this book, we treat knowledge in the more general setting, namely, in the context of epistemic structures. Knowledge items are epistemic structures. Beliefs are epistemic structures associated with descriptive knowledge. However, beliefs are related only to declarative or descriptive knowledge while there are also other epistemic structures, to which knowledge is intrinsically attached. In particular, there is operational knowledge and representational knowledge. To understand knowledge, it is important to know that there are various types, sorts, and kinds of knowledge. That is why we start Chapter 2 (Section 2.1) with exposition and exploration of diverse classifications, taxonomies and typologies of knowledge. In Section 2.2, different approaches to knowledge characterization are discussed and analyzed from the perspective of the existential characteristics of knowledge. In Section 2.3, dimensions of knowledge are described and investigated. Section 2.4 contains knowledge about metaknowledge and metadata, where metaknowledge is knowledge about knowledge, while metadata provide information about data. However, it is necessary to not only know properties of knowledge but also be able to evaluate and justify these properties. This is the main topic of Chapter 3, where Section 3.1 tells the reader about knowledge evaluation and Section 3.2 narrates knowledge justification issues. It is especially significant to appraise and justify consistency of knowledge. That is why in Section 3.3, we explain
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how to work with knowledge that has been traditionally considered inconsistent giving an overview of existing approaches to this problem and an exposition of some parts of the theory of logical varieties. This section is based on the papers (Burgin and de Vey Mestdagh, 2011; 2015). In the next part of the book, we separate and study three key levels of knowledge: the microlevel, macrolevel, and megalevel (Burgin, 1997). Chapter 4 describes the microlevel, or the quantum level of knowledge, its structures, properties, and processes. This level contains “bricks” and “blocks” of knowledge that are used for construction of other knowledge systems. We call such minimal “bricks” knowledge quanta and study them in Section 4.1. Two fundamental theories of the knowledge quanta are presented — the Quantum Theory of Knowledge (QTK) created by Burgin (1995a; 1997; 2004) and the Semantic Link Network Theory (SLNT) developed by Zhuge (2002; 2004; 2010; 2012). Relations between these two theories are established. “Blocks” of knowledge are identified with structured quantum knowledge items and we consider such quantum knowledge items as signs and symbols discussing different approaches and models in Section 4.2. Operations with and relations between knowledge quanta and other quantum knowledge units representing dynamics and structural organization of the quantum level of knowledge are constructed and explored in Section 4.3. On the macrolevel, or the level of average knowledge, considered in Chapter 5, researchers study knowledge representation used by people and artificial systems for practical purposes. Section 5.1 explains utilization of languages, such as natural, mathematical, programming, and scientific languages, for knowledge representation, preservation, and processing. Section 5.2 presents means of logics, which are used for knowledge representation, validation, preservation, and processing, while Section 5.3 describes elements of the theory of abstract properties, which is a synthesis of logic and qualitative physics providing even more powerful means for knowledge representation, validation, acquisition, preservation and processing. Next three sections of Chapter 5 are dedicated to
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knowledge representation in AI. Semantic networks and ontology are the topics of Section 5.4. Scripts and productions are exposed in Section 5.5. Frames and schemas are studied in Section 5.6 with the emphasis on the new direction in this area called mathematical schema theory. On the megalevel, or the global level of knowledge, researchers consider the immense knowledge systems such as mathematics, physics, biology, advanced mathematical, and physical theories. Chapter 6 contains an exposition of the global level of knowledge describing structure and organization of such knowledge systems. Knowledge production, acquisition, engineering and application are studied in Chapter 7. Section 7.1 analyzes knowledge production and acquisition as basic cognitive processes. Section 7.2 is concerned with problems of knowledge organization and engineering. Section 7.3 treats issues of knowledge application and management. Relations between information and knowledge are studied in Chapter 8. Section 8.1 presents structural aspects of knowledge–information duality exploring different opinions about the triad Data–Information–Knowledge. Section 8.2 considers relations between epistemic structures and cognitive information. Dynamic aspects of knowledge, data, and information interaction are the main concern of Section 8.3. Section 8.4 analyzes information as a source of knowledge, while Section 8.5 investigates knowledge as a measure of information in the context of mathematical stratum of the general theory of information. The last Chapter 9 contains some conclusions and directions for future research. Exposition of material is aimed at different groups of readers. Those who want to know more about history of knowledge studies and get a general perspective of the current situation in this area can skip proofs and even many theoretical results given in the strict mathematical form. At the same time, those who have a sufficient mathematical training and are interested in formalized knowledge theories can skip preliminary deliberations and go directly to the sections that contain mathematical exposition. Thus, a variety of readers will be able to find interesting and useful issues in this book
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if each reader chooses those topics that are of interest to her or to him. It is necessary to remark that the research in the area of knowledge studies and application is extremely active, while knowledge is related almost to everything. Consequently, it is impossible to include all ideas, issues, directions, and references to materials that exist in this area, for which we ask the reader’s forbearance.
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Chapter 2
Knowledge Characteristics and Typology In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. Galileo Galilei
2.1. The differentiation and classification of knowledge There is a strong tendency to reduce the many to the few, the complex to the simple, the various to the uniform. Richard Pring
For millennia, philosophers, who were the first to study the problems of knowledge, have asserted that knowledge is a kind of beliefs reducing all knowledge to declarative or descriptive knowledge and actively imposing this opinion on all others. Even now, the majority of philosophers believe in this declaration. For instance, such experts in contemporary philosophical theories of knowledge as John Pollock and Joseph Cruz write: “Epistemology might better be called “doxastology”, which means the study of beliefs” (Pollock and Cruz, 1999). However, this understanding was challenged. At first, physicists discovered operational knowledge. It was Nobel laureate Percy Williams Bridgman (1882–1961), who insisted that conceptual knowledge is, in essence, operational. He wrote that “any concept 45
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[is] nothing more than a set of operations; the concept is synonymous with the corresponding set of operations” (Bridgman, 1927). Many scientists, especially, physicists and psychologists, became enthusiasts of this methodological approach bringing into being operationalism, also called operationism, as a direction in methodology of science based on the idea that to know the meaning of a concept is to have a method of measurement for it (Bridgman, 1936; 1959; Boring et al., 1945; Chang, 2009). Behaviorist psychologists, such as Edwin Boring (1886–1968), Stanley Smith Stevens (1906–1973), and Edward Chace Tolman (1886–1959), became ardent adherents of operationalism. They used operationalism as a weapon in their fight against more traditional psychologists (Feest, 2005). Nevertheless, despite the initial popularity of Bridgman’s approach, by the middle of the 20th century, the common attitude among philosophers and philosophically-minded scientists towards operationalism was strongly critical (Chang, 2009), although operational knowledge was explicitly used by logical positivism in its verification theory of meaning (Frank, 1956). Another kind of knowledge — representational knowledge — was elucidated in methodology of science. Namely, the structuralist direction represented scientific knowledge in the form of a scientific theory as a system of models (Sneed, 1971; Stegm¨ uller, 1976; 1979; Balzer et al., 1987). The computational approach treats scientific knowledge in the form of a scientific theory as complex data structures in computational systems, which contain organized packages of rules (operational knowledge), concepts (representational knowledge), and problem solutions (operational and descriptive knowledge) (Thagard, 1988). Lobovikov included questions and problems (erotetic knowledge) into scientific knowledge in the form of a scientific theory (Lobovikov, 1984). Pearce and Rantala combined representational and descriptive knowledge in their model of a scientific theory (Pearce and Rantala, 1981).
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Later the structure-nominative direction in methodology of science included descriptive, representational knowledge and operational knowledge as specific components of general knowledge systems in general and scientific theories in particular (Burgin and Kuznetsov, 1988; 1989; 1991; 1992; 1993; 1994; Balzer et al., 1991; Burgin, 2011). Namely, in the structure-nominative model of scientific knowledge, operational knowledge constitutes the pragmaticprocedural subsystem, while representational knowledge constitutes the model-representing subsystem (Burgin and Kuznetsov, 1988; 1989; 1991; 1992a; 1992b; 1994). In addition, operational knowledge also called procedural knowledge, has become popular in knowledge management (Valente and Rigallo, 2002; 2003). Some philosophers also made a distinction between “know-that” as descriptive/declarative knowledge and “know-how” as operational/procedural knowledge. In general, they interpreted operational/procedural knowledge as knowledge that is manifested in the use of a skill, whereas descriptive/declarative knowledge as explicit knowledge of a fact (Fantl, 2012). Although there is a discussion whether ancient Greeks considered “know-how” as a specific kind of knowledge, it is assumed that Gilbert Ryle was the first philosopher to treat “know-how” as knowledge distinguishing it from propositional knowledge or “know-that.” He identified “know-how” with a disposition whose “exercises are observances of rules or canons or the application of criteria” (Ryle, 1949). His main argument was that it was possible to have lots of knowledge-that, without possessing any knowledge-how. In addition, he insisted that “knowledge-how is a concept logically prior to the concept of knowledge-that” (1971/1946). Later in her analysis of behavior from the epistemological perspective, Katherine Hawley came to the conclusion that “know-how” was a matter of successful actions plus warrant (Hawley, 2003). In a similar way, Thorkelson (2008) writes: “What is knowledge? The time-worn and widely criticized philosophical definition is “justified true belief ” (Gettier, 1966; Goldman, 1967; Lewis, 1996); for anthropological purposes it suffers from three major
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Theory of Knowledge: Structures and Processes problems centered especially around the term “belief.” First, the definition reduces knowledge to propositional knowledge, “knowing-that,” thus occluding other knowledge types like practical “know-how” (knowledge embodied in routinized dispositions), affective states (knowledge embodied in emotion and sentiment), and phenomenological acquaintance (conferred, for instance, by sensory experience or artistic representation).”
Although many philosophers started to understand that knowledge-how and knowledge-that are distinct kinds of knowledge and consequently, procedural knowledge is non-propositional, others tried to reduce procedural knowledge to propositions. For instance, Bruce Kogut and Udo Zander write: “Procedural knowledge consists of statements that describe a process, such as the method by which inventory is minimized” (Kogut and Zander, 1992). Some educators also rejected reduction of knowledge to descriptive knowledge and especially, to propositions or beliefs. For instance, Scheffler (1965) discerns “know-that” as descriptive knowledge and “know-how” as operational knowledge. There are also an extreme comprehensions of operational knowledge. For instance, Nonaka and Takeuchi (1995) assume that all knowledge is about action as any knowledge must be used to some end. Thus, methodological and sociological analysis shows that there are three basic categories of knowledge: ∗ Representational knowledge about an object is representations of this object by knowledge structures, such as models and images, e.g., when Bob has an image of his friend Ann, it is representational knowledge about Ann. ∗ Descriptive knowledge also called declarative knowledge or sometimes propositional knowledge is knowledge about properties and relations of the objects of knowledge, e.g., “a swan is white”, “a lion is an animal” or “three is larger than two”. However, objectively, declarative knowledge is only a kind of descriptive knowledge, while propositional knowledge is a kind of declarative knowledge.
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∗ Operational knowledge also called procedural knowledge consists of rules, procedures, algorithms, etc., and serves for organization of behavior of people and animals, for control of system functioning and for performing actions. A more exact categorization shows that procedural knowledge is a kind of operational knowledge. Note that in the literature, there is no agreement as to the definition of operational (procedural) knowledge. For instance, Lewicki et al. (1987) equate procedural knowledge either with cognitive skills or with processing rules. At the same time, Kogut and Zander claim that operational (procedural) knowledge consists of statements that describe the process (Kogut and Zander, 1992). In addition, each basic category consists of three subcategories. ∗ Representational knowledge can represent statics or/and dynamics, while dynamics knowledge, in turn, is divided into two more exact categories — representation of functions or of processes. ∗ Descriptive knowledge can be either informal, e.g., linguistic, i.e., represented by texts in natural languages; or semiformal such as the conventional mathematical language; or formal, e.g., logical, i.e., represented by logical expressions (formulas). ∗ Operational knowledge can be either procedural, e.g., in the form of instructions, algorithms, programs, plans and scenarios; or instrumental, e.g., descriptions of tools of operations, operators and performers; or axiological, e.g., in the form of goals, tasks, values, estimates, norms or judgments. Moreover, the basic categories of knowledge contain other subcategories. For instance, existential knowledge, i.e., knowledge about existence of the object of knowledge, is a kind of descriptive knowledge because existence and its characteristics are properties of the object. Representational knowledge also comprises structural knowledge, which is basic to problem solving in creation of plans and strategies, setting conditions for different procedures, and in determining structures of different systems.
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Differentiation of knowledge into three types allows solving the problem of relations between art and knowledge. Different thinkers have been seriously puzzled by the following situation (cf., for example, (Pring, 1976; Reid, 1985; Bender, 1993; John, 1998)). On one hand, art definitely gives knowledge. On the other hand, if we assume that knowledge is only descriptive, e.g., in the form of logical propositions, then we all know that art goes beyond descriptions not speaking about logical propositions. For instance, each proposition in logic has its negation but as Pring writes, what conceivably could be the negation of the Mona Lisa (Pring, 1976). As a result of this confusion, philosophers even suggested “to reconceive knowledge in such a way that we may eventually come to understand how it can be gained non-propositionally” (Worth, 2010). In contrast to this, art can convey representational and operational knowledge. Indeed, on the one hand, art is representation of different things. It can imitate (represent or reflect) states of the external world — nature, people, society, etc., as well as the inner state of the artist. “Art as a representation of outer existence (admittedly “seen through a temperament”) has been replaced by art as an expression of humans’ inner life” (Worth, 2010). In such a way, art gives representational knowledge. On the one hand, art can teach people providing models of different actions, behavior, and attitudes. In such a way, art gives operational knowledge. It is interesting that descriptive and representational knowledge have operational representations, descriptive and operational knowledge have representational (model) representations and representational and operational knowledge have descriptive representations. For instance, productions (cf., Section 5.5) give operational representation of descriptive (declarative) knowledge in the form of conditional propositions. Algorithms serve as models of processes and actions giving operational representation of representational (model) knowledge. Programs in declarative programming languages (cf., Section 5.1.3) give descriptive representation of algorithms as a form of operational knowledge. Propositions and predicates as forms of descriptive knowledge have model representations in the structuralist model (reconstruction) of a scientific theory (cf., Chapter 6), as
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well as in the possible-world semantics, also called Kripke semantics (Kripke, 1963). Processing of different types of knowledge in the human brain involves corresponding types of memory. Namely, declarative memory and procedural memory are two major parts of the long-term memory. Experimental evidence for a distinction between declarative memory and procedural memory was demonstrated by Milner (1962). Declarative memory (descriptive memory) is the memory that stores declarative knowledge, such as knowledge of facts and events. It is also called explicit memory because knowledge it accumulates is explicitly stored and retrieved. Procedural memory (operational memory) is the memory that stores procedural (operational) knowledge in the form of skills and knowledge how to do things, such as the utilization of things or movements of the body. Procedural memory is also called implicit memory, because knowledge it accumulates is typically acquired through repetition and practice and used without explicit and conscious awareness. Cohen and Squire (1980) coined the term procedural memory for storing and using skills and procedures. These parts of long-term memory involve different regions of the brain and function in a different manner. Declarative memory uses the hippocampus, entorhinal cortex and perirhinal cortex as the coding system and is mostly situated in the temporal cortex. Procedural memory uses the cerebellum, putamen, caudate nucleus and the motor cortex as the coding system and is situated in different parts of the brain. For instance, learned skills such as driving are stored in the putamen, while instinctive procedures such as sleeping are stored in the caudate nucleus and the cerebellum is involved with timing and coordination of body skills. In addition, researchers demonstrated that declarative memory can be further sub-divided into episodic memory and semantic memory (Tulving, 1972). Episodic memory is the memory of experiences and specific events in time in a serial form, from which we can reconstruct the actual events that took place at any given point in our lives.
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Semantic memory is a structured representation of general factual knowledge, such as facts, meanings, concepts and knowledge about the external world. Knowledge in semantic memory often is abstract and relative. It is possible to suggest that representational knowledge is related to eidetic memory. As we know, there is a big variety of properties and characteristics of knowledge, as well as many different types and kinds of knowledge. To organize this huge diversity into a system, it is worthwhile to classify knowledge with respect to various criteria. These criteria are based on five types of attributes: — — — — —
Characteristics of knowledge and its representations. Features of processes that are related to knowledge. Parameters of systems that produce or/and use knowledge. Properties of the knowledge domain. Traits of the knowledge carriers.
Here are some examples. The most popular feature of knowledge is truthfulness, which takes two values True and False in the classical interpretation. In fuzzy logics, truthfulness, takes values in the interval [0, 1] where the value 0 means absolutely false and the value 1 means absolutely true. Truthfulness of knowledge depends both on of characteristics, knowledge and its representations and on the properties of the knowledge domain. It is necessary to remark that from ancient times, many researchers have thought that knowledge and information cannot be false assuming that being true is an inherent characteristic of knowledge. They believe that if a belief is false, then it is not knowledge as knowledge is a justified true belief. Other thinkers admit that knowledge can be true and can be false. For instance, in his description of the World 3, Popper (1972; 1979) asserted that this world contains all the knowledge and theories of the world, both true and false. Thus, Popper assumed existence of false knowledge. Burgin (2010) gives a detailed explanation why information and hence knowledge can be false.
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An important feature of knowledge from the perspective of related processes is complexity, for example, complexity of acquisition, complexity of integration or complexity of learning. We see that complexity of knowledge depends on such processes as knowledge acquisition, knowledge integration or learning. There are different types of professional knowledge, for example, professional knowledge of a lawyer, professional knowledge of an architect or professional knowledge of a physicist. There is knowledge about specific domains, for example, mathematical, physical, biological, or sociological knowledge. It is possible to consider four kinds of knowledge based on information characteristics introduced by Collins (1993): 1. 2. 3. 4.
Symbol-type knowledge. Embodied knowledge. Embrained knowledge. Encultured knowledge.
Symbol-type knowledge is represented by symbols and can be transferred without loss on flashcards, hard drives, CD-ROM drives, floppy disks, and so forth. Embodied knowledge depends on properties and functioning of the human body. For instance, embodied knowledge of the notion chair depends on the ability to put the body in the sitting position on a chair. Embodied knowledge is a kind of embedded knowledge. In this context, a general understanding treats embedded knowledge as the knowledge that is set in processes, products, culture, routines, artifacts, or structures (Horvath, 2000; Gamble and Blackwell, 2001). Knowledge is embedded either formally, for example, through a management initiative to formalize a certain useful technique, or informally as the organization uses or people behavior. Embrained knowledge depends on the physical characteristics of the brain. For instance, Collins (1993) explains, there are kinds of knowledge dependent on the way neurons are interconnected or on the chemistry the brain or on the formation of solid shapes in the brain.
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Encultured knowledge depends on the social environment. For instance, natural languages are the model example of encultured knowledge. Thus, the right way to use a language, e.g., to speak, is the sanction of a social group but not of a separate individual and those who do not remain in contact with the social group will soon cease to know how to speak properly according to the rules of the group. Researchers also consider three kinds of knowledge, which form a representational classification: 1. Symbolic knowledge is represented by symbols. 2. Subsymbolic knowledge is constructed from knowledge elements that are not symbols. 3. Wired knowledge is a part of a physical system. Let us consider some examples. Example 2.1.1. Images on the screen of a computer or TV are units of representational knowledge. These images are formed from pixels (points of different colors and brightness) on the screen. Thus, these images are units of subsymbolic representational knowledge. Example 2.1.2. An algorithm in the form of a computer program is a unit of symbolic operational knowledge, while an algorithm implemented in the hardware of a computer is a unit of wired operational knowledge. Application of the representational knowledge classification allows researchers to solve some methodological problems. For instance, in the theory of computations, many think that computations of neural networks are not algorithmic because they assume that algorithms can be only symbolic. However, algorithms as a kind of operational knowledge can be not only symbolic but also wired, and a neural network is just a wired algorithm. This understanding is supported by the fact that artificial neural networks are modeled by conventional computers where these networks are represented by conventional symbolic algorithms in the form of computer programs.
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It is possible to consider three important types of knowledge related to information characteristics suggested by Banathy (1995): 1. Referential knowledge has meaning in the system R. 2. Non-referential knowledge has meaning outside the system R, e.g., knowledge that reflects mere observation of R. 3. State-referential knowledge reflects an external model of the system R, e.g., knowledge that represents R as a state transition system. Philosophers usually consider two kinds of knowledge, which form a cognitive classification: • A priori knowledge, which is known independently of experience. For instance, Kant interpreted a priori knowledge as a “transcendental” form of knowledge coming from “rational insight”. • A posteriori knowledge is knowledge that people get from experience that can be of two types: ◦ Empirical experience, which is accumulated from practical activity, e.g., experimentation, giving empirical knowledge. ◦ Theoretical experience, which is accumulated from mental activity, e.g., thinking, giving theoretical knowledge. Separating knowledge with respect to the knower, i.e., to the system that has knowledge, we come to the system classification: — Personal knowledge. — Communal knowledge. — Network knowledge. Michael Polanyi (1891–1976) explicated two sorts of personal knowledge (Polanyi, 1966; 1974): Explicit knowledge is codified knowledge, such as knowledge found in documents. Tacit knowledge is intuitive, hard to define knowledge that is mostly experience based.
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This accessibility classification is extremely popular within business and knowledge management where the following descriptions of these sorts of knowledge were elaborated. Explicit knowledge can be articulated in formal language and records, communicated by people or through information technology and stored. Tacit (or implicit) knowledge is personal knowledge embedded in individuals based on their experience and involving such intangible factors as personal emotions, beliefs, procedures, perspectives, goals and values. People can know about something but be unable to explain the process that led to their knowledge and even explain what they know. Tacit knowledge is difficult to articulate, communicate and store, although it can be communicated through face-toface contact and storytelling. According to Polanyi, tacit knowledge that underlies explicit knowledge being more fundamental in that all knowledge is either tacit or it was initially rooted in tacit knowledge, which cannot be objective. Tacit knowledge located exclusively in the human mind constitutes the invisible part of organizational knowledge including organizational culture, experience, feelings, confidence, relationships being the principal driving force of the organization. Thus, it is possible to compare the organizational knowledge base to an iceberg, the explicit knowledge is the visible part of this iceberg, codified and classified knowledge integrated into documents, procedures and business processes, and codified in informational systems. However, Botha et al. (2008) pointed out that tacit and explicit knowledge should be seen as a spectrum (the accessibility spectrum) rather than as two separate points. Tacit and explicit knowledge are the endpoints of the accessibility spectrum. Thus, knowledge is mostly a mixture of tacit and explicit elements rather than being one or the other. Taking into account this issue, it is possible to formalize the accessibility spectrum defining a measure of knowledge explicitness with the scale from 0 to 1. In this scale, the measure of tacit knowledge will be 0, while the measure of explicit knowledge will be 1. Logical tools for adequate description, valid transformation and effective generation of explicit and implicit knowledge are developed
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in (Burgin and Rybalov, 2003). As explicit and implicit knowledge contain propositions that can be inconsistent for two reasons: (1) due to utilization of different systems of knowledge generation, e.g., strong entailment in one case and weak entailment in the other, and (2) because of application of these mechanisms to different parts of new knowledge depending on level of conflict the latter generated. Thus, there is no consistent classical calculus that can generate both explicit and implicit knowledge. Therefore, the logical representation of explicit and implicit knowledge together forms a non-trivial logical variety (Burgin, 1997). Here we slightly extend the accessibility classification considering three following sorts of personal knowledge: — Externally explicit or articulated (codified) knowledge. — Internally explicit (unarticulated) knowledge. — Tacit (incommunicable) knowledge. Two latter categories form implicit knowledge in contrast to explicit knowledge. In this extended accessibility classification, internally explicit knowledge is situated in the accessibility spectrum between externally explicit knowledge and tacit knowledge. People very often have internally explicit but unarticulated knowledge. This is well known to teachers, who habitually encounter situations when their students can apply definite rules to solve problems but cannot explain and sometimes even repeat these rules. There are also three gradations of implicit knowledge: — Instincts are a form of operational knowledge. — Unconscious knowledge belongs to the knower but the knower is not aware of it. — Conscious but explicitly inexpressible knowledge — the knower knows that she/he has this knowledge but cannot explicitly express it. The following situations apparently demonstrate the difference between unconscious knowledge and conscious but explicitly inexpressible knowledge. Imagine students in a class who are
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performing arithmetical operations with numbers. Always there are students who correctly use the commutative law in their calculations. However, some of them do this without remembering this law. They have unconscious knowledge of this law. Other students do this remembering this law, but if the teacher asks them to formulate the law, they would be unable to do this. They have conscious but explicitly inexpressible knowledge of this law. Finally, there are students who have externally explicit knowledge of the commutative law. There are three gradations of explicit knowledge: — Personal knowledge. — Shared knowledge. — Personalized (internalized) knowledge. When an engineer invents some new device, her knowledge about this device is personal. If she writes a paper or tells a colleague about it, her knowledge about this device becomes shared. For the colleague who hears this new device and understands, this knowledge becomes personalized (internalized). In addition, there is a carrier-based classification suggested in (Nonaka and Takeuchi, 1995): — — — —
Individual knowledge. Group knowledge. Organizational knowledge. Knowledge of a group of organizations or super-organizational knowledge
Amalgamating three latter classes into one class, we obtain the following classification: — Personal knowledge. — Public knowledge. One more knowledge classification is considered in (Ekinge and Lennartsson, 2000): — Individual knowledge. — Shared knowledge. — Objectified knowledge.
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As it is sometimes happens in science, the same word tacit is used for denoting different concepts because it is also used in the following classification also suggested by Polanyi. — Focal knowledge is about the object or phenomenon that is in focus. — Tacit knowledge is the general background knowledge used as a tool to handle or improve what is in focus. The focal and tacit dimensions are complementary. Tacit knowledge varies from one situation to another. It functions as a background knowledge assisting in accomplishing a task which is in focus. By the level of externalization, knowledge is broken up into three classes: 1. Personal knowledge is knowledge that belongs to an individual. 2. Shared knowledge is knowledge that is shared by a group of individuals. 3. External knowledge is knowledge that does not belong to an individual. For instance, knowledge of what a person is going to do during the day is personal. Knowledge of mathematics is shared knowledge. For a non-mathematician, knowledge of category theory is external knowledge. By the level of internalization, knowledge is divided into three categories: 1. Subconscious knowledge is knowledge of an individual such that the individual is not aware of its existence. 2. Implicit knowledge is knowledge of an individual such that the individual is aware of its existence but cannot express it, e.g., verbalize it. 3. Explicit knowledge is knowledge of an individual such that the individual can express it, e.g., verbalize it. Aristotle’s very influential three-fold classification of disciplines as theoretical, productive, or practical is used as the base for classification of forms of knowledge in (Smith, 1999).
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— Theoretical knowledge is pursuing truth for its own sake. — Productive knowledge is knowledge for making things. — Practical knowledge displays in ability of making judgments and decisions. Theoretical knowledge is related to the form of thinking appropriate to theoretical activities, which according to Aristotle, is contemplative involving meditating over facts and ideas that the person already possesses (knows). Productive knowledge and enquiry is used in productive disciplines for performing an action or operation. Aristotle associated this form of thinking and doing, with the work of craftspeople or artisans. Practical knowledge was originally associated with ethical and political life. Their purpose was the cultivation of wisdom and knowledge involving decision-making and human interaction. For instance pure mathematics is theoretical knowledge, toolmaking procedures are productive knowledge, and social work training methods are practical knowledge. Piaget (1967/1971) identified three kinds of knowledge: 1. Physical knowledge consists of facts about the features of something such as “the window is transparent,” “the crayon is white,” “the cat is grey” and “the air was cold and dry yesterday.” Physical knowledge directly reflects the objects and can be obtained by exploring objects and noticing their qualities. 2. Social knowledge consists of names and conventions made up by people. Here are some examples: “The name of this dog is Bounty,” “New Year is on January 1,” or “It is polite to say thank you for a gift.” Social knowledge may be arbitrary and is knowable by being told or demonstrated by other people, found in books, journals, and on the Internet. 3. Logico-mathematical knowledge, according to Piaget, consists of relations and structures. Logico-mathematical knowledge is constructed by each individual, inside his or her own head or learned from people, books, journals, and the Internet.
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There is also a problem-oriented knowledge classification. • Know-what is the fundamental form of knowledge, e.g., people/ group/organizations know what they know (perhaps through their formal education) but do not necessarily know when and how to apply the knowledge to solve problems. • Know-how is the ability to solve various problems. • Know-why is explanatory knowledge enabling individuals to move a step beyond know-how and create opportunities to deal with unknown interactions and unseen situations. Sveiby (1997) analyzes two types of knowledge: • Agentive knowledge is mostly oriented towards using the body as a tool. • Intellective knowledge is oriented towards using the mind as a tool. In the domain of religion and mysticism, usually two types of knowledge are considered: • Esoteric knowledge is preserved and/or understood by a small group of those specially initiated, or of rare or unusual interest. • Exoteric knowledge is, to the contrary, open to everybody although it does mean that anybody can understand it. By its representation of the domain (object), knowledge has three types: — Exact knowledge. — Fuzzy or vague knowledge. — Indeterminate knowledge including probabilistic knowledge. To understand the difference between these types of knowledge, let us consider the following examples. Imagine we take an urn with ten balls. If all balls in the urn are definitely blue and we know this, then we have exact knowledge that if we take one ball from the urn at random, then it will be a blue ball. If the color of the balls is between blue and green, then because there is no strict boundary between blue and green, we have fuzzy knowledge that if we take one ball from the urn at random, then it will be a blue ball. If five balls in the urn are definitely blue, five balls in the urn are definitely red
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and we know this, then we have probabilistic knowledge that if we take one ball from the urn at random, then it will be a blue ball, namely, we assume that the probability will be 1/2. Probabilistic knowledge is knowledge for which only the probability of being correct (true) is given. On one hand, this is contrary to the established through millennia approach to knowledge, which has to be always true. On the other hand, probability theory was created by Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665) in an attempt to describe uncertain knowledge in mathematical terms. However, the crucial incursion of probabilistic knowledge happened with the advent of quantum mechanics, which persuasively demonstrated that in many situations, people could have only probabilistic knowledge about nature. This changed understanding of the role of probabilistic knowledge. The chief proponent of the new approach was Hans Reichenbach (1891–1953). He assumed that probability is the pillar of knowledge systems and without this understanding, the structure of the world cannot be correctly represented and interpreted because, according to Reichenbach, knowledge about the future cannot be as accurate as knowledge about past events (Reichenbach, 1949). Consequently, knowledge about the future is inevitably probabilistic. Moreover, descriptions of the past events also are not completely accurate and thus, they demand probabilistic representation. In essence, all knowledge can be only probabilistic in such a way that for each knowledge unit, there is the probability of being true or correct. It is interesting that mathematics, which is traditionally treated as the most exact discipline because, as many think, mathematical proofs establish, in some sense, absolute knowledge, is also coming to probabilistic knowledge. Namely, some mathematicians suggest using probabilistic proofs of mathematical results. In this case, a theorem is asserted as true only with some high probability p (cf., for example, (Bass and Burdzy, 1989; Alon and Spencer, 2000)). Traditionally probability is considered as a function that takes values in the interval [0, 1] although each value of this function is also called the probability of an event. All conventional interpretations of probability support this assumption about the range of probability, while all popular formal descriptions, e.g., axioms for probability,
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such as Kolmogorov’s axioms, canonize this premise. However, scientific research and practical problems brought researchers to the necessity to use more wide-ranging concepts of probability in general and negative probabilities, in particular. Negative probabilities have been extensively used in physics (Dirac, 1930; 1942; 1974; Heisenberg, 1931; Wigner, 1932; Pauli 1943; 1956; Feynman, 1950) and mathematical finance (Jarrow and Turnbull, 1995; Duffie and Singleton, 1999; Forsyth et al., 2001; Haug, 2004; Burgin and Meissner, 2011; 2012), their mathematical theory is developed in (Bartlett, 1945; 1986; Allen, 1976; Burgin, 2009; 2010; Khrennikov, 2009). Applications of negative probability show that it has been useful for knowledge evaluation in physics and mathematical finance. However, negative probability could be a useful tool for representation, exploration and utilization of probabilistic declarative and representational knowledge in general and not only in these areas. This possibility is based on the existence of opposite knowledge, namely, if a statement r contains some knowledge, than it is natural to assume that the statement r (not r) contains opposite knowledge. Let us take a statement r and assume that it is true with the probability p(r). In classical logic, the Law of Excluded Middle tells us that when r is not true, then the negation r of r is true. This implies the equality p(r) = 1 − p(r). However, in real life, there is often a possibility when different options exist in the case when r is not true. For instance, it is possible that r is not defined for some objects. Thus, the probability p(r) is not uniquely defined by the probability p(r). In this situation, it is beneficial to use probabilities that can take both positive and negative values treating r as the statement opposite to r. Then a negative value of the probability p(r) can be interpreted as positive probability for the opposite statement r. It is possible to read more about interpretation of negative probability in (Burgin, 2010e). Probabilistic operational knowledge is represented by probabilistic algorithms and automata, while more general indeterminate operational knowledge is represented by non-deterministic algorithms and automata. A non-deterministic algorithm is an algorithm where the result and/or the way the result is obtained depend on chance. Examples of
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non-deterministic algorithms are non-deterministic finite automata or non-deterministic Turing machines (Burgin, 2005). In turn, non-deterministic algorithms are examples of non-deterministic operational knowledge. A probabilistic algorithm also called randomized algorithm is an algorithm where the result and/or the way the result is obtained depend on chance with the known probability. Examples of probabilistic algorithms are probabilistic finite automata or probabilistic Turing machines (Burgin, 2005). Probabilistic algorithms are examples of probabilistic operational knowledge. There is also a domain-oriented classification: — General knowledge. — Domain-specific knowledge. Note that general and domain-specific knowledge are the endpoints of the knowledge domain spectrum. Pollock and Cruz (1999) divide knowledge into several areas: — — — — — —
Perceptual knowledge is knowledge from perception. A priori knowledge is what is known independently of experience. Moral knowledge is knowledge of ethical principles. Memorized knowledge is knowledge from the memory. Inductive knowledge is knowledge of inductive generalizations. Knowledge of other minds.
Here are some more classifications of knowledge. Reif (1997) use the following classification of knowledge: — — — — — — —
General knowledge. Procedural interpretation knowledge Declarative knowledge. Procedural knowledge. Special knowledge. Compiled knowledge. Coherent knowledge.
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Reif and Allen (1992) use the following classification of knowledge: — — — — — — — —
General knowledge. Main interpretation knowledge. Definitional knowledge. Ancillary knowledge. Supplementary knowledge. Case-specific knowledge. Entailed knowledge. Concept knowledge.
To categorize knowledge, Wiig (1993) constructs a threedimensional classification. The first is the possession dimension with three categories: — Public knowledge. — Shared knowledge. — Personal knowledge. The second is the dynamic dimension with two categories: — Active knowledge. — Passive knowledge. The third is the typological dimension with four categories: — — — —
Factual knowledge. Conceptual knowledge. Expectational knowledge. Methodological knowledge.
To categorize knowledge, Boisot (1998) constructs a threedimensional classification. The first is the codification dimension with two categories: — Codified knowledge. — Uncodified knowledge. The second is the abstraction dimension with two categories: — Abstract knowledge. — Concrete knowledge.
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The third, diffusion dimension has two categories: — Diffused knowledge. — Undiffused knowledge. Ueno et al. (1987) consider two types of knowledge: — Factual knowledge (facts). — Knowledge for decision-making (rules). van Dijk (2004) introduces several classifications of knowledge. The social classification: — — — —
Personal knowledge. Interpersonal knowledge. Social (group) knowledge. Cultural knowledge. The hierarchical classification:
— Specific/particular knowledge. — General knowledge. The ontological classification: — — — — — —
Real knowledge. Concrete knowledge. Abstract knowledge. Fictitious knowledge. Historical knowledge. Future knowledge. The confidence classification:
— Absolutely certain knowledge. — More or less certain knowledge. Tuomi (1999) suggested an eight-fold bidirectional classification of knowledge, which is presented in Table 2.1 and has eight classes of knowledge.
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Table 2.1. The eight-fold bidirectional classification of knowledge. Dynamic typology Acquisition typology
Self-referential, i.e., active, knowledge
Stockpiled (or sediment), i.e., passive, knowledge
Ontogenetic, i.e., learned, knowledge
Cognitive knowledge
Habitual knowledge
Phylogenetic, i.e., transgenerational, knowledge
Socio-cultural knowledge
Instinctive knowledge
Tuomi (1999) remarks that habits, i.e., habitual knowledge and its expression in behavior, bridge the mind and body by imbedding meaning into the body. Besides, in contrast to active knowledge, passive (sediment) knowledge does not change or changes very slowly. On the individual level, passive (sediment) knowledge is wired into the structure of the personality, while on the organizational (social) level, it is embedded in the organizational (respectively, social) practice. De Jong and Ferguson-Hessler (1996) use the following classification of knowledge: — Situational knowledge is knowledge about situations as they typically appear in a particular domain. — Procedural knowledge contains actions and operations that are valid within a domain helping problem solver to make transitions from one problem state to another. — Conceptual knowledge is static knowledge about facts, principles, and concepts that apply within a domain. — Strategic knowledge helps organizing problem-solving processes providing a general plan of solution activities. In addition, De Jong and Ferguson-Hessler (1996) define levels of knowledge introducing the hierarchical classification: — Surface or superficial knowledge. — Deep or deep-level knowledge. There are different approaches to discern surface and deep-level knowledge. The accessibility hardship differentiates these types in the
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following way: — Surface or superficial knowledge is easily accessible knowledge. — Deep or deep-level knowledge is knowledge that is hard to obtain. For instance, according to the accessibility hardship, knowledge that the Sun gives light is surface knowledge, while knowledge that the Sun radiates its energy due to thermonuclear processes is deep knowledge. The representability trait sets these two types apart on the different foundation: — Surface or superficial knowledge is knowledge about outward attributes of the knowledge object (domain). — Deep or deep-level knowledge is knowledge about imperative properties of the knowledge object (domain). For instance, according to the representability trait, knowledge that the Earth is big is surface knowledge, while knowledge that the Earth is a planet is deep knowledge. One of the criteria for knowledge classification is the nature of the carrier of knowledge. According to this criterion, we have the following types: digital knowledge, printed knowledge, written knowledge, symbolic knowledge, molecular knowledge, quantum knowledge, and so on. For instance, digital knowledge is represented by digits, printed knowledge is contained in printed texts, and quantum knowledge is contained in quantum systems. Another criterion is the type of the system that acquires information used in knowledge formation. According to this criterion, we have the following types: visual knowledge, auditory knowledge, olfactory knowledge, cognitive knowledge, genetic knowledge, and so on. For instance, according to neuropsychological data, 80% of all information that people get through their senses is visual information, 10% of all information is auditory information, and only 10% of information that people get through other senses. One more criterion for knowledge classification is the specific domain this knowledge is about. According to this criterion, we have the following types: physical knowledge biological knowledge, genetic
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knowledge, social knowledge, physiological knowledge, ethic knowledge, esthetic knowledge, weather knowledge, car knowledge, emotional knowledge (in the sense of (Sanders, 1985; Keane and Parrish, 1992; George and McIllhagga, 2000; Bosmans and Baumgartner, 2005; Knapska et al., 2006)), author knowledge, political knowledge, health care knowledge, quality knowledge, geological knowledge, economical knowledge, stock market knowledge, and so on. One more criterion is the area to which knowledge is applied. This criterion determines orientations of knowledge. It is possible to discern the following orientations of knowledge: ∗ Cognitive knowledge provides information about different objects and domains. ∗ Procedural knowledge serves for organization of behavior of people and animals, functioning of various systems and performing actions. ∗ Educational knowledge helps learning and becoming educated. ∗ Pragmatic knowledge serves for gaining something. ∗ Economic knowledge serves for getting profit. Machlup (1992) introduced five types of knowledge: ◦ practical knowledge; ◦ intellectual knowledge, which includes knowledge related to general culture and knowledge that satisfies of intellectual curiosity; ◦ pastime knowledge, i.e., knowledge that satisfies non-intellectual curiosity or the desire for light entertainment and emotional stimulation; ◦ spiritual and religious knowledge; ◦ unwanted knowledge, which is accidentally acquired and aimlessly retained. Kesh and Ratnasingam (2007) use the following knowledge classification: • Declarative knowledge as know-about; • Procedural knowledge as know-how; • Individual knowledge as knowledge created and inherent in the individual;
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• Social knowledge as knowledge created and inherent in the collective actions of the group; • Conditional knowledge know-when; • Relational knowledge know-with; • Pragmatic knowledge as useful knowledge of an organization. Knowledge usually has gradations in its orientation. Let us consider some of them. Cognitive levels of knowledge about an object A reflect the intent of knowledge. These levels, or grades, are ordered from the lowest to highest: 1. Knowledge about existence of the object A, which includes naming when this knowledge is explicit. 2. Knowledge as a description of the object A and/or of what is related to the object A, which includes knowledge of properties of A and/or of what is related to A. 3. Knowledge as understanding (of a description) of properties of the object A and/or of what is related to the object A. 4. Knowledge as holistic understanding of the object A. 5. Knowledge as an ability to explain properties of the object A and/or of what is related to the object A. 6. Knowledge as an ability to explain the structure of the object A. 7. Knowledge as an ability to explain the object A as a system with its internal and external connections. As an example of levels of knowledge, we can take knowledge about such an object as the Earth. At first, people only knew that there was something where they lived, that is, the Earth (the first level). Then they found some properties of the Earth by observation (the second level). For instance, they found that different plants grow on the Earth and different animals live on the Earth. Later people began to understand some properties of the Earth (the third level). For instance, they understood how to use soil to grow useful plants and how seasons are changing. However, holistic understanding of the Earth was achieved only when it was demonstrated that the Earth is a planet, which rotates around the Sun (the fourth level). Later
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scientists explained some properties of the planet Earth (the fifth level). For instance, it was explained why the Earth rotates around the Sun. Finding the configuration of the Solar System brought the knowledge about the external structure of the Earth, while geological studies explained (to some extent) the inner structure of the Earth (the sixth level). However, the seventh level is not yet achieved as science still does not have a good explanation of the Earth as a multifaceted system, which includes ecological, geological, and physical (not only mechanical) explanations. It is possible to compress cognitive levels of knowledge into three epistemological gradations of knowledge: — Xerox knowledge is knowledge without understanding. — Understandable knowledge is knowledge that is understood by its owner. — Explainable knowledge is knowledge that its owner can explain to others. Relation to the knowledge domain, i.e., the domain this knowledge is about, gives us one more classification: • Complete knowledge completely describes its domain. • Partial knowledge only partially describes its domain. • Irrelevant knowledge does not at all describe its domain. Operational levels of knowledge about an object A reflect the potency of knowledge. These levels are also ordered from the lowest to highest: 1. Knowledge as an ability to perceive the object A, which usually includes naming. 2. Knowledge as an ability to do something with the object A. 3. Knowledge as an ability to use the object A for some purpose. For instance, at first, people were able to perceive electricity in the form of a lightning (the first level). Then they invented the lightning rod to divert lightning from people and buildings (the second level). Later they learned how to use electricity (the third level) and as
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we know, now electricity is one of the material pillars of the human civilization. Relations between knowledge and a given system Q determine three important types of knowledge: — knowledge K is accessible for Q if Q has access to K, — knowledge K is available for Q if Q can get (obtain) K, — knowledge K is acceptable for Q if given K, the system Q can accept K. There is essential difference between these classes. For instance, even if a person has access to some knowledge, it does not mean that this person can get this knowledge. Imagine you come to a library that has one million books. You know that knowledge you need is in some of these books but you do not know in which one. If you do not have contemporary search tools to get this knowledge and can only read books to find it, then it will not be available to you. You cannot read all million books. Here is one more example. Knowledge about Lebesgue integration, which is a high-level mathematical concept with a developed theory, is accessible to anybody who has a book on Lebesgue integration but it is available only to those who know mathematics. Some laws of physics, e.g., Heisenberg’s uncertainty principle, state that there is knowledge about physical reality unavailable to anybody. Some mathematical results, e.g., G¨odel’s incompleteness theorems, claim that there is knowledge about mathematical structures unavailable to anybody. In computer science, it is proved (cf., for example, (Sipser, 1997; Burgin, 2005)) that for a universal Turing machine, knowledge whether this machine halts given arbitrary input is unavailable. As we know, when a person can get some knowledge, it does not mean that this person accepts this knowledge. Imagine you read about some unusual event in a newspaper, but you do not believe that it is possible. Then knowledge about this event is available to you, but you cannot accept it because you do not believe that it is possible. There are many historical examples of such situations. For millennia, mathematicians tried to directly prove that it is possible to deduce the fifth postulate of the Euclidean geometry
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from the first four postulates. Being unable to achieve this, mathematicians were becoming frustrated and tried some indirect methods. Girolamo Saccheri (1667–1733) tried to prove a contradiction by assuming that the first four postulates were valid, while the fifth postulate was not true (Burton, 1997). To do this, he developed an essential part of what is now called a non-Euclidean geometry. Thus, he was able to become the creator of the first non-Euclidean geometry. However, Saccheri was so sure that the only possible geometry is the Euclidean geometry that at some point, he claimed a contradiction and stopped further reasoning. Actually, his contradiction was only applicable in Euclidean geometry. Of course, Saccheri did not realize this at the time and he died thinking he had proved Euclid’s fifth postulate from the first four. Thus, knowledge about non-Euclidean geometries was available but not acceptable to Saccheri. As a result, he missed an opportunity to obtain one of the most outstanding results in the whole mathematics. A more tragic situation due to biased comprehension involved such outstanding mathematicians as Niels Henrik Abel (1802–1829) and Carl Friedrich Gauss (1777–1855). As history tells us (Bell, 1965), there was a famous long-standing problem of solvability in radicals of an arbitrary fifth-degree algebraic equation. Abel solved this problem proving impossibility of solving that problem in a general case. In spite of being very poor, Abel himself paid for printing a memoir with his solution. This was an outstanding mathematical achievement. That is why Abel sent his memoir to Gauss, the best mathematician of his time. Gauss duly received the work of Abel and without deigning to read it he tossed it aside with the disgusted exclamation “Here is another of those monstrosities!” Moreover, people often do not want to hear truth because truth is unacceptable to them. For instance, the Catholic Church suppressed knowledge that the Earth rotates around the Sun because people who were in control (the Pope and others) believed that this knowledge contradicts to what was written in the Bible. Relations between these three types of knowledge show that any available knowledge is also accessible. However, not any accessible knowledge is available and not any acceptable knowledge is available
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or accessible. For instance, there are many statements that a person can accept but they are inaccessible for this person. A simple example gives theory of algorithms. It is known that given a word x and a Turing machine T , it is impossible, in general, to find whether T accepts x or not using only recursive algorithms (cf., for example, (Burgin, 2005)). Thus, knowledge about acceptance of x by T is acceptable to any computer scientist because this knowledge is neutral. At the same time, this knowledge is in principle inaccessible by recursive algorithms for infinitely many words. Exploring accessibility, we find that it is possible to have access to some knowledge to a different extent. For instance, imagine that you need two books A and B. The first one, A, is in your library at your home. You can go to the shelf where the book is and take it any time you want. At the same time, the second book, B, is only in the library of another city. You can get it, but only by the interlibrary exchange. Thus, both books are accessible, but the first one is much easier to retrieve. This shows that accessibility is a property of knowledge, which can be estimated (measured) and used in the knowledge quality assessment. As knowledge may be available to a different extent, availability is a graded property of knowledge, which can be estimated (measured) and used in the knowledge quality assessment. There are different levels at which knowledge may be acceptable. For instance, knowledge about yesterday’s temperature is acceptable as knowledge, while knowledge about tomorrow’s temperature is acceptable only as a hypothesis. Thus, acceptability is a graded, fuzzy, or linguistic property (Zadeh, 1973) of knowledge, which can be estimated (measured) and used in the knowledge quality assessment (cf., Section 6.2). In addition, it is possible to distinguish conditional counterparts of accessible, available, and acceptable knowledge. — knowledge K is conditionally accessible for Q if Q has access to a carrier of K; — knowledge K is conditionally available for Q if Q can get (obtain) a carrier of K;
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— knowledge K is conditionally acceptable for Q if given K, the system Q can accept a carrier of K. To see the difference between accessible knowledge and conditionally accessible knowledge, imagine a book in English. For a person E who knows English and has the book, knowledge in this book is accessible. At the same time, for a person D who does not know English and has the book, knowledge in this book is only conditionally accessible. There are different conditions for accessibility. One condition is that D learns English. Another condition is that D finds an interpreter. One more condition assures that, that the book is translated from English to the language D knows. To see the difference between available knowledge and conditionally available knowledge, imagine a book in English on Lebesgue integration. For a person D who knows English, basic calculus and can get this book, knowledge in it is available. At the same time, for a person C whose knowledge of mathematics is very low but she can buy this book, knowledge in it is only conditionally available. Location characteristics generate the following classification of knowledge: — Individual knowledge is knowledge to which only one person has access. — Group knowledge is knowledge to which only people from a certain group have access. — Public knowledge is knowledge accessible to everybody. Modalities reflect definite aspects of knowledge. There are three existential modalities of knowledge: — Existential knowledge reflects what is, i.e., the existing state of the knowledge domain. — Potential knowledge reflects what can be, i.e., possible state of the knowledge domain. — Compulsory knowledge reflects what must be, i.e., the necessary state of the knowledge domain.
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There are three confidential modalities of knowledge: — Assertoric knowledge affirms the content of an epistemic structure meaning that the epistemic structure is knowledge, which asserts something about its domain (object). — Hypothetic knowledge conjectures the content of an epistemic structure suggesting that an epistemic structure may be knowledge. — Erotetic knowledge inquires about the content of an epistemic structure expressing lack of knowledge and having the form of a question, problem or puzzle. For instance, an assertoric proposition asserts that something is (or is not) the case, in contrast to a hypothetic proposition, which asserts the possibility of something being (or not being) the case, or to an apodeictic proposition, which assert that something is (or is not) necessarily the case, e.g., something is necessarily or selfevidently true or false. Note that the division of propositions into these three classes is rather subjective depending on the opinion. For instance, for some people, e.g., for the majority of philosophers and mathematicians, the proposition “two plus two always equals four” is apodeictic. For other people, e.g., for the majority of physicists, this proposition is assertoric, while for those who know about non-Diophantine arithmetics studied in (Burgin, 1977; 1997g; 2007; 2010c) this proposition is hypothetic. Researchers also used other names for these types of knowledge. For instance, LaDuke called erotetic knowledge by the name antiknowledge (LaDuke, 2002). There are three temporal modalities of knowledge, which reflect directions in knowledge time-orientation: — Knowledge about the future has anticipation modality. — Knowledge about present has current modality. — Knowledge about the past has bygone modality. It is easy to comprehend these modalities for descriptive knowledge. For instance, propositions stating something about the past,
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e.g., “Yesterday was cold”, have bygone modality. Propositions stating something about present, e.g., “Today is warm”, have current modality. Propositions stating something about the future, e.g., “According to the weather prognosis, tomorrow will be hot”, have anticipation modality. Note that knowledge with any modality can be true or false although knowledge with anticipation modality is more often false in comparison with two other modalities as people rarely have the ability to predict future. However, operational and representational knowledge also have temporal modalities. For instance, taking operational knowledge, we see that now the majority of quantum algorithms (Deutsch, 1985) have anticipation modality because there are no quantum computers, which can perform these algorithms. At the same time, algorithms for counting using abacus have bygone modality, while algorithms used in contemporary operating systems have current modality. Taking representational knowledge, we see that, for example, the Ptolemaic model of the Solar system has bygone modality, while the Copernican model of the Solar system has current modality. 2.2. Existential characteristics of knowledge Knowledge is an unending adventure at the edge of uncertainty. Jacob Bronowski
Existence is an important property of anything. Properly inquiring whether an object A exists, it is necessary to define or at least, to describe this object. Thus, discussing existence of knowledge, we need to explain what knowledge is and here we come to a big problem. From ancient times, as we have seen in the previous chapter, philosophers and other researchers have tried to build a comprehensive definition of knowledge and still different opinions exist causing a lot of controversy in this area. There were many suggestions but in spite of this, the diversity of essences called knowledge evades any exact and comprehensive definition. In spite of many unsuccessful efforts to figure out the unique definition of knowledge, there are various descriptions of knowledge, some of which are essentially disparate.
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For instance, Froehlich writes, “for some philosophers, validated, true information is that which coheres with other truths (coherence theory of truth). For others, what corresponds to reality (correspondence theory of truth). For others, it is what works or is functional (pragmatic theory of truth). At any event it is always contextual” (cf., ((Zins, 2007)). Even larger diversity of understandings and interpretations is reflected in dictionaries and encyclopedias. For instance, in the Webster’s Revised Unabridged Dictionary (1998), knowledge is defined as: 1. A dynamic process: (a) (b) (c) (d) (e) (f)
the act or state of knowing; clear perception of fact, truth, or duty; certain apprehension; familiar cognizance; learning; cognition.
2. An object: (a) that which is or may be known; (b) the object of an act of knowing; (c) a cognition. 3. An object: (a) (b) (c) (d) (e) (f)
that which is gained and preserved by knowing; instruction; acquaintance; enlightenment; scholarship; erudition.
4. A property: (a) that familiarity which is gained by actual experience; (b) practical skill; as, a knowledge of life. 5. As a domain: (a) scope of information; (b) cognizance; (c) notice; as, it has not come to my knowledge.
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In the Oxford English Dictionary, knowledge is defined as: (i) expertise, and skills acquired by or education; the theoretical or subject, (ii) what is known in a particular information, (iii) awareness or familiarity gained situation.
a person through experience practical understanding of a field or in total; facts and by experience of a fact or
In monographs on knowledge engineering (Osuga et al., 1990), we find the following definitions: 1. Knowledge is a result of cognition. 2. Knowledge is a formalized information, to which references are made or which is utilized in logical inference. The most popular approach in artificial intelligence is to define knowledge functionally as it was suggested by Allen Newell (1927– 1992). It means that if an external observer O can ascribe to some system A, e.g., to an agent, definite goals and if O witnesses that A is behaving so as to achieve these goals in systematic, rational mode, i.e., according to the principle of rationality, the observer O assumes that A has knowledge (Newell, 1982). One of the problems with this methodology is that as it is demonstrated in (Burgin and Krymsky, 1985), there is no one unique concept of rationality — different people and different systems interpret rationality in their own way. It implies that rationality is relative and what seems rational to one person can be completely irrational to another one. This makes the functional definition of knowledge essentially dependent on the observer and instead of unification, it generates a multiplicity of concepts of knowledge. Herbert Simon (1916–2001) suggested that the development of information technology changes the meaning of the term “to know.” While traditional meaning is to have knowledge in ones memory, now it is understood as to have knowledge where to find the necessary knowledge (Simon, 1971).
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All approaches to knowledge discussed above give some general ideas about knowledge but these definitions, or better, to say, descriptions, are not sufficiently constructive even to unmistakably distinguish knowledge from knowledge representation and from information. The following example demonstrates differences between knowledge and knowledge representation. Some event may be described in several articles written in different languages, for example, in English, Spanish, and Chinese, but by the same author. These articles convey the same semantic information and contain the same knowledge about the event. However, representation of this knowledge is different. As in the case with information, there is also a distinction between knowledge representation and knowledge carrier. For instance, an individual can be a carrier of knowledge but representation of this knowledge is only in his brain. That is why here we do not strive to obtain an encompassing precise definition of knowledge. There are many books and papers, in which this goal is pursued. As an example, we can take such an interesting and well-written book as (Pollock and Cruz, 1999). In contrast to this, we follow the steps of scientists, who build models of studied phenomena instead of explanation of what these phenomena are in layman terms. Thus, our goal is to build efficient and flexible models of knowledge on different levels of its existence. To achieve this goal, we implement the pragmatic approach to knowledge, which is adopted in artificial intelligence (AI) where there is often no “attempt to define knowledge in the philosophical or even the popular view” (Fayyad et al., 1996). That is, we do not try to give a complete characterization of knowledge precisely discerning it from other epistemological structures. Our goal is to describe knowledge structure, acquisition, behavior, relations, and utilization. Many researchers assume that in contrast to data and information, knowledge, exists only in some knowledge system, such as an individual, society, or a knowledge base. Consequently, we apply the observer-oriented approach to knowledge. Namely, we do not try to exactly define knowledge in general or to describe it in an absolute manner. In contrast to this, we presume that an observer
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(user or knower) characterizes and utilizes some epistemic structures as knowledge. That is, dealing with epistemic structures, an observer makes use of an assembly K of knowledge properties as a knowledge criterion labeling epistemic structures that satisfy criteria from the assembly K by the name knowledge (Chisholm, 1989; Pollock and Cruz, 1999). Criteria (properties) from the arrangement K are called existential characteristics of knowledge because to find whether knowledge with respect to K exists, we need to find an object that satisfies these criteria. For instance, Fayyad et al. (1996) write “we can consider a pattern to be knowledge if it exceeds some interestingness threshold . . .” However, in (Burgin, 1989a, 2010), it is assumed that these criteria are subjective and can be different for different individuals and different societies. For instance, what was treated as knowledge three millennia ago, e.g., that the Sun rotates around the Earth, now is often considered as being a misconception. In a similar way, Fayyad et al. (1996) write “knowledge . . . is purely user oriented and domain specific and is determined by whatever functions and thresholds the user chooses.” Let us look at some examples of knowledge criteria (existential characteristics of knowledge). It is possible to consider the following assembly K of knowledge properties as an objective knowledge criterion: 1. To be a belief. 2. To be true. 3. To be justified. This criterion corresponds to the time-worn and widely criticized philosophical definition of knowledge as a “justified true belief” (Russell, 1912; 1948; Gettier, 1966; Goldman, 1967; Lewis, 1996). As Thorkelson shows, this definition suffers from three major problems centered especially around the term “belief” (Thorkelson, 2008). First, this definition reduces knowledge to propositional knowledge, “knowing-that.” As a result, other types of knowledge such as operational knowledge in the form of practical “know-how” (knowledge embodied in actions, behaviors, and procedures), as well as
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representation knowledge in the form of affective states (knowledge embodied in emotion and sentiment) and phenomenological acquaintance (conferred, for instance, by sensory experience or artistic representation) are excluded. Second, insofar as “belief” is considered as a mental state of the individual, the definition directs towards an egocentric rather than socio-centric theory of knowledge (Silverstein, 2004). Third, since a belief is an isolated, singular entity, it is possible to think of knowledge as an unordered aggregate of isolated epistemic items, e.g., propositions, instead of as a coordinated, though not necessarily total epistemic system. Scheffler (1965) suggests the following criteria of descriptive/ declarative knowledge: 1. To be a belief. 2. To be supported by an adequate evidence. To efficiently treat descriptive/declarative knowledge, it is also possible to use the following assembly K of knowledge properties as a subjective knowledge criterion: 6. To be a belief. 7. To be believed (assumed) as being true. 8. To be believed (assumed) as being justified. Here is one more similar assembly K of knowledge properties, which can be used as a knowledge criterion: 1. To be a belief. 2. To give correct information about the knowledge object (domain). 3. To give exact information about the knowledge object (domain). However, knowledge criteria do not need to include the condition of being a belief. For instance, operational knowledge, as a rule, has the form of instructions and not of beliefs. Consequently, it is possible to use other knowledge criteria such as “to be useful,” “to be correct” or “to be constructive.” The observer-oriented approach makes it possible to solve some paradoxes related to knowledge. For instance, let us imagine that
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two millennia ago somebody was asked what the Sun was doing. Historical sources allow us to presume that he or she would tell: The Sun is giving light and is rotating around the Earth.
(2.1)
Contemporary knowledge supports the first statement but rejects the second one as according to the present-day celestial mechanics, the Earth rotates around the Sun. It means that what people considered as knowledge two millennia ago, now is treated as a misconception. This example and many others bring people to the skeptical view on knowledge, which claims that it is impossible to have knowledge or to correctly discern knowledge from other epistemic structures. However, the observer-oriented approach to knowledge gives a different answer to this riddle. It says that the statement (2.1) was a temporal subjective knowledge about the Sun. In our approach, we distinguish three basic types of observers: internal observers and two kinds of external observers — real external observers and abstract external observers. An external observer can be either a real system (real external observer), such as a scientist, or an abstract system (abstract external observer), such as a scientific theory. An internal observer is a knowing system (knower), i.e., the system such as a reader of a book or a computer that stores knowledge. Usually, an observer is treated as a physical system that in the same way, i.e., physically, interacts with the observed object. Very often, an observer is interpreted as a human being. However, here we use a broader perspective, allowing an abstract system also to be an observer because in our case the observed object is knowledge, i.e., it is an abstract system itself. In addition, it is necessary to understand that interaction between abstract systems involves representations of these systems and the performing system, which perform interaction and is usually physical. An interesting approach to knowledge posits it as a process of knowing. For instance, Polanyi (1974) regards knowledge as both
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static “knowledge structure” or “knowledge item” and dynamic “knowing”. When people discuss something, they usually either suppose that what they discuss exists or try to show that it does not exist. It means that existence is treated as a dualistic possibility — either something exists or it does not exist. However, this is very inexact and in many cases even wrong because there are different kinds, types, and modes of existence (Burgin, 2012). For instance, does a fictional hero of some novel exist? Many think that he does not exist but a more correct answer is that this fictional hero exists as a mental entity but does not exist as a material object. This kind of existence was considered by Alexius Meinong (1853– 1920) in his analysis of language. He assumed that language, when properly understood, is a guide to ontology and this ontology permits two kinds of existence: genuine existence of physical objects and generalized existence of objects in imagination, e.g., fictional characters, square circles, and golden mountains. To have a complete picture of reality, we come to the conclusion that forms of existence are determined by the world stratification and structuration (Burgin, 2012). Taking the structuration determined by the Existential Triad of the world (Burgin, 1997; 2010), which stems from the long-standing tradition in philosophy and is presented in Figure 2.1, we come to the three existential forms — material/ physical existence, mental existence, and structural existence. In this stratification, the Physical (material) World is interpreted as the physical reality studied by natural sciences (cf., for example, (Born, 1953)), the Mental World encompasses different levels of mentality, and the World of Structures consists of various forms and types of structures. World of Structures
Physical World
Mental World
Figure 2.1. The existential triad of the world
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Usually people comprehend the Mental World as individual mentality. Science extended this picture exploring the Mental World on three levels, all of which are included in the Mental World of the Existential Triad: • The first level treats mentality of separate individuals and is the subject of psychological studies. As in the case of physical reality, now psychology knows a lot about mentality/psyche of people. It is necessary also to remark that in the same way as physics does not study the physical reality as a whole but explores different parts, levels and aspects of it, psychology also separates and investigates different parts and aspects of the individual mental reality, such as intelligence, emotions, conscience, or unconscious. However, there are components of individual mentality that yet lay beyond the studies of contemporary psychology. • The second level deals with group mentality of various groups of people and is the subject of social psychology, which bridges sociology and conventional psychology. In particular, this level includes group conscience, which incorporates collective memory (Durkheim, 1984), collective intelligence (Brown and Lauder, 2000; Nguen, 2008a) and is projected on the collective unconscious in the sense Jung (see (Jung, 1969)) by the process of internalization. • The third level encompasses mental issues of society as a whole. Social mentality includes social memory, social intelligence, and social conscience. Social psychology also studies some features of this level. However, these three levels do not exhaust the whole Mental World. In fact, the Mental World from the Existential Triad comprises higher (than the third) levels of mentality although they are not yet studied by science (Burgin, 1997; 2010). It is possible to relate higher levels of the Mental World to the spiritual mystical worlds described in many religious esoteric teachings. Some thinkers, following Descartes, consider the individual mental world as independent of the physical world. Others assume that individual mentality is completely generated by physical systems of the organism, such as the nervous system and brain as its part. However,
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in any case, the mental world is different from the physical world and constitutes an important part of our reality. Psychological experiments and theoretical considerations show that the Mental World is stratified into three spheres: cognitive or intellectual sphere, affective or emotional sphere and effective or regulative sphere. This stratification is based on the extended theory of triune brain and the concept of the triadic mental information (Burgin, 2010). The Mental World has elements and components, which are similar to elements and components of the Physical World. In a natural way, the Mental World has its mental space, mental objects (structures), and mental representations (Burgin, 1998a). It is also necessary to explain that the World of Structures directly corresponds to Plato’s World of Ideas/Forms because ideas or forms might be associated with structures. Indeed, on the level of ideas, it is possible to link ideas or forms to structures in the same way as atoms of modern physics may be related to atoms of Democritus and Leucippus. Only recently, modern science came to a new understanding of Plato ideas, representing the global world structure as the Existential Triad of the world, in which the World of Structures is much more comprehensible, exact, and explored in comparison with the World of Ideas/Forms. When Plato and other adherents of the World of Ideas/Forms were asked what idea or form was, they did not have a satisfactory answer. In contrast to this, many researchers have been analyzing and developing the concept of a structure (Ore, 1935; 1936; Bourbaki, 1948; 1957; 1960; Bucur and Deleanu, 1968; Corry, 1996; Burgin, 1997; 2010; 2011; 2012; Landry, 1999; 2006). It is possible to find the most thorough analysis and the most advanced concept of a structure in (Burgin, 2012). As a result, in contrast to Plato, mathematics and science has been able to elaborate a sufficiently exact definition of a structure and to prove existence of the world of structures, demonstrating by means of observations and experiments, that this world constitutes the structural level of the world as a whole. Informally, a structure is a collection of symbolic (abstract) objects and relations between these objects. Each system, phenomenon or process in nature, technology or society has some
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structure. These structures exist like material things, such as tables, chairs, or buildings do, and form the structural level of the world. When it is necessary to learn or to create a system or to start a process, it is done, as a rule, by means of knowledge of the corresponding structure. Structures mould things in their being and comprehension. If we say that structures exist only embodied in material things, then we have to admit that material things exist only in a structured form, i.e., matter (physical entities) cannot exist independently of structures. For instance, atomic structure influences how the atoms are bonded together, which in turn helps one to categorize materials as metals, ceramics, and polymers and permits us to draw some general conclusions concerning the mechanical properties and physical behavior of these three classes of materials. Even chaos has its structure and not a unique one. The three worlds from the Existential Triad are not separate realities: they interact and intersect. Individual mentality is based on the brain, which is a material thing, while in the opinion of many physicists mentality influences physical world (see, for example, (Herbert, 1987)). At the same time, our knowledge of the physical world essentially depends on interaction between mental and material worlds (see, for example, (von Bayer, 2004)). Moreover, our mentality influences the physical world and can change it. We can see how ideas change our planet, create many new things and destroy existing ones. Even physicists, who research the very foundation of the physical world, developed the, so-called, observer-created reality interpretation of quantum phenomena. A prominent physicist, Wheeler, suggests that in such a way it is possible to change even the past. He stresses (Wheeler et al., 1983) that elementary phenomena are unreal until observed. In addition, there is a projection of the Mental World into the Physical World in the form of creations of human mentality (creativity), such as books, movies, magazines, newspapers, cars, planes, computers, and computer networks. This projection determines the Extended Mental World, which consists of the Mental World and its projection. The Extended Mental World correlates with the World 3 from the General Popper Triad of the world.
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Structural and material worlds are even more intertwined. Actually, no material thing exists without a structure. Even chaos has its chaotic structure. Structures make things what they are. For instance, it is possible to make a table from different material: wood, plastics, iron, aluminum, etc. What all these tables have in common is not their material; it is specific peculiarities of their structure. In a similar way, according to Poincar´e (1908), space is in reality amorphous, and it is only the things in space that give it a structure (form). As some physicists argue, physics studies not physical systems as they are but structures of these systems, or physical structures. In some sciences, such as chemistry, and areas of practical activity, such as engineering, structures play the leading role. For instance, the spatial structure of atoms, chemical elements, and molecules determines many properties of these chemical systems. Contemporary physics treats the physical world as a net of interacting components (systems), where there is no physical meaning to the state of an isolated object. A physical system (or, more precisely, its contingent state) is represented by the net of relations with the surrounding objects it retains. As a result, the physical structure of the world is identified with such a global net of system relationships. As North (2009) writes, physics is supposed to be telling us about the nature of the world, while physical theories are formulated in a mathematical language, using mathematical structures, which implies that mathematics is somehow telling us about the physical make-up of the world. Plato postulated independent existence of his world of ideas, and it was demonstrated that it is possible to consider structures as scientific counterparts of Plato ideas (Burgin, 2011). Thus, it is natural to ask the question whether structures exist without matter. Here we are not going into detailed consideration of this fundamental problem. It is important that as a coin has two sides, material things always have two aspects — substance and structure. Like atoms studied by contemporary physics were prefigured by ancient thinkers, such as Democritus from Abdera (460–370 B.C.E.) and Leucippus of Miletus (ca. 480 – ca. 420 B.C.E.), the Existential
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Triad also has several precursors suggested by various thinkers such as Plato, Aristotle, Popper, or G¨odel. John Locke also suggested a similar triadic stratification of the world and knowledge about the world. He wrote: “All that can fall within the compass of human understanding, being either, first, the nature of things, as they are in themselves, their relations, and their manner of operation: or, secondly, that which man himself ought to do, as a rational and voluntary agent, for the attainment of any end, especially happiness: or, thirdly, the ways and means whereby the knowledge of both the one and the other of these is attained and communicated; I think science may be divided properly into these three sorts.” (Locke, 1690)
This gives us the structures presented in Figures 2.2 and 2.3. Naturally, the structure of science, according to Locke, is structurally isomorphic to his structure of the World. Note that Locke triads are similar (but not exactly) to the Existential Triad. The difference is that: (1) nature is only a part of the physical world, e.g., people and machines are elements of the physical world but do not belong to nature; (2) signs are only one kind of structures; and (3) although mentality is a pivotal characteristics of human beings, not only human beings have mentality, while a human being cannot be reduced to her/his mentality.
World of Signs
Nature
Human beings
Figure 2.2. The Locke triad of the world Doctrine of Signs
Natural sciences
Social sciences and Humanities
Figure 2.3. The Locke triad of science
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According to the existential stratification of reality, knowledge exists as structure in the world of structures but has many representations in two other worlds. Namely, there are different physical representations of knowledge, e.g., printed texts in books and journals, written algorithms, software of computer and networks, written manuscripts or states of the computer memory in knowledge bases. There are also mental representations of knowledge in the mentality (mind) of people and in the mentality of society. However, even having correct knowledge representation, the knowing system (knower) does not always possess knowledge. Indeed, imagine a situation when you have a book written in a language you do not know, e.g., in Hindu or Japanese. This book can have a lot of knowledge but you do not possess this knowledge because you cannot read the book. Thus, knowledge exists but is not accessible to you. This and other examples show that there are many modes, modalities, kinds, types, gradations, and dimensions of existence (Burgin, 2012). However, when existence is treated as a property, it usually means that some object exists, at least, as a mental entity, i.e., it has a name (sometimes several names) and different ascribed properties. Although we can ask the question whether there is a material object that has these properties. For instance, after Dirac in 1930’s suggested a new particle — positron, the majority of physicists believed that such a particle did not exist. However, after several years existence of positron was proved by experiments. Knowing is frequently related to existence. Very often people assume that what they do not know does not exist. This exhibits the subjective form of existence, that is, existence in mentality of people. For instance, for a long time, people did not know that the Solar system has such a planet as Neptune. Consequently, this planet did not exist for them although it existed as a celestial body, i.e., it had physical existence. Moreover, existence is a property not only of material things but also of theoretical constructs. For a long time, mathematicians did not know about negative numbers. Consequently, negative numbers did not exist for them. Even after negative numbers had been discovered in the East, namely, in China and India, and then brought to Europe, many European mathematicians tried to argue
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that negative numbers did not exist (Martinez, 2006). In a similar way, some notable mathematicians of the 19th century insisted that irrational numbers did not exist (Burton, 1997). For a long time, it was assumed that knowledge is something that exists only in the mentality of people. Some researchers believe that this is the crucial difference between knowledge and information, which exists in anything. However, the technological development changed the situation. Indeed, because knowledge is vital to the whole existence of people, various artificial tools have been invented for knowledge acquisition, storage, transmission, and transformation. Among other things, people invented and developed oral and written languages, papyrus, clay tablets, paper, printing, books, and computers. This brought an understanding that knowledge also existed not only in people’s mentality but also in various physical things but not in all in contrast to information. As a result, researchers started to explore knowledge in artificial systems only after computers came into being and the research area called AI emerged. Later when knowledge became crucial for organizations, researchers began studying knowledge not only on the level of individual mentality but also in the group (collective) mentality, performing research in social epistemology and knowledge management. One of the important results of this research was explication of distinctions between tacit knowledge, which exists in individual mentality, and explicit knowledge, which usually belongs both to individual mentality and to collective mentality. 2.3. Descriptive properties of knowledge and corresponding typology Beware of false knowledge; it is more dangerous than ignorance. Bernard Shaw
Descriptive properties (characteristics) of knowledge can be formally described by a function assigning values from the property scale to knowledge items or by an abstract property (cf., Chapter 5 and (Burgin, 1985)). For instance, truthfulness is a descriptive property with the scale {True, False}. Note that it is possible to represent
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relations between knowledge items as abstract properties (Burgin, 2010). Relational properties (characteristics) of knowledge characterize relations of knowledge items, e.g., represented by texts or computer files, to physical or mental objects (systems). For instance, the denotation theory of meaning is based on relations between knowledge items and objects they denote (Russell, 1905; Ryle, 1957; Parkinson, 1968). In other words, relational characteristics of knowledge are molded by relations between knowledge and systems that are related to knowledge. Contextual properties (characteristics) of knowledge are properties of objects related to knowledge. It is possible also to formally describe them by a function assigning values from the property scale to these objects or by an abstract property (cf., Chapter 5 and (Burgin, 1985)). Internal characteristics of knowledge are innate to knowledge as a structural phenomenon. We start with the structure of knowledge, which is the basic integral characteristic of knowledge. Note that we make a distinction between a knowledge structure and the structure of knowledge. A knowledge structure is one or several knowledge items in their structural form, while the structure of knowledge is a structural description of knowledge organization displaying its internal and external relations in a general form, that is, relations that are innate to knowledge in general. To find the structure of knowledge, we observe that the indispensable trait of knowledge indicates that each element or system of knowledge refers to some object domain because knowledge is always knowledge about something. It means that for any knowledge system (element) K, there is a domain D of real or abstract objects and K describes the whole D or its part. Note that it is possible to treat a domain as one object. In such a way, we come to the following diagram, which is a specific case of named sets (cf., Appendix). representation Knowledge Item K
Object (Domain) D. reflection
(2.2)
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Diagram (2.2) is also related to Diagram (2.3) because knowledge is a cognitive representation of knowledge objects and knowledge domains, which is intrinsically related to their symbolic representations. Besides, cognitive representation is often symbolic. Object (Domain) D
Cognitive (symbolic) representation of D
(2.3) For instance, a symbolic representation of an object A may be a sentence in a natural language (English, Spanish, or French), a logical formula, a mathematical expression and so on. Thus, according to this new approach, the statement “People live on the Earth” is not knowledge per se. It is a cognitive structure, which becomes knowledge only when it is connected to the object (system) that consists of people and the celestial body called the Earth. However, Diagram (2.2) does not give a complete structure of knowledge because knowledge does exist by itself but belongs to a knowledgeable system or a knower. This gives us Diagram (2.4). Knower (knowledgeable system) knowing
possession
(2.4)
representation Knowledge item K
Domain D
The structures of knowledge presented by Diagrams (2.2)–(2.4) reflect the surface level of knowledge organization. An exposition of further details about knowledge organization and structure is given in Chapters 4–6. We discern the following kinds of knowledge systems: — Knowledge item is a knowledge system that is contemplated separately of other knowledge systems. — Knowledge unit is a knowledge item that is used for constructing other knowledge systems and treated as a unified entity. — Knowledge quantum is a minimal, in some sense, knowledge unit. — Knowledge element is an element of a knowledge system (structure).
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By definition, any knowledge unit is also a knowledge item but the converse is not always true. To understand the difference between a knowledge unit and a knowledge item, let us look at some examples. Knowledge in one story (novel) is a knowledge unit. Knowledge in several unrelated stories (novels) is a knowledge item. Knowledge about one person is a knowledge unit. Knowledge about several unrelated persons is a knowledge item. However, knowledge about members of one family is a knowledge unit. All of the above are knowledge structures or knowledge systems. Here we utilize two meanings of the term knowledge system: 1. Knowledge system as a concise representation of knowledge. 2. Knowledge system as a structure consisting of knowledge elements and relations between them. An example of a knowledge item: “This book is about knowledge. Its title is “Theory of Knowledge: Structures and Processes.” It has many pages. It has nine chapters.” An example of a knowledge unit: “This book is about knowledge.” This statement is also a knowledge quantum. An example of a knowledge element: “a book” At the same time, there are also composite knowledge elements, e.g., “an interesting book.” 2.3.1. Dimensions and other characteristics of knowledge Dimensions are basic descriptive characteristics of epistemic structures in general and knowledge in particular. Each dimension has
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different gradations or/and modalities. Some of these gradations are discrete, while others can be considered as continuous. It is possible to discern the following dimensions of epistemic structures (of knowledge): 1. The correctness dimension reflects adequacy of to its object or domain representation by the knowledge item. 2. The confidence dimension reflects confidence of the knower (knowledge user) in the knowledge item property estimation, including certainty of the knower (knowledge user) in knowledge item correctness as its component. 3. The validation dimension reflects confirmation of confidence of the knower (knowledge user) in the knowledge item property estimation, including justification of the correctness estimation of a knowledge item. 4. The complexity dimension reflects complexity of a knowledge item including several components such as clarity. 5. The significance dimension reflects value and significance of a knowledge item. 6. The efficiency dimension reflects the role of a knowledge item in achieving some goals. 7. The reliability dimension reflects reliability of a knowledge item. 8. The abstractness/generality dimension reflects the level of abstraction of a knowledge item, as well as the degree of generalization achieved by a knowledge item. 9. The completeness/exactness dimension reflects completeness of a knowledge item with respect to its object (domain), as well as exactness with which a knowledge item reflects/represents its object (domain). 10. The meaning dimension reflects meaning of a knowledge item. The first three dimensions are the separation dimensions as these traits have been traditionally used to separate knowledge from other epistemic structures, e.g., from beliefs (cf., Section 2.2). The next six dimensions are the feature dimensions. The 10th dimension is the integration dimension as it can include any other dimension and is the primary feature for knowledge utilization.
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Each dimension integrates specific knowledge properties usually containing these properties as its components. There are also other characteristics of knowledge. Novelty shows the extent to which the content is original, or its relation to other works, e.g., whether it repeats, duplicates, adds to, or contradicts the previous work. Knowledge domain is what this knowledge is about. It is expressed as the subject matter as well as named or implied persons, places, institutions, devices, etc. Specificity or depth refers to depth of coverage or degree of detail of the knowledge in a message. Amount of knowledge has many different meanings and measures. One of such measures is the number of characters, pages or other physical characteristics of a text, which is a carrier of knowledge. Another measure of knowledge amount is the Hartley–Shannon entropy (Burgin, 2010). One more estimate for knowledge amount is the recipient’s sense of number of known facts or ideas although there is not yet a formal measure for this. Algorithmic complexity gives a possibility to measure amount of knowledge about constructive objects (Burgin, 2010). 2.3.2. Correctness, relevance, and consistency of knowledge Here we give the most general definition of correctness treating it as a relational property. In essence, correctness reflects relation of knowledge to some system C of correctness conditions. Definition 2.3.1. A knowledge system (unit) K is correct with respect to a system C of conditions or simply C -correct if it satisfies all conditions from C . Definition 2.3.2. Conditions from the system C are called components of C -correctness. Let us consider some examples. Example 2.3.1. Let us look at such procedural/operational knowledge as programs (BK). This knowledge informs computer what to
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do or how to perform computations. Correctness becomes a critical issue in software and hardware production and utilization as a result of the increased social and individual reliance upon computers and computer networks. A study of the National Institute of Standards found that only software defects resulted in $59.5 billion annual losses (Tassey, 2002; Thibodeau, 2002). It is possible to find a detailed analysis of the concept software correctness and the most comprehensive development of this concept in (Burgin and Debnath, 2006; 2007). There are different forms of software correctness, such as functional, descriptive, procedural, temporal, and resource correctness. Example 2.3.2. Let us contemplate propositional knowledge, i.e., knowledge expressed by propositions in some logical language. This may be, the most popular form of knowledge representation (cf., (Bar-Hillel and Carnap, 1952; 1958; Halpern and Moses, 1985)). Then a knowledge system is usually represented by a propositional calculus. Traditionally, it is assumed that such knowledge is correct if this calculus is consistent. Software correctness introduced in (Burgin and Debnath, 2006) is one more example of knowledge correctness as a software system is a representation of operational knowledge. Consistency of descriptive knowledge in the sense of (Nuseibeh et al., 2001) is also an example of knowledge correctness. Existence of various correctness conditions results in a variety of correctness types. Types of correctness: — Truth — Correlation — Consistency In addition, correctness of knowledge may be higher or lower. For instance, the system K can satisfy only of conditions from C or some conditions can be satisfied only partially. As a result, correctness is, in essence, a gradual, often fuzzy, property because in many cases, conditions from the set C can be satisfied only partially. This shows that it is possible to introduce different measures of correctness, which are
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functions on knowledge items that can take numerical values, vector values, or even values in a partially ordered set. This brings us to degrees of correctness, which turn correctness into a fuzzy property or more generally, into an abstract property (cf., (Burgin, 1985) and Chapter 6). Let us take a system of conditions C and consider a measure m of condition satisfaction by knowledge items. Definition 2.3.3. A knowledge system (unit) K is correct to the degree n with respect to a system C of conditions if the measure m of satisfaction of conditions from C is equal to n. Note that n may be not only a number but also a vector when we separately consider correctness components or an element from a partially ordered set. Let us consider some examples. Example 2.3.3. Let us consider propositional knowledge, i.e., knowledge represented by propositions. Defining correctness of propositions, it is possible to take into consideration three aspects of propositions — syntactic, semantic, and pragmatic aspects. For each of these aspects, we take one criterion of knowledge correctness. The syntactic criterion of correctness c1 : The proposition has the form of a syntactically correct English sentence. The semantic criterion of correctness c2 : The proposition is true, e.g., true in the sense of classical logic. The pragmatic criterion of correctness c3 : The proposition has a model in the real world. This allows us to formally define knowledge correctness taking as the system C = {c1 , c2 , c3 } of correctness conditions and evaluate some propositions. We denote the weight of a proposition p relative to the correctness criterion ci by wi (p), i = 1, 2, 3.
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Proposition p1 : 7 ≥ 3. Correctness weights: w1 (p1 ) = 0 because it is not a syntactically correct English sentence. w2 (p1 ) = 1 because it is true that 7 ≥ 3. w3 (p1 ) = 1 because the conventional (Diophantine) arithmetic is a model for p1 . Proposition p2 : A bear is an animal. Correctness weights: w1 (p2 ) = 1 because it is a syntactically correct English sentence. w2 (p2 ) = 1 because it is true that a bear is an animal. w3 (p2 ) = 1 because the set of all animals is a model for p2 . Proposition p3 : In 2000, the King of France was blue. w1 (p3 ) = 1 because it is a syntactically correct English sentence. w2 (p3 ) = 0 because it is not true that in 2000, the King of France was blue. w3 (p3 ) = 0 because there is no model for p3 in the real world. Thus, we have three elements of a weighted knowledge space (cf., Section 3.1): (p1 ; 0, 1, 1) (p2 ; 1, 1, 1) (p3 ; 1, 0, 0) Note that Example 2.3.3 shows that the most popular attribute of knowledge — truth — is only one kind of knowledge correctness. Example 2.3.4. Let us consider operational knowledge, i.e., knowledge represented by automata or algorithms.
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The model criterion of correctness c1 : The automaton A is a Turing machine (Burgin, 2005; or Section 2.3). The termination criterion of correctness c2 : The automaton A is defined for all words in its alphabet. This allows us to formally define knowledge correctness taking as the system C = {c1 , c2 } of correctness conditions and evaluate some propositions. We denote the weight of an automaton A as an operational knowledge relative to the correctness criterion ci by wi (p), i = 1, 2. Automaton A1 is a deterministic finite automaton (Burgin, 2005). Correctness weights: w1 (A1 ) = 0 because it is not a Turing machine. w2 (A1 ) = 1 because any deterministic automaton A is defined for all words in its alphabet (Burgin, 2005). Automaton A2 is a universal Turing machine (Burgin, 2005). Correctness weights: w1 (A2 ) = 1 because it is a Turing machine. w2 (A2 ) = 0 because a universal Turing machine is defined for all words in its alphabet. Thus, we have two elements of a weighted knowledge space (cf., Section 3.1): (A1 ; 0, 1) (A2 ; 1, 1) Definition 2.3.1 allows us to introduce three modalities of knowledge correctness: — Description modality of knowledge correctness reflects how well this knowledge represents or describes its domain. — Attribution modality of knowledge correctness reflects how well this knowledge is connected to the domain that this knowledge is attributed to.
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— Logical modality of knowledge correctness reflects how well this knowledge satisfies some logical rules, such as, for example, absence of contradictions. Description and attribution modalities are definitely relational properties of knowledge because they depend on relations between knowledge and some domains. Logical modality can be relational property of knowledge in some cases and can be internal property of knowledge. For instance, such a component of logical modality as (inner) consistency is an internal property of knowledge systems, while consistency of one knowledge system with respect to another knowledge system is a relational property of knowledge systems. Each of knowledge correctness modalities has its components. Accuracy, exactness and precision are components of the description modality of knowledge correctness. Definition 2.3.4. Accuracy of descriptive knowledge reflects to what extent the given description is sufficient and does not include unnecessary issues. For instance, if a man is tall and thin, then the proposition “He is a big man” is less accurate than the proposition “He is a tall man.” Usually informal descriptions are less accurate than formal descriptions. For instance, the proposition “His height is seven feet” is more accurate than the proposition “He is a tall man.” Definition 2.3.5. Accuracy of representational knowledge shows to what extent the given representation is sufficient and does not include unnecessary features. For instance, the Copernican model of the Solar system is more accurate than the Ptolemaic model of the Solar system. Definition 2.3.6. Accuracy of operational knowledge shows to what extent the given procedures, tools, goals, norms are sufficient and necessary for reaching the desired goal. For instance, a car mechanic has more accurate operational knowledge about cars than an average person.
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Note that knowledge represented by a statement can be clear but not accurate, as in the case of the statement “The weight of a dog is between one and one thousand pounds.” Definition 2.3.7. Precision of descriptive and representational knowledge reflects whether it is possible to give more details or if a knowledge item could be more specific. For instance, representation of real numbers has different levels of exactness or precision. For floating point numbers, there are three commonly used levels of precision: single precision, double precision and long double, or extended, precision (Sauer, 2006). In the single precision representation, the exponent of a number has 8 bits and the mantissa of a number has 23 bits. In the double precision representation, the exponent of a number has 11 bits and the mantissa of a number has 52 bits. In the long double precision representation, the exponent of a number has 15 bits and the mantissa of a number has 64 bits. Definition 2.3.8. Precision of operational knowledge shows how close it allows to approach the desired goal. For instance, two ways of number truncation are usually used — chopping and rounding (Sauer, 2006). By construction, rounding gives more precise results than chopping. Note that knowledge represented by a statement can be both clear and accurate, but not precise, as in the case “Bob is overweight” because we do not know how overweight Bob is — one pound or 500 pounds. Definition 2.3.9. Exactness of descriptive and representational knowledge shows how a knowledge item matches the knowledge domain, i.e., whether a descriptive knowledge item describes and representational knowledge represents a larger or a smaller domain in comparison with its assigned domain and the extent of the existing difference. For a long time, philosophers assumed that knowledge is always absolutely true and completely exact. However, a little by little,
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new generations of researchers has begun to understand that knowledge can be only partially true and moderately exact. The first idea to mathematical treatment of partially true and moderately exact knowledge dates back at least to the middle of the 19th century when George Boole aimed to reconcile the classical logic, which tends to express complete knowledge or complete ignorance, and probability theory as an extension of the classical logic such as it tends to express partial or/and imprecise knowledge or ignorance (Boole, 1854). His approach represented subjective interpretation of probabilities because people often do not have enough information to assign definite numbers to probabilities of given events. Keynes formulated and applied an explicit interval estimate approach to probability, further developing the theory of imprecise probability and describing its applications (Keynes, 1921). However, probability represents only one aspect knowledge vagueness and inexactness. To reflect these properties of knowledge in a better way, researchers developed fuzzy set theory, which has become one of the most popular mathematical approaches to problems of uncertainty and imprecision is fuzzy set theory. Fuzzy sets were introduced by Lofti Asker Zadeh in 1965 and approximately at the same time, Salii (1965) defined a more general kind of structures called L-relations, which were studied by him in an abstract algebraic context. Fuzzy relations, which are used now in different areas, such as decision-making (Kuzmin, 1982) and clustering (Bezdek, 1978), are special cases of L-relations when L is the unit interval [0, 1]. The aim of Zadeh was to get better mathematical models for reallife systems and processes, as well as better techniques for human reasoning and decision-making, than the conventional set theory allowed by constructing a more realistic set theory. To achieve this goal, Zadeh considered generalizations of sets that allow graded membership of their elements. Thus, he assumes that elements can have different grades of membership in a set. His main argument was that “classes of objects encountered in real physical world do not have precisely defined criteria of membership” (Zadeh, 1965). This approach also reflects situations in which our knowledge about membership is incomplete. Fuzzy set theory replaces the two-valued membership
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function used for sets with a real-valued membership function. As a result, membership may be treated as a probability, or as a degree of truthfulness. In a similar way, it is possible to assign a real value to assertions as an indication of their degree of truthfulness. To represent imprecise, vague, inexact or fuzzy knowledge, Lofty Zadeh also suggested using linguistic variables as a second-level structure based on fuzzy sets (Zadeh, 1973). To understand knowledge correctness, let us consider its components and facets. Relevance, domain interpretability and domain describability are components of the attribution modality of knowledge correctness. Relevance of knowledge is a pivotal component of knowledge correctness, which has three basic types: • Knowledge K is domain relevant if it is related to the domain (object) it is attributed to. • Knowledge K is problem relevant if it is related to the problem under consideration. • Knowledge K is goal relevant if it is useful in achieving a definite goal. Definition 2.3.10. The domain relevance of a knowledge item shows the extent to which this knowledge item is related to some issue of a considered domain or how does that bear on the issue. As relations can be stronger or weaker, domain relevance may be higher or lower. For instance, if the domain is a forest in the U.S., then knowledge about the river near this forest is more relevant to this domain than knowledge about some river in Australia. To represent these distinctions in the quantitative form, it is possible to introduce different measures of domain relevance, the scale of which can be either the two-element set {0, 1} or the interval [0, 1] or the set of all non-negative real numbers. True knowledge about a domain A can be irrelevant to a domain B, which is not related to A. For instance, knowledge about elementary particles is irrelevant to music or art. Knowledge of geometry is irrelevant to moral issues.
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Definition 2.3.11. The problem relevance of a knowledge item shows the extent to which this knowledge item is related to some problem. As relations can be stronger or weaker, problem relevance may be higher or lower. To represent these distinctions in the quantitative form, it is possible to introduce different measures of problem relevance, the scale of which can be either the two-element set {0, 1} or the interval [0, 1] or the set of all non-negative real numbers. Definition 2.3.12. The goal relevance of a knowledge item shows the extent to which this knowledge item is helpful (useful) in achieving some goal. As in the case of other types of relevance, it is possible to introduce different measures of domain relevance, the scale of which can be the two-element set {0, 1} or the interval [0, 1] or the set of all nonnegative real numbers. For instance, if the goal is to know weather in Los Angeles, then knowledge about weather in Santa Monica is more relevant to this goal than knowledge about weather in New York. A knowledge item represented by a statement can be clear, accurate, and precise, but not relevant to the question at issue. For instance, students often think that the amount of effort they put into a course should be used in raising their grade in a course. Often, however, the “effort” does not measure the quality of student learning, and when this is so, effort is irrelevant to their appropriate grade (Elder and Paul, 2009). It means that if the goal of a student is to get good knowledge or a high grade, then the operational knowledge of how to apply student’s effort can be weakly goal relevant or even goal irrelevant. Relevance of knowledge influences knowledge correctness in general because if a knowledge unit K is irrelevant to some issue K of the domain D of its attribution, then this knowledge cannot be treated as correct with respect to the issue K. It is possible to consider relevance as a binary property with only two values — relevant and irrelevant. However, a more exact representation of this property treats relevance as a fuzzy property allowing different degrees of relevance.
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Let us consider two other components of the attribution modality of knowledge correctness. Definition 2.3.13. Domain interpretability of knowledge reflects how an item of knowledge can be interpreted in the domain this knowledge is attributed to. The third component of the attribution modality of knowledge correctness is not the same for different types of knowledge. Definition 2.3.14. Domain descriptability of descriptive knowledge reflects how well this knowledge can describe the domain (object) it is attributed to. For instance, the statement “The Mars has two satellites” has higher domain descriptability than the statement “The Mars has satellites.” Definition 2.3.15. Domain representability of representational knowledge reflects how well this knowledge can represent the domain (object) it is attributed to. For instance, when the domain (object) of knowledge is number π, then the number 3.14159 has higher domain representability (i.e., gives a better approximation) than the number 3.14. Definition 2.3.16. Domain applicability of operational knowledge reflects how well this knowledge can be applied to the domain (object) it is attributed to. For instance, when the domain of knowledge consists of computations, then knowledge in the form of Turing machines has higher domain applicability than knowledge in the form of finite automata, while knowledge in the form of inductive Turing machines has higher domain applicability than knowledge in the form of Turing machines (Burgin, 2005). The logical modality of knowledge correctness also has three components. Namely, consistency, provability, and testability are components of the logical modality of knowledge correctness. Consistency is an important relational characteristic of a knowledge system. The traditional approach to knowledge consistency
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implies separation (and elimination) elements of knowledge that are called contradictions. For instance, a standard example of a logical contradiction is the expression p∧p where p is a proposition, e.g., the expression “It is a table and it is not a table” is contradictory by the rules of classical logic. However, as we have seen in Chapter 1 and will see in Chapter 6, Indian logic accepts statements of the form “S is and is not P ,” which are unacceptable, for example, in Aristotle’s syllogistics. It is interesting to know that in the 20th century, fuzzy logic also made such statements logically correct (Bandemer and Gottwald, 1996). For instance, a ball that is partially green and partially yellow is green and is not green. Studying system consistency in logic and beyond, researchers came to the conclusion that consistency is not an absolute property quality as in classical logic but is a relative property of various systems, including logical systems. The most general definition of consistency is given in (Nuseibeh et al., 2001). Namely, at first, a system C of consistency conditions is determined in a class of systems K. Then we have the following concept. Definition 2.3.17. A system R from K is consistent (inconsistent) with respect to C if satisfies (does not satisfy) all conditions from C . The most popular example of consistency is logical consistency when a system of propositions or predicates is consistent when it does not allow inference (deduction) of expressions A and A. A weaker kind of consistency is weak logical consistency when a system of propositions or predicates is consistent when it does not contain expressions A and A. In logical calculi and in logics, weak logical consistency coincides with logical consistency. Here we are mostly interested in consistency of knowledge systems. When we are dealing with propositional knowledge, the most popular is the conventional consistency, the basic condition for which is absence of contradictions. Another reason for exclusion of contradiction is the situation when any false statement (contradiction) implies any other statement in classical logics. As a result, people
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always considered contradictions as abhorrent irregularities of thinking, which have to be eliminated from correct thinking and valid logic. However, as we have seen before, what is a contradiction by classical criteria can provide useful knowledge. As, a result, a new kind of logical consistency — paraconsistency — was introduced in the 20th century by weakening conditions of logical consistency (Priest et al., 1989). Namely, a logic is paraconsistent if and only if its logical consequence relation is not explosive. Here explosiveness means that using axioms and consequence relations of the logic, it is possible to deduce any formula, e.g., proposition, in the language of this logic. Paraconsistent logics accommodate inconsistency in a manner that treats inconsistent information and inconsistent knowledge as informative. There are different systems of paraconsistent logics, e.g., discussive logics, non-adjunctive systems, preservationism, adaptive logics, logics of formal inconsistency, many-valued logics, and relevant logics. As it is possible to have several consistency conditions and/or some of the conditions can be satisfied only partially, in general, consistency and inconsistency are fuzzy properties. In general, consistency is a particular case of correctness. Namely, comparing Definitions 2.3.1 and 2.3.17, we see that correctness becomes consistency when correctness conditions actually are consistency conditions. However, consistency in general and logical consistency, in particular, is only one type of correctness. There are also other types such as provability and testability. Provability as a component of the logical modality of knowledge correctness reflects how and to what extent it is possible to prove, e.g., support by evidence or infer, correctness of a given knowledge item. For instance, we can assume that a system of statements, e.g., a formal theory, is correct only when it is consistent and it is possible to prove its consistency. From this point of view, any sufficiently powerful mathematical theory U , i.e., a theory that includes the formal arithmetic, is not correct by itself because by the second G¨ odel’s incompleteness theorem, it is impossible to prove consistency of U
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using only means from the theory U . Nevertheless, the theory U may be internally incorrect but externally correct if there are other means to prove its consistency. Testability as a component of the logical modality of knowledge correctness reflects how and to what extent it is possible to estimate, e.g., support by evidence or infer correctness of a given knowledge item. Testability is essentially important for operational knowledge. For instance, it is possible to treat a computer program as a potentially correct operational knowledge item if it is possible to test it finding and correcting all deficiencies. In this case, the possibility of deficiency correction is also a correctness condition. An important type of knowledge correctness is truthfulness. There are different ways to introduce truthfulness of knowledge. All of them involve two types of truthfulness functions: domain-oriented, reference-oriented, and attitude-oriented. According to the first approach, we have the following model of truthfulness. A system T , or as it is now fashionable to call it now, an agent A that has knowledge K about the domain (object) D is considered. Then the truthfulness K means that (condition from C) the description that K gives for D is true. Thus, the truthfulness tr(K, D) of the knowledge K about the domain D is a function of two variables that takes two values — true and false. In addition, the function tr(T , D) gives conditions for differentiating knowledge from similar structures, such as beliefs, descriptions or fantasies. Knowledge truthfulness, or domain related correctness, shows absence of distortions in knowledge representation of its domain. Thus, truthfulness is closely related to accuracy of knowledge, which reflects how close is given knowledge to the absolutely exact knowledge. However, truthfulness and accuracy of knowledge are different properties. For instance, statements “π is approximately equal to 3.14” and “π is approximately equal to 3.14159” are both true, i.e., their truthfulness is equal to 1. At the same time, their accuracy is different. The second statement is more accurate than the first one. We see that conventional truthfulness can indicate only two possibilities: complete truth and complete falsehood.
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To measure truthfulness in a better way, we utilize a measure tr of truthfulness of knowledge in the system T . Such a measure, allows one to develop the comprehensive approach to true knowledge. However, in many situations, it is impossible to verify (or validate) truthfulness or falsehood and we come to three basic types of knowledge: — correct knowledge, — incorrect knowledge, — unverified knowledge. To formalize these concepts, let us take a number k from the interval [0, 1]. Definition 2.3.18. (a) A portion of knowledge I is called true or genuine knowledge about D with respect to a measure cor if cor (I, TA ) > 0. (b) A portion of knowledge I is called true or genuine to the degree k knowledge about D with respect to a measure cor if cor (I, TA ) > k. This definition looks natural and adequately works in many situations. However, there are some problems with it. Imagine that information that gives correct knowledge about some domain (object) D comes to A but it does not change the knowledge system TA because correct knowledge about D already exists in TA . In this case, cor(I, TA ) = cor (I(TB ), TA ) — cor(TB , TA ) = 0. This implies that it is necessary to distinguish relative, i.e., relative to a knowledge system TA , knowledge correctness and absolute correction. To define absolute correction, we take a knowledge system T 0D that has no a priori knowledge about the domain (object) D. Definition 2.3.19. A portion of knowledge I is called purely true knowledge about D with respect to a measure cor if cor(I, T0D ) > 0. It is necessary to understand that it is not a simple task to find such a knowledge system T0D that has no a priori knowledge about the domain (object) D. Besides, truthfulness depends on other properties of the knowledge system T0D , e.g., on algorithms that
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are used for conversion of received information into knowledge. It is possible to get true information, which eventually transformed into incorrect knowledge. For instance, during the Cold War, witty people told the following joke. “Russia challenged the United States to a foot race. Each country sent their fastest athlete to a neutral track for the race. The American athlete won. The next day, the major Soviet newspaper “Pravda”, which means truth in Russian, published an article with the following title: Russia takes second in international track event, while the United States comes in next to last.” We see that literally the article of “Pravda” is true, but people’s a priori knowledge makes them to assume a big race with many participants and in such a way, they get a wrong impression and false knowledge if they did not know how many participants was in the race. It is possible to treat truthfulness and correctness as linguistic variables in the sense of (Zadeh, 1973). For instance, we can separate such classes as highly correct/true knowledge, sufficiently correct/true knowledge, weakly correct/true knowledge, weakly false knowledge, and highly false knowledge. From this point of view, we come to three fuzzy types of knowledge: — true or genuine knowledge, — partially true knowledge, — false knowledge. However, it is possible to separate these types of knowledge in a more general situation, utilizing the concept of knowledge measure. There are different ways to do this. Analyzing different publications, we separate two classes of approaches: relativistic definitions and universal definitions. The latter approach is subdivided into objectdependent, reference-dependent, and attitude-dependent classes. At first, we consider the relativistic approach to this problem.
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To have genuine knowledge in the conventional sense, we take such measure m as correctness of knowledge or such measure as validity of knowledge or of knowledge acquisition. Let us specify the relativistic approach in the case of cognitive knowledge, taking a measure m that reflects such property as truthfulness. Thus, to get more exact representation of the convenient meaning of the term true knowledge, we consider only cognitive knowledge and assume that true cognitive knowledge gives true knowledge, or more exactly, make knowledge truer than before. However, it is necessary to understand that the truth of knowledge and the validity of its acquisition are not always the same. For instance, the truth of knowledge represented by propositions and the validity of reasoning are distinct properties, while there are relations between them (cf., for example, (Suber, 2000)). This relationship is not entirely straightforward. It is not true that truth and validity, in this sense, are utterly independent because the impossibility of “case zero” (a valid argument with true premises and false conclusion) shows that one combination of truth-values is an absolute bar to validity. According to the classical logic, when an argument has true premises and a false conclusion, it must be invalid. In fact, this is how we define invalidity. However, in real life, people are able to take a true statement and to infer something false. An example of such a situation gives the Cold War Joke considered in this section. To formalize the concept of knowledge truthfulness, we use the model developed in (Burgin, 2004) and described in Chapter 4. According to this model, general knowledge K about an object F has the structure represented by Diagram (2.5) and high level of validation.
g W
L
t
p
U
C f
(2.5)
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This diagram has the following components: (1) some class U containing an object F ; (2) an intrinsic property that is represented by an abstract property T = (U, t, W ) with the scale W , which is defined for objects from U (cf., Chapter 5); (3) some class C of names, which includes a name “F ” of the object F ; (4) an ascribed property that is represented by an abstract property P = (C, p, L) with the scale L, which is defined for names from C (cf., Chapter 5); (5) the correspondence f assigns names from C to objects from U where in general case, an object has a system of names or more generally, conceptual image (Burgin and Gorsky, 1991) assigned to it; (6) the correspondence g assigns values of the property P to values of the property T . In other words, the correspondence g relates values of the intrinsic property to values of the ascribed property. For instance, when we consider a property of people such as height (the intrinsic property), in measuring the height, we can get only an approximate value of the real height, or height with some precision (the ascribed property). In more detail, the basic structure of knowledge is discussed and described in Chapter 4. According to the attitude-dependent approach, we have the following definitions. Definition 2.3.20. General knowledge T about an object F for a system R is the entity that has the structure represented by Diagram (2.5) that is estimated (believed) by the system R to represent with high extent of confidence true relations. Consequently, we come to three main types of knowledge about some object (domain): — objectively true knowledge, — objectively neutral knowledge, — objectively false knowledge.
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Taking some object domain D and tr as the measure m in Definition 2.3.3, we obtain unconditional concepts of true and false knowledge. Definition 2.3.21. A portion of knowledge I is called objectively true knowledge about D if tr(I, D) > 0. To adequately discuss a possibility of false knowledge existence from a methodological point of view, it is necessary to take into account three important issues: multifaceted approach to reality, historical context, and personal context. Thus, we come to the following conclusion. First, there is a structural issue in this problem. Namely, the dichotomous approach, which is based on classical two-valued logic, rigidly divides any set into two parts, in our case, true and false knowledge. As a result, the dichotomous approach gives a very approximate image of reality. Much better approximation is achieved through the multifaceted approach based on multivalued logics, fuzzy reasoning, and linguistic variables. Second, there is a temporal issue in this problem. Namely, the problem of false knowledge has to be treated in the historical or, more exactly, temporal context, i.e., we must consider time as an essential parameter of the concept. Indeed, what is treated as true in one period of time can be discarded as false knowledge in another period of time. Third, there is a personal issue in this problem, i.e., distinction between genuine and false knowledge often depends on the person who estimates this knowledge. For instance, for those who do not know about non-Diophantine arithmetics (Burgin, 1997d; 2001b; 2010c), 2 + 2 is always equal to 4. At the same time, for those who know about non-Diophantine arithmetics, it becomes possible that 2 + 2 is not equal to 4. In light of the first issue of our discussion about false knowledge, we can see that in cognitive processes, the dichotomous approach, which separates all objects into two groups, A and not A, is not efficient. Thus, if we take the term “false knowledge”, then given
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a statement, it is not always possible to tell if it contains genuine or false knowledge. To show this, let us consider following statements: 1. 2. 3. 4. 5.
“π “π “π “π “π
is is is is is
equal equal equal equal equal
to to to to to
3.” 3.1.” 3.14.” 3.1415926535.” (4/3)2 .”
According to the definition of pi and our contemporary knowledge that states that pi is a transcendent number, all these statements contain false knowledge. In practice, they are all true but with different exactness. For example, the statement (4) is truer than the statement (1). Nevertheless, in the ancient Orient, the value of pi was frequently taken as 3 and people were satisfied with this value (Eves, 1983). Archimedes found that pi is equal to 3.14. For centuries, students, and engineers have used 3.14 as the value for pi and had good practical results. Now calculators and computers allow us to operate with much better approximations of pi, but nobody can give the exact decimal value of this number. Importance of the temporal issue is demonstrated by the following example from the history of science that helps to better understand the situation with false knowledge. Famous Greek philosophers Leucippus (fl. 445 B.C.E.) and Democritus (460–360 B.C.E.) suggested that all material bodies consist of small particles, which were called atoms. “In reality,” said Democritus, “there are only atoms and the void.” We can ask the question whether this idea about atoms contains genuine or false knowledge. From the point of view of those scientists who lived after Democritus but before the 15th century, it contained false knowledge. This was grounded by the fact that those scientists were not able to look sufficiently deep into the matter to find atoms. However, the development of scientific instruments and experimental methods made it possible to discover microparticles such that have been and are called atoms. Consequently, now it is a fact,
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which is accepted by everybody, that all material bodies consist of atoms. As a result, now people assume that the idea of Leucippus and Democritus contains genuine or true knowledge. This shows how people’s comprehension of what is genuine knowledge and what is understood as false knowledge changes with time. Lakatos (1976) and Kline (1980) give interesting examples of similar situations in the history of mathematics, while Cartwright (1983) discusses analogous situations in the history of physics. All these examples demonstrate that it is necessary to consider false knowledge as we use negative numbers, as well as not to discard false knowledge as we do not reject utility of such number as 0. History of mathematics demonstrates that understanding that 0 is a number and a very important number demanded a lot of hard intellectual efforts from European mathematicians when Arab mathematicians brought to them knowledge about 0 from India. Going to the third point of the discussion about false knowledge related to the personal issue, let us consider other examples from the history of science as here we are studying knowledge by scientific methods. In his lectures on optics, Isaac Newton (1642–1727) developed a corpuscular theory of light. According to this theory, light consists of small moving particles. Approximately at the same time, Christian Huygens (1629–1695) and Robert Hook (1635–1703) built a wave theory of light. According to their theory, light is a wave phenomenon. Thus, it was possible to ask the question who among them, i.e., was it Newton or Huygens and Hook, gave genuine knowledge and who gave false knowledge. For a long time, both theories were competing. As a result, the answer to our question depended whether the respondent knew physics and was an adherent of the Newton’s theory or of the theory of Huygens and Hook. However, for the majority of people who lived at that time both theories did not provide knowledge because those people did not understand physics. A modern physicist believes that both theories contain genuine knowledge. So, distinction between genuine and false knowledge in some bulk of knowledge depends on the person who estimates this knowledge.
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Existence of false knowledge is recognized by the vast majority of people, but some theoreticians insist that false knowledge is not knowledge. There is persuasive evidence supporting the opinion that false knowledge exists. For instance, readers find a lot of false or inaccurate knowledge in newspapers, books, and magazines. Recent studies found a considerable amount of inaccurate knowledge on the Internet (Hernon, 1995; Connell and Triple, 1999; Bruce, 2000; Berland et al., 2001). The new and truly wonderful medium, the Internet, unfortunately has one glaring downside. Namely, along with all the valid knowledge it provides, the Internet also contains much misleading knowledge, false knowledge, and outright hype. This is also true for the fields of science and science criticism. Certainly many so-called “discussion groups” and informal “book review” sites are good examples of the blind leading the blind (Fallis, 2004). On an Internet blog, the author of this book once encountered an assertion of one of the bloggers that there was a critique of a book A in a paper D. Finding the paper D, the author did not come across any such a critic in it and was very surprised even thinking about such a possibility because the paper D had been published in 2003, while the book A had been published only in 2005. Another example of the situation when “the blind leads the blind” we can take the critique of the professor P aimed at the book B. Indeed, this critique was irrelevant and contained essential logical and factual mistakes. However, being asked if he read the book he criticized, P answered that he did not need to do that because he saw an article by the same author and understood nothing. Naturally, in this situation, his critique was an example not only of incompetent but also indecent behavior because using Internet and other contemporary means of communication, professor P transmitted this false knowledge to those who read his writings on this topic. Thus, we see that the problem of false knowledge is an important part of knowledge studies and we need more developed scientific methods to treat these problems in an adequate manner. To have genuine knowledge relevant to usual understanding, we take such measure as correctness of knowledge or such measure as
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validity of knowledge. Thus, we can call cognitive knowledge false when it decreases validity of knowledge it gives. Definition 2.3.8 shows that we have false knowledge when its acceptance makes our knowledge less correct. For instance, let us consider people who lived in ancient Greece and accepted ideas of Leucippus and Democritus that all material bodies consist of atoms. Then they read Aristotle’s physics that eliminated the idea of atoms. Because we now know that the idea of atoms is true, Aristotle’s physics decreased true knowledge about such an aspect of the world as the existence of atoms and thus, gave false knowledge about atoms. We can ask the question whether this idea contains true or false knowledge. From the point of view of those scientists who lived after Democritus but before the 15th century, it contained false knowledge. This was grounded by the fact that those scientists were not able to go sufficiently deep into the matter to find atoms. We see that false knowledge is also knowledge because it has a definite impact on the infological system. Only this impact is negative. It is necessary to understand that the concept of false knowledge is relative, depending on the chosen measure. Let us consider the following situation. A message M comes, telling something completely incorrect. Thus, it will be wrong knowledge with respect to a semantic measure of knowledge (cf., Chapter 4). At the same time, if all letters in the message M were transmitted correctly, it will contain genuine knowledge with respect to the (statistical) Shannon’s measure of information (cf., Chapter 3). It is interesting that there is no direct correlation between false knowledge and meaningless knowledge. Bloch in his book “Apology of History” (1949) gives examples when false knowledge was meaningful for people, while genuine knowledge was meaningless for them. Moreover, a knowledge unit can be true knowledge with respect to another measure and false knowledge with respect to the third measure. For instance, let us consider some statement X made by a person A. It can be true with respect to what A thinks. Thus, knowledge in X is genuine with respect to what A thinks (i.e., according to the measure m2 that estimates correlation between the statement
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X and beliefs of A). The statement X can be false with respect to the real situation. Thus, knowledge in X is false with respect to the real situation (i.e., according to the measure m3 that estimates correlation between the statement X and reality). At the same time, knowledge in X can be pseudo knowledge from the point of view of the person D who does not understand it (i.e., according to the measure m1 that estimates correlation between the statement X and knowledge of D). The opposite property to correctness is incorrectness. Definition 2.3.22. (a) A knowledge system K is incorrect with respect to a system C of conditions if it violates at least one condition from C. (b) A knowledge system K is strongly incorrect with respect to a system C of conditions if it violates all conditions from C. For instance, taking the following system of conditions for formal logics C = {(1) a logic L is not trivial, i.e., it is not empty and does not contain all formulas from the logical language; (2) a logic L does not contain expressions of the form A&A}, we see that any classical logic is incorrect with respect to C if and only if it is strongly incorrect with respect to C (Shoenfield, 2001). However, there are paraconsistent logics that are incorrect with respect to C validating the second condition from C but not strongly incorrect with respect to C because they still may satisfy the first condition from C (Priest et al., 1989). 2.3.3. Confidence in and certainty of knowledge Confidence is an essentially psychological characteristic of knowledge, which shows the extent to which an individual or a group strongly believes (is convinced) that some epistemic structures are knowledge. In a more general interpretation, confidence, as a knowledge characteristic, reflects the knower’s (knowledge user) mental state of being without doubt about estimation of the epistemic structure, e.g., knowledge item, property. For instance, a person can be confident that her belief, e.g., belief that the Venus is a planet of
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the Solar system or that Venus is a goddess, is knowledge, i.e., it is true. We can see that these two beliefs are essentially different. To reflect these differences, it possible to call conventional confidence by the name psychological confidence and consider another type of confidence called epistemic confidence, which reflects epistemological status of knowledge items, i.e., whether they are indeed knowledge items and not other epistemic structures. It is useful to distinguish three kinds of psychological confidence — internal, external, and exterior confidence. Internal confidence is confidence of an individual or a group in epistemic structures of the same individual or group. For instance, self-confidence is having confidence in oneself or more exactly, confidence in own abilities and qualities, which are certain epistemic structures. External confidence is confidence of an individual or a group in epistemic structures of another individual or group. Exterior confidence is confidence of an individual or a group in epistemic structures stored in some knowledge carrier, such as a book, journal, or knowledge base. Certainty is a more restricted characteristic of knowledge, reflecting higher levels of confidence. Thus, it is a psychological characteristic of the knower (knowledge user). For instance, it can be certainty of the knower (knowledge user) in knowledge item correctness, e.g., knowledge item is certain for the knower (knowledge user) when this knower is supremely convinced of its truth. However, there are kinds, or more exactly, interpretations of certainty. Another kind is epistemic certainty, which is not a psychological but an epistemological characteristic estimating that an epistemic structure (knowledge item) has the highest possible epistemic status. This status has to be validated in epistemology as the theory of knowledge. Epistemic certainty often but not always correlates with psychological certainty. For instance, it is possible that a knower (knowledge user) has epistemically certain knowledge, e.g., a belief that enjoys the highest possible epistemic status, but does not have psychological
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certainty either being unaware of epistemic certainty or having doubt about its validity. An opposite situation is also possible when a knower (knowledge user) has psychological certainty about certain knowledge item strongly believing in its truthfulness, but in spite of this, the knowledge item does not have epistemic certainty. For instance, Aristotle was (psychologically) certain that all swans were white but as it was discovered later, this belief was not epistemically certain. A more general than epistemic certainty is source certainty, which is a knowledge characteristic reflecting the highest possible status of the knowledge item coming from some source. For instance, epistemic certainty gets the status from epistemology. Moral certainty discussed by some philosophers gets the status from God or from tradition. External certainty, as a highest degree of external confidence, gets the status from an authoritative group or individual, e.g., a knowledge item can be certain because Aristotle or Kant said so. Psychological confidence and certainty can be based on different reasons — on assurance, validation, groundedness, or even on persuasion. For instance, groundedness by evidence reflects the extent of the of knowledge evaluation. When confidence is ungrounded, it is called arrogance or hubris. It is natural to consider degrees of confidence and certainty. For instance, Carnap treated epistemic certainty as a having some degree, which could be objectively measured. Definite techniques for measuring confidence and consequently, certainty have been developed in statistics where such concepts as confidence level, confidence interval, confidence coefficient and confidence bounds have been introduced for this purpose. A confidence interval is an interval estimate of the confidence that a sample characteristic or parameter gives confident (reliable) knowledge of the same characteristic (parameter) for the whole population (Fisher, 1956). How frequently the calculated confidence interval contains the parameter is determined by the confidence level or confidence coefficient, which is a numerical estimate of the confidence. For instance, a 90% confidence level means that it is possible to expect the corresponding confidence interval to include 90%
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of the sample characteristic or sample parameter estimated for different samples. While two-sided confidence limits form a confidence interval, their one-sided counterparts are called lower or upper confidence bounds and are also numerical estimates of the confidence (Fisher, 1956). There are also methods of discriminating degrees of psychological confidence and certainty. For instance, in the legal practice, the following degrees of certainty are used:
no credible evidence, some credible evidence, a preponderance of evidence, clear and convincing evidence, beyond reasonable doubt, beyond any shadow of a doubt, what is usually recognized as an impossible standard to meet.
Degrees of certainty are related to degrees of confidence but they are not the same. For instance, a reasonable degree of confidence can correspond to a low degree of certainty. It also happens that the degree of psychological confidence is different from the degree of epistemological confidence. When the degree of psychological confidence is essentially larger than the degree of epistemological confidence, people speak about overconfidence or presumptuousness. For instance, overconfidence is an excessive belief in someone or something, e.g., a plan, succeeding, without any regard for possible failure. 2.3.4. Complexity and clarity of knowledge Complexity has become a buzzword in contemporary science. This term utilized in a variety of scientific fields and from them, it entered popular usage on a new level of credibility. Trying to explain, why complexity is so important and why it is more important now than it was before, we come to three following issues. First, people have to deal with more and more complex systems. On one hand, the development of science is bringing cognition to
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more and more complex systems. On the other hand, the development of engineering and social organization resulted in building more and more complex technical systems and developing more and more complex social systems. All this is directly related to knowledge. Studying complex systems in nature, society, and technology, scientists, as a rule, need adequately complex knowledge systems to represent and model systems they study. To create and invent complex systems, engineers, including software and social engineers, need sufficiently complex operational knowledge. Second, complexity serves as a measure of needed resources. In turn, needed resources correlate with system efficiency. Indeed, when two systems give the same results but the first one demands less resources than the second system, then the fist system is more efficient than the second. Thus, complexity becomes a measure of efficiency. For instance, knowledge that demands less time or less efforts for understanding is more efficient for learning. At the same time, usually simple knowledge demands less time and efforts for understanding than complex knowledge. For instance, it is easier to understand that 2 + 2 = 4 than the statement that there are infinitely many prime numbers. Pager (1970) defines efficiency of computation as the value that is inversely proportional to complexity of the same computation. In the same way, it is possible to define efficiency of any process as the value that is inversely proportional to complexity of this process. Efficiency is a clue problem and a pivotal characteristic of any activity. Inefficient systems are ousted by more efficient systems. Consequently, problems of efficiency are vital to any society and any individual. Many great societies, Roman Empire, British Empire and others perished because they had become inefficient. However, there are many different criteria of efficiency, and to understand this importance and, at the same time, complex phenomenon, it is necessary to use mathematical methods. Such methods are provided by the mathematical theory of complexity. Moreover, many other properties of systems are related to complexity. For example, Carlson and Doyle (2002) investigate relations
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between complexity and robustness in biological, social, economical, and engineering systems. They show a non-trivial interplay of these important properties. Third, complexity helps to comprehend what is practically possible to achieve and what is not. Many things that are possible to build or compute in theory are not constructible in reality because there are no means to do this. That is why, in particular, in the theory of algorithms, the field oriented at operational knowledge, difference is made between computable and tractable problems. Manin (1991) suggests that the development of mathematical knowledge, and we would like to add, also of scientific knowledge is directed by complexity issues. The reason is that simpler systems are more feasible for cognition. Therefore, cognition goes from simple to more and more complex systems of knowledge. In the past, mankind has learned to understand reality mostly through simplification and analysis, ignoring a huge number of factors and details. That is why, for example, physics is more developed than biology: biological systems are much more complex than physical systems. However, in spite of all its importance, there is no generally accepted, formalized, and unique definition of complexity in general and knowledge complexity, in particular. Complexity has proved to be an elusive concept. Different researchers in different fields are bringing new philosophical and theoretical tools to deal with complex phenomena in complex systems. “What is complexity?” is a basic question of Gell-Mann (1995). However, after many elaborate considerations and creative insights, he comes to the conclusion that “a variety of different measures would be required to capture all our intuitive ideas about what is meant by complexity and by its opposite, simplicity.” Going back to the origin of the word complexity, we find the Latin word “complexus”, which means “entwined” or “twisted together”. That is why, in mathematics (more exactly, in topology), topological complex is a structure built from simplexes (Spanier, 1966). This also reflects the situation when a system that consists of many parts is considered complex. However, this is not always true. For instance, the sequence 11 . . . 1 that consists of a thousand of symbols, 1 is not
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complex. At the same time, the sequence that consists of a thousand of symbols from the number “pi” taken from the left side is complex. In general, as Heylighen (1996) writes, complexity can be characterized by lack of some symmetry or “symmetry breaking”, that is, by the fact that no part or aspect of a complex entity can provide sufficient information to actually or statistically predict the properties of the others parts. In other words, complexity is connected to the difficulty of modeling associated with complex systems. If we analyze what it means when we say that some system or process is complex, we come to a conclusion that it is complex to do something with this system or process: to study it, to describe it, to build it, to control it, and so on. Here are two examples of types of complexity taken from the world of business and industry (Paul, 2002). The first one is complexity of functioning that reflects a high number of operations to be performed. The second one is complexity of integration that reflects a high number of problems encountered in integration processes. The same is true for complexity of knowledge, which is estimated from the perspective of related processes. For instance, complexity of knowledge depends on such processes as knowledge acquisition, knowledge transmission, knowledge integration, teaching, and learning. Thus, complexity is always complexity of doing something. Being related to activity and functioning, complexity allows one to represent efficiency in a natural way: when a process has high efficiency, it is simple from the point of view of demanded resources, and when a process has low efficiency, it is complex from the point of view of demanded resources. For example, we can take time as a measure of efficiency: what is possible to do in one hour is efficient, while what is impossible to do even in 1,000 years is inefficient. To estimate temporal efficiency of processes and procedures, such measure as computational complexity is utilized. It estimates time of computation or any other algorithmic process. At the same time, as there are many resources, there are many corresponding measures of complexity. There are various relations
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between different measures of complexity. In particular, an important and interesting relation is trade-off between different kinds of complexity. For example, in computation, it is possible in some cases to use more memory for program execution, decreasing the time of execution, or to use less memory, paying for with more time. Highly optimized tolerance (HOT) is one recent attempt, in long history of efforts, to develop a general framework for studying complexity in such fields as biology and engineering (Carlson and Doyle, 2002). The main idea of HOT is that higher structural complexity of a system (more complex for construction, modeling or understanding) is aimed at decreasing behavioral/functioning complexity of a system (simpler maintenance, less changes under external influence, etc.). This shows how a trade-off between structural and behavioral complexity can inspire the development of systems. Here we use the informal definition of complexity from the book (Burgin, 2005). Definition 2.3.23. Complexity of a system R with respect to a process (or a group of processes) P is the quantitative or qualitative characteristic (measure) of resources necessary for (used by) the process P involving R. There are different kinds of involvement. P may be a process in the system R. For instance, R is a scientific domain, e.g., physics, as a dynamic knowledge system, P is a process of the development of a scientific theory in R, and the resource is researchers who work in this area. P may be a process that is realized by the system R. For instance, R is a computer, P is a computational process in R, and the resource is memory. P may be a process controlled by the system R. For instance, R is operational knowledge in the form of a program, P is a computational process controlled by R, and the resource is time. P may be a process that builds the system R. For instance, R is operational knowledge in the form of a software system, P is the process of its design, and the resource is programmers.
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P may be a process that operates with R, e.g., transforms, utilizes, models, and/or predicts behavior of the system R. For instance, R is operational knowledge in the form of a program, P is the process of writing the program R, and the resource is programmers who are writing this program. Definition 2.3.24. Complexity of a system R with respect to a process (or a group of processes) P is the quantitative or qualitative characteristic (measure) of resources necessary for (used by) the process P involving R. Definition 2.3.25. A complexity measure on a set of systems U is a (partial) numerical function that assigns higher numbers to systems with higher complexity. In cognitive processes, complexity is closely related to information and knowledge, representing specific kind of information and knowledge measures. In turn, processes use different kinds of resources: Natural resources consumed by a process P : time, space, information, energy, power, minerals, and so on. Social resources consumed by a process P : people involved, their time, efforts, expertise, knowledge, and so on. Artificial resources consumed by a process P : system time, system space, data, knowledge, memory, system units, system actions, computers, experimental devices, e.g., telescopes or microscopes, and so on. The general definition of system complexity gives us definition of knowledge complexity. Definition 2.3.26. Complexity of a knowledge system K with respect to a process (or a group of processes) P is the quantitative or qualitative characteristic (measure) of resources necessary for (used by) the process (the processes from) P involving K.
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Different kinds of processes determine different kinds of complexity: — If processes from P are processes of utilization of K, then we have utilization complexity of K. — If processes from P are processes of transformation of K, then we have transformation complexity of K. — If processes from P are using K for solving some problems, then we have problem complexity of K. — If processes from P are processes for obtaining, e.g., acquisition of, K, then we have cognitive complexity of K. Note that in the first three cases, knowledge K plays the role of the used resource and its complexity is a significant characteristic of this resource. Problem complexity is very important because problems represent a pivotal form of erotetic knowledge. People solve problems all the time and solution of some of these problems is vital for individuals, organizations, and communities. Thus, complexity of some problems is essentially important for people. If it is impossible to solve a problem with given resources, we assume that it has infinite complexity. The halting problem for Turing machines is an example of a problem with infinite complexity for operational knowledge in the form of Turing machines since we know that it has no solution in the class of all Turing machines. However, for operational knowledge in the form of inductive Turing machines, this problem has finite complexity. This shows that, in general, problem complexity is a relative property, which essentially depends on knowledge used for solving the problem. Definitions 2.3.25 and 2.3.27 imply that complexity is always complexity of doing something and although complexity is attributed to a system, it is a principal characteristic of a process and of the operational knowledge in the form of an algorithm if the process is determined by an algorithm. However, it is possible to extend the constructions of such measures to complexity of arbitrary processes and through processes to arbitrary systems. For instance, if we take some non-algorithmic process, such as cognition, then it is possible
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to measure its complexity by the amount of resources this process needs. To make Definitions 2.3.25 and 2.3.27 constructive, it is necessary to build mathematical models of efficiency. One of such models is complexity of algorithms. Complexity is a mirror reflection of efficiency: the more efficient is an algorithm (system of algorithms) A for a problem P (problems from some class K ), the less complex is P (problems from K ) for the algorithm (system of algorithms) A. Mathematical models of complexity allow researchers to measure efficiency of various algorithms. It is necessary to have different complexity measures to estimate complexity of knowledge from different perspectives. For instance, complexity of working with such a representational knowledge as a model depends on the coarse graining (level of detail) of the description of the entity, on the previous knowledge and understanding of the world that is assumed, on the language employed, on the coding method used for conversion from that language into a string of bits, and on the particular ideal computer chosen as a standard (Gell-Mann, 1995). It is possible to consider these characteristics separately assigning to each a specific complexity measure or to build an integral complexity measure for estimation of the overall complexity. Complexity of a system, e.g., of a knowledge system, or a process, e.g., of cognition depends on the system making estimation. It may have an external observer, a user, or a designer. Relativity of complexity in general and of knowledge complexity, in particular, is perfectly demonstrated by the following joke. A Mathematician (M) and an Engineer (E) attend a lecture by a Physicist. The topic concerns Kaluza–Klein theories involving knowledge about spaces with dimensions of 11, 12 and even higher. M is sitting, clearly enjoying the lecture, while E is frowning and looking generally confused and puzzled. By the end, E has a terrible headache. After the lecture ends, M comments about the wonderful lecture. E says, “How do you understand this stuff?” M: “I just visualize the process.”
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E: “How can you POSSIBLY visualize something that occurs in 11dimensional space?” M: “Easy, you first visualize it in an n-dimensional space, then let n go to 11.” Mathematics makes subjective complexity objective introducing various criteria for complexity. For instance, a problem A is complex because its solution demands a huge amount of memory, while a problem B is complex because its solution involves performance of a huge amount of operations. Consequently, the problem A is complex for a computer with small memory, but it is simple for a computer with big memory. At the same time, the problem B is simple for a high performance computer, but is complex for an ordinary computer. Clarity and comprehensibility show easiness of understanding and often vary with the individual user of knowledge, e.g., the reader of a book. It is a very important property because if a knowledge item is unclear, it is hard to determine other properties of this item, e.g., whether it is accurate or relevant. In fact, it is impossible to tell anything about it without understanding what information it conveys. There are several practical criteria of clarity: — — — —
It It It It
is is is is
possible possible possible possible
to to to to
elaborate further on that issue. express that issue in another way. give an illustration for that issue. give an example for that issue.
Accessibility of knowledge is a kind of knowledge complexity. There are different measures of knowledge accessibility. Here we reflect on two of them. In developed knowledge systems, there are different levels of knowledge storage, which have different complexity, e.g., time, of knowledge access. This situation is modeled by the measure of actuality — the easier is the access the higher is the measure of actuality. On the other hand, complexity, e.g., time, of extraction and/or production of potential knowledge from actual knowledge can be
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rather different for different knowledge items. This situation is modeled by the measure of potentiality — the easier is the extraction/ production the less is the measure of potentiality. The size of knowledge representation is a kind of complexity. For instance, an algorithm or a program is an operational knowledge item, while the length of an algorithm or a program is a popular measure of complexity in the theory of algorithms and in algorithmic information theory (Burgin, 2005). Other examples of operational knowledge complexity measures are: — — — — — — — — —
Kolmogorov or algorithmic complexity; Time complexity; Space complexity; Average time complexity; Average space complexity; Static complexity measures; Dynamic complexity measures; Direct complexity measures; Dual complexity measures.
These and other complexity measures are used in various areas. Axiomatic approach to complexity of operational knowledge in the form of algorithms and abstract automata is developed in (Blum, 1967; Burgin, 2005; 2010d; Cˆ ampeanu, 2012). Axiomatic approach to complexity of operational knowledge in the form of computer software is developed in (Prather, 1984; Bollmann and Zuse, 1987; Burgin and Debnath, 2003). 2.3.5. Significance of knowledge Significance of knowledge is a relative characteristic, which depends on the person or system that evaluates this knowledge. What is significant for one person can be insignificant for another. For instance, a scientist makes observation, which is not important or even significant for him but is very important for his colleague. In a similar way, social knowledge is, as a rule, not important for a physicist but rather essential for a sociologist.
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It is possible to find an interesting example of knowledge significance in the areas of real number representations and computer arithmetic. In the standard decimal or binary representation, a real number is represented by a signed sequence of digits. These digits give knowledge about numbers. For instance, if the last digit of a number in the standard decimal representation is 4, then it is an even number. In a similar way, if a number has three digits in the standard decimal representation, then this number is larger than 99. Note that by tradition, the positive sign + is not usually displayed. For instance, 246.159 is the standard decimal representation of a real number. It is also possible to represent the same number by the sequence 00246.159000. However, zeroes in this sequence are insignificant because they do not change the number and they are omitted, as a rule, in the standard representation. Besides, what is significant in one type of representations can become insignificant in another type of representations. For instance, there is the scientific notation or scientific representation of real numbers, which consists of three parts: the sign of the number, the mantissa of the number, and the exponent of the number. The mantissa of the number is a real number, which is less than 10 and larger or equal to one, and the exponent is an arbitrary integer number. For instance, 3.159 × 1011 is the scientific notation. In science, very large and very small numbers frequently occur in many fields. So, to represent these numbers in a much shorter form, scientists invented scientific notation. For instance, the mass of a proton is approximately 0.00000000000000000000000165 gram. This is the standard representation of a decimal number. In scientific notation, this number has a much shorter representation. Namely, it is equal to 1.65 × 10−24 gram. Here is one more example. France’s national debt at the end of 2012 was $5,200,000,000,000. In scientific notation, we have $5.2 × 1012 . We see that only digits in the mantissa are significant with respect to scientific notation, while in the standard notation are significant. Later scientific notation was used as the floating point representation of real numbers in computers. This technique allows computers to operate in a much larger range of numbers than the fixed point,
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i.e., standard representation. However, the floating point representation has its pitfalls. One of the significant problem that arises in different situations is the loss of significant digits. For instance, numbers 246.1593 and 246.1592 have seven significant digits and thus, seven-digit accuracy, while their difference 246.1593 − 246.1592 = 0.0001 = 1 × 10−4 has only one significant digit and thus, only onedigit accuracy. Being relative, significance, nevertheless, is an important characteristic of knowledge because for an individual to successfully function, this individual needs knowledge that is significant with respect to her or his functioning. Sometimes absence of the necessary knowledge can cause very bad consequences and even bring to disaster. At the same time, the whole amount of knowledge accumulated by society and its members is so huge that one individual cannot acquire all of it and it is necessary to make the right selection. Significance is one of the most important criteria for this selection. It is possible to treat significance as a binary property with two values — significant and insignificant. However, a more accurate approach regards significance as a gradual property of knowledge. It is possible to represent graduality in different scales. The most exact are numerical scales, in which it is possible to have significance of order 5 or of order 3. There are also ordered scales. For instance, for a student it is more significant what she will get as a final grade than what will be her grade at an intermediate test. There are also nominal scales. For instance, it is possible to use such a scale {extremely significant, very significant, moderately significant, sufficiently significant, slightly significant, almost insignificant, insignificant, completely insignificant}. High orders or levels of significance are called importance. It is possible to use the threshold for importance in the scale of significance — knowledge significance of which is larger than the threshold for importance is important. Otherwise, it is unimportant. In this case, importance of knowledge can be a binary property as significance can be. However, it is more natural to consider different grades of importance treating it as a gradual property. Importance
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of knowledge as a category of knowledge significance can be represented in numerical scales, ordered scales, and nominal scales. Value of knowledge also reflects its significance. Starting with Plato, philosophers have discussed what is it about knowledge (if anything) that makes especially valuable for people. With the development of human civilization, the value of knowledge has continuously grown. However, different thinkers expressed diverse opinions on this issue. For instance, in 1775, Samuel Johnson (1709–1784) wrote, “All knowledge is, of itself of some value,” while Samuel Taylor Coleridge (1772–1834) stated, “The worth and value of knowledge is in proportion to the worth and value of its object” connecting value to the knowledge object or domain. Pragmatic approach, which is already present in Plato dialogues, asserts that the value of knowledge depends how this knowledge helps people in their activity aimed at achieving definite goals, e.g., Francis Bacon (1561–1626) declared “Knowledge Itself Is Power” (ipsa scientia potestas est), while Alvin Toffler (1990) proposed that knowledge is a wealth and force multiplier, in that it augments what is available or reduces the amount needed to achieve a given purpose. 2.3.6. Efficiency of knowledge The efficiency dimension reflects the role of a knowledge item in achieving certain goals or solving particular problems. It means that efficiency of knowledge is a relative property — the same knowledge item can be highly efficient for one goal and have low efficiency for another goal. For instance, knowledge in a textbook in mathematics is efficient for learning mathematics and is not efficient for learning music. In addition, efficiency also depends on the knowledge user. One individual can use the same knowledge more efficiently than another one. For instance, knowledge in a monograph on category theory will be useless, and thus, inefficient, for a non-professional but it may be very efficient for a mathematician who works in this area. Let us consider efficiency of operational knowledge. One kind of efficiency is related to the number of problems that can be solved using this operational knowledge.
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Definition 2.3.27. The more problems can be solved using an operational knowledge item K, the more potentially efficient K is. However, it is important not only to know that it is possible to solve some problem P in principle, but also to be able to find a relevant solution in practice. Such problems for which the latter is possible are called tractable. Tractability of a problem is a relative property, being dependent on the operational knowledge that is used for solution. This gives us a definition for pragmatic efficiency of operational knowledge. Definition 2.3.28. The more problems are tractable with respect to an operational knowledge item K, the more pragmatically or functionally efficient K is. Thus, pragmatic efficiency of operational knowledge depends on two parameters: power of the provided by operational knowledge means for solving problems and resources that are used in the process of solution. If it is impossible to get the necessary resources, it is unattainable to solve the problem under consideration. Thus, we come to the concept of resource efficiency of algorithms. Definition 2.3.29. The fewer resources are used for solution of a problem (of problems from some class) by means provided by operational knowledge, the more resource efficient this operational knowledge is with respect to this problem (class of problems). One more kind of efficiency is related to the quality of solution. Definition 2.3.30. The better solution for problems is provided by operational knowledge, the more mission efficient this operational knowledge is. Reliability, exactness, and relevance are examples of mission efficiency demonstrating that different dimensions of knowledge may have common components as, for example, relevance is a component of both correctness and efficiency. This analysis shows that knowledge efficiency is a function E(K, G, U ) where K is a knowledge item, G is a goal and U is a knowledge
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user. Note that not only an individual but also a software system or a robot can be a knowledge user. 2.3.7. Reliability of knowledge What Ketelaar wrote about information (Ketelaar, 1997) can be even more applied to knowledge, especially, because information is the source of knowledge: Why do we demand more of the quality of food or a car than we demand of that other essential resource — knowledge? Reliability and authenticity determine the credibility and the usefulness of knowledge. These concepts, developed in different cultures and at different times, are essential for our information society in its dependence on trust in knowledge and information. When new knowledge is created and distributed, conditions should be met to ensure the reliability and authenticity of this knowledge.
Reliability of knowledge shows to what extent it is possible to rely on some knowledge item. For instance, if a person remembers the telephone number of her friend, but when she dials this number, she finds that the number is incorrect. This shows that her knowledge of the telephone number is not very reliable. There are different reasons for this unreliability — she may simply forget the right number, her friend can change her number or she can dial a wrong number by mistake. Reliability of knowledge has three dimensions: Reliability of knowledge content depends on several attributes such as accuracy, veracity, credibility, correctness, and validity. Reliability of knowledge source is obtained when the attributes of content reliability are applied to the origin of knowledge, e.g., the author or corporate source of knowledge in the same way as to its content. This may be, for example, a rating of the previous content reliability of knowledge from this source, or of the circumstances under which a particular message originated. Reliability of knowledge transmission or/and production is obtained when the way (technique or process) of knowledge transmission or/and production is estimated.
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All these three dimensions are important for knowledge. First, people often judge reliability of knowledge they receive from some source by the reliability of this source. For instance, people are more inclined to believe authoritative individuals. Second, reliability of knowledge transmission may be crucial because invalid transmission can convert true knowledge into false knowledge. This sometimes happen in newspapers when truth is intentionally or unintentionally distorted. For instance, all newspapers in the Soviet Union provided false knowledge about Western countries to their readers. Third, humankind strived to develop reliable methods of knowledge production. One of the most (if not the most) reliable in this field is science. Unfortunately, even in science, some researchers faked their results demonstration that it is possible to corrupt even reliable knowledge production by an unreliable source. 2.3.8. Abstractness and generality of knowledge Abstractness and generality as properties of knowledge stem from two cognitive operations — abstraction and generalization. In the abstractness/generality dimension, abstractness reflects the level of abstraction of a knowledge item. It is possible to compare knowledge items with respect to their generality. It is possible to measure the level of abstraction of a knowledge item by the quantity of features in describing/representing the knowledge object or knowledge domain. When fewer features are taken into account, then the level of abstraction increases. Analyzing abstraction, it is possible to introduce levels of abstractness or levels of abstraction for epistemological structures in general and knowledge, in particular. Definition 2.3.31. (a) Knowledge (an epistemological structure) the domain of which consists of material objects has the first level of abstractness. (b) Knowledge (an epistemological structure) the domain of which consists of knowledge (epistemological) structures of the level n of abstractness has the (n + 1)th level of abstractness.
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Note that the level of abstractness is not an absolute characteristic of a knowledge item but reflects the angle of contemplation. In mathematics, levels of abstraction are formalized, for example, in type theory developed by Whitehead and Russell (1910–1913). In this theory, a set may contain only sets that have types lower than this set as its elements. Therefore, each type determines the corresponding level of abstraction. Another attempt to describe levels of abstraction was done by Hayakawa (1949), who introduced the concept Abstraction Ladder based on the approach from (Korzybski, 1933). In the Abstraction Ladder, individual (proper) names of material things form the first (verbal) level of abstraction, while common names, such as a dog or a plane, form the second level of abstraction. Higher level are constructed by taking concept with less and less defining properties. As a characteristic related to abstractness, generality reflects degree of generalization achieved by a knowledge item. The degree of generalization of a knowledge item reflects the scope of the described/ represented knowledge domain. The degree of generalization increases when a broader domain is described/represented or more aspects of the domain are mirrored. These two options are revealed in two components of generality — depth and breadth. Depth of a knowledge item reflects how many and to what extent aspects of an issue related to the knowledge item domain are taken into account. This shows that depth is the aspect of generality. Breadth of a knowledge item reflects the scope of the knowledge item domain, i.e., whether the knowledge content is applicable to a broad domain or to highly specific one. Note that a line of reasoning may be clear accurate, precise, relevant, and deep, but lack breadth as in an argument from either the conservative or liberal standpoint, which gets deeply into an issue, but only recognizes the insights of one side of the question (Elder and Paul, 2009). It is possible to compare knowledge items with respect to their generality. The more abstract knowledge is usually more general.
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Definition 2.3.32. A knowledge item (an epistemological structure) K the domain of which contains the domain of a knowledge item (an epistemological structure) H is more general than H. Analyzing generality, it is useful to introduce degrees of generality for epistemological structures in general and knowledge, in particular. Let us consider two knowledge items (epistemological structures) K and H. Definition 2.3.33. (a) If the domain of the knowledge item (the epistemological structure) K contains the domain of the knowledge item (the epistemological structure) H and there is no knowledge item (the epistemological structure) G such that the domain of K contains the domain of G, which, in turn, contains the domain of H, then K has the first degree of generality over H. (b) If the knowledge item (the epistemological structure) K has the first degree of generality over a knowledge item (the epistemological structure) G and G has the degree n of generality over the knowledge item (the epistemological structure) H, then K has the degree n + 1 of generality over H. As we can see, the level of generality is not an absolute characteristic of a knowledge item but reflects the angle of contemplation. Mathematical formalization of knowledge generality and abstraction are elaborated in Section 4.3.1. 2.3.9. Completeness of knowledge versus precision of knowledge Completeness of a knowledge item (system) K with respect to a domain D characterizes to what extent all essential aspects of the domain D are represented by K. For instance, knowledge that most birds fly but there are birds, e.g., penguins, that do not fly is more complete with respect to birds than knowledge that all birds fly. Knowledge that there are white and black swans is more complete with respect to birds than knowledge that all swans are white. There is a direction in epistemology that focuses on partial knowledge. In most cases, it is impossible to have complete knowledge and
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what we have is, as rule, incomplete or partial knowledge. It is possible to find complete knowledge about the state of affairs only in mathematical problems from a textbook. In real life, knowledge is only more or less complete. Thus, people use partial knowledge to solve real-life problems and epistemology studies partial knowledge. One of the consequences of this situation is that only bounded rationality is possible as complete knowledge and comprehensive information are inaccessible when people make decisions in real life situations. Precision of a knowledge item (system) K with respect to an issue (aspect) A characterizes the difference (or ratio) between (of) the issue (aspect) A and (to) its representation by K. For instance, correct knowledge of time in minutes is more precise than correct knowledge of time in hours. Knowledge that number π is equal to 3.14 is more precise than knowledge that number π is equal to 3. There is an ongoing conflict between completeness of knowledge versus precision of knowledge in information retrieval in databases and in search engines on the Internet. In information retrieval, precision (also called positive predictive value) is the ratio of the number of retrieved relevant instances, e.g., documents, to the number of all retrieved instances (documents). Precision ratios are often used in evaluation of the search engine quality. Completeness (also called recall) is the ratio of the number of retrieved relevant instances, e.g., documents, to the number of all relevant instances in the system. While it is possible to estimate this characteristic in database information retrieval, it is unrealistic even to try to calculate this value in the Internet search because search engines are unable to index or retrieve all the potentially available information. However, it is possible to make estimates of the precision of a given search engine for some area of knowledge.
2.3.10. Meaning of knowledge Knowledge exists in different forms and shapes. The most popular form is symbolic expressions, which are carriers of this knowledge.
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However, as we discussed in Section 2.1, practical knowledge or “know-how” often is embodied in customized dispositions, affective states such as emotions and sentiments, and phenomenological acquaintances conferred, for example, by sensory experience or artistic representation. Various images can be carriers of knowledge. Therefore, speaking about meaning of knowledge, it is necessary to take into account, knowledge carriers, and representations. We start with the meaning of symbolic expressions for three reasons. First, it is the most popular representation of knowledge. Second, the theory of meaning is mostly developed for expressions in natural languages, logical languages, and programming languages. Third, it is much easier to assign meaning to expressions than to other forms of knowledge representations, e.g., for emotions or feelings. When knowledge exists in the form of an expression, it is possible to derive its meaning by finding meaning of this expression. To do this, it is possible to use a special discipline called semantics, which has been developed in semiotics, linguistics, computer science and logic and studies meaning of expressions (in linguistics and logic) and signs (in semiotics). Often researchers do not discern semantics and theory of meaning. However, a more exact approach separates semantics as an operational theory of meaning that assigns semantic contents to signs and expressions of a language and the foundational theory of meaning, which explores the reason in virtue of which signs and expressions have the semantic contents that they have (Speaks, 2014). Here we consider only the operational theory of meaning but in a much broader sense, namely, as a theory the goal of which is to assign meaning to arbitrary objects. We begin with semantics in the classical sense. Semanticists generally recognize two sorts of meaning that an expression (such as the sentence “Knowledge is power”) or a sign (such as the sign ∫ ) may have: extensional meaning and intentional meaning. Extensional meaning is the relation that the expression (the sign) has to things and situations in the real world, as well as in possible worlds.
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Intentional meaning is the relation the expression (the sign) has to other expressions (signs). In semiotics, extensional meaning is presented by the relation between a sign and its object, which is traditionally called denotat or denotation. Intentional meaning presented by the relation between a given sign and the signs that serve in practical interpretations of the given sign is traditionally called sign connotation. However, many theorists prefer to restrict the application of semantics to the denotative aspect, using other terms or completely ignoring the connotative aspect. As a result, semiotic semantics related to symbols (signs), e.g., letters, words and texts, consists of two components. One of them, relational semantics, expresses intentional meaning and represents relations between symbols (signs). The other one, denotational semantics, expresses extensional meaning and represents relations between symbols and objects these symbols (signs) represent. In knowledge theory, extensional meaning of a knowledge item is presented by the relation between the knowledge item and its domain, which we call denotation of the knowledge item. Intentional meaning of a knowledge item is presented by the relations between of the knowledge item and other knowledge items, which we call connotation or content of the knowledge item. As a result, we get the Epistemological Triad presented in Figure 2.4. For instance, the domain of procedural knowledge consists of procedures and experiences in a field of work or behavior. At the same time, the corresponding content of procedural knowledge contains relations between descriptions of such procedures and experiences, as well as the way they can be applied, e.g., protocols in the medical sector, acceptation rules in the insurance branch, and methods of portfolio analysis in the business world. Knowledge Item
Denotation/Domain
Connotation/Content
Figure 2.4. The Epistemological Triad
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The domain of descriptive knowledge consists of different objects, their properties and relations between these objects. At the same time, the corresponding content of descriptive knowledge contains properties of and relations between knowledge items that describe such properties and relations of objects. The domain of representational knowledge consists of different objects with their properties and relations, while the corresponding content of representational knowledge contains relations between knowledge items that describe such objects with their properties and relations. Practical experience of people in the knowledge domain shows that there are gradations of meaning of knowledge items, e.g., concepts. In particular, Langacker (1991a) considers two levels of meaning — profile and base. The profile of a knowledge item K is the direct, e.g., literal, interpretation of K. For instance, a definition of a concept is its profile, while the base is the encyclopedic knowledge that the concept presupposes. In a general case, it is possible to understand the meaning of a knowledge item K as a knowledge system associated with K, e.g., a semantic network of a concept is treated as the concept meaning. Then the larger is the associated knowledge system the deeper meaning it reveals. Types of knowledge induce forms of meaning in semantics as a whole. It gives us three basic form of meaning: ∗ Descriptive meaning of knowledge reflects how this knowledge describes its domain. For instance, the statement “Aristotle was a philosopher” means that Aristotle had high intelligence. ∗ Operational meaning of knowledge reflects processes, actions, rules, procedures, and algorithms related to this knowledge. For instance, the statement “Aristotle was a philosopher” means that Aristotle developed philosophical theories and ideas. ∗ Representational meaning of knowledge reflects what this knowledge represents. For instance, the statement “Aristotle was a philosopher” means that there was a philosopher with the name Aristotle.
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In addition, there are various kinds of semantics with their specific conception of meaning. Logical semantics is related to propositions and involves truth values of these propositions. There is much philosophical discussion about the nature of “truthbearers” — the kinds of things that can be true or false. Various writers have suggested such things as propositions, statements, assertions, utterances, sentence-types, sentence-tokens, beliefs, opinions, theories, doctrines, facts, etc. Linguistic semantics is related to words and texts and is expressed by relations between them. There are different directions in linguistic semantics. Let us consider some of them. Formal semantics studies the logical aspects of meaning, such as sense, reference, implication, and logical form. Lexical semantics is a subfield of linguistic semantics and studies word meanings and word relations. According to this methodology, words either denote things in the world or concepts, depending on the particular approach to lexical semantics. Conceptual semantics studies the cognitive structure of meaning. Cognitive semantics is part of the cognitive linguistics movement and is based on the following assumptions. First, grammar is a conceptualization of meaning. Second, conceptual structure is embodied in and motivated by the usage of words in a language. Third, the ability to use language draws upon general cognitive resources and not on a special language module. However, there are psychologists and neurophysiologists who claim that linguistic abilities are based on very specific structures in the brain. Structural semantics, as logical positivists maintain, is the study of relationships between the meanings of terms within a sentence, the meanings of sentences within a text, and how meaning of larger systems can be composed from meanings of smaller systems. Frame semantics developed by Charles J. Fillmore (1929–2014) attempts to explain meaning in terms of their relation to general understanding, asserting that it is impossible to understand the meaning of a word or a text without access to all the essential
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knowledge that relates to that word or text. This essential knowledge is called the semantic frame of the corresponding word or text. In other words, a semantic frame of a word or text can be defined as a coherent structure of concepts that are related to this word or text such that without knowledge of all of them, one does not have complete knowledge of this word or text. Frames are based on recurring experiences (Fillmore, 1976; 1982). For instance, the writing frame is based on recurring experiences of writing. Semantics for computer applications falls into three categories (Nielson and Nielson, 1995): — Operational semantics is the field where the meaning of a construct is specified by the computation it induces when it is executed on a machine. In particular, it is of interest how the effect of a computation is produced. — Denotational semantics is the field where meanings are modeled by mathematical objects that represent the effect of executing the constructs. Thus, only the result is of interest but not how it is obtained. — Axiomatic semantics is the field where specific properties of the effect of executing the constructs represent meaning and are expressed as assertions. Thus, there are always aspects of the executions that are ignored. Another dimension of meaning is studied by pragmatics, which is the study of the ability of natural language users (e.g., speakers or writers) to communicate not only the general meaning but also their intentions, goals, and feelings, what the users mean and not the text. To distinguish the semantic meaning from the pragmatic meaning of a message (or sentence), communication researchers use the term the informative intent, also called the sentence meaning, and the term the communicative intent, also called the sender meaning or speaker meaning when it is an oral communication (Sperber and Wilson, 1986). In semiotics, pragmatics represents relations of signs to their impacts on those who use them, e.g., relations of signs to interpreters. This impact exists when a sign makes sense to the interpreter. Sense of a message (information) is defined by Vygotskii (1956) as a system
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of psychological facts emerging in the brain caused by the received message (information). The ability to understand another sender intended meaning is called pragmatic competence. A statement about pragmatic functions belongs to metapragmatic. While pragmatics deals with the ways people reach their goals in communication, metapragmatics explains how it is possible to reach the same goal with different syntax and semantics. Suppose, a person wants to ask someone else to stop smoking. This can be achieved by using several utterances. It is possible to say directly, “Stop smoking, please!” This utterance has straightforward and clear semantic and pragmatic meaning. Alternatively, one could say, “Oh, this room needs more air conditioners”, or “We need more fresh air here.” In the given context, these utterances imply a similar meaning but are indirect requiring pragmatic inference to derive the intended meaning. A popular assumption in the philosophy of mind and cognitive science is that the propositional attitudes of subjects are underwritten by an internal language of thought and comprised of mental representations. In other words, linguistic meaning is explained directly in terms of the contents of mental representations. Here are two theories of meaning based on these conjectures. The picture theory of meaning is a theory of linguistic reference and meaning verbalized by Ludwig Wittgenstein (1889–1951). In his Notebooks 1914–16 Wittgenstein wrote that language is first and foremost a representational system, with which people make (logical or mental) pictures of facts and these pictures are models of reality. Elements of the picture are combined with one another in a definite way by relations and connections. According to Wittgenstein, sentence is meaningful if and only if it is a fact, which corresponds to a possible fact in the world. To be a picture, a fact must have something in common with what it pictures. Thus, a meaningful proposition pictures a state of affairs or atomic fact in the world. In other words, the picture theory of meaning asserts that statements are meaningful if they can be defined or pictured in the real world. Wittgenstein compared the concept of logical/mental pictures (German Bild) with spatial pictures.
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A similar approach is the image theory of meaning, which is a theory in which meaning of linguistic expressions derived from mental images associated with these expressions. There are different kinds and types of mental images. Conceptual networks and diagrammatic schemas are mental images of structural knowledge. These images increase efficiency of knowledge processing and utilization. Specific mental images form the foundation of image theory, which is a descriptive theory of decision-making based on the assumption that decision makers represent meaning of knowledge as images (Beach and Mitchell, 1987). One image consists of principles that recommend pursuit of specific goals. A second image represents the future state of events that would result from attainment of those goals. A third image consists of the plans that are being implemented in the attempt to attain the goals. A fourth image represents the anticipated results of the plans. Decisions consist of (1) adopting or rejecting potential candidates to be new principles, goals, or plans, and (2) determining whether progress toward goals is being made, i.e., whether the aspired-to future and the anticipated results of plan implementation correspond. Decisions are made using either (1) the compatibility between candidates and existing principles, goals and plans, as well as the compatibility between the images of the aspiredto and the anticipated states of events; or (2) the potential gains and losses offered by a goal or plan. The denotation meaning of a knowledge item is the domain described or represented by this item. The relation meaning of a knowledge item is the structure (network of relations) with this item in the knowledge space. The estimate/significance meaning of a knowledge item is the significance of this item to the knower or to the knowledge observer. The relation meaning of a knowledge item includes the implicational meaning of this knowledge item, which consists of the knowledge implied by this item. The relation meaning of a knowledge item also includes the contextual meaning of this knowledge item, which consists of all contexts in which this item appears.
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Usually, the concept of a context is applied only to languages. Namely, a context of a word w (a text t) is a text T that includes w (respectively, t). Then the contextual meaning of a word w (a text t) is the set of all contexts in which this word (text) appears. However, it is possible to define context for knowledge. Definition 2.3.34. A context of a knowledge item k (a knowledge system K) is a knowledge system N that includes k (respectively, K). For instance, if a knowledge item is a mathematical theorem th from a textbook, then one context of th is the theory to which this theorem belongs, e.g., if th states that the derivative of the sum of two functions is the sum of the derivatives of these two functions, then the context is the Calculus. Another context to this knowledge item is the content of the textbook that contains th. Many mentalist theories of meaning have in common that they analyze one sort of representation — linguistic representation — in terms of another sort of representation — mental representation. It means a reduction of structural entities to mental entities. Grice developed an analysis of meaning based on two assumptions (Grice, 1989): (1) facts about what expressions mean are to be explained, or analyzed, in terms of facts about what speakers mean by utterances of them; and (2) facts about what speakers mean by their utterances can be explained in terms of their intentions. These two theses allow one reducing meaning of expressions (utterances) to the contents of the intentions of speakers. Another approach to analysis of meaning is based on the concept of belief, i.e., beliefs are taken into account rather than intentions of speakers. An interesting approach to meaning was elaborated by Osgood et al. (1978), introduced a measure of meaning and constructed a technology for measurement of meaning.
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2.3.11. Other descriptive properties of knowledge External characteristics of knowledge are molded by systems that are related to knowledge. Different authors explicated and discussed various external characteristics of knowledge. Such characteristics (properties) as availability, accessibility, being personal, being implicit, being explicit, being operational, being representational, being descriptive, and some others are considered in Section 2.1. Let us depict additional characteristics. The location of a knowledge item describes the place where the knowledge carriers of this item are situated. For instance, within the company or organization, carriers of a knowledge item can be situated in the front or the back-office, but also on the other side of the world. There are different types of knowledge carriers: people, documents on paper in the form of books, newspapers or reports, documents in computer files, web sites, pictures, paintings, etc. The form of a knowledge item describes the form of the representation of this knowledge item. For instance, it is possible to represent one knowledge item by a text and another knowledge item by speech. The material form of a knowledge item describes the carrier of this knowledge item. For instance, the material form of one knowledge item is a book and the material form of another knowledge is a computer. The content of a knowledge item describes what aspects of the domain (object) of this knowledge item it reflects. There are several temporal characteristics (properties) used in databases and knowledge bases. The generation time of a knowledge item describes the date when the knowledge item was generated. The acquisition time of a knowledge item describes the date when the knowledge item was obtained. Acquisition time is very important for temporal databases (Snodgrass and Jensen, 1999; Burgin, 2008a). The transformation time of a knowledge item describes the last date when the knowledge item was transformed.
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The accessibility time of a knowledge item describes what time it takes to access this knowledge item. The availability time of a knowledge item notifies at what periods of time this knowledge item is available. For instance, people can call 24 hours a day to get information from one organization or can obtain information from another organization only during its working hours. Such a property as novelty of knowledge has three gradations: • New knowledge is knowledge that the knower (knowledge system) did not have before. • Old knowledge has two meanings — it is either knowledge that the knower (knowledge system) obtained long ago or knowledge that is not new. • Contemporary knowledge is knowledge the knower (knowledge system) has at the considered period. All novelty gradations are relative. For instance, knowledge can be new for one individual but not new for another one or for a group. Knowledge about non-Euclidean geometries was new in the middle of the 19th century but it is old at the beginning of the 21st century. The contemporary knowledge at the end of the 20th century is very different from the contemporary knowledge at the end of the 10th century. Let us consider some more of knowledge properties. A knowledge item (knowledge) is outdated if a more recent knowledge item gives a better representation of the knowledge domain (object). Knowledge is safe when it cannot be distorted by some process. Knowledge is shareable because it does not decrease when it is given to others. This shows that knowledge differs from material resources to a great extent. Knowledge is private when only the person to whom it belongs has access to this knowledge. Confidentiality of a knowledge item means that access to it is restricted only to authorized people or systems. Note that privacy implies confidentiality.
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Integrity of knowledge involves maintaining the consistency, accuracy, and trustworthiness of this of a knowledge item over its entire life cycle. It means that integrity is a compositional property, which includes such components as the consistency, accuracy, and trustworthiness of a knowledge item. Depth of knowledge characterizes to what degree of details knowledge describes or represents its domain. Knowledge scope is the property, the value of which is the knowledge domain. Such property as knowledge generality considered above reduces to the relation between the scope of different knowledge items. Knowledge can be interesting or not. For instance, Alfred North Whitehead (1861–1974) wrote, “It is more important that a proposition (i.e., propositional knowledge) be interesting than that it be true.” Thus, we see that an extremely active research in the knowledge studies domain has allowed researchers to find many properties of knowledge and knowledge processes and to use these properties in artificial intelligence, education, and psychology. 2.4. Metaknowledge and metadata A library may be very large; but if it is in disorder, it is not so useful as one that is small but well arranged. In the same way, a man may have a great mass of knowledge, but if he has not worked it up by thinking it over for himself, it has much less value than a far smaller amount which he has thoroughly pondered. Arthur Schopenhauer
The Greek word meta means “beside”, “after”, “later than” or “in succession to”. Often people understand that something with the name “metaX” occurs later on the timeline than X. However, a more popular meaning in contemporary languages is “beside” or “after.” For instance, carpus is the wrist, while metacarpus is the part of the human hand between the wrist and the fingers or we may say, after the wrist and before the fingers. In a similar way, metatarsus is the part of the human foot after the tarsus and before the toes.
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An important part of philosophy is metaphysics. It came to philosophy through the common name for several of Aristotle’s works. However, Aristotle himself did not call the subject of these works by the name metaphysics but referred to it as “first philosophy”. The name metaphysics comes from the editor of Aristotle’s works, Andronicus of Rhodes. He placed the books on first philosophy right after the work called Physics, and called them Metaphysics meaning “the book that come after the [books on] physics”. Due to this, later generations of philosophers called the subject metaphysics thinking it meant “the science of what is beyond the physical”. However, now metaphysics is one of the pivotal branches of philosophy concerned with explaining the fundamental nature of being and the world as the manifestation of being and clarifying the fundamental notions by which people understand the world. Due to fewer restrictions in philosophical exploration in comparison with research in physics, metaphysics often goes beyond physics in many aspects. In contemporary understanding, the word “metaX” means above X. However, the word above means after in a hierarchy when you go in the direction bottom-up. This exactly relates to metaknowledge and metadata. Metaknowledge or meta-knowledge is knowledge about knowledge. For instance, it may be a cluster of definitions and methods aiming to guide you in gathering the pertinent knowledge with regard to your activity. The metaknowledge is often used to guide functioning of a system including goal formation and future planning. Metaknowledge is intrinsically connected to metadata. Metadata (also called metacontent) are data that provide information about one or more aspects of the data. For instance, data with the standard file information such as file size, type, location, and date of creation are metadata for data in the file. If data are organized as a text, then their metadata usually contain information about: — the language of the text, — the number of words in the text,
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— — — — —
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the number of pages in the text, the number of symbols in the text, the number of lines in the text, who the author is, when the text was written,
and in some cases, — a short summary of the text. However, a summary of a text actually is metaknowledge as it contains knowledge about the text. Another example of metaknowledge is an annotation of a book. An important form of metadata is represented by named sets (Burgin, 1990; 1995; 2011). Indeed, it is demonstrated that all main data structures and models are efficiently represented in a form of named sets or chains of named sets (Burgin, 1992a). In addition, named sets and their chains also give a unifying data model for data structures used in programming languages, operating systems, and computer hardware: vectors, lists, arrays, trees, strings, tables, records, streams and the like. The same is true for such forms of data on the Web as digital imagery (in the form of frames or pictures) and audio data. In such a way, we come to a unified data meta-model that is suitable data models on all levels: from high-level or conceptual to representational or implementation to low-level or physical models. In turn, named sets and their chains form efficient high-level metadata (cf., (Siegel and Madnick, 1991; Tannenbaum, 2002)) for different purposes, in particular, for XML documents and schemas (Nocedal et al., 2011). Such general representation allows us to introduce various operations with named sets and their chains oriented to data preparation, processing, search, and acquisition. Some operations are similar to those found in relational algebras (Codd, 1990). Other operations, such as sequential composition of named sets and their chains, are different from operations in relational databases. All these operations provide means for working with data models that are different from
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the relational ones. For instance, sequential composition of named set chains represents extensions of hierarchies. Although metadata have been used from the beginning of computer era, the term metadata was introduced in (Bagley, 1968). Since then it became popular in many areas such as information management, information science, information technology, librarianship, and databases. For instance, library catalogs are metadata in libraries, which are created in the process of cataloging resources, such as books, journals, newspapers, magazines, manuscripts, DVDs, web pages, or digital images. Here are more examples of information contained in metadata for some data: • • • • • • • •
Means of creation of the data; Purpose of data creation and utilization; Time and date of creation; Source, creator or author of data; Location where the data was created; Location where the data are stored; Standards used; Data model for representing data structure.
Metadata research emerged as a discipline crosscutting many areas and domains. It has been directed at the provision of structural descriptions (often called annotations) to Web resources or applications. Descriptions in the form of metadata function as a basis for advanced services in many application areas, including search and location, personalization, federation of repositories, and automated delivery of information. For instance, the HTML format for defining web pages has means for inclusion of various metadata, from basic annotations, dates and keywords to further advanced metadata schemas (Nocedal et al., 2011). In addition, metadata are used in database servers, data virtualization servers, and application servers. Metadata in these servers are used for describing business objects in various enterprise systems and applications. Structural metadata commonality is also important to support data virtualization.
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In information systems, metadata are often attached to data in the form of labels and tags. For instance, a digital tag is a simple system of keywords or terms assigned to a compound data (knowledge) item such as a computer file, digital image, or Internet bookmark. The main goal of tagging is identification of a data (knowledge) item in a system for finding this item by browsing or searching. Usually creators or users of data (knowledge) items informally choose tags for these items. Tagging and labeling on the Internet has achieved wide popularity due to the growth of social networks, blogging, photography sharing and bookmarking sites. These sites allow users to create and manage labels or tags in the form of keywords. Often labels and tags provide semantic information about data (knowledge) items to which they are attached. For instance, triple tags have three parts — a namespace, a predicate, and a value — for purpose of their meaningful interpretation by computer programs. Metadata has become even more important for the Semantic Web with its technological framework for ontology-based metadata. The central idea of the Semantic Web is to extend the existing Web by adding semantic means to the web management resources allowing better search, processing, integration, and presentation of the Web information in a meaningful, intelligent manner. Different new means have been developed for the Semantic Web. An example is the distributed intelligent managed element (DIME) network architecture described in (Burgin and Mikkilineni, 2014). Although it is possible to write books about metadata (cf., for example, (Siegel and Madnick, 1991; Tannenbaum, 2002)), our main concern here is metaknowledge. Thus, the first step in this direction is understanding that the main feature of metadata is that they contain knowledge about data. Some of this knowledge is related only to the corresponding data, while other describes knowledge contained in the corresponding data, For instance, taking data in the form of a text, we see that metadata that inform us about the size of the text, e.g., the number of pages, give knowledge about data, while metadata in the form of annotation give knowledge about knowledge in the text and this knowledge is metaknowledge.
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Let us consider other kinds of metaknowledge that have existed from the times long before the first computers were created. A kind of operational metaknowledge is represented by metarules. There are many metarules in logic, i.e., rules about how to use other rules. Metarules instruct how to manipulate expressions or formulas that are well formed or explain how to use deduction rules or how to perform deductions. For instance, “the pair A → B, A implies B”is a deduction rule because it represents a variety of deduction rules such as “If it is a rain, the trees are wet” and “It is a rain” imply “The trees are wet”. Models of systems and processes contain representational knowledge about these systems and processes. In the same way as models contain knowledge about their domains, metamodels contain knowledge about models. It means that metamodels contain metaknowledge. Mathematics gives an abundance of examples of metamodels. Let us consider some of these examples. Differential equations in a general form, e.g., ∂ m ui (t, x) = ∂tm
Pα (aαji (t, x), u(t, x), Dxα Dtk ui (t, x))
|α|+k≤m,k m; and d((e; w1 , . . . , wn ), (l; u1 , . . . , um )) = dv ((w1 , . . . , wn , 0, . . . , 0), (u1 , . . . , um )) + d(e, l)
(3.5)
when n < m. In finite-dimensional vector spaces, we can take the Euclidean metric as dv , defining the distance dv ((x1 , . . . , xn ), (y1 , . . . , yn )). However, as we discussed before, it is natural to assume that the fiber F is an infinite-dimensional vector space. In this case, we simply postulate existence of a metric in it. Usually, metrics in vector spaces are defined by norms (Rudin, 1991; Burgin, 2013). Note that in this case, we use only formula (3.3) because all fibers Fa have the same dimension. Proposition 3.1.3. The distance d((e; w1 , . . . , wn ), (l; u1 , . . . , um )) defines a metric in the space Ww . Proof . By definition, d((e; w1 , . . . , wn ), (e; w1 , . . . , wn )) = 0. When d((e; w1 , . . . , wn ), (l; u1 , . . . , um )) = 0, then d(e, l) = 0 and, thus,
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e = l because d is a metric in W . Besides, dv ((w1 , . . . , wn ), (u1 , . . . , um )) = 0 and thus, (w1 , . . . , wn ) = (u1 , . . . , um ) because dv is a metric in a vector space. Consequently, d((e; w1 , . . . , wn ), (l; u1 , . . . , um )) = 0 if and only if (e; w1 , . . . , wn ) = (l; u1 , . . . , um ). The function d is symmetric because the function d is symmetric in W , while the function dv is symmetric in the vector space F . In addition, let us take arbitrary weighted (symbolic) epistemic structures (e; w1 , . . . , wn ), (l; u1 , . . . , um ) and (h; v1 , . . . , vp ) from E and denote d(e, l) = a, d(l, h) = b, d(e, h) = c, dv ((w1 , . . . , wn ), (u1 , . . . , um )) = d, dv ((u1 , . . . , um ), (v1 , . . . , vp )) = k and dv ((w1 , . . . , wn ), (v1 , . . . , vp )) = r. Then we have: c ≤ a + b because d is a metric in W , r ≤ d + k because dv is a metric in the vector space F . Consequently, d((e; w1 , . . . , wn ), (h; v1 , . . . , vp )) = r + c ≤ (d + a) + (k + b) = d((e; w1 , . . . , wn ), (l; u1 , . . . , um )) + d((l; u1 , . . . , um ), (h; v1 , . . . , vp )) i.e., the third axiom of metric spaces is true. Proposition is proved. Corollary 3.1.1. d((e; w1 , . . . , wn ), (h; v1 , . . . , vn )) ≥ d(e, h) and d((e; w1 , . . . , wn ), (h; v1 , . . . , vn )) ≥ dv ((w1 , . . . , wn ), (v1 , . . . , vn )) Corollary 3.1.2. d((e; w1 , . . . , wn ), (h; v1 , . . . , vn )) d(e, h) < k.
m; and d((e; w1 , . . . , wn ), (l; u1 , . . . , um )) = dv ((w1 , . . . , wn , 0, . . . , 0), (u1 , . . . , um ))2 + d(e, l)2
(3.8)
when n < m. Structures in the spaces Wes , Wses , Wesw , and Wsesw are inherited by epistemic spaces and their states. In particular, a weighted epistemic space E and each its state is a vector bundle E = (E, pE , Ee ) with the metric d in the base E. We remind that a set X in a metric space E with a metric d is called bounded if there is a number k such that for any points a and b from X, d(a, b) < k. To study bounded sets in metric spaces that are spaces of vector bundles, we need additional concepts. Example 3.1.16. Osgood, Suci and Tannenbaum (1978) define distance in semantic spaces (cf., Example 3.1.8) by the formula from the m-dimensional Euclidean spaces: m d(e, l) = d2elj . j=1
In this formula, m is the number of factors and delj is the difference between the coordinates of the elements e and l with respect to the same factor (dimension) j. In the most refined models, the number m is equal to 3 (Osgood et al., 1967). Let us consider a vector bundle E = (E, pE , Ee ) with the fiber F . Definition 3.1.20. A set X ⊆ E is called rectangular in E if X = {(b, u)|b ∈ Xe = pE (X), u ∈ F and for any a ∈ Xe and v ∈ F ((b, v) ∈ X ⇒ (a, v) ∈ X)}.
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Example 3.1.17. Let us consider a trivial vector bundle H = (H = {a, b, c} × R, ph , He = {a, b, c}). Then the set X = {(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)} is rectangular in H, while the set Z = {(a, 1), (a, 3), (b, 1), (b, 5), (c, 1), (c, 3)} is not rectangular in H. Definition 3.1.21. If X ⊆ E, then the minimal rectangular in E set R(X) that contains X is called the rectangular closure of X in E. Proposition 3.1.4. The rectangular closure of a set in E always exists and is unique. Indeed, we can put R(X) = {(a, u)|a ∈ Ee = pE (X), u ∈ F and there is b ∈ Xe ((b, u) ∈ X)}. Proposition 3.1.5. The operation of taking the rectangular closure of a set in E is a closure operation in the sense of (Kuratowski, 1966) on sets in metric spaces. In particular, the operation of taking the rectangular closure of a set in E is idempotent, i.e., R(R(X)) = R(X) for any X ⊆ E. Proposition 3.1.6. A set X in E is rectangular if and only if X = R(X). Proof is left as an exercise. Definition 3.1.22. If X ⊆ E, then the fiber projection σ(X) of X is defined as follows: σ(X) = {u | ∃b ∈ Xe ((b, u) ∈ X}. For instance, taking sets X and Z from Example 3.1.17, we see that: σ(X) = {1, 2}, and σ(Z) = {1, 3, 5}. Definitions imply the following result.
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Proposition 3.1.7. A set X in E is rectangular if and only if X = Xe × σ(X). Some properties of sets and their rectangular closures are the same. Proposition 3.1.8. A subset X of the space E of the vector bundle E = (E, pE , Ee ) is bounded if and only if its rectangular closure is bounded. Proof . Sufficiency. By definition, any subset of a bounded set is bounded. Necessity. Let us assume that X is bounded. It means that there is a positive number k such that d(x, z) < k for any two points x = (a, u) and z = (b, v) from X where a, b ∈ Xe . Let us take two points p and q from R(X). Then p = (c, w) and q = (d, y) with c, d ∈ Xe . By the definition of the rectangular closure R(X), there are points x and z from X such x = (a, w) and z = (b, y) with a, b ∈ Xe . By the properties of metric, d(p, q) ≤ d(p, x) + d(x, z) + d(z, q). By initial conditions, d(x, z) < k. At the same time, by the definition of the metric d and Corollary 3.1.2, we have: d(p, x) = d((c, w), (a, w)) = d(c, a) < k, and d(z, q) = d((b, y), (d, y)) = d(b, d) < k. Consequently, d(p, q) < 3k. Proposition is proved because p and q are arbitrary points from R(X). Reducing the problem of boundedness to rectangular sets, now we find conditions of boundedness for rectangular sets. Proposition 3.1.9. A rectangular subset X of the space E of the vector bundle E = (E, pE , Ee ) is bounded if and only if the projection
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Xe = pE (X) of X and the fiber projection σ(X) of X are uniformly bounded. Proof . Necessity. Let us assume that the projection Xe = pE (X) of X is unbounded. It means that for any positive number k, there are two points a and b in Xe such that d(a, b) > k. As Xe is the projection of X, there are two points x and z in X such that a = pE (x) and b = pE (z). By the definition of the metric in the space E, d(x, z) ≥ d(a, b) > k. Consequently, X is also unbounded. Now, let us suppose that the fiber projection σ(X) of X is not uniformly bounded. It means that for any positive number k, there are two points u and v in σ(X) such that dv (u, v) > k. As σ(X) is a projection of X, there are points x = (a, u) and z = (b, v) from the space X. By Corollary 2, d(x, z) ≥ k as by choice of the points u and v, dv (u, v) > k. Thus, the space X is not bounded. Then by the Law of Contraposition, if the space X is bounded, then the projection Xe = pE (X) of X and the fiber projection σ(X) of X are bounded. Sufficiency. Let us suppose that the projection Xe = pE (X) of X and the fiber projection σ(X) of X are bounded. It means that there is a positive number k, such that for any two points a and b in Xe , we have d(a, b) < k and there is a positive number h, such that for any two points u and v from σ(X), we have d(x, z) < h. Let us take two points x and z from X. Then x = (a, u) and z = (b, v) where a, b ∈ Xe , while u and v belong to the fiber F of the bundle E. As X is a rectangular set, points (a, u) and (b, v) belong to X and by definition, d(x, y) = d(a, b) + d(u, v) < k + h. Consequently, the set X is bounded. Propositions 3.1.7 and 3.1.9 imply the following result. Corollary 3.1.3. A subset X of the space E of the vector bundle E = (E, pE , Ee ) is bounded if and only if the projection Xe = pE (X) of X and the fiber projection σ(X) of X are uniformly bounded. Note that here we study weighted epistemic structures with real number weights and weighted epistemic space in which weights form
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real vector spaces. However, using the same technique, it is possible to obtain similar results for weighted epistemic structures with complex number weights or vector weights and for weighted epistemic space in which weights form complex vector spaces. 3.2. Knowledge evaluation, justification, and testing It is better to learn late than never. Publilius Syrus
In the context of epistemic structures in general and knowledge, in particular, evaluation means finding properties or values of properties of these structures (knowledge). In a more strict sense, knowledge evaluation also means evaluation of epistemic structures properties related to knowledge, e.g., properties that allow discerning knowledge from other epistemic structures. In the context of epistemology, (knowledge) justification means giving sound grounds for holding a belief (Pollock, 1974). Epistemological justification is a normative notion. Thus, we see that justification is evaluation of the property “to be justified” and justification is a kind of evaluation. In spite of this, we separately consider justification because of the long-standing philosophical tradition related to the notion of justification. Testing an epistemic structure (knowledge) is a process, or a procedure that controls such a process, aimed at answering some question about properties of this structure (knowledge) (Burgin and Debnath, 2007). Thus, testing represents a class of processes or procedures for evaluation. 3.2.1. Knowledge evaluation As knowledge evaluation is evaluation of knowledge properties in the context of epistemic structures, it means finding whether a knowledge item has some properties (in the classical (set-theoretical) interpretation of properties) or what is the value of a property (in the sense of the theory of abstract properties, cf., Chapter 5 and (Burgin, 1985; 1986)) for this knowledge item.
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There are different types and kinds of evaluation in general and knowledge evaluation, in particular. Taking into account processes of (knowledge) evaluation, we get the procedural typology of evaluation: • Testing or direct evaluation (of knowledge properties) is a procedure performed with the object of evaluation (with a knowledge item/system) aimed at evaluation of properties of this object. • Representational evaluation (of knowledge properties) is finding properties of the object of evaluation (of a knowledge item/system) by performing operations with a representation of this object. • Relational evaluation (of knowledge properties) is finding properties of the object of evaluation (of a knowledge item/system) by comparing this object (knowledge item) to another object of the same type (knowledge item/system). An example of knowledge testing is finding whether a given scientific theory or knowledge base is consistent by comparing postulates and theorems of this theory (units of this knowledge base). An example of representational knowledge evaluation is finding whether a given scientific theory or knowledge base is consistent by representing this theory (knowledge base) as a logical calculus (logical variety) and testing its consistency. An example of relational knowledge evaluation is finding whether a given scientific theory or knowledge base is consistent by comparing this theory (knowledge base) to a consistent scientific theory (knowledge base). Social and individual practice of dealing with epistemic structures in general and knowledge, in particular, shows that there are three operational types of testing (direct evaluation):
➢ Testing by computation; ➢ Testing by inference; ➢ Testing by application. When a computer or a network is used to find knowledge about a given object, e.g., some person, in a knowledge base, it is testing by computation.
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interpretation knowledge system
knowledge domain
similarity another knowledge system
Figure 3.9. The Representational Tetrad
When a scientist tries to find if an already known law of nature follows from his theory, it is testing by inference. When an engineer tries to find he can use some physical theory for engineering problems, he applies this theory to his problems and it is testing by application. Similar to testing, representational knowledge evaluation also has three types because it is possible to consider three types of representations for a knowledge system — a model of this knowledge system (metaknowledge, the domain of reference of this knowledge system, i.e., the domain in which this knowledge system is interpreted, and representation by a similar knowledge system (cf., Figure 3.9). It gives us the representational classification: • Representational evaluation by a model; • Representational evaluation by the domain of reference; • Representational evaluation by a similar knowledge system. When a scientist formalizes his theory as a logical calculus (logical variety) to find its formal properties, it is evaluation by a model. When a physicist checks his theory against experimental data, it is evaluation by the domain of reference. When a scientist wants to show that his new theory is better than the existing theory, he compares both theories and this is evaluation by a similar knowledge system. In addition to procedural types, there are also three target types of (knowledge) evaluation:
➢ General evaluation is finding values of definite properties of a given object (of a knowledge item in our case); ➢ Validation is finding whether a given object (a knowledge item in our case) has definite properties;
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➢ Cause evaluation is finding causes for definite properties of a given object (of a knowledge item in our case). An example of general evaluation is finding computational complexity of an algorithmic problem as an algorithmic problem is a kind of operational knowledge. An example of validation is finding whether a scientific theory or knowledge base is complete with respect to a given domain. To do this, scientists perform various experiments with the domain and explore if a given theory explains all encountered phenomena. An example of cause evaluation is finding why a scientific theory or knowledge base is inconsistent. There are also three aspect types of (knowledge) evaluation:
➢ Meaning evaluation is evaluation of meaning of a given object (of a knowledge system in our case); ➢ Representation evaluation is evaluation of a representation of a given object (of a knowledge system in our case); ➢ Structure evaluation is evaluation of the structure of a given object (of a knowledge system in our case). An example of meaning evaluation is finding meaning of theoretical terms, which was one of basic tasks for logical positivism (Carnap, 1928). An example of representation evaluation is finding why a scientific theory or knowledge base is inconsistent. An example of structure evaluation is finding whether a model of a scientific theory or knowledge base is adequate. As the general theory of evaluation shows, the process of evaluation has three main stages: preparation, realization, and analysis (Burgin and Kavunenko, 1994). Preparation demands the following operations to achieve correct and sufficiently exact evaluation: 1. Choosing evaluation criteria. 2. Corresponding characteristics (indices) to each of the chosen criteria. 3. Representing characteristics by indicators (estimates).
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Indicator
Figure 3.10. The Attributive Triad
This shows that a complete process of evaluation preparation has the structure of the Attributive Triad (cf., Figure 3.10), in which nomological tools for evaluation — evaluation criteria, indices, and indicators — are prepared. A specific realization of this process is the GQM (GoalQuestion-Measurement) approach to software measurement (Basil and Rombach, 1988), which is a case of operational knowledge evaluation and in which software metrics play the role of evaluation indices. According to GQM, creation of software indicators (software metrics) for software evaluation has to include the following three stages: 1. Setting goals specific to needs in terms of purpose, perspective, and environment. 2. Refinement of goals into quantifiable tractable questions. 3. Deducing metric and data to be collected (as well as the means for their collection) to answer the questions. Thus, the first stage in evaluation demands to determine a specific criterion for evaluation. This criterion gives the goal of evaluation. For instance, criteria of good software include such properties as reliability, adequacy, exactness, completeness, convenience, user friendliness, etc. However, such properties are also directly immeasurable and to estimate them, it is necessary to use corresponding indices and indicators. However, a criterion can be too general for direct estimation. This causes necessity to introduce more specific properties of the evaluated object. To get these properties, quantifiable tractable questions are formulated. Such properties play the role of indices for this criterion. Thus, the second stage of evaluation consists of index selection that reflects criteria. Sometimes an index can coincide with the corresponding criterion, or a criterion can be one of its indices. However, in many cases, it is impossible to obtain exact values for the chosen indices. For instance, we cannot do measurement with absolute precision. What is possible to do is only to get some estimates of
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indices. Consequently, the third stage includes obtaining estimates or indicators for selected indices. In the case of software, these indicators have form of software measures. With respect to software quality such indicators are called software metrics. Similar approach was suggested by Belchior, Xex´eo, and da Rocha (1996) in their hierarchical software quality evaluation model (SQEM). This model is based on four main concepts: objectives or goals, factors, criteria, and evaluation processes. Quality objectives or goals form criteria and represent important properties that a product should possess. Each goal is decomposed into definite factors, which are sometimes further decomposed into more detailed subfactors. Factors and subfactors play the role of indices and define different users’ perspectives about the quality of a software product. After this, it is necessary to convert obtained indices into corresponding indicators. However, a software metric (indicator) is useful only when there are corresponding procedures/algorithms of measurements. Thus, we need to reflect more stages for the metric development, which includes six stages — three listed above and three new ones. 4. Designing procedures/algorithms for data collection. 5. Designing procedures/algorithms for computing metrics values. 6. Designing procedures/algorithms for analysis of measurement results. It is necessary to remark that measuring algorithms are usually recursive, while operational knowledge, e.g., computer programs, is a system of dynamic objects, which are usually changing during their life cycle. For instance, to facilitate the necessary changes for computer software, many software update services are suggested on the Web. That is why super-recursive algorithms would be more efficient and flexible for evaluation (measurements) of operational knowledge in software engineering (Burgin and Debnath, 2009). The dynamics of knowledge evaluation is reflected by the Evaluation Triad (cf., Figure 3.11). Let us give an example of this triad. Assume that a ball is taken from the urn with green and blue balls. Then it is possible to consider
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Measure
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Measurement
Figure 3.11. The Evaluation Triad
such a property of a ball as “to be blue.” Then the corresponding measure is the probability that a randomly taken ball is blue. In this case, the measurement procedure is extraction of a ball and observation to determine its color. The Evaluation Triad goes after the Attributive Triad because the property with which the Evaluation Triad starts is in essence the indicator from the Evaluation Triad. Studying epistemic structures and their important case — knowledge, researches evaluated them from different perspectives. The most popular one has been the representation perspective because epistemic structures in general and knowledge in particular always represent or reflect some domain. Thus, it is possible to estimate relevance of this representation (reflection) to the represented (reflected) domain, e.g., to physical reality. This brings us to the cognitive scale, in which epistemic structures form the representational continuum situated in the relevance dimension and presented in Figure 3.12. All elements from the representational continuum are called impressions. Informally an impression is an epistemic structure with an estimate, such as true, or false, relevant or irrelevant, correct or incorrect, grounded or ungrounded. Note that impressions are not only statements or propositions, but they may be, for example, images or procedures. In the representational continuum, impressions are ordered by the degree of relevance so that impressions with the higher relevance in them lie to the right of impressions with the lower relevance in them. When the degree of relevance becomes sufficiently high, the corresponding structures from the representational continuum are Knowledge complete relevance
Impressions
Misconceptions/Delusions complete irrelevance
Figure 3.12. The representational continuum
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called knowledge. When the degree of relevance becomes very low, the corresponding structures from the representational continuum are called misconceptions and delusions. This shows that it is possible to discern these groups of impressions either based on the factual situation or on the evaluation that exists in the corresponding knowledge system. For instance, a person can think that he knows something while by objective estimates it will be only a bunch of misconceptions. When a man in a desert, deprived of water and thirsty, has a mirage of a well, very often this man thinks that he knows that very soon he will be able to drink, while in reality, it is a misconception that he is seeing water. Examples of truth-valued impressions are propositions, statements, assertions, utterances, sentence-types, sentence-tokens, beliefs, opinions, theories, doctrines, facts, etc. Examples of beneath-truth-valued impressions are words, naming expressions (such as big mountain, three monkeys, or pretty dolls), exclamations (“Ouch”, “Ups”), numbers (seventy seven, three hundred and five) There are different evaluation criteria for impressions: — Correctness, which is traditionally called truth, reflects relation of knowledge or more general, of an impression to the domain this impression (knowledge) reflects. — Confidence by assurance reflects to what extent the agent is certain in the validity of an impression. — Groundedness by evidence reflects the extent of the impression evaluation. For instance, we have an impression that it will be a good weather tomorrow. This prognosis is justified by weather forecast but only tomorrow we will know if this impression is correct and to what extent. If we strongly believe in the weather forecast, our confidence by assurance is high. If we know probability of correct weather forecasting, then the groundedness by evidence is equal to this probability, e.g., if this probability is 70%, then the groundedness by evidence is equal to 70%.
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There are external impressions formed under the impact of information coming from external sources and there are internal impressions formed under the impact of information coming from internal sources. For instance, according to contemporary science, dreams give internal impressions, while vision gives external impressions. It is necessary to remark that there are other opinions that assume existence of external dreams, i.e., dreams that convey information from external sources. Functional typology and other classifications of impressions are considered in Chapter 2. Here we describe only epistemological taxonomy, which is related to the epistemological scale. Another approach to epistemic structures is represented by the confidence perspective because epistemic structures in general and knowledge in particular are evaluated by cognitive systems, such as people, by believing in these epistemic structures. Thus, it is possible to estimate the degree of believing building the confidence/certainty scale, in which epistemic structures form the confidence/certainty continuum presented in Figure 3.13. All elements from the confidence/certainty continuum are called beliefs. They are ordered by the certainty (confidence) that they are true, correct or relevant as a degree of believing. In this ordering, beliefs with the higher certainty (confidence) lie to the right of beliefs with the lower certainty (confidence). When the degree of believing (confidence or certainty) that the belief is true, correct or relevant becomes sufficiently high, the corresponding structures from the confidence/certainty continuum are called knowledge. The maximal uncertainty is achieved in the middle of the scale where certainty of truthfulness, correctness or relevance reaches zero. When the degree of believing (confidence or certainty) that the belief is Knowledge
Beliefs
as complete believing or complete certainty of (confidence in) truthfulness, correctness or relevance
maximal uncertainty
Fantasies as complete disbelieving or complete certainty of (confidence in) falseness, incorrectness or irrelevance
Figure 3.13. The confidence/certainty continuum
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untrue, incorrect or irrelevant becomes sufficiently high, the corresponding structures from the confidence/certainty continuum are called misconceptions or delusions. Note that it is also possible to treat certainty of (confidence in) falseness, incorrectness, or irrelevance as negative certainty of truthfulness, correctness, or relevance. When certainty is represented by subjective probability, we come to negative probabilities (Dirac, 1930; 1942; 1974; Heisenberg, 1931; Wigner, 1932; Pauli 1943; 1956; Feynman, 1950; 1987; Burgin and Meissner, 2010; 2012). As elements in the confidence continuum are also impressions, it is possible to classify them according to this scale. Namely, we have the following descriptions: Beliefs are impressions that are grounded (justified) to some extent. Knowledge consists of impressions that are (or are considered) highly grounded (justified). Fantasies are impressions that are (or are considered) almost or completely ungrounded (baseless). It is possible to define these concepts in a more exact and detailed way discerning three approaches — the objectdependent, reference-dependent, and attitude-dependent approaches. The object-dependent approach differentiates beliefs according to their correspondence to the domain (object). Definition 3.2.1. A unit B of individual validated knowledge (of collective validated knowledge) about a domain D (objects from the domain D) is the epistemic structure from Figure 3.1 or 3.2 that is highly justified and true. This is the traditional perspective on knowledge as a specific kind of beliefs. Definition 3.2.2. A unit B of specific validated belief (of general validated belief) about a domain D (objects from the domain D) is the epistemic structure from Figure 3.1 or 3.2 that is sufficiently justified. As Bem (1970) writes, “beliefs and attitudes play an important role in human affairs. And when public policy is being formulated,
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beliefs about beliefs and attitudes play an even more crucial role.” As a result, beliefs are thoroughly studied in psychology and logic. Belief systems are formalized by logical structures that introduces structures in belief spaces and calculi, as well as by belief measures that evaluate attitudes to cognitive structures and are built in the context of fuzzy set theory. There are developed methods of logics of beliefs (cf., for example, (Munindar and Nicholas, 1993) or (Baldoni et al., 1998)) and belief functions (Shafer, 1976). Logical methods, theory of possibility, fuzzy set theory, and probabilistic technique form a good base for building CIF systems in computers. Definition 3.2.3. A unit M of specific validated fantasy (of general validated fantasy) about a domain D (objects from the domain D) is the epistemic structure from Figure 3.1 or 3.2 that is not justified or even justified that it is not true. For instance, looking at the Moon in the sky, we know that we see the Moon. We can believe that we will be able to see the Moon tomorrow at the same time and we can fantasize how we will walk on the Moon next year. According to the attitude-dependent approach, we have the following definition. Definition 3.2.4. A unit B of specific assumed knowledge (of general assumed knowledge) for a system R about a domain D (objects from the domain D) is the epistemic structure from Figure 3.1 or 3.2 that is estimated (believed) by the system R to represent D (objects from D) with the high extent of confidence. We see that in contrast to the object-dependent approach, in the attitude-dependent approach, knowledge is relative depending on the system it belongs to. What is knowledge for one system may be fantasy to another one. For instance, what was knowledge about atoms for Democritus is a pure fantasy for a contemporary physicist. In a similar way, we define basic structures for belief units and fantasy units. Definition 3.2.5. A unit B of specific assumed belief (of general assumed belief) for a system R about a domain D (objects from the
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domain D) is the epistemic structure from Figure 3.1 or 3.2 that is estimated (believed) by the system R to represent D (objects from D) with the moderate extent of confidence. We see that what is assumed as belief by a system does not necessarily coincide with a validated belief and vice versa. Definition 3.2.6. A unit B of specific assumed fantasy (of general assumed fantasy) for a system R about a domain D (objects from the domain D) is the epistemic structure from Figure 3.1 or 3.2 that is estimated (believed) by the system R to represent D (objects from D) with the low extent of confidence or without any confidence. Note that assumed fantasy for some people may be validated knowledge for other people. For instance, Knowledge of contemporary people about radio and television would be an assumed fantasy for people who lived in the 17th century. If confidence depends on some validation system, then there is a correlation between the first and the second stratification of epistemic structures into three groups — knowledge, beliefs, and fantasy. According to the reference-dependent approach, distinctions between knowledge, beliefs and fantasies depend not on the object domain but is estimated by comparison to another epistemic, e.g., knowledge, system. For instance, to show validity of their theories, physicists often compare new theories to existing theories validity of which has been justified by a diversity of experiments. One more perspective on epistemic structures is represented by the activity perspective because epistemic structures in general and knowledge in particular are assessed by their role in achieving some goals. Thus, it is possible to estimate the measure of efficiency building the operational scale, in which epistemic structures form the pragmatic continuum presented in Figure 3.14. Knowledge complete efficiency
Schemas
Wrong Schemas complete inefficiency
Figure 3.14. The pragmatic continuum
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All elements from the pragmatic continuum are called schemas. They are ordered by the degree of efficiency so that schemas with the higher believing in them lie to the right of schemas with the lower efficiency. When the degree of efficiency becomes sufficiently high, the corresponding structures from the pragmatic continuum are called knowledge. When the degree of efficiency becomes very low, the corresponding structures from the pragmatic continuum are called wrong schemas. Organization of knowledge evaluation is based on three knowledge perspectives: — The objective knowledge perspective is defined by the correlation with (closeness to) the real (true) situation. — The subjective knowledge perspective is defined by the strength of the belief in truthfulness. — The verification knowledge perspective is defined by the extent of the supportive evidence. Each perspective can be estimated by a corresponding measure or by a system of measures. For instance, it is possible to estimate knowledge from the objective knowledge perspective by the measure of domain related truthfulness considered later in this chapter. For measures reflecting the subjective knowledge perspective, it is viable using various psychological techniques for measuring the strength of beliefs. Supportive evidence in the verification knowledge perspective can be measured by its strength, groundedness, and extent. There are different systems of evaluation, validation, and justification: — science, for which the main validation technique is experiment; — mathematics, which is based on logic with its deduction and induction; — religion with its postulates and creeds; — history, which is based on historical documents and archeological discoveries.
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Knowledge has been always connected to truth. Namely, the condition, that a person knows something, P , always implied that P is true (cf., for example, (Pollock and Cruz, 1999)). However, the history of the humankind shows that knowledge is temporal, that is, what is known at some time can be disproved later. For instance, for a long time people believed that the Earth was flat. However, several centuries ago it was demonstrated that this is not true. In a similar way, for a long time, philosophers and scientists knew that the universe always existed. However, in the 20th century it was demonstrated that this is not true. Modern cosmology assumes that the universe erupted from an enormously energetic, singular event metaphorically called “big bang”, which spewed forth all the space and all of matter. As time passed, the universe expanded and cooled. A more recent example tells us that before 1994, physicists knew that the gravitational constant GN had a value between 6.6709 · 10−11 and 6.6743 · 10−11 m3 kg−1 s−2 (Montanet et al., 1994) Since then three new measurements of GN have been performed (Kiernan, 1995) and we now have four numbers as tentative values of the gravitational constant GN , which do not agree with each other. These examples show that knowledge has the temporal modality — what is true at some time can become false later, while what is considered false can become valid knowledge in the future. We see that it is possible to speak about false knowledge and knowledge of some time, say knowledge of the 19th century. It means that there is subjective knowledge, which can be defined as beliefs with the high estimation that they are true, and objective knowledge, which can be defined as beliefs that correspond to what is really true. Different approaches to epistemic structure differentiation, e.g., founding difference between knowledge, beliefs and fantasies, demand different criteria, measures, indicators, procedures, and techniques for evaluation. For instance, in the reference-dependent approach, we measure truthfulness of knowledge not with respect to an object domain, but with respect to another knowledge system, e.g., thesaurus. For instance, let us consider a situation when one person A gives some information I to another person B. Then it is viable to
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measure truthfulness of I with respect to A or more exactly, to the system of knowledge TA of A. To achieve this goal, we use some measure cr(T, D) of correlation or consistency between knowledge systems T and D. Let TA be the system of knowledge of A and TB be the system of knowledge of B. Definition 3.2.7. The function cr(I, TA ) = cr(I(TB ), TA ) − cr (TB , TA ) is called a measure of relative truthfulness of knowledge. Object-related truthfulness works when the real situation is known. However, it is impossible to compare knowledge directly with a real system. So, in reality, we always compare different systems of knowledge. It means that the object-dependent approach can be reduced to the reference-dependent approach. However, one system of knowledge can be closer to reality than another system of knowledge. In this case, we can assume that the corresponding truthfulness is object related (at least, to some extent). For instance, in the objectdependent approach, it is possible to compare theoretical knowledge to experimental data. This assumption is used as the main principle of science: it is presupposed that correct experiment gives true knowledge about reality and to find correctness of a model or theory, we need to compare the model or theory with experimental data. Such an approach to knowledge is developed in the externalist theories of knowledge (Pollock and Cruz, 1999). Example 3.2.1. Let us consider the system TA of knowledge of a person A. The system TSA is generated from all (some) statements of A. Then the value of the function tr(TSA , TA ) reflects sincerity of A, while the value of the function tr(TSA , TA ) reflects sincerity of information I given by A. Example 3.2.2. System related truthfulness is useful for estimating statements of witnesses. In this case, the measure of inconsistency incons(TA , TB ) between TA and TB is equal to the largest number of contradicting pairs (pA , pB ) of simple statements pA from TA and pB from TB when pA and pB are related to the same object or event. The measure of inconsistency incons(TA , TB ) between TA and TB
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determines several measures of consistency cons(TA , TB ) between TA and TB . One of such measures of consistency cons(TA , TB ) between TA and TB is defined by the formula cons(TA , TB ) =
1 . (1 + incons(TA , TB ))
(3.9)
It is possible to normalize the measure of inconsistency incons(TA , TB ), defining inconsN (TA , TB ) as the largest ratio of the number of contradicting pairs (pA , pB ) and number of all pairs (pA , pB ) of simple statements pA from TA and pB from TB when pA and pB are related to the same object or event. This measure generates the corresponding normalized consistency measure consN (TA , TB ) by the formula (3.9). Another normalized consistency measure consN (TA , TB ) is defined by the formula (3.10). cons(TA , TB ) = 1 − incons(TA , TB )
(3.10)
Relations of a portion of knowledge I to its object OI reflect important properties of this knowledge. One of the most important relations of this kind is truth, i.e., correct representation of OI by I. We can say that knowledge is true, or factual, if it allows one to build true knowledge. However, it is necessary to understand that in the context of the general theory of information, truth is only one of many properties of knowledge. Thus, we consider here a more general situation and distinguish true knowledge as a specific kind of genuine knowledge (cf., Definition 3.2.2). It is reasonable to call cognitive knowledge true when it gives knowledge with a high level of validity. However, genuine knowledge is not always true knowledge. For instance, the Hartley measure of uncertainty (entropy) of an experiment E that can have k outcomes is equal to log2 k. Then it is possible to measure knowledge I in a message M that tells us that the experiment E can have h outcomes as m(I) = log2 k − log2 h. The message and consequently, knowledge I can be false, i.e., it is not true that the experiment E can have h outcomes, but if h is less than k, knowledge I is genuine with respect to the chosen measure. Treating knowledge as the quality of a message that is sent from the sender to the receiver, we can see that knowledge does not have
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to be accurate. It may be conveying a truth or a lie, or just be something intangible. Even a disruptive noise used to inhibit the flow of communication and create misunderstanding would in this view be a form of knowledge. That is why, the problem of measuring knowledge accuracy is so important for knowledge and library science (cf., for example, (Fallis, 2004)). The measure tr(T , D) of correctness validity allows us to define truthfulness for knowledge. If we have a thesaurus T and a unit of knowledge K, then the truthfulness of K about an object domain D is given by the formula tr (K, D) = tr (K(T ), D) − tr (T , D). Here K(T ) is the thesaurus T after it receives/processes knowledge K. The difference shows the impact of K on T . Definition 3.2.8. The function tr(K, D) is called a measure of domain related truthfulness or domain related correctness of knowledge. Domain related truthfulness or correctness of a knowledge item K reflects changes K makes in a thesaurus. This property is related to accuracy of knowledge, which measures the changes in the distance of the initial knowledge to the absolutely exact knowledge. To measure domain related truthfulness, we explicate the structure of a knowledge item using Diagrams (3.1) and (3.2), which give the following diagram for knowledge structure. e QA
PA
q
p
A
(3.11)
NA n
Here NA is a name of the object A and QA is a feature (an intrinsic property) of A, e.g., if A is a book, then NA is usually the title of A, the intrinsic property QA may be the year of its publication or the author, while the attribute (ascribed property) PA is the cognitive representation of QA . In our case, when QA is the year of publication,
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then PA is the number that represents this year, e.g., 2012, or if QA is the author, PA is the first and the last names of the author. The essential part of the knowledge structure is the cognitive or symbolic component of knowledge. It is the ascribed property of the object A or in general, of objects from the knowledge domain U . The cognitive component of knowledge renders information about the object A or the knowledge domain U . It is reflected by Diagram (3.12). p NA
PA.
(3.12)
In addition, any knowledge system has two parts: informational and substantial. The component (A, q, QA ) as its substantial part and all other elements, that is, the component (NA , p, PA ) and two relations e and n, as its informational part. Cognitive parts of knowledge form knowledge systems per se as abstract structures, while adding to substantial parts forms extended knowledge systems. To evaluate the degree to which a given knowledge is true, we use Diagram (3.11), validating correspondences e, q, and p. There are different systems of knowledge validation. Science, for which the main validation technique is experiment, is a mechanism developed for cognition of nature and validation of obtained knowledge. Mathematics, which is based on logic with its deduction and induction, is a formal device constructed for empowering cognition and knowledge validation. Religion with its postulates and creeds is a specific mechanism used to validate knowledge by its compliance with religious postulates and creeds. History, which is based on historical documents and archeological discoveries, is also used to obtain and validate knowledge about the past of the society. Each system of validation induces a corresponding validation function tr(T, D) that gives a quantitative or a qualitative estimate of the truthfulness of knowledge T about some domain D. Note that the domain D can consist of a single object F . If a validation function tr(T, F ) is defined for separate objects, then it is possible to take some unification of all values tr(T, F ) ranging over all objects F in the domain D to which knowledge T can be related. This allows us to obtain a truthfulness function tr(T, F ) for
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systems of knowledge. In general, such unification of values tr(T, F ) is performed by some integral operation in the sense of (Burgin and Karasik, 1976; Burgin, 1982). Definition 3.2.9. (a) An integral operation W on the set R of real numbers is a mapping that corresponds a number from R to a subset of R, and for any x ∈ R, we have W ({x}) = x. (b) A finite integral operation W on the set R of real numbers is a mapping that corresponds a number from R to a finite subset of R, and for any x ∈ R, we have W ({x}) = x. As a rule, integral operations are partial. That is, they attach numbers only to some subsets of R. At the same time, it is possible to define integral operations in arbitrary sets. Examples of integral operation are: summation, multiplication, taking the minimum or maximum, determining the infimum or supremum, evaluating integrals, taking the first element from a given subset, taking the sum of the first and second elements from a given subset, and so on. Examples of finite integral operation are: summation, multiplication, taking minimum, determining maximum, calculating the average or finite weighted average for finite sets, taking the first element from a given finite subset, and so on. The following integral operations are the most relevant to the problem of knowledge truthfulness estimation: (1) taking the average value; (2) taking the minimal value; (3) taking the maximal value. Let us consider some measures of correctness. Example 3.2.3. One of the most popular measure tr(T, D) of truthfulness is correlation between the experimental data related to an object domain D and data related to D that are stored in the knowledge system T . Correlation r is a bivariate measure of association (strength) of the relationship between two sets of corresponded numerical data.
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It often varies from 0, which indicates no relationship or random relationship, to 1, which indicates a strong relationship or from −1, which indicates a strong negative relationship to 1, which indicates a strong relationship. Correlation r is usually presented in terms of its square (r 2 ), interpreted as percent of variance explained. For instance, if r 2 is 0.1, then the independent variable is said to explain 10% of the variance in the dependent data. However, as experts think, such criteria are in some ways arbitrary and must not be observed too strictly. The correlation coefficient often called Pearson product–moment correlation coefficient is a measure of linear correlation and is given by the formula: n (xi − x ¯)((yi − y¯) . r = i=1 n n
(xi − x ¯)2 ((yi − y¯)2 i=1
i=1
In the context of knowledge validation, numbers xi are numerical data about objects from the domain D obtained by measurement and experiments, while numbers xi are numerical data about objects from the domain D obtained from the knowledge K. In general, the correlation coefficient is related to two random variables X and Y and is defined in the context of mathematical statistics as Cov(X, Y ) E((X − µX )(Y − µY )) = . rX,Y = σX σY σX σY Here µX (µY ) is the mean and σX (σY ) is the standard deviation of the random variable X(Y ). Beside correlation coefficient r, which is the most common type of correlation measure, other types of correlation measures are used to handle different characteristics of data. For instance, measures of association are used for nominal and ordinal data. One more measure of correlation is given by the formula n (xi − x ¯)(yi − y¯). i=1
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This formula has the following interpretation: n
(xi − x ¯)(yi − y¯) > 0
means positive correlation;
(xi − x ¯)(yi − y¯) < 0
means negative correlation;
(xi − x ¯)(yi − y¯) = 0
indicates absence of correlation.
i=1 n i=1 n i=1
Example 3.2.4. It is possible to take validity of stored in the system knowledge T about the object domain D as a measure tr(T, D). There are different types of validity. For instance, researchers have introduced four types of validity for experimental knowledge: conclusion, internal, construct, and external validity. They build on one another, and each type addresses a specific methodological question. For instance, Bertrand Russell (1926) uses an attitude-dependent approach in his definition of knowledge. Consideration of beliefs forms the starting point for Russell’s definition of knowledge. Belief relations in a system are estimated by belief and plausibility measures (cf., (Klir and Wang, 1993)) or by extended belief measures that are described below. It makes possible to take some belief or plausibility measure as a measure tr(T, D) of truthfulness of knowledge in the system T . A belief and plausibility measures are kinds of fuzzy measures introduced by Sugeno (1974). Let P(X) be the set of all subsets of a set X and A be a Borel field of sets from P(X), specifically, A ⊆ P(X). Definition 3.2.10. A fuzzy measure on A in X (in the sense of Sugeno) is a function g : A → [0, 1] that assigns a number in the unit interval [0, 1] to each set from A so that the following conditions are valid: (FM1) X ∈ A, g(Ø) = 0, and g(X) = 1, i.e., the function g is normed.
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(FM2) the function g is monotone, i.e., for any A and B from A, the inclusion A ⊆ B implies g(A) ≤ g(B). (FM3) For any non-decreasing sequence A1 ⊆ A2 ⊆ · · · ⊆ An ⊆ An+1 ⊆ · · · of sets from A, the following equality is valid
∞ An = lim g(An ). g n=1
n→∞
(FM4) For any non-increasing sequence A1 ⊇ A2 ⊇ · · · ⊇ An ⊇ An+1 ⊇ · · · of sets from A, the following equality is valid
∞ An = lim g(An )). g n=1
n→∞
If A ∈ A, then the value g(A) is called the fuzzy measure of the set A. Then this definition was improved further through elimination of the condition (FM4) (Sugeno, 1977; Zimmermann, 2001). Many measures with infinite universe studied by different researchers, such as probability measures, belief functions, plausibility measures, and so on, are fuzzy measures in the sense of Sugeno. A more general definition of a fuzzy measure is used in (Klir and Wang, 1993), as well as in (Burgin, 2005c) where these measures are applied to exploration of fuzzy dynamical systems. Let P(X) be the set of all subsets of a set X and B be an algebra of sets from P(X), in particular, B ⊆ P(X). Definition 3.2.11. A fuzzy measure on B in X is a function g : B → R+ that assigns to each set from B a positive real number, is monotone (FM2): (FM1) g(Ø) = 0. (FM2) For any A and B from B, the inclusion A ⊆ B implies g(A) ≤ g(B). In what follows, we call g simply a fuzzy measure and call B the algebra of fuzzy measurable sets with respect to the fuzzy measure g.
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Popular examples of fuzzy measures are possibility, belief and plausibility measures. Possibility theory is based on possibility measures. Let us consider some set X and its power set P(X). Definition 3.2.12. (Zadeh, 1978; Zimmermann, 2001). A possibility measure in X is a partial function Pos : P(X) → [0, 1] that is defined on a subset A from P(X) and satisfies the following axioms: (Po1) Ø, X ∈ A, Pos(Ø) = 0, and Pos(X) = 1. (Po2) For any A and B from A, the inclusion A ⊆ B implies Pos(A) ≤ Pos(B). (Po3) For any system {Ai ; i ∈ I} of sets from A,
Ai = sup Pos(Ai ) Pos i∈I
i∈I
Possibility can be also described by a more general class of measures (cf., (Oussalah, 2000; Zadeh, 1978)). Definition 3.2.13. A quantitative possibility measure in X is a function P : P(X) → [0, 1] that satisfies the following axioms: (Po1) Ø, X ∈ A, P (Ø) = 0, and P (X) = 1; (Po2a) For any A and B from A, P (A ∪ B) = max{P (A), P (B)}. A quantitative possibility measure is a fuzzy measure. Dual to a quantitative possibility measure is a necessity measure (Oussalah, 2000). Definition 3.2.14. A quantitative necessity measure in X is a function N : P(X) → [0, 1] that satisfies the following axioms: (Ne1) N (Ø) = 0, and N (X) = 1. (Ne2) For any A and B from P(X), N (A ∩ B) = min{N (A), N (B)}. A quantitative necessity measure is a fuzzy measure. Possibility and necessity measures are important in support logic programming (Baldwin, 1986), which uses fuzzy measures for reasoning under uncertainty and approximate reasoning in expert systems, based on the logic programming style.
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In addition to ordinary measures, fuzzy measures encompass many kinds of measures introduced and studied by different researchers. For instance, beliefs play an important role in people’s behavior. To study beliefs by methods of fuzzy set theory, the concept of a belief measure was introduced and studied (Shafer, 1976). Definition 3.2.15. A belief measure in X is a partial function Bel : P(X) → [0, 1] that is defined on a subset A from P(X) and satisfies the following axioms: (Be1) Ø, X ∈ A, Bel (Ø) = 0, and Bel(X) = 1. (Be2) For any system {Ai ; i = 1, 2, . . . , n} of sets from A and any n from N , Bel(A1 ∪ · · · ∪ Ai ) ≥
n i=1
Bel (Ai ) −
Bel (Ai ∩ Aj )
i 0, it is a projective syntactic (K, F)-variety with the depth k. (4) A (full) syntactic K-variety if for any k > 0, it is a K-variety with the depth k. We see that the collection of the intersections kj=1 fij (Aij ) and k j=1 gij (Tij ) makes a unified system called a variety out of separate logical calculi C i . For instance, these intersections can contain norms/laws that were the same in one country during different periods of time or norms/laws common for different countries. Projective syntactic (K, F)-varieties add one important feature to properties of projective syntactic (K, F)-prevarieties. Namely, not only components C i of the cover {C i ; i ∈ I} of M are calculi from K but also all intersections of the component images in L are presented by calculi C i from K. Syntactic K-varieties properties of projective syntactic (K, F)varieties and syntactic K-prevarieties. Namely, they are unions of these calculi C i from K and intersections of these calculi C i are also calculi from K. The main goal of syntactic logical varieties, quasi-varieties and prevarieties is in presenting knowledge in the form of sets of logical formulas as a structured logical system using logical calculi, which have means for inference and other logical operations. Semantically, it allows one to describe the domain of interest, e.g., a database, knowledge of an individual or the text of a novel, by a syntactic logical variety dividing the domain in parts that allow representation by calculi. A fragmentation of a set of formulas in the process of logical variety formation allows separation of contradictory formulas making each calculus consistent and restricting interference of contradictory formulas. Note that there are projective syntactic varieties and syntactic varieties that are not logical. This happens when not all calculi from K used for building these varieties are logical calculi. For instance, we have differential and integral calculi in mathematics, which are not logical calculi. The calculus as a mathematical discipline is a variety with two components — differential calculus and integral calculus.
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Each of these components is subdivided into two subcomponents — calculus of a single variable and multivariable calculus. The extended calculus as a mathematical discipline contains one more component — time-scale calculus, which is a unification of difference equations with differential equations, combining integral and differential calculi with the calculus of finite differences and offering a formalism for studying hybrid discrete–continuous dynamical systems (Agarwal et al., 2002). It is possible to use different properties of logical systems for stratification of systems of formulas and building logical varieties, prevarieties and quasi-varieties. The most popular property is consistency, which is used as a guideline for structuring a set of logical formulas into a logical variety, prevariety or quasi-variety with consistent logical calculi as its components. However, it is also possible to use other properties of logical calculi, such as completeness, axiom independence or safety for dynamic logics, as guidelines for structuring. Now let us reflect on components and levels of logical varieties, prevarieties, and quasi-varieties. Definition 3.3.12. The set A is called the lower level or the axioms A(M) of the prevariety (quasi-variety or variety) M = (A, H, M ). Indeed, formulas from A are used as the basis for inference in M, i.e., they play the same role as axioms in logical calculi or formal theories. Definition 3.3.13. The set M is called the upper level or the theorems T (M) of the prevariety (quasi-variety or variety) M = (A, H, M ). Indeed, formulas from M are deduced in M as theorems are deduced in logical calculi or formal theories. Definition 3.3.14. The set H is called the intermediate or inferential level D(M) of the prevariety (quasi-variety or variety) M = (A, H, M ). Lemma 3.3.1. For any logical prevarieties (quasi-varieties or varieties) M = (A, H, M ) and N = (B, G, N ), inclusions A ⊆ B and H ⊆ G imply the inclusion M ⊆ N .
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Proof is left as an exercise. In comparison with varieties and prevarieties, logical quasivarieties, and quasi-prevarieties are not necessarily globally closed under logical inference. This trait provides higher flexibility in knowledge representation and management. An example of a logical variety is a distributed database or knowledge base, each component of which consists of consistent knowledge/ data. In this case, the components of this knowledge/database are naturally represented by components of a logical variety. Besides, in one knowledge base different object domains may be represented. In these domains some object may have properties that contradict properties of an object from another domain. As an example let us consider a knowledge base containing mathematical information. Suppose that this information concerns some large mathematical field like algebra or even its part — theory of groups. Mathematical logics are frequently considered as the constructive basis of mathematics, while logical calculi are viewed as precise models and formalizations of real mathematical theories. However, the theory of groups does not coincide with elementary (logical) theory of groups that is a deductive calculus. The field that is called in mathematics “the Theory of Groups” contains various sub-theories (Hall, 1959). In the theory of groups, such mathematical objects as finite and torsion-free groups are studied. In any finite group, the formula ∀x∃n(xn = e) is valid where e is the identity element. At the same time, in torsion-free groups another formula ∀x∀n(xn = e) is true. Thus, if theory of groups with its sub-theories, such as the theory of finite groups and theory of torsion-free groups, is represented as a single calculus, then both these formulae produce a contradiction. At the same time, a relevant logical variety in which sub-theories are represented by its components provides means for consistent representation of the theory of groups. Another example of a mathematical field that cannot be represented by a consistent logical calculus but is naturally described by a consistent logical variety is geometry. Indeed, geometry has many subtheories — the Euclidean geometry, Riemannian geometry, a variety of non-Euclidean geometries, projective geometry, differential
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geometry, analytic geometry, affine geometry, and metric geometry. Impossibility of their immersion in a consistent logical calculus is caused by the situation that axioms of some of these geometries contradict axioms of others. For instance, the fifth postulate is basic in the Euclidean geometry, while in non-Euclidean geometries, this postulate is false. Inference in a logical variety M is restricted to inference in its components because at each step of inference, it is permissible to use only rules from one set Hi applying these rules only to elements from the set Ti . This allows one to better model non-monotonicity of human thinking. Indeed, the main difference between monotonic and nonmonotonic reasoning arises from the different kinds of knowledge used in the process of inference. For instance, in the case of nonmonotonic reasoning, an inference rule of the following type can be used: “A is true if B cannot be proved”, i.e., to prove A the system relies on its ignorance of B. The statement B is not included in the system of initial axioms. That is why by the above given rule of inference, the statement A becomes true in the intellectual system. However, it is possible that B becomes proved at some stage of the inference. So, in this situation, the intellectual system must invalidate A and even more — to revise each piece of knowledge depending on A. In this way, the monotonic property of the consequence relation is violated. Usually, the statement A is excluded and the knowledge/belief revision takes place. Logical varieties allow database users and other intelligent systems not to eliminate knowledge/beliefs in the process of revision but to build a new component from which all knowledge/beliefs that contradict B are eliminated. In such a way, all previously obtained knowledge/beliefs are preserved. Definition 3.3.15. If a logical quasi-variety (quasi-prevariety, prevariety, or variety) M is built from the calculi C i , then these calculi used in the formation of M are called components of M. For instance, when a logical quasi-variety (prevariety or variety) is used for knowledge representation in multi-agent environment, it is natural to represent knowledge of each agent by a component of this
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logical quasi-variety (prevariety or variety). In a similar way, when a logical quasi-variety (prevariety or variety) is used for modeling and management of distributed knowledge and databases, it is natural to represent knowledge (data) in each local component by a component of this logical quasi-variety (prevariety or variety). We consider two types of logical varieties, prevarieties and quasivarieties. Definition 3.3.16. (a) If all components of a logical quasi-variety (quasi-prevariety, prevariety, or variety) M are pure logical calculi, then M is a pure logical quasi-variety (quasi-prevariety, prevariety, or variety). (b) If at least one component of a logical quasi-variety (quasiprevariety, prevariety, or variety) M is an applied logical calculus, then M is a applied logical quasi-variety (quasi-prevariety, prevariety, or variety). Lemma 3.3.2. Being a pure or applied logical quasi-variety (quasiprevariety, prevariety, or variety) does not depend on its cover. Indeed, if a cover of a logical quasi-variety (quasi-prevariety, prevariety, or variety) M consists only of pure logical calculi, then the one used in M language is logical and consequently, there are no non-logical axioms in M. At the same time, if a cover of a logical quasi-variety (quasi-prevariety, prevariety, or variety) M has at least one applied logical calculus, then there is, at least, one non-logical axiom in M and it has to belong to, at least, one component of any cover of M and this component will be an applied logical calculus. Although any logical calculus is a logical variety, this particular case does not give anything new in logic because logical calculi already exist in logic. A non-trivial example of logical varieties is given by many-sorted logics (Turner, 1984: Manzano, 1993; Meinke and Tucker, 1993; Abadi et al., 2010). In these logics, the variables range over different domains. Consequently, logical variables are “typed” as variables in many computer programming languages. Many-sorted logics allow one not to work with the domain of discourse as a homogeneous collection of objects, but to partition
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this domain into several parts with various functions and relations connecting them. In this case, these parts being formalized form a model variety, while the system of logics that describe these parts forms a syntactic variety. For instance, semantics of computer languages employ different types (domains) of data, such as the integers and the real numbers. Each domain has its own equality, relations, identities, and arithmetical operations. The logical language that describes the union of these domains will have two sorts of variables, real variables, and integer variables. The meaning of a quantifier would be determined by the type of the variable it binds. The corresponding logic will be a logical variety built of two calculi. Intersection of these calculi will include such formulas as the commutative law x + y = y + x, and the associative law x + (y + z) = (x + y) + z. Towers of calculi introduced by Maslov (1983) for representation of dynamic aspects of formal theories are an example of logical varieties. One more example of naturally formed logical varieties is the technique Chunk and Permeate built by Brown and Priest (2004). This technique suggests to begin reasoning from inconsistent premises proceeds by separating the assumptions into consistent theories (called by the authors chunks). These chunks are components of the logical variety shaped by them. After this, appropriate consequences are derived in one component (chunk). Then those consequences are transferred to a different component (chunk) for further consequences to be derived. This is exactly the way how logical varieties are used to realize and model non-monotonic reasoning (Burgin, 1991d). Brown and Priest suggest that Newton’s original reasoning in taking derivatives in the calculus, was of this form. Concepts of logical varieties and prevarieties provide further formalization for local logics of Barwise and Seligman (1997), manyworlds model of quantum reality of Everett (Everett, 1957; 1957a;
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1973; DeWitt, 1971; Davies, 1980; Herbert, 1987), and pluralistic quantum field theory of Smolin related to the many-worlds theory (Smolin, 1995). As history of physics tells us, to avoid some contradictions of quantum theory, Everett suggested that the indeterminism of quantum systems generates and is a consequence of a multifoliate reality. In it, the universe is continually branching into myriads of ‘parallel universes,’ which are physically disconnected but equally real assuming that every branch actually occurred. However, in later days, Everett suggested that the many of these worlds were not actual but presented many views of the same world. In any case, it is possible to interpret each component of a logical variety as a description of a separate world or a definite view (perspective) of such a world. Logical variety approach well correlates and complements the general methodology of labeled deductive systems used for generalization of modal logics to enable reasoning about structures of actual worlds, where each world has an arbitrary associated modal theory and the whole system has a logical variety as its theory. Let us consider a syntactic quasi-prevariety (quasi-variety, prevariety, or variety) M = (A, H, M ) with the cover {C i = (Ai , Hi , Ti ); i ∈ I} and the set N of formulas, i.e., N is a subset of a logical language L. Definition 3.3.17. (a) The set V = {Ti ; i ∈ I} is called the flat map of the syntactic quasi-prevariety (quasi-variety, prevariety, or variety) M and a quasi-prevariety (quasi-variety, prevariety, or variety) flat map of the set N . (b) If N = M , then the set V = {C i ; i ∈ I} is called a quasiprevariety (quasi-variety, prevariety, or variety) map of the set N . Note that one set N of formulas can have different maps. Proposition 3.3.1. If N ⊆ P, then any quasi-prevariety (quasivariety) (flat) map of the set P is a quasi-prevariety (quasi-variety) (flat) map of the set N . Proof is left as an exercise. For prevariety and variety maps, this result is not always true.
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Proposition 3.3.2. (a) Any quasi-variety (flat) map of an arbitrary set of formulas N is a quasi-prevariety (flat) map of N . (b) Any prevariety (flat) map of an arbitrary set of formulas N is a quasi-variety (flat) map of N . (c) Any variety (flat) map of an arbitrary set of formulas N is a prevariety (flat) map of N . Proof is left as an exercise. Let us consider a class of syntactic quasi-prevarieties (quasivarieties, prevarieties or varieties) H. Definition 3.3.18. (a) The set W of all flat maps of a set of formulas N corresponding to quasi-prevarieties (quasi-varieties, prevarieties, or varieties) from H is called the flat H-atlas of N . (b) The set W of all quasi-prevariety (quasi-variety, prevariety, or variety) maps of set of formulas N corresponding to quasi-prevarieties (quasi-varieties, prevarieties, or varieties) from H is called the quasiprevariety (quasi-variety, prevariety, or variety) H-atlas of N . Proposition 3.3.3. If N ⊆ P, then any quasi-prevariety (quasivariety) (flat) H-atlas of the set P is a subset of the quasi-prevariety (quasi-variety) (flat) H-atlas of the set N . Proof is left as an exercise. For prevariety and variety maps, this result is not always true. Proposition 3.3.4. (a) Any quasi-variety (flat) H-atlas of an arbitrary set of formulas N is a subset of the quasi-prevariety (flat) H-atlas of N . (b) Any prevariety (flat) H-atlas of an arbitrary set of formulas N is a subset of the quasi-variety (flat) H-atlas of N . (c) Any variety (flat) H-atlas of an arbitrary set of formulas N is a subset of the prevariety (flat) H-atlas of N . Proof is left as an exercise. Definition 3.3.19. If {C i ; i ∈ I} is the cover of the syntactic quasiprevariety (quasi-variety, prevariety, or variety) M, then the cardinality |I| of the set I is called the weight of M.
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Let us consider a class of syntactic quasi-prevarieties (quasivarieties, prevarieties, or varieties) H. Definition 3.3.20. The cardinality min{|V|; V is an H-map of N } is called the H-weight of the set of formulas N . Proposition 3.3.5. If H is a class of syntactic quasi-prevarieties (quasi-varieties) and N ⊆ P, then the H-weight of N is less than or equal to the H-weight of P . Proof is left as an exercise. Let us consider two logical calculi C = (A, H, T ) and B = (B, G, P ). Definition 3.3.21. (a) The logical calculus C is a subcalculus of the logical calculus B if A ⊆ B and T ⊆ P . (b) The logical calculus C is a strict subcalculus of the logical calculus B if A ⊆ B and H ⊆ G. For instance, the applied calculus (formal theory) of the Zermelo– Fraenkel set theory ZF, without the Axiom of Choice is a strict subcalculus of the Zermelo–Fraenkel set theory ZF with the Axiom of Choice (Fraenkel and Bar-Hillel, 1958). Proposition 3.3.6. Any strict subcalculus of a logical calculus B is a subcalculus of B. Indeed, if a logical calculus C = (A, H, T ) is a strict subcalculus of the logical calculus B = (B, G, P ), then whatever is deducible in C is also deducible in B, and thus, T ⊆ P . Proposition 3.3.7. If a logical calculus A is a (strict) subcalculus of a logical calculus B and B is a (strict) subcalculus of a logical calculus C, then A is a (strict) subcalculus of C. Indeed, if A = (A, F, Q), C = (C, H, T ), B = (B, G, P ), A ⊆ B, B ⊆ C, Q ⊆ P and P ⊆ T , then by properties of sets, A ⊆ C and Q ⊆ T . In a similar way, if A ⊆ B, B ⊆ C, F ⊆ G and G ⊆ H, then by properties of sets, A ⊆ C and F ⊆ H. Let us consider two syntactic logical varieties M = (A, H, M ) and Q = (B, G, N ).
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Definition 3.3.22. (a) The logical variety M is a subvariety of the logical variety Q if all components of M are subcalculi of the components of Q. (b) The logical variety M is a strict subvariety of the logical variety Q if all components of M are strict subcalculi of the components of Q. (c) The logical variety M is a direct subvariety of the logical variety Q if all components of M are also components of Q. Note that any component of Q is also a direct subvariety of the logical variety Q. Definitions imply the following result. Proposition 3.3.8. (a) Any direct subvariety M of the logical variety Q is a strict subvariety of Q. (b) Any strict subvariety M of the logical variety Q is a subvariety of Q. Proposition 3.3.7 implies the following result. Proposition 3.3.9. If a logical variety M is a (direct or strict) subvariety of a logical variety N and N is a (direct or strict) subvariety of a logical variety Q, then M is a (direct or strict) subvariety of Q. Proposition 3.3.10. If a logical variety M is a direct subvariety of a logical variety N, then the flat map of M is a subset of the flat map of N. Proof is left as an exercise. Proposition 3.3.11. If a logical variety M is a subvariety of a logical variety N, then the weight of M is less than or equal to the weight of N. Proof is left as an exercise. Corollary 3.3.1. If a logical variety M is a direct (strict) subvariety of a logical variety N, then the weight of M is less than or equal to the weight of N.
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Formation of a unified logical system from given logics is an important problem of logic as a discipline. In particular, it is useful to be able to include a system of calculi into one calculus. Gabbay writes that “the problem of combining logics and systems is central for modern logic, both pure and applied. The need to combine logics starts both from applications and from within logic itself as a discipline. As logic is being used more and more to formalize field problems in philosophy, language, artificial intelligence, logic programming, and computer science, the kind of logics required becomes more and more complex.” (Gabbay, 1999). Logical varieties give a relevant context for solving this problem. Let K be a class of logical calculi and M = {Ci ; i ∈ I} be a deductive variety (K-variety). Definition 3.3.23. A logical variety M is called: (a) Discrete if its components are disjoint; (b) Classical if all its components are classical deductive calculi; (c) Connected if any two of its components have a non-void intersection; (d) Compatible if it is a subset of a consistent calculus; (e) K-compatible if it is a subset of a calculus from K; (f) Provably compatible if it is possible to prove by classical methods that it is a subset of a consistent calculus; (g) Consistent if all its components are consistent calculi. Example 3.3.1. Let us take the axiom system AG of group theory, axioms AM for metric spaces and postulates PE of the Euclidean geometry. Each of these systems gives birth to a logical calculus when we apply classical deduction rules. These calculi form a discrete classical logical variety. Example 3.3.2. Taking the calculus C(AG) generated by the group axioms AG, calculus C(AAG) generated by the abelian group axioms AAG, and calculus C(AS) generated by the semigroup axioms AS, we obtain a connected classical logical variety. It is provably compatible because C(AS) ⊆ C(AG) ⊆ C(AAG).
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Definition 3.3.24. Components {Ci ; i ∈ J} of M are called: (1) (Provably) compatible if the subvariety of M generated by these components is (provably) compatible. (2) K-compatible if the subvariety of M generated by these components is K-compatible. (3) Connected if the subvariety of M generated by these components is connected. Lemma 3.3.3. A compatible deductive variety M is consistent. Indeed, a subcalculus of a consistent calculus is also consistent. Lemma 3.3.4. For any deductive variety M, there is a discrete deductive variety DM such that their upper levels are isomorphic, i.e., T (M) ≈ T (DM). Proof . To build DM, we change variables in all components of M so that any two new calculi built from the components of M do not have common variables. In addition, we prohibit using the substitution rule for inference and eliminate all connecting mappings that exist in M. The union of the obtained calculi forms the discrete deductive variety DM. As deduction goes separately in each component, T (M) ≈ T (DM) where the isomorphism is obtained by renaming the variables in formulas. Lemma is proved. Lemma 3.3.4 directly implies the following result. Proposition 3.3.12. The discrete counterpart DM of a variety M preserves consistency, i.e., if M is a consistent variety, then DM also is a consistent variety. Transition to the discrete counterpart of a logical variety preserves compatibility. Proposition 3.3.13. If M is compatible (K-compatible), then DM is compatible (K-compatible).
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Proof is based on the construction of DM described in the proof of Lemma 3.3.4. Theorem 3.3.1. For any number n > 1 there is a classical consistent connected deductive logical variety M with n components such that any n − 1 components of M are compatible, but M is not compatible. Proof. Let us consider a consistent set A1 , A2 , A3 , . . . , An−1 of independent well-formed formulas from the classical propositional (or first-order predicate) calculus. We remind that formulas are independent if neither of them can be deduced from all others in this set. Then there is a consistent deductive logical variety M of the form M = {Ci ; Ci = (Ai , d, Ti ) and i = 1, 2, 3, . . . , n}, where An =A1 ∨A2 ∨A3 ∨ · · · ∨An−1 , d is the set of all deduction rules of classical the first-order predicate calculus and Ti is the set of all formulas deducible from Ai by rules from d. Note that as all formulas A1 , A2 , A3 , . . . , An−1 belong to the classical propositional (or first-order predicate) calculus, each of them is self-consistent (non-contradictory), implying that each C1 , C2 , C3 , . . . , Cn−1 is a consistent classical calculus. In addition, the calculus Cn is also consistent because otherwise formulas A1 , A2 , A3 , . . . , An−1 would not be independent. Thus, all Ci are classical calculi (i = 1, 2, 3, . . . , n) and by the definition of compatibility, it is possible to include any n − 1 components of M in a consistent classical calculus. At the same time, the set of all formulas A1 , A2 , A3 , . . . , An−1 , An is inconsistent because the formula D = A1 ∧ A2 ∧ A3 ∧ · · · ∧ An−1 is deduced from the formulas A1 , A2 , A3 , . . . , An−1 and = ¬D. Theorem is proved. Remark 3.3.1. The condition that the variety is classical is essential. Now let us explore provable compatibility. Theorem 3.3.2. For any number n > 1 there is a classical connected deductive logical variety M with n components such that any n − 1
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components of M are provably compatible, M is compatible, but it is not provably compatible. Proof . Let us consider a classical logical calculus C such that it has a finite number of axioms (axiom schemas) and it is consistent but it is impossible to prove its consistency. The formal arithmetic is an example of such a calculus as by the first theorem it is impossible to prove its consistency by classical methods (G¨odel, 1931; 1932), while consistency of arithmetic was proved using more powerful methods (Gentzen, 1936; Ackermann, 1940; Schutte, 1960). It is possible to presume that each axiom (axiom schema) from C generates a provably consistent classical logical calculus. If L is the language of calculus C, then there is a consistent but not provably consistent calculus D in the language C such that it has the least number of axioms each of which is provably non-contradictory. Let us assume that D has m axioms A1 , A2 , A3 , . . . , Am and prove that for any number n ≥ 2 there is a classical connected deductive logical variety M with n components such that any n − 1 components of M are provably compatible, M is compatible, but it is not provably compatible. At first, we do this for n ≤ m. In this case, we take n disjoint groups G1 , G2 , G3 , . . . , Gn of axioms A1 , A2 , A3 , . . . , Am and consider the deductive variety M = {Ci ; i = 1, 2, 3, . . . , n} where Ci is the classical logical calculus with axioms from the group Gi . When n = m, each group Gi consists of one axiom Ai . In a general case, this variety M has the following properties. It is compatible because it is a subset of the consistent calculus D. It is not provably compatible because D is not a provably consistent calculus and as M contains all axioms of D, there is no other provably consistent calculus that contains M. At the same time, any n − 1 components of M are provably compatible because D is consistent but not provably consistent calculus with the least number of axioms. So, for n ≤ m, Theorem 3.3.2 is proved. Now let us consider the case n > m. Each classical logical calculus Ci with one axiom Ai has infinitely many formulas
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B1 , B2 , B3 , . . . , Bn , . . . . For instance, it has all conjunctions of this axiom with itself. Thus, it is possible to consider the deductive variety M = {Ci ; i = 1, 2, 3, . . . , n} where each of first m classical logical calculus Ci is generated using the single axiom Ai , all next n − m classical logical calculus Ci are generated consecutively using the formulas Bi . Note that it is the same variety as in the previous case only with a different map. The variety M is compatible because it is a subset of the consistent calculus D. It is not provably compatible because D is not a provably consistent calculus and as M contains all axioms of D, there is no other provably consistent calculus that contains M. At the same time, any n − 1 components of M are provably compatible because it is possible to include any n − 1 components of M into a consistent calculus that has not more than n − 1 axioms from the set A1 , A2 , A3 , . . . , Am and D is a consistent but not provably consistent calculus with the least number of axioms. Thus, Theorem 3.3.2 is proved. Remark 3.3.2. The condition that the variety is classical is essential. Remark 3.3.3. For limit ordinals, similar results are not valid as the following result demonstrates. Theorem 3.3.3. For any classical deductive logical variety M, if any finite subset of components of M is compatible, then M is compatible. Proof. Let us consider a classical deductive logical variety M in which any finite subset of components of M is compatible. When M has a finite weight, then all its components are compatible, i.e., M is compatible. Now let us assume that the weight of the variety M is infinite and consider a finite set X of formulas from M. Then there is a finite number of components {Ci ; = 1, 2, 3, . . . , n} of M such that X is a subset of the union ∪ni=1 Ci . As the set {Ci ; i = 1, 2, 3, . . . , n} is compatible, then it is possible to include all formulas from X into a consistent logical calculus, i.e., the set X is consistent. As this is true for any finite set of formulas from M, by the compactness theorem
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(cf., (Shoenfield, 2001)), the whole set is consistent, i.e., it is possible to include M into a consistent logical calculus. Theorem is proved. Corollary 3.3.2. For any classical deductive logical variety M with a countable number of components, if any finite subset of components of M is compatible, then M is compatible. A set of formulas in a classical logical system is consistent if and only if it has a model (Shoenfield, 2001). It gives us the following result. Proposition 3.3.14. A deductive logical variety M is compatible if and only if all its components have a common model. This result is not constructive in a general case because it is not always possible to find a model for a set of formulas. However, it is possible to derive a constructive criterion for compatibility of logical varieties. Theorem 3.3.4. A classical consistent connected deductive logical variety M = {Ci ; i = 1, 2, 3, . . . , n} is compatible if and only if for each component Ci , the following condition is valid: (* ) There are no formulas A1 , A2 , A3 , . . . , Ai−1 , Ai+1 , . . . , An from the corresponding calculi C1 , . . . , Ci−1 , Ci+1 , . . . , Cn , i.e., Aj ∈ Cj , such that the formula A1 ∨A2 ∨A3 ∨ · · · ∨Ai−1 ∨Ai+1 · · · ∨An belongs to the calculi Ci . Proof. We prove this result for i = 1. For all other values of i, proof is the same. 1. Let us assume that there are formulas A2 , A3 , . . . , An from the corresponding calculi C2 , C3 , . . . , Cn such that the formula A2 ∨A3 ∨ · · · ∨An belongs to the calculi C1 . Then the set of formulas A2 , A3 , . . . , An , A2 ∨A3 ∨ · · · ∨An is inconsistent in a classical calculus and this set belongs to any classical deductive calculus that contains the variety M. As such calculus will be inconsistent, the variety M is incompatible.
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2. Let us consider an incompatible classical deductive logical variety M = {Ci ; i = 1, 2, 3, . . . , n}. Then any classical calculus C that contains T(M) also contains the classical calculus C1 [C2 , . . . , Cn ], which is obtained by adding all formulas from C2 , . . . , Cn to the calculus C1 and taking the deductive closure. In such a way, we obtain classical calculus that contains T(M). As M is incompatible, the calculus C1 [C2 , . . . , Cn ] is inconsistent. By the reduction theorem for consistency (cf., (Shoenfield, 2001; Section 4.1)), if the calculus C1 [C2 , . . . , Cn ] is inconsistent, then there are formulas A2 , A3 , . . . , An from the corresponding calculi C2 , C3 , . . . , Cn such that the formula A2 ∨A3 ∨ · · · ∨An is deducible in C1 . As C1 is a calculus, this formula belongs to the calculi C1 . Theorem is proved. The compatibility of a logical variety means that it is possible to immerse all components of this variety into one calculus from the class K of logical calculi. Thus, the obtained results show that the possibility of logic system immersion into one calculus is undecidable for a finite number of logics (Theorem 3.3.2), while for an infinite number of logics, the decidability problem is reducible to the finite case (Theorem 3.3.3) and thus, it is undecidable in general. It is necessary to explain that logical varieties, prevarieties, and quasi-varieties implicitly perform various functions in knowledge organization and management. One of these functions is stratification of knowledge systems. Stratification is a popular technique in knowledge base theory and practice. For instance, Hunter and Liu (2009) introduce knowledge base stratification to solve the problem of merging multiple knowledge bases. Benferhat and Baida (2004) use stratified first order logic for access control in knowledge bases. Benferhat and Garcia (2002) employ stratification for handling inconsistent knowledge bases. Lassez et al. (1989) show how stratification can be useful as a tool in the interactive model-building process, demonstrating that it is possible to reduce the computational complexity of the process by the use of stratification that limits consistency checking to minimal strata.
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There are different kinds of knowledge stratification. In physical stratification, each stratum is a separate or at least, closed physical system. For instance, any distributed database is physically stratified. In analytical stratification, each stratum is determined by a specific name (label) and all elements from this stratum have this label (name). Knowledge base stratification used for handling inconsistent knowledge bases (Benferhat and Garcia, 2002; Brewka, 1989; Cholewinski, 1994) for constructing models of a knowledge base (Lassez et al., 1989), representation of information in temporal databases (Burgin, 2008a) and for merging multiple knowledge bases and information integration (Benferhat et al., 2004; Burgin, 2004a; Hunter and Liu, 2010) is analytical. Different classes of knowledge form corresponding strata of knowledge systems. There are different principles of knowledge classification, which allow us to build several types of knowledge system stratifications. Time is an important characteristic of knowledge, giving different stratifications. The temporal stratification. 1. The past stratum of knowledge consists of knowledge obtained/ accepted in the past. 2. The current stratum of knowledge consists of the actual (used now) knowledge. 3. The future stratum of knowledge. For instance, the knowledge “the Earth is flat” is past knowledge, while the knowledge “the Earth is round” is current knowledge. The past stratum of knowledge consists of three substrata: the forgotten past knowledge, outdated but preserved past knowledge and still actual past knowledge. The current stratum of knowledge consists of three substrata: the disappearing current knowledge, consolidated current knowledge, and emergent current knowledge. The future stratum of knowledge consists of three substrata: the tentative/ potential future knowledge, realizable future knowledge, and emergent future knowledge.
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More precise temporal stratifications are used in temporal knowledge and databases. A temporal knowledge/database is a database with built-in time aspects. In particular, it supports a temporal knowledge/data model and has a temporal version of the query language (Snodgrass and Jensen, 1999; Date et al., 2002). Temporal knowledge/data stored in a temporal knowledge/database are different from the knowledge/data stored in non-temporal knowledge/database in that a time coordinate is attached to the knowledge/data. This is different from the conventional knowledge/data, which are usually considered to be valid now. Past and future knowledge/data are not stored. Usually past knowledge/data are modified, overwritten with new (updated) knowledge/data or deleted to achieve their temporal relevancy. Future knowledge/data are not considered because it is assumed that we do not receive information about the future. There are many complexity measures of algorithms, methods, and procedures. Taking a complexity measure C, it is possible to partition all algorithms (methods or/and procedures) into separate classes that have different complexity measures. Each such a partition induces a corresponding stratification of knowledge with respect to such knowledge characteristics as accessibility, inference, and generation, which are specific forms of knowledge acquisition. Here are some examples of such stratifications. The accessibility stratification. 1. Directly or one-step accessible knowledge. 2. Two-step accessible knowledge. .......... 3. n-step accessible knowledge. Another stratification is based on complexity of knowledge inference. The inference stratification. 1. Directly implied knowledge. 2. Two-step inferable knowledge. ....... . n. n-step inferable knowledge.
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One more stratification is based on complexity of knowledge generation. The generation stratification. 1. Directly generable/computable knowledge. 2. Two-step generable/computable knowledge. ....... . n. n-step generable/computable knowledge. Steps in generation, inference, and access may be determined by: — Time slicing when each step is assigned some period of time for realization. — Elementary operations. For instance, it is possible to assume that knowledge acquisition is direct if it demands less than 3 seconds. The first step of knowledge acquisition can be estimated as an interval from 3 seconds to 30 seconds. The second step of knowledge acquisition can be estimated as an interval from 30 seconds to 1 minute. The third step of knowledge acquisition can be estimated as an interval from 1 minute to 3 minutes and so on. It is also possible to measure complexity, e.g., effort in generation, by the power of algorithms (Burgin, 2010d). In this case, we have an algorithmic ladder, which consists of classes of algorithms with increasing computing power. The traditional algorithmic ladders have one of the following forms: (1) Finite automata, deterministic pushdown automata, pushdown automata, and Turing machines. (2) Regular, or linear grammars, context-free grammars, contextsensitive grammars, and unrestricted, or phrase-structure grammars. New achievements of the theory of algorithms and computation extend these ladders: (1) Finite automata, deterministic pushdown automata, pushdown automata, Turing machines, inductive Turing machines (Burgin,
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2005), and infinite-time Turing machines (Hamkins and Lewis, 2000). (2) Regular, or linear grammars, context-free grammars, contextsensitive grammars, unrestricted, or phrase-structure grammars, grammars with prohibition (Burgin, 2005b), and Boolean grammars (Okhotin, 2003). Inductive Turing machines give an example of an algorithmic ladder. Namely, the n-th strata of the inductive algorithmic ladder consists of inductive Turing machines with the structured memory that have order n but do not have order n + 1 (Burgin, 2005). It is also possible to build an algorithmic ladder using inductive or limit. Turing machine have structured program (rules for computation) or structured (heads) operating devices (Burgin, 2005). Stratified Boolean grammars (Wrona, 2005) and grammars with exclusion (Burgin, 2015) give more examples of stratified operational knowledge. Different hierarchies studied in the theory of recursive functions, such as arithmetical hierarchy or analytical hierarchy, represent stratified declarative and operational knowledge about sets and relations (Rogers, 1987; Burgin, 2005). When we have a stratified system of knowledge represented in a logical form, it is natural to treat this system as a logical variety in which each stratum is a component of the variety.
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Chapter 4
Knowledge Structure and Functioning: Microlevel or Quantum Theory of Knowledge Real knowledge is to know the extent of one’s ignorance. Confucius
Here we consider the microlevel or quantum level of knowledge with three main goals: • construction of an adequate mathematical model of knowledge units; • explication of elementary knowledge units; • exploration and description of elementary knowledge unit integration into complex knowledge systems. This study provides means for discerning data and knowledge and finding links between them, as well as for improving efficiency of information processing by computers. The quantum level of knowledge, or more exactly, of the knowledge universe, contains “quantum bricks” and “quantum blocks” of knowledge that are used for construction of other knowledge systems. For instance, knowledge macrosystems, such as logical calculi and varieties, as well as formal theories in logic and mathematics, are constructed using knowledge microsystems or quantum elements, 307
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Table 4.1. A portion of a relational database where information about students is stored. Here N/T means that the student did not take this course. Course student Alex K Ben X Costas H Dan R Eddi T Frank S
Math 180
Math 211
Cs 100
Hist 200
A A B C C D
A B N/T D C D
A A A N/T D B
C B A N/T A F
such as propositions and predicates. Primitive propositions and predicates, such as “Knowledge is power”, 2 + 2 = 4 or “Information can give knowledge”, serve as “bricks” for composite propositions, logical functions and predicates, such as “(2 + 2 = 4) ∨ (2 + 2 = 2)” or “If A is a Frenchman, then A is a European” or “If X is a metric space, then X is a topological space”, while composite propositions, logical functions, and predicates, are “blocks” or compound logical units. All these “bricks” and “blocks” are used for logical inference, as well as for building logical calculi, logical varieties, and formal theories in logic and mathematics. Units of quantum knowledge or knowledge quanta find another application in relational databases where data are represented by a collection of relations often in the form of tables (flat relations), in which object data form rows — one row for one object — and attributes form columns — one column for one attribute. Rows in the database relations are usually called tuples. When people interpret relational data, they form knowledge. Therefore, with interpretation, the rows are symbolic knowledge items, or more exactly, symbolic knowledge quanta, which are symbolic components of extended quantum knowledge units. As a simple example of a relational database, let us consider such a table with information about students and their grades. Table 4.1 is composed from knowledge quanta such as: has Student Alex K
grade A in Math 180,
(4.1)
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has Student Frank S Student Alex K
309
grade D in Math 180,
(4.2)
grade A in Cs 100,
(4.3)
grade B in Math 211.
(4.4)
has Student Ben X
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It is demonstrated that a normalized datum in the relational data model has a complimentary nature to its structure that can be expressed by semantic primitives associated with its storage and retrieval operations. These semantic primitives are symbolic knowledge quanta, which form the foundation for understanding much larger retrieval issues, such as how data relations form logical navigation paths and how to express navigation paths in a tangible manner as a list of nested lists. Symbolic knowledge quanta have the structure of named sets providing a powerful insight into how navigation paths represent the natural intersection of data and the structure that organizes it. This shows the importance of knowledge quantization for efficient knowledge and data storage and retrieval. In this chapter, we describe three approaches to knowledge on the quantum level: • Quantum theory of knowledge (QTK), which studies knowledge quanta of different types; • Semantic link theory of knowledge (SLTK), which studies semantic links of different types; • Semiotics, which studies signs and symbols as quantum units of knowledge. 4.1. Basic structures of knowledge units on the quantum level — knowledge quanta and semantic links Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Richard Courant
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We start with QTK, which studies knowledge quanta in two forms — symbolic and substantial. While symbolic knowledge quanta are symbolic expressions, substantial knowledge quanta also include knowledge objects or domains. This complete representation is essential because knowledge is always about something, i.e., about the knowledge object or knowledge domain. 4.1.1. Quantum theory of knowledge (QTK) The superior man understands what is right; the inferior man understands what will sell. Confucius
To study knowledge as an essence, we have to begin with an important observation that as we have already discussed, there is no knowledge per se but we always have knowledge about something — about weather, about the Sun, about the size of a parameter, about occurrence of an event, and so forth. In other words, knowledge is always related to some real or abstract object or domain. Plato was may be the first to formulate this explicitly in his dialogue Republic. The object to which knowledge is related may be a galaxy, planet, point, person, bird, letter, novel, action, operation, event, process, love, surprise, picture, song, sound, light, square, triangle, etc. We call such an object knowledge domain or knowledge object. However, to discern a knowledge object, as well as to be able to speak and to write about it, we have to name it. A name may be a label, number, idea, text, process, and even a physical object of a relevant nature. For instance, a name may be a state of a cell in computer memory or a sound or a sequence of sounds when you pronounce somebody’s name. Note that a name can be very complex. For instance, a long text can be used as a name of the object described by this text. In particular, this book may be used as a name of the object called knowledge. In the context of named set theory (cf., Appendix and (Burgin, 2011)), any object can be used as a name of another object. Knowledge by its essence is knowledge of something, namely, about the knowledge object or knowledge domain, which forms the external structure of knowledge per se. Consequently, any knowledge
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system has two parts: cognitive, that is, knowledge per se, and substantial, which consists of the knowledge object (knowledge domain) with its structure (internal or external). These two parts are connected by a relation (correspondence), which is conveyed by the word “about” in English. Cognitive part of a knowledge system is also called symbolic because, as a rule, it is represented by a system of symbols. Cognitive/symbolic parts of knowledge form knowledge systems per se as abstract structures, while addition of substantial parts to them forms extended knowledge systems. At first, let us consider descriptive knowledge as the most typical category of knowledge (cf., Chapter 2). In this case, the simplest knowledge about an object gives some property of this object. As Aristotle wrote, we can know about things, nothing but their properties. The simplest property is existence of the object in question. However, speaking about properties, we have to discern intrinsic and ascribed properties of objects. In this, we are following the longstanding tradition of attributive realism, in which it is assumed that objects have intrinsic properties. Taking an object A and its feature (intrinsic property) QA , we come to an inherent descriptive quantum (IKQ) of knowledge K = (A, q, QA ), the graphical form of which is represented by Diagram (4.5). A
q
QA.
(4.5)
Note that it is possible to treat the property QA as a traditional property of an object represented by a value of an attribute or as the attribute itself represented by a predicate in the conventional description of properties or by an abstract or natural property in the advanced portrayal of properties (Burgin, 1985; 1986; 2010). For example, taking a physical body B, we know that it can have such an intrinsic property as 10 kg of mass. At the same time, it can have such an intrinsic property as “being a rigid object” (an attribute), as well as intrinsic property mass (a natural property). Definition 4.1.1. When the object A and the property QA are indecomposable, the inherent quantum of knowledge (4.1) is called an elementary inherent descriptive knowledge unit (EIKU).
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According to the contemporary understanding of reality, people do not have direct access to intrinsic properties of natural objects (Frieden, 1998; Burgin, 2010; 2012). It is only possible to receive information about intrinsic properties (Burgin, 2010). Consequently, intrinsic properties (features) of natural, e.g., physical, objects are reflected by ascribed properties (attributes). Ascribed properties of natural objects are obtained by observation, experiments, measurement, calculation or inference. Intrinsic properties of abstract, e.g., mathematical, objects are given in the form of assumptions, axioms, postulates, etc. It is possible to consider any known intrinsic property of an abstract object also as an ascribed property of this object. There are also other ascribed properties of abstract objects. In particular, approximations of intrinsic properties (features) of abstract objects are ascribed properties (attributes) of these objects. For instance, when we say that the length of a circle with the radius 1 ft is 6.28 ft, we are speaking about an ascribed property (attribute) of this circle because the length is equal to 2π ft and this is an irrational number. Intrinsic and ascribed properties are similar to primary and secondary qualities discerned by the prominent English philosopher John Locke (1632–1704). He assumed that primary qualities are objective features of the world, such as shape and size. By contrast, secondary qualities depend on mind. Examples of secondary qualities are color, taste, sound, and smell. At the same time, unlike primary properties, intrinsic properties of physical objects are inaccessible in a direct way. As we have mentioned above, to get knowledge about intrinsic properties of physical objects, people obtain information that makes it possible to represent intrinsic properties by ascribed properties. Besides, people and other intelligent systems, as a rule, use names when they deal with various objects, either natural or abstract. As a result, when names are separated from objects themselves properties, the cognitive representation of A splits into two components — a name NA of A and an attribute PA , as a value of a property P of A, while the object A is paired with its feature (property) QA . These observations allow us to build the next level of knowledge structure by formalizing the concept of descriptive knowledge
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in the form of an individual descriptive extended knowledge quantum (IDEKQ) or simply knowledge ide-quantum DK = [(A, q, QA ), (n, e), (NA , p, PA )], the graphical form of which is displayed in Diagram (4.6). e QA
PA
(4.6)
p .
q A
NA n
Here NA is a name of the object A and QA is a feature (an intrinsic property) of A, e.g., if A is a book, then NA is usually the title of A, the intrinsic property QA may be the year of its publication or the author, while the attribute (ascribed property) PA is the cognitive representation of QA . In our case, when QA is the year of publication, then PA is the number that represents this year, e.g., 2012, or if QA is the author, PA is the first and the last names of the author. For instance, it is possible to understand PA in Diagram (4.2) as a classical property such as being white, as a fuzzy property such as being 50% white, as a physical property such as weight or height, and as a value of a physical property, e.g., having weight 100 lb. Table 4.1 represents a system of descriptive extended knowledge quanta. Let us consider their structural portrayal. knowledge of Math 180 content
A ,
(4.7) person with the name Alex K knowledge of Hist 200 content
Alex K
B ,
(4.8) person with the name Ben X
Ben X
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knowledge of taken courses
(C, C, D, A) .
(4.9) person with the name Eddi T
Eddi T
Note that a compound object can be represented by a single name and several intrinsic properties can be described by one ascribed property. For instance, under definite conditions, a multitude of stars is called by the name a galaxy. In the considered example, an annotation of the book or the title and the name of the author(s) may be also used as the book names. From the perspective of the general theory of structures (Burgin, 2012), the knowledge ide-quantum DK represents the external structure of individual quantum units of descriptive knowledge. Let us perform structural analysis of the knowledge ide-quantum DK = [(A, q, QA ), (n, e), (NA , p, PA )] and its graphical form in Diagram (4.6). The knowledge ide-quantum DK has specific components. The first one is the attributive or estimate component of the knowledge ide-quantum DK. It reflects relation of the intrinsic property QA to the ascribed property PA and is represented by Diagram (4.10). e PA .
QA
(4.10)
For instance, it is possible to understand e in Diagram (4.10) as being observable meaning that the intrinsic property QA is observable and observation gives us the ascribed property PA . The second one is the naming component of the knowledge idequantum DK. It reflects the process of naming of the object A when this object is separated from other objects, discovered or constructed in the knowledge domain. The naming component is represented by Diagram (4.11). n A
NA .
(4.11)
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The third one is the substantial component of the knowledge idequantum DK. It is the intrinsic property of the object A from the knowledge domain U . The substantial component is reflected by Diagram (4.12). q A
QA .
(4.12)
The substantial component may be material, for example, U consists of all elementary particles or of computers, or may be ideal. For instance, A is a text or a real number, while the domain U consists of texts or of real numbers. The fourth one is the cognitive or symbolic component of the knowledge ide-quantum DK. It is the ascribed property of the object A from the knowledge domain U . The symbolic component of individual knowledge renders information about the object A from the knowledge domain U . Diagram (4.13) reflects the cognitive (symbolic) component of DK. p NA
PA .
(4.13)
From the perspective of the general theory of structures (Burgin, 2012), the cognitive (symbolic) component of the knowledge idequantum DK represents the inner structure of individual quantum units of descriptive knowledge. The theory of named sets (Burgin, 2011) provides efficient means for structural analysis in general and for structural analysis of knowledge in particular. According to the theory of named sets, this structure is a second order named set M (Burgin, 2011), which is built of two named sets. The first one is described by Diagram (4.12), which represents the objective or substantial part of Diagram (4.6) playing the role of the support in the named set M , and the second one is portrayed by Diagram (4.13), which represents the subjective or cognitive/symbolic part of Diagram (4.6) playing the role of the reflector in the named set M . The morphism (n, e) between Diagram (4.12) and Diagram (4.13) forms the component called reflection of this named set M of the second order, the graphical form of which is represented by Diagram (4.6) (Burgin, 2011).
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At the same time, it is possible to consider Diagram (4.6) as a named set morphism (n, e) from the named set (A, q, QA ) into the named set (NA , p, PA ) (Burgin, 2011). Each interpretation has its advantages, for example, when we build operations with knowledge quanta — in some cases one interpretation is better, while in other cases another interpretation is more useful. When all nodes and arrows — the object A, its property QA , the name NA and the value PA in Diagram (4.6) — are elementary, then the IDEKQ is also elementary. However, the name can consist of several words or the value can include several numbers. In this case, the IDEKQ is compound and not elementary. Definition 4.1.2. An individual descriptive extended knowledge quantum (IDEKQ) K is elementary if its structure is not decomposable into two or more knowledge quanta. Taking Diagram (4.6), we see that a quantum of descriptive knowledge IDEKQ resembles the structure of an atom, where Diagram (4.7) plays the role of the nucleus of this “atom” and Diagram (4.8) forms its symbolic shell similar to the electronic shell of an atom. Utilizing this analogy, we call the knowledge quantum K = (A, q, QA ), the graphical form of which is presented by Diagram (4.7), by the name a nuclear (or intensional) knowledge quantum, while the knowledge quantum H = (NA , p, PA ), the graphical form of which is presented by Diagram (4.8), we identify by the name a symbolic knowledge quantum. In logic, descriptive symbolic knowledge quanta are represented by propositions in the form of declarative sentences from natural languages or logical formulas, while in natural languages, descriptive symbolic knowledge quanta are represented by declarative sentences from these languages, e.g., “Everest is a high mountain,” or by expressions such as “a blue ball” or “a high mountain,” which represent knowledge quanta (ball, is, blue) and (mountain, is, high), correspondingly. Diagram (4.7) represents the structure of the base of an individual extended descriptive knowledge quantum (IEDKQ) or objective descriptive knowledge quantum (ODKQ). In other words, the base is
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the object of knowledge and its intrinsic property, or the value of an intrinsic property of this object. Diagram (4.8) represents the structure of an individual abstract descriptive quantum (IADKQ) of knowledge or subjective descriptive knowledge quantum (SDKQ). In other words, IADKQ is a nominal (symbolic) representation of the object of knowledge and the ascribed property that represents the intrinsic property of this object, or the value of such a property. It is possible that connections or relations in Diagram (4.6) are specified and named according to this specification. This gives us an individual valued descriptive extended knowledge quantum (IVDEKQ) or simply knowledge ide-quantum. For instance, we can specify Diagram (4.6) in the following way: the object A is a car, its name NA is Cadillac, its property QA is “the number of doors” and its value PA is 4. This gives us the following diagram. value number of doors
4
nd
number
.
(4.14)
car Cadillac vehicle make
Another possible specification of Diagram (4.6) preserves the naming component of Diagram (4.14), i.e., the object A is a car, its name NA is Cadillac, but changes the symbolic component of Diagram (4.14), i.e., the considered property QA is “color” and its value PA is “white”. This gives us the following diagram. value color nd
white number
car Cadillac vehicle make
.
(4.15)
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As a result, we have a nuclear valued knowledge quantum represented by Diagram (4.16) and a symbolic valued knowledge quantum portrayed by Diagram (4.17). painted car
color.
(4.16)
In this case, we interpret the morphism e in Diagram (4.9) as painted. painted Cadillac
white.
(4.17)
For instance, it is possible to understand PA in Diagram (4.9) as a classical property such as being white and p as its fuzzy counterpart, e.g., p = 0.5 or p = 50% meaning being 50% white. In a similar way, we find that individual quanta of representational knowledge, which depicts the knowledge object (domain) by models (images), and of operational knowledge, which represents the knowledge object (domain) by procedures, algorithms, instructions or processes, have a similar structure, which is also a second order named set constructed of two named sets and a morphism between them (Burgin, 2011). Namely, we have the following diagram, which represents an individual extended representational knowledge quantum (IREKQ) or simply knowledge ire-quantum RK = [(A, q, SA ), (n, a), (NA , p, MA )]. a MA
SA q
p . A
n
(4.18)
NA
Here NA is a name of the object A and SA is an intrinsic structure of A, while MA is a representing structure (model) of A. For instance, if A is an atom, then NA is usually the type of A, e.g., an atom of hydrogen. According to contemporary physics, people do not have access to the intrinsic structure SA but there are various models
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(ascribed structures) MA of the atom A. One of them called the Solar model represents the structure of an atom consisting of the nucleus the electronic shell, which consists of electrons rotating around the nucleus. For instance, we can specify Diagram (4.18) in the following way: the object A is the Solar System, its name NA is the linguistic expression “the Solar System”, its structure SA is the intrinsic structure of the Solar System, i.e., the Sun and planets with relations between them, and its model MA is a Copernican or Ptolemaic model of the Solar System. This gives us the following diagram. modeling the intrinsic structure
a model of the Solar System
structuring
representation
the Solar System
.
(4.19)
“the Solar System” naming
From the perspective of the general theory of structures (Burgin, 2012), the knowledge ire-quantum RK represents the external structure of individual quantum units of representational knowledge. In a similar way, we have the following diagram, which represents an individual extended operational knowledge quantum (IEOKQ) or simply knowledge ioe-quantum OK = [(A, q, SA ), (f , g), (NA , p, TA )]. g TA
SA q
p .
A
(4.20)
NA f
Here the object A is an action, operation or process, such as behavior or functioning; NA is its name and SA is an intrinsic structure of A; while TA is a procedural (operational) structure (model) of
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A. Here TA can be a system of instructions, an algorithm, a program, a method, a technique, or a procedure. For instance, A is a process of searching information on the Internet; its name NA is “search on the Internet”, “Yahoo search” or “Google search”; SA is the type of processes of search on the Internet; and is the program of the search engine that perform the search A. From the perspective of the general theory of structures (Burgin, 2012), the knowledge ioe-quantum OK represents the external structure of individual quantum units of operational knowledge. We also have an individual extended valued operational knowledge quantum (IVOEKQ), the graphical form of which is represented by Diagram (4.21).
the intrinsic structure
description a program (algorithm) for computation of π
structuring
description computation of π.
computational process
(4.21)
naming
As in the case of descriptive knowledge, we split extended representational knowledge quantum RK = [(A, q, SA ), (n, a), (NA , p, MA )] and operational knowledge quantum OK = [(A, q, RA ), (f, g), (NA , p, TA )] into two parts. In the first case, it gives us the individual substantial (nuclear) representational knowledge quantum K = (A, q, SA ), the graphical form of which is presented by Diagram (4.22), and the individual symbolic (cognitive) representational knowledge quantum H = (NA , p, MA ), the graphical form of which is presented by Diagram (4.23). q A
SA ,
(4.22)
MA .
(4.23)
p NA
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Here NA is a name of the object A and SA is an intrinsic structure of A, while MA is a representing the ascribed structure (assigned model) of A. Note that the substantial component can also be symbolic when A is a symbolic object, e.g., a mathematical expression (formula) or a scientific concept, and its intrinsic property SA is also represented in a symbolic form, e.g., if A is the concept car, then SA can be its weight. In addition, there are two more components. The first one is the attributive or estimate component of the knowledge ire-quantum RK. It reflects relation of the intrinsic structure SA to the ascribed model MA and is represented by Diagram (4.24). f SA
MA .
(4.24)
The second is the naming component of the knowledge irequantum RK. It reflects the process of naming of the object A when this object is separated from other objects, discovered or constructed in the knowledge domain. It is represented by the Diagram (4.25). g A
NA .
(4.25)
Splitting of the extended operational knowledge quantum OK gives us the individual nuclear operational knowledge quantum K = (A, q, RA ), the graphical form of which is presented by Diagram (4.26), and the individual symbolic operational knowledge quantum H = (NA , p, TA ), the graphical form of which is presented by Diagram (4.27). q A
SA
(4.26)
TA .
(4.27)
and p NA
Here the object A is an action, operation or process, such as behavior or functioning; NA is its name and SA is an intrinsic
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structure of A; while TA is a procedural (operational) structure (model) of A. In addition, there are two more components. The first one is the attributive or estimate component of the knowledge ioe-quantum OK. It reflects relation of the intrinsic structure SA to the ascribed model TA and is represented by Diagram (4.28). f TA .
SA
(4.28)
The second is the naming component of the knowledge ioequantum OK. It reflects the process of naming of the object A when this object is separated from other objects, discovered or constructed in the knowledge domain. It is represented by the Diagram (4.29). g A
NA .
(4.29)
In our study, we discern two types of knowledge quanta — individual and collective. Diagram (4.6) represents the structure of individual descriptive knowledge or knowledge about an individual object, e.g., one person or a star. However, a similar structure characterizes descriptive knowledge about concepts or about classes of individual objects. We call such knowledge collective or general. To model collective knowledge quanta explicating their structure, let us consider a domain U , which consists of a class of objects, e.g., all object that are unified by some concept (for models of concepts, see Section 5.2), for example, a class of cats or a class of dogs, relations between these objects and operations with these objects. Objects can be people, animals, systems, processes, actions, symbols, elementary entities, etc. In essence, an object is anything that can be considered (may be only in an abstract/ideal way) as distinct from anything else, e.g., from other things or beings (either real or abstract or ideal). We build a general definition of an object in Section 5.2 classifying objects relative to the Existential Triad considered in Section 2.2. Note that it is possible that the set U consists of a single object. We call this set U the collective knowledge domain (collective
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knowledge universe). Then knowledge about objects from U involves: (1) (2) (3) (4)
The domain U . An intrinsic property Q0 of objects from U . A class C of names for objects from U . An ascribed property P0 of objects from U .
In the formalized representation, the intrinsic property Q0 is represented by an abstract property Q = (U, q, W ) with the scale W (elements of the theory of abstract properties are considered in Section 5.2), while the ascribed property P0 is represented by an abstract property P = (C, p, L) with the scale L and the domain C, i.e., P is defined for names from C. In many cases, it is possible to assume that P = P0 although in contrast to P , P0 is a natural property in a general case. In this context, the property P is ascribed to objects from U , although not directly, but through their names. Thus, we come to the following definition. Definition 4.1.3. A collective extended descriptive knowledge quantum (CEDKQ) or simply knowledge cde-quantum about objects from the class U is the structure, CDK = [(U , q, W ), (f , g), (C, p, L)], the graphical form of which is represented by Diagram (4.30). g W
L
q
p . U
(4.30)
C f
In Diagram (4.30), the correspondence f relates each object H from U to its name «H » = f (H) from C (or to its system of names or, more generally, to its conceptual representative or conceptual image in the sense of (Burgin and Gorsky, 1991)) and the correspondence g assigns values of the property Q to values of the property P . In other words, g relates values of the intrinsic property to values of the ascribed property. For instance, when we consider such property of material things as weight, it is an intrinsic property. In weighting
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any thing, we can get only an approximate value of the real weight, or weight with some precision. It is the ascribed property. That is, when we measure an intrinsic property, we obtain values of the corresponding ascribed property. Relation f in Diagram (4.30) can have the form of some algorithms/procedures of object recognition, construction, or acquisition. Relation g can have the form of some algorithms/procedures of measurement, evaluation, or prediction. Note that in general, any object from U can be a big system that consists of other objects. For instance, it can be a galaxy or the whole physical universe. In this case, the knowledge quantum K about U is not elementary. Remark 4.1.1. It is possible that objects from U are characterized by a system of properties. However, this does not demand to change our representation of a collective descriptive knowledge quantum because the system of properties is equivalent to one property (Burgin, 2010). Only in this case, the knowledge K about U is not elementary. Similar to the knowledge ide-quantum, the knowledge cdequantum CDK has specific components. The first one is the attributive or estimate component of the knowledge cde-quantum CDK. It reflects relation of the intrinsic property Q to the ascribed property P and is represented by Diagram (4.31). g W
L.
(4.31)
For instance, it is possible to understand g in Diagram (4.31) as being observable. The second one is the naming component of the knowledge cde-quantum DK. It reflects the process of naming of the objects from the domain U . The naming component is represented by the Diagram (4.11). f U
C.
(4.32)
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The third one is the substantial component of the knowledge idequantum K. It is the intrinsic property of the objects from the knowledge domain U . The substantial component is reflected by the Diagram (4.12). q U
W.
(4.33)
The substantial component may be material, for example, when U consists of all elementary particles or of computers, or may be ideal. For instance, U is a collection of texts or a set of real numbers. The fourth one is the cognitive or symbolic component of the knowledge cde-quantum CDK. It is the ascribed property of the objects from the knowledge domain U . The symbolic component of collective knowledge renders information about the knowledge domain U . Diagram (4.34) reflects the cognitive (symbolic) component of CDK. p NA
PA .
(4.34)
From the perspective of the general theory of structures (Burgin, 2012), the cognitive (symbolic) component of the knowledge cde-quantum CDK represents the inner structure of individual quantum units of descriptive knowledge. On the quantum level, collective extended representational knowledge is similar to individual extended descriptive knowledge and naturally has similar components. Definition 4.1.4. A collective extended representational knowledge quantum (CERKQ) or simply knowledge cre-quantum about objects from the class U is the structure CRK = [(U , q, S), (f , g), (C, p, M )], the graphical form of which is represented by Diagram (4.35). g S
M
q
p. U
C f
(4.35)
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In Diagram (4.35), U is the knowledge domain, S is the class of structures of objects from the domain U , the symbol C denotes the class of names of objects from U , and M is the class of models of objects from U . On the quantum level, collective extended procedural knowledge is similar to individual extended representational knowledge and has similar components. Definition 4.1.5. A collective extended operational knowledge quantum (CEPKQ) or simply knowledge coe-quantum about objects from the class U is the structure COK = [(U , q, S), (f , g), (C, p, P )], the graphical form of which is represented by Diagram (4.36). g S
P
q
p . U
(4.36)
C f
In Diagram (4.36), U is the knowledge domain, which consists of actions, operations or processes, S is the class of structures of actions, operations, and processes from the domain U , the symbol C denotes the class of names of these objects from U , and P is the class of symbolic representations of objects from U , such as systems of instructions, algorithms, or procedures. Diagrams (4.6), (4.18), (4.20), (4.30), (4.35), and (4.36) explicate the structure of different types of extended knowledge quanta. Using knowledge quanta of these types and their components, it is possible to aggregate all knowledge systems from such units. Some suggest that knowledge does not exist outside some knowledge system. Quantum units of knowledge form such minimal knowledge systems where knowledge dwells. Knowledge may range from general to specific (Grant, 1996). General knowledge is broad, often publicly available, and independent of particular events. Specific knowledge, in contrast, is context-specific. General knowledge, its context commonly shared, can be more easily and meaningfully codified and exchanged, especially among different
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knowledge or practice communities. Codifying specific knowledge so as to be meaningful across an organization or group requires its context to be described along with the focal knowledge. This, in turn, requires explicitly defining contextual categories and relationships that are meaningful across knowledge communities. Three types of collective knowledge determine three types of focal knowledge. Taking Definition 4.1.6 as a base, we define specific knowledge in the following way. Definition 4.1.6. A focal descriptive knowledge quantum (SQ) K about an object F is represented by Diagram (4.37). gW DW
BL pC .
tU DU
(4.37)
BC fU
Here DU is a subset of U that contains F ; DW is the set of values of the property T on objects from DU , i.e., DW = {t(u); u ∈ DU }; BC is a subset of C that consists of the names of objects from DU , i.e., BC =}f (u); u ∈ DU }; BL is the set of values of the property P on the names of objects from DU ; fU , tU , pC , and gW are corresponding restrictions of relations fU , tU , pC , and gW . Focal knowledge resembles individual knowledge. The difference is in the perspective: individual knowledge is taken by itself, while focal knowledge is treated in the system of collective knowledge. Such an approach to focal knowledge results in the commutative cube (4.38), in which all mappings rW , rL , rU , and rC are inclusions. gW DW
BL
rW
gW
W t
pC .
L
g
DU rU
U
p C
f
BC rC
(4.38)
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Any system of knowledge is built from such elementary units by means of relations, which glue these “knowledge bricks” together. However, it is possible that some larger blocks are constructed from elementary units and then systems of knowledge are built from such blocks. Often elementary units of knowledge are expressed by logical propositions of the type: An object F has the property P or The value of a property P for an object F is equal to a. These propositions correspond to the forms of elementary data in the operator information theory of Chechkin (1991). Being transparent, the structure presented by Diagram (4.6) is also the basic structure of descriptive knowledge on any level — from the quantum level through the macrolevel to the megalevel. Indeed, any knowledge system K is about some domain D of objects, which is called the knowledge domain. When the domain consists of one object, it is also called the knowledge object. Names of objects from the knowledge domain D form the system ND . Besides, the domain D is structured. This structure SD is represented by a knowledge structure RD . This knowledge structure RD can be a logical system, differential equation, difference equation, system of equalities and/or equalities, text in a natural language, binary code, and so on. As a result, we come to the following diagram: g RD
SD t
p .
D
(4.39)
ND f
Besides, data and knowledge can themselves be objects, which have names, as well as intrinsic and ascribed properties. In this case, Diagrams (4.6), (4.18), (4.20), (4.30), (4.35), and (4.36) describe the structure of metaknowledge. However, using specific names for
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data and knowledge provides new opportunities in different areas. An interesting example of this application is given by named data, which are now so popular in the Internet technology defining its future development (Jacobson et al., 2012; Ntuli and Han, 2012). 4.1.2. Semantic link network theory (SLNT) and Semantic link theory of knowledge (SLTK) Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information on it. Samuel Johnson
Another representation of quantum knowledge is developed in the semantic link theory of knowledge (SLTK) based on semantic link network theory (SLNT) elaborated by Hai Zhuge and his collaborators (Zhuge, 2004; 2010; 2012; Zhuge and Shi, 2003; 2004; Zhuge and Sun, 2010; Zhuge and Xu, 2011; Zhuge and Zhang, 2010). The goal of SLNT is to create a semantic map of the Web, representing complex systems as semantic networks. As we will see in Chapter 5, semantic networks form one of the basic classes of knowledge representation. The advantage of SLNT is that an elementary unit of semantic networks is delineated and a system of operations with these units is developed allowing development of complex networks. The SLNT elementary unit is called a semantic link, which is a triad α = (X, α, Y ) where X and Y are called semantic nodes and can be any objects, e.g., texts, people, computers, semantic links, etc., while α is the connection (link) between X and Y , which indicates a relation between these semantic nodes. The graphical representation of the semantic link α has the following form: X
α
Y. α
Besides, Zhuge also calls the labeled arrow , as well as the inner component α of the semantic link α = (X, α, Y ), by the
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name semantic link (Zhuge, 2012). In addition, α is called the semantic indicator of the semantic link α (Zhuge, 2012). To discern these three entities, we employ the following terminology: — the triad α = (X, α, Y ) is called a complete semantic link, — the relation α is called an inner semantic link, or semantic indicator, and — the corresponding labeled arrow is called an arrow semantic link. Note that a semantic link is, like many other basic structures, a kind of fundamental triads (named sets) (cf., Appendix). The general nature of nodes in a semantic link implies that it is possible to use semantic links for building not only semantic networks but also networks in which physical objects are connected by semantic links. A semantic link network is a triad (N , L, ) where N is a set of nodes, L is a set of semantic links and is a semantic space, which consists of a concept hierarchy ℘ and a set of rules . The extended representation of a semantic link network also includes a mapping from nodes and links into the semantic space . As we can see, semantic links in the sense of Zhuge can connect physical objects. Here we are interested in knowledge, which is a structural essence. That is why we consider here only symbolic semantic links to build the SLTK. The SLTK elementary unit of knowledge is called a knowledge link and is a symbolic triad α = (X, α, Y ) where X and Y are called knowledge nodes and can be names of any symbolic objects, e.g., texts, words, symbols, pictures, semantic links, etc., while α is a connection (link) between the knowledge nodes X and Y , which represents a semantic relation between the objects with the names X and Y . Thus, a knowledge link is a kind of complete semantic links in which nodes are arbitrary symbolic objects. We will discern individual knowledge/semantic links and type knowledge/semantic links. In an individual knowledge/semantic link, α is an individual name of a certain relation. For instance, the knowledge/semantic link ({1, 2, 3}, {(1, 3)}, {1, 2, 3}), where {(1, 3)} is a binary relation in the set {1, 2, 3}, is individual.
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In a type knowledge/semantic link, α is an individual name of a certain relation. For instance, if ord denotes an order relation in the set {1, 2, 3}, then the knowledge/semantic link ({1, 2, 3}, ord, {1, 2, 3}) is a type knowledge/semantic link. At the same time, if nord denotes the natural order relation in the set {1, 2, 3}, i.e., nord = {(1, 2), (2, 3), (1, 3)}, then the knowledge/semantic link ({1, 2, 3}, nord, {1, 2, 3}) is an individual knowledge/semantic link. Note that an individual semantic relation can connect only two objects, while a type semantic relation can connect many different objects. For instance, the order relation ord can be defined in many distinctive sets, and even in one set, it can be defined differently. Zhuge describes 12 general types of complete, arrow, and inner semantic links, which are also types of knowledge links when they involve only symbolic objects (Zhuge, 2012): 1. The cause–effect link, in which the inner semantic link is denoted by ce indicating that the left node is the cause of the right node, i.e., the complete semantic link (X, ce, Y ) means the node X is the cause of the node Y . 2. The implication link, in which the inner semantic link is denoted by imp indicating that the left node implies the right node, i.e., the complete semantic link (X, imp, Y ) means the node X implies the node Y . 3. The subtype link, in which the inner semantic link is denoted by stOf indicating that the features of the left node include all features of the right node, i.e., the complete semantic link (X, stOf, Y ) means the features of the node X include all features of the node Y . 4. The similar link, or similarity link, in which the inner semantic link is denoted by sim indicating that the semantics of the right node is similar to the semantics of the left node, i.e., the complete semantic link (X, imp, Y ) means the semantics of the node X is similar to the semantics of the node Y . 5. The instance link, in which the inner semantic link is denoted by insOf indicating that the left node is an instance of the right
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node, i.e., the complete semantic link (X, insOf, Y ) means the node X is an instance of the node Y . The sequential link, in which the inner semantic link is denoted by seq indicating that the right node follows the left node, i.e., the complete semantic link (X, seq, Y ) means the node Y follows the node X. The reference link, in which the inner semantic link is denoted by ref indicating that the right node is a further explanation of the left node, i.e., the complete semantic link (X, ref, Y ) means the node Y is further explanation of the node X. The equal link, or equality link, in which the inner semantic link is denoted by e indicating that the meaning of the right node is the same as the meaning of the left node, i.e., the complete semantic link (X, e, Y ) shows the meaning of the node X is same as the meaning of the node Y . The empty link, in which the inner semantic link is denoted by Ø indicating that the right and the left nodes are completely irrelevant to one another, i.e., the complete semantic link (X, Ø, Y ) means the nodes Y and X are completely irrelevant to one another. The null or unknown link, in which the inner semantic link is denoted by Null or by N indicating that the relation between the two nodes is unknown or uncertain, i.e., the complete semantic link (X, N , Y ) shows the relation between X and Y is unknown or uncertain. The semantic equivalence link, in which the inner semantic link is denoted by equiv, indicating that the connected nodes can substitute for one another wherever they occur, i.e., the complete semantic link (X, equiv, Y ) shows X and Y can substitute for one another wherever they occur.
The last of general semantic links considered by Zhuge (2012) represents a unary operation with semantic links and will be described in Section 4.3.2. 12. The non-α relation link, in which the inner semantic link is denoted by Non (α) or by αN indicating that there is no relation
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α between the two nodes, i.e., the complete semantic link (X, αN , Y ) shows there is no relation α between X and Y . One more operation considered by Zhuge (2012), reversion, is described in Section 4.3.2. Here we describe other general types of knowledge/semantic links. 13. The property link, in which the inner semantic link is denoted by prOf indicating that the left node is a property (feature) of the right node, i.e., the complete semantic link (X, prOf, Y ) means the node X is a property (feature) of the node Y . For instance, the complete semantic link (blue, prOf, blue ball) means the color blue ball is a property (feature) of a blue ball. 14. The part link, in which the inner semantic link is denoted by ptOf indicating that the left node is a part of the right node, i.e., the complete semantic link (X, ptOf, Y ) means the node X is a part of the node Y . For instance, the complete semantic link (an arm, ptOf, a woman) means an arm is a part of a woman. 15. The element link, in which the inner semantic link is denoted by elOf indicating that the left node is an element of the right node, i.e., the complete semantic link (X, elOf, Y ) means the node X is an element of the node Y . For instance, the complete semantic link (the Earth, elOf, the Solar System) means the Earth is an element of the Solar System. 16. The name link, in which the inner semantic link is denoted by nmOf indicating that the left node is a name of the right node, i.e., the complete semantic link (X, nmOf, Y ) means the features of the node X is a name of the node Y . For instance, the complete semantic link (Michael, nmOf, the man) means Michael is the name of the man. 17. The before link, in which the inner semantic link is denoted by be indicating that on the time scale, the left node is before the right node, i.e., the complete semantic link (X, be, Y ) means the node X is before the node Y . For instance, the complete semantic link (Winter, be, Spring) means Winter is before the Spring.
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18. The after link, in which the inner semantic link is denoted by af indicating that on the time scale, the left node is after the right node, i.e., the complete semantic link (X, af, Y ) means the node X is after the node Y . For instance, the complete semantic link (Summer, af, Spring) means that Summer is after the Spring. 19. The function link, in which the inner semantic link is denoted by fnOf indicating that the features of the left node is a function of the right node, i.e., the complete semantic link (X, fnOf, Y ) means the features of the node X is a function of the node Y . For instance, the complete semantic link (moving people, fnOf, a car) means that a function of a car is moving people. 20. The relation link, in which the inner semantic link is denoted by rn indicating that the left node and the right node are in some relation, i.e., the complete semantic link (X, rn, Y ) means the features of the node X and the node Y are in some relation. For instance, the complete semantic link (10, rn, 5) means numbers 5 and 10 are in some relation, in particular, in the relation of divisibility, i.e., 10 is divisible by 5. 21. The in link, in which the inner semantic link is denoted by in indicating that the left node is in the right node, i.e., the complete semantic link (X, in, Y ) means the node X is in the node Y . For instance, the complete semantic link (Michael, in, the house) means Michael is in the house. 22. The better link, in which the inner semantic link is denoted by bt indicating that on some scale, the left node is better that the right node, i.e., the complete semantic link (X, bt, Y ) means the node X is better that the node Y . For instance, the complete semantic link (honesty, bt, deception) means honesty is better than deception. 23. The bigger link, in which the inner semantic link is denoted by bg indicating that for some scale, the left node is bigger than the right node, i.e., the complete semantic link (X, bg, Y ) means the node X is bigger than the node Y . For instance, the complete semantic link (the Sun, bg, the Earth) means the Sun is bigger than the Earth.
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24. The subclass link, in which the inner semantic link is denoted by scOf indicating that the left node is a subclass of the right node, i.e., the complete semantic link (X, scOf, Y ) means the node X is a subclass of the node Y . For instance, the complete semantic link (all dogs, scOf, all animals) means the class of all dogs is a subclass of the class of all animals. As Zhue writes (Zhuge, 2012), it is impossible to list all possible semantic relations and corresponding semantic links. Moreover, even taking a sufficiently complete set of link (relation) operations, it is usually impossible to build a semantic link base. The same is true for knowledge links. However, it is possible to develop such a base for semantic networks representing many specific domains. Theory and practice of semantic networks shows that very often knowledge/semantic links indicate connections not absolutely but only to some degree. Thus, it is useful to build and utilize graded counterparts for all general knowledge/semantic links: 1. The graded cause–effect link, in which the inner semantic link is denoted by (ce, cdg) indicating that the left node is a partial cause of the right node, i.e., the complete semantic link (X, (ce, cdg), Y ) means the node X is the cause of the node Y to the degree cdg. For instance, the complete semantic link (genetic inheritance, (ce, 80%), length of life) means the length of life of an individual depends on her genetic inheritance by 80%. 2. The probabilistic cause–effect link, in which the inner semantic link is denoted by (ce, pr) indicating that the left node is the cause of the right node with the probability pr, i.e., the complete semantic link (X, (ce, pr), Y ) means the node X is the cause of the node Y with the probability pr. 3. The graded implication link, in which the inner semantic link is denoted by (imp, dg) indicating that the left node implies the right node to the degree dg, i.e., the complete semantic link (X, imp, Y ) means the node X implies the node Y to the degree dg. 4. The graded subtype link, in which the inner semantic link is denoted by (stOf, ext) indicating that the features of the left node include all features of the right node to the extent ext, i.e.,
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the complete semantic link (X, (stOf, ext), Y ) means the features of the node X all features of the node Y to the extent ext. The graded similar link, or graded similarity, link, in which the inner semantic link is denoted by (sim, sd) indicating that the semantics of the right node is similar to the semantics of the left node to the degree sd, i.e., the complete semantic link (X, imp, Y ) means the semantics of the node X is similar to the semantics of the node Y to the degree sd. The graded instance link, in which the inner semantic link is denoted by (insOf, id) indicating that the left node is an instance of the right node to the degree id, i.e., the complete semantic link (X, (insOf, id), Y ) means the node X is an instance of the node Y to the degree id. The graded sequential link, in which the inner semantic link is denoted by (seq, pr) indicating that the right node follows the left node with the probability pr, i.e., the complete semantic link (X, (seq, pr), Y ) means the node Y follows the node X with the probability pr. The graded reference link, in which the inner semantic link is denoted by (ref, rext) indicating that the right node is a further partial explanation of the left node, i.e., explanation to the extent ext. So, the complete semantic link (X, (ref, rext), Y ) means the node Y is an explanation of the node X to the extent rext. The graded equal link, or graded equality, link, in which the inner semantic link is denoted by (e, eext) indicating that the meaning of the right node is almost the same as the meaning of the left node, i.e., the complete semantic link (X, by (e, eext), Y ) shows the meaning of the node X is same to the extent eext as the meaning of the node Y . The graded empty link, in which the inner semantic link is denoted by (Ø, ed) indicating that the right and the left nodes are partially irrelevant to one another to the degree ed, i.e., the complete semantic link (X, (Ø, ed), Y ) means the nodes Y and X are irrelevant to one another to the degree ed. The graded null or graded unknown link, in which the inner semantic link is denoted by (Null, nd) or by (N , nd) indicating
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that the relation between the two nodes is unknown or uncertain to the degree nd, i.e., the complete semantic link (X, (N , nd), Y ) shows the relation between X and Y is unknown or uncertain to the degree nd. The graded semantic equivalence link, in which the inner semantic link is denoted by (equiv, pt) indicating that the connected nodes can substitute for one another in the pt percent of situations, i.e., the complete semantic link (X, (equiv, 25%), Y ) shows X and Y can substitute for one another in 25% of the situations. The graded non-α relation link is denoted by (Non (α), pr) or by (αN , pr) indicating that there is the probability pr that there is no relation α between the two nodes, i.e., the complete semantic link (X, (αN , 0.3) Y ) shows there is the probability 0.3 that there is no relation α between X and Y . The graded property link, in which the inner semantic link is denoted by (prOf, prext) indicating to what extent the left node is a property (feature) of the right node to the extent prext, i.e., the complete semantic link (X, (prOf, prext), Y ) means the node X is a property (feature) of the node Y to the extent prext. For instance, the complete semantic link (blue, (prOf, 0.5), ball B) means the color blue ball is a property (feature) of the ball B to the extent 0.5. The graded part link, in which the inner semantic link is denoted by (ptOf, ptext) indicating that the left node is a part of the right node to the extent ptext, i.e., the complete semantic link (X, (ptOf, ptext), Y ) means the node X is a part of the node Y to the extent ptext. For instance, the complete semantic link (woman Y , (ptOf, 0.7), family X) means the woman Y is a part of the family X to the extent 0.7. The graded element link, in which the inner semantic link is denoted by (elOf, elext) indicating that the left node is an element of the right node to the extent elext, i.e., the complete semantic link (X, (elOf, elext), Y ) means the node X is an element of the node Y to the extent elext. For instance, the complete semantic link (the Earth, (elOf, 1), the Solar System) means the Earth is an element of the Solar System to the extent 1.
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17. The graded name link, in which the inner semantic link is denoted by (nmOf, next) indicating what part of a name of the right node is the left node, i.e., the complete semantic link (X, (nmOf, next), Y ) means the features of the node X is next of a name of the node Y . For instance, if the name of a bridge is the Golden Bridge, then the complete semantic link (Golden, (nmOf, 12 ), bridge) means that Golden is one half of the name of this bridge. 18. The graded before link, in which the inner semantic link is denoted by (be, bdg) indicating to what degree on the time scale, the left node is before the right node, i.e., the complete semantic link (X, (be, bdg), Y ) means the node X was bdg years before the node Y . For instance, the complete semantic link (Columbus, (be, Washington)) means Columbus lived years before Washington. 19. The graded after link, in which the inner semantic link is denoted by (af, adg) indicating to what degree on the time scale, the left node is after the right node, i.e., the complete semantic link (X, (af, adg), Y ) means the node X is (was) bdg years after the node Y . For instance, the complete semantic link (Washington, (be), Columbus) means Washington lived years after Columbus. 20. The graded function link, in which the inner semantic link is denoted by (fnOf, fgext) indicating that the features of the left node is a partial function of the right node, i.e., the complete semantic link (X, (fnOf, fgext), Y ) means the features of the node X is a function of the node Y . For instance, the semantic link (moving things, (fnOf, 30%), a car) means that 30% of car functions is moving things. 21. The graded relation link, in which the inner semantic link is denoted by rn indicating that the left node and the right node are in some relation, i.e., the complete semantic link (X, rn, Y ) means the features of the node X and the node Y are in some relation. For instance, the complete semantic link (10, rn, 5) means numbers 5 and 10 are in some relation, in particular, 10 is divisible by 5. 22. The graded in link, in which the inner semantic link is denoted by in indicating that the left node is in the right node, i.e., the
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complete semantic link (X, in, Y ) means the node X is in the node Y . For instance, the complete semantic link (Michael, in, the house) means Michael is in the house. 23. The graded better link, in which the inner semantic link is denoted by (bt, btext) indicating that on some scale, the left node is better that the right node to the extent btext, i.e., the complete semantic link (X, (bt, ext), Y ) means the node X is better that the node Y to the extent btelext. For instance, the complete semantic link (honesty, (bt, much), deception) means honesty is much better than deception. 24. The graded bigger link, in which the inner semantic link is denoted by (bg, bgext) indicating that for some scale, the left node is bigger than the right node to the extent bgext, i.e., the complete semantic link (X, (bg, bgext), Y ) means the node X is bigger than the node Y to the extent bgext. For instance, the complete semantic link (the Sun, (bg, much), the Earth) means the Sun is much bigger than the Earth. 25. The graded subclass link, in which the inner semantic link is denoted by (scOf, sext) indicating that the left node is a subclass of the right node to the extent sext, i.e., the complete semantic link (X, (scOf, sext), Y ) means the node X is a subclass of the node Y to the extent sext. For instance, the complete semantic link (penguins, (scOf, 0.7) birds) means the class of all penguins is a subclass of the class of all birds. When the grades of a semantic knowledge/link take values in the interval [0, 1], as it is, for example, for the probability pr, we have fuzzy general knowledge/semantic links. Semantic links are used for construction of semantic link networks. Definition 4.1.7 (Zhuge, 2012). A semantic link network (SLN) is a relational network that consists of the following parts: a set of semantic nodes, a set of (arrows) semantic links between the nodes, and a semantic space. In contrast to their name, semantic nodes can be any objects. Semantic links between nodes are regulated by attributes of nodes or
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generated by interactions between nodes. A semantic space includes a classification hierarchy of concepts and a set of rules for reasoning and inferring semantic links, for networking and transformations of the network. Knowledge flows in semantic link networks as well as in physical networks with people and computers as nodes. According to the theory of knowledge flow developed in (Zhuge, 2012), Knowledge Grid environment has three flows: 1. knowledge flow 2. information flow 3. service flow Parallel to these flows, the cyber-physical society has three more flows: 4. material flow 5. energy flow 6. money and other symbolic goods, e.g., stocks or bonds, flow In the resource-mediated mode, knowledge flows through four types of links: • • • •
Question answering links Citation links Hyperlinks Semantic links
4.1.3. QTK–SLTK connection He who learns but does not think, is lost! He who thinks but does not learn is in great danger. Confucius
Here we consider relations between the knowledge representation in semantic link network theory (SLNT) described in Section 4.1.2 and the knowledge representation in the theory of quantum knowledge (QTK) described in Section 4.1.1.
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Definition 4.1.8. We call two knowledge representations R1 and R2 equivalent if all knowledge representable (expressible) in R1 is also representable (expressible) in R2 and vice versa. In computer science, there are many theorems in which equivalence of different representations of operational knowledge is proved. For instance, it is proved that the representation of functions (or more exactly, of operational knowledge about functions) given by Turing machines is equivalent to the representation of functions given by partial recursive functions or that the representation of formal languages (or more exactly, of operational knowledge about formal languages) given by deterministic finite automata is equivalent to the representation of formal languages given by non-deterministic finite automata (Sipser, 1997). However, not all representations are equivalent. For instance, it is proved that the representation of formal languages given by Turing machines is not equivalent to the representation of formal languages given by finite automata (Sipser, 1997). Representation Equivalence Theorem. Knowledge representations in the cognitive (symbolic or information) component of the quantum theory of knowledge and in SLTK are equivalent. Proof. (a) From QTK to KLTK: We have to show that any knowledge that can be represented in QTK can be also represented in KLTK. In QTK, the general representations of knowledge quanta and their components are given in Diagrams (4.6), (4.18), (4.20), (4.30), (4.35) and (4.36). Thus, we need to demonstrate that it is possible build all these diagrams using complete semantic/knowledge links. At first, we take Diagram (4.6). As the KLTK element of knowledge — a complete semantic/knowledge link (X, α, Y ) — is a fundamental triad with arbitrary symbolic support X and reflector Y , it can naturally represent the knowledge quantum (A, q, QA ). Going to Diagram (4.6), we see that it is possible to represent all its components — the attributive (estimate) component (QA , e, PA ), the naming component (A, n, NA ), substantial component (A, q, QA ) and cognitive (symbolic) component (NA , q, PA ) — as complete
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semantic/knowledge links in a similar way to Diagram (4.5). Consequently, it is possible to build Diagram (4.6) using complete semantic/knowledge links. As all other Diagrams (4.18), (4.20), (4.30), (4.35), and (4.36) have the same structure, we can apply to then the same constructions, building these knowledge quanta from complete semantic/knowledge links. Thus, we see that whatever can be represented in QTK can be also represented in KLTK. (b) From KLTK to QTK: Let us consider a knowledge link in a general form X
α
Y
or (X, α, Y ). QTK representation of an arbitrary semantic or knowledge link in the form of abstract property (X, α, Y) Object/object name
{0, 1}
.
property scale
In this representation, objects (object names) are knowledge links from SLTK, while properties (property values) are relations that reflect membership of the triad (X, α, Y ) in a semantic network. It is a classical membership relation with the scale {0, 1}, which describes membership of the triad (X, α, Y ) in a semantic network. This membership relation may be fuzzy, having the form (X, α, Y)
[0, 1] .
There is another QTK representation of an arbitrary semantic or knowledge link in the following form of abstract property (cf., Section 5.3) (X, Y)
α.
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In this representation, objects (object names) are pairs of objects (object names) from KLTK, while properties (property values) are connection/link (connection/link names). Thus, we see that whatever can be represented in KLTK can be also represented in QTK. Consequently, both systems — KLTK and QTK — are equivalent. Theorem is proved. Because general descriptive knowledge units from QTK can include arbitrary objects, a similar proof gives us the following result. General Equivalence Theorem. Object representations in QTK and in semantic link network theory are equivalent. The obtained results show the equivalence of quantum units of knowledge in different theories implicating uniqueness of the inner structure of such units.
4.2. Signs and symbols as quantum units of knowledge In life, particularly in public life, psychology is more powerful than logic. Ludwig Quidde
In natural languages, concepts are treated as quantum units of knowledge. At the same time, concepts are kinds of symbols, while symbols are types of signs. Thus, all of them, i.e., signs, symbols and concepts, are quantum units of knowledge. Two main interpretations of the word “sign” are used by people: (1) sign as a physical object with some meaning, and (2) sign as a conceptual (theoretical) structure. We will call a sign by the name material sign when we have in mind the first interpretation and by the name conceptual sign when we bear in mind the second interpretation. Examples of material signs are digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 used in everyday calculations or letters, such as a, b, and c, from alphabets of natural languages. Icons that we can see on the screen of a computer are also material signs. Here we are mostly interested in conceptual signs, treating material signs as names of conceptual
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signs. Thus, in what follows, the word sign denotes a conceptual sign, while a name of a sign usually means a material sign. In a similar way, there are three different, however, connected, meanings of the word “symbol”. In a broad sense, symbol is the same as sign. For example, the terms “symbolic system” and “sign system” are considered as synonyms, although the first term is used much more often. Another understanding identifies symbol with a physical sign. Theoretical models of the structure of conceptual signs has been constructed in the discipline, which is called semiotics and is a general theory of signs. Semiotics studies structures and functions of signs and their communicative operation, including sign processes (semiosis), indication, designation, signification, likeness, analogy, metaphor, symbolism, signification, and communication. The term semiotics comes from the Greek word σηµε˜ιoν (meaning a sign or a mark) and it was first used in English by Henry Stubbes (1670) in the form semeiotics denoting the branch of medical science related to the interpretation of signs and by John Locke (1690) in the form semeiotike as “the doctrine of signs”. The importance of signs and signification has been recognized throughout much of the history of philosophy, and in psychology as well. For instance, Umberto Eco (1986) argues that semiotic theories are implicit in the work of most, perhaps all, major thinkers. Plato and Aristotle both explored the relationship between words (as signs) and the real world. Much later, Augustine of Hippo (354– 430) (Saint Augustine) considered the nature of the sign in society (St. Augustine, 1974). The general study of signs was popular in scholastic philosophy and logic. For instance, Peter Abelard (1079– 1142) noted that linguistic signification does not cover the whole range of sign processes instructed that arbitrary things might function as signs, too, if they were connected to each other in such a way that the perception of one led to the cognition of the other (Abelard, 1927; 1956). The unknown author, now commonly named Ps.-Robert Kilwardby, in his work written somewhere between 1250 and 1280, strengthens Augustine’s renowned dictum that “all instruction is either about things or about signs” stating that “every science is
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Material Sign
Sign Object
Sign Observer
Figure 4.1. The Bacon/Augustine Sign Triad
about signs or things signified.” In the context of named set theory where signs are kinds of names (Burgin, 2011), this statement tells us that every science is about names or things named. William of Ockham (ca. 1285–1347/49) brought the concepts of sign and signification to logic restricting the general concept of sign to what later became a propositional sign. Roger Bacon (ca. 1214–1293) was probably the most important medieval philosopher of sign being the author of the most extensive medieval treatise on signs known so far (Bacon, ca. 1267). He developed a general conception of signification, as well as a detailed theory of the linguistic sign integrated into a broader theory of sign in general. According to Bacon, a sign, as it was already pointed out by Augustine, is a triadic relation, such that it is — in principle — a sign of something to someone. This gives us the following model of sign (see Figure 4.1). In addition, Bacon elaborated a detailed classification of signs by adapting up, combining, and modifying elements of several prior sign typologies. According to Bacon, all signs belong to the following classes: 1. Natural signs 1.1. Signs signifying by inference, concomitance, consequence 1.1.1. Signs signifying necessarily 1.1.1.1. Signs signifying something in present 1.1.1.2. Signs signifying something in the past 1.1.1.3. Signs signifying something in the future 1.1.2. Signs signifying with probability 1.1.2.1. Signs signifying something in present 1.1.2.2. Signs signifying something in the past 1.1.2.3. Signs signifying something in the future
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1.2. Signs signifying by configuration and likeness, e.g., images or pictures 1.3. Signs signifying by causality 2. Signs given and directed by a soul 2.1. Signs signifying instinctively without deliberation 2.2. Signs signifying with deliberation, e.g., words 2.3. Interjections This classification was complemented in the following way: 1. Natural signs signifying unintentionally by their essence 1.1. Inferential signs based on a more or less constant concomitance of sign and what it signifies 1.2. Iconic signs, based on similarity in appearance 1.3. Signs based on a causal relation between the sign and the signified thing. 2. Signs of inference 2.1. Necessary signs 2.2. Probable signs It is necessary to remark that the division of all signs into two main classes of natural and given signs was taken from Augustine, the distinction between necessary and probable signs was adopted from Aristotle and their subdivision according to their temporal reference was a traditional element in the theories of the sacramental sign. The next major step in semiotics was done when John Poinsot published his Tractatus de Signis in 1632. It was, perhaps, the earliest, fully systematized treatise in semiotics. Contemporary semiotics, was independently originated by the American logician and philosopher Charles Sanders Peirce (1839– 1914), who originally called it semeiotic, and by the French linguist Ferdinand de Saussure (1857–1913), who originally called it semiology. Saussure (1916) defined semiology as a discipline that includes linguistics as a special case. At the same time, Peirce included in semeiotic both language studies and logic defining its three branches:
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1. Syntax as a discipline that studies how signs interact with one another. 2. Semantics as a discipline that studies how signs are related to things in the world. 3. Pragmatics as a discipline that studies how signs are employed by the agents who use them. As signs exist in a huge diversity of situations, the founder of semiotics, Charles Sanders Peirce and his follower Charles William Morris (1901–1979) defined semiotics very broadly predicting that it would influence a variety of disciplines. For instance, Morris wrote (1938): “The sciences must look to semiotic for the concepts and general principles relevant to their own problems of sign analysis. Semiotic is not merely a science among other sciences but an organon or instrument to all sciences.”
Indeed, semiotics has become an important tool in communication research, information theory, linguistics and the fine arts, as well as in psychology, sociology, and aesthetics. At the same time, we have the following situation. Although many other disciplines recognize the potential importance of semiotic paradigms for their fields, they have not yet found a satisfying way of integrating them into their domain. It is possible to find the first theoretical approach to the concept sign in works of Ferdinand de Saussure, who is sometimes called the father of theoretical linguistics. Saussure studied linguistic signs and according to Saussure, the basic property of a sign is that it points to something different from itself, transcendent to it (Saussure, 1916). To represent this property, Saussure introduced a structural model of sign in the form of the dyadic sign triad (see Figure 4.2). As in other cases, this triad is a kind of a fundamental triad (named set) described in Appendix. signification signifier
signified
Figure 4.2. The dyadic sign triad of de Saussure
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Related to a word, a signifier, also called signal or signifiant, may be understood as the sound or written pattern of the word or in actual, physical realization as part of a speech act. At the same time, a signified, also called signifie, may be treated as the conception or meaning of the signifier. According to contemporary views, human reality is a construction and the product of signifying activities which are culturally specific, culturally determined and often unconscious. According to Saussure, signs can exist only in relation to other signs. In this context, a linguistic sign is not a link between a thing and a word, but between a concept and a sound/written pattern, where the pattern is the hearer/reader’s psychological impression given to him by the evidence of her/his senses. This was rather different from previous approaches focused on the relationship between words and the things they designate. A dyadic model of sign was also supported by Eco (1976), who defined a sign as anything which may be interpreted as standing for something else. Note that it is necessary to make a distinction between sign as a structure (a conceptual sign) and material sign as a component name of this structure, e.g., letters or digits are such material signs. This understanding is represented by Figure 4.3, which is a structural model of sign: material sign (sign name)
object
Figure 4.3. The dyadic sign triad of Eco
However, in a more detailed model adopted from Hjelmslev (1963), Eco explained that any given (material) sign is required to be an element of the expression-plane, and must therefore be conventionally correlated to one or more elements of the content-plane (cf., Figure 4.4). Developing this approach, Eco also assumed that it was possible for the expression of a sign to have more than one content (cf., Figure 4.5), while the content of a sign may have more than one expression (cf., Figure 4.6). For instance, such symbol (material sign) as 1 can denote a digit of a decimal numerical system (Content 1),
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expression plane • • content plane
Figure 4.4. The Hjelmslev–Eco model of sign with one expression and one content expression plane • •
• •
content plane
Figure 4.5. The Hjelmslev–Eco model of sign with one expression and several contents expression plane • • • • content plane
Figure 4.6. The Hjelmslev–Eco model of sign with several expressions and one content
a digit of a binary numerical system (Content 2) or the number one (Content 3). At the same time, the number one can be denoted by the symbol 1 (Expression 1), by the letter α (Expression 2), by the letter ℵ (Expression 3), by the word “one” (Expression 4) or by the word “ein” (Expression 5).
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content plane B • expression plane • •
•
content plane A
Figure 4.7. The Hjelmslev–Eco model of sign with one expression plane and two content planes Representamen or Sign Vehicle (Sign Name)
Denotat (Sign Object)
Interpretant (Sign Meaning)
Figure 4.8. The Balanced Sign Triad of Peirce
Besides, in some cases, a sign may have expressions in more than one expression plane, and may have contents in more than one content plane (cf., Figure 4.7). For instance, such symbol (material sign) as 1 can denote a digit of a decimal numerical system (Content 1), a digit of a binary numerical system (Content 2) or the number one (Content 3), while its expressions can be such material symbols as “1”, “1”, “1” or “1”. Peirce extended the dyadic model by further splitting the signified (or content) into essentially different parts: the sign’s object and interpretant (the meaning of the sign), and thus, coming to the triadic model of a sign, the balanced sign triad: As Alp (2010) writes, historically, dyadic and triadic sign models form different cultures, and they are applied to semiotic targets independently. In the Peirce’s model (cf., Figure 4.8), a sign is understood as a structure that consists of three elements: the Name (Peirce called it Sign Vehicle), Object and Meaning of the sign (Peirce, 1931–1935). Sign Vehicle is conceived as a physical representation of a sign, that
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is, a material sign, which is called sign in everyday life. In other words, a sign is often comprehended as some elementary image inscribed on paper, clay tablet, piece of wood or stone, presented on the screen of a computer monitor, and so on. This material representation plays the role of a name of the sign. Peirce implied that signs establish meaning through recursive relationships that arise in sets of three main semiotic elements: • Representamen, also called Sign or Sign Vehicle by Peirce or Sign Name in a general context, is the component of the sign that represents the denoted object or objects and is similar to Saussure’s signifier. Note that in this context, the sign name is not necessarily a single word. It can be a quite elaborated object. • Object, also called extent, is what the sign represents or encodes. • Interpretant is the meaning formed into a further sign by interpreting or decoding a sign. The object of a sign can be anything thinkable, for example, a law, fact, possibility, or idea. Peirce considered two kinds of sign objects: • The immediate object is represented in the sign name (representamen). • The dynamic object is the object as it really is. In addition, Peirce considered three kinds of the interpretants of a sign: ◦ The immediate interpretant is the meaning that is already in the sign, or more exactly, the meaning that is without delay ascribed to the sign by the interpreter when she/he receives the sign. ◦ The dynamic interpretant is the meaning as formed in a process of sign comprehension by the interpreter. ◦ The final interpretant is the meaning that would be reached if formation process were to be pushed far enough, namely, it is a kind of an ideal meaning, with which an actual, that is, dynamic, interpretant may, at most, coincide. Note that it is also possible to consider the final object, which is an ideal collection of the object with all its possible changes.
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Besides, as the sign name, representamen or sign vehicle is also an object, it is possible to consider the immediate sign name, dynamic sign name, and final sign name. It is necessary to remark that each component the components of the structure of a sign — the name (sign vehicle), referent and interpretant/sense — is also an object and has a name, the name of this object. In addition, these objects also have an interpretant, i.e., sense and meaning, associated with their names. As a name is itself an object, it has a name and interpretant, i.e., sense and meaning, as any other object. In an intensional context, the names that occur denote the meaning or sense of the objects for the reader or listener. It means that each component of a sign can acquire the role of another component. As a result, the structure of a sign has the property called fractality, which tells that the structure of the whole is repeated/reflected in the structure of its parts. This property was envisioned by Peirce, who constructed combinations of Balanced Sign Triads (triangles) connecting them together in different ways by attaching a vertex of one to a vertex of another. These combinations determine operations with signs. In such a way, Peirce explicated structures of higher-level (secondlevel in Figure 4.9) and higher-order (third-order in Figure 4.10) signs, that is, signs of signs, or metasigns. An example of vertical escalation of Balanced Sign Triads is given in Figure 4.9, while an example of horizontal expansion of Balanced Sign Triads is given in Figure 4.10. Meaning (Interpretant) of the representation
Meaning (Interpretant) of the object
object
name (symbol) of the concept
name (symbol) of the object
Figure 4.9. The two-level Balanced Sign Triad of Peirce
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meaning (Interpretant)
of the object
object
name (symbol) of the object
of the name
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meaning (Interpretant) of the string
name (symbol) of the name
string of letters
Figure 4.10. The third-level Balanced Sign Triad of Peirce
Note that if the meaning (Interpretant) of the object is knowledge of the first level, then the meaning (Interpretant) of the representation is knowledge of the second level, i.e., it metaknowledge (cf., Section 2.3). Another operation with signs introduced by Peirce makes it possible to connect Balanced Sign Triads side by side in order to represent signs of signs of signs as we see in Figure 4.10. Note that if the name of the object contains knowledge of the first level, then the name of the name contains knowledge of the second level, while the corresponding string of letters that serves as the name of the name of the name contains knowledge of the third level (cf., Section 2.3). Peirce explained that signs mediate the relationship between their objects and their interpretants in a triadic mental or mind-like process, which goes through three stages: firstness, secondness, and thirdness. Firstness is a universal category of phenomena and is associated with a vague state of mind in which there is awareness of the environment, a prevailing emotion, and a sense of the possibilities when the mind is in neutral, waiting to formulate thought. Secondness is a category associated with moving from possibility to greater certainty expressed by actions, reactions, causality, or reality when the mind identifies what message is to be communicated. Thirdness is the category of signs and is associated with generality, representation, continuity, and purpose. This process is reversed in the receiver. At first, the active mind receives the sign name as a material sign. Then the mind acquires from the memory the mental object associated with the sign and in such a way produces the interpretant of the sign, which is the result of signification. Even when a sign represents by a resemblance or
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factual connection independent of interpretation, the sign is a sign only insofar as it is, at least, potentially interpretable. Peirce also refers to the “ground” of a sign, which is the pure abstraction of the sign quality. It is interesting that if we analyze what Peirce wrote about his model of sign, we see that he described the structure not of a triangular triad but the structure of a fundamental triad defined in Appendix. Indeed, Peirce (1903) wrote: “A Sign, or Representamen, is a First which stands in such a genuine triadic relation to a Second, called its Object, as to be capable of determining a Third, called its Interpretant . . .”
This gives us the following diagram, which is a fundamental triad because the Object connects the Representamen with the Interpretant. Representamen
Object
Interpretant .
Being a thorough taxonomist, Peirce offered a taxonomic scheme of signs demonstrating there could be 59,049 types of signs. However, at the top level of this classification, according to Peirce, there were three basic types of signs listed here in decreasing order of conventionality: — Symbols are highly conventional, — Icons (iconic signs) always involve some degree of conventionality, — Indices (indexical signs) directly bring our attention to their objects. These properties are implied by the following definitions. Definition 2.4.1. An icon looks like what it signifies. An example of icons is a photograph. Definition 2.4.2. An index has a causal and/or sequential relationship to its signified. For instance, indices are directly perceivable events that can act as a reference to events that are not directly perceivable. You may not see a fire, but you do see the smoke and that indicates to you that
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a fire is burning. Words this, that, these, and those are also examples of indices. Definition 2.4.3. A symbol represents something in a completely arbitrary relationship. Thus, in the majority of cases, symbols are subjective. Their relation to the signified object can be dictated either by social and cultural conventions or by habit or by creative thinking. Words are a prime example of signs. This shows that each of the terms — icon, index, and symbol — has two interpretations: (1) it is a physical object with some meaning, and (2) it is a conceptual (theoretical) structure. We are interested here in the second meaning of these terms. Besides, each of these concepts — icon, index and symbol — has the same structure as the concept sign. One more triadic sign model was built by Morris. In contrast to de Saussure and Peirce, Morris (1938) defines sign in a dynamic way relative to some interpreter. He writes that S is a sign (M.B. more exactly, the sign name) of an object or objects D for an interpreter I to the degree that I takes the account of D in virtue of the presence of S. Thus, the object S becomes a sign only if somebody (an interpreter) interprets S as a sign (M.B. the sign name). This gives us the diagram in Figure 4.11. In addition to these basic components of sign, Morris also includes relations of the sign with other signs. This adds one more dimension, transforming the triad into a tetrahedron (Figure 4.12). According to Morris, the sign name is what supports the triadic relation of the sign with other signs, with designated objects and with the subjects using the sign. These relations are represented by the Interpreter (I)
Sign/Sign Name (S)
Object (D)
Figure 4.11. The Dynamic Sign Triad of Morris
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Interpreter
Sign System
Object
Figure 4.12. The Dynamic Sign Tetrahedron of Morris Sign vehicle
Sense
Referent
Figure 4.13. The Semiotic Triangle
corresponding fields of semiotics. Syntactics deals with sign relations to other signs in the Sign System, semantics is concerned with the study of designated objects and pragmatics is oriented at the subjects (interpreters) who use signs, whereas semiotics or semiology as a scientific discipline is occupied with the general study of sign. Note also that Morris considers fuzzy relations because he takes into account the degree that the interpreter I takes into account of the object(s) D in virtue of the presence of the sign name S. This brings us to a variant of the Balanced Sign Triad of Peirce called the Semiotic Triangle (Figure 4.13) and has the following components (N¨ oth, 1990): • Sign vehicle is the form of the sign. • Sense is the sense made of the sign by the interpreter of the sign. • Referent is what the sign “stands for.” In this context, sense denotes the concept meaning for the interpreter of the sign. Eco (1976) discerns three kinds of the sign vehicles, which are material signs: 1. Signs for which there may be any number of tokens (replicas) of the same type, for example, a printed word.
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2. Signs whose tokens are different but similar, for example, a word which someone speaks or which is handwritten. 3. Signs whose token is their type, or signs in which type and token are identical, for example, a unique original oil-painting or sculpture. Another semiotic triangle (Figure 4.14) was suggested by Ogden and Richards (1953). Note that not only this but also other versions of the Balanced Sign Triad of Peirce have been called by the name semiotic triangle. Sowa introduced another model of sign, which is similar to the Dynamic Sign Triad of Morris. In his model, a sign has three aspects: (1) an entity that represents (2) another entity to (3) an agent (Sowa, 2000a). It is represented in Figure 4.15. In (Vetrov, 1968), another model of a sign is considered (cf., Figure 4.16). There is one more model of a sign, in which the name and the object are connected by the triadic relation. Namely, according to Symbol (Material Symbol)
Thought (reference)
Referent
Figure 4.14. The Semiotic Triangle of Ogden and Richards agent
entity
entity
Figure 4.15. The sign model of Sowa Material Sign (Word or Text) represents
actualizes in the receiver
actualizes in the sender Object Mental image reflects
Figure 4.16. The Functional Semiotic Triangle
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Type
Object
Reliability
Figure 4.17. The sign model of Foucault
Foucault (1966), signs are determined by three parameters of the classical Port-Royal logic. These parameters characterize the connection between the name of the sign and object denoted by the sign: — Origin of the connection shows whether the sign is natural or conventional. — Type or form of the connection explains whether the sign belongs to the object it denotes or does not belong. — Reliability of the connection illustrates whether the sign connection between the sign and its object is certain, i.e., without any doubt, or only plausible. It gives us the model in Figure 4.17. The variety of sign models show that signs as units of quantum knowledge have a sophisticated inherent structure. 4.3. Operations with and relations between quantum knowledge units To attain knowledge, add things every day. To attain wisdom, remove things every day.” Lao Tzu
In this section, we describe some operations with and relations between knowledge quanta. The goal is to provide constructive tools for building new knowledge quanta from given knowledge quanta, as well as to illustrate relations that exist in knowledge systems. Taking these operations, we can build a knowledge algebra (Kset, oK) where: (1) Kset — is a set of knowledge items, (2) oK — is a set of epistemic operations with knowledge items.
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Note that (Kset, oK) may be a conventional universal algebra or a multibase universal algebra. Besides, taking operations with and relations between knowledge items (units), we can construct a knowledge algebraic system (Kset, oK, rK) where: (1) Kset — is a set of distinguishable classes/types of knowledge items, (2) oK — is a set of available operations, which has to be specialized according to the assumed Ic and recognized/assumed specific knowledge structures (classes). We may notice that at the present time, any knowledge conceptualizations and knowledge meta-ontologies do not enable yet to develop a formal full-size algebra of knowledge although algebraic models in the Semantic Link Network theory are constructed in (Zhuge, 2012), while general operations with knowledge systems are studied in (Burgin, 2011a; 2011b; 2014). Before starting our exposition of relations and operations, we remind some definitions in this area. A unary relation on a set of objects is a property of these objects. For instance, being white is a unary relation in the set (property) of tables and being interesting is a unary relation in the set (property) of books. A binary relation on a set of objects indicates pairs of these objects. For instance, to be taller is a binary relation on the set of people, for example, we can say that Alex is taller than Bob. A ternary relation on a set of objects indicates triples of these objects. For instance, to be a student of a professor P who works in a university A is a ternary relation on the set of people, for example, we can say that Alex is a student of the professor Pascal, who is working at UCLA. An n-ary relation on a set of objects indicates groups of n objects. An integral relation on a set of objects indicates some groups of these objects. For instance, company as a group of people who work in this company is an integral relation on the set of people because different companies usually have different numbers of employees.
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A unary operation in a set of objects assigns one object to another object. For instance, any function can be treated as a unary operation. A binary operation in a set of objects assigns one object to pairs of objects. For instance, addition and multiplication are binary operations in the set of numbers. A ternary operation in a set of objects assigns one object to pairs of objects. An n-ary operation in a set of objects assigns one object to triples of objects. An integral operation in a set of objects assigns one object to sets of objects. For instance, summation in mathematics is an integral operation in the set of numbers (Burgin, 2008). 4.3.1. Properties of and relations between nodes and links in SLN and knowledge quanta in QTK The following properties (unary relations) are considered in SLN. Definition 4.3.1. If {(X, α, Y ) and (Y , α, Z)} imply (X, α, Z), then the type link α or the type α-link is called transitive. For instance, the sequential link, equal link, reference link, cause– effect link, implication link, and subtype link are transitive. Definition 4.3.2. If the relation α in the semantic link α = (X, α, X) is symmetric, then the link α is called symmetric. For instance, the equal link, empty link, unknown link, semantic equivalence link, and similarity link are symmetric. There are also binary relations between semantic links. The reachability relation is defined in the following way. Definition 4.3.3. (a) If there is a semantic link (X, α, Y ), then the node Y is called directly semantically reachable from the node X by the link α. (b) The node Y is called directly semantically reachable from the node X if it is semantically reachable by some semantic link. (c) The node Y is called semantically reachable from the node X if there is a sequence of nodes X0 , X1 , X2 , X3 , . . ., Xn such that
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X0 = X, Xn = Y , and each Xi is directly semantically reachable from the node Xi−1 (i = 1, 2, 3, . . ., n). Let us consider some examples. Example 4.3.1. By Cantor’s theorem (cf., (Fraenkel and Bar-Hillel, 1958)), the set Q of all natural numbers is countable, i.e., it is equivalent to the set N of all natural numbers. It means that the set Q is semantically reachable from the set N by the equivalence link. Example 4.3.2. As it is proved in the theory of algorithms, automata and computation, the class of all non-deterministic finite automata is semantically reachable from the class of all deterministic finite automata by the equivalence (or more exactly, by linguistic equivalence) link (Sipser, 1997; Burgin, 2010d). Example 4.3.3. As it is proved in the theory of algorithms, automata and computation, the class of all Turing machines with one tape is semantically reachable from the class of all Turing machines with n tapes by the equivalence (or more exactly, by functional equivalence) link (Sipser, 1997; Burgin, 2010d). Definition 4.3.3.c implies the following result. Proposition 4.3.1. If a node Y is semantically reachable from a node X and a node Z is semantically reachable from the node Y , then the node Z is semantically reachable from the node X. An important binary relation is implication of links. Definition 4.3.4. A semantic link α = (X, α, Y ) implies a semantic link β = (X, β, Y ), it is also said that α implies β and is denoted by α ⊆ β or (X, α, Y ) ⇒ (X, β, Y ) or α ⇒ β, if correctness (validity) of the semantic link α = (X, α, Y ) implies correctness (validity) of the semantic link β = (X, β, Y ). Definitions 4.3.1 and 4.3.4 imply the following result. Proposition 4.3.2. Implication of semantic links is a transitive relation.
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Let us consider some examples. Example 4.3.4. The equal link implies the similarity link because if two objects are equal (in some sense), then they are naturally similar. Example 4.3.5. According to the conventional understanding, the element link implies the part link because elements are usually treated as parts. However, in mereology, the concept of an element is not used and thus, such an implication is not true (Slupecki, 1958). Example 4.3.6. If a set is treated as a kind of a class, then the subset link implies the subclass link. An important concept in the operation of a semantic network is the inheritance of relations, which determines relations between semantic links. The following properties and relations are studied in the quantum theory of knowledge. At first, we consider binary relations “more abstract” and “more general”. Let us consider two descriptive collective knowledge quanta (units): KG described by Diagram (4.40) and HG described by Diagram (4.41): l L
M ,
p
q C
(4.40)
D f k
W
M .
t
q U
(4.41)
D h
Definition 4.3.5. The knowledge unit HG is more abstract than the knowledge unit KG if both Diagrams (4.40) and (4.41) can be
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included into the commutative combined diagram (4.42), in which l and d are projections: k
M l
W
L
q .
g t
p
U
h
(4.42)
D d
C f
For example, average knowledge about a group of people, e.g., their average salary, is more abstract than knowledge about some individual from this group, e.g., her salary. Average salary of a person for a year is more abstract than the salary of this person for some month. As the correct combination of commutative diagrams gives a commutative diagram, we have the following result. Proposition 4.3.3. If a knowledge unit HG is more abstract than a knowledge unit KG and a knowledge unit FG is more abstract than the knowledge unit HG, then the knowledge unit FG is more abstract than the knowledge unit KG. Having the combined diagram for FG and HG and the combined diagram for HG and KG, we can build the combined diagram for FG and KG because composition (product) of projections is a projection. Let us consider two descriptive collective knowledge quanta (units): KG described by Diagram (4.40) and PG described by Diagram (4.43): n Z
N .
r
l V
(4.43)
E m
Definition 4.3.6. The knowledge unit HG is more general than the knowledge unit KG if both Diagrams (4.40) and (4.43) are included
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into the commutative joined Diagram (4.44), in which u and care injections:
.
(4.44)
For example, knowledge about properties of birds is more abstract than the corresponding knowledge about eagles. As correct combination of commutative diagrams gives a commutative diagram, we have the following result. Proposition 4.3.4. If a knowledge unit HG is more general than a knowledge unit KG and a knowledge unit FG is more general than the knowledge unit HG, then the knowledge unit FG is more general than the knowledge unit KG. Having the joined diagram for FG and HG and the joined diagram for HG and KG, we can build the joined diagram for FG and KG because composition (product) of injections is an injection. As we can see from Chapter 2, another important relation between knowledge quanta (knowledge units) is consistency. For instance, the knowledge quanta (A, is, a man) and (A, is, a student) are consistent, while the knowledge quanta (A, is, a man) and (A, is, a building) are inconsistent. Computability and measurability are useful properties of knowledge quanta (knowledge units). Definition 4.3.7. The knowledge unit HG is computable if the attribute q in Diagram (4.40) is computable. In general, computability is a relative property, which depends on the class of algorithms or automata that are used for computation
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(Burgin, 2005). Consequently, computability of knowledge quanta (units) is also a relative property. Definition 4.3.8. The knowledge unit HG is measurable if the intrinsic property p in Diagram (4.40) is measurable. Similar to computability, measurability is a relative property, which depends on the class of measuring devices. For instance, now we can measure much more than scientists were able in the 19th century. Consequently, measurability of knowledge quanta (units) is also a relative property. To define the inclusion relation, we consider two descriptive collective knowledge quanta (units): KG given by Diagram (4.45) and HG given by Diagram (4.46): l P
L
p
,
(4.45)
.
(4.46)
q V
D f k
W
M
t
r U
C h
Definition 4.3.9. The extended knowledge unit KG is included in the extended knowledge unit HG if the domain V is a subdomain of the domain U, the set D is a subset of the set C, while the properties (V, p, P) and (D, q, L) are components of the properties (U, t, W) and (C, r, M), correspondingly. For instance, if we have U = {car 1, car 2}, V = {car 1}, D = {BMW1}, C = {BMW1, BMW2}, W = {color}, L = {white},
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M = {white, gray}, r(BMW1) = q(BMW1) = white and q(BMW2) = grey, (cf., Diagrams (4.47) and (4.48)) then the knowledge unit KG is included in the knowledge unit HG. l {color}
{white} ,
p
(4.47)
q
{car 1}
{BMW1} f k
{color}
{white, gray}, .
t
r
{car 1, car 2}
(4.48)
{BMW1, BMW2} h
As inclusion of sets is a transitive relation, we have the following result. Proposition 4.3.5. If a knowledge unit HG is included in a knowledge unit KG and a knowledge unit FG is included in the knowledge unit HG, then the knowledge unit FG is included in the knowledge unit KG, i.e., inclusion of extended descriptive (representational or operational) knowledge units is a transitive relation. It is also possible to define other relations between descriptive individual and collective knowledge quanta (units), such as inferentiability, reducibility, and equivalence, as well as relations between representational and operational, individual and collective knowledge quanta (units), such as simulation relation, representability, reducibility, and equivalence. Now construct some relations between symbolic knowledge units. To define the inclusion relation, we consider two symbolic descriptive collective knowledge quanta (units): SKG given by
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Diagram (4.49) and SHG given by Diagram (4.50): l C
L ,
(4.49)
.
(4.50)
f D
M
Definition 4.3.10. The symbolic knowledge unit SKG is included in the symbolic knowledge unit SHG if the domain C is a subdomain of the domain D and the property (C, l, L) is a component of the property (D, f , M). Computability and measurability are also useful properties of symbolic descriptive knowledge quanta (symbolic knowledge units). However, in this case, measurability is interpreted not as in physics or in any natural science where characteristics of physical systems are measured by specific devices but as it is comprehended in mathematics where there is a special discipline — the theory of measure and integration (cf., for example, (Konig, 2009)). Definition 4.3.11. The symbolic descriptive knowledge unit (C, l, L) is computable if it is computable as an abstract property. As in the case of extended knowledge quanta (units), computability of symbolic knowledge quanta (units) is a relative property. Definition 4.3.12. The symbolic descriptive knowledge unit (C, l, L) is measurable if it is a measurable function in the sense of the theory of measure and integration. Similar to computability, measurability of symbolic knowledge units is a relative property, which depends on the utilized measure. A useful binary relation is reducibility of symbolic knowledge units, which is based on the concept of reduction studied in the theory of abstract properties (cf., Chapter 5). Definition 4.3.13. The symbolic knowledge unit SKG is reducible to the symbolic knowledge unit SHG if the abstract property (C, l, L) can be reduced to the abstract property (D, f , M). When we consider relative reducibility of abstract properties, i.e., reducibility that depends of admissible functions used for reduction, then reducibility of symbolic knowledge unit also becomes relative.
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Reducibility of symbolic knowledge unit implies equivalence relation between symbolic knowledge units. Definition 4.3.14. The symbolic knowledge unit SKG is equivalent to the symbolic knowledge unit SHG if SKG is reducible to SHG and SHG is reducible to SKG. Another useful binary relation between symbolic knowledge units is “to be an extension of”. There are two kinds of this relation — “to be a right extension of” “to be a left extension of”. Definition 4.3.15. The individual symbolic knowledge unit (C, l, L) is a right extension of the individual symbolic knowledge unit (C, f , M) if L = M ∪ K. For instance, if C is the word car and functions f and l describe properties of this car, namely, M = {white} and L = {white, four doors, automatic, 2010} where 2010 is the year of its production, then the individual symbolic knowledge unit (C, l, L) is a right extension of the individual symbolic knowledge unit (C, f , M). Definition 4.3.16. The collective symbolic knowledge unit (C, l, L) is a left extension of the collective symbolic knowledge unit (D, f , L) if C = D ∪ E. For instance, if C = {car1, car2, car3}, D = {car1} and functions f and l describe color of these cars, namely, L = {white}, then the individual symbolic knowledge unit (C, l, L) is a left extension of the individual symbolic knowledge unit (D, f , M). Proposition 4.3.7. Relations “to be a right extension of” “to be a left extension of” are transitive. Projections are dual relations to extensions. Definition 4.3.17. The individual symbolic knowledge unit (C, l, L) is a right projection of the individual symbolic knowledge unit (C, f , M) if (C, l, L) is a right extension of the individual symbolic knowledge unit (C, f , M). By this definition, right extension is the inverse relation to right projection. This is an example of a relation between relations, which are also included in the structure of a system (cf., Section 5.1).
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Definition 4.3.18. The individual symbolic knowledge unit (C, l, L) is a left projection of the individual symbolic knowledge unit (C, f , M) if (C, l, L) is a left extension of the individual symbolic knowledge unit (C, f , M). By this definition, right extension is the inverse relation to left projection. Proposition 4.3.8. Relations “to be a right projection of” “to be a left projection of” are transitive. Proof is left as an exercise. It is also possible to define other relations between symbolic descriptive individual and collective knowledge quanta (units), such as inferentiability, “to be a renaming” and “to be a reinterpretation”, as well as relations between symbolic representational and operational, individual and collective knowledge quanta (units), such as simulation relation, representability, reducibility, and equivalence. 4.3.2. Operations with extended knowledge quanta Extended knowledge quanta with operations form epistemic quantum algebras. Here we consider only some of these operations and study their properties. 4.3.2.1. Unary operations with extended knowledge quanta Operation 4.3.1. The operation renaming of an individual knowledge quantum K is represented by the following transformation: Diagram (4.51) is changed to Diagram (4.52) e QA
PA
q
p
A
NA n
,
(4.51)
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e QA
PA
q
po
A
.
(4.52)
MA no
Informally, renaming means changing the name of the object A. For instance, when number 1 is considered as an element of the set {1, 2, 3, . . . }, it is called a natural number. However, when the same number is considered as an element of the set {0, 1, 2, 3, . . . }, it is called a whole number. In such a way, moving from natural numbers to whole numbers, we rename number 1. As extended knowledge quanta are built of named sets, we can apply results from the theory of named sets (Burgin, 2011) obtaining the following result. Proposition 4.3.6. A renaming of a renaming of an extended knowledge quantum K is a renaming of K. Renaming can be pure, when only the name is changed, and combined, when, for example, the ascribed property (attribute) is also changed. Operation 4.3.2. The operation re-evaluation of an individual knowledge quantum is represented by the following transformation: Diagram (4.53) is changed to Diagram (4.54). e QA
PA ,
q
p
A
NA n
(4.53)
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e QA = QB
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PA .
qo
po
B
(4.54)
NB = N A no
Informally, reevaluation means giving the same name to another object B. For instance, taking the number 1 with the name a natural number, we can give the same name to the number 3. As extended knowledge quanta are built of named sets, we can apply results from the theory of named sets (Burgin, 2011) obtaining the following result. Proposition 4.3.7. A re-evaluation of a re-evaluation of an extended knowledge quantum K is a reevaluation of K. Let us consider combined operations. Operation 4.3.3. Re-evaluative renaming of an individual knowledge quantum is represented by the following transformation: Diagram (4.55) is changed to Diagram (4.56) e PA
QA q
p
A
,
(4.55)
NA n e
QB
PB
qo
po
B
NB no
.
(4.56)
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Re-evaluative renaming is a kind of combined renaming. Note that re-evaluation, renaming, and re-evaluative renaming are multivalued operations. There are operations with extended knowledge quanta induced by binary relations between extended knowledge quanta. Let us consider two of these operations — abstraction and generalization, which are very popular in science and mathematics. Operation 4.3.4. Abstraction transforms a knowledge unit into a more abstract knowledge unit. Proposition 4.3.3 implies the following result. Proposition 4.3.8. Abstraction of an abstraction of an extended knowledge quantum K is abstraction of K. Operation 4.3.5. Generalization transforms a knowledge unit into a more general knowledge unit. Proposition 4.3.4 implies the following result. Proposition 4.3.9. Generalization of generalization of an extended knowledge quantum K is a generalization of K. 4.3.2.2. Binary operations with extended knowledge quanta There are different kinds of unions of extended quantum knowledge units. Here we define some of them for descriptive collective knowledge quanta. Operation 4.3.6. In the disjunctive union of collective extended quantum knowledge units, which is denoted by the symbol , objects and their names are combined together with their properties as separate objects. Namely, let us take two collective extended quantum knowledge units K and H and build their disjunctive union K H.
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K: g W
L
q U
p ,
(4.57)
n ,
(4.58)
C f
H: h V
M
r T
D t
K H: g∪h {W, V}
{L, M}
q∪r
p∪ n .
(4.59)
{C, D}
{U, H} f ∪t
As the union of sets is a commutative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.10. The disjunctive union of collective extended quantum knowledge units is a commutative operation. As the union of sets is an associative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.11. The disjunctive union of collective extended quantum knowledge units is an associative operation. As the union of sets is an idempotent operation (Fraenkel and Bar-Hillel, 1958), we have the following result.
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Proposition 4.3.12. The disjunctive union of collective extended quantum knowledge units is an idempotent operation, i.e., K K = K for any collective extended quantum K. The rich structure of collective extended knowledge quanta allows defining other types of unions for them. Operation 4.3.5. In the union with name amalgamation of collective extended quantum knowledge units, which is denoted by the symbol ∪n , objects and their names are also combined together with their properties. However, while objects and their properties are combined together as separate objects forming multisets in a general case, names are combined by the operation of union of sets. Namely, let us take two collective extended quantum knowledge units K and H and build their union with name amalgamation K ∪n H. K: g W
L
q
p , U
(4.60)
C f
H: h V
M
r
n , T
(4.61)
D t
K ∪n H: g∪h {W, V}
{L, M}
q∪r {U, H}
u w
C∪D
.
(4.62)
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For instance, the domain U has one object with the name ball and the domain H has one object with the name ball. Then due to the name amalgamation, the domain {U , H} has two objects with the same name ball. That is, C = D = {ball} and C ∪ D = {ball}. As the union of sets is a commutative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.13. The union with name amalgamation of collective extended quantum knowledge items is a commutative operation. As the union of sets is an associative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.14. The union with name amalgamation of collective extended quantum knowledge items is an associative operation. As the union of sets is an idempotent operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.15. The union with name amalgamation of collective extended quantum knowledge units is an idempotent operation, i.e., K ∪n K = K for any collective extended quantum K. One more operation is feature union. Operation 4.3.6. In the feature union, or union with feature merger, denoted by the symbol ⊕, objects and their names are combined together by the set-theoretical union and stay the same in this operation but their intrinsic properties (features) are merged, namely, when the same object has different properties, these properties are combined (merged) into one property. For instance, let us consider two knowledge units K and H. In K, the (abstract) property is color, e.g., U has an object b, which is called a ball and is white. At the same time, in H, the (abstract) property is weight, e.g., H has the same object b, which is called a ball and has weight 10 kg. If K ⊕ H is the feature union of K and H, then the (abstract) property is color and weight, e.g., the object b, which is called a ball has the property {white, 10 kg}.
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Merging of relational database schemas involves feature union of quantum knowledge items. As the union of sets is a commutative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.16. The feature union of collective extended quantum knowledge items is a commutative operation. As the union of sets is an associative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.17. The feature union of collective extended quantum knowledge items is an associative operation. As the union of sets is an idempotent operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.18. The feature union of collective extended quantum knowledge units is an idempotent operation, i.e., K ⊕ K = K for any collective extended quantum K. One more operation is attributive union. Operation 4.3.7. In the attributive union, or union with attribute merger, denoted by the symbol ⊗, objects and their names are combined together by the set-theoretical union and stay the same in this operation but their assigned properties (attributes) are merged, namely, when the same object has different assigned properties, these properties (attributes) are combined (merged) into one property. For instance, let us consider two knowledge units K and H, elements of which are cities. In K, the attribute (assigned property) is population, e.g., a city a with 10 million population, while in H, the attribute (assigned property) is the place on the Earth, e.g., the city a situated in Europe. If K ⊗ H is the attributive union of K and H, then the property is population and the place on the Earth, e.g., the attribute of the city a is {10 million, Europe}. Merging of relational database schemas attributive union of quantum knowledge items. As the union of sets is a commutative operation (Fraenkel and Bar-Hillel, 1958), we have the following result.
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Proposition 4.3.19. The attributive union of collective extended quantum knowledge items is a commutative operation. As the union of sets is an associative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.20. The attributive union of collective extended quantum knowledge items is an associative operation. As the union of sets is an idempotent operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.21. The attributive union of collective extended quantum knowledge units is an idempotent operation, i.e., K ⊕ K = K for any collective extended quantum K. One more operation is union with object amalgamation. Operation 4.3.8. The union with object amalgamation of collective extended quantum knowledge units, which is denoted by the symbol ∪O , is applied when an object has several names. In this case, these names and the object properties are merged into one set and these names and the object properties are assigned to their common object. For instance, a geometrical object I has two names — a segment and an interval and the property “the length of I is 25 inches.” It is possible to consider two quantum items of knowledge K1 and K2 about the object I, which are represented by Diagrams (4.63) and (4.64), respectively. Then when the union with object amalgamation of these two quantum items of knowledge is performed, Diagrams (4.63) and (4.64) that represent this quantum knowledge are merged into Diagram (4.65). The result of the operation represented by Diagram (4.64) is denoted by K = K1 ∪O K2 . g length
25 in
q
p I
segment
,
(4.63)
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g length
25 in
q
r I
,
(4.64)
interval g
length q
25 in {p, r}
I
.
(4.65)
{segment, interval}
Here is one more example. A person A has two names — Barbara and Barbie, as well as two heights — 5 ft at the age 12 when she was called Barbie and 6 ft at the age 30 when she was called Barbara. As the union of sets is a commutative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.22. The disjunctive union with object amalgamation of quantum knowledge items is a commutative operation. As the union of sets is an associative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.23. The disjunctive union with object amalgamation of quantum knowledge items is an associative operation. As the union of sets is an idempotent operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.24. The attributive union of collective extended quantum knowledge units is an idempotent operation, i.e., K ⊕ K = K for any collective extended quantum K. One more operation is Cartesian product. Operation 4.3.9. In the Cartesian product of collective extended quantum knowledge units, which is denoted by the symbol ×, the Cartesian products of object domains and of name domains are formed together with their properties. Namely, let us take two
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quantum knowledge units K and H and build their Cartesian product K × H. K: g W
L
q
p , U
(4.57)
C f
H: h V
M
r
n , T
(4.58)
D t
K × H: W×V
g×h
p×n .
q×r U×H
L×M
f×t
(4.59)
C×D
As the Cartesian product of sets is a commutative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.25. The Cartesian product of collective extended quantum knowledge units is a commutative operation. As the Cartesian product of sets is an associative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.26. The Cartesian product of collective extended quantum knowledge units is an associative operation. There are also operations with higher than two arity, as well as integral operations with extended knowledge units (knowledge quanta) but they are studied elsewhere.
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4.3.3. Operations with symbolic knowledge quanta and complete semantic links Operations with symbolic knowledge quanta and complete semantic links are induced by the union of named sets studied in (Burgin, 2011) because these knowledge items are special cases of named sets. Here we consider only some of these operations in the domains of descriptive knowledge quanta and of complete semantic links. 4.3.3.1. Unary operations with symbolic knowledge quanta Having a symbolic knowledge quantum (N , c, P ), it is possible to invert its relation c obtaining the symbolic knowledge quantum (P , c−1 , N ). Logically, such an operation is called inversion. Operation 4.3.10. If K = (N, c, P ) is a symbolic knowledge quantum, then the symbolic knowledge quantum invK = (P, c−1 , N ) where c−1 is the inverse of the relation c is the inversion of K, which is denoted by inv K. Inversion is a very natural operation for representational knowledge. For instance, let us consider a (formal) theory T and its model M . They are included in the symbolic knowledge quantum U = (T , int, M ), in which M represents T . At the same time, it is possible to consider T as a representation of M obtaining the inverse symbolic knowledge quantum invU = (M , model, T ). Inversion also works for descriptive knowledge because in the theory of abstract properties, values of properties may be arbitrary objects (Burgin, 1985). Then what was the object becomes a property and what was the property becomes an object, while what was an object becomes a property as the result of inversion. The same is true for operational knowledge where inversion is an innate operation. For instance, in the theory of algorithms, automata, and computation, a Turing machine is represented by a collection T = (A, Q, q0 , F , R) where A is its alphabet, Q is its set of states, F is the set of the final states, q0 is the start state and R is its system of rules. This gives the operational symbolic knowledge quantum W = (Turing machine, r, (A, Q, q0 , F , R)). Inverting this knowledge
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quantum, we obtain the inverse knowledge quantum invW = ((A, Q, q0 , F , R), t, Turing machine), which gives knowledge that collection T = (A, Q, q0 , F ,R) is a Turing machine. Proposition 4.3.27. The inversion of symbolic quantum knowledge units is an idempotent operation, i.e., inv(invU ) = U for any symbolic knowledge quantum U . Note that it is possible to apply inversion both to individual and collective symbolic knowledge quanta. Operation 4.3.11. The renaming operation rn changes the right component in a symbolic knowledge quantum, i.e., if K = (N, c, P ) is a symbolic knowledge quantum, then rnK = (N, b, T ) for some name T . For instance, taking the operational symbolic knowledge quantum V = ((A, Q, q0 , F, R), t, Turing machine) where (A, Q, q0 , F , R) is the mathematical description of some Turing machine (Hopcroft et al., 2007) and applying the renaming operation, which changes the name “Turing machine” to the name “abstract automaton”, we can obtain the operational symbolic knowledge quantum rnV = ((A, Q, q0 , F, R), l, abstract automaton). As symbolic knowledge units are a special kind of named sets, we can apply results from the theory of named sets (Burgin, 2011) obtaining the following result. Proposition 4.3.28. A renaming of a renaming of a knowledge quantum K is a renaming of K, i.e., the sequential composition of renamings is a renaming. Attributes in symbolic knowledge quanta also have names. For instance, symbolic knowledge quantum K = (N, c, P ) has the attribute with the name “c”. Therefore, we have one more operation of renaming. Operation 4.3.11. The attribute renaming operation arn changes the right component in a symbolic knowledge quantum, i.e., if K = (N, c, P ) is a symbolic knowledge quantum, then arnK = (N, b, P ) for some attribute name b.
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For instance, it is possible to call some plant by the name “a tree” or by the name “an oak”. As symbolic knowledge units are a special kind of named sets, we can apply results from the theory of named sets (Burgin, 2011) obtaining the following result. Proposition 4.3.28. An attribute renaming of an attribute renaming of a knowledge quantum K is an attribute renaming of K, i.e., the sequential composition of attribute renamings is an attribute renaming. Attribute renaming is a widespread unary operation in relational databases where it is called rename and denoted by the letter ρ (Elmasri and Navathe, 2000). It is used to change names of attributes in a relation. Operation dual to renaming is called reinterpreting. Operation 4.3.12. The reinterpreting operation rt changes the left component in a symbolic knowledge quantum, i.e., if K = (N, c, P ) is a symbolic knowledge quantum, then rtK = (M, b, P ) for some object M . For instance, taking the operational symbolic knowledge quantum V = ((A, Q, q0 , F, R), t, Turing machine) where (A, Q, q0 , F, R) is the mathematical description of a universal Turing machine (Hopcroft et al., 2007) and applying the reinterpreting operation, which changes the description (A, Q, q0 , F, R) to the description (X, P, q0 , H, D) of a Turing machine that computes the identity function, we can obtain the operational symbolic knowledge quantum rtV = ((X, P, q0 , H, D), h, Turing machine). As symbolic knowledge units are a special kind of named sets, we can apply results from the theory of named sets (Burgin, 2011) obtaining the following result. Proposition 4.3.29. Reinterpreting of reinterpreting of a knowledge quantum K is reinterpreting of K, i.e., composition of reinterpretings is a reinterpreting. Some unary operations with symbolic knowledge units are induced by binary relations between symbolic knowledge units. For
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instance, the relations “to be a right extension of” and “to be a left extension of” give us two operations — right extension and left extension. Operation 4.3.13. A right extension of an individual symbolic knowledge unit (A, l, L) is an individual symbolic knowledge unit (A, f , M) for which the equality M = L ∪ K where K consists of properties of A is true. If an individual symbolic knowledge unit (C, l, L) describes us some properties of an object with the name C, then its right extension gives more properties of the same object. For instance, if in the individual symbolic knowledge unit (C, f , L), C is the word car and function l describes the color of this car, namely, M = {white}, then extending this knowledge unit from the right, we can get the individual symbolic knowledge unit (C, l, L), in which L = {white, four doors, automatic, 2010} where 2010 is the year of its production. Proposition 4.3.29. Right extension of a right extension of an individual symbolic knowledge quantum K is a right extension of K, i.e., composition of right extensions is a right extension. It is also possible to apply right extension to collective symbolic knowledge units. The operation left extension acts in the realm of collective symbolic knowledge units. Operation 4.3.14. A left extension of a collective symbolic knowledge unit (C, l, L) is a collective symbolic knowledge unit (D, f , L) for which the equality D = C ∪ E is true. Left extension combines different knowledge domains of objects with the same property together. For instance, if C = {car1, car2, car3}, D = {car1} and functions f and l describe color of these cars, namely, L = {white}, then the individual symbolic knowledge unit (C, l, L) is a left extension of the individual symbolic knowledge unit (D, f , M). Proposition 4.3.29. Left extension of a left extension of a collective symbolic knowledge quantum K is a left extension of K, i.e., composition of left extensions is a left extension.
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Note that both extensions are multivalued operations. However, it is possible to make right extensions a regular operation by assigning parameters to extensions, which can determine what attributes have to be added. In such a way, we obtain a parametric family of right extensions. Adding parameters indicating objects to be added, we can also make left extensions a regular operation. Operations dual to extensions are called projections. Operation 4.3.15. A right projection of an individual symbolic knowledge unit (A, l, M) is a collective symbolic knowledge unit (A, f , L) for which the equality M = L ∪ H where H consists of properties of A is true. By this definition, right extension is the inverse operation to right projection. Proposition 4.3.29. Right projection of a right projection of an individual symbolic knowledge quantum K is a right projection of K, i.e., composition of right projections is a right projection. In a similar way, we define left projection. Operation 4.3.14. A left projection of a collective symbolic knowledge unit (C, l, L) is a collective symbolic knowledge unit (D, f , L) for which the equality C = D ∪ E is true. By this definition, left extension is the inverse operation to left projection. Proposition 4.3.29. Left projection of a left projection of a collective symbolic knowledge quantum K is a left projection of K, i.e., composition of left projections is a left projection. Note that both projections are multivalued operations. However, it is possible to make right projections a regular operation by assigning parameters to projections, which can determine what attributes have to be eliminated (truncated). In such a way, we obtain a parametric family of right projections. Adding parameters indicating objects to be eliminated, we can also make left projections a regular operation.
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4.3.3.2. Unary operations with semantic links Zhuge introduces the following operation with semantic links (Zhuge, 2012). Operation 4.3.13. The reverse relation operation, or simply inversion, R changes α to αR and a semantic link (X, α, Y ) to the semantic link (Y , αR , X). As Zhuge writes, a semantic link (semantic relation) and its reverse declare the same thing, but one of them may be useful in some situations, while the other may be useful in other situations (Zhuge, 2012). Example 4.3.7. The before link is the reverse of the after link. Example 4.3.8. The worse link is the reverse of the better link. Example 4.3.9. The smaller link is the reverse of the bigger link. The reverse relation operation of semantic links is the involution operation of named sets (Burgin, 2011). This allows us to use named set theory to prove the following proposition obtaining properties of the reverse relation operation. Proposition 4.3.30 (Zhuge, 2012). (Idempotent Law) αRR = α for any semantic link α. Proposition 4.3.31. (Symmetry Law) αR ≡ α if and only if α is a symmetric semantic link. For instance, for symmetric relations, the relation link coincides with its reverse. Corollary 4.5.3 (Zhuge, 2012). (1) simR = sim (2) ØR = Ø (3) N R = N (4) equiv R = equiv
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Proposition 4.3.32. (Monotone Law) A semantic link α implies the semantic link β if and only if the semantic link αR implies the semantic link β R . 4.3.3.3. Binary operations with symbolic knowledge quanta At first, we describe some operations with knowledge units. Operation 4.3.6. The union of symbolic collective quantum knowledge units K = (C, l, L) and H = (D, h, M ) combines names and properties as separate objects and is defined as K ∪ H = ({C, D}, g ∪ h, {L, M }) where l ∪ h|C = l and g ∪ h|D = h. Note that {C, D} and {L, M } are multisets in a general case. As the union of sets is a commutative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.10. The union of symbolic collective quantum knowledge units is a commutative operation. As the union of sets is an associative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.11. The union of symbolic collective quantum knowledge units is an associative operation. As the union of sets is an idempotent operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.12. The union of symbolic collective quantum knowledge units is an idempotent operation, i.e., K ∪ K = K for any symbolic collective quantum K. It is also possible to define the union with properties amalgamation of symbolic collective quantum knowledge units and the union with name amalgamation of symbolic collective quantum knowledge units in a similar way as it is done for collective extended quantum knowledge units (cf., Section 4.3.2.2). One more operation with symbolic collective quantum knowledge units is Cartesian product.
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Operation 4.3.9. In the Cartesian product of collective extended quantum knowledge units K = (C, l, L) and H = (D, h, M ) forms the Cartesian products of name domains C ×D and of property scales L × M and is defined as K × H = (C × D, g × h, L × M ). As the Cartesian product of sets is a commutative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.25. The Cartesian product of symbolic collective quantum knowledge units is a commutative operation. As the Cartesian product of sets is an associative operation (Fraenkel and Bar-Hillel, 1958), we have the following result. Proposition 4.3.26. The Cartesian product of symbolic collective quantum knowledge units is an associative operation. There are also operations with higher than two arity, as well as integral operations with symbolic collective knowledge units (knowledge quanta) but they are studied elsewhere. 4.3.3.4. Binary operations with semantic links Operation 4.3.7. The semantic addition + of two complete semantic links α = (X, α, Y ) and β = (X, β, Y ) is performed by merging the inner semantic links α and β (Zhuge, 2012). It means that α + β = (X, α & β, Y ). The inner semantic link (semantic indicator) α&β is also denoted by α + β. Example 4.3.10. If α = (X, sim, Y ) and β = (X, ptOf, Y ), which means that X is a part of Y and X is similar to Y , then α + β = (X, sim&ptOf, Y ), which means that X is a part of Y and is similar to Y . This inner semantic link sim & ptOf is pivotal for the theory of fractals when some parts are similar to the whole (cf., for example, (Flake, 1998)). Example 4.3.11. If α = (X, ce, Y ) and β = (X, elOf, Y ), which means that X is an element of Y and X causes Y , then α + β = (X, ce&elOf , Y ), which means that X is an element of Y and causes Y .
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Note that semantic addition is a partial operation on the class of all complete semantic links. There are two cases when semantic addition is not defined. First, we cannot add complete semantic links if their left nodes are different or/and their right nodes are different. Seconds, some semantic links are incompatible and it is impossible to consistently add them. For instance, inner semantic links Ø and e are incompatible because it is impossible that two objects are equal and at the same time, completely irrelevant to one another. Thus, we cannot add complete semantic links, in which inner semantic links are Ø and e. However, even when some semantic links are incompatible, they can be partially compatible and it might be possible to consistently add graded counterparts of these semantic links. Let us consider some properties of semantic addition. Proposition 5.5.4 (Zhuge, 2012). (Commutative Law) If the semantic addition α + β is defined, then α + β = β + α for any complete semantic links α and β. Proposition 5.5.5 (Zhuge, 2012). (Associative Law) If the semantic addition α+β, α+γ and β+γ is defined, then (α+β)+γ = α + (β + γ) for any complete semantic links α, β, and γ. Proposition 5.5.6 (Zhuge, 2012). (The Law of Zero) α + N = N + α = α for any complete semantic link α and the corresponding complete semantic link N with the inner semantic link N . Note that semantic addition α + N and N + α is always defined. Proposition 5.5.4 (Zhuge, 2012). (Implicative Law) If a complete semantic link α implies a complete semantic link β, then α + β = α. Corollary 5.5.3 (Zhuge, 2012). (Idempotent Law) α+α = α for any complete semantic link α. Proposition 5.5.4 (Zhuge, 2012). (Distributive Law for Reversion) If the semantic addition α + β is defined, then (α + β)R = αR + β R for any complete semantic links α and β.
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Proposition 5.5.4 (Zhuge, 2012). (Distributive Law for Implication) If semantic addition β + γ is defined and a complete semantic link α implies a complete semantic link β and a complete semantic link γ, then α implies β + γ. Proposition 5.5.4 (Zhuge, 2012). (Conjunction Law) If the semantic addition α + β is defined, then the complete semantic link α+β implies the complete semantic link β and the complete semantic link α. Generalizing the concept of semantic addition, we can build a parametric family of semantic operations. Let us consider a partial binary operation ◦ on the totality of all inner semantic links. If this operation is defined for some pairs of complete semantic links in which left nodes are the same and their right nodes are the same, then we can build the parallel ◦–composition of complete semantic links. Operation 4.3.7. The parallel ◦–composition ◦ of two complete semantic links α = (X, α, Y ) and β = (X, β, Y ) is performed by performing the operation ◦ with the inner semantic links α and β, i.e., α ◦ β = (X, α ◦ β, Y ). Example 4.3.12. If ◦ = &, then α ◦ β = α + β. Note that parallel ◦–composition of complete semantic links is a special case of parallel composition of named sets (Burgin, 2011). Parallel ◦–composition of complete semantic links inherits properties of the operation ◦. Proposition 5.5.4. (Commutative Law) If the operation ◦ is commutative and α ◦ β is defined, then α ◦ β = β ◦ α for any complete semantic links α and β. Proposition 5.5.5. (Associative Law) If the operation ◦ is associative and α ◦ β, α ◦ γ and β ◦ γ are defined, then (α ◦ β) ◦ γ = α ◦ (β ◦ γ) for any complete semantic links α, β and γ.
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Semantic addition of complete semantic links is not the only special case of parallel ◦–composition of complete semantic links. Operation 4.3.7. The semantic disjunction ∨ of two complete semantic links α = (X, α, Y ) and β = (X, β, Y ) is performed by taking the disjunction of the inner semantic links α and β. It means that α ∨ β = (X, α ∨ β, Y ). Example 4.3.13. If α = (X, elOf , Y ) and β = (X, ptOf , Y ), then α ∨ β = (X, elOf ∨ ptOf , Y ), which means that X is either a part of Y or an element of Y . Let us consider some properties of semantic disjunction. Proposition 5.5.4. (Commutative Law) If semantic disjunction α∨ β is defined, then α ∨ β = β ∨ α for any complete semantic links α and β. Proof is left as an exercise. Proposition 5.5.5. (Associative Law) If the semantic disjunction α ∨ β, α ∨ γ and β ∨ γ is defined, then (α ∨ β) ∨ γ = α ∨ (β ∨ γ) for any complete semantic links α, β and γ. Proof is left as an exercise. Proposition 5.5.4. (Implicative Law) If a complete semantic link α implies a complete semantic link β, then α ∨ β = α. Proof is left as an exercise. Corollary 5.5.3. (Idempotent Law) α ∨ α = α for any complete semantic link α. Proposition 5.5.4. (Distributive Law for Reversion) If the semantic addition α∨β is defined, then (α∨β)R = αR ∨β R for any complete semantic links α and β. Proof is left as an exercise. Proposition 5.5.4. (Distributive Law for Implication) If the semantic disjunction β∨γ is defined and a complete semantic link α implies
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a complete semantic link β and a complete semantic link γ, then α implies β ∨ γ. Proof is left as an exercise. Proposition 5.5.4. (Conjunction Law) If the semantic disjunction α ∨ β is defined, then the complete semantic link α implies the complete semantic link α ∨ β and the complete semantic link β implies the complete semantic link α ∨ β. Proof is left as an exercise. Corollary 5.5.3. If semantic disjunction α ∨ β and the semantic addition α + β are defined, then α + β implies α ∨ β. Taking a partial binary operation ◦ on the totality of all inner semantic links, which is defined for some pairs of complete semantic links in which the right node of the first complete semantic link is equal to the left node of the second complete semantic link, we can build the sequential ◦–composition of complete semantic links. Operation 4.3.7. The sequential ◦–composition ◦ of two complete semantic links α = (X, α, Y ) and β = (Y, β, Z) is performed by performing the operation ◦ with the inner semantic links α and β, i.e., α ◦ β = (X, α ◦ β, Z). A special case of sequential ◦–composition of complete semantic links is semantic multiplication × of complete semantic links studied in (Zhuge, 2012) and defined in the following way. Operation 4.3.7 (Zhuge, 2012). Given two complete semantic links α = (X, α, Y ) and β = (Y, β, Z), if it is possible to find semantic indicators γ1 , γ2 , γ3 , . . . , γk that connect X and Z by reasoning, then the reasoning process is called semantic multiplication, which is denoted by α × β = γ where γ = γ1 + γ2 + γ3 + · · · + γk . In turn, this defines the semantic multiplication × of two complete semantic links α = (X, α, Y ) and β = (Y, β, Z), which is equal to α × β = (X, α × β, Y ).
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Zhuge (2012) also formulates laws of semantic multiplication: (1) (2) (3) (4) (5)
α×e=α =e×α α×N =N =N ×α α × Ø = N = Ø × α and Ø × Ø = N (α + β) × γ = α × γ + β × γ and α × (β + γ) = α × β + α × γ (α × β)R = αR × β R
There are also operations with higher arity than two and integral operations with semantic links but they are studied elsewhere. Operations with semantic links and symbolic knowledge quanta play an important role in construction and functioning of semantic networks. Although people are more accustomed to operations with a fixed arity, e.g., to binary operations integral operations with knowledge quanta find various applications. For instance, as it is explained at the beginning of this chapter, a relation in a relational database is an individual knowledge quantum if such a relation describes one object and is a collective knowledge quantum when it describes several objects. Then the basic operations in relational databases — projection and selection — are examples of integral operations with knowledge quanta. Projection is a basic operation in relational databases (Codd, 1970). It is applied to rows (tuples) in a relation R from a relational database and has a set of attribute names {a1 , a2 , a3 , . . ., an } as its parameters or arguments. Projection transforms the relation R in such a way that all rows in the result are restricted to the set {a1 , a2 , a3 , . . ., an }. As projection can be applied to any number of rows (tuples) in a relation and rows are symbolic individual knowledge quanta, it is an integral operation on symbolic individual knowledge quanta. Being applied to one row, database projection coincides with the right projection of symbolic knowledge quanta. Selection, sometimes called restriction, is another basic operation in relational databases (Codd, 1970). It is applied to rows (tuples) in a relation R from a in relational database and has a relation Q between two of the attributes as its parameter or argument. Selection selects all those rows in R for which the relation Q holds for the
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values of the chosen attributes. As selection can be applied to any number of rows (tuples) in a relation and rows are symbolic individual knowledge quanta, it is an integral operation with symbolic individual knowledge quanta. One more integral operation with symbolic individual knowledge quanta is used in relational databases. It is generalized selection, which is also a parametric operation as selection but it allows one to use for selection of rows an arbitrary propositional formula. To conclude, it is necessary to remark that on one hand, binary relations, or more exactly, set-theoretical binary relations are symbolic knowledge quanta. In addition, it is possible to represent arbitrary relations as a set or list symbolic knowledge quanta. On the other hand, symbolic knowledge quanta and their systems are very convenient objects for mathematical modeling and manipulation as they form regular structures and allow many operations. However, there are some problems with their utilization as a basis structure for data mining theory. First, relations from relational databases are not exactly set-theoretical relations. Actually, they are named-set-theoretical relations because rows and columns in these relations always have some names. Usually, columns are named by attributes, while rows are named by objects information about which is stored in the database. Named relations are some kinds of named sets. So, relational approach to databases implicitly uses named sets. Second, if we want to be able to apply theoretical techniques and methods to search on the Internet, we have to take into account how data on the Internet are organized. Looking at web pages, we see that those data scarcely have the structure of a relation (a settheoretical one or even, a named-set-theoretical relation). Those data are texts in a very general sense, which are often multimedia texts with the hierarchical structure. The structure of the Internet objects is so sophisticated that they often are called unstructured objects. As a result, the relational model for these data appears inadequate. In contrast to this, named sets give very powerful means for modeling such structures. Named sets allow one to represent arbitrary multimedia texts, as well as hierarchical structures (Burgin, 2008a; Nocedal et al., 2011).
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Chapter 5
Knowledge Structure and Functioning: Macrolevel or Theory of Average Knowledge Knowledge is that which, next to virtue, truly raises one person above another. Joseph Addison
Although philosophers paid a lot of attention to knowledge of people developing logic as a tool for knowledge acquisition (cognition) and justification, the structures used in logic for these objectives were rather limited. So, the real interest to problems of knowledge structures and functioning emerged when people tried to make their computers intelligent. When researchers started to teach computers to solve problems people can solve, they found that while computers can solve some problems (mostly calculations with numbers) even better than people could do, many problems that were easy for people were not doable for computers. To overcome these limitations of computers, researchers created a scientific direction called artificial intelligence (AI). In the Handbook of Artificial Intelligence, it is possible to find the following definition of AI (Barr and Feigenbaum, 1981): “Artificial Intelligence is the part of computer science concerned with designing intelligent computer systems, that is, systems that exhibit 395
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Theory of Knowledge: Structures and Processes the characteristics we associate with intelligence in human behavior — understanding language, learning, reasoning, solving problems, and so on.”
This definition describes AI as a research area. At the same time, AI has another meaning. Many people understand AI as interactive intelligence realized by artificial means, having in mind computers as the most appropriate means for this purpose. We call it technological AI. For instance, Alan Turing (1912–1954) suggested the following test for AI (Turing, 1956). If people who communicate with a computer in one room and a human being in another room are not able to tell in which room there is a computer and in which a person is situated, then it would be possible to say that that computer with its software embodies AI. Consequently, the aim of researchers in AI has been creation of computer programs that would allow computer to demonstrate intelligent behavior. However, in scientific community in general and in AI, in particular, there is a big controversy about definitions of intelligence and what behavior, it is possible to call intelligent. Here we are not going to discuss these questions. We only want to understand how using mathematical technique, researchers have been trying to build AI. There are three main approaches to this problem. The first one is empirical based on the technological development. Its essence is verbalized as follows. Let us build more and more sophisticated computers and their software, observe their functioning, and analyze the results of our observations. If we will persist, sometimes computers will become intelligent.
This approach is represented by the famous Turing’s test for AI (Turing, 1956), as well as by knowledge engineering aimed at the development of expert systems (Giarratano and Riley, 1998). The second approach is philosophical and methodological. It is based on speculative reasoning about computers, human intelligence, and their interrelations. As an example of a problem considered in this approach, we can take the problem whether mind (or brain) is some kind of a computer or it is something much
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more complex. Different aspects of this approach were presented by Dreyfus (1973). The third approach is theoretical with essential utilization of mathematics. Its target is the construction of mathematical models for computers and their software. This provides for explication of AI capabilities through a study of these models and evaluation of those aspects that can be considered intelligent. In modern mathematics, computers, and their software are modeled by different types of abstract automata and algorithms. Consequently, the power of algorithms has been used as a measure for AI capabilities. It gives a framework within which fairly sophisticated systems of AI may be specified, designed, analyzed, and verified in a systematic rather than ad hoc manner. A coherent application of this approach would result in the most promising payoffs for practitioners. It is essential to remark that although the third approach is based on rigorous methods of mathematics, its adherents explicitly speak and write, as a rule, what computers in themselves are, while in reality they are dealing with mathematical models or even with informal images of actual computers. Consequently, they substitute computers for their models (finite automata, Turing machines, RAM, etc.) and estimate power of computers by capabilities of one or another mathematical model. However, any model gives only some approximation to a real computer. That is why, before using a model as a measurement device, it is necessary to find adequacy of the model. This adequacy is a measure of precision of the estimations that are made by means of this model. Another remark serves to attract attention of the AI researchers to the fact that evaluation of computers has to take into account their hardware, software, and utilization. What concerns hardware and software, now these components are included in the evaluated mainframe, while their utilization is often neglected. As Wing (1998) states, formal methods are used to describe only properties of hardware and software. At the same time, it is a great difference between using a computer as a calculator and simulating complex processes in atmosphere with the help of the same computer. Consequently, if a new way of utilization of computers is elaborated, it can change
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drastically their capabilities. However, we still do not have technological AI, e.g., in the form of intelligent computers, and research in the area of AI continues. Studies in this area attracted attention of many researchers to knowledge representation and modeling because it was reasonably assumed that knowledge is the base of intelligence. The main attention was to the knowledge used by people in science and everyday life. This average knowledge constitutes the macrolevel of the knowledge universe. Different models of knowledge on the macrolevel have been introduced. The most known models are relational and logical structures (cf., for example, (Thayse et al., 1988)), semantic networks (cf., for example, (Minsky, 1968; Findler, 1979; Ueno et al., 1987; Burgin and Gladun, 1989; 1990; 1990a; Sowa, 1987; 1991; Lehmann, 1992)), systems of frames (Minsky, 1974), informal and formal schemas (Anderson, 1977; Arbib, 1989; 1992; 1994; Nebel, 1994; Armbruster, 1996; Brewer, 1999; Burgin, 1973; 2005a; 2006; 2010a; Rohrer, 2006; Zhuge and Sun, 2010), scripts or formal scenarios (cf., (Schank and Abelson, 1977)), and productions (cf., for example, (Ueno et al., 1987)). Theoretical studies of these structures form the theory of average knowledge. As a result, a special area in AI called knowledge representation (KR) emerged, research in which is aimed at representing knowledge in symbolic structures to facilitate inference and construction of new knowledge items from given knowledge systems. To achieve these goals, KR research involves construction of efficient symbolic structures that represent knowledge, operations with knowledge, such as inference and knowledge integration, and relations between knowledge items. For instance, in 1980s, the predominant representational paradigms were semantic networks, frames, predicate logic, and production systems. Researchers in KR analyze how to accurately and effectively reason and process knowledge using different systems of knowledge representations. Knowledge representation involves three kinds of semantics: formal semantics, which explains correct transformation of knowledge structures; content semantics, which enables knowledge interpretation; and operational semantics, which assigns
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meaning to the processes, actions, rules, procedures, and algorithms related to this knowledge. Knowledge representation is the primary concern for AI as the goal of AI is to create a machine that is ‘truly’ intelligent, i.e., the behavior of which is similar to the behavior of intelligent human being. Developing AI, most researchers assume that intelligent behavior is based on knowledge. Therefore, it is necessary to represent knowledge in a form acceptable for a machine and then to teach the machine to use this knowledge. Similar to the concept of AI, knowledge representation has also another interpretation. Namely, it is comprehended as a definite system, e.g., logic or schemas, used for representing definite knowledge about some domain. Researchers found key characteristics of knowledge representations: • Expressivity or expressiveness of a given knowledge representation means how much knowledge it can represent. However, more expressive representations are likely to require more complex means for construction and operation. • Understandability of a given knowledge representation, e.g., of logic, means how well humans can understand knowledge in this form. Such properties as modularity and hierarchies of classes allow achieving higher understandability. • Efficiency in representing and processing knowledge means how well humans (or computers) can build knowledge items and operate with them. For instance, a given knowledge representation can provide better or worse means for knowledge acquisition or for elimination of redundant or conflicting knowledge. It is usually assumed that efficient knowledge representation supports intellectual activity and creativity. • Hardship of a given knowledge representation characterizes easiness of knowledge modifying and updating when this representation is used. It is possible to classify different knowledge representations.
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There are three substantial classes of knowledge representations, which correspond to the Existential Triad of the world considered in Chapter 2: — Abstract (structural) representations, such as formal logic or semantic networks — Material representations, such as the human brain or computer memory — Mental representations, such as concepts, ideas, or mental schemas For instance, representation of operational knowledge by algorithms and representation of representational knowledge by differential equations are abstract representations. Representation of knowledge in a book or in a database is material representation, while knowledge in the heads of people is mentally represented. There are also three stylistic classes of knowledge representations: — Formal representations — Informal representation — Semiformal representations The most widespread formal approach to knowledge representation on the macrolevel is based on utilization of logical languages and calculi, such as the propositional calculus or first-order predicate calculus. For instance, Halpern and Moses (1985) consider knowledge of an agent (a person or an artificial system) as a set of propositions. The conceptual world of such an agent consists of formulas of some logical language concerning the real world and agent’s knowledge. Systems of propositions (propositional calculi) describe actual and possible worlds in theories of Bar-Hillel and Carnap (1952; 1958), Hintikka (1970; 1971; 1973; 1973a) and many other researchers. The semi-formal methods include Minsky’s frames (1975), Schank and Abelson’s scripts (1977), and related methods for showing typical, default, or expected information. The main weakness of the semiformal methods is that they are not defined precisely. For instance, Minsky and Schank presented a variety of stimulating examples in
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their papers, but they never formulated exact definitions that completely characterized them. However, without exact definition, semiformal methods are more flexible and adaptive. Informal methods are very flexible and adaptive but not efficient for many purposes. For instance, natural languages do not allow precise representation of scientific knowledge. Informal methods are also very hard for computer processing. Natural languages form the most popular informal knowledge representations. It is interesting that similar to knowledge, knowledge representations also have the same three forms: • Descriptive, e.g., declarative, representations • Operational, e.g., instructional, representations • Representational representations Each of these types can epitomize any type of knowledge. For instance, programs written in functional and logic-based programming languages are descriptive/declarative representations of operational knowledge (cf., Section 5.1.3). Objects in object-oriented programming languages are operational representations of descriptive knowledge (cf., Section 5.1.3). Formulas that describe dynamic processes are representational representations of operational knowledge (cf., Section 5.1.2). Models of theories are representational representations of descriptive knowledge. At the same time, formal theories are descriptive representations of representational knowledge. Specification of tools used for knowledge representation determine instrumental classes of knowledge representations: — — — — — — — —
Mathematical representations, Logical representations, Scientific representations, Metaphoric representations, Linguistic representations, Digital representations, Schematic representations, Iconic representations,
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— Symbolic representations, — Algorithmic representations. There are also perceptional classes of knowledge representations such as: — Visual representations, — Vocal representations, — Tactile representations, e.g., the tactile writing system Braille. Writing about representation of descriptive knowledge and defining its task as construction of computable models for some domain, Sowa (2000) suggests it is a multidisciplinary subject, which applies theories and techniques from three other fields: 1. Logic provides formal structures and rules of inference. 2. Ontology defines the kinds of things that exist in the application domain. 3. Computation supports the applications that distinguish knowledge representation from pure philosophy. For instance, without logic, knowledge representation is vague and uncertain lacking criteria for determining whether statements are redundant or contradictory. Without ontology, the terms and symbols are ill-defined, confused, and confusing, and only computation provides means for implementing knowledge representation in computable models. 5.1. Language as a universal tool for knowledge representation Language shapes the way we think, and determines what we can think about. Benjamin Lee Whorf
Language is a highly complex phenomenon. People made many efforts to study existing languages and to create new languages for different purposes.
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There are three structural classes of languages: — Informal languages, which include natural languages, such as English, Spanish, or Chinese. — Semiformal languages, which include languages of mathematics, physics, biology, and other sciences. — Formal languages, which include logical languages and programming languages. Formal languages have formalized syntax (generating algorithms), semantics (interpretation), and utilization. Note that the concept of a formal language in linguistics, mathematics, and logic is essentially different from the concept of a formal language in the theory of algorithms, automata, and computation where a formal language is simply any subset of the set of all strings in some alphabet (cf., for example, (Hopcroft et al., 2007)). There are two basic classes of informal languages — natural languages and artificial languages, such as Esperanto. There are also three referential classes of languages: — All-purpose languages such as natural languages. — Wide-ranging languages such as languages of mathematics or philosophy. — Specialized languages such as logical languages and programming languages. Here we consider languages as a tool for knowledge representation and processing analyzing three classes of languages: natural languages, languages of mathematics and science, and algorithmic and programming languages. 5.1.1. Natural languages Do not say a little in many words but a great deal in a few. Pythagoras
The term language has two basic meanings: an abstract concept studied by linguistics and a specific linguistic system, e.g., “English”. There are various interpretations of language as a linguistic system.
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It may be understood as a formal system of signs governed by grammatical rules of combination and aimed at meaning communication. Another approach treats language as the mental faculty that allows humans to undertake linguistic behavior by learning languages and utilizing them to produce and understand utterances. Another characterization is based on the social functions of language representing language as a system of communication that enables humans to exchange verbal or symbolic utterances. Natural languages evolved naturally as a means of communication among people becoming a powerful tool for information transmission as well as for storage of information and knowledge. According to Renzl (2007), (natural) languages and meaning play a crucial role in knowledge construction, with language as a vehicle of knowing aiming at improvement of the efficiency of people interaction. When people came to the necessity of information transmission, they developed different natural languages. Various linguistic texts became symbolic units of knowledge. For instance, the sentence “This is a book about knowledge” is in English and represents knowledge of the content of this book. To be an efficient tool for communication and discourse, as well as for informing, modeling, rejection, influence, formation, and expression of ideas, languages in general and natural languages in particular, have to reflect definite reality containing enough information about its domain. Natural languages function as all-purpose tools of communication. Consequently, they are aimed at reflection of the whole world known to people. As Wittgenstein (1922) wrote, the internal structure of reality shows itself in language. On the one hand, language reflects the real world in which people live. On the other hand, it is formed under the influence of nature and social forces in the mental space of people as individuals and society as a whole. As a consequence, the structure of language reflects basic features of society and nature. For instance, the lexicon of language contains only such words that are used in society. Natural languages are linear as texts in natural languages are formed as linear sequences of words. The cause of this peculiarity is, certain characteristics of the nervous system and the mentality of a person. A similarity between language
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and the world is explicated in the structural similarity (isomorphism) between a natural language and the real world. As we know (cf., Chapter 2), the world is structured as the Existential Triad. This structure is exactly reflected in the large-scale structure of language. Indeed, in the opinion of experts in the field of linguistics and philosophy of language, the following regularity can be explicated in the history of poetic and philosophy of language: the language imperceptibly directs theoretical ideas and poetic impulses on one of the axes of it three-dimensional space, in which the first direction, which is taken, goes along semantics, then it turns to syntax, and at last, to pragmatics (Stepanov, 1985). This directly implies that the global structure of the world (the Existential Triad) is reflected in the triadic structure of language (syntax, semantics, and pragmatics), where reflection is a structural similarity in the sense of (Burgin, 2010; 2012). In this similarity, the World of Structures corresponds to syntax, which covers purely linguistic structures. The Physical World is embodied in semantics, which connects language structures and the real objects. The Mental World (as the complex essence) gives birth to the pragmatics of language as pragmatics refers to goals and intentions of people. Being a tool for communication and discourse, any language is used for information and knowledge transmission as the primary goal of language utilization since communication is information exchange, e.g., information transmission that goes from one person to another and back repeating this cycle many times. Thus, specific aspects of information transmission influence many features of language. The three-dimensional structure of language (cf., Figure 5.1) — syntax, semantics, and pragmatics — is a manifestation of this influence (Burgin, 2010). Syntax
Semantics
Pragmatics
Figure 5.1. The Language Triad
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From the pragmatics point of view, there are four types of sentences in languages that have the same structure as English, French, or Spanish: — — — —
A declarative sentence makes a statement. An interrogative sentence asks a question. An imperative sentence makes a command. Exclamations (an exclamatory sentence) expresses emotions.
For instance, the sentence “It is an apple” is declarative, the sentence “What is this?” is interrogative, the sentence “Read this book” is imperative and “Vow!” is an exclamation. This classification reflects how sentences of natural languages represent knowledge. Declarative sentences play the main role in representing descriptive assertoric knowledge. For instance, the sentence “Tomorrow it will be warm” gives knowledge about weather. Interrogative sentences principally contain descriptive erotetic knowledge. For instance, the sentence “What time is now?” expresses lack of knowledge about time. Imperative sentences convey operational knowledge. For instance, the sentence “Get up!” instructs what is necessary to do. Exclamatory sentences (exclamations) carry representational knowledge. For instance, the exclamation “Great!” reflects emotions of the speaker. Forms of linguistic expressions are correlated with the types of sentences: 1. Statements or propositions describe objects or situations and are expressed by declarative sentences. 2. Questions ask for information and are expressed by interrogative sentences. 3. Instructions or tasks describe or demand some actions and are expressed by imperative sentences. 4. Exclamations describe emotions and are expressed by exclamatory sentences.
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Besides, exploring properties of information transmission, researchers found six factors (components) of this process: • • • • • •
the addresser (sender or source); the addressee (recipient or receiver or receptor or target); the message (carrier); the context (environment); the code used in the message; contact (interaction).
These factors (components) of information transmission form elaborated structures called information triads, which are analyzed in the context of fundamental triads (Burgin, 1993b; 2010). As language is a tool for information transmission, these factors also influence functions of language. Extending Karl B¨ uhler’s Organon-Model, Roman Jakobson (1896–1982) defined six communication functions of language, according to which an effective act of verbal communication can be described (Jakobson, 1960; 1971): 1. The Referential Function describes (conveys information about) some real phenomenon, e.g., a situation, object or mental state and is usually expressed by declarative sentences, e.g., “This book is about knowledge”, performing representation of descriptive assertoric knowledge. 2. The Expressive (alternatively called Emotive or Affective) Function is usually expressed by exclamatory sentences, interjections, and other sound changes that do not alter the denotative meaning of an utterance but provide information about the Addresser’s (Speaker’s or Writer’s) internal state (feelings or emotions), e.g., “Wow, what a book!” 3. The Conative Function engages the Addressee (Recipient or receiver or receptor) directly trying to elicit some behavior (action) from the Addressee (Recipient or receiver or receptor) and is usually expressed by imperative sentences, e.g., “Bill! Please give me this book!”, performing representation of operational knowledge.
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4. The Poetic Function focuses on “the message for its own sake” (the code itself, and how it is used) performing representation of descriptive assertoric knowledge about the text (code) and is the operative function in poetry as well as in slogans. 5. The Phatic Function is utilization of language for the sake of interaction, e.g., building a relationship between both parties in a conversation or dialogue. We can observe the Phatic Function in greetings and casual discussions of the weather, particularly with strangers. It also provides the keys to start, maintain, verify, or finish the communication process with such words as “Hello?”, “Ok?”, “Hummm”, “Goodbye”, etc. 6. The Metalingual (also called Metalinguistic or Reflexive) Function is the use of language (or of code by Jakobson) to discuss or describe itself, i.e., it involves self-reference. For instance, the sentence “The previous sentence is declarative” performs metalingual function presenting knowledge about the text. Each of these six functions has a pair of associated factors, in which the first one is the source factor and the second one is the target factor: 1. The referential function is associated with the pair (message, context), e.g., as in the message “Water is a liquid”; 2. The emotive function is oriented toward the pair (message, the addresser), e.g., as in the interjections and exclamations such as “Bah!” and “Oh!”; 3. The conative function points toward the pair (message, the addressee) and is usually expressed in the form of imperatives and apostrophes; 4. The poetic function restricts its focus on the message for its own sake being associated with the pair (message, message); 5. The phatic function is associated with the pair (message, contact) assisting in establishment, prolongation or discontinuation of the contact (communication); 6. The metalingual function is associated with the pair (message, code) to establish mutual agreement on the code (for example, to provide a definition);
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All language functions are related to information transmission and reception and thus, are typical features of information flows. Namely, we have: • The referential function of an information flow conveys cognitive information about some real phenomenon. • The expressive function of an information flow conveys affective information and describes (or provides cognitive information about) feelings of the information sender. • The conative function of an information flow attempts to elicit some behavior from the addressee by conveying effective information. • The poetic function of an information flow focuses on information independent of reference. • The phatic function of an information flow builds a relationship between both parties, the sender and receiver, in a communication by conveying cognitive and affective information between both parties. • The metalingual function of an information flow reflects selfreference establishing connections of the information item inside the information flow. These parallel traits of natural languages and information flow show that information, in general, realizes the same functions as language, while information studies reflect all three dimensions of information — statistical information theory is based on information syntax, semantic information theory portrays information semantics, and pragmatic information theory portrays information pragmatics (Burgin, 2010). All considered language functions involve information transmission and reception and thus, are typical to information flow. Namely, we have: • The referential function of information flow conveys cognitive information about some real phenomenon.
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• The expressive function of information flow conveys affective information and describes (or provides cognitive information about) feelings of the information sender. • The conative function of information flow attempts to elicit some behavior from the addressee by conveying effective information. • The phatic function of information flow builds a relationship between both parties, the sender and receiver, in a communication by conveying cognitive and affective information between both parties. • The metalingual function of information flow self-references, i.e., establishes connections of the information portion to itself. • The poetic function of information flow focuses on information independent of reference. It is possible to discern different types of knowledge representation in languages: 1. Direct representation by text and linguistic expressions. 2. Relational representation by references to texts and linguistic expressions. 3. Procedural representation by derivation of texts and linguistic expressions. For instance, the sentence “It is an apple” is a direct representation, the sentence “It is possible to acquire knowledge about people from their biographies” is a relational representation, while the sentence “If you want to know about knowledge, read this book” is a procedural representation. To represent different types of knowledge, natural languages developed various means. For instance, a metaphor is a figure of speech in which an implicit comparison is made between two unlike things that actually have something in common. At the same time, another interpretation treats a metaphor as an analogy between two objects or ideas, conveyed by the use of a word instead of another. Both interpretations are unified when we understand that a metaphor gives representational knowledge and it is possible to build
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a named set representation (Burgin, 2011) of a metaphor, which is specified in Diagram (5.1) in the graphical form analogy object 1
object 2
(5.1)
or in Diagram (5.2) in the analytical form (object 1, analogy, object 2).
(5.2)
Metaphors are closely related to allegory. Allegory (from Greek: αλλoς (allos) which means other, and αγoρε´vειν, (agoreuein) which means to speak) is a figurative mode of knowledge representation conveying a meaning that is different from the literal meaning. Allegory is represented by the following diagram. expression A ↓ meaning 1
⇒
expression A ↓ . meaning 2
(5.3)
Metaphors and allegories are frequently used in literature. They are also used in linguistics and semiotics, for example, in analyzing myths (Griffin, 2006; 2008; 2009). 5.1.2. Languages of science and mathematics The book [of nature] is written in mathematical language . . . Galileo Galilei
Languages of science and mathematics have been mostly used for portrayal of operational and representational knowledge. Mathematicians and scientists have elaborated a lot of mathematical structures and theoretical scientific models for modeling, and study of a variety of natural phenomena. As mathematical language is an essential part of the majority of scientific languages, here we have our attention on the language of mathematics as a tool for knowledge representation. Mathematics was created as a tool for solving practical problems such as counting, measuring, business transactions, building calendars, and buildings. Thus, at the beginning, mathematics was the
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discipline that studied numbers and geometrical shapes/forms, consisting of two parts — arithmetic and geometry (Burton, 1997). Then mathematicians created many new mathematical fields, which expanded far beyond numbers and geometrical shapes. Such an enormous expansion brought a new vision of mathematics as a discipline. A little by little, mathematicians started to understand that mathematics is the formalized (abstract) science of structures (Lautman, 1938; Bourbaki, 1957; 1960; Burgin, 1998; 2012). For instance, Bourbaki from the first volume of the Elements in 1939 hunted for a general theory of all the ways a set can be structured in mathematics discerning between operations on the set, finitary relations among its members and infinitary relations on the set such as define convergence or topology. Bourbaki asserted that mathematics is a discipline that studies mathematical structures and only such structures (Bourbaki, 1957; 1960). As a result, Bourbaki is chiefly identified with the idea of a mathematical structure being the cornerstone of mathematics. To proceed in this direction, the leaders of the initial Bourbaki’s group Dieudonn´e and Weil led the group in codifying ways that a few kinds of structures recur throughout mathematics and explicating three basic classes of mathematical structures: order structures, algebraic structures, and topological structures. Some philosophers and other thinkers also came to the same conclusion (Piaget, 1971; Resnick, 1997; Shapiro, 1997). Numbers and geometrical shapes/forms are still mathematical objects actively studied by mathematicians but they are treated as only special kinds of structures among many others. Reflecting this peculiarity, the language of mathematics has become the primary language of formalized structures. Therefore, to understand how this language is used for knowledge representation, we need to know what a structure is. As North (2009) writes, the idea of a mathematical structure is relatively straightforward. There are number structures, geometric structures, topological structures, algebraic structures, and so forth. Mathematical structure tells us how simpler mathematical objects are organized to form more complex mathematical objects. That is why, Dieudonn´e (1970) wrote: “I do not say it [M.B., structure] was
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an original idea of Bourbaki — there is no question of Bourbaki’s containing anything original.” For instance, natural numbers are related by means of different operations, such as addition or multiplication, e.g., 1 + 2 = 3 or 7×5 = 35. These operations form an algebraic structure on the set N of all natural numbers. One more structure on the set N determines order between natural numbers. For instance, 1 is less than 2, while 7 is larger than 5. The traditional approach to a mathematical structure is represented by the following definition. Definition 5.1.1. A mathematical structure is a set with relations between its elements. However, this definition is incomplete defining only structures of the first order. In their book on set theory, the group of mathematicians who used the pseudonym Bourbaki (1960) constructed a much more sophisticated definition of a structure, which is extremely formalized involving many other mathematical constructions and concepts. Being too intricate, it were not used even by Bourbaki themselves in their other works. Moreover, the Bourbaki’s theory of structures was not general enough to apply to all the mathematical objects they needed, and it was too complicated to use when it did apply. In particular, after working with Grothendieck, the leader of the initial Bourbaki’s group Dieudonn´e acknowledged that their theory of structures “has since been superseded by that of category and functor, which includes it under a more general and convenient form” (1970). However, categories and functors are only some special kinds of structures according to the most comprehensive definition of a structure is elaborated in the general theory of structures (Burgin, 2012) and formulated below. Definition 5.1.2. A mathematical structure Q consists of parts/elements of Q and relations of Q, which form three groups: (1) relations between parts/elements of Q; (2) relations between parts/elements of Q and relations of Q; and (3) relations between relations of Q.
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Note that relations between relations of Q are also relations of Q and thus, relations between relations between relations of Q, relations between relations between relations between relations of Q and so on belong to the structure Q. As a result, we have a potentially infinite hierarchy of relations instead of only relations of the first level taken into account in the conventional definition of a mathematical structure. The new concept of structure with all its forms allowed solving the long-standing problem of understanding ideas of Plato and forms of Aristotle. Structure gives a scientific explication of both concepts (Burgin, 2012). In addition, it provides other insights. For instance, it is possible to consider Aristotle as the forefather of structural realism. Indeed, in his theory of thinking, Aristotle asserts that the realization of a form of an object in one’s mind is as real the instantiation of the corresponding form in external reality and both forms exhibit the same powers and properties and the same necessary relations to other forms. As forms are special kinds of structures (Burgin, 2012), this means that mental structures correctly represent structures in nature and society. Definition 5.1.2 comprises the concepts of a mathematical structure elaborated by Bourbaki and other mathematicians. The order of a structure Q is determined by its relations. Definition 5.1.3. If there are no relations in Q, then Q is a structure of the zero order. In this context, any set is a structure of the zero order with respect to its inner structure (cf., Chapter 6). Definition 5.1.4. (a) If Q has only relations between parts/elements of Q, then Q is a structure of the first order. (b) All relations between parts/elements of Q have the first order. With respect to its internal structure (cf., Chapter 6), any set is a structure of the first order. Any conventional structure is a structure of the first order. Definition 5.1.5. (a) Relations that involve relations of order n have order n + 1.
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(b) If Q has relations of order n, then Q is a structure of order n. Note that a structure of order n can also have order n − 1. An emphasis on mathematical structures also had a marked influence outside mathematics. For instance, Bourbaki introduced three basic types of mathematical structures (algebraic structures, order structures, and topological structures) and Piaget (1964) found a close correspondence between these types and the first operations through which a child interacts with the world. Besides, in many disciplines such as linguistics, sociology, anthropology, esthetics, economics, and psychology, the structural approach became very popular shaping the direction of research called structuralism (Piaget, 1971; Burgin, 2012). However, the traditional approach to mathematical structures has been treated as too primitive and different researchers criticized it. For instance, based on this understanding, Carter (2008) argues that a set taken by itself may not be regarded as a structure. In addition, she demonstrates that the relation that exists between the structure and the set on which this structure is defined (as a system of relations) is important for mathematical practice. In his critic of the structuralist view of mathematical objects, Parsons (1990) demonstrates that mathematical objects include relations not only to objects from the same set but also relations to other objects, which Parsons calls external relations. Thus, the more general concept of structure elaborated in (Burgin, 2010; 2011; 2012) eliminates all objections to structural understanding of mathematical objects. In particular, the general theory of structures (Burgin, 2012) discerns five types of structures: — — — — —
Internal structures Inner structures External structures Intermediate structures Outer structures
Moreover, a limited understanding of the concept of structure had existed in all fields of science, mathematics, and humanities.
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To improve this understanding, the general theory of structures was created in (Burgin, 2012). This theory demonstrates that any system has not a single structure as it has been believed before but several structures of three types — inner structures, intermediate structures, and outer structures. In addition, the new theory introduced the new complete concept of structure and its mathematical formalization. As a result, the theory of mathematical structures developed by Bourbaki (1960) became a special formalized subtheory of the general theory of structures. Moreover, it is interesting to know that in mathematics there are situations when objects with different structures are identified, that is, treated as one and the same object. For instance, the natural number 3 is considered the same as the rational number 3. However, the outer structure of the natural number 3 is the structure of the object N of all natural numbers, while the outer structure of the rational number 3 is the structure of the object Q of all rational numbers. The inner structure of the natural number 3 is the named set (X, f , 3) where X consists of (all) sets that have three elements and f connects all these sets to the symbol 3, while the inner structure of the rational number 3 is the named set (Z, r, 3) where Z consists of (all) fractions 3/1, 6/2, 9/3, . . . and f connects all these fractions to the symbol 3. This shows that mathematical objects are not structures, at least, in that oversimplified sense that is traditionally assigned to the concept of structure. It is natural to ask a question why in many cases, different structures are treated in mathematics as the same object. We can answer this question explaining that mathematics is a tool (mechanism) for simplification through a rigorous generalization and unification. Thus, when the rational number 3, the whole number 3, the integer number 3, the rational number 3, and the real number 3 are comprehended as the same object, it provides a useful unification, simplifying many cases of mathematical reasoning. In 1970 in his thesis written at Moscow State University, Burgin added a new dimension to this understanding demonstrating that mathematics is a science of abstract structures in the same sense as physics is a science of material structures because mathematics is
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similar to any natural science having the theoretical, experimental, and applied parts (cf., (Burgin, 1998)). Namely, while the domain of physics is a part of the physical world, the domain of mathematics is a part of the world of structures, which consists of structures described and formalized in the mathematical language. However, recent research shows that this is only an approximate understanding. Achieving the next level of understanding, it is natural to conclude that mathematics is a science of systems of structures although systems of structures are also structures. To grasp the essence of the new understanding, we have to uncover the difference between a (pure or abstract) structure and a system. In a structure, elements/parts do not have other properties and relations except those that belong to this structure. In a system, elements/parts may have other properties and relations. For instance, an algebraic (abstract) category is a (pure) structure, while the category of groups is a mathematical system (of structures) because its elements — groups — have inner structure. An abstract group is a (pure) structure, while the group of n × n orthogonal matrices or n × n unitary matrices is a mathematical system (of structures) because its elements — matrices — have inner structure. It is possible to read more about mathematical structures in (Bourbaki, 1948; 1957; 1960; Shapiro, 1997; Resnick, 1997; 1999; Burgin, 1998; 2011; 2012). Besides, any mathematical book introduces and studies many specific mathematical structures such as numbers, groups, spaces, triangles, graphs, formulas, operations, fields, and measures. Mathematics is intrinsically connected to logic. On one hand, the main tool of formal mathematics is reasoning, which is formalized and studied by logic. On the other hand, logics on its higher level is formalized and developed as a mathematical discipline, which is called mathematical logic (Russell, 1908; Church, 1956; Kleene, 2002). Thus, there are many structures in logic in general and in mathematical logic, in particular. Examples of logical structures are propositions, predicates, deductions rules, well-formed formulas and calculi. Mathematical logic is mostly used for working with descriptive knowledge although the area in mathematical logic called model
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theory deals with representational knowledge, while another area in mathematical logic called recursion theory studies operational knowledge. In comparison with mathematical logic, mathematics provides extensive tool for representational knowledge in the form of mathematical models, which give knowledge about their domains. May be the most well known, popular, and useful mathematical models are differential equations, which represent knowledge about a host of diverse phenomena in nature, technology, economy, and society. Many fundamental laws of physics and chemistry are presented in the form of differential equations. Biologists, physiologists, and economists make use of differential equations modeling behavior of complex systems. Let us consider some examples. Physicists use the following models having the form of differential equations: • Maxwell equations form the foundation of classical electrodynamics, classical optics, and electric circuits underlying modern electrical and communications technologies (Maxwell, 1865). • The Boltzman equation describes behavior of gases. • The Poisson equation has many applications in electrostatics, mechanical engineering, and theoretical physics. • The Poisson–Boltzmann equation finds diverse applications in physiology, polymer science, theory of biomolecular systems, and semiconductor technology, describing the distribution of the electric potential in solution or electron interactions in a semiconductor (Fogolari et al., 1999; Gruziel et al., 2008). • The Einstein field equations (also known as Einstein equations) form the base of general theory of relativity that describes the gravitational interactions as a result of spacetime being curved by matter and energy (Einstein, 1915). • The Schr¨ odinger equation describes how the quantum state of a physical system changes with time. • Equations of motion, which include the famous Newton’s second law of motion, describe how the position of a physical system changes with time.
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• The Korteweg–de Vries equation first introduced by Boussinesq (1877) and rediscovered by Diederik Korteweg and Gustav de Vries (1895) is a mathematical model of waves on shallow water surfaces describing the string in the Fermi–Pasta–Ulam problem in the continuum limit, shallow-water waves with weakly nonlinear restoring forces, long internal waves in a density-stratified ocean, ion acoustic waves in a plasma, and acoustic waves on a crystal lattice. • Hamiltonian mechanics is based on Hamiltonian equations. • Lagrangian mechanics is based on Lagrangian equations. Chemists use the following models having the form of differential equations: • The rate equation describes links between the reaction rate with concentrations or pressures of reactants and constant parameters such as normally rate coefficients and partial reaction orders (Connors, 1991). Biologists and physiologists use the following models having the form of differential equations: • The Lotka–Volterra equations describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. • The Verhulst equation describes biological population growth. • The Bertalanffy growth equation describes individual growth. • The Hodgkin–Huxley equations give a mathematical model that describes how action potentials in neurons are initiated and propagated. Economists use the following models having the form of differential equations: • The Black–Scholes equation (also known as Black–Scholes–Merton equation) is a mathematical model of a financial market with certain derivative investment instruments. • The Solow–Swan equation is an exogenous growth model, an economic model of long-run economic growth.
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• The Sethi equation is an advertising model describing a salesadvertising dynamics (Sethi, 1983). Such mathematical structures as groups represent important knowledge in quantum physics (cf., Appendix for a definition of a group). Physicists use the subtheory of group theory known as the theory of group representations, which allows them, for example, to classify all the observed spectroscopic states of atoms and molecules (cf., for example, (Wigner, 1959; Schonland, 1965; Schensted, 1967)). Another mathematical structure called a Lie algebra is also used as a practical model in particle physics (Georgi, 1999). One more mathematical structure called a manifold provides means for building different kinds of representational knowledge in physics (Choquet-Bruhat and DeWitt-Morette, 1982) Such mathematical constructions as abstract automata and algorithms represent operational knowledge in computer science and network technology (Burgin, 2005). Operational knowledge is also represented by such mathematical structures as process algebras and process calculi. In computer science, the algebraic approach is a popular and effective way to study processes in multicomponent systems, such as the Internet, multicore computers and clusters of multi-core computers. To efficiently deal with such problems computer science developed a concurrency theory, which extensively and successfully utilizes algebraic structures and techniques. Researchers built many powerful models of concurrency, which, to mention but a few, include: — — — — — — — —
Petri nets (Petri, 1962), communicating sequential processes (CSP) (Hoare, 1985), calculus of communicating systems (CCS) (Milner, 1989), event-signal-process model (ESP) (Lee and SangiovanniVincentelli, 1996), view-centric reasoning (VCR) model (Smith, 2000), extended view-centric reasoning (EVCR) (Burgin and Smith, 2006), event-action-process model (EAP) (Burgin and Smith, 2006), process networks (Kahn, 1974),
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— — — — — —
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process algebras (Baeten, 1990; Baeten and Bergstra, 1991), dataflow process networks (Lee and Parks, 1995), discrete event simulators (Fishman, 1978), synchronization trees (Winskel, 1985), labeled transition systems (Sassone et al., 1996), grid automata (Burgin, 2003).
Looking at science, we see a big variety of scientific languages. Although some hold that science is an applied mathematics (cf., (Grene, 1974)), this ideal model for science is not adequate and each scientific discipline has its own language, which often includes a mathematical language. While science has been developing with its scientific languages, philosophers of science tried to explore and explain these languages. Usually the beginning of this process is ascribed to the beginning of the 20th century when logical positivists such as Friedrich Albert Moritz Schlick (1882–1936), Otto Neurath (1882–1945), Herbert Feigl (1902–1988) and Rudolf Carnap (1891–1970) started their exploration of science with the emphasis on scientific languages. In the picture of science developed by logical positivists, the language of a scientific theory consists of a set of symbols, which form the alphabet and include terms of the theory, and of the rules to generate formulas, which are correct with respect to syntax and called well-formed formulas (wff). Each scientific theory has two types of terms — logical/ mathematical and non-logical. It is assumed that the set of logical/mathematical terms of a scientific theory includes logical symbols, e.g., connectives and quantifiers, and mathematical expressions and objects, e.g., numbers, derivatives, and integrals. At the same time, non-logical terms of a scientific theory belong to two classes — observational and theoretical terms. They are used for denoting (naming) physical objects, their properties and relations, such as white, warm, longer than, electron, electromagnetic field, or quark. Formulas and expressions of a scientific theory, as well as the corresponding statements, are divided into five classes: (i) Logical and mathematical expressions (statements), which do not contain non-logical and non-mathematical terms.
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(ii) Purely observational expressions (statements), which contain observational terms but not theoretical terms . (iii) Purely theoretical expressions (statements), which contain theoretical terms but not observational terms. (iv) Expressions (statements) that contain both theoretical and observational terms. (v) Descriptions or rules of correspondence between observational and theoretical terms, which contain both observational and theoretical terms. However, this picture of scientific languages is oversimplified. For instance, science often uses metaphors as representational knowledge because, as MacCormac writes, without them it would be impossible to pose a new hypothesis intelligibly (MacCormac, 1976). Some scientific metaphors are reflected in scientific terms. For instance, only vectors grow in vector fields, which are abstract mathematical structures useful for physics. Groups in mathematics are essentially different from groups of people and abstract rings in algebra have nothing in common with rings, which people are wearing. We can take the planetary model of an atom introduced by Ernest Rutherford (1871–1937) as an example of a more complex metaphor in science. In this model, the Solar System is used as a metaphor for explaining the structure of an atom giving a picture of a few planet-like electrons rotating around the nucleus of the atom. Although later, it was demonstrated that this model cannot be adequate; this metaphor suggested by Rutherford and developed by Bohr is still utilized for educational purposes and as a symbol for atoms and atomic energy. However, scientific metaphors play an important role not only in explanation of scientific results but also in obtaining these results, that is, in scientific cognition (Leatherdale, 1974; Rothbart, 1997). Metaphors give creative insights for understanding unknown phenomena, as well as generate the basis for concept formation. Examples are black holes and white dwarfs, which are popular objects in cosmology. A formal theory of metaphor based on the theory of abstract properties presented in Section 5.3 is developed in (Burgin and Rothbart, 1998).
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To conclude, we consider the role of mathematics in human cognition. As many know, physicists are trying to create a theory of everything. However, this is impossible because physics studies only physical, that is, material systems. Therefore, in the best case, physicists will be able to create a theory of everything on the level of physics. However, for example, people are studied not only on this level but primarily on the biological, psychological, and social levels. This shows that a physical theory of everything can describe the world only from its material side. At the same time, we already know that everything has a structure and, as a rule, not even one. Thus, the general theory of structures is a theory of everything as its studies everything. Mathematics, for example, is such a general theory of formal structures. That is why mathematics studies everything by formalized methods and techniques, being a theory of everything on its structural level and providing structural models (as units of representational knowledge) for a diversity of various phenomena in nature, society, and technology.
5.1.3. Algorithmic and programming languages By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Alfred North Whitehead
Creation of electronic computers instigated creation of algorithmic and programming languages for representation of operational knowledge. If for a long time operational knowledge had been expressed informally by means of natural languages and semi-formally by means of mathematical languages, computers demanded more exact description of actions and operations. Besides, mathematics also required exact sufficiently general representation of operational knowledge. To satisfy these needs, the general concept of algorithm was introduced and formalized. There are different approaches to defining algorithm. As an informal notion, algorithm has a variety of interpretations and definitions.
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For instance, a popular in mathematics point of view on algorithm is presented by Rogers (1987): Algorithm is a clerical (i.e., deterministic, bookkeeping) procedure which can be applied to any of a certain class of symbolic inputs and which will eventually yield, for each such input, a corresponding output.
In this definition, “procedure” is interpreted as a system of rules that are arranged in a logical order, and each rule describes a specific action, while an algorithm is treated as a kind of procedure. “Clerical” or “bookkeeping” means that it is possible to perform according to these rules in a mechanical way, so that a device is actually able to carry out these actions. Unfortunately, this definition is incomplete because algorithms used by people do not produce an output for each relevant input and it is even proved that there is no algorithm discerning when it is possible to obtain such an output. Donald Knuth (1971), a well-known computer scientist, defines algorithm as follows: An algorithm is a finite, definite, effective procedure, with some output.
In this definition, “finite” means that it has a finite description and there must be an end to the work of an algorithm within a reasonable time. “Definite” means that it is precisely definable in clearly understood terms, no “pinch of salt” type vagaries, or possible ambiguities. “Effective” means that a device is actually able to carry out actions prescribed by an algorithm. The concluding words “with some output” allow different interpretations. Some understand the condition to give output assuming that an algorithm always gives a result. As we have seen, this is not true for all practical algorithms. It is also possible to comprehend these words as the condition that algorithm in some cases gives the output. This better correlates with computing practice and theory, which came to a broader understanding, according to which algorithms are aimed at producing results, but in some cases cannot do this. This shows that Knuth’s definition is too imprecise.
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In the Free Online Dictionary of Computing (http://foldoc. doc.ic.ac.uk/) algorithm is defined as a detailed sequence of actions to perform to accomplish some task. A more exact definition consistent with the computational practice, is given in (Burgin, 2005), where it is written: An algorithm is an unambiguous (definite) and adequately simple to follow (effective) prescription (organized set of instructions or rules) for deriving necessary results from given inputs (initial conditions).
Programmers, mathematicians, and computer scientists have developed a variety of special languages for representing various algorithms. An algorithmic language is a formal language oriented at describing algorithms in a formal way. It is possible to separate two classes of algorithmic languages: abstract or theoretical algorithmic languages and programming languages. Languages from the first class are used in the theory of algorithms, automata, and computation for mathematical representation of algorithms (cf., for example, (Hopcroft et al., 2007; Burgin, 2005)). Examples of such languages are: — — — — —
The The The The The
language language language language language
of of of of of
Turing machines; neural networks; finite automata; formal grammars; automata with structured memory;
A programming language is formal language designed for describing computational processes in a compressed, exact form or, equivalently, for writing down algorithms to be executed by computers. To this day, hundreds of programming languages have been created but the majority of them are not used for practical programming. There are two basic classes of programming languages: high-level programming languages, which are not related to any specific computer, and low-level or machine-oriented programming languages,
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which take the specific features of a given type of computers, such as instruction set or addressing modes, into account. In turn, high-level programming languages are usually divided into two classes: problem-oriented and universal programming languages. Universal programming languages also include several subclasses. There are several main types of programming languages: — Array programming languages, also called vector or multidimensional programming languages, utilize operations with vectors, matrices, and higher-dimensional arrays. — Assembly programming languages use machine instructions coded in a form understandable by people. — Message-passing programming languages are designed for programming concurrent processes. Dataflow programming languages organize computations with respect to the flow of data often specifying the process in a visual form. — Decision tables as programming languages express the logic of computation in the form of a decision table — Declarative programming languages, which include functional and logic-based programming languages, describe a problem rather than defining the process of its solution. — Imperative programming languages construct programs as serial orders (imperatives) given to a computer. — Functional programming languages define programs and subroutines as mathematical functions. — Semantic programming languages represent all programs and data as one distributed interconnected graph with the ‘nod-edge’ as the basic building block. — Logic-based programming languages specify a set of attributes that a solution must have, rather than prescribing the process to obtain a solution. — Procedural programming languages describe the procedure of computation with programs composed of one or more units or modules that encode procedures. — Object-oriented programming languages combine data, and methods of manipulating the data in a single unit called an object.
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— Symbolic programming languages describe programs that are able to manipulate formulas and program components as data. Similar to natural languages, a programming language is constructed over an alphabet of basic symbols, in which programs are written down in the form of a hierarchical system of grammatical elements, between which relations are given (similarly to the words, phrases, and sentences in a natural language, whose connections are given by syntactic rules). The lowest level elements, formed by chains of basic symbols, are called lexemes, or lexical units. For the lexemes occurring in a program the class to which they belong is defined, and for certain classes of lexemes (e.g., identifiers) also their scope — some uniquely identifiable part of the program to which all occurrences of a given lexeme belong (a block). Exactly one occurrence of such a lexeme is said to be defining; the other occurrences of the lexeme in its scope are called applied. The further levels of elements of an algorithmic language are formed by notions, or non-terminals. The relation that may hold between the notions of an algorithmic language is that of being a (direct) constituent of (i.e., an immediate constituting part), while the individual constituents of a given notion are related to each other by concatenation (textual sequence). The transitive closure of the constituent relation uniquely assigns to each notion some subword of the text of the program, which is said to be the (terminal) production of this notion. There is one initial notion, the production of which is the entire program text. A tree whose root is the initial notion, whose terminal vertices (leafs) are lexemes and basic symbols, whose internal vertices are concepts and whose branches are constituent relations, is called a production or syntax tree of a program. The construction of such a tree is known as the syntactic analysis or parsing of a program. Sometimes, practitioners do not make a specific distinction between the concepts of “programming language” and “algorithmic language” For instance, one of the first programming languages was called Algorithmic Language (Algol). To conclude, it is important to understand that now due to proliferation of computers, the most popular way of operational
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(procedural) knowledge representation are computer programs and algorithms. It is possible to read more about the classical approach to algorithms and computer programs, for example, in (Machtey and Young, 1978; Shoenfield, 2001; Knuth, 1997; Berlinski, 2001; Manin, 1991) and about a more advanced approach to algorithms in (Burgin, 2005). 5.2. Logic as a tool for knowledge representation and production The logic of the world is prior to all truth and falsehood. Ludwig Wittgenstein
As a separate discipline, logic has origins in philosophy and goes back, at least, to Aristotle (384–322 B.C.E.). Therefore, it is natural that the word logic originates from the Greek word logos, which has acquired three meanings in the process of its development. According to one of them, logic is a specific research (cognitive) field, while the other interpretation exposes logic as a definite structure, which can be formal or informal. The third (mundane) understanding regards logic as a way of reasoning (cf., for example, (Agre, 2000; McCarthy et al., 2004; Smehov, 1987)). As a field, logic exists in philosophy and mathematics having the form of the discipline of formal principles of reasoning and correct inference. As per modern understanding, logic as a discipline is the study of correct reasoning by structural (often formal) methods. Some understand logic as an analysis of laws of thought. The term logic as a formal (mathematical) structure has two meanings. On one hand, a logic L consists of a logical language L together with a deductive system (logical calculus) and/or truth semantics. Sometimes a model-theoretic semantics (interpretation) is also included. The language corresponds to a part of a natural language like English or Spanish. The deductive system (logical calculus) is developed to record, capture, and codify, the inferences of which are correct for the given language, and the truth semantics
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is built to reflect, capture, and codify the meanings, in the form of truth-conditions, or possible truth conditions, for at least part of the language L. It is also called a logical semantics. On the other hand, there is an opinion that a logic L consists only of inference/deduction rules and rules of interpretation. What is done according to these rules is called logical, while what is in dissonance with these rules got the name illogical. However, there are different systems of logical rules and what is called logical with respect to one system may be illogical with respect to another system of logical rules. Logic as a way of reasoning emerged millennia ago. It was the first stage of logic as a discipline. We can see argumentation and inference in such ancient texts as the Bible and Indian Vedas. The second stage of logic as a discipline is characterized by explicit formulation of rules for logical reasoning. The third stage was formation of logic as a separate discipline, which happened in ancient Greece and ancient India. Although the roots of contemporary logic are in the logic of ancient Greeks and especially in the logic developed by Aristotle and called syllogistics, even before Aristotle, the development of logics started in ancient India. The first known Indian logician Medhatithi Gautama (ca. 6th century B.C.E.) founded the anviksiki school of logic. The classical text Mahabharata, originated around the 5th century B.C.E., refers to the anviksiki and tarka schools of logic. Another Indian logician P¯ an.ini, who lived around 5th century B.C.E., developed a form of logic in the process of formalization of Sanskrit grammar. In his work Arthashastra, philosopher Chanakya (ca. 350–283 B.C.E.) described logic as an independent field of inquiry anviksiki (Ganeri, 2001). However, some authors conceive that in India, logic was never developed as a distinct discipline but was always embedded in epistemology (Matilal, 1998). Jain logic developed and flourished from 6th century B.C.E. to th 17 century C.E. making its own unique contribution to Indian logic and taking logic as the foundation for the theory of knowledge. According to Jain theory of knowledge, ultimate principles should always be logical and no principle can be devoid of logic or reason.
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In essence, the development of logic in India continued from ancient times to our days. It is possible to mention the analysis of logical inference by Gotama (ca. 2nd century C.E.), who founded of the Nyaya school of Hindu classical philosophy, and the tetralemma of Nagarjuna (ca. 2nd century C.E.). Indian Buddhist logic (called Pramana) flourished from about 5th century C.E. up to 13th century C.E. The three main authors of Buddhist logic are Vasubandhu (between 4th and 8th centuries C.E.), Dign¯ aga (480–540 C.E.), and Dharmak¯ırti (600–660 C.E.). Indian logic had many results that surpassed European logic not only at that time but even much later. For instance, Buddhist logic used four forms of predication: 1. S is P , e.g., “a square is a rectangle” or “English is a language”. 2. S is not P , e.g., “a square is not a circle” or “Poland is not a language”. 3. S is and is not P , e.g., if a ball is partially green and partially yellow, we can say “the ball is green and is not green” or “there is and is no mathematical straight line”. 4. S neither is nor is not P , e.g., if a ball is partially green and partially yellow, “the ball neither is green nor is not green” or “the world of ideas neither is real nor it is not real”. One more example is syllogism developed in Indian, or more exactly, the Nyaya, logic and called Prayoga. It has five parts (Rao, 1998): 1. 2. 3. 4. 5.
Pratij˜ n¯ a (proposition or hypothesis); Hetu (reason); Ud¯ aharana or Drstanta (example); Upanaya (application); Nigamana (conclusion). Here is an example of application of the Nyaya syllogism.
1. The proposition (hypothesis): the house is on fire; 2. The reason: the smoke is around the house; 3. The example: fire is accompanied by smoke, as in the kitchen;
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4. The application: as in kitchen, so for the house; 5. The conclusion: therefore, the house is on fire. At the same time, Aristotle’s syllogism, which was later transformed into the deduction rule modus ponens, has only three parts: 1. Major premise; 2. Minor premise; 3. Conclusion. Here is an example of application of Aristotle’s syllogism. 1. Major premise: People who lived in Athens spoke Greek. 2. Minor premise: Aristotle lived in Athens. 3. Conclusion: Aristotle spoke Greek. In China, Mo Di (ca. 470–ca. 391 B.C.E.), also called Mozi or Mo Tzu, who lived shortly after Confucius during the Hundred Schools of Thought period, founded the Mohist school of philosophy, which, in particular, studied problems of valid inference and rules for correct conclusions. Mo Di introduced the “three-prong method” for testing the truth or falsehood of statements. His followers later expanded on this approach founding the School of Names, representatives of which are often called logicians. As its name shows, philosophers from this classical direction in Chinese philosophy were interested in language, disputation, and metaphysics, focusing on names (ming in Chineese, which also means words) and their relation to “stuff” (shi, objects, events, situations). These thinkers lived mostly in the 2nd century B.C.E. So, in essence, Chinese as many other cultures used various logical forms in their reasoning and studied some problems of logic, e.g., related to names, without developing explicit logical systems (Weimin, 2009). In Europe, the development of logic continued after Aristotle created syllogistics. For instance, Theophrastus of Eresus (ca. 371–ca. 286 B.C.E.), who was Aristotle’s successor as the head of his school at Athens, studied hypothetical syllogisms. Stoic philosophers made an essential contribution to logic starting with Euclid of Megara (ca
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435–ca 365 B.C.E.), a pupil of Socrates and slightly older contemporary of Plato. The ancient Stoics developed modality in logic, built a theory of the material conditionals, studied relations between and truth introducing tense operators (cf., (Mates, 1953)). Besides, the first employment in the Western philosophical tradition of the notion of proposition, in roughly modern sense, is also found in the writings of the Stoics (McGrath, 2014). In the 3rd century B.C.E., Zeno and his followers distinguished the material aspects of texts from their content, which was called lekta. In turn, lekta included axiomata, or the meanings of declarative sentences. For the Stoics, only axiomata, and not the texts used to represent them, were able to be true or false. In spite of its bright beginning, the European logic stopped its development for many centuries, while the mainstream of logic as a discipline shifted to the East, where an interesting development of logic could be attributed to Islamic thinkers. At first, some forms of analogical reasoning, inductive reasoning and categorical syllogism were introduced in Islamic jurisprudence, law) and theology. This process started in the 7th century C.E. before the Arabic translations of Aristotle. Later Arabs incorporated mathematics, logic and philosophy of ancient Greeks translating their works into Arabic and started developing their own logic. The great Islamic polymath Ab¯ u Al¯ı al-Husayn ibn AbdAll¯ ah ibn Al-Hasan ibn Ali ibn S¯ın¯ a (980– 1037), who was known as Avicenna (Latinate form of Ibn-S¯ın¯ a) in Europe, created his own logic known as “Avicennian logic” as an alternative to Aristotelian logic. In addition, he developed a theory of definition and classification, elaborated an original theory on “temporal-modal” syllogism with temporal modifiers such as “at all times”, “at most times”, and “at some time” and introduced quantification of the predicates in categorical propositions. By the 12th century, Avicennian logic was predominantly used in the Islamic world replacing Aristotelian syllogistics. It is interesting to know that Ibn S¯ın¯ a was a prolific writer as by historical evidence, he wrote around 450 works, from which 240 has survived, including 150 works in philosophy and 40 works in medicine.
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Other Islamic thinkers also developed logic. For instance, Ibn Hazm (994–1064) wrote the Scope of Logic, in which he discussed sense perception as a source of knowledge. Al-Ghazali (1058–1111) applied Avicennian logic to Islamic theology Kalam. Fakhr al-Din al-Razi (b. 1149) developed a form of inductive logic. Ibn Taymiyyah (1263–1328) argued that inductive reasoning was more useful than the syllogism. Judaic logic as a form of grounded reasoning is already present in The Bible. Later advanced hermeneutic and inference called midot rules were developed and extensively used by the authors of Talmud (2nd –3rd century C.E.).These rules went much further than the syllogistics of Aristotle but were informal. Historians discern three basic sets of rules: the seven rules of Hillel, the thirteen rules of Rabbi Ishmael b. Elisha and the thirty two rules of Rabbi Eliezer ben Jose HaGelili (Sion, 2010; Klein, 2013). Rabbi Akiva also introduced some rules of inference (exigesis). Hillel, also called Hillel HaGadol, or Hillel HaZaken, or Hillel HaBavli, (ca. 60 B.C.E.–20 C.E.) was a famous Jewish religious leader and sage, whose views were recorded in the Talmud and many other books. According to some sources, he lived approximately from 110 B.C.E. to 10 C.E., while other sources give another period — approximately from 60 B.C.E. to 20 C.E., Jewish scholars used the Seven Rules of Hillel long before Hillel was born but he was the first to write them down. That is why they have his name. Here is the contemporary understanding of the Seven Rules of Hillel: 1. Kal wa-homer (which means Light and heavy in English): It is possible to apply a law stated for one case to another case when the second case gives even more reasons for application of this law. This situation is often indicated by the phrase “how much more. . .” The Rabbinical sources describe two forms of this rule: • Kal wa-homer meforash when the argument appears explicitly. • Kal wa-homer satum when the argument is only implied.
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2. Gezerah shawah (which means Equivalence of expressions in English): It is possible to apply the same rules (considerations) to two separate texts when they are analogical based on a similar phrase, word or root, e.g., when the same words describe two separate cases. 3. Binyan ab mi-katub ehad (which is translated as Building up a “family” from a single text): If several passages have a common characteristic, then it is possible to apply a law stated for one of them to all of them. 4. Binyan ab mi-shene ketubim (which is translated as Building up a “family” from two or more texts): When a law is deduced by comparing two passages, then it is possible to apply this law to other passages that have characteristics common with the two initial passages. 5. Kelal u-Perat (which is translated as The general and the particular): A general principle may be restricted by a particularization of it in another passage — or, conversely, a particular rule may be extended into a general principle. 6. Ka-yoze bo mi-makom aher (which means Analogy made from another passage): Application of laws may seem to conflict in two passages and to resolve the contradiction, these passages compared to a third one, which has general though not necessarily verbal similarity to them. 7. Davar ha-lamed me-’inyano (which means Explanation obtained from context): The total context, not just the isolated statement must be considered for an accurate inference (exegesis). Rabbi Ishmael “Ba’al HaBaraita” or Ishmael ben Elisha (90-135) was a rabbinic sage, whose views were recorded in the Talmud and who extended the seven rules of Hillel obtaining what is now called
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the thirteen rules of Rabbi Ishmael: 1. It is possible to apply a law stated for one case to another case when the second case gives even more reasons for application of this law. This rule is actually the same as the first rule of Hillel and is also called Kal wa-homer in Hebrew. 2. If the same words and expressions are used in two parts of the text, then what is possible to apply to the situation described in one part, it is also possible to apply to the situation described in the other part. This rule is actually the same as the second rule of Hillel and is called Gezerah shawah in Hebrew. 3. Rules deduced from a single passage of the text or from two passages. This rule is a combination of the third and fourth rules of Hillel and is called Binyan av in Hebrew. 4. Application of a general law is clarified (e.g., explained and supported) by its particular case. This rule is similar to the fifth rule of Hillel and is called Kelal u-Perat in Hebrew. However, there is another rendering of this rule: If at first, a law is given in a general form and after this, in its particular form, then this law is applied only in the particular form. 5. It is possible to extend a particular law to a general law if the general case goes after the particular case. This rule is also similar to the fifth rule of Hillel and is called u-Perat u-Kelal in Hebrew. 6. When a law is applied to a general case, then applied to a particular case, and then again applied to the general case, it follows that it is possible to apply this law only for a part of a general case that is similar to the particular case. This rule is called Kelal u-Perat u-Kelal in Hebrew. 7. Elucidation of a general application of a law is performed by a particular application and clarification of the particular application is performed by the general application. Utilization of this rule excludes employment of the three previous rules. This rule is called Kelal she-hu tzarik le-Perat u-Perat she-hutzarik le-Kelal
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8.
9.
10.
11.
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in Hebrew, which means the general requires the particular and the particular requires the general. When a particular instruction implied by a general law is considered separately based on a special regulation, then the general law has to be applied according to the application of the particular instruction. This rule is called cal davar shehayah bechlal v’yatza in Hebrew. When a general law is applied to a special case without some conditions used in this law, then in this application, the omitted conditions are not taken into account. This rule is called v’yatza leton . . . ch’enyon in Hebrew. When a general law is applied to a special case with additional conditions but without some conditions used in this law, then in this application, the new conditions are used, while the omitted conditions are not taken into account. This rule is called v’yatza leton . . . shelo ch’enyon in Hebrew. When a particular case is excluded from a general law, then in future, the general law is applied to this case only when it is directly prescribed. This rule is called v’yatza ledon in Hebrew. Inference (exegesis) is made either from the context or from a later reference in the passage containing this law. This rule is an extension of the seventh rule of Hillel and is called Davar ha-lamed me-’inyano ve davar ha-lamed mi-sofo in Hebrew. When application of laws may seem to conflict in two passages, then this contradiction must be solved by reference to a third passage, which has general though not necessarily verbal similarity to the first two. This rule is the same as the sixth rule of Hillel and is called Ka-yozebo mi-makomaher in Hebrew.
Eliezer ben Jose, also called Eliezer ben Yose HaGelili, was a renown Jewish rabbi, who lived in Judea in the 2nd century, was a student of the famous Rabbi Akiva (ca. 50–135) and whose views were recorded in the Talmud. In particular, Rabbi Eliezer ben Jose elaborated the thirty two rules of Rabbi Eliezer intended for inference of haggadic interpretation.
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As we can see, inference rules of Jewish sages include deduction and abduction, e.g., analogy, shaping a semiformal operational logical calculus, and implying non-monotonicity of inference in the case of inconsistent knowledge. This shows that parallel to the logic of propositions, operational logic has been developing in hermeneutics and legal area. It is necessary to remark that in addition to non-conventional reasoning, Jewish logic essentially employed not only logically consistent calculi but also logical varieties (cf., Section 3.1) representing opinions of different sages. As these opinions often contradicted one another, a logical formalization of such a fundamental text as Talmud, which is the focal treatise of Rabbinic Judaism, was possible exclusively in the form of a logical variety. From the perspective of traditional logic, Talmud is a collection of contradictions, and only elaboration of new logical constructions — logical varieties, prevarieties and quasi-varieties — makes it possible to represent Talmud in a logically coherent way. In spite of the highly developed informal logic, which existed in Jewish tradition, the first authored by a Jew work specifically on logic in the classical understanding appeared in the 12th century. It was the “Makalah fi Sana’at al-Mantik” written by the great Jewish sage Moshe ben Maimon (1135–1204), also known as Maimonides or Rambam or M¯ us¯ a ibn Maym¯ un. Later (in 1319) another great Jewish philosopher and mathematician Levi ben Gerson (1288-1344), mostly known as Gersonides or the Ralbag, wrote a logical tract Sefer Ha-heqesh Ha-yashar (On Valid Syllogisms), in which he examined problems associated with Aristotle’s modal logic and which was translated into Latin without Gersonides’ name attached to it (Manekin,1992; Simonson, 2000; Rudavsky, 2015). In Europe, the study and then the further development of logic resumed after the Dark Ages. The first known philosopher who contributed to logic was Peter Abelard (1079–1142). In his works, he discussed problems of conversion, opposition, quantity, quality, tense logic, a reduction of de dicto to de re modality, and some others. Abelard also clearly formulated several semantic principles such
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as the bi-conditional for the theory of truth, which was introduced by Tarski in the 20th century. However, Abelard rejected this approach. Philosophers think that the most important contribution to logic made by Abelard was the clear formulation of a pair of relevant criteria for logical consequences. Later such philosophers as William of Ockham (ca. 1287–1347), Jean Buridan (ca. 1295–after 1358), and Albert of Saxony (ca. 1320– 1390) developed a supposition theory, which studied how predicates (e.g., “being a cat”) range over a domain of individuals (e.g., the set of all cats). Buridan also elaborated a theory of consequences based on a synthesis of entailments and inference rules. In his investigation of syllogistics, Buridan suggested some kind of completeness proofs. Besides, Jean Buridan, William of Ockham and Albert of Saxony built the theory of consequences, which employed hypothetical, conditional propositions, i.e., two propositions connected by the term ‘if . . . , then’. In particular, Ockham made a clear distinction between material consequences similar to the modern material implication and formal consequences similar to the modern logical implication. Albert of Saxony also examined various sentences that were difficult for interpretation due to the presence of words (terms) that, according to medieval logicians, did not have a proper and determinate signification but only modified the signification of the other words (terms) in the propositions where they appeared. In his analysis of epistemic verbs and infinity, Albert explained that a proposition had its own signification signifying a “mode of a thing” distinctive from signification of its terms. Based on this approach, he analyzed paradoxes of self-reference demonstrating that since every proposition, by its very form, signified that it was true, an “insoluble” proposition would turn out to be false because it would signify at once both that it was true and that it was false. After this, logic was successfully developed in Europe. Philosophers wrote treatises and published books on logic. Traditional logic started with Antoine Arnauld’s and Pierre Nicole’s Logic, or the Art of Thinking, better known as the Port-Royal Logic (Arnauld and
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Nicole, 1683). It was probably the most influential work on logic in England until the 19th century. In essence, almost all outstanding European philosophers contributed to logic. Approximately, at the same time, John Poinsot (1589–1644) published his book Logic, Francisco Suarez (1548–1617) published his book Metaphysical Disputations and Giovanni Girolamo Saccheri (1667–1733) published his book Logica Demonstrativa. Gottfried Wilhelm Leibniz (1646–1716) had many revolutionary ideas in logic developed mostly between 1670 and 1690 although the majority of them were not published during his life. The central aim of Leibniz in logic was to extend the traditional syllogistic to a Universal Calculus, in which “all mistakes in reasoning will at once show up in a wrong combination of characters, and therefore the application of the characteristic script provides a means to discover the mistake in a disputed point like in every other calculation”. Although researchers found several drafts of candidates for such a calculus, none of these writings was eventually sent to press. Philosophers divide logical works of Leibniz into four directions: 1. 2. 3. 4.
Studies in the theory of the syllogism. Works on the Universal Calculus. Development of propositional logic. Advancement of modal logic.
When separate writings and fragments of Leibniz were joined together, researchers found an impressive system of four calculi: • The algebra of concepts LC , which is deductively equivalent to the Boolean algebra of sets; • The quantificational system LQ with “indefinite concept” functions playing the role of quantifiers ranging over concepts; • A propositional calculus of strict implication obtained from LC by the strict analogy between the containment relation between concepts and the deduction relation between propositions; • The so-called “Plus-Minus-Calculus” as abstract system of “real addition” and “subtraction” in the sense of Leibniz.
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Thus, we can see that although logic was developed in different countries, the mainstream of contemporary logic has been formed under the influence of the classical Greek logic. Traditionally contemporary logic as a discipline is divided into two big areas: informal logic and formal logic. Informal logic studies natural language argumentation, separating errors in reasoning, which are called fallacies. Different types of fallacies have been found and studied. For instance, A Red Herring is a fallacy in which an irrelevant topic is presented in order to divert attention from the original issue. Here are some examples of informal logics: — — — — — — — — — —
Popper’s logic of scientific discovery (Popper, 1972); scholastic logic (cf., (Perrier, 1909)); dialectic or dialectical logic (Hegel, 1813); universal logic (Beziau, 2012); Logic in Reality (Lupasco, 1951; Brenner, 2008); logic of science or logic of physics (Bridgman, 1927); logic of epistemology (Hintikka and Hintikka, 1988); logic of diagnosis (Tarasov et al., 1989); market logic (Agre, 2000); logic of distributed systems (Barwise and Seligman, 1997).
Philosophers also treat formal (symbolic) logic and dialectical logic as separate logics independent of each other (Routley, 1979). Formal logic studies inference with purely formal content and consists of two subdisciplines: symbolic logic and non-symbolic logic. Aristotle’s syllogistics is an example of formal non-symbolic logic. Symbolic logic as a discipline takes its origin in the works of George Boole (1815–1864), Augustus De Morgan (1806–1871), Friedrich Ludwig Gottlob Frege (1848–1925), Giuseppe Peano (1858–1932), and Charles Sanders Peirce (1839–1914). The core systems studied by symbolic logic are symbolic structures called logics and calculi. Typically, a logic, as a symbolic system, consists of a specific formal, or sometimes informal, language together with an inference system, which generates (forms) a calculus (logic’s syntax), and/or a model-theoretic semantics. The language is, or corresponds to, an
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enhanced and often formalized part of a natural language, such as English, Spanish, or French. The inference system is to capture, codify, or simply record, what reasoning is correct for the given language. Often, inference is, or is called, deduction although there are other kinds of inference — induction, abduction, and analogy. The goal of the logical semantics is to represent meaning, or truth-conditions, or possible truth conditions, for the logical language. An important sub-discipline of symbolic logic is mathematical logic. There are three meanings of the term mathematical logic: — Logic as a discipline that uses mathematical techniques; — Symbolic logic as a discipline applied to mathematics; — Logic as a manner of reasoning used in mathematics. Mathematical logic as a discipline that uses mathematical techniques consists of three parts: — The part called formal (usually, axiomatic) theories studies formal symbolic systems such as axiomatic set theories or formal arithmetic; — proof theory investigates deduction and deductive systems exploring dynamical processes in the syntax of mathematical logic; — model theory studies of interpretations of formal systems and represents semantics of mathematical logic. Besides, such a discipline as recursion theory, which explores constructive representations of functions and processes, is sometimes considered as a part of logic, while in other cases, it is treated as a part of the theory of algorithms. There is a big diversity of various logics (as structures) introduced and studied by different authors. Some of them are given in the following list: — syllogistics (Aristotle); — classical logic, which includes the classical propositional logic and classical predicate logic (Boole; De Morgan; Peirce); — algebraic logic (cf., (Halmos, 1962; Plotkin, 1991)); — algebraic polymodal logic (Goldblatt, 2000);
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autoepistemic logic (Moore, 1985); belief logic (Levesque, 1984); business logic (cf., (Bohrer, 1997)); categorical logic (cf., (Goldblatt, 1984; Lambek and Scott, 1988)); combinatory logic (cf., (Fitch, 1974; Hindley et al., 1972)); complex logic (Zinoviev, 1973); computability logic (Japaridze, 2003; 2006); conclusion logic (Shoesmith and Smiley, 1978); conditional logic (cf., (Nute, 1980)); conservative logic (Fredkin and Toffoli, 1982); constructive logic (Bridges and Richman, 1987); continuous logic (cf., (Chang and Keisler, 1966)); cumulative logic (cf., (Degen, 1984); cumulative default logic (Brewka, 1991); default logic (cf., (Reiter, 1980; Konolige, 1988; Brewka, 1989)); deontic logic (also called logic of norms) (cf., (Hilpinen, 1971; von Wright, 1951; 1963; 1968)); dependence logic (V¨ a¨ an¨ anen, 2007) deviant logic (Haack, 1974); discussive (also called discursive) logic (Ja´skowski, 1948/1999;1949/1999; Ciuciura, 2008); dividing and conquering logic (Amir, 2002); dynamic logic (cf., (Harrel, 1979; Harel et al., 2000)); epistemic logic (cf., (Hintikka and Hintikka, 1988; Schlesinger, 1985)); equational logic (Clouston, 2009); erotetic logic (also called logic of questions and answers) (cf., (Prior and Prior, 1955; Harrah, 2002; Belnap and Steel, 1976)); fibring logics (Gabbay, 1999); first-order logics (cf., (Shoenfield, 2001; Kleene, 2002)); free logics (Leonard, 1956; Lambert, 2003); fuzzy logic (cf., (Zadeh, 1975; McNeill and Freiberger, 1993; Nguyen and Walker, 1996; Bandemer and Gottwald, 1996)); higher-order logics (cf., (Lambek and Scott, 1988)); hybrid logic (Areces et al., 2001);
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hypermodal logics (Gabbay, 2002); illocutionary logic (Searle and Vanderveken, 1985); inclusive logic (Mostowski, 1951; Quine, 1954); independence friendly logic (IF logic) (Hintikka and Sandu, 1989); inductive logic (cf., (Carnap, 1952; Kyburg, 1970)); infinitary logic (Barwise, 1969); information-theoretic logic (Corcoran, 1998); intensional logic (cf., (Anderson, 1984)); intermediate or superintuitionistic logics (Umezawa, 1959); interpretability logics (cf., (Japaridze and de Jongh, 1998)); intuitionistic logic (cf., (Dummett, 1973)); labeled logics (Gabbay, 1996); linear logic (cf., (Girard, 1987; 1998)); local logics (Barwise and Seligman, 1997); logic of decision (Jeffrey, 1966); logic of discovery (H´ajek and Havranek, 1978); logics of formal inconsistency (Carnielli et al., 2007); logics of names (Simons, 2001; Cocchiarella, 2005); many-sorted logic (cf., (Turner, 1984: Manzano, 1993; Meinke and Tucker, 1993)); many-valued logic (cf., (Lukasiewicz, 1920; Post, 1921; Ackermann, 1967)); metalogic (cf., (Hunter, 1971; Burgin, 2007a)); monadic predicate logic (cf., (Tharp, 1973)); modal logics (cf., (Chellas, 1980; Hughes and Cresswell, 1968)); neutrosophic logic (Smarandache, 2002); nominal logic (Pitts, 2003); non-monotonic logics (cf., (Brewka, 1991; Burgin, 1991d; Turner, 1984)); operational logic (cf., (Luchi and Montagna, 1999)); operational quantum logic (cf., (Coecke et al., 2000)); paraconsistent logic (cf., (Ja´skowski, 1948; 1949; Arruda, 1980; Priest, 1986)); polar logic (Birnbaum, 1980); polyadic logic (Banerji, 1988);
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polymodal logic (Japaridze, 1988); possibilistic logic (Benferhat et al., 1992); predicate functor logic (cf., (Quine, 1976; Kuhn, 1983)); probabilistic logics (cf., (Boole, 1854; Reichenbach, 1932; 1935; Hailperin, 1984; Russell, 2014)); process logic (Harel et al., 1982); prohairetic logic (Moutafakis, 1987; Wright, 1963a); provability logics (Solovay, 1976; Boolos, 1993); quantum logic (cf., (Mittelstaedt, 1978)); relevant logic (also called relevance logic) (cf., (Anderson and Belnap, 1975; Read, 1988; Restall, 2000)); resource logics (Gabbay and Queiroz, 1992); restrictive access logics (Gabbay, 1993); second-order logics (cf., (V¨ a¨an¨anen, 2001; Rossberg, 2004)); slash logic (Hodges, 1997; 1997a); spatial logics (cf., (Aiello et al., 2007); stationary logic (Barwise et al., 1978); substructural logics (Restall, 2000); temporal logic (also called tense logic) (cf., (Allen, 1984; van Benthem, 1991; 1995)); triadic logic (cf., (Fisch and Turquette, 1966)); type-free logic (cf., (Menzel, 1986)).
We see a great diversity of logics created by researchers. However, all logics listed here as the most important and/or popular form only a little part of all possible logics. For instance, there exists a continuum of different intermediate logics (Umezawa, 1959). According to Kant, logic studies three main essential structures of thinking: concepts, judgments, and conclusions. This division of logical description of mental activity and knowledge corresponds to three traditional basic levels of logic as a tool of knowledge representation, grounding and acquisition with their specific basic structures: — On the first (conceptual) level, reality (the knowledge domain) is reflected by a system of concepts, terms, and names as the basic structures of this level.
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— On the second (representational) level, concepts are used in statements (judgments) about the knowledge domain and its constituents, while reality (the knowledge domain) is reflected by a system of statements such as propositions and predicates, which are the basic structures of this level. This is the level of the logic language. — On the third (inferential) level, statements (judgments) are used in reasoning and knowledge transformation, e.g., in inference, aimed at obtaining new knowledge and grounding existing knowledge. As a result, the system of statements becomes dynamic, and the knowledge domain (reality) is reflected by logical calculi as the basic structures of this level. Each level has its own basic structures and rules of their composition. However, the development of logic and extension of its domain brought forth a new (organizational) level of knowledge where reality (the knowledge domain or knowledge object) is described by logical varieties, prevarieties and quasi-varieties (Burgin, 1991d; 1995b; 1997d; 2004a; 2008a; Burgin and de Vey Mestdagh, 2011; 2015; de Vey Mestdagh and Burgin, 2015). On this level, all systems from the previous levels, such as concepts, statements, languages, rules of inference, and logical calculi, are organized as the higher-order structures called logical varieties, prevarieties and quasi-varieties. However, this description of four logical levels reflects the classical approach to logic representing only descriptive or assertoric knowledge, which asserts something about its domain (object). At the same time, people use not only statements for reasoning, but also questions, queries, and conjectures. To represent these forms of reasoning in a formal way, probabilistic, hypothetic, and erotetic logics have been created in the 20th century (Boole, 1854; Reichenbach, 1932; 1935; Hailperin, 1984; Halpern, 1999; Russell, 2014; Kleiner, 1970; Belnap and Steel, 1976; Harrah, 2002). In addition, dynamic and operational logics have been elaborated for operational knowledge representation (cf., for example, (Allen,
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1984; van Benthem, 1991; Luchi and Montagna, 1999; Harrel, 1979; Harel et al., 2000)). In non-assertoric logics, the first level is similar to conceptual level of the classical logic employing names, terms, and concepts. Only the basic names, terms, and concepts from these logics represent other epistemological objects forming foundation for the second level, which is essentially different. For instance, on the second level, erotetic logics employ questions, queries and problems instead of statements; probabilistic logics assign probabilities to statements; dynamic logics use instructions and other names of processes, actions, and events; while hypothetic logics use conjectures and hypotheses instead of statements. Consequently, the third level of these logics consists of hypothetic, dynamic and erotetic logical calculi, while the fourth level encompasses hypothetic, dynamic, and erotetic logical varieties. Below we consider the first three levels in more detail, while the fourth level is in depth described in Section 3.3.
5.2.1. Concepts, names, terms, and objects By “object” is meant some element in the complex whole that is defined in abstraction from the whole of which it is a distinction. John Dewey
Although the mainstream literature in cognitive science regards the concept as a kind of mental particular, we treat concepts as structures, which in many cases have mental representation. In addition, concepts can have physical representation, for example, by texts in textbooks and monographs. Concepts used in logic, mathematics and science, as well as notions utilized in commonplace reasoning are important types of symbols. Indeed, a concept, or notion, is defined as a general idea derived from specific instances, i.e., a concept is a symbolic (usually, linguistic) representation of these instances. This implies the fact that concepts/notions are signs/symbols of ideas and have structures similar to the general structure of a symbol. Often a concept is
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treated as a unit of knowledge and considered as a constituent of a proposition. The first known model of a concept belongs to Friedrich Ludwig Gottlob Frege (1848–1925). According to him, concepts are ways of thinking of objects, properties and relations (Frege, 1891; 1892; 1892a). To explicate the structure of a concept, Frege assumed that any concept has a name and this name is related to an object or collections of objects, which he called by the name denotation. Besides, Frege suggested that in addition to the denotation, names or descriptions of a concept also express a sense, which accounts for the cognitive significance of the concept, and people develop the denotation of a concept through its sense. This gives us the following diagram as a model of concept. This model is similar to the Semiotic Triangle of N¨ oth, which is a model of sign analyzed in Chapter 4. In addition, Frege defined concept also as a function, the value of which is always a truth-value (Frege, 1891). Another model of concept was suggested by Bertrand Russell (1872–1970). He regarded concepts as constituents of propositions, while his model of concept is similar to the model of Frege. Considering concepts, Russell explained that some words (for example, proper names) indicate particular objects. This indication is represented by
Concept name
Denotation
Sense
Figure 5.2. The Concept Triangle of Frege
Name
Denotation
Meaning
Figure 5.3. The Concept Triangle of Russell
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following structure. indicates Proper name
. Meaning of the proper name (object)
(5.4)
Thus, for Russell a concept has a name and two more constituents. On one hand, concepts symbolize objects that are their exemplifications. Russell calls the relation between concepts and their particular exemplifications denotation (Russell, 1905). This relation is objective or, as Russell also says, logical. On the other hand, concepts have a part formed by meanings of corresponding linguistic expressions that mean objects denoted by the concept. This gives us the following structure of a concept. Each component of the structure of a concept — the name, denotation and meaning in the Concept Triangle of Russell or the name, denotation and sense in the Concept Triangle of Frege — is also an object and has a name, the name of this object. In addition, these objects also have a denotation and sense (or meaning), associated with their names. As a name is itself an object, it has a name and a denotation and sense (or meaning), as any other object. In an intensional context, the names that occur denote the meaning or sense of the objects for the reader or listener. It means that each component of a concept can acquire the role of another component. It means that each component of a concept can acquire the role of another component. As a result, the structure of a concept has the property called fractality, which tells that the structure of the whole is repeated/reflected in the structure of its parts. It is interesting that explicating the structure of concepts, proper names and propositions in the theories of Russell and Frege, Tatievskaia (1999) demonstrates that they have the form of a fundamental triad, which is called a named set, or named set chain (cf., Appendix). She portrays these structures as the diagrams in Figure 5.4. This portrayal allows us to see that in the first diagram in Figure 5.4, Sense connects Sentence with Reference, while in the second diagram, Sense connects Proper Name with Reference. At the same time, the third diagram is a composition of two fundamental
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Sentence
Proper name
Concept-word
Sense of the sentence (thought)
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Reference of the sentence (truth-value)
Sense of the proper name (thought)
Reference of the proper name (object)
Sense Reference Referent of the concept-word of the concept-word of the concept-word (thought) (object) Figure 5.4. Conceptual structures of Frege
triads: in the first one, Sense connects Concept-word with Reference, while in the second one, Reference connects Sense with Referent. Cohen and Murphy (1984) consider five types of concept models: extensional, fuzzy set-theoretical, prototypical and semantic. The extensional or set-theoretical model is based on set theory and represents a concept as a set of objects that belong to the concept, rather than as some form of mental representation. It is a classical model of concepts (cf., Diagram (5.5)). t Concept name
Set of objects .
(5.5)
The fuzzy extensional or fuzzy set-theoretical model is based on fuzzy set theory and represents a concept as a fuzzy set of objects that to some extent belong to the concept (cf., Diagram (5.6)). The structure of this model is similar to the structure of the extensional model but in contrast to the relation t, which simply assigns each object to the concept name, the relation r shows to what extent each object belongs to the concept with the given name. r Concept name
Set of objects .
(5.6)
The semantic or knowledge-representation model is based on componential semantics and usually represents a concept as a system of
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attributes in feature-based models and by conceptual networks in network models (cf., Diagram (5.7)). p System of attributes
Concept name
.
(5.7)
These models of concepts are considered classical. There are also non-classical models of concepts. The prototypical or prototype model is based on prototype theory and has two forms — extensional and attributive. The principal idea of prototype theory is graded categorization of objects, according to which some members of a category are more central or typical than others. In prototype theory, the prototype is defined it as the most central member or members of a category. It is assumed that graded categorization is likely to be present in all cultures. Psychologists found experimental evidence that some members of a category are more privileged than others forming gradation of categories: 1. Response time shows that queries involving prototypical members elicited faster response times than for non-prototypical members. 2. Priming of objects from the same category, prototypical members come to the top of the list. 3. Exemplars named by people more frequently include prototypical items. In the attributive prototype model, a concept is represented by a system of the concept role-values as its most common attributes. Such a system is treated as a prototype base (cf., Diagram (5.8)). These role-values of a concept are not assigned by definition because the concept does not specify necessary and sufficient criteria for membership. Objects belong to the concept if they sufficiently agree with the concept role-values (Rosch and Mervis, 1975). u Concept name
System of role-values
.
(5.8)
In the extensional prototype model, a concept is represented by an object called a prototype, which plays the role of the most common concept representative (cf., Diagram (5.9)). Objects that have
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the most typical role-values are considered “good” prototypes, while objects that have atypical or unacceptable values are considered “borderline” or “poor” prototypes. For instance, for the concept a bird, robin is a “good” prototype, while penguin is “poor” prototype. Other objects belong to the concept if they are sufficiently similar to the prototypes (Rosch and Mervis, 1975). w Concept name
Prototype .
(5.9)
According to (Hampton, 2011), researchers have proposed five broad classes of concept models, which are described below. The classical model assumes that each concept is clearly and entirely defined by a system of necessary and sufficient as the system common attributes (Armstrong et al., 1983; Osherson and Smith, 1981). Diagram (5.10) is a representation of the structure of this model. l Concept name
Set of attributes .
(5.10)
Later the classical view was extended by dividing representation attributes into two groups: defining features, which form the core definition of the concept extension, and characteristic features, which are true only for typical category members and which may form the basis of a recognition procedure for quick categorization. Thus, the classical concept model generates the structure of the Attributive Concept Triangle (cf., Figure 5.5). The second model is the prototype model, two forms of which are considered above. The third model is the theory-based model rooted in the cognitive development tradition (Murphy and Medin, 1985). Concepts Concept name
Defining features
Characteristic features
Figure 5.5. The attributive concept triangle
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are framed by the theoretical understanding of the world. While a prototype representation of a concept is a system of unconnected attributes, the theory-based representation has the form of a structured frame or schema (cf., Diagram (5.11)), comprising theoretical knowledge about the relations between these attributes, as well as their causal and explanatory links. k
(5.11)
Concept schema .
Concept name
The fourth model belongs to the psychological essentialism (Medin and Ortony, 1989). It is constructed as the development of the classical and theory-based models aimed at synthesis of psychological models with the philosophical intuitions. The model employs a classical “core” definition for concepts, but one in which the core definition may frequently contain empty “place holders”, for example, for additional still unknown attributes (cf., Diagram (5.12)). In such a way, these “place holders” allow adding additional knowledge to concept definitions. f Concept name
Concept schema with variables
.
(5.12)
The fifth one is the exemplar model is similar to the prototype model. In it, the concept is based not on one prototype, but on a number of different exemplar representations (Medin and Shoben, 1988). This system of exemplar representations is regarded as a prototype base (cf., Diagram (5.13)). h Concept name
System of prototypes
.
(5.13)
The exemplar model of concept is similar to a part of the educational model of concept developed in education (Merrill and Tennyson, 1977). Its structure is portrayed by the following triangle (cf., Figure 5.6). Name
Examples
Attributes
Figure 5.6. The educational concept triangle
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However, there are essential differences between the exemplar and educational models of concept. In the exemplar model, only examples of objects comprised by the concept are included. In contrast to this (cf., Figure 5.7), when people in general and children, in particular, learn concepts, they use are given both examples of objects comprised by the concept (positive examples) and examples of objects that do not belong to the concept (negative examples). The situation with concept attributes is the same — when people in general and children, in particular, learn concepts, they learn attributes of objects comprised by the concept (positive attributes) and attributes that objects from the concept do not have (negative attributes). This peculiarity of concept learning shows that it would be reasonable to use negative probabilities ((Dirac, 1942; Feynman, 1950; Forsyth et al., 2001; Burgin, 2009; Burgin and Meissner, 2010; 2012), in probabilistic models of learning and teaching. Another educational model of concept was elaborated by Tar´ abek (2006; 2007), who called it the Triangular Model of concept, which is presented in Figure 5.8. Name
Positive examples
Examples
Attributes
Negative examples
Positive attributes
Negative attributes
Figure 5.7. The extended educational concept model
Core
Meaning
Sense
Figure 5.8. The triangular concept model
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This model has the following components (Tar´ abek, 2006; 2007). The core of a concept has three appearances: • Linguistic in a form of a word or expression. • Symbolic in a form of a physical symbol. • Visual in a form of an image. The meaning of a concept has three levels: 1. More concrete concepts as particular cases of this concept. 2. Very concrete concepts as particular cases of this concept. 3. Objects, phenomena, actions that belong to this concept. The sense of a concept is a list of meaningfully related concepts and corresponding links. The Triangular Model of concept also employs the following concept architecture (cf., Figure 5.9). core
periphery
semantic frame
Figure 5.9. The triangular concept architecture
In this model, the periphery of a concept plays the role of the meaning, while the semantic frame of the concept plays the role of the sense. One more model of concept has been developed in formal concept analysis (FCA) (Ganter and Wille, 1999; Stumme and Wille, 2000). This model is basic for FCA being formed from three components: objects and attributes related by an incidence relation (cf., Diagram (5.14)), which are defined as rigorous mathematical structures. To build this model, a formal context K is defined as a triad (named set) K = (G, I, M ), where • G is a set of objects, • M is a set of attributes,
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Concept intent
Figure 5.10. The FCA model of a concept
• I is a relation between G and M , in which the relation (g, m) ∈ I means, the object g has the attribute m, i.e., I is the connection between objects and attributes. I objects
attributes .
(5.14)
Thus, we can see that formal contexts are named sets and it is possible to apply to them different named set operations (Burgin, 2011). Definition 5.2.1. A formal concept C in a context K = (G, I, M ) is a pair C = (A, B), where (cf., Figure 5.10). • A is a set of objects called the extent of the concept and A ⊆ G; • B is a set of attributes called the intent of the concept and B ⊆ M ; and the following condition (M) is satisfied: A × B is a maximal rectangle in the binary relation I, i.e., sets A and B are maximal with A × B ⊆ I. There are also various relations between formal contexts that come from the named set theory. For instance, a formal context K = (G, I, M ) is a subcontext of a formal context D = (H, J, L) if G ⊆ H, M ⊆ L and I is the restriction of J on G and M , i.e., I = J|(G,M ) . Relations between formal contexts imply relations between formal concepts. For instance, it is possible to define two kinds of subconcepts. Definition 5.2.2. A formal concept (A, B) in a context K is an attributive subconcept of a formal concept (D, E) in the context K if B ⊆ E. For instance, the concept of a computer is an attributive subconcept of the concept of a notebook. Definition 5.2.3. A formal concept (A, B) in a context K is an extensive subconcept of a formal concept (D, E) in the context K if A ⊆ D.
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For instance, the concept of a lion is an extensive subconcept of the concept of an animal. Proposition 5.2.1. A formal concept (A, B) in a context K is an attributive subconcept of a formal concept (D, E) in the context K if and only if the formal concept (D, E) in the context K is an extensive subconcept of the formal concept (A, B) in the context K. Proof . Let us assume that a formal concept (A, B) in a context K = (G, I, M ) is an attributive subconcept of a formal concept (D, E) in the context K. It means that B ⊆ E, i.e., if b ∈ B, then b ∈ E. If B = E, then A = D because by Condition (M), A is maximal with respect to the inclusion A × B ⊆ I and D is maximal with respect to the inclusion D × B ⊆ I. If B = E, then D × B ⊆ D × E ⊆ I because B ⊆ E. As A is maximal with respect to the inclusion A × B ⊆ I, we have D ⊆ A. It means that the formal concept (D, E) in the context K is an extensive subconcept of the formal concept (A, B) in the context K. Now let us assume that a formal concept (A, B) in a context K = (G, I, M ) is an extensive subconcept of a formal concept (D, E) in the context K. It means that A ⊆ D, i.e., if a ∈ A, then a ∈ D. If A = D, then B = E because by Condition (M), B is maximal with respect to the inclusion A × B ⊆ I and E is maximal with respect to the inclusion D × B ⊆ I. If A = D, then A × E ⊆ D × E ⊆ I because A ⊆ D. As B is maximal with respect to the inclusion A × B ⊆ I, we have E ⊆ B. It means that the formal concept (D, E) in the context K is an attributive subconcept of the formal concept (A, B) in the context K. Later developments of FCA by Lehmann and Wille (1995) was based on the pragmatic philosophy of Charles S. Peirce with his three universal categories and the general triadic approach. The new idea was to use the category conditions in addition to the basic categories objects and attributes. The basic concept is a triadic context, which is defined as a quadruple (G, M, B; Y) where G is a set of objects, M is a set of attributes, and B is a set of conditions, while Y is a ternary relation
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Conceptual representative
Figure 5.11. The first specification rank of the representational model of a concept
between G, M, and B, i.e., Y ⊆ G × M × B. Then the formula (g, m, b) ∈ Y means that the object g has the attribute m under (or according to) the condition b. A triadic concept of a triadic context (G, M, B; Y) is defined as a triple (A1 , A2 , A3 ) denoting the relation A1 × A2 × A3 ⊆ Y, which is maximal with respect to componentwise inclusion. The triadic concepts are structured by three quasiorders given by the inclusion order within each of the three components. Burgin and Gorsky (1991) introduced a new model called the representational model of a concept. Its surface structure is presented in Figure 5.11, in which concept name can be one word, e.g., “cat”, an expression, e.g., “an infinitely small number”, or a text, and which is a specific kind of fundamental triads. We see that the representational model of a concept comprises all other models of a concept as a structure having higher level of abstraction. For instance, in the extensional prototype model, the prototype is the conceptual representative, while in the theory-based model, the concept schema is the conceptual representative. It may look that triadic models of concept are different. However, in the Concept Triangle of Frege, it is possible to consider the components Denotation and Sense as the conceptual representative of the concept. In a similar way, it is possible to consider the components Denotation and Meaning as the conceptual representative of the concept in the Concept Triangle of Russell. This model is further developed in (Burgin, 2012) by explicating several structural levels, which we call the specification ranks of the concept. The first specification rank of a concept encloses the surface structure of the representational model, which is portrayed in Figure 5.11. Going to the second stratum, the component Conceptual Representative, which constitutes the first stratum of the model is divided into three parts — Interpretant, Connotation (also called
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Connotat), and Denotation (also called Denotat), which form the second stratum of the model. Combination of the first and second strata generates the concept structure of the second specification rank, which is similar to the Russell model of a concept. In turn, each of the components Interpretant and Denotation/ Denotat is divided into three parts, forming the third stratum. The component Interpretant is split into Intention, Sense, and Meaning, while the component Denotation/Denotat is split into Scope/Extent, Exemplars, and Prototype. At the same time, Connotation/Connotat is split into Associative/indirect (e.g., Metaphorical) Interpretant and Associative/indirect Denotat. Combination of three described strata forms the concept structure of the third specification rank. To understand the representational model, let us look what meaning all these terms have. In conventional models, the Denotation/Denotat of a concept, which is also called the Referent by some authors, is the collection of entities that are unified by this concept and denoted by the concept name. For instance, the denotation of a concept cat consists of all cats. In the representational model, the component Denotation/ Denotat of a concept consists of three parts — the Prototype, Exemplars and Scope (also called Extent) — and only the Scope/Extent is the collection of entities that are unified by this concept. The Prototype is the most typical representative of the Scope/Extent, while the Exemplars include representatives from the Scope (Extent) such that give its sufficiently complete description of the Extent. The term Interpretant goes back to the Peirce’s model of a sign (cf., Chapter 4). In the sign model, Interpretant of a sign is the meaning of this sign and because a concept is a kind of signs, we assume that in classical models, such as the Concept Triangle of Russell, Interpretant coincides with the meaning of the concept. In the representational model, the Interpretant of a concept consists of three components — Intention, Sense, and Meaning. As the component of the representational model, the term Meaning has the conventional
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meaning, which is analyzed in Section 2.3.10. The component Sense of a concept accounts for its cognitive significance, by which people conceive the denotation of this concept. The component Intention indicates the internal content of a concept, which is expressed in its explicit definition. The component Connotation/Connotat of a concept is a commonly understood cultural or emotional association that the concept carries, in addition to its explicit or literal meaning. A connotation is frequently described as either positive or negative, taking into consideration the emotional response. For instance, it is possible to characterize a stubborn person by two concepts — strong-willed or pig-headed. While these concepts have the same literal meaning (stubborn), the concept strong-willed connotes respect and admiration for person’s will (a positive connotation), while the concept pigheaded connotes frustration in dealing with this person (a negative connotation). As an example of Metaphorical Denotat, we can take the mythological connotation of Barthes, which shows how examples can be exploited to communicate more than just the one allusion to the concept in a certain context (Barthes, 1977). Thus, a picture or a diagram used as an example of a concept in a discourse on what it denotes will not only be subject to certain representational conventions. In addition, it can also contain clues that it should be understood as describing the concept in a certain way — or conversely contain strategies to avoid such a description. We can consider Diagrams (5.9)–(5.14) as examples of a Metaphorical Denotat. In the representational model, the component Connotation/ Connotat of a concept includes two components — the Associative/ Indirect Interpretant (e.g., Metaphorical Interpretant) and Associative/indirect Denotat (e.g., Metaphorical Denotat). The Associative Interpretant is the meaning indirectly associated with the concept, while the Associative Denotat consists of images and emotions caused by the concept. In the example considered above, the Associative Interpretant of the concept strong-willed can include such properties as strong, persistent, successful, and/or reliable. The Associative
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Denotat of the same concept can include images George Washington, Napoleon, or Alexander the Great, as well as emotions of admiration, respect and/or appreciation. We come to the concept structure of the fourth specification rank when a distinction is made between literal components of the third stratum — literal Sense/Intention, literal Meaning, literal Extent, and literal Prototype (the first part of the fourth stratum) — and metaphorical components of the third stratum — metaphorical Sense/Intention, metaphorical Meaning, metaphorical Extent, and metaphorical Prototype (the second part of the fourth stratum). For instance, taking the concept lion, we have a literal meaning “a member of the Felidae family of carnivorous mammals” and a metaphorical meaning “the king of animals”. With respect to its literal components — the literal meaning, literal extent, literal sense, and literal prototype, the concept plays the role of an index. With respect to its metaphorical components — the metaphorical meaning, metaphorical extent, metaphorical sense, and metaphorical prototype, the concept plays the role of a symbol. Continuing this procedure of the concept structure explication, we come to the concept structure of the fifth rank when a distinction is made between strict components of the fifth stratum — strict Sense, strict Intention and strict Meaning — and fuzzy components of the fifth stratum — fuzzy Sense, fuzzy Intention and fuzzy Meaning; as well as between exact components of the fifth stratum — exact Extent, exact Exemplars and exact Prototype; graded components of the fifth stratum — graded Extent, graded Exemplars and graded Prototype; and approximate components of the fifth stratum — approximate Extent, approximate Exemplars and approximate Prototype. For instance, the concept “the Sun” has the exact Extent, which consists of the definite star. At the same time, the concept “a star” has the approximate Extent because first, we do not know all stars and second, new stars are formed and some existing stars explode and stop being stars. In a similar way, the meaning of the concept “the Earth” is strict, while the meaning of the concept “a planet” is fuzzy.
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Conceptual Representative
Interpretant
Connotat/Connotation
Denotat/Denotation
attributive Intention
Sense
Meaning
Exemplars
Prototype
Scope/Extent
operational relational literal/direct
metaphorical/indirect
literal/direct individual
metaphorical/indirect group
all individual objects united strict
fuzzy
by the concept graded
Associative/indirect Interpretant
exact
approximate
Associative/indirect Denotat
Figure 5.12. The multilevel representational model of a concept
The concept structure of the fifth rank, which is composed of all strata from the first to the fifth, is presented in Figure 5.12. Note that graded categorization of the concept extent is a central notion in many models of cognitive science and cognitive semantics (cf., for example, Langacker, 1987; 1991)). It is possible to continue the procedure of the concept structure explication clarifying that there are three forms of meaning: — Attributive meaning has the form of a system or schema of attributes. — Network or relational meaning has the form of a conceptual network or schema. — Operational meaning has the form of a system or schema of operations/procedures/algorithms. In an attributive system, all attributes are given. In contrast to this, an attributive schema may contain slots for yet unknown attributes. Thus, an attributive system provides an exact attributive meaning, while an attributive schema provides an approximate attributive meaning.
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In a conceptual network, all nodes (i.e., concepts) and links are fixed. In contrast to this, a conceptual schema may contain node (concept) variables and link variables (Burgin, 2006). Thus, a conceptual network provides an exact network/relational meaning, while a conceptual schema provides an approximate network/relational meaning. In a system of operations, procedures or algorithms, all of these elements are fixed. In contrast to this, an operational schema may contain node (i.e., operation, procedure, and algorithm) variables and link variables (Burgin, 2006). Thus, a system of operations, procedures, or algorithms provides an exact operational meaning, while an operational schema provides an approximate operational meaning. We see that the representational model of a concept comprises all other models of a concept. Metaphorical Denotation (metaphorical Denotat) and metaphorical Interpretant belong to the Concept Connotation (Connotat). With respect to its literal components — the literal meaning, literal extent, literal sense, and literal prototype - the concept plays the role of an index. With respect to its metaphorical components — the metaphorical meaning, metaphorical extent, metaphorical sense, and metaphorical prototype — the concept plays the role of a symbol. Besides, there are three forms of meaning: — Attributive meaning has the form of a system of attributes. — Network or relational meaning has the form of a conceptual network. — Operational meaning has the form of a system of operations/ procedures/algorithms. It is necessary to remark that each component of the structure of a concept — the name, interpretant, connotat, and denotat — is also an object and has a name, i.e., the name of this object. In addition, these objects play the role of denotat and themselves have a connotat and an interpretant, i.e., meaning, sense and intention, associated with their names. As a name is itself an object, it has a name and interpretant, i.e., meaning, and connotat, i.e., sense,
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as any other object. In the intensional context, the names that occur denote the meaning or sense of the objects for the reader or listener. It means that each component of a concept can acquire the role of another component. As a result, the structure of a concept has the property called fractality, which tells that the structure of the whole is repeated/reflected in the structure of its parts. The epistemological category concept is intrinsically connected to two other epistemological categories — name and object. For instance, we can see that there is no concept without a name — concepts do not exist without names. Exact names of concepts are called terms. This reflects the situation that logic, like any other discipline, extensively uses names (cf., for example (Kripke, 1972; McCulloch, 1989; Grove and Halpern, 1993; Haas, 1995; Grove, 1995)). Alonzo Church (1903–1995) describes importance of names for mathematical logic, contemplating the concept name as one of the basic terms used in logic when at the beginning of his book Introduction to Mathematical Logic, he clarifies the main concepts of logic starting with the concept name and making clear the prevalent importance of names for logic, although he takes into account only proper names (Church, 1956). As another logician Stephen Cole Kleene (1909–1994) writes, it is better to think about variables in logic not as some placeholders for appropriate objects but as names from a warehouse used for denoting different objects (Kleene, 2002). In his book, one more logician Shoenfield also describes how names for individuals and expressions are constructed and used (Shoenfield, 2001). Moreover, the great mathematician Jules Henri Poincar´e (1854– 1912) wrote in his methodological studies that without a name, no object exists in science or mathematics (Poincar´e, 1908). The word name stems from the Greek word ωνoµα (onoma) and its successors — the Latin word nomen, Old English word nama, and Old High German word namo. In a social context, a name is defined as a word or words by which an entity is designated (American Heritage Dictionary, 2009). A more general definition of name is given in (Gorsky et al., 1991). Namely, a name is defined as an expression
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of a natural language that denotes a separate object, collection of similar objects, properties, relations, etc. A name usually is treated as a label for an object, such as a human being, animal, thing, place, service, and even an idea or theory. It is used to identify and distinguish one or several objects from anything different. Names can single out a class or category of objects, or a particular object, either uniquely, or within a given context. In the theory of named sets (Burgin, 2011), the term name has a much broader meaning and scope. People even developed names for names, or more exactly, names for classes of names. Here are some of them. A name of a person is called an anthroponym. For instance, names Andrew, Bill, and Ann are anthroponyms. A name of a place is called a toponym. For instance, names America, Sahara, and Malta are toponyms. A name of a body of water is called a hydronym. For instance, names Amazon, Nile, and Black Sea are hydronyms. A name of an ethnic group is called an ethonym. For instance, names Indians, Spanish, and Jews are ethonyms. An assumed name of a person is called a pseudonym. For instance, names Mark Twain, Lewis Carroll, and Napoleon are pseudonyms. An assumed name of a writer or journalist is called a pen name. An assumed name of an actor is called a stage name. Psychologists also studied how infants perform naming and learn names of things (cf., for example (Baldwin and Markman, 1989; Markman, 1989; Hall, 1999; Waxman, 1998; Baillargeon, 2000; Waxman, 2002; 2003)). As Waxman (2002) writes, “Even before they can tie their own shoes, human infants spontaneously form concepts to capture various relations among the objects and events they encounter, and they learn words to express them. Name learning, more than any other development achievement, stands at the very center of the crossroad of human cognition and language.” However, usage of names in such a rigorous discipline as logic is not always rigorous. For instance, as Grove (1995) writes, modal epistemic logics for many agents sometimes ignore or simplify the distinction between the agents themselves, and the names these agents use
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when reasoning about each other. At the same time, problems motivated by practical computer science applications show that utilized theories of naming are often inadequate. For instance, their main concern is proper names while other names, i.e., common names for many objects, are also extremely important. Thus, practical applications demand logic to pay more attention to names and naming. Logicians have developed special tools for working with names in logic, and all of them involve building new or transforming existing named sets. For instance, Gabbay and Malod (2002) extend predicate modal and temporal logics introducing a special predicate W (x), which names the world under consideration. Such a naming allows one to compare the different states the world (universe or individual) can be in after a given period of time, depending on the alternatives taken on the way. As Gabbay and Malod (2002) remark, the idea of naming the worlds and/or time points goes back to Prior (1967). Labeling is a kind of naming andlabeled logics and labeled deductive systems form a new and actively expanding direction in logic (cf., (Basin et al., 2000; Chau, 1993; Gabbay, 1994; 1996; Gabbay and Malod, 2002; Vigan` o and Volpe, 2008)). Labeled logics use labeled signed formulas where labels (names of the formulas) are taken from information frames. As the result, the set of formulas in a labeled logic becomes an explicit named set, the support of which consists of logical formulas, while the set of names is an information frame, i.e., the system of labels. The derivation rules act on the labels as well as on the formulae, according to certain fixed rules of propagation. It means that derivation rules are morphisms (mappings) of the corresponding named sets. In default logics, there is even an algorithm for grounded naming (labeling) (Roos, 2000). Besides, the set of names is the level of types, and the naming relation connects objects and types. The elementary theory of types and names developed by J¨ ager (1988) is aimed at application in computer science. Objects are the entities the computer can directly manipulate. They are promptly accessible and explicitly represented in suitable form, e.g., as bitstrings in a computer memory. In contrast to this, types are abstract collections of objects. In order to address them, computers have to use their names. Hence the name nX of
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a type X has to provide enough information such that X can be determined from nX . If the object a is the name of type X, then the extension ext(a) of a is this (unique) type X. In the context of the theory of named sets, the extension ext(a) of a is the interpretation of a. Understanding importance of names and naming, philosophers and logicians have developed different theories of names (cf., for example, (Mill, 1862; Frege, 1892; Donnellan, 1972; Evans, 1973; Putnam, 1975; Kripke, 1972; Geurts, 1997; Cumming, 2009)). The most renowned are Mill’s theory of names (Mill, 1862), description (descriptivist) theory of names, causal theory of names (Evans, 1973; 1985), and proper theory of names (Katz, 1977). It is necessary to remark that contemporary philosophers are mostly interested in proper names, creating their theories of names. For instance, in the Stanford Encyclopedia of Philosophy, the article Name (Cumming, 2009) starts with the statement that proper names are familiar expressions of natural language and then deals only with proper names. In the 19th century, John Stuart Mill (1806–1873) developed the predominant theory of names. The basic tenet of his theory suggests that the meaning of a proper name is simply its bearer (direct referent) in the external world. Now one of the most influential is the descriptivist (description) theory of names generally attributed to Frege (1892) and Russell (1905) and further developed by Church (1956) and Searle (1958). This theory says that naming, and reference in general, goes on by mentally connecting a set of properties with a name, identifying something as having each of these properties, and applying the name to the object according to this identification. Frege (1892; 1892a) develops his approach on the distinction between sense and reference. In the case of proper names, the sense (or Sinn) of a name consists in the (usually) definite description that speakers associate with it. This sense is objective (it is an abstract object) for Frege and must not be confused with its subjective representation in the mind of each individual speaker. At the same time, a proper name can have more than one sense associated with it.
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name
referent
sense
Figure 5.13. The descriptivist model of a name
Following Frege, the descriptivist theory of names associates the following triadic structure with a (proper) name (cf., Figure 5.13). In this structure, the meaning (semantic content) of the name is identical to the descriptions associated with the name by speakers, while the name referent consists of the objects that satisfy these descriptions. Besides, we see that this structure is, in essence, the structure of a concept (cf., Figure 5.2). Russell suggested a slightly different approach, making an important distinction between what he calls “ordinary” proper names and “logically” proper names. Logically proper names are indexicals such as this and that, which are directly connected to sensual data or other objects of immediate acquaintance. In contrast to this, ordinary proper names are abbreviated definite descriptions. The causal theory of names advanced by Saul Kripke and Hilary Putnam assumes that a currently used name names a certain object depends on whether current use of the name causally depends on its use by people who originally dubbed the object with that name. In this context, the causal theory of proper names is the view that: — the meaning of a proper name is simply the individual object to which, in the context of its use, the name refers; — the name’s referent is originally fixed by the naming, in which the name becomes a rigid designator of the referent; — later uses of the name succeed in referring to the referent by being linked by a causal chain to that original naming act. Thus, the causal theory implies that all proper names get their meaning by an initial act of naming, while their reference is fixed. Whether it is a name of a person, a ship, a town, a planet, or whatever, the original act of naming always exists, and the object named is rigidly connected to its name. In such a way, the name becomes a
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rigid designator of what it refers to where a name is a rigid designator if and only if it denotes its reference in all possible worlds. The prominent philosopher Edmund Gustav Albrecht Husserl (1859–1938) paid special attention to the explanation of the difference between names and objects. He separated different kinds of names: Meaningfully indicating names play the role of properties forming a group that uniquely identifies an object and expresses a meaning. Purely indicating names also form a group that uniquely identifies an object but express no meaning other than being a name of some object. Universal names denote sets of objects united by a concept, which is the meaning of a universal name. For instance, “the victor in Jena” and “the loser in Waterloo” are meaningfully indicating names of Napoleon Bonaparte although they express different meanings. In a similar way, “the equilateral triangle” and “the equiangular triangle” are universal names that express different meanings, but designate the same class of object, namely, the class of triangles. In contrast to this, such names as Aristotle, Socrates, Napoleon Bonaparte, Edmund Gustav Albrecht Husserl and so on, have no meaning, but have the role of designating an object, namely, a definite person. Thus, we can see that logic and philosophy have paid a significant attention to names, treating name as one of the basic objects of the logical and philosophical enquiry. Along these lines, specific logics of names have been also developed. For instance, Abadi (1998) constructed a logic to explicate and work with the meaning of local names the Simple Distributed Security Infrastructure (SDSI) used to provide security for distributed computer systems. Halpern and van der Meyden (2001) built the Logic of Local Name Containment, which provides a more exact characterization of SDSI name resolution. Its semantics is closely related to that of logic programs, leading to efficient implementation of queries concerning local names. Meanwhile, it was discovered that names and naming are intrinsically connected to named sets (cf., Appendix and (Burgin, 2011)).
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For instance, all models of concepts considered above either are named sets (fundamental triads), e.g., the set-theoretical model (Diagram (5.2)), the fuzzy set-theoretical model (Diagram (5.3)) and the knowledge-representation model (Diagram (5.4)), or are built of named sets, e.g., the Frege model (Figure 5.2), the Russell model (Figure 5.3) and multilevel representational model (Figure 5.12). Utilization of named sets as a foundation for naming in logic provides means for making logical studies more rigorous and concise. The standard model of a named set consists of three components: (1) people or some objects, (2) their names, and (3) connections between people (objects) and their names. However, it is necessary to understand that named set theory (Burgin, 2011) is not a theory of names. It is a mathematical theory of fundamental mathematical structures and the unified foundation of mathematics. At the same time, named sets rigorously explicate the structure of the concept name and consequently provide tools for building mathematical foundations for different theories of names. Terms are a special case of names. There are two meaning of the word term. In general, a term is a name (word) that has exact definition. In mathematics and logic, formal expressions are called terms. When we speak about a name, it is, as a rule, a name of some object. Besides, the word object is very popular. Nevertheless, there is no exact definition of an object. For instance, Gaede writes: “. . . the study of Physics is first and foremost the study of objects. Indeed, the most important topics of contemporary Physics revolve around physical objects. Hawking (in his book “A brief history of time”) states that space-time is an object, a black hole is widely considered to be a dynamic object, and Particle Physics is defined as the study of motion of subatomic objects known as ‘particles’. Therefore, in Physics, we have no alternative but to define what we mean by object. It turns out, however, that in the entire history of Physics no one has ever bothered to define this fundamental term. Not a single textbook begins by defining what an object is.” (Gaede, 2003)
There are also particular theories of object, in which there are definitions or, at least, description of the concept object. However, in
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each case, the concept of an object in these theories is very specific and restricted to a particular area. For instance, object relations theory is a derivative of psychoanalytic theory that emphasizes interpersonal relations, primarily in the family, e.g., between mother and child. In this theory, object has a very specific meaning being a person that is the target of another’s feelings or intentions, and representation refers to the way the person has or possesses an object and is the mental image of an object. There are two positional types of objects: — An external object is an actual person, place or thing that a person has invested with emotional energy. — An internal object is one person’s mental representation of another person, such as a reflection of the child’s way of relating to the mother, idea, or fantasy about a person, place, or thing. In addition, there are two structural types of objects: — A part-object is an object that is part of a person, such as a hand or breast. — A whole-object is another person who is recognized as having rights, feelings, needs, hopes, strengths, weaknesses, and insecurities just like one’s own. In computer programming, there are object-oriented programming languages based on the concept of an object. A software object consists of two parts: the internal object state and the object behavior. The state is written in fields, which are variables in the object-oriented programming language, while the object behavior is represented through methods, which are functions in the objectoriented programming language. Methods operate on the internal object states and serve as the primary mechanism for object-toobject communication. Object-oriented programming provides a number of benefits, such as: 1. Modularity means that the source code for any object can be written and maintained independently of the source code for other
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2.
3.
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objects in the same program. In addition, once created, an object can be easily passed around inside the software system. Reusability means that it is possible to use an object from one program in any number of other programs. Thus, a software developer is able to use already implemented and tested complex, taskspecific objects, created by expert programmers. Testability and debugging ease means that it is possible to test and debug each object separately from other objects and the whole software system. Information-hiding means that interacting only with an object’s methods, it is not necessary to use (disclose) the details of its internal implementation. Exchangeability means that it is possible to change one object by another one if the first one, for example, turns out to be problematic.
An interesting theory of objects was developed by Alexius Meinong Ritter von Handschuchsheim (1853–1920), who was an Austrian philosopher and psychologist from the University of Graz (Meinong, 1904; 1904a; 1907). Assuming that the concept of an object cannot be defined in terms of a general type and its variations, Meinong introduces only a vague understanding of the concept object. So, whatever can be experienced in some way, i.e., be the target of a mental act, and be denoted by any grammatically correct phrase is an object (Gegenstand in German). Consequently, not only physically existing things but also all kinds of items of imagination and thought are objects including even impossible objects such as the round square, as well as paradoxical objects, or as Meinong calls them, defective objects, such as the person who says “Whatever I say is a lie.” For instance, the phrases “the present King of France,” or “the round square” are supposed to denote genuine but paradoxical objects. It is admitted that such objects do not subsist, but nevertheless they are supposed to be objects. This is in itself a difficult view; but the chief objection is that such objects, admittedly, are apt to infringe the law of contradiction.
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To explain possibility of such objects, Meinong analyzes the seemingly paradoxical sentence “There are objects of which it is true that there are no such objects” using two closely related principles: (1) the principle of the independence of so-being from being, and (2) the principle of the indifference of the pure object to being (Meinong, 1904). The independence principle states that an object with properties is independent of whether it has being or not. The indifference principle that an object is by nature indifferent to being. Meinong expressed this in the form that “the pure object stands ‘beyond being and non-being’ ?”, meaning that neither being nor non-being belongs to the make-up of an object’s nature. In his theory, Meinong introduces the notion of incompletely determined objects, i.e., objects that are undetermined with respect to at least one property. An example of incomplete objects is a triangle. In relation to defective objects, Meinong separates absurd objects as examples of nonsense such “lvap” or “bnuv de”, from inconsistent objects such as the round square. In this context, inconsistency is something that involves incompatible properties but is still understandable, whereas nonsense is something that cannot be understood at all. Here it is necessary to make two remarks. First, while people are expanding their knowledge, what is absurd for one generation may become consistent knowledge for another generation. For instance, the word quark or the sentence (1) “Neutrons are built of quarks” was an absurdity for all people living in the 10th century but now physicists know that a quark is a legal physical object, while the sentence (1) describes a meaningful physical relation between two kinds of physical objects — neutrons and quarks. Second, what is an absurdity for one person can be meaningful for another individual. For instance, the word boson means nothing for a farmer but is a very important term for a physicist who studies the quantum reality. The same is true for inconsistencies. For instance, from ancient times to our days, philosophers are using the term round square as
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an archetypal example of inconsistency, which does not define any object. Meinong, for example, assumed that such objects as a round square have no category of being at all as they are “homeless objects”, to be found not even in Plato’s heaven of ideas. However, mathematicians were able to ascribe meaning to this term building such a mathematical object, which is naturally called a round square. To do this, they build a geometrical object, which is a circle and a square at the same time. As any circle is round, this object is a round square. Let us describe how we can build such a round square. In the Euclidean metrics, the geometrical shape ABCD in Figure 5.14 is a square because all its sides are equal and all its angles are equal to 90◦ . At the same time, taking the Manhattan metrics, the figure ABCD is a circle. Indeed, a circle is a geometric figure in which all point are at the same distance from one point, which is called the center of the circle. The Manhattan, or taxicab, metrics (cf., (Krause, 1987)) defines the distance d between two points in the coordinate plane in the
y B
K
1
H
A
O
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C 1
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Figure 5.14. A round square
x
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following way: d((x, y), (u, v)) = |x − u| + |y − v|. In particular, the distance between the point (0, 0) and the point (x, y) is equal to |x| + |y|. Consequently, the distance from the point (0, 0) to any point of the figure ABCD is equal to 1. Thus, the figure ABCD is a circle in the Manhattan metrics and a square in the Euclidean metrics. Consequently, it is a round square. In his theory, Meinong gives the following classification of objects. Objects, which always have outside-being, are separated into two classes: 1. Objects that have being are separated into two classes: a. Real objects, which exist as well as subsist. b. Ideal objects, which only subsist. 2. Objects that do not have being are separated into two classes: a. Objects that have non-being are separated into two classes: i. Non-contradictory objects. ii. Contradictory objects. b. Objects that are not determined with respect to being. Defining an object as what can be experienced in some way, Meinong analyzes experiences. By his approach, all experiences, even the most elementary ones, are complex mental phenomena, containing, at least, three constituents: (1) the action, (2) the psychological (mental) content, and (3) the object of the experience. This gives us the following diagram (cf., Figure 5.15). While the first two components, (1) and (2), must exist if the experience exists, the third components (3) need not. If somebody object
action
content
Figure 5.15. The structure of a Meinong object
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has a strong hope for total peace, for example, then (1) the action of hope exists, (2) the psychological content, i.e., hope, exists, but (3) no total peace may occur. It means that there is only the non-existent object the total peace. Meinong believes that experiences can have different objects for two reasons. First, different kinds of acts correspond to different kinds of objects. For instance, “objects” correspond to representations, while “objectives” are related to thoughts. Second, it is possible to assume that inside an action, any variation of the objects is dependent on a variation of some mental component that is the psychological content of the experience. The difference between the objects must somehow come down to an internal difference between the representations in question. If you have two different representations, one of red and another of green, for example, the difference between the objects is founded on a genuinely mental difference, namely the difference between the psychological red-content and the psychological green-content. Meinong calls the relation of a content to its corresponding object the “adequacy relation” (Meinong, 1910), and he takes it to be an ideal relation. Ideal relations, in contrast to real relations, subsist necessarily between the terms of the relation. If one color, say red, is different from another, say green, than they must be different. If you compare colors located somewhere, the relation between a color spot and its location is called real because the color, say red, could be located elsewhere, or another color could be in the place of the red color spot. Ideal relations, however, attach once and for all and with necessity to their terms. Meinong just postulates the adequacy relation and offers only negative determinations of it. He stresses the point that adequacy is not a relationship of sameness or of similarity. Since it is an ideal relation, real relations — for example pictorial or even causal relationships — are excluded. A positive hint is given by a kind of metaphorical use of the word “fitting”: the mental content and its object must be fitted to each other. As Meinong supposes that the different kinds of acts are coordinated with the different kinds of presented objects, his classification
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of mental elementary experiences allows a categorization of all objects. Namely, two kinds of experience — intellectual represented by thoughts, and emotional represented by feelings and/or desires. It results in three kinds of objects: objects of thought, objects of feeling, and objects of desire. We see that Meinong elaborated an extensive theory of objects but did not a sufficiently clear definition of an object. He was interested what objects existed and what objects did not exist but considered as the majority of philosophers and scientists that the concept of object is so clear that it does not need any definition. However, even a rigorous discussion of object existence demands a sufficiently exact definition of object. Contemporary theories of objects, such as substance theory and bundle theory, also do not provide adequate definitions of object. Substance theory of objects maintains that an object is a substance, which stands under the change. However, ideal entities, such as concepts, ideas, algorithms or names, can also be objects. Bundle theory of objects implies that an object is a collection of properties and relations. However, physical objects have independent being and cannot be reduced to their properties and relations. To eliminate these deficiencies and limitations, we suggest the following definition. Definition 5.2.4. An object is anything that is considered as a whole, i.e., in its entirety, and has a name. This definition well correlates with the definition of John Dewey, who wrote (Dewey, 1938): “By “object” is meant some element in the complex whole that is defined in abstraction from the whole of which it is a distinction.”
However, in his definition, Dewey uses other concepts, such as element or whole, which themselves have loose (if any) definitions. In essence, we can see objects everywhere. According to the existential structuration of the world (cf., Section 2.2), there are physical objects, mental objects, and structural objects.
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Structural component (e.g., name)
Physical component
Mental component
Figure 5.16. The structure of a total object
At the same time, people make distinction between material (physical) objects, ideal objects, and abstract objects. This classification is relative. For instance, the Earth as a planet is a material object. The Earth as a goddess in Greek mythology is an ideal object, while the Earth as an element of the Copernican model of the Solar system is an abstract object. It is possible to introduce the concept of a total object as an object that has all three components — physical, mental, and structural. For instance, when we see a tree in a forest or garden, it is a total object, for which the tree itself is the physical component of this total object. The image of this tree in mentality of a person is the mental component of this total object and the structure of this tree is the structural component of this total object. Note that the name is the simplest structural component of a total object. As a result, the model of a total object (cf., Figure 5.16) is similar to the Russell model of a concept (Figure 5.3). Now having a definition of an object and the corresponding typology, it is possible to explore the problem of object existence as it was done by Meinong, Russell, and several other philosophers, who investigated this problem. In contrast to what has been done before, we can base our exploration on the given definition and the multidimensional theory of existence developed in (Burgin, 2012). The problem of existence has been in the center of philosophy from the very beginning of its existence. Philosophers suggested different approaches to this problem and created theoretical systems and philosophical directions based on these approaches. In Western philosophy, the main of these directions are: idealism (objective, subjective, and pluralistic), materialism, realism, pragmatism, social constructivism, and existentialism. All these systems assumed that there is only one unique reality and it is necessary to understand and
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described it. Each of these directions gives its own description of this unique reality. In contrast to this, the multidimensional theory of existence postulates three pure forms (dimensions) of reality: 1. The actual reality consists of natural objects and processes directly or indirectly perceived by senses and reflected in the central nervous system (CNS). 2. The virtual reality, often called virtuality, is created, simulated or reflected by some technological system (device or machine), e.g., computer games, movies, and videos. 3. The imaginary reality is created by mentality, e.g., heroes in prose and poems, characters in movies and plays. In addition, there are four combined forms of reality: 1. The mixed reality is a combination of actual and virtual reality, i.e., it is situated between actual reality and virtual reality forming the Virtuality Continuum and including augmented reality and augmented virtuality. 2. The materialized reality is a combination of imaginary and virtual reality. 3. The actualized reality is a combination of imaginary and actual reality. 4. The enhanced reality is a combination of actual, imaginary and virtual reality. Understanding multiplicity of realities allows obtaining natural solutions to problems of object existence, which bothered philosophers for a long time. For instance, actively discuss whether such objects as “the king of France in 2000” or “Pegasus” or “a golden mountain” exist. The multidimensional theory of existence tells us that all these objects do exist but “the king of France in 2000” and “a golden mountain” exist in the imaginary reality of the philosophical discourse, while “Pegasus” exists in the imaginary reality of Greek legends. It has been an active discussion in philosophical circles in what sense mathematical objects in general and numbers, in particular,
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exist. The multidimensional theory of existence explains that numbers and other mathematical structures exist in the world of structures, as well as in mentality of people. The world of structures is the actual reality, while mentality forms imaginary reality for numbers and other mathematical structures (Burgin, 2012). An important part of the concept meaning and sometimes the whole meaning of the concept is its definition. Usually it places the concept in a system of other concepts. According to Tappenden (2008), the discovery of a proper definition is rightly regarded in practice as a significant contribution to mathematical knowledge. For instance, the progress of algebraic geometry is reflected as much in its definitions as in its theorems (Harris, 1988). In a similar way, Arnauld and Nicole (1683) assume that nothing is more important in science than classifying and defining in a good way. To achieve better understanding of the physical world, people elaborate definitions of different things or objects. To better understand one another, people define words they use, or in other words, they give definitions to names. Naturally, studies of definitions have a long history, which begins with ancient Greek philosophers Plato (427–347 B.C.E.), Archytas (428–347 B.C.E.), Aristotle (384–322 B.C.E.) and Antipater of Tarsus (2nd century B.C.E.) and continued with some Roman intellectuals such as Marcus Tullius Cicero (106–43 B.C.E.) and Gaius Marius Victorinus (4th century). Later many philosophers and mathematicians, including such thinkers as Peter Abelard (1079–1142), Thomas Hobbes of Malmesbury (1588–1679), John Locke (1634–1704), Blaise Pascal (1623–1662), Joseph Diaz Gergonne (1771–1859), Augustus De Morgan (1806–1871) and John Stuart Mill (1806–1873), discussed now to build correct definitions, what types of definitions exist and what is necessary to define. For instance, Victorinus distinguished the following types of definitions (cf., (Popa, 1976)): — Substantial definitions; — Conceptual definitions; — Substitution definitions;
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— — — — — —
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Differentiating definitions; Causal definitions; Rhetoric definitions; Exemplifying definitions; Relative definitions; Listing definitions.
Definitions are a specific kind of knowledge and therefore, they also have three forms: • Descriptive definitions; • Operational definitions; • Representational definitions. The majority of researchers starting with Plato and Aristotle acknowledged only descriptive definitions in the form of statements or propositions (Popa, 1976). Only in the 20th century, Bridgman, (1927) introduced operational definitions as sets of operations with the defined objects. Representational definitions were not explicitly introduced or used in philosophy and methodology of science. However, such definitions are often employed in early childhood. Indeed, to learn concepts, children observe representatives, e.g., exemplars or prototypes of some concept and these representatives shape representational definitions of the learned concepts. Note that representatives can be real objects or their images, e.g., pictures or photographs. For instance, to learn the concept dog, a child observes different dogs connecting them with the word dog and in such a way, learns the concept. Besides, representational definitions correspond to listing definitions described by Marius Victorinus. The question what is necessary to define separated two classes of definitions. Some philosophers presumed that people define things (physical objects). For instance, Aristotle in his Topics wrote: “A definition is a proposition describing the essence of a thing.”
However, starting with Hobbes, philosophers changed their attitude to definitions assuming that it was necessary to define words (names) and not things. Now many think the main objective is to define concepts.
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5.2.2. Statements, queries, and instructions It is more important that a proposition be interesting than that it be true. Alfred North Whitehead
The next level of logic consists of classical logical objects such as statements, propositions, and predicates, and new logical objects — abstract properties (in assertoric logics), queries and questions (in erotetic logics) and instructions, operators and operations (in operational logics). These logical objects as formal constructions use elements from the first level as building blocks. As the majority of logics, including classical logics, are assertoric, we begin with assertoric logics. Looking at the second level of contemporary assertoric logics, we see statements, propositions, and predicates. In logic and philosophy, the term proposition, which is derived from the word proposal, refers to both (a) the “content” or “meaning” of a meaningful declarative sentence or (b) the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence. In the classical logic, a proposition is a sentence that is either true or false. For a long time, propositions have been studied as minimal elements (atoms) of logic that are either true or false. Recently, logicians started to study structured propositions (King, 2011). However, even before the famous mathematician and philosopher Bertrand Russell (1872–1970) ascribed definite structure to propositions, regarding concepts as constituents of propositions (Russell, 1903). That is why many current structured propositions theorists attribute the idea of structured propositions to Russell. Statements and predicates also have developed structures, which represent corresponding statements and predicates (Church, 1956; Burgin, 2011). According to contemporary logic, there are two meanings of the term statement: (1) A statement is a well-formed declarative sentence. (2) A statement is the information content of a well-formed declarative sentence.
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While in the first case, statements, and sentences coincide, in the second case, a sentence is only a linguistic carrier of a statement, whereas there may be many other linguistic carriers of the same statement. For instance, it is natural to assume that sentences “Alex wrote this letter” and “This letter is written by Alex” express the same statement. Thus, in the second case, a sentence, which is a linguistic object bearing a statement, is related to this statement, which is a logical object, like a numeral, which is physical entity, to the number it refers to, which is a structural entity. This shows that the second approach to statement meaning is more arguable than the first one. Often a statement is viewed as a truth bearer. However, many understand that it involves the binary valued truth function, meaning that a statement can be either true or false. This misconception comes to us from classical logic. Now when we know various manyvalued logics, such as fuzzy logic, in which the truth function takes values in the interval [0, 1], or intuitionistic logic, in which the truth function takes values in the set “{true, false, unknown}”, it is necessary to admit truth values of sentences cannot restricted only to two values — true and false. Under the condition of being a truth bearer in classical logic, the sentence “In 2000, the King of France was wise” is not (does not contain) a statement because this sentence is neither true nor false as in 2000, there was no king in France. However, taking a logic with three truth-values “true, false, undefined”, we conclude that this sentence is (contains) a statement. The concept statement is intrinsically related to the concept proposition, which is fundamental in contemporary logic and has the following properties. As the statements, propositions are represented by affirmative sentences reflecting the meaning and/or intention. They are the primary bearers of truth-values and the objects of belief and other “propositional attitudes” (i.e., what is believed, doubted, etc.), as well as the referents of that-clauses and the meanings of declarative sentences. This situation involves the following named set (fundamental triad).
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representation {propositions}
{affirmative sentences}
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.
Aristotelian logic treats a proposition as a sentence that affirms or denies a predicate of the subject of this sentence. Let us consider the following examples: “All stars are big” and “The Sun is a star”. In the first example, the subject is “All stars” and the predicate “are big”. In the second example, the subject is “The Sun” and the predicate is “is a star”. Some philosophers do not distinguish statements and propositions in natural languages although formal logic uses only the term proposition. However, others conceive these terms as different. For instance, Strawson advocated the use of the term statement with the second meaning in preference to the term proposition (Strawson, 1950). At the same time, some philosophers argue that some (or all) kinds of speech, actions and other objects besides the declarative sentences also have propositional content being carriers of propositions. For instance, some signs can convey propositions without forming a sentence nor even being linguistic, e.g., traffic signs express definite meaning, which is either true or false. Bertrand Russell assumed that propositions were structured systems composed of objects and properties as constituents. Wittgenstein and some other philosophers held that a proposition is the set of possible worlds/states of affairs in which it is true. One important difference between these views is that the Russell’s approach allows one to differentiate two propositions that are true in all possible worlds/states. It is possible to theorize four relations between concepts statement and proposition: 1. A proposition is a special case of statements. Indeed, this will be the case if we assume that only declarative sentences are bearers (carriers) of propositions, while both declarative and affirmative sentences are bearers (carriers) of statements. 2. A statement is a special case of propositions. Indeed, this will be the case if we assume that only declarative sentences are bearers (carriers) of statements, while both declarative and interrogative
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sentences with yes–no answers are bearers (carriers) of propositions. 3. Neither statements form a subclass of propositions nor do propositions form a subclass of statements. 4. Proposition and statement are different names of the same concept. Propositions are used in formal logic as units of a formal language. Logics based on propositions are called propositional, sentential, or statement logics. They include only operators (actually, names of operators) and propositional constants as symbols in their languages. The propositions in this language are either propositional constants, which are considered atomic propositions, or composite propositions, which are built by recursively applying operators to propositions. It means that proposition names are treated as propositions. In contemporary logic, knowledge represented by propositions acquires meaning through the possible-world semantics. The base of this semantics for a propositional calculus C is a logical (formal) universe W , states of which are fixed assignments of truth values to primitive propositions from C. Such states of the universe W are interpreted as the worlds that a cognitive system, e.g., an intelligent agent, assumes as possible, i.e., a possible world is described by the propositions that are true and by those that are false in this world. In such a way, systems of propositions describe situations in the world W . Other type of logics called predicate or quantificational logics include names of variables and operators, predicate and function symbols, and quantifiers as symbols in their languages. The propositions in these logics are more complex. First, terms are defined in the following way. A term is either a variable or a function symbol applied to the number of terms equal to the number of variables in the function (its arity). For instance, if x, y, and z are variables and + is a binary function symbol, i.e., the function + has two variables, then x+y and x+(y+z) are terms. Propositions are constructed from terms.
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A proposition is either a predicate symbol applied to the number of terms equal to the number of variables in it (its arity), or an operator applied to the number of propositions equal to the number of variables in it (its arity), or a quantifier applied to a proposition. For instance, taking a binary predicate symbol = , a the quantifier ∀, the symbol N as a name of the set of all natural numbers, symbol ∈, operator ⇒ and three variables x, y, and z, we can build the proposition ∀x, y, z ∈ N ((x = y) ⇒ (x + z = y + z)). This more complex structure of propositions allows predicate logics to build finer inferences achieving greater expressive power. Thus, it is possible to suggest denoting informational content of sentences in natural languages by the term statement and calling sentences in formal logical languages by the name proposition. According to historical evidence, in the Western philosophical tradition, it is possible to find the first employment of the notion of proposition as the informational content of sentences in the writings of the Stoics. Namely, in the third century B.C., Zeno and his followers distinguished the material aspects of words and sentences from lekta that denoted what was expressed by words and sentences. In turn, lekta included axiomata, or the meaning of declarative sentences, while only axiomata, and not the words used to articulate them, were properly said to be true or false. In symbolic logic as a discipline, statements are formalized as propositions and well-formed formulas. Here we describe how this formalization is performed in symbolic logic and in mathematical logic as the central sub-discipline of symbolic logic. The majority of logics as systems have many formation layers. The first formation layer consists of logics of sentences (also called propositional or sentential logics). The second formation layer consists of the first order predicate logics, which are logics of objects. The third formation layer consists of the second order predicate logics and so on. There are different predicate logics (or predicate calculi): the classical predicate logic (predicate calculus), monadic predicate logic (predicate calculus) (Tharp, 1973), predicate functor logic (Quine, 1976; Kuhn, 1983), etc.
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Predicate logics use various quantifiers. The most popular quantifiers are: — the existential quantifier ∃ means “there is” or “there are”; — the universal quantifier ∀ means “for any” or “for all”. If A = {ai ; i ∈ I} is an infinite set, then the expression “a predicate P (x) is true for almost all elements from A”, or “almost all elements from A have a property P ” or in logical notation ∀∀xP (x), means that P (x) can be untrue only for a finite number of elements from A. For instance, if A = ω, then almost all elements of A are bigger than 10, or another example is that conventional convergence of a sequence l to x means that any neighborhood of x contains almost all elements from l. Church (1956) describes binary formal equivalence quantifier and several implication quantifiers: singular-binary implication quantifier, binary formal implication quantifier, ternary implication quantifier and so on. One more quantifier, ∃! , means “there is (a) unique”. It is called the uniqueness quantifier. There are also quantifiers ∃many , which means “there are many”, and ∃few , which means “there are few”. All these quantifiers are linear. To achieve higher expressibility of logical languages, logicians invented branching quantifiers. A typical example of such quantifiers is the simplest Henkin quantifier ∃x∀y, ∃a∀b. In quantum logics and relational databases, quantifiers are mappings that satisfy specific axioms (cf., for example (Halmos, 1962; 2000; Leblanc, 1962; Plotkin, 1991)). For instance, in relational databases, the existential quantifier ∃ is represented by the database operator selection and the universal quantifier ∀ is represented by the database operator projection. As a result, quantifiers are represented by named sets as mappings and operators are also represented by named sets (cf., Appendix).
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Besides, whenever we have a representation, it involves a fundamental triad (named set). Thus, in the case of quantifiers, we have two fundamental triads: represented by ∀
projection operator
and represented by ∃
selection operator
.
Basic structures of logic, such as propositions, predicates, logical calculi, and logical varieties, are constructed using different logical languages. It is possible to show that any language, formal or natural, is built of different named sets. All this demonstrates that any logic is formed using different named sets as construction blocks. Indeed, a language in a constructive representation/definition is a triad (named set) of the form L = (X, R, L) where X is the alphabet, R is the set of constructive algorithms/rules for building well-formed (correct) expressions (e.g., words or texts) from L, and L is the set of words of the language L. Logical languages are artificial languages developed intentionally for representation of logical reasoning within a culture. The typical feature of logical languages is that their structure (inner relations) and grammar (formation rules) are intended to express the logical information within linguistic expressions in a clear and effective way. Languages used in logic have, as a rule, constructive definitions in a form of production rules. Elements of logical languages are logical expressions or formulas constructed in a proper way and often called wff. Elements of the language of the classical propositional or sentential logic/calculus represent propositions in a formal way using variables and constants. Propositional variables and constants are denoted (named) by letters, which are considered as atomic formulas and form a part of the alphabet of the language, while another part is formed by symbols of the following logical operations: negation is denoted either by or by ∼; for example, A means “not A”;
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conjunction also called “logical and” is denoted either by ∧ or by & or by ·; for example, A ∧ B means “A and B”; disjunction also called “logical or” is denoted by ∨; for example, A ∨ B means “A or B”; implication is denoted either by → or by ⇒ or by ⊃; for example, A → B means “A implies B”; equivalence is denoted either by ↔ or by ≡ or by ⇔; for example, A ↔ B means “A is equivalent to B”. If P (x) and Q(x) are some propositions or predicates, then: Their conjunction is equal to P (x) & Q(x) or P (x) ∧ Q(x). Their disjunction is equal to P (x) ∨ Q(x). The negation of P (x) is equal to P (x). It is possible to use fewer operations (and thus, a smaller alphabet) by expressing some of these operations by mean of others, e.g., P → Q is equivalent to P ∨ Q. For example, Church (1956) uses only one logical operation ⊃. In addition, the left and right parentheses, (and) or the left and right brackets [and] are included in the alphabet. It is possible to use fewer operations (and thus, a smaller alphabet) by expressing some of logical operations by means of others, e.g., P → Q is equivalent to P ∨Q. For example, Church (1956) uses only one logical operation ⊃ to build the classical propositional calculus. In addition to constants, variables and operation symbols, the left and right parentheses, (and) or/and the left and right brackets [and] are included in the alphabet. It is interesting that there are logics that use many other operations. For instance, linear logic has the following system of operations: • ⊗ is called multiplicative conjunction or multiplicative and or times (or sometimes tensor); • ⊕ is called additive disjunction or additive or or plus; • & is called additive conjunction or additive and or with; • $ is called multiplicative disjunction or multiplicative or or par; • ∧ is called classical conjunction; • ∨ is called classical disjunction;
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is called top; ⊥ is called bottom; ! is interpreted as of course (or sometimes bang); ? is interpreted as why not; −◦ is called linear implication;
Elements of the language LP of the classical propositional (or sentential) logic/calculus are called wffs and are traditionally built by the following rules: 1. Letters of the alphabet are wffs from LP . 2. If ϕ is a wff, then ϕ is a wff from LP . 3. If ϕ and ψ are wffs, then (ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ → ψ), and (ϕ ↔ ψ) are wffs from LP . These rules form the set of algorithms R that are used to build the language LP . Elements of the language LCP C of the classical predicate logic/calculus of the first order give a formal representation of binary properties. The predicate calculus language has a developed alphabet and elaborated symbolic notation. Lower-case letters a, b, c, . . ., x, y, z, . . . are traditionally used to denote individuals (variables or constants). Upper-case letters M , N , P , Q, R, . . . are traditionally used to denote (variable or constant) predicates. The alphabet LCP C of the language LCP C consists of six parts: — A set F of function symbols (e.g., +, × or ·). — A set P of predicate symbols (e.g., P1 or Pa ). — A set C of logical connectives or symbols of operations (usually, it is: , ∧, ∨, →, and ↔). — A set S of punctuation symbols (usually, it is: ( , ), : and , ). — A set Q of quantifiers (usually, they are ∀ and ∃, although sometimes other quantifiers such as ∀∀ and ∃! are also used). — A set V of variables. Note that non-classical logics often use not only logical operations but also logical operators. For instance, the modal operator
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expresses necessity, while the modal operator ♦ expresses possibility in modal logics. Every function symbol, predicate symbol, and connective has its arity. Namely, an n-ary function has the form f : X n → X and an n-ary predicate has the form P (x1 , x2 , . . ., xn ). As a rule, 0-ary predicates and/or 0-ary functions are called constants. Another way to deal with constants is to include their names in the alphabet of the language. The language LCP C of the classical predicate calculus, as a general structure, encompasses the language LP of the classical propositional calculus because propositions may be constructed by juxtaposition of a predicate with an individual constant or variable and using quantifiers. Elements of the language LCP C of the classical predicate logic/calculus are also called wffs and built by the following rules: (1) Letters of the alphabet A of the language LCP C are wffs from LCP C . (2) Expressions P (x1 , x2 , . . ., xn ) where P is an n-ary predicate symbol is a wff from LCP C . (3) If ϕ is a wff, then ϕ is a wff from LCP C . (4) If ϕ and ψ are wffs, then (ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ → ψ), and (ϕ ↔ ψ) are wffs from LCP C . (5) If H(x1 , x2 , . . ., xn ) is a wff containing a free variable x, then ∃ xH(x1 , x2 , . . ., xn ) and ∀ xH(x1 , x2 , . . ., xn ) are wffs from LCP C . Here a variable is free if it is not related to a quantifier. Consequently, the rule (5) makes any instance of x bound (that is, not free) in the formulas ∃xH(x1 , x2 , . . . , xn ) and ∀xH(x1 , x2 , . . . , xn ). In logic, symbol ∃ means “there is” or “there are” and symbol ∀ means “for any” or “for all”. If A = {ai ; i ∈ I} is an infinite set, then the expression “a predicate P (x) is true for almost all elements from A”, or “almost all elements from A have a property P ” or in logical notation ∀∀xP (x), means that P (x) can be untrue only for a finite number of elements from A. For instance, if A is the set N of all natural numbers, then
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almost all elements of A are bigger than 10. Taking the mathematical calculus (cf., for example, (Larson and Edwards, 2006; Burgin, 2008), we have another example of the quantifier ∀∀, namely, the conventional convergence of a sequence l = {ai ; i = 1, 2, 3, . . .} to number a means that any neighborhood O a of a contains almost all elements from l, or in the formal language, we have: lim ai = a if and only if ∀Oa∀∀i ∈ N (ai ∈ Oa).
i→∞
There are many other logics and they have different languages. For instance, conventional logics are extended to probabilistic logics by assigning probabilities to statements, i.e., to propositions and predicates. This allows representation of probabilistic knowledge (cf., for example, (Boole, 1854; Reichenbach, 1932; 1935; Hailperin, 1984; Russell, 2014)). This direction in formal logic was initiated by Leibniz, who envisioned that it would be necessary to estimate likelihood of propositions and a way of proof leading not to certainty but only to probability of propositions. However, Leibniz did not develop such a logic and it was George Boole, who introduced a mathematical concept of imprecise probability aiming to reconcile classical logic, which tends to express complete knowledge or complete ignorance, and probability theory, which has a propensity to express partial or/and imprecise knowledge or ignorance (Boole, 1854). It is necessary to remark that some logics have an essentially different structure. For instance, the “logic” of (non-relativistic) quantum mechanics is thought of as being the lattice of closed subspaces of a separable infinite dimensional Hilbert space (Mackey, 1963). 5.2.3. Logical systems of inference There is a tradition of opposition between adherents of induction and of deduction. In my view, it would be just as sensible for the two ends of a worm to quarrel. Alfred North Whitehead
Looking at the third level of logic, we see deductions and other inference operations and processes. A deduction system consists of
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deduction rules. Each deduction rule and consequently, each step of deduction has the form A→B
(2.1)
A B.
(2.2)
or Here A and B are some expressions or sets of expressions from the language of the corresponding logic. For instance, taking as A two propositions “We live in the USA” and “The USA are situated in the Western Hemisphere of the Earth,” we can deduce as B the proposition “We live in the Western Hemisphere of the Earth.” Thus, a deduction rule is (has the structure of) the fundamental triad (A, →, B). A deduction is a sequence (of applications) of deduction rules. A → B → C → D → E,
(2.3)
where A, B, C, D, and E are groups of logical statements. As a result, any deduction incorporates the structure of a deduction rule in a more elaborate structure of deduction, which is usually a sequence of named sets. The logical symbol |= is usually interpreted as truth-functional entailment (cf., for example, (Bergman et al., 1980) or (Kleene, 1967)). That is, if a and b are some propositions or predicates, then a |= b means “if a is true in some interpretation, then b is true in the same interpretation”. In this context, |= b means that b is always true. Dynamic properties of logics are realized by logical processes based on reasoning operations, such as inference or deduction. Deduction is always formal, while there are three classes of inference: formal inference, semiformal inference and informal inference. For instance, in practical mathematics, inference is, as a rule, semiformal, while in symbolic logic it is sometimes semiformal and sometimes formal, e.g., deduction in mathematical logic is a kind of formal inference. In turn, inference is a kind of reasoning when reasoning is performed step by step and each step is a transition from premises to conclusions. Another kind of reasoning is argumentation, the goal of
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which is not formula derivation but persuasion of the opponent or the audience. Here we consider dynamical structures in mathematical logic. The subject of mathematical logic has origins in philosophy. Philosophers explained how and why the logical rules for inference or deduction (such as the modus ponens or law of excluded middle) are valid. Later philosophical arguments were formalized and then put into the mathematical form. It is also a legacy from philosophy that mathematical logic distinguishes semantic analysis, answering the question “What is true?” from syntactic considerations, answering the question “How to express something true?” In such a way, logicians have developed two traditional parts of mathematical logic: its syntax (proof theory) and semantics (model theory). Basic dynamic structures of logic are logical calculi and logical varieties. The idea of the concept of logical calculus comes from Leibniz, who also introduced names differential calculus and integral calculus. He wrote that in future informal and vague arguments of philosophers would be changes for formal and exact calculations with formulas (Leibniz, 1989). Such calculations would allow one to find who of those philosophers was right and who was wrong. To make such generalized calculations with formulas, people use definite rules. Systems of rules form algorithms when they are precise, exactly realizable and sufficiently simple to be performed by a mechanical device. Otherwise, such systems are called procedures. Generalized calculations are performed with symbolic expressions, which are elements of definite languages, usually, formal languages. The goal of this work is to study relations between algorithms, procedures, languages, and calculi as a part of metalogic. On the syntactic level, dynamics of a formal logic goes on in an appropriate logical calculus in the traditional representation or by the corresponding syntactic logical variety in the advanced representation (Burgin, 2010). In mathematics, the word calculus has two meanings. The most popular in general mathematics understanding is that Calculus is a name that is now used to denote the field of mathematics that
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studies properties of functions, curves, and surfaces. As this is the most popular meaning in mathematics, we call it the Calculus. It is usually subdivided into two parts: differential calculus and integral calculus. The main tool of the Calculus is operating with functions to study properties of these functions. This operation can be regarded as a generalized calculation with these functions. This explains the name calculus used for this field, which originated from the Latin word meaning pebble because people many years ago used pebbles to count and do arithmetical calculations. The Romans used calculos subducere for “to calculate”. Thus, the Calculus is called so because it provides analytic, algebra-like techniques, or means of computing, which apply algorithmically to various functions and curves. Many mathematical problems that had very hard solutions or even such problems that mathematicians had not been able to solve, after the calculus had been developed, became easily solvable by mathematics students. Later the Calculus developed into analysis, or mathematical analysis. There are also other calculi in analysis, for example, operational calculus and calculus of variations. Another mathematical meaning of the word calculus comes from mathematical logic where calculus is a formal system used for logical modeling of mathematical and scientific theories. In a traditional setting, a logical calculus consists of three parts: axioms, rules of deduction (inference), and theorems (cf., (Kleene, 2002; Mendelson, 1997). There are two meanings of the term logical calculus. According to the classical understanding, a logical calculus consists of a set A of axioms or a priori statements, a set T of theorems or deduced statements, and a set R of rules of deduction or inference. We call this system (A, R, T ) a standard logical calculus or simply, a logical calculus. If such a system (A, R, T ) contains non-logical axioms, it is called a formal theory. In another interpretation, a logical calculus consists only of a logical language M and a set R of rules of deduction or inference. We call this system (M , R) a free logical calculus. Axioms form the foundation of a logical calculus because the axiomatic approach provides efficient tools for knowledge
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compression, storage, extraction and production. Axioms store knowledge of the calculus, as well as of the represented system, in the compressed form. Inference rules allow one to extract knowledge from axioms and previously inferred knowledge items producing in such a way new knowledge. There is a long-standing debate whether a system that holds axioms also possesses all knowledge inferred from these axioms. To adequately answer this question and clarify the situation, we need to make a distinction between potential knowledge and actual knowledge. What is stored in the form of data in a knowledge system is actual knowledge of this system. What a knowledge system can extract or produce from its actual knowledge is inner potential knowledge of this system. What a knowledge system can extract or produce from accessible to it knowledge is outer potential knowledge of this system. Taking a logical calculus, it is natural to assume that axioms form the actual knowledge of this calculus, while theorems give the inner potential knowledge of this calculus. Potential knowledge of a logical calculus is produced by inference (deduction) algorithms. This naturally brings us to the levels of inner potential knowledge. Definition 5.2.5. If C = (A, R, T ) is a logical calculus, then the nth level of inner potential knowledge consists of all theorems from T for which there is an inference by rules from R with not more than n steps. By this definition, all axioms from A are situated at the zero level. Levels of inner potential knowledge allow us to solve the Problem of Omniscience. Namely, Omniscience in this context means that a person who has some knowledge K in the logical form and logical rules of (inference) also knows all knowledge deducible from K. The Problem of Omniscience asks whether it is possible for a human being really to have infinite knowledge. The solution to this problem is that a person who has some knowledge K in the logical form and logical rules of (inference) knows only some number of levels of knowledge deducible from K.
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Logical calculi are constructed using different logical languages. A language in a constructive representation/definition is a fundamental triad (named set) of the form L = (X, R, L) where X is the alphabet of L, R is the set of constructive algorithms/rules for building well-formed (correct) expressions (e.g., words or texts) from L, and L is the set of words of the language L. Logical languages are a special kind of artificial languages developed intentionally within a culture. The typical feature of logical languages is that their structure (inner relations) and grammar (formation rules) are intended to express the logical information within linguistic expressions in a clear and effective way. Languages used in logic have, as a rule, constructive definitions in a form of logical operations considered in the previous section. The rules of these operations form the set of algorithms R that build the language L. Elements of logical languages are logical expressions or formulas. To emphasize that these formulas are constructed in a proper way, they are often called wffs. The language LP of the classical propositional (or sentential) logic/calculus is considered in the previous section. Elements of the language of the classical propositional or sentential logic/calculus are formulas that give a formal representation of propositions. Propositional variables are denoted by letters, which are considered as atomic formulas and form a part of the alphabet of the language. The language LCPC of the classical predicate logic/calculus is considered in the previous section. Elements of the language of the classical predicate logic/calculus are formulas that give a formal representation of binary properties. Now we give a general formal definition of a logical calculus. Let L be formal (logical) language or a language of well-formed formulas and R be an algorithmic language, procedural language or a language of rules of inference in L. Definition 5.2.6. A (syntactic or deductive) logical calculus, usually called calculus, in the pair of languages (L, R) is a triad of the form C = (A, H, T ).
(5.15)
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where H ⊆ R, A, T ⊆ L, A is the set of axioms, H consists of inference rules (rules of deduction) by which from axioms the theorems of the calculus are deduced, and the set of theorems T is obtained by applying algorithms/procedures/rules from H to elements from A. When L is a logical language and H consists of rules of logical deduction, C is a deductive calculus. The Principle or Law of Excluded Middle, or in Latin, tertium non datur (a third is not given) is one of the basic axioms of the classical logic It is formulated as “A has a property B or A does not have a property B.” Categorical syllogisms of Aristotle gives examples of classical inference rules (cf., for example, (Timothy, 1973)). The set A in the formula (3.3.1) is called the base (an axiom system or generating expressions) of the calculus C . The set H is called the system of inference rules of the calculus C . The set T is called the body (the set of theorems or deducible expressions) of the calculus C and is constructed by applying algorithms from H to expressions from A. They are denoted as follows: A = A(C ),
H = H(C ),
T = T (C ).
The named set (A, R, T ) is called the basic named set of the calculus C . The same calculus may be represented by another (deduction) named set (A, d, T ) where the relation d connects any axiom a from A with such theorems t from T that a is used in a process of deduction of t. Note that there calculi that are not logical. For instance, differential calculus is a purely mathematical construction. That is, when L contains descriptions and denotations of real/complex numbers and functions, while H consists of rules of differentiation/integration, C is the differentiation/integration calculus. A deduction system consists of deduction rules. Each deduction rule and consequently, each step of deduction has the form A→B
(5.16)
A B.
(5.17)
or
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Here A and B are some expressions or sets of expressions from the language of the corresponding logic. For instance, A consists of two formulas ϕ and ϕ → ψ, while B is the formula ψ. Thus, a deduction rule is the named set (A, →, B). Often the system of inference rules has only one rule called modus ponens: ϕ, ϕ → ψ ψ.
(5.18)
In the programming language notation, it means If ϕ
and ϕ → ψ,
then ψ.
(5.19)
Other rules are derived from modus ponens and then used in formal proofs to make proofs shorter and more understandable. These rules serve to directly introduce or eliminate connectives, e.g., “If ϕ and χ,
then ϕ ∧ χ“(or ϕ, χ ϕ ∧ χ)
or “If ϕ,
then ϕ ∨ χ“(or ϕ ϕ ∨ χ).
A standard transformation rule is substitution. This rule is necessary because axiom schemas demand substitution to become axioms and be applied. Different systems of axioms for the classical propositional calculus have been devised to achieve consistency, completeness, and independence of axioms. All these systems are logically equivalent. For instance, Kleene (2002) suggests the following list of axioms (axiom schemas) for the classical propositional calculus, in which Greek letters denote propositions: ϕ → (χ → ϕ),
(5.20)
(ϕ → (χ → ψ)) → ((ϕ → χ) → (ϕ → ψ)),
(5.21)
ϕ → (χ → (ϕ ∧ χ)),
(5.22)
ϕ → ϕ ∨ χ,
(5.23)
χ → ϕ ∨ χ,
(5.24)
ϕ ∧ χ → ϕ,
(5.25)
ϕ ∧ χ → χ,
(5.26)
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(ϕ → ψ) → ((χ → ψ) → (ϕ ∨ χ → ψ)),
(5.27)
(ϕ → χ) → ((ϕ →χ) → ¬ ϕ),
(5.28)
ϕ → ϕ.
(5.29)
Usually the system of inference/deduction rules of the classical propositional calculus has only one rule called modus ponens (5.18). A standard transformation rule is substitution. This rule is necessary because axiom schemas demand substitution to become axioms and be applied. Shoenfield (1967) suggests the following list of axiom schemata of the classical first-order predicate calculus: Propositional Axiom: ϕ ∨ ϕ Identity Axiom: x = x Substitution Axiom: ϕx [a] → ∃xϕ Equality Axioms: (a) If F is a symbol of an n-ary function from F, then x1 = y1 ∧ x2 = y2 ∧ . . . ∧ xn = yn → F (x1 , x2 , . . . , xn ) = F (y1 , y2 , . . . , yn ); (b) If P is a symbol of an n-ary predicate from P, then x1 = y1 ∧ x2 = y2 ∧ . . . ∧ xn = yn → P (x1 , x2 , . . . , xn ) = P (y1 , y2 , . . . , yn ). Usually the system of inference/deduction rules of the classical predicate calculus has two basic rules. One is modus ponens (5.18). The other is called the substitution rule. It states: If t is a term, ϕ is a formula possibly containing the variable x, and ϕ[t/x] is the result of replacing all free instances of x by t in ϕ, then ϕ implies ϕ[t/x], or as a formal expression: t . (5.30) ϕϕ x Shoenfield (1967) also suggests the following inference rules: Extension rule: ϕ implies ϕ ∨ ψ Cancellation rule: ϕ ∧ ψ implies ϕ
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Associative rule: (ϕ ∨ ψ) ∨ χ = ϕ ∨ (ψ ∨ χ) Cut rule: (ϕ ∨ ψ) and (ϕ ∨ χ) imply (ψ ∨ χ) ∃-introduction rule: ϕ → ψ implies ∃xϕ → ψ if x is not a free variable in ψ. While inference/deduction rules represent microsteps of reasoning, there are also macrosteps of reasoning. One of the most important macrosteps of reasoning, which is extensively used in mathematics, is the, so-called, proof from contradiction. The essence of this proof is that trying to prove that some system (object) A has a property P , we make an assumption that A does not have this property P . Then we show that this contradicts the initial conditions. This allows us to conclude that our assumption was not true and due to the Principle or Law of Excluded Middle, the system (object) A has the property P . There are many other logics and they have different calculi. For instance, proof nets used in linear logic give another example of macrosteps of reasoning. It is possible to read more about classical logical systems and structures, for example, in ((Kleene, 2002; Shoenfield, 1967; Bergman et al., 1980; Manin, 1991; Mendelson, 1997). Syntactic logical calculi provide functional formalization to the notion of a formal theory (Smullyan, 1962). In turn, a formal theory formalizes some source theory from a scientific discipline (e.g., from mathematics, physics, or economics). In order to specify a formal theory, predicates, functions, and relations, which are regarded as basic for a given field of study, are chosen. These predicates delimit the scope of the formal theory and are the primitives of the theory and together with logical and punctuation symbols form the alphabet of the theory language.
5.3. Theory of abstract properties Democracy is when the indigent, and not the men of property, are the rulers. Aristotle
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There are different generalizations of logical systems. One of the most far-reaching generalizations is the theory of abstract properties developed in (Burgin, 1985a; 1986; 1989) and applied to social sciences (Burgin, 1990a), to epistemology in (Burgin and Rothbart, 1998) and to methodology of science in (Burgin and Kuznetsov, 1992; 1992b; 1993; 1994). Properties are very important. As the great Aristotle wrote, we can know about things nothing but their properties. Thus, it is natural that properties play an important role in mathematics, logic and all sciences. However, concepts of properties in mathematics, logic and science are basically different. In mathematics and logic, a property is represented by the predicate that defines the set of all objects that have this property in common. For instance, if P is a predicate on a set X, it is traditionally said P is a property on X, while the notation P (x) is used to denote a sentence or statement P that an object x has the property P . Then the set of all the objects that have this property P is denoted by the formula {x|P (x)}, meaning that it is just a set of all x for which P is true. By definition, a predicate P on X is a Boolean-valued function P : X → {true, false} and thus, it is an abstract property. Similar situation exists in philosophy and linguistics where the ontological fact that something has a property is typically represented in language by applying a predicate to a subject. For instance, Swoyer and Orilia write: “Properties (also called ‘attributes,’ ‘qualities,’ ‘features,’ ‘characteristics,’ ‘types’) are those entities that can be predicated of things or, in other words, attributed to them. For example, if we say that that thing over there is an apple and is red, we are presumably attributing the properties red and apple to it. Thus, properties can be characterized as predicables. Relations, e.g., loving and between, can also be viewed as predicables and more generally can be treated in many respects on a par with properties. Indeed, they may even be viewed as kinds of properties.” (Swoyer and Orilia, 2014)
Indeed, relations are also a kind of multivariable abstract properties (Burgin, 1985).
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Based on understanding property as a predicate, logicians developed formal theories of properties, which are formalized systems that aim at formulating “general non-contingent laws that deal with properties” (Bealer and M¨ onnich, 1989). To do this, such theories construct terms corresponding to properties, e.g., variables with systems of properties as their domain. There are two main approaches in this area: either the terms standing for properties are predicates (Cocchiarella, 1986) or such terms are subject terms that can be linked to other subject terms by a special predicate, e.g., pred, that is meant to express a predication relation similar to representation of the membership relation in standard set theory by the special predicate ∈ (Bealer, 1982). For instance, in the first approach, the formula ∃P (P (j) & P (m)) represents the statement “there is a property that both Alex and Bob have,” while in the second approach, this statement is represented by the formula ∃x (pred(x,j) & pred(x,m)). In Menzel (1986; 1993), both approaches are combined. With its development, logic created models for properties truthfulness of which had more than two values. For instance, fuzzy predicates and propositions were introduced into logic in the 20th century to represent fuzzy properties. Truthfulness of fuzzy properties may be partial with values in the interval [0, 1]. Consequently, fuzzy properties are not physical or other scientific properties. Nevertheless, fuzzy predicates and propositions are also abstract properties, the scale of which is equal to the interval [0, 1]. A very different concept of property exists in science. For instance, a physical property is any measurable property, quality or attribute whose values describe states of physical systems. The changes in the physical properties of a system reflect transformations of this system. Physical properties are often called observables. Thus, in science, properties are not predicated but measured and in contrast to predicated properties, they have different scales and diverse values. That is why the goal of creation of the theory of abstract properties was a mathematical synthesis of the concept of property in mathematics and logic with the concept of property in science. The main concept in this theory is abstract property.
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Let us take a class (universe) U of objects, an abstract class M of partially ordered sets, i.e., such a class that with any partially ordered set contains all partially ordered sets isomorphic to it, and consider a partially ordered set L from M. Definition 5.3.1. (a) An abstract property P of objects from the universe U is a named set P = (U , p, L) where p : U → L is a partial function (L-predicate). (b) The partial function p is called the evaluation function Ev(P ) (or the functional component) of the property P . (c) The partially ordered set L is called the scale Sc(P ) of the abstract property P . We write P (u) = ∗ when the property P is undefined for the object u. Abstract properties are used as mathematical models of real properties. For instance, when you want to know something about a computer, you look into a list of its specifications, which contains many properties of a computer, which are examples of abstract properties. The universe U where these properties are defined consists of computers. Here is one of such specifications: Table 5.1. A specification of a computer. System memory installed Hard disk capacity Monitor diagonal size Graphics memory amount Graphics chipset Supported operating systems Graphics card Monitor type Processor model Processor clock speed
128 MB 30 GB 18 in. 64 MB nVidia GeForce II GTS Windows 98 SE Guillemot three-dimensional Prophet II GTS CRT Athlon 1000 MHz
Some properties of computers are now incorporated into their names. For instance, when you see the name “Dell Dimension L866r
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Pentium III 866 MHz,” you know that this computer has the processor clock frequency 866 MHz. In turn, the processor clock frequency determines how many instructions it can execute per second. An important property of physical bodies is mass. The scale of this property is the infinite interval [0, ∞) of the real line. The most popular property in logic is truth defined for logical expressions. In classical logics, only one such property as truth with the scale L = {T, F} is considered, i.e., predicates and propositions take one of two value T (true) and F (false). Thus, the set {T, F} is the scale of the abstract properties that represent predicates and propositions. For modal logics, which have only one truth property that is determined for logical expressions, modalities are expressed by means of modal operators. At the same time, modal operators are abstract properties defined for well-formed formulas and taking values in modal well-formed formulas. For instance, the modal operator of :f → f for any well-formed formula f . necessity is defined as Another possibility to express modality is to determine different modal truth properties: “truth,” “necessary truth”, and “possible truth”. These truth properties are also abstract properties. Valued sets give one more example of abstract properties (Dukhovny and Ovchinnikov, 2000; Ovchinnikov, 2000; Frascella and Guido, 2008). We remind that a valued set is a function from a given set into a given linearly ordered set L. Thus, we see that valued sets also are particular cases of abstract properties and thus, they are represented by named sets (Burgin, 2011). There are several methods to represent concepts by abstract properties. For instance, we can take an abstract property P of names with the scale that consists of conceptual representatives. It is also possible to represent concepts by abstract properties by an abstract property P so that P assigns the Extent of a concept C to the name of C. We obtain another representation when the Meaning of a concept C is assigned to the name of C. Taking the next level of logic, we see that it is possible to represent statements and propositions by abstract properties. Indeed, a statement or a proposition is the information content of a well-formed
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declarative sentence. It means that statements and propositions are represented by declarative sentences, for example, in English. At the same time, there are two basic parts to every sentence in English: the subject and the predicate (Webster’s English Language Desk Reference, 1999). The simple subject is the noun or pronoun that identifies the person, place, or thing the sentence is about. The complete subject is the simple subject and all the words that modify it. The predicate contains the verb that explains what is going on with the subject. The simple predicate contains only the verb, while the complete predicate contains the verb and any complements and modifiers. Thus, taking the property of subjects that takes values in predicates, we obtain an abstract property, which represents all sentences in English. Consequently, this abstract property also represents all statements and propositions expressed by English sentences. An important construction in the theory of abstract properties is reduction (Burgin, 1989; 2007b). Informally, reduction of an abstract property P1 to an abstract property P2 means that knowing values of the abstract property P2 , we can find values of the abstract property P1 . This idea is formalized in the following way giving several types of abstract property reductions. Let us consider a pair of universes U and V , a pair of classes of mappings Φ and Ψ, and a pair of properties P = (U, p, L) and R = (V, r, M ). Definition 5.3.2. A property P = (U, p, L) is reduced to a property R = (V, r, M ) in the pair (Φ, Ψ) if there is there are mappings g : U → V from Φ and h : M → L from Ψ such that h preserves the partial order in M and p = h ◦ r ◦ g, i.e., the following diagram is commutative. p U
L
g
h V
M r
Reduction allows establishing relations between properties of objects from different universes.
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Note that it is possible to treat reduction either as a relation on or as an operation with abstract properties. There are many reductions in the theory of algorithms, automata, and computation. Let us consider some of them. Example 5.3.1. In the axiomatic theory of algorithms, automata and computation, the most general concept of operational reduction is introduced (Burgin, 2007b). Let us assume that all algorithms (automata) are defined and take values in a set X, e.g., in the set Σ∗ of all words in the alphabet Σ, and consider two classes of algorithms (automata) R and Q, which play the role of the classes of mappings Φ and Ψ from Definition 5.3.2. Definition 5.3.3. An algorithm A is (R,Q) — reducible to an algorithm B if there is an algorithm D from R and an algorithm H from Q such that for any element x ∈ X, we have A(x) = H(B(D(x))), or rD◦ rB ◦ rH where rX is the mapping determined by an algorithm (automaton) X. Reduction of algorithms (automata) is represented by the following commutative diagram: A X
X
H
D
X
X B
For instance, any Turing machine is reducible to a universal Turing machine. This reduction and other reductions of algorithms and automata have been very useful in computer science allowing researchers to prove many useful and important results such as undecidability of many algorithmic problems, NP-completeness of a variety of practical problems or equivalence of different classes of algorithms and automata (Minsky, 1967; Sipser, 1997; Burgin, 2005; 2007b; 2010d). Another example of reduction of properties is reduction of problems, which is defined in the following way (Burgin, 2010d).
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Definition 5.3.4. A problem P can be reduced to a problem Q if knowing a solution to the problem Q, it is possible to find a solution to the problem P . To describe this relation by property reduction, we consider a property Sol = (Pr, solproc, sol) where Pr is a set of problems, sol is a set of problem’s solutions and solproc is a set of solving processes, each of each assigns (finds) a solution to a problem. Reduction of problems is extensively used in mathematics and computer science. For instance, in arithmetic, division of fractions is reduced to multiplication of fractions by the rule c b d b ÷ = · . a d a c This is reduction of the property division Div : F 2 → F to the property multiplication Mlt : F 2 → F where F is the set of all fractions and the properties (operations) are defined in the following way Div((b/a), (c/d)) = (b/a) ÷ (c/d) and Mlt((b/a), (c/d)) = (b/a)· (c/d). The reduction mapping is g: F 2 → F 2 where g((b/a), (c/d)) = ((b/a), (d/c). The following commutative diagram corresponds to this reduction. F2
Div F
g 2
F
Mlt
In addition, in arithmetic, subtraction of signed numbers is reduced to addition by the rule a − b = a + (−b). This is reduction of the property subtraction Sn : R2 → R to the property addition An : R2 → R where R is the set of all real numbers and the properties (operations) are defined in the following way: Sn(a, c) = a − c and Mn(a, c) = a + c. The reduction mapping is g : R2 → R2 where g(a, c) = (a, −c).
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The following commutative diagram corresponds to this reduction. R2
Sn
g
R R
2
Mn
Besides, multiplication of numbers is reduced to addition, squaring, and subtraction by the rule a × b = ((a + b)2 − a2 − b2 )/2. This is also reduction of properties. Let us consider some properties of reduction. Proposition 5.3.1. If a property P is reduced to a property R and the property R is reduced to a property T , then the property P is reduced to the property T . Proof is left as an exercise. As it is possible to reduce any property to itself, Proposition 5.2.1 implies the following result. Proposition 5.3.2. Reduction of properties induces a partial preorder on properties. Reduction of properties also generates an equivalence relation. Definition 5.3.5. Properties P and R are r-equivalent if the property P is reduced to the property R and the property R is reduced to the property P . Proposition 5.3.2 implies the following result. Proposition 5.3.3. r-equivalence of properties is an equivalence relation on properties. Example 5.3.2. The famous formula E = mC 2 induces r-equivalence of properties mass m and energy E. Indeed, E is reduced to m by the mapping g(x) = xC 2 and m is reduced to E by the mapping h(x) = xC −2 . Remark 5.3.1. r-equivalence of properties is a special case of real equivalence of properties studied in (Burgin, 1985).
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Remark 5.3.2. Reduction of properties is defined by a contravariant morphism of named sets that represent these properties (Burgin, 2011). Two special cases of extended reduction — direct and inverse reductions — are especially important for various applications. We obtain direct reductions by restricting the class of mappings Ψ exactly to identical mappings and acquire inverse reductions by restricting the class of mappings Φ exactly to identical mappings. Let us consider a universe U and two properties P = (U , p, L) and R = (U , r, M ). Definition 5.3.6. A property P = (U , p, L) is directly reduced to a property R = (U , r, M ) if there is there is a mapping h : M → L where h preserves the partial order in M such that p = h◦r, i.e., the following diagram is commutative. U
p
r
L M
h
Definition 5.3.7. A direct reduction of a property P = (U , p, L) to a property R = (U , r, M ) is strict if M = L. For instance, if we take such properties as the weight P1 of an individual in kilograms and the weight P2 of an individual in pounds, then P1 is reducible to P2 and P2 is reducible to P1 . Example 5.3.3. The concept of the right reduction from the axiomatic theory of algorithms, automata and computation is an example of a strict direct reduction of properties (Burgin, 2007b). Indeed, let us assume that all algorithms (automata) are defined and take values in a set X, e.g., in the set Σ∗ of all words in the alphabet Σ, and consider a class of algorithms (automata) Q. Definition 5.3.8. An algorithm A is right Q-reducible to an algorithm B if there is an algorithm H from Q such that for any element x ∈ X, we have A(x) = H(B(x)), or rB ◦ rH where rX is the mapping determined by an algorithm (automaton) X.
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Reduction of algorithms (automata) is represented by the following commutative diagram: X
A
B
X X
H
Properties of reduction in general imply corresponding properties of right reduction. Proposition 5.3.1 implies the following results. Corollary 5.3.1. If a property P is a direct reduction of a property R and the property R is a direct reduction of a property T , then the property P is a direct reduction of the property T . Corollary 5.3.2. If a property P is a strict direct reduction of a property R and the property R is a strict direct reduction of a property T , then the property P is a strict direct reduction of the property T . Proposition 5.3.2 implies the following results. Corollary 5.3.3. Direct reduction of properties induces a partial preorder on properties. Corollary 5.3.4. Strict direct reduction of properties induces a partial preorder on properties, which correlates with the preorder induced by direct reduction. Direct reduction of properties also generates an equivalence relation. Definition 5.3.9. Properties P and R are (strictly) dr-equivalent if the property P is a (strict) direct reduction of the property R and the property R is a (strict) direct reduction of the property P . Proposition 5.3.3 implies the following results. Corollary 5.3.5. dr-equivalence of properties is an equivalence relation on properties. Corollary 5.3.6. Strict dr-equivalence of properties is an equivalence relation on properties.
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There is also another kind of abstract property reduction. Definition 5.3.10. A property P = (U , p, L) is inversely reduced to a property R = (V , r, L) if there is there is a mapping h: M → L where h preserves the partial order in M such that p = h◦ r, i.e., the following diagram is commutative. U
p L
g V
r
Definition 5.3.11. An inverse reduction of a property P = (U , p, L) to a property R = (V , r, L) is strict if U = V . Example 5.3.4. An enumeration v is a mapping from the set N of all natural numbers onto an arbitrary countable set (Ershov, 1977). It means that an enumeration v of a set A is an abstract property Ev = (N , v, A). An enumeration v of a set R is reduced to an enumeration u of the same set if there is a mapping f : N → N such that v = uf , that is, when the following diagram is commutative. f N
N
v
u
(5.31)
R
We see that reduction of enumerations is a strict inverse reduction of abstract properties. Example 5.3.5. The concept of the left reduction from the axiomatic theory of algorithms, automata and computation is an example of inverse reduction of properties (Burgin, 2007b). Indeed, let us assume that all algorithms (automata) are defined and take
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values in a set X, e.g., in the set Σ∗ of all words in the alphabet Σ, and consider a class of algorithms (automata) R. Definition 5.3.12. An algorithm A is left R-reducible to an algorithm B if there is an algorithm D from R such that for any element x ∈ X, we have A(x) = B(D(x)), or rD◦ rB where rX is the mapping determined by an algorithm (automaton) X. Reduction of algorithms (automata) is represented by the following commutative diagram: X
A X
D X
B
Definition 5.3.13. An algorithmic problem P can be m-reduced or is m-reducible to an algorithmic problem Q in a class K if there is an algorithm/automaton A from K and an injection m : D(P) → D(Q) realized in K by an algorithm/automaton M such that given an element x from D(P) and a solution SQ (M (x)) of the problem Q for the element M (x) from D(Q), the value A(M (x)) is a solution to the problem P for the element x from D(P), and in this way, it is possible to obtain all solutions to the problem P, i.e., if the problem Q does not have a solution for the element M (x), then P does not have a solution for the element x. We remind that the domain D(Q) of a problem Q is the set of all tentative initial conditions for this problem. Reduction of problems helps finding characteristics of these problems, such as decidability or recognizability. Let us consider a class of automata/algorithms K, automata from which determine total functions and which is closed with respect to sequential composition (cf., (Burgin, 2010d)). Proposition 5.3.4. If a problem P is m-reducible in the class K to a problem Q and the problem Q is decidable in K, then the problem P is also decidable in K. Indeed, if an algorithm B from K decides the problem Q and an algorithm A from K reduces the problem P to the problem Q, then
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the sequential composition of A and B decides the problem Q and is in K. Properties of reduction in general imply corresponding properties of right reduction. Proposition 5.3.1 implies the following results. Corollary 5.3.7. If a property P is an inverse reduction of a property R and the property R is an inverse reduction of a property T , then the property P is an inverse reduction of the property T . Corollary 5.3.8. If a property P is a strict inverse reduction of a property R and the property R is a strict inverse reduction of a property T , then the property P is a strict inverse reduction of the property T . Proposition 5.3.2 implies the following results. Corollary 5.3.9. Inverse reduction of properties induces a partial preorder on properties. Corollary 5.3.10. Strict inverse reduction of properties induces a partial preorder on properties, which correlates with the preorder induced by inverse reduction. Direct reduction of properties also generates an equivalence relation. Definition 5.3.14. Properties P and R are (strictly) ir-equivalent if the property P is a (strict) inverse reduction of the property R and the property R is a (strict) inverse reduction of the property P . Proposition 5.3.3 implies the following results. Corollary 5.3.11. ir-equivalence of properties is an equivalence relation on properties. Corollary 5.3.12. Strict ir-equivalence of properties is an equivalence relation on properties. To include logic into the theory of abstract properties, it is necessary to define operations with abstract properties because there
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different operations in logic. The basic logical operations are: negation denoted either by or by ∼; conjunction denoted either by ∧ or by & or by ·; disjunction denoted by ∨; implication denoted either by → or by ⇒ or by ⊃; and equivalence denoted either by ↔ or by ≡ or by ⇔. So, we need similar operations with abstract properties. We build these operations in a more general way. Let us consider n abstract properties Pi = (U , pi , Li ) (i = 1, 2, 3, . . . , n), a partially ordered set L and a mapping ω : L1 × L2 × L3 × · · · × Ln → L of the Cartesian product L1 × L2 × L3 × · · · × Ln into L. Definition 5.3.15 (Burgin, 1985). A property P = (U , p, L) is called the ω–composition of the properties Pi = (U , pi , Li ) (i = 1, 2, 3, . . ., n) if for any object u from U , the following conditions are satisfied ω(P1 (u), P2 (u), P3 (u), . . . , Pn (u)) for all i = 1, 2, 3, . . . , n P (u) = ∗ otherwise
when Pn (u) = ∗
Proposition 5.3.5. All basic logical operations with classical propositions and predicates–, &, ∨, →, and ↔–are compositions of abstract properties in the form of propositions and predicates. Proof . We show what mappings of the scales generate basic logical operations representing these mapping by corresponding mapping tables. We use here the scale {T, F } although it is also possible to use the scale {0, 1}. The mapping table for negation : x
x
T
F
F
T
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The mapping table for conjunction &: x
y
x&y
T
T
T
F
T
F
T
F
F
F
F
F
The mapping table for disjunction ∨: x
y
x∨y
T
T
T
F
T
T
T
F
T
F
F
F
The mapping table for implication →: x T F T F
y T T F F
x→y T T F T
The mapping table for equivalence ↔: x T F T F
y T T F F
x↔y T F F T
It is easy to check that the compositions defined by these mapping define corresponding logical operations. In a similar way, we prove the following result.
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Proposition 5.3.6. All basic logical operations with fuzzy propositions and predicates–, &, ∨, → and ↔– are compositions of abstract properties in the form of propositions and predicates. There are also other compositions of abstract properties. Proposition 5.3.7. Any arithmetical operation, e.g., addition, multiplication or division, determines a composition of abstract properties with number scales. For instance, the property ma where m(u) is the mass of the body u and a is its acceleration is the multiplicative composition of properties m and a. According to the Newton’s law F = ma, is r-equivalent to the property F , which is the force acting on the body. The Newton’s gravitational law F = G(mM/r 2 ) gives a more complicated composition of properties. To immerse logic into the theory of abstract properties, we also need partial reductions of abstract properties, which allow one to represent logical deduction. Let us consider a universe U , two properties P = (U, p, L) and R = (U , r, M ) and elements a from L and b from M . Definition 5.3.16 (Burgin, 2010). A property P = (U , p, L) is (a, b)-reduced to a property R = (U , r, M ) if the value of P is b whenever the value of R is a, i.e., R(u) = a implies P (u) = b for any u from U . Partial reduction of abstract properties is intrinsically related to logical deduction. Deduction is a technique obtaining new true statements from given true statements. It means that if we consider such a property as truthfulness with two values T and F , then the conclusion of a deduction rule has to be true if the premises are true. This exactly means that the conclusion of a deduction rule is (T , T )-reduction of the conjunction of the premises. The most popular deduction rule is modus ponens, which has the form A, A → B ⇒ B, where A are B are propositions, or equivalently, A&(A → B) ⇒ B.
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Table 5.2. A truth table. A
B
A→B
A & (A → B )
T F T F
T T F F
T T F T
T F F F
According to modus ponens, if A and A → B are true, then B is true as the truth Table 5.2 shows. The truth-value of a proposition is a property of this proposition with the scale {0, 1}. Therefore, from Table 5.2, we see that the property B is (T , T )-reduced to the property A & (A → B), while the property A & (A → B) is (F , F )-reduced to the property B. This shows that the mathematical theory of abstract properties includes logic as its subtheory (Burgin, 1989). It is possible to regard the theory of abstract properties as a synthesis of logic and qualitative physics (Bobrow, 1984). Using operations in the domain of abstract properties, it is possible to reduce systems of abstract properties to one abstract property. This reduction is based on the following concept of equivalence. Definition 5.3.17. An abstract property P = (U, p, L) is equivalent to a system Z of abstract properties {Pi = (U, pi , Li ); i ∈ I} if the validity of the inequality Pi (a) = Pi (b) for some i ∈ I implies P (a) = P (b) for any a, b ∈ U, and vice versa. Composition of properties makes it possible to prove the following result (Burgin, 1990a)). Theorem 5.3.1. For any system Z = {Pi = (U, pi , Li ); i ∈ I} of abstract properties, there is a property P = (U, p, L) equivalent to Z. Abstract properties allow us to define natural properties. Let us consider two sets of transformations TU and TL . The first one, TU , consists of natural transformations of the universe U , while the second one, TL , consists of admissible transformations of the scale L.
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Definition 5.3.18. An abstract property P = (U, p, L) is called invariant with respect to TU , if for any transformation H ∈ TU and any object A from U, we have p(A) = p(H(A)). For instance, such property of a physical body as speed is invariant with respect to linear transformations of the physical space. Definition 5.3.19. An abstract properties P = (U, p, L) and Q = (U, q, M ) are called transformationally equivalent with respect to TL , if there is a transformation G ∈ TL such that M = G(L) and for any object A from U, q(A) = G(p(A)). For instance, such properties as the weight of a physical body in kilograms is transformationally equivalent to the weight of a physical body in grams and to the weight of a physical body in pounds. Definition 5.3.20. The equivalence class with respect to admissible transformations TL of abstract properties invariant with respect to natural transformations TU is called a natural property. For instance, such property of a physical body as weight is a natural property. 5.4. Semantic networks and ontology I learned very early the difference between knowing the name of something and knowing something. Richard Feynman
Semantic network or semantic net is a knowledge representation formalism that is based on a mathematical concept called a graph (cf., Appendix) and describes objects and their relationships in the form of a network consisting of nodes and (usually directed) links between nodes in the form of arcs or arrows. The nodes represent objects or concepts by their names, while the links represent relations between nodes also by their types (names). Both nodes and arcs are labeled by the names of these nodes and arcs although when a semantic network contains arcs of one type, these arcs are not labeled. In such a way, a semantic network defines a set of binary relations on a set of nodes. In the mathematical context, a semantic network is a labeled
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(usually directed) graph. The structure of a semantic network defines its meaning. Typical semantic relations (links in a semantic network): • is a is a relation that indicates that one object is a subset of another object in the network, e.g., “Elephant is a mammal”. • is an instance of is a relation that indicates that one object is an element of another object in the network, e.g., “Jumbo is an instance of elephant”. • is a prototype of is a relation that indicates that one object is a special case or a model of another object. • is a part of is a relation that indicates that one object is a physical part of another object, e.g., “Wheel is a part of a car”. It is possible to find other semantic relations in Section 4.1.2. Semantic networks became very popular when researchers started to use them in artificial intelligence and machine translation although earlier versions have long been used in philosophy, psychology, and linguistics. The oldest known semantic network was drawn in the 3rd century C.E. by the Greek philosopher Porphyry in his commentary on Aristotle’s categories. Porphyry used this network to illustrate Aristotle’s method of defining categories by specifying a genus or general type, which encompasses the special cases as the subtypes of the general type. Then this procedure was iterated for introduced subtypes and so on. The structure of this semantic network is a special kind of graphs called a forest. For computers, semantic networks were first introduced by Richard H. Richens of the Cambridge Language Research Unit in 1956 as an “interlingua” for machine translation of natural languages because semantic networks allow spreading activation, imposing inheritance, and using nodes as representations of objects.
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Most semantic networks are cognitively based and can be organized into a taxonomic hierarchy. The declarative graphic form of semantic networks provides for their efficient utilization aimed at knowledge representation and support of automated systems for reasoning about the knowledge. Network form and linear notation are both capable of expressing equivalent knowledge, but certain representational mechanisms are better suited to one form or the other. For instance, network form is more efficient for operation while linear notation is more suitable for derivation. Some semantic networks are highly informal, while others are formally defined, e.g., as systems in logic. For formal definitions, it is possible to use either languages of network theory, such as mathematical schema theory (Burgin, 2005; 2006; 2010a) and semantic link network theory (Zhuge, 2004; 2010; 2012; Zhuge and Shi, 2003; 2004), or languages of logic. Sowa (1991) discerns six types of semantic networks. 1. Definitional networks are utilized to define concepts utilizing the subtype or “is-a” relation between concept types and their subtypes. Definitional networks are also called generalization or subsumption hierarchies because they support and graphically represent the rule of inheritance for transferring properties defined for a type to all of its subtypes. Since being a subtype is true by definition, the knowledge these networks represent is often assumed to be necessarily true. 2. Assertional networks are designed to assert propositions. Unlike definitional networks, the information in an assertional network is assumed to be contingently true, unless it is explicitly marked with a modal operator. Some assertional networks have been utilized as models of the conceptual structures underlying natural language semantics. The distinction between definitional and asserting networks is similar to the distinction between semantic memory and episodic memory (Tulving, 1972). 3. Implicational networks are based on implication as the primary relation for connecting nodes. They may be used to represent patterns of beliefs, causality, statements, or inferences.
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4. Executable networks include some mechanism, such as marker passing or attached procedures, which can perform inferences, pass messages, or search for patterns and associations. 5. Learning networks build or extend their representations by acquiring knowledge from examples. The new knowledge may change the old network by adding and deleting nodes and arcs or by modifying numerical values, called weights, associated with the nodes and arcs. 6. Hybrid networks combine two or more of the previous techniques, either in a single network or in separate, but closely interacting networks. Some semantic networks have been explicitly designed to implement and test hypotheses about human cognitive mechanisms as computational reasons may lead to the same conclusions as psychological evidence, while others have been designed primarily for increasing computer efficiency of knowledge representation and processing. However, there are some problems with semantic network utilization. First, they are intractable for large domains. Second, they do not represent performance or meta-knowledge very well. In spite of this, semantic networks proved very useful in different areas including AI. It is possible to distinguish two broad classes of semantic networks — static and dynamic semantic networks. Static semantic networks are usually stable in the process of functioning (utilization) but can be changed by those systems that have permission to do this. • Linguistic networks which include conceptual networks and definitional networks; • Statement networks which include assertional networks; • Implicational or causal networks. An example of a linguistic semantic network is the lexical database of English called WordNet. It provides short, general definitions of words in English and exhibits various semantic relations between these words, or more exactly, between the concepts named by these
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words. Some of the most common semantic relations defined are: — meronymy (A is part of B, i.e., B has A as a part of itself), — holonymy (B is part of A, i.e., A has B as a part of itself), — hyponymy (or troponymy) (A is a subordinate of B, i.e., A is a kind of B), — hypernymy (A is superordinate of B), — synonymy (A denotes the same as B), — antonymy (A denotes the opposite of B). Let us consider examples of these relations. 1. The word “hand” is connected to the word “body” by the meronomy relation. 2. The word “computer” is connected to the word “keyboard” by the holonymy relation. 3. The word “car” is connected to the word “vehicle” by the hyponymy relation. 4. The word “animal” is connected to the word “dog” by the hypernymy relation. 5. The words “plane” and “aircraft” are connected by the synonymy relation. 6. The words “light” and “heavy” are connected by the antonymy relation. Figure 5.17 gives an example of a conceptual (linguistic) network, which represents knowledge in chemistry, namely, a chemical classification. Figure 5.18 gives one more example of a conceptual (linguistic) network, which represents knowledge in particle physics. Now we give examples of statement networks in Figures 5.19 and 5.20. In computer science, graphical representations of finite automata are very popular (Burgin, 2005). They are examples of implicational or causal networks. Indeed, the schema in Figure 5.21 informs us that if the automaton is in the state q, then the input 1 causes transition to the state p; if the automaton is in the state p, then the input 0
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Chemical elements
Metalloids
Alkali metals
Other Metals
Alkaline Earth Metals Transition Metals
Non-metals
Halogens
Noble Gases
Rare Earth Elements
Figure 5.17. A conceptual network representing chemical elements in which all relations (links) are to be a subclass
causes transition to the state t; if the automaton is in the state t, then the input 1 causes transition to the state r, and so on. Dynamic semantic networks are changing itself in the process of functioning (utilization). • Executable networks; • Learning networks; • Formation networks. A useful form of dynamic semantic networks are growing semantic networks, which are efficiently used as learning and formation networks (Gladun, 1986; 1987). In the process of their functioning, growing semantic networks form new concepts from information about objects from some class of objects from the environment. The structure of a growing semantic network is a labeled pyramidal graph, i.e., a labeled acyclic oriented graph in which a vertex either have no entering edges or have more than one entering edge (cf., Figures 5.17, 5.18, 5.19 and 5.22). Vertices with no entering edges are called receptors. Other vertices are called conceptors. A subgraph of the pyramidal graph including some conceptor C and all vertices from which there are paths to this conceptor is called the pyramid of the conceptor C. Receptors of growing semantic networks correspond to the names of the relations, properties, states, actions, objects, and classes of objects. Conceptors represent descriptions of objects or situations
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524
Physical microobjects
particles
antiparticles
elementary particles
bosons
composite particles/hadrons
fermions
gauge bosons
baryons
nucleons charmed baryons
quarks
mesons
bottom baryons
pentaquarks
leptons
up quark down quark charm quark strange quark top quark bottom quark
protons
gluon W boson Higgs boson
hyperons
neutrons
positron
Λ particles Σ particles Ξ particles Ω particles
Z boson photon
antiproton
graviton
electron neutrino electron muon neutrino muon tau neutrino
tau
Figure 5.18. Modern classification of physical particles
is A cat A fish
an animal is not
Figure 5.19. A statement network for the proposition “A cat is an animal, while a fish is not an animal”
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goes to Andy
his school
Alice
the theater is in
Figure 5.20. A statement network for the proposition “Andy goes to school and Alice is in the theater”
0, 1 q
p
ε
0 1
r
t 0, 1
Figure 5.21. A non-deterministic finite automaton A with ε-transitions and the alphabet {0, 1}”
Figure 5.22. A growing semantic network where is a receptor and is a conceptor
and repeating fragments of such descriptions in the form of a concept. Conceptors corresponding to intersections of descriptions of objects or situations represent conjunctive definitions (concepts) of classes of environment elements. A new mathematical model for growing semantic network is developed in (Burgin and Gladun, 1989; 1990; 1990a). In contrast to the traditional mathematical model where growing semantic network are represented in the form of labeled graphs, the new approach
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formalizes semantic network as hierarchical systems of named sets. This model has significant advantages because named set theory (Burgin, 2011) provides much more operations for manipulation with semantic networks in comparison with graph theory (Berge, 1973). Another form of dynamic semantic networks are extended Petri nets (Zhou, 2013). As an alternative to semantic networks, contemporary artificial intelligence and related areas, such as databases, use ontology as a kind of knowledge representation although it is necessary to remark that some ontologies have the form of semantics networks. The notion of ontology came to artificial intelligence and information science from philosophy. In philosophy, ontology is a branch of metaphysics, which is concerned with the nature of reality and existence analyzing various types or modes of existence. However, coming to artificial intelligence, database and web applications, the term ontology changed its meaning. The main difference is that for a philosopher, ontology is a theoretical discipline, while for a computer scientist, ontology is a system of description of some domain or representation of knowledge about a domain. For instance, Gruber defines ontology in the following way: “An ontology is a description (like a formal specification of a program) of the concepts and relationships that can formally exist for an agent or a community of agents. This definition is consistent with the usage of ontology as set of concept definitions, but more general. And it is a different sense of the word than its use in philosophy” (Gruber, 1993).
As descriptions of domains, ontologies usually include the following elements: • Individuals called instances or “ground level” objects; • Classes, sets, collections, concepts, types of objects, or kinds of things; • Attributes, aspects, properties, features, characteristics, or parameters that various objects and classes possess or may possess; • Relations, which describe how classes and individuals from the domain are related to one another; • Complex structures, e.g., complex terms, formed from simpler structures;
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• Restrictions, e.g., axioms, which in contrast to axioms in logic and mathematics, includes all true statements about the domain, defining what must be true and what must not be true in the described domain; • Rules often in the form of if-then statements used for logical inferences and ontology transformations; • Operations, which include both ontological operations and representation of domain operations; • Events, such as the changing of attributes or relations. Usually, two kinds of ontologies are discerned — domain-specific ontologies and upper or foundation ontologies (Maedche and Staab, 2001). A domain-specific or simply domain ontology represents knowledge about a specific domain as part of the world. For instance, an ontology about the domain of computer software would model the meaning of such terms as program, programmer, programming language, etc. In contrast to this, an upper ontology (or foundation ontology) represents knowledge common to many domain ontologies. There are special ontology languages to build ontologies (Uschold and Gruninger, 1996). To increase expressive abilities and efficiency of websites, developers use tags for structuring information. However, tags essentially increase the complexity of the format of content available on websites. Ontologies help to harness this complexity taking account of appropriate combinations of tags and content. 5.5. Scripts and productions Resistance to change is as natural as change itself — but history tells us that change is inevitable. Matt Sakey
A script is a structured representation describing a stereotyped sequence of events or actions in a particular context (Schank and Abelson, 1977).
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Developed by Roger Schank, Robert Paul Abelson (1928–2005) and their research group, scripts have been used in natural language understanding systems to organize a knowledge base in terms of the situations that the system should understand. Scripts provide operational (procedural) knowledge about systems, actions, and situations. A somewhat different understanding treats script as a procedural structure, which prescribes a set of circumstances expected to follow on from one another in a definite environment, i.e., as an imaginary sequence or chain of situations, which could be anticipated in some setting. A script is composed of following components: Entry conditions, which must be satisfied for the script starts. Results or facts are conditions that will be true after the script has terminated. Events and actions which are represented in conceptual dependency form using variables which are then concretized in the script. Props are slots representing objects involved in events of the script. Roles are the systems of actions that the individual participants perform. Scenes are sequences of events that occur according to the script based on the temporal aspects of the script. Scripts also can have include tracks, which are versions of the script and may share the same components of the script. Social schema in the form of a script is generated by an event, e.g., going to a restaurant, which includes scenes, e.g., booking a table, arriving at the restaurant ordering food etc. props such as food and menu; enabling conditions such as having money; roles such as a waiter and a client; and outcomes such as not feeling hungry. Social cognition researchers are particularly interested in studying what happens when scripts involve conflicts with existing norms. The classical example is the script “Restaurant” (Schank and Abelson, 1977), which has the following informal description: 1. Go to a restaurant; 2. Be seated; 3. Get menu;
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Read menu; Order food; Eat food; Pay for meal; Exit the restaurant.
After some formalization, we obtain a semi-formal script “Restaurant” Props: Tables; Menu; Food; Money. Roles: Customer; Waiter; Cook. Scene 1: Entering Customer PTRANS Customer into restaurant; Customer ATTEND eyes to tables; Customer MBUILD where to sit; Customer PTRANS Customer to table; Customer MOVE Customer to sitting position. Scene 2: Ordering Customer PTRANS menu to Customer (menu already on table); Customer MBUILD choice of food; Customer MTRANS signal to Waiter; Waiter PTRANS to table; Customer MTRANS ‘I want food’ to Waiter; Waiter PTRANS to Cook.
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Scene 3: Eating Cook ATRANS food to Waiter; Waiter PTRANS food to Customer; Customer INGEST food. Scene 4: Exiting Waiter MOVE write check; Waiter PTRANS to Customer; Waiter ATRANS bill to Customer; Customer ATRANS money to Waiter; Customer PTRANS out of restaurant. The final formalization gives us the formal script “Restaurant”: Props Tables; Menu; F = Food; Bill; Money. Roles P = Customer; W = Waiter; C = Cook; K = Cashier; O = Owner. Entry conditions P is hungry; P has money. Results P has less money; P is not hungry; P is pleased (optional).
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Scene 1: Entering P PTRANS P into restaurant; P ATTEND eyes to tables; P MBUILD where to sit; P PTRANS P to table; P MOVE P to sitting position. Scene 2: Ordering (Menu on table) P PTRANS menu to P
(S asks for menu) O MTRANS signal to W WPTRANS W to table PMTRANS “need menu” to W W PTRANS W to menu W PTRANS W to table
P MTRANS food list to P * P MBUILD choice of F P MTRANS signal to W W PTRANS W to table P MTRANS ‘I want F’ to W W PTRANS W toV W MTRANS (ATRANS F) to C
C MTRANS ‘no F’ to W
C
DO (prepare F script)
W PTRANS W to P W MTRANS ‘no F’ to P (go back to*) or to (go to Scene 4 at no pay path)
Scene 3: Eating C ATRANS F to W W ATRANS F to P P INGEST F (Option: Return to Scene 2 to order more; otherwise go to Scene 4)
(O brings menu) W PTRANS menu to P
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532 Scene 3: Eating
C ATRANS F to W; W ATRANS F to P; P INGEST F. (Option: Return to Scene 2 to order more; otherwise go to Scene 4). Scene 4: Exiting S MTRANS to W W MOVE (write bill) W PTRANS W to P W ATRANS bill to P P ATRANS tip to W P PTRANS P to K
P ATRANS money to K
P ATRANS check to K
P PTRANS P to out of restaurant (No pay path)
This script utilizes several conceptual dependencies: PTRANS represents a physical motion from one place to another. MTRANS represents information transmission. ATRANS represents transmission of symbolic entities, e.g., transmission of ownership. A behavioral script is a type of frame that describes behavior of people in definite situations. Scripts allow different solutions for the same situation or scene, which form the so-called semantic spectrum of actions. Script theory was primarily intended to explain language processing and higher thinking skills in the context of storytelling and the development of intelligent tutors (Schank, 1991). A variety of computer programs have been developed based on scripts, for example, educational software (Shank and Cleary, 1995).
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A useful kind of operational (procedural) knowledge representation is productions, which are rules for obtaining knowledge, making decisions and/or performing actions. Productions are intensively used to representing (procedural) knowledge in expert systems. In some sense, productions are elementary scripts. The term production and the corresponding construction were introduced and studied by Emil Post (1897–1954) in (Post, 1943). There are two basic forms of productions: If ,
then
(5.32)
or If ,
then .
(5.33)
The first form is called premise-conclusion rules and the second form is called condition-action rules. In an abstract representation, both forms are described by the formula A → B, which means A implies B. Conditions, premises and conclusions can be in the form of simple or compound statements. For instance, we can consider the following productions If you want to go from America to Europe, then you have to use a plane or ship,
(5.34)
If you love art and want to spend vacations in Europe, then go to Italy.
(5.35)
The first of them represents a premise-conclusion rule and the second one represents a condition-action rule. Some of the benefits of if-then rules are that they are modular, each defining a relatively small and, at least in principle, independent piece of knowledge. New rules may be added and old ones deleted usually independently of other rules. Premise-conclusion rules are utilized for deriving new knowledge from given knowledge. In this case, the technique called forward
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chaining is used. For instance, we have the following productions (5.36) and (5.37) and initial knowledge (5.38): If a celestial body rotates about the Sun, then it gets light from the Sun.
(5.36)
If a celestial body is a planet from the Solar System, then it rotates about the Sun.
(5.37)
The Earth is a planet from the Solar System.
(5.38)
Then we can derive the following inference: Applying rule (5.37), we have: The Earth rotates about the Sun. Applying rule (5.36), we have: The Earth gets light from the Sun. Premise-conclusion rules are also utilized for decision justification. In this case, the technique called backward chaining is used. For instance, we have the following backward chaining: I am going to buy tickets for a plane to Rome. Why? Because “I want to go to Italy and Rome is in Italy”. Why? Because “If you love art and want to spend vacations in Europe, then go to Italy”. Why? Because I live in Canada, love art and want to spend vacations in Europe. Condition-action rules are utilized for decision-making and organization of functioning. For instance, we have the following inference: I live in Canada, love art and want to spend vacations in Europe.
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Applying rule (5.35), we have: I will go to Italy. In addition, we have: Rome is in Italy. Applying rule (5.34), we have: I will buy tickets for a plane to Rome. Pospelov (1990) suggests a more detailed form of a production: (N , D, C), where N is the name, D represents the domain, i.e., those objects to which this production can be applied, and C is the core of the production. The core of a production has the following form: (S, A → B, P ), where S denotes preconditions, i.e., conditions of the production applicability, and P denotes postconditions, i.e., conditions that must be fulfilled when the production has been applied. The component A → B is called the nucleus of the production. Note that all forms of productions and their components have the structure of a fundamental triad (named set) (cf., Appendix A). Usually rule-based systems, such as expert systems, consist of a set of rules in the form of productions, a knowledge base and an inference engine. The rules encode active domain knowledge as premise-conclusion and/or condition-action pairs. The knowledge base contains initial knowledge and previously deduced knowledge. The inference engine works in the context of a non-monotonic logic applying a conflict resolution strategy to deal with inconsistencies and handle cases where more than one rule is suitable for application. Comparing scripts and productions with abstract automata and algorithms, which are popular means of operational knowledge representation, we see that productions are on the level of finite
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automata, while scripts are on the level of recursive and superrecursive algorithms. 5.6. Frames and Schemas All truths are easy to understand once they are discovered; the point is to discover them. Galileo Galilei
A frame is a data structure introduced by Marvin Minsky in the 1970s for knowledge representation that allows imitating the way in which people keep information in the brain and make use of it when the need arises. Minsky frames were intended to help Artificial Intelligence systems recognize and utilize patterns and their specific instances (Minsky, 1974). Frames support inheritance and are often used to capture knowledge about typical objects or events from some class, such as cars, planes, organizations, people or triangles. A frame is a labeled graph in which the nodes are called slots. Each frame is a structured knowledge item that contains information about a given object. A frame also has a name (identification), by means of which this frame with all its slots can be retrieved. Each slot has a name and characterizes a property or relation of the object represented by the frame. Aspects of the relation represented by a slot are described by facets of the slot. Slots may contain default values (subject to override by detecting a different value for an attribute), refer to other frames (component relationships) or contain methods for recognizing pattern instances being able to hold declarative and procedural information. Slots may have the following facets: — — — — — — —
Type of a value; Default value determined by default fillers; Current value determined by current fillers; Constraints on a value in the form of properties or axioms; Salience reflecting measure of the slot’s importance; Cardinality determining minimum and maximum values; Methods or Procedures determining actions on values and other facets;
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Here are examples of standard methods: — The method IF NEEDED is applied to acquire a slot value; — The method IF CHANGED is applied to change a value of a slot; — The method IF ADDED is applied to add a value to a slot; — The method IF REMOVED is applied to delete a value of a slot; Facets of a slot may be variable or constant. Variables in slots of a frame allow adaptation of the frame to different objects and situations by assigning constant values to these variables. Consequently, frame languages are usually focused on the recognition and description of objects and classes. Simple slots are pairs or a triple . However, in the majority of frame systems, slots are complex structures that have more facets, which describe the properties of the relation represented by the slot. The value of a slot may be elementary, e.g., a text or a number, or it may be another frame. Some frame systems allow multiple values for slots and support procedural facets (methods) used either to compute the slot value or to make consistency checking or to update values of this or other slots. The meaning of a frame is not defined by any systematic rules from logic and language or by relations to the real world. It is determined only by the procedures that manipulate the slots and their aspects, while there are no formalized restrictions on these procedures. Let us consider an example of a frame for the object car. Example 5.5.1. A frame F with the name “Car”, which stores knowledge about cars has the following slots: 1. The slot with the name “Vehicle” informs that the class of cars is a subclass of the class of vehicles and is related to the frame “Vehicle”.
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2. The slot with the name “Number of wheels” informs how many wheels the car has, e.g., this slot may have the value 4. 3. The slot with the name “Number of doors” informs how many doors the car has, e.g., this slot may have the value 2 or 4. 4. The slot with the name “Make” informs what company produced the car, e.g., this slot may have the value “GMC”, “Honda” or “Toyota”. 5. The slot with the name “Model” describes the model of the car, e.g., this slot may have the value “Accord” when the previous slot has the value “Honda” or “Toyota”. Collections of related frames are linked together into framesystems, which are used to make knowledge processing economical representing changes of emphasis and attention and accounting for the effectiveness of the representation. Frame-systems provide additional means for description of the frame to different objects and situations performing this process in two stages: at first, the most relevant frame is selected from the frame-system and then this frame is adapted to the current object or situation. For visual scene analysis, the different frames of a system describe the object, e.g., a system, from different viewpoints, and the transformations between one frame and another stand for the effects of object changes, e.g., moving the object from place to place. In turn, frame-systems are linked by an information retrieval network performing selection of frames and their adaptation. Frames became popular because they were assumed being powerful based on associated procedural languages and easy to learn due to the arbitrariness of slot names and the values. Frame technology was further developed in (Burgin and Slyusar, 1980) based on the block-schema language elaborated in (Burgin, 1973; 1976). Some consider frames as a specific kind of semantic networks. Others assume that semantic networks and frames have the following differences: 1. In a semantic network, nodes hold no associated information except their names, while in a frame, a node is a slot, which may contain a lot of information.
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2. In a semantic network, all kinds of relations are automatically managed according to a set of methods of inheritance that can be defined by the constructor of the network, while in frame systems, inheritance of properties is described by a single class of relations. Frames, semantic networks and scripts are kinds of the structure called schema or scheme, which is very popular in the field of knowledge representation. Note that the plural of schema or scheme is schemas, correspondingly, schemes in U.S. or schemata in UK. Kant was perhaps the first to introduce the word schema into philosophy, making brief remarks about schemata and describing the “schematism” as “an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover” (Kant, 1781/1929). As an example, he described the “dog” schema as a mental pattern that could delineate the figure of a four-footed animal in a general manner, without limitation to any single determinate figure from experience, or any possible image that a person can represent directly. In neuroscience, the notion of a schema was introduced by Head and Holmes (1911) who discussed body schemas in the context of brain damage. Bartlett (1932) implemented the notion of a schema as part of a study of remembering. Another important use of schemas in psychology was initiated by Piaget (1952), who viewed cognitive development from biological perspective and described it in terms of operation with schemas. With respect to adaptation, Piaget believed that humans desire a state of cognitive balance or equilibration. When the child experiences cognitive conflict, such as a discrepancy between what the child believes the state of the world to be and what she or he is experiencing, adaptation to the new situation is achieved through assimilation and/or accommodation. Assimilation involves making sense of the current situation in terms of previously existing mental structures called schemas. Accommodation requires construction of new schemas when new information does not fit into existing schemas. For instance, when a child encounters a bird for the first time and learns that it is different from dogs or cats, he must create a new schema for birds. Piaget characterized schemas, or schemes, as general characteristics
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of an action that allow the application of the same action to a different context by means of the mind’s natural tendency to organize information into related, interconnected structures — schemas. However, Bartlett’s research was neglected in America as the behaviorist psychology prevailed and only later Ulric Gustav Neisser (1928–2012) returned schemas to psychology (Neisser, 1967). In recent years, researchers extensively explored human cognition and knowledge structure in terms of schemas treated as an efficient tool for representation of human knowledge (cf., for example, (Anderson, 1977; Rumelhart, 1980)). It has been experimentally demonstrated that schemas from the person’s mentality influence the way incoming information is acquired and interpreted (cf., for example, (Pichert and Anderson, 1977)). Some researchers assume that a schema is defined as an organized body of knowledge, conceived theoretically as a set of interconnected propositions centering on a general concept, and linked peripherally with other concepts (cf., for example, (Gagne, 1986)). However, this is a too restrictive definition and the majority of cognitive psychologists utilize different conceptions and models of schemas. That is why in psychology and education, a schema is treated as a mental structure people use to organize, compress and simplify their knowledge of the world in which they live. There are schemas about people, nature, artificial devices, animals, and in fact, about almost everything. Schemas also represent knowledge about concepts and objects with the relationships they have with other objects, situations, events, sequences of events, actions, processes and sequences of actions. Researchers often deal with schemas as the basic units of human knowledge and cognition (Suzuki, 1987). As Rumelhart writes: “Schemata can represent knowledge at all levels-from ideologies and cultural truths to knowledge about the meaning of a particular word, to knowledge about what patterns of excitations are associated with what letters of the alphabet. We have schemata to represent all levels of our experience, at all levels of abstraction. Finally, our schemata are our knowledge. All of our generic knowledge is embedded in schemata.” (Rumelhart, 1980)
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In this context, a mental schema is an abstract structure of knowledge, a mental representation stored in memory upon which all information processing depends. It may represent knowledge at different levels, e.g., cultural truths, linguistic knowledge, or ideologies. They are mental templates that represent a person’s knowledge about people, situations or objects, and which originate from prior knowledge or experiences. People use schemas to organize their knowledge and provide a framework for future understanding. All social stereotypes and roles, scripts, worldviews, and archetypes are schemas. Schemas influence our attention, as we are more likely to notice things that fit into schemas we have. If our schemas do not allow to represent some incoming information, then this information is either ignored or encoded into a new schema. Often (but not always) schemas are prone to distortion. They influence what we look for in different situations. They are conservative having a tendency to remain unchanged, even in the face of contradictory information. All biases have their schemas. People are inclined to place others who do not fit their schema in a “special” or “different” category, rather than to consider the possibility that the schema they have may be faulty. Dynamical schemas of situations are called interaction schemas, which are applied to the visuomotor coordination of the frog, highlevel visual recognition, hand control, language processing, and perceptual robotics (Arbib, 1992; 1995). Interaction schemas, which include motor schemas and perceptual schemas, are defined by the execution of tasks involving the physical environment. A set of basic motor schemas is hypothesized to provide simple prototypical patterns of interaction with the world, whereas perceptual schemas recognize certain possibilities of interaction and regularities of the physical world. Motor schemas are similar to control systems but distinguished in that they can be combined to form coordinated control programs that control the phasing in and out of patterns of co-activation, with mechanisms for the passing of control parameters from perceptual to motor schemas and forming coordinated control programs to mediate complex behaviors. Various schema parameters
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represent properties of physical objects such as size, location, time, and motion. Many schemas may be abstracted from the perceptual-motor interface. Schema activations are largely task-driven, reflecting the goals of the organism and the physical and functional requirements of the task. Theoretical representation posits schemas as “programs” (in a generalized sense) or some kind of procedures for a system that has continuing perception of, and interaction with, its environment, with concurrent activity of many different schemas passing messages back and forth for the overall achievement of some goal. At the same time, it is possible to treat schemas as self-contained computing agents (objects) with the ability to communicate with other similar agents, and whose functionality is specified by some behavior. In addition, brain theory further requires schemas to be implemented in specific neural networks. A schema is both a store of knowledge and the description of a process for applying that knowledge. Consequently, a schema is often instantiated to form multiple schema instances as active copies of the process to apply operational knowledge. Namely, given a schema that represents generic knowledge about some object, an individual may need several active instances of the schema, each suitably tuned, to subserve our perception of a different instance of the object. Schemas can become instantiated in response to certain patterns of input from sensory stimuli or other schema instances that are already active. The alternative view (Arbib and Liaw, 1995) is that there is a limited set of schemas (maybe only one) and that only the schemas can be active. By contrast a schema instance is rather a record in working memory that records that a certain “region of space time” R activated a specific schema S with certain parameters {P} and confidence level C. On the latter view, processes of attention phase the activity of a schema in and out for different regions. Presumably, however, the working memory provides top-down activation of a schema when attention returns to those regions where the schema was recently active (cf., (Itti and Arbib, 2005)).
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Each instance of a schema has an associated activity level. That of a perceptual schema represents a “confidence level” that the object represented by the schema is indeed present; while that of a motor schema may signal its “degree of readiness” to control some course of action. The activity level of a schema instance may be but one of many parameters that characterize it. Thus the perceptual schema for “ball” might include parameters to represent size, color, and velocity. The use, representation, and recall of knowledge is mediated through the activity of a network of interacting computing agents, the schema instances, which between them provide processes for going from a particular situation and a particular structure of goals and tasks to a suitable course of action (which may be overt or covert, as when learning occurs without action or the animal changes its state of readiness). This activity may involve passing of messages, changes of state (including activity level), instantiation to add new schema instances to the network, and deinstantiation to remove instances. Moreover, such activity may involve self-modification and self-organization. The key question is to understand how local schema interactions can integrate themselves to yield some overall result without explicit executive control, but rather through cooperative computation, a shorthand for “computation based on the competition and cooperation of concurrently active agents”. For instance, in interpretation of visual scenes, schema instances are used to represent hypotheses that particular objects occur at particular positions in a scene, so that instances may either represent conflicting hypotheses or offer mutual support. Cooperation yields a pattern of “strengthened alliances” between mutually consistent schema instances that allows them to achieve high activity levels to constitute the overall solution of a problem; competition ensures that instances which do not meet the evolving consensus lose activity, and thus are not part of this solution (though their continuing subthreshold activity may well affect later behavior). In this way, a schema network does not, in general, need a top-level executor, since schema instances can combine their effects by distributed
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processes of competition and cooperation, rather than the iteration of an inference engine on a passive store of knowledge. This may lead to apparently emergent behavior, due to the absence of global control. In brain theory, a given schema, defined functionally, may be distributed across more than one brain region; conversely, a given brain region may be involved in many interaction schemas. A top-down analysis may advance specific hypotheses about the localization of (sub)-schemas in the brain and these may be tested by lesion experiments, with possible modification of the model (e.g., replacing one schema by several interacting schemas with different localizations) and further testing. Schemas, and their connections within a schema network, must change so that over time they may well be able to handle a certain range of situations in a sufficiently adaptive way. In a general setting, there is no fixed repertoire of basic interaction schemas. New schemas are formed as assemblages of old schemas and being formed, a schema are usually tuned by certain adaptive mechanism. This tunability of schema assemblages allows them to become units of behavioral control, much as a skill is honed into a unified whole from constituent parts. Such tuning may be expressed at the level of schema theory itself, or may be driven by the dynamics of modification of unit interactions in some specific implementation of the schemas. The theory of interaction schemas is consistent with a model of the brain as an evolving self-configuring system of interconnected units. Once an interaction schema as a theoretical model of animal behavior has been refined to the point of hypotheses about the localization of schemas, it is possible to model a brain region by seeing if its known neural circuitry can indeed be shown to implement the posited schema. In some cases, the model involves properties of the circuitry that have not yet been predicted and tested, thus laying the ground for new enhancements and experiments. In AI, it is possible to implement individual schemas using artificial neural networks or appropriate programming language for developing the corresponding software system on a computer.
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Schemas affect what we notice, how we interpret things and how we make decisions and act. They act like filters, accentuating and downplaying various elements. We use them to classify things, such as when we ‘pigeon-hole’ people. They also help us forecast, predicting what will happen. We even remember and recall things via schemas, using them to ‘encode’ memories. Schemas are often shared within cultures, allowing short-cut communications. Every word is, in effect, a schema, as when you read it you receive a package of additional inferred information. We tend to have favorite schema which we use often. When interpreting the world, people will try to use these first, going on to others if they do not sufficiently fit. Often when something does not match the schema, it is ignored. Some schemas are easier to change than other schemas, and some people are more open about changing any of their schemas than other people. Schema theory assumes that when individuals obtain information, they attempt to fit it into some structure in memory that helps them making sense of the received information. Schema theory proposes that the individuals breaks information into suitable chunks, which are then coded, structured by existing schemas or organized in new schemas and stored in the brain for later recall. Schema theory suggests efficient tools and an active strategy for coding technique necessary for facilitating the recall and utilization of knowledge. Schemas are hierarchically organized mental structures, which allow the learners to understand and associate what is being presented to them. Information processing based on schema theory is far removed from the serial symbol-based computation and artificial intelligence actively contributes to schema theory, even when it does not use this term. For instance, Minsky (1986) promoted a Society of Mind analogy in which “members of society”, the intelligent agents, are analogous to schemas. The study of interactive “agents” in a more general context has become an established theme in artificial intelligence when actions are mediated through a network of schemas. Here are some examples of interaction schemas in human mentality.
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Example 5.6.1. The schema of face recognition is tentatively acquired at around two or three months of age (which succeeds to a previous scheme already present at birth). This schema corresponds to the mental structure which connects the various states of a face defined by configurations of perceptual indices (front view, side view, etc.) related to actions-transformations (head rotations, subject’s or object’s rotation). Example 5.6.2. The schema of (shape or) size constancy is the insertion of the various sizes of an object related to its distance from the perceiver in a transformational system (system of transformations) governing the moves of the object. Present at birth, it could be reconstructed during the first months of life. Example 5.6.3. The schema of object’s permanence (the “objective” form), the one achieved according to Piaget at around 16–18 months of age, is the mental structure which connects the various successive states of a set of objects (their different localizations or relative positions) to their successive displacements (transformations), even bridging across periods when the object disappears from the view. To achieve better comprehensibility, interaction schemas are usually represented in a graphical form as in Figure 5.23. Example 5.6.4. A hypothetical coordinated control schema for reaching and grasping is given in Figure 5.23 (Arbib, 1989). Interaction schemas form the base for the Computational Neuroscience, structure and function of which are given in Figure 5.24. It is interesting to note that even subneural modeling brings us to grid automata from the automata theory (Burgin, 2005). One more important type is image schema defined as a recurring structure within our cognitive processes which establishes patterns of understanding and reasoning (Johnson, 1987; Lakoff, 1987; Rohrer, 2006). Image schemas are formed from our bodily interactions, from linguistic experience, and from historical context. In contemporary cognitive linguistics, an image schema is treated as an embodied prelinguistic structure of experience that motivates
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recognition criteria
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547 visual input
visual input
visual input
Visual activation
Location
of visual search
Size target
Orientation
Recognition
Recognition
location
size
orientation
visual, kinesthetic
activation
visual and
and tactile
of reaching
kinesthetic input
input
Fast Phase
Hand
Hand
Movement
Preshape
Rotation
Slow Phase
Actual
Movement
Grasp
Hand Reaching
Grasping
Figure 5.23. Dashed lines — activation of signals (i.e., control links, in our terminology); solid lines — transfer of data (i.e., information links, in our terminology)
conceptual metaphor mappings. Experimental evidence supporting existence of image schemas in mentality of people is drawn from several disciplines, including cognitive psychology, neuroscience and studies in spatial cognition in both linguistics and psychology. Naturally, schemas are used in a variety of areas. Mental schemas, also called mental models, concepts, mental representations, and knowledge structures, are used by people for organization of their behavior, thinking, writing, and speaking.
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Computational Neuroscience via Structure and Function
Brain / Behavior / Organism
Schemas
Brain Regions
Functional Decomposition
Components / Layers / Modules Structural Decomposition in a form of grid automata
Generalized Neural Networks Structure meets Function
Subneural Modeling in a form of grid automata
Figure 5.24. A version of the schema for the Computational Neuroscience suggested by M. A. Arbib
A social schema of behavior is generated by an event (going to a restaurant), that consists of a script and scenes (booking a table, arriving at the restaurant ordering food etc.); props (menu); enabling conditions (money); roles (waiter, client); and outcomes (not feeling hungry). Social cognition researchers are particularly interested in studying what happens when the schema activated conflicts with existing norms. In general, social schemas represent general social knowledge, e.g., describing the structure of organizations. An ideological schema is generated by attitudes or opinions on relevant social or political issues, for example, on energy and ecology. A formal schema is related to the rhetorical structure of a written text, such as differences in genre or between narrative styles and their corresponding structures.
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A linguistic schema includes the decoding features a person needs in order to understand how words are organized and fit together in a sentence (be it spoken or written discourse). A content schema refers to knowledge about the subject matter or content of a text. An orienting schema of the nearby environment, or “cognitive map”, guides the organism around the environment. A cognitive map contains schemata of the objects in the environment and spatial relations between the objects. A schema for oneself is called a self-schema. People also hold schemas for idealized or projected selves, or possible selves. A schema of another person is called a person schema. Person schemas can be mental or represented by physical objects, e.g., by texts or graphs. A role schema depicts roles or occupations of people, while a schema for events or situations is called an event schema. Event schemas are formally represented by scripts. People use schemas to organize their knowledge and provide a framework for future understanding, e.g., according to Piaget, in their development, children adopt a variety of schemas to understand the world. Examples of schemas include organization of scientific disciplines, library classifications, social schemas, stereotypes, social roles, worldviews, and archetypes. We can see that frames, scripts and semantic networks are special case of schemas. It is natural to consider three types of schemas: • static schemas describe (give knowledge about) classes of objects. • function schemas describe (give knowledge about) classes of changes or relations. • process schemas describe (give knowledge about) classes of processes, such as behavior, interaction, or communication. In contrast to static schemas, function schemas, and process schemas are dynamic schemas. For instance, frames are usually static schemas, while scripts are dynamic schemas. In a natural way, schema theory came from psychology to education where it was introduced by Anderson (1977). He pointed out
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that schemas is a form of representation for complex knowledge providing a principled model of how old knowledge might influence the acquisition of new knowledge. Following this principle, schema theory was applied to understanding the reading process serving as an important counterweight to purely bottom-up approaches to reading. The schema theory emphasizes that reading involves both the bottom-up information from the perceived letters coming into the eye and the use of top-down knowledge to construct a meaningful representation of the content of the text. A notion of a schema rather different in emphasis and properties from those we have just been considering has been very popular in programming, where it was formalized and extensively used for theoretical purposes. At the beginning, program schemas, or program schemata, were introduced by Lyapunov in 1953 and published later in (Lyapunov, 1958) under the name operator schema. Afterwards Ianov, a graduate student of Lyapunov, transformed operator schemas into a logical form called a logical schema of algorithm (later named Ianov program schemata) and proved many properties of these schemas (Ianov, 1958; 1958a; 1958b). The main result of Ianov is a theorem about the decidability of equivalency of schemas that use only one-argument functions. Approximately at the same time, Kaluznin (1959) introduced the concept of a graph-schema of an algorithm. Subsequently, this concept was generalized by Bloch (1975) and applied to automaton synthesis, discrete system design, programming, and medical diagnostics. Program schemas were later studied by different authors, who introduced various kinds of program schemas: recursive, push-down, free, standard, total schemas (cf., for example, (Karp and Miller, 1969; Paterson and Hewitt, 1970; Garland and Luckham, 1971; Logrippo, 1978)). Fischer (1993) introduced the mathematical concept of a lambda-calculus schema to compare the expressive power of programming languages. The theory of program schemas has been considered as a base for (Yershov, 1977) or one of the main directions (Kotov, 1978) in theoretical programming. In the 1960s, program schemas were used to create programming languages and build translators. To study parallel computations, flow graph and dataflow
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schemas have been introduced and utilized (Slutz, 1968; Keller, 1973; Dennis, Fossen, and Linderman, 1974). Dataflow schemas are formalizations of dataflow languages. Program schemas and dataflow schemas formed an implicit base for the development of the first programming metalanguage — the block-schema (flow-chart) language (Burgin, 1973; 1976). Moreover, the advent of the Internet and introduction of the Extensible Markup Language, abbreviated XML, started the development of schema languages (cf., for example, (Duckett et al., 2001; Van Der Vlist, 2004)). As developers know, the advantage of XML is that it is extensible, even to the point that you can invent new elements and attributes as you write XML documents. Then, however, you need to define your changes so that applications will be able to make sense of them and this is where XML schema languages come into play. In these languages, schemas are machine-processable specifications that define the structure and syntax of metadata specifications in a formal schema language. There are many different XML schema languages (W3C Schema, Schematron, Relax NG, and so on). They are based on schemas that define the allowable content of a class of XML documents. Schema languages form an alternative to the DTD (Document Type Definition), and offer more powerful features including the ability to define data types and structures. XML schemas from these languages provide means for defining the structure, content and semantics of XML documents, including metadata. A specification for XML schemas is developed and maintained under the auspices of the World Wide Web Consortium. The Resource Description Framework (RDF) is an evolving metadata framework that offers a degree of semantic interoperability among applications that exchange machine-understandable metadata on the Web. RDF schema (Resource Description Framework schema) is a specification developed and maintained under the auspices of the World Wide Web Consortium. The Schematron schema language differs from most other XML schema languages because it is a rulebased language that uses path expressions instead of grammars. This means that instead of creating a grammar for an XML document, a Schematron schema makes assertions applied to a specific
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context within the document. If the assertion fails, a diagnostic message that is supplied by the author of the schema can be displayed. RELAX NG is a grammar-based schema language, which is both easy to learn for schema creators and easy to implement for software developers. A natural tool for providing flexible data structures is to create schemas for describing an object and any of the interrelationships that exist within a data structure. There are many different kinds of schemas used in different areas of information technology. For instance, relational databases such as SQL Server use schemas to contain their table names, column keys, and provide a repository for trigger and stored procedures. In addition, when a developer creates a class definition, he or she can define schemas to provide an efficient object-oriented interface for properties, methods, and events. An XML schema is by definition a well-formed XML document with a set of namespaces, which consist of declarations providing a unique set of identifiers and establishing a definite structure on XML elements and their attributes. The original namespace in the XML specification used URI (Universal Resource Identifiers) as a base for differentiating various XML vocabularies. Later this namespace was extended under the XML schema specification to include schema components in the structure and not just single elements and attributes. The unique identifier was changed from a URI to a security boundary that is owned by the schema author because URI does not point to a physical location. The whole namespace consists of two components — the XML schema namespace and target namespace. Special kind of XML schemas has been developed for energy simulation data representation (Gowri, 2001). Another application of XML schemas is e-business. For instance, the ebXML specification schema developed by UN/CEFACT and Oasis provides a standard framework by which business systems may be configured to support execution of business collaborations, which consist of business transactions. Such schemas are called business process specification schemas. Transactions can be implemented using one
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of many available standard patterns of business interaction. These patterns determine the ongoing exchange of business documents and signals between the partners to achieve the required electronic commerce transactions. Another example of business process specification schemas is the XML schema definition developed by the Danish Broadcasting Corporation for business-to-business exchange interface with the DR metadata standard. XML schemas are also used for modeling and exploration of business objects (Daum, 2003). Star schemas determine methods of organizing information in a data warehouse that allows the business information to be viewed from many perspectives. In addition, an important tool in database theory and technology is the notion of the database schema, which gives a general description of a database, is specified during database design, and is not expected to change frequently (Elmasri and Navathne, 2000). Database schemas are represented by schema diagrams. Database management system (DBMS) architecture is often specified utilizing database schemas. Three important tasks of databases are (Elmasri and Navathne, 2000): 1. Insulation of program and data (program-data and programoperation independence). 2. Support of multiple user views. 3. Use of a catalog to store the database description (schema). To realize these tasks, the three-schema architecture, or ANSI/SPARC architecture, of DBMS was developed (Tsichridsis and Klug, 1978). In this architecture, schemas are defined at three levels: 1. The internal level has an internal schema, which describes the physical storage structure of the database. 2. The conceptual level has a conceptual schema, which describes the structure of the whole database for a community of users. 3. The external or view level includes a number of external schemas or user views. Each external schema describes the part of the database that a particular user group is interested in.
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Most DBMS do not separate the three levels completely, but support the three-schema architecture to some extent. An interesting special kind of schemas was recently introduced by Google in 2012. It is called a knowledge graph (Google: Knowledge Graph, 2012; Kohs, 2014). It is organized around objects called entities, which include individuals, societies, places, events, organizations, countries, sports teams, books, works of art, movies and so on, with facts connected to them and relations between these different objects. This structure is growing very fast: in May 2012, it included 3.5 billion facts connected to 500 million entities. By December of that year, it had grown to include 570 million entities with 18 billion facts connected to them. The knowledge management system also called Knowledge Graph organizes and manages this gigantic knowledge schema — a labeled graph of knowledge — collecting and merging information about entities from many data sources. Based on this knowledge schema, Knowledge Graph provides structured and detailed information about the topic in addition to a list of links to other sites. Now other Internet companies, such as Yahoo or Diffbot, are developing their own knowledge graphs. It is interesting to know that the term knowledge graph appeared in computer science much earlier. In 1982, Hoede and Stokman started building a theory of knowledge graphs to use it for extracting knowledge from medical and sociological texts and building corresponding expert systems (cf., (Zhang, 2002)). Later several researchers in the Netherlands continued to develop and apply this theory (Hoede and Willems, 1989; Smit, 1991; van den Berg, 1993; Zhang, 2002; Wang et al., 2010). In this context, a knowledge graph is also a schema with the variable called a token, which is a node in a knowledge graph and denotes a perception of an individual from the real world. Note that according to the Existential Triad, the real world consists of three components: the physical world, the mental world and the structural world (Burgin, 2012). Perceptions of an individual from the mental world are usually called concepts or conceptions.
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Perceptions have different types, which are constants represented by marks and are also nodes in knowledge graphs. Token and marks are connected by links (edges or arcs) representing various relations. Here is an example of a knowledge graph Rose → ← cat Here Rose and cat are marks and Rose is the name of a cat. In this example, relations are represented by arrows (directed arcs). However, according to the theory of knowledge graphs, a relationship between two concepts a and b is a graph in which both a and b occur. It gives an interesting example of a named set A = (a, f, b), in which the naming relation f is a graph (cf., Appendix). It is demonstrated that the mathematical schema theory (Burgin, 2005; 2006; 2010a) encompasses all types of schemas used in programming, database theory, and computer science, as well as on the Internet and real-world databases. A notion of a schema has been also used in mathematical logic, metamathematics, and set theory. In 1927, John von Neumann (originally, J´ anos Lajos Margittai von Neumann) (1903–1957) introduced the concept of an axiom schema. It has become very useful in axiomatic set theories (for instance, the axiom of subsets is, according to the conceptions of Thoralf Skolem (1887–1963), Wilhelm Friedrich Ackermann (1896–1962), Willard Van Orman Quine (1908–2000) and some other logicians, an axiom schema) and other axiomatic mathematical theories (Fraenkel and Bar-Hillel, 1958). Logicians studied axiomatizability by a schema in the context of general formal theories (Vaught, 1967). In addition to axiom schemas, schemas of inference (e.g., syllogism schemas) have been also studied in mathematical logic (cf., for example, (Fraenkel and Bar-Hillel, 1958)). Actually, syllogisms introduced by Aristotle, as well as deduction rules of modern logic are schemas for logical inference and mathematical proofs. Another mathematical field where the concept of schema is used is category theory. This concept was introduced by Alexander Grothendieck (1928–2014) in a form equivalent to a multigraph and later generalized to the form of a small category (Grothendieck,
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1957). From categories, the concept of a schema came to algebraic geometry, where now it plays an essential role. In contrast to logic, mathematics and computer science, for a long time, exploration and utilization of mental schemas, such as interaction schemas, has yielded no efficient formalism. The first step to formalization of mental schemas in general and interaction schemas, in particular, was made by creation of the RS (Robot Schema) language (Lyons, 1986; Lyons and Arbib, 1989) and NSL (Neural Simulation Language) (Weitzenfeld, 1989; Weitzenfeld et al., 2002). RS is a language designed to facilitate sensory-based task-level robot programming. RS uses port automata (Arbib et al., 1983) to provide semantics of schemas. NSL was developed to aid the exploration of neural network simulations through interactive computer graphics. Arbib and Ehrig (1990) made two first attempts at providing a rapprochement between a methodology for parallel and distributed computation in the context of brain theory and perceptual robotics based on RS-schemas and an algebraic category theory of the specification of modules and their interconnections developed in (Ehrig and Mahr, 1985; 1990). However, as Arbib (2005) writes: “It must be confessed that [that work] was more a program for research than a presentation of results, and that research remains to be done”. The first resourceful formalization of all-purpose schemas in general and mental schemas, in particular, was achieved in mathematical schema theory developed by Burgin (2005; 2006; 2010a). This theory encompasses all types of schemas used in psychology, sociology, logic, meta-mathematics, set theory, programming, and many other areas. Here we present elements of mathematical schema theory, which, in particular, enables us to represent not only external features of interaction schemas and their functioning but also essential structural peculiarities of interaction schemas and their assemblages. Schemas can have ports, which are specific schema elements. They belong (are assigned) to schema nodes and through which information/data come into (output ports or outlets) and are sent outside the schema (input ports or inlets). Thus as before, any system
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P of ports is the union of its two disjunctive subsets P = P in ∪ P out where P in consists of all inlets from P and P out consists of all outlets from P . If there are ports that are both inlets and outlets, we combine such ports from couples of an input port and an output port. To formalize schemas, we consider, at first, those elements from which schemas are built of. There are three types of schema elements: nodes or vertices, ports, and ties or edges. Elements of all types belong to three classes: 1. Object/node, port, and connection/edge constants. 2. Object/node, port, and connection/edge variables. 3. Objects, ports, and connections with variables. In formal schemas, variables are represented by their names in a conventional manner (cf., Examples 5.6.1–5.6.4). In informal schemas, variables are represented by their descriptions or specifications in the form of a text (cf., Example 5.6.3), picture, text with pictures, etc. Example 5.6.5. The symbol T can be used as an automaton variable the range of which is the class of Turing machines. The expression NN can be used as an automaton variable the range of which is the class of neural networks. The symbol P can be used as an automaton variable the range of which is the class of port automata. Thus, variables T for Turing machines, A for finite automata, N for neural networks, etc. in the schema from Example 5.6.10 are automaton/node variables. Example 5.6.6. Information connections denoted by solid lines and process connections denoted by dashed lines in the schema from Example 5.6.10 are connection/edge variables. It is also possible to use different connection variables for links implemented on physical media, such as coaxial cable, twisted pair, or optical fiber. Example 5.6.7. The expression T [with x tapes] can be used as a denotation for an automaton with the variable x the range of which is the number of Turing machines tapes.
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Example 5.6.8. The expression c [with bandwidth x] can be used as a denotation for a connection/link with the variable x the range of which is the bandwidth (throughput) of the link. Remark 5.6.1. Each variable x is determined by its name x and range Rg x. Types of ranges determine types of variables. For instance, a variable whose range encompasses some class of neural networks has the neural network type. Remark 5.6.2. Variables in a schema form in a general case not a set but a multiset (cf., for example, Knuth, 1997) because the same variable x may be assigned to different nodes, links or ports. In addition to variables, we need variable functions. A variable function takes values in variables. For instance, a linear real function f is a variable function as it has the form f (x) = ax + b where a and b are arbitrary real numbers. Another example of a variable function is the function that takes any value xn for a given argument x. Variable functions can be of different types: • fuzzy functions in the sense of fuzzy set theory when values of the function have estimates, e.g., to what extent this value is correct, true or exact (Zimmermann, 1991); • non-deterministic functions when values of the function are not uniquely determined by the argument; • probabilistic functions in the sense of fuzzy set theory when values of the function have probabilities showing, e.g., to what extent this value is correct, true or exact. Wave function in quantum mechanics is an example of a probabilistic function. Remark 5.6.3. There is one-to-one correspondence between nondeterministic functions and set-valued functions, in which values are some sets. Remark 5.6.4. There are fuzzy functions with fuzzy domain and/or range. However, here we do not consider such functions.
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Remark 5.6.5. There is one-to-one correspondence between fuzzy functions in the above sense and fuzzy-set-valued functions, in which values are some fuzzy sets. All these structures make possible to define basic schemas. Definition 5.6.1. A basic schema R is the following system that consists of two sets, two multisets, and one mapping: R = (AR , VN R , C R , VCR , cR ). Here: The set AR is the set of all object names (node constants) from R; the multiset VN R consists of all object variables from R; the set C R is the set of all connections/links (link constants) from R; the multiset VCR consists of all link variables from R; and cR : C R ∪ VCR → ((AR ∪ VN R ) × (AR ∪ VN R )) ∪ (A R ∪ V N R ) ∪ (A R ∪ V N R ) is a (variable) function, called the node-link adjacency function, that assigns connections to nodes where A R and A R are disjunctive copies of AR , while V N R and V N R are disjunctive copies of VN R . In some cases, we need more information about schemas. A specific kind of such information is related to ports of the nodes. Ports are used to provide necessary connections between nodes inside the schema and between the schema and other systems. In this case, we consider port schemas. Definition 5.6.2. A port schema B is the following system that consists of three sets, three multisets, and three mappings B = (AB , VNB , P B , VPB , CB, VCB , pIB , cB , pEB ). Here: The set AB is the set of all object names (node constants) from B used as nodes of the schema; the multiset VN B consists of all object variables from B also used as nodes of the schema; the set C B is the set of all connections/links (link constants) from B; the multiset
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VCB consists of all link variables from B; the set P B = P IB ∪ P EB (with P IB ∩ P EB = ∅) is the set of all ports of B, P IB is the set of all ports (called internal ports) of the nodes from AB , and P EB is the set of external ports of B, which are used for interaction of B with different external systems; the multiset VP B consists of all port variables from B and is divided into two disjunctive submultisets VP Bin that consists of all variable inlets from B and VP Bout consists of all outlets from B; pIB : P IB ∪ VP B → AB ∪ VN B is a (variable) total function, called the internal port assignment function, that assigns ports to nodes; cB : C B ∪ VCB → ((P Ibout ∪ VP Bout ) × (P Ibin ∪ VP Bin )) ∪ (P IBin ∪ V P Bin ) ∪ (P IBout ∪ V P Bout ) is a (variable) function, called the port-link adjacency function, that assigns connections to ports where P IGin , P Igout , V P Bin and V P Bout are disjunctive copies of P IGin , P Igout , V P Bin and V P Bout , correspondingly; and pEB : P EB ∪ VP B → AB ∪ P IB ∪ C B ∪ VN B ∪ VP B ∪ VCB is a (variable) function, called the external port assignment function, that assigns ports to arbitrary elements from B, e.g., ports can be assigned to links or to other ports. However, in what follows, we assume that all studied port schemas do not have external nodes. Note that totality of the internal port assignment function pIB means that each internal port is assigned to some node. Usually, basic schemas are used when the modeling scale is big, i.e., at the coarse-grain level, while port schemas are used when the modeling scale is small and we need a fine-grain model. Schemas without ports, i.e., basic schemas, give us the first approximation to cognitive structures, while schemas with ports, i.e., port schemas, is the second (more exact) approximation. In some cases, it is sufficient to use schemas without ports, while in other situations to build an adequate, flexible and efficient model, we need schemas with ports. For instance, interaction schemas (Arbib, 1985), schemas of programs (cf., Garland and Luckham, 1969; Dennis et al., 1974; Fischer, 1993), or flow-charts (cf., Burgin, 1976; 1985; 1996) do not traditionally have ports. Even schemas of computer hardware are usually presented without ports (Heuring and Jordan, 1997).
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Definition 5.6.3. Internal ports of a port schema B to which no links are attached are called open or free. External ports of a port schema B to which no links or nodes are attached are called free. External ports of a port schema B, being open when the schema is considered by itself, are used for connecting B to external systems. Remark 5.6.6. It is possible to consider two representations of schemas: planar or graphical and linear or symbolic. To achieve better comprehension, schemas are usually represented in a graphical form as in Figures 5.21 and 5.22. Example 5.6.5. A basic schema of a grid automaton. In the schema from Figure 5.25, variables form the multiset, which contains: two variables Tm, one variable NN, two variables RAM, five variables FA, one variable CA, six variables m, and one variable GA. Here is the semantics of these variables: Tm is a variable the range of which is the class of all Turing machines; RAM is a variable the range of which is the class of all random access machines; S is a variable the range of which is the class of all servers; m is a variable the range of which is the class of all modems; NN is a variable the range of which is the class of all neural networks; FA is a variable the range of which is the class of all finite automata; CA is a variable the range of which is the class of all cellular automata; GA is a variable the range of which is the class of all grid automata. Example 5.6.11. A formal basic schema that formalizes the interaction schema from Figure 5.23 is given in Figure 5.26. This schema has connections/links of two types: links for activation of nodes and for transfer of data. Such a formalization of the schema from Figure 5.23 allows us to better study its properties and transformations. It demonstrates that this schema has realizations not only by the brain neural structures but also by computer programs.
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GA Tm1
c1
c3
c5
c2
c4
c6
CA1 FA1
m2 FA2 RAM1
m1
S1
m3
NN1
m5
Tm2
FA3 FA4 FA5
m4 RAM2
m6
GA1 Tm1
CA1
c1
c3
c5
c2
c4
c6 FA1
m2 FA2 RAM1
m1
S1
m3
NN1
m5
Tm2
FA3 FA4 FA5
m4 RAM1
m6
GA3
Figure 5.25. A schema of a grid automaton GA
Here is the list of variables and their meaning in this schema: X1 is a variable the range of which is the class of all schemas (algorithms or neural assemblages) for visual location; X2 is a variable the range of which is the class of all schemas (algorithms or neural assemblages) for size recognition;
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X1
X4
X7
X2
X3
X5
X6
X8
Figure 5.26. Dashed lines represent activation of signals, while solid lines represent transfer of data
X3 is a variable the range of which is the class of all schemas rithms or neural assemblages) for orientation recognition; X4 is a variable the range of which is the class of all schemas rithms or neural assemblages) for fast phase movement; X5 is a variable the range of which is the class of all schemas rithms or neural assemblages) for hand preshape; X6 is a variable the range of which is the class of all schemas rithms or neural assemblages) for hand rotation; X7 is a variable the range of which is the class of all schemas rithms or neural assemblages) for slow phase movement; X8 is a variable the range of which is the class of all schemas rithms or neural assemblages) for actual grasp.
(algo(algo(algo(algo(algo(algo-
We discern three classes of constants and variables in schemas: • static constants and variables; • function constants and variables; • process constants and variables. Static constants are representations (names) of objects that do not represent changes.
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Static variables are variables the range of which consists of representations (names) of objects that do not represent changes. Function constants are representations (names) of functions and relations. Function variables are variables the range of which consists of representations (names) of functions and relations. Process constants are representations (names) of actions, events, and processes. Process variables are variables the range of which consists of representations (names) of actions, events, and processes. With respect to this classification, there are: • static schemas with only static constants and variables; • function schemas that have function and may be, static constants and variables; • process schemas that have process constants and variables. Schemas that have variables are called variable. There is a natural connection between basic schemas from port schemas. The following algorithm shows how to get basic schemas from port schemas. Let us consider a port schema B = (AB , VNB , P B , VPB , C B , VCB , pIB , cB , pEB ) and build a basic schema DB. The internal port assignment function pIB and port-link adjacency function cB determine the node-link adjacency function ncB of a basic schema DB ¯ IBin = P IBin ∪ in the following way. Let us take l ∈ C B , P ¯ IBout = P IBout ∪ VPB , A ¯B = AB ∪ VNB , A ¯ and A ¯ VPB , P B B ¯ are disjoint copies of AB , and pIB ∗ = (pIB × pIB ) ∗ pIB ∗ pIB : ¯ IBout ) ∪ P ¯ IBin ∪ P ¯ IBout → (A ¯B × A ¯B ) ∪ A ¯ ∪ A ¯ . ¯ IBin × P (P B B Here × is the product and ∗ is the coproduct of mappings in the sense of category theory (cf., for example, (Herrlich and Strecker, 1973)). Then nc B is a composition of functions pIB and cB , namely, nc B (l) = pIB ∗ (cB (l)). The node-link adjacency function nc B determines a schema in which links are adjusted directly to nodes, ignoring ports. Thus, it is possible to exclude ports from the schema, obtaining a schema without ports or a basic schema. Consequently, this algorithm gives
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us a basic schema, which is denoted by DB where DB = (AB , VNB , C B , VCB , ncB ) and called the projection of the port schema B. At the same time, it is possible to consider any basic schema as a special kind of port schemas, in which any node has exactly one port and all connections go through this port. Example 5.6.7. Let us consider the basic schema (adapted from (Burgin, 2005; Ch. 2)) of a Turing machine AT,w that reduces the problem of deciding whether a given Turing machine has some nontrivial property P to the halting problem for Turing machines (cf., Figure 5.27). This example shows that it is possible to build schemas of algorithms and automata. Here is the list of variables in this schema and their properties: T denotes some Turing machine from a given class K . M denotes a Turing machine that does not have the property P . G denotes a finite automaton with two inputs. One input comes from the outside through M , while the second input of G is the output of the machine T . The automaton G can be in two states: closed and open. Initially G is closed until it receives some input from T , which makes it open. When G is closed, it gives no output. When G is open, it gives the word that comes to G from M as its output. The structure of the Turing machine AT,w . When an informal schema, such as an interaction schema or flowchart of a program, is formalized, its formal representation is a
G u
M
w
T
AT,w
Figure 5.27. The basic schema of a Turing machine AT w
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mathematical model of this schema. This model allows one to study, build, and apply schemas utilizing powerful tools of mathematics. The procedure of formalization is rather simple. To get a formal representation of an interaction schema, we denote descriptions by variables, properly assign ranges of these variables, and make relevant substitutions in the schema. Remark 5.6.7. It is possible to consider schemas with zero variables. Remark 5.6.8. If an automaton (system) is given by its specification in the sense of Blum, Ehrig, and Parisi-Presicce (1987) and Ehrig and Mahr (1985; 1990), then components and their compositions become a special kind of schemas. This allows a more rigorous development of a component-based technology similar to one developed by these same authors. A port schema P is described by three grid characteristics, three node characteristics, and three edge characteristics. The grid characteristics are: 1. The space organization or structure of the schema P . This space structure may be in physical space, reflecting where the corresponding information processing systems (nodes) are situated, or it may be a mathematical structure defined by the geometry of node relations. Besides, we consider three levels of space structures: local, region, and global space structures of a schema. Sometimes these structures are the same, while in other cases they are different. The space structure of a schema can be static or dynamic. The dynamic space structure can be of two kinds: persistent or flexible. However, the space structure of a schema may be variable. Inherent structures of the schema are represented by its grid and connection grid (cf., Definitions 5.6.8 and 5.6.9). Due to a possible non-determinism in the port assignment functions and port-link adjacency function, there is a possibility of non-determinism in inherent structures of the schema.
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2. The topology of the schema P is a complex structure that consists of node topology determined by the type of the node neighborhood and port topology determined by the type of the port neighborhood. A neighborhood of a node (port) is the set of those nodes (ports) with which this node directly interacts (is directly connected). As the port assignment functions and port-link adjacency function may be non-deterministic, the topology of the schema P also may be non-deterministic. In particular, a schema may have fuzzy or probabilistic topology. For deterministic schemas, we have three main types of topology: — A uniform topology, in which neighborhoods of all nodes of the schema have the structure. — A regular topology, in which the structure of different node neighborhoods is subjected to some regularity. — An irregular topology where there is no regularity in the structure of different node neighborhoods. An example of a regular but non-uniform schema topology is the schema of a cellular automaton in the hyperbolic plane or on a Fibonacci tree (Margenstern, 2002). In this schema, nodes are variables ranging over finite automata, while all edges/links are fixed. Non-deterministic schemas can also be regular and irregular. 3. The dynamics of the schema P determines by what rules its nodes exchange information with each other and with a tentative environment of P and in particular, how nodes use ports and corresponding links. This dynamics is usually an algorithmic function that depends on values of its variables because some of nodes and/or links are variables and there is a permissible non-determinism in the port assignment functions and port-link adjacency function. Interaction with the environment separates two classes of schema: open schemas allow interaction (accepting and transmitting information) with the environment through definite connections, while closed schemas do not have means for such interaction. For instance,
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traditional schemas representing concepts and logical propositions are closed. Existence of free ports makes a closed schema potentially open as it is possible to attach connections to these ports. The node characteristics are: 1. The type and structure of the node, including structures of its ports. There are different levels of node typology. On the highest level, there are two types of nodes: an object node and a variable node. Each of these types has subtypes, e.g., a neural network, Turing machine or finite state machine. These subtypes form the next level of the type hierarchy. Subtypes of these subtypes (e.g., a Turing machine with one linear tape) form one more level of the type hierarchy and so on. 2. The external dynamics of the node determines interactions of this node. According to this characteristic, there are three types of nodes: accepting nodes that only accept or reject their input; generating nodes that only produce some output; and transducing nodes that both accept some input and produce some output. Note that nodes with the same external dynamics can have different dynamics when they work in a grid. For instance, let us take two nodes: a transducing node B and a generating node B. Initially they have different dynamics. However, as parts of a schema P , they both work as generating nodes because the schema dynamics prescribes this. For nodes of the schema that are variables, we have not a definite dynamics but a type of dynamics. Primitive ports do not change node dynamics. However, compound ports are able to influence processes in the whole schema and in the node to which they belong. For instance, a compound port can be an object, e.g., an automaton, or even a schema. 3. The internal dynamics of the node determines what processes go inside this node. For nodes of the schema that are variables, we have not a definite dynamics but a type of dynamics. For instance, it may be given that the node with number 3 in a schema computes function f (x). Such nodes are usually used in program schemas
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(which are traditionally called program schemata (cf., for example, (Fischer, 1993))). The edge characteristics are: 1. The external structure of the edge. According to this characteristic, there are three types of edges: a closed edge (a link or link variable) both sides of which are connected to ports of the schema; an ingoing edge in which only the end side is connected to a port of the port schema; and an outgoing edge in which only the beginning side is connected to a port of the port schema. 2. Properties and the internal structure of the edge. There are different levels of edge typology. On the highest level, there are two types: constant and variable links. Each of these types has subtypes that form the next level of the type hierarchy. According to the internal structure, there are three subtypes of edges: a simple channel that only transmits data/information; a channel with filtering that separates a signal from noise; and a channel with data correction. Subtypes of these subtypes form the next level and so on. 3. The dynamics of the edge determines edge functioning. For instance, two important dynamic characteristics of an edge are bandwidth as the number of bits (data units) per second transmitted on the edge and throughput as the measured performance of the edge. In schemas, these characteristics may be variable. Properties of links/edges separate all links into three standard classes: 1. Information link/connection is a channel for processed data transmission. 2. Control link/connection is a channel for instruction transmission. 3. Process link/connection realizes control transfer and determines how the process goes by initiation of a node in the grid by another node (other nodes) in the grid. Process links determine what to do, control links are used to instruct how to work, and information links supply automata with data in a process of schema or its instantiation functioning.
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There are different types and kinds of schema variables. The dynamic typology discerns three types of basic variables: 1. System variables. 2. Function variables. 3. Process variables. The schema from Example 5.6.5 uses system variables (Ti for Turing machines, Ai for finite automata, N for neural networks, etc.). The schema from Example 5.6.6 uses function variables, e.g., X2 is a variable for such function as size recognition. The scaling classification discerns three types of variables: 1. Individual variables that are used in one node, port or link from the schema. 2. Local variables that are used in a group of nodes, ports or links from the schema. 3. Global variables that are used for the whole schema. Difference between constants and variables in schemas results in existence of special classes of schemas: — — — — — — — — —
Basic/port schemas with constant nodes; Basic/port schemas with constant links; Port schemas with constant ports; Port schemas with constant port assignment; Basic/port schemas with constant node-link adjacency. Port schemas with dynamic port assignment; Basic/port schemas with dynamic node-link adjacency. Port schemas with deterministic port assignment; Basic/port schemas with deterministic node-link adjacency.
Let us consider operations on schemas. Utilization of different schemas usually involves various operations. There are three basic vertical unary operations in the hierarchy of both basic and port schemas: abstraction, concretization, and determination. Definition 5.6.4. Changing a variable to a constant from the range of this variable is called an interpretation of this variable.
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Definition 5.6.5. An operation of changing (interpreting) some of the variables in a schema R to constants is called a concretization operation Con applied to R, while the result Con R of this operation is called a concretization of R. Example 5.6.8. An instantiation of a schema in the sense of (Arbib, 1989) is its maximal concretization. Example 5.6.9. The grid automaton from Figure 5.23 is a concretization of the schema from Figure 5.26. Definitions imply the following result. Lemma 5.6.1. Concretization of a schema preserves the schema topology and structure. Definition 5.6.6. If a concretization Con R of a schema R is a grid automaton, then Con R is called a realization of R, while the corresponding operation is called a realization operation. Remark 5.6.9. We have noted that a schema may involve the cooperative activity of multiple brain regions. In particular, then, a schema becomes a “mode of activity” for grid automata and — depending on input and context — a grid automaton can support many different schemas as their realization. Proposition 5.6.1. If P is a realization of a schema R, and R is a concretization of a schema Q, then P is a realization of the schema Q. Indeed, P is obtained from R by changing all variables from R to constants and R is obtained from Q by changing some variables from Q to constants. Thus, P does not have variables and so it is a grid automaton obtained by changing all variables from Q to constants. Corresponding to a schema Q its realization RQ is an operation that is also called realization. Corollary 5.6.1. Realization of a schema is an idempotent operation.
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Definition 5.6.7. A realization Rea R of a schema R becomes an instantiation of R when Rea R starts functioning. Abstraction is an operation opposite to concretization. Definition 5.6.8. An operation of changing some of the constants in a schema P to variables is called an abstraction operation Con applied to P , while the result Abs P of this operation is called a abstraction of P . Remark 5.6.10. Abstraction and concretization are operations with a set of tentative results in contrast to conventional arithmetical and algebraic operations such as addition or multiplication, which give only one result (if any). Example 5.6.10. The schema from Figure 5.26 is an abstraction of the grid automaton from Figure 5.23. In some sense, operations of abstraction and concretization of schemas are reciprocal (inverse) with respect to one another. Namely, they have the following property. Lemma 5.6.2. (a) If a schema P is obtained from a schema R by abstraction, then it is possible to get schema R by concretization of P . (b) If a schema R is obtained from a schema P by concretization, then it is possible to get schema P by abstraction of R. As abstractions and concretizations are transformations of schemas, it is natural to introduce their composition as consecutive performance of corresponding transformations. Definitions imply the following result. Proposition 5.6.2. (a) Composition of schema concretizations is a schema concretization. (b) Composition of schema abstractions is a schema abstraction. Definition 5.6.9. (a) A schema P is (strongly) equivalent to a schema R if they have the same realizations (concretizations). (b) A schema P is (strongly) equivalent to a schema R with respect to a class A of grid automata if they have the same realizations (concretizations) in A.
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For instance, taking an interaction schema, we are interested in its realization in the class A of neural networks or even more exactly, in the class B of neural ensembles in the brain. Remark 5.6.11. There are other interesting equivalencies of schemas. Proposition 5.6.3. Two schemas are strongly equivalent if and only if it is possible to obtain one from the other by renaming the variables in the first schema. Proof is left as an exercise. Operations of abstraction and concretization define corresponding relations in the set of all schemas. Definition 5.6.10. If a schema P is obtained from a schema R by abstraction (or a schema R is obtained from a schema P by concretization), then P is called more general than R and R is called more concrete than P . It is denoted by Pa > R and R >c P , respectively. Example 5.6.11. The schema from Figure 5.26 is more abstract than the schema from Figure 5.23. We remind that a strict partial order on a set X is a reflexive and transitive relation. Lemma 5.6.3. Both >c and a > are strict partial orders. Proof . By the definition, a strict partial order is a transitive antisymmetric relation (cf., Appendix). Thus, we have to test these properties. 1. Transitivity. By Proposition 5.6.2.a, the relation >c is transitive and by Proposition 5.6.2.b, the relation a > is transitive. 2. Antisymmetry. (a) If Pa > R, then a schema P is obtained from a schema R by abstraction. By definition, it means that the number nvP of variables in P is larger than the number nvR of variables in R. Because whole numbers are linearly ordered, it is impossible to have the relation Ra > P , which involves nvP < nvR .
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As P and R are arbitrary schemas, the relation a > is antisymmetric. b) If P >c R, then a schema P is obtained from a schema R by concretization. By definition, it means that the number nvP of variables in P is less than the number nvR of variables in R. Because whole numbers are linearly ordered, it is impossible to have the relation R a > P , which involves nvP > nvR . As P and R are arbitrary schemas, the relation >c is antisymmetric. Lemma is proved. Concretization is a special kind of a more general operation on schemas. Definition 5.6.11. An operation of decreasing (delimiting) the range of a variable function f is called a determination operation Det applied to f , while the result Det f of this operation is called a determination of f . In a similar way, it is possible to change port assignment and port-link adjacency functions. Definition 5.6.12. An operation of decreasing (delimiting) the range of the variable port assignment functions and/or port-link adjacency function of a schema R is called a determination operation Det applied to R, while the result Det R of this operation is called a determination of R. Determination operation defines specific relations between schemas. Definition 5.6.13. If a schema P is obtained from a schema R by determination, then P is called more determined than R. It is denoted by Pdet ≥ R. Lemma 5.6.4. Pdet ≥ R is a relation of partial order. Proof is similar to the proof of Lemma 5.6.3. It is interesting to study properties of this partial order relation for some well-known classes of schemas, e.g., for program schemata.
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To represent structures of schemas, we use multigraphs, directed multigraphs, partially directed multigraphs, generalized multigraphs, generalized directed multigraphs, and generalized partially directed multigraphs. Schema grids can be not only conventional or stable directed multigraphs and generalized directed multigraphs, but also variable directed multigraphs and generalized directed multigraphs. Definition 5.6.14. (a) A multigraph G has the following form: G = (V, E, c). Here V is the set of vertices or nodes of G; E is the set of edges of G; the edge-node adjacency or incidence function c : E → V × V may be constant or variable. This function assigns each edge to a pair of vertices. (b) An oriented or directed multigraph G has the following form: G = (V, E, c). Here V is the set of vertices or nodes of G; E is the set of edges of G, each of which has the beginning and the end, i.e., they are arrows; the edge-node adjacency or incidence function c : E → V × V may be constant or variable. This function assigns each edge to a pair of vertices so that the beginning of each edge is connected to the first element in the corresponding pair of vertices and the end of the same edge is connected to the second element in the same pair of vertices. (c) If in the structure G = (V, E, c), some edges are arrows and some are not, this structure is a partially oriented or partially directed multigraph. A multigraph is a graph when c is an injection (Berge, 1973). When the adjacency function is variable, the multigraph (directed multigraph or partially directed multigraph) is called variable. It is possible to consider bi-directed edges but they play the same role in graphs as edges that are not directed. Usually multigraphs, directed multigraphs, partially directed multigraphs, generalized multigraphs, generalized directed multigraphs, and generalized partially directed multigraphs are represented in the geometrical (graphical) form. For instance, the
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geometrical structure in Figure 5.28(a) is a graph, the geometrical structure in Figure 5.28(b) is a multigraph, the geometrical structure in Figure 5.28(c) is a directed graph, the geometrical structure in Figure 5.28(d) is a directed multigraph, the geometrical structure in Figure 5.28(e) is a partially directed graph, and the geometrical structure in Figure 5.28(f) is a partially directed multigraph. Schemas of open systems demand more general constructions as their characteristics. Definition 5.6.15. (a) A generalized multigraph G has the following form: G = (V, E, c). Here V is the set of vertices or nodes of G; E is the set of edges of G; the edge-node adjacency or incidence function c : E → (V ×V ∪V ), which assigns each edge either to a pair of vertices or to one vertex, may be constant or variable. In the latter case, when the image c(e) of an edge e belongs to V , it means that e is connected to only to one vertex c(e). (b) A generalized oriented or directed multigraph G has the following form: G = (V, E, c : E → (V × V ∪ Vb ∪ Ve )). Here V is the set of vertices or nodes of G; E is the set of edges (arrows) of G (with fixed beginnings and ends); Vb ≈ Ve ≈ V ; the edge-node adjacency function c: E → (V ×V ∪Vb ∪Ve ), which assigns each edge either to a pair of vertices or to one vertex, may be constant or variable. In the latter case, when the image c(e) of an edge e belongs to Vb , it means that e is connected to the vertex c(e) by its beginning. When the image c(e) of an edge e belongs to Ve , it means that e is connected to the vertex c(e) by its end. Edges that are mapped to the set Vb ∪ Ve are called open. (c) If in the structure G = (V, E, c) some edges are arrows and some are not, this structure is a generalized partially oriented or partially directed multigraph. Note that the difference between graphs (multigraphs) and generalized graphs (generalized multigraphs) is that in graphs
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(a)
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Figure 5.28. Examples of graphs (a), multigraphs (b), directed graphs (c), directed multigraphs (d), partially directed graphs (e), and partially directed multigraphs (f)
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(multigraphs), all edges are connected to vertices by both sides, while in generalized graphs (generalized multigraphs) edges can be connected to vertices only by one side. Graphs, multigraphs, generalized graphs and generalized multigraphs are used to represent inner structure of schemas. Definition 5.6.16. The grid G(P ) of a (basic or port) schema P is the (variable) generalized oriented multigraph that has exactly the same vertices and edges as P , while its adjacency function cG(B) is nc B . Note that the grid of a variable schema can be constant if in this schema, there are only node variables. Edge variables in a schema also do not make its grid variable if domain of each edge variable contains edges of a single type. Example 5.6.12. Figure 5.29 gives the graphical form of the grid G(P ) of the schema P from Example 5.6.9. We can see that the grid G(P ) of a schema P is a partially directed graph. Example 5.6.13. Figure 5.30 gives the graphical form of the grid G(R) of the schema R from Example 5.6.12. o
o
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o Figure 5.29. The grid G(P ) of a schema P
o o o o o
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Figure 5.30. The grid G(R) of the schema R from Example 5.6.12
Any port schema has the same grid as its projection on the corresponding basic schema. Proposition 5.6.4. For any port schema P , we have G(P ) = G(DP ) where DP is the basic schema built from P . Grids of schemas allow one to characterize definite classes of schemas. Proposition 5.6.5. A schema B is closed if and only if its grid G(B) satisfies the condition Im c ⊆ V × V , or in other words, the grid G(B) of B is a conventional multigraph. Proposition 5.6.6. A schema B is an acceptor only if it has external input ports or/and its grid G(B) has edges connected by their end, or Im c ∩ Ve = Ø. Proposition 5.6.7. A schema B is a transmitter only if it has external output ports or/and its grid G(B) has edges connected by their beginning, or Im c ∩ Vb = Ø. Proposition 5.6.8. A schema B is a transducer only if it has external input and output ports or/and its grid G(B) has edges connected by their beginning and edges connected by their end.
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Definition 5.6.17. The connection grid CG(B) of a port schema B is the (variable) generalized oriented multigraph nodes of which bijectively correspond to the ports of B, while edges and the adjacency function cCG(B) are the same as in B. Proposition 5.6.9. The grid G(B) of a port schema B with constant port assignment is a homomorphic image of its connection grid CG(B). Indeed, by the definition of a schema, ports are uniquely assigned to nodes, and by the definition of the grid G(B) a schema B, the adjacency function cG(B) of the grid G(B) is the composition of the port assignment function pB and the adjacency function cB of the schema B. Proposition 5.6.10. For each schema, there is its maximal with respect to the relations g ≥ and det ≥ abstraction. Proof . It is possible to change object names, ports, and connections with variables to variables with corresponding ranges so that the new schema is equivalent to the initial one. Besides, if we have a set of variables, we can change this set to one variable with the corresponding range so that the new schema is equivalent to the initial one. Let R be a schema. We transform it in the following way. We assign to each node of its connection grid CG(R) a variable the range of which encompasses all possible automata. We assign to each port of its connection grid CG(R) a variable the range of which encompasses all possible ports. We assign to each edge of G(R) another variable the range of which encompasses all possible links. In addition, we take as the port assignment functions and port-link adjacency function non-deterministic functions that allow maximal flexibility of assignments and adjustments, i.e., any port may be assigned to any node or link from R in a permissible way and any link may be adjacent to any node or a pair of nodes. In such a way, we obtain a maximal abstraction of R.
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Dynamics of schemas is represented not only by operations but also by different kinds of homomorphisms. Definition 5.6.18. A structural homomorphism f of a basic schema P into a basic schema R is a mapping of nodes and connections of P such that nodes of P are mapped into nodes of R, connections of P are mapped into connections of R, and the node-link adjacency function is preserved. For port schemas, we have two kinds of structural homomorphisms. Definition 5.6.19. A (weak) structural homomorphism f of a port schema P into a port schema R is a mapping of nodes and connections of P such that nodes of P are mapped into nodes of R, connections of P are mapped into connections of R, and the (node-link adjacency function) port assignment functions and port-link adjacency function are preserved. It is possible to consider a structural homomorphism f of a basic schema P into a basic schema R as a pair of mappings: one of them f(n) maps nodes of P into nodes of R and the other one f(e) maps links of P into links of R. In addition, assignment functions and adjacency relation are preserved. A structural homomorphism f of a port schema P into a port schema R uniquely corresponds to and determines the homomorphism f g : CG(P ) → CG(R) of the corresponding connection grids, while a weak structural homomorphism f of a port schema P into a port schema R uniquely corresponds to and determines the homomorphism f g : G(P ) → G(R) of the corresponding grids. In a natural way, compositions of structural homomorphisms and weak structural homomorphisms of port schemas are introduced as conventional sequential composition of mappings. Definitions imply the following results. Proposition 5.6.11. Any concretization Con P (abstraction Abs P ) of the schema P defines a structural homomorphism fcon : Con P → P (fabs : Abs P → P ).
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Proposition 5.6.12. Composition of (weak) structural homomorphisms of schemas is a (weak) structural homomorphism of schemas. In some situations, it is useful to have more restrictions on mapping of schemas. Definition 5.6.20. A (weak) structural homomorphism f of a schema P into a schema R is called a (weak) typed homomorphism if the following conditions are satisfied: (a) variables from P are mapped into variables and constants of the same type from R; (b) constants from P are mapped into constants of the same type from R. In a natural way, compositions of typed homomorphisms of schemas are introduced as conventional sequential composition of mappings. Proposition 5.6.13. Composition of typed homomorphisms of schemas is a typed homomorphism of schemas. Proposition 5.6.14. (a) Any concretization Con R (abstraction Abs P ) of a schema R is defined by a VE-homomorphism c : R → Con R of schemas. (b) Any abstraction Abs P of a schema P is defined by an inverse VE-homomorphism h : P → Abs P of schemas. Proposition 5.6.15. The transformation of a schema R into the corresponding basic schema DR is a weak typed homomorphism of schemas. Utilizing typed homomorphisms and structural homomorphisms, as well as Propositions 5.6.11 and 5.6.12, we build four categories of schemas: the category TSC in which objects are schemas and morphisms are their typed homomorphisms; the category GSC in which objects are schemas and morphisms are their structural homomorphisms; the category WTSC in which objects are schemas and morphisms are their weak typed homomorphisms; and the category
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WGSC in which objects are schemas and morphisms are their weak structural homomorphisms. Proposition 5.6.16. TSC is a subcategory of GSC, while WTSC is a subcategory of WGSC. Indeed, typed homomorphisms are a special kind of structural homomorphisms. Consequently, the category TSC contains all objects from the category GSC but only some of its morphisms. In a similar way, the category WTSC contains all objects from the category WGSC but only some of its morphisms. Definitions imply the following result. Proposition 5.6.17. WTSC is a quotient category of TSC, while WGSC is a quotient category of GSC. It is possible to separate in both categories some special classes of morphisms useful in schema theory. Such morphisms represent formation and transformations of schemas. Definition 5.6.21. (a) A (structural) homomorphism of schemas f : R → P is called a (structural) V-monomorphism [E-monomorphism] if images of any two vertices [links] from R do not coincide. (b) A (structural) homomorphism of schemas f : R → P is called a (structural) VE-monomorphism if it is both a (structural) V-monomorphism and E-monomorphism. Definition 5.6.22. (a) A (structural) homomorphism of schemas f : R → P is called a (structural) V-epimorphism [E-epimorphism] if any vertex [link] from P is an image of some vertex [link] from R. (b) A (structural) homomorphism of schemas f : R → P is called a (structural) VE-epimorphism if it is both a (structural) V-epimorphism and E-epimorphism. Definition 5.6.23. The image Im f of a (structural) homomorphism of schemas f : R → P is the largest subschema of P such that any its vertex from P is the image of some vertex [link] from R and any its link is the image of some link from R. Let f : R → P be a (structural) homomorphism of schemas.
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Lemma 5.6.5. Im f is the largest subschema of P such that f defines an (structural) VE-epimorphism of R onto Im f . Let f : R → P be a (structural) E-epimorphism of schemas. It is possible to derive properties of R from properties of P and vice versa. Proposition 5.6.18. (a) If the grid G(R) is connected (full) and f : R → P is a (structural) E-epimorphism of schemas, then the grid G(P ) is also connected (full). (b) If fan-in (fan-out) of all edges from the grid G(P ) is larger than n and f : R → P is a (structural) E-epimorphism of schemas, then fan-in (fan-out) of all edges from the grid G(R) is larger than n. Corollary 5.6.2. (a) If the grid G(P ) is disconnected and f : R → P is a (structural) E-epimorphism of schemas, then the grid G(R) is also disconnected. (b) If fan-in (fan-out) of all edges from the grid G(R) is smaller than n and f : R → P is a (structural) E-epimorphism of schemas, then fan-in (fan-out) of all edges from the grid G(P ) is smaller than n. Corollary 5.6.3. If f : R → P is a (structural) E-epimorphism of schemas, then the number of components of the grid G(R) is larger than or equal to the number of components of the grid G(P ). The concept of a subschema is important in schema theory. Definition 5.6.24. A schema P is a (strong) structural subschema of a schema R if the grid G(P ) is a generalized oriented submultigraph of the grid G(R) (the connection grid CG(P ) is a generalized oriented submultigraph of the connection grid CG(R)). It is denoted by P ⊆S R and P ⊆SS R, respectively. Lemma 5.6.6. Any strong structural subschema of a schema R is its structural subschema, i.e., P ⊆SS R implies P ⊆S R. Example 5.6.14. The schema R given in Figure 5.31 is a structural subschema of the schema from Figure 5.23 and of the schema from Figure 5.26. However, the schema R is neither a subschema of
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Figure 5.31. Dashed lines represent activation of signals, while solid lines represent transfer of data
the schema from Figure 5.23 nor a subschema of the schema from Figure 5.26. Here is the list of variables in this schema: T1 and T3 are Turing machines; NN is a neural network; FA and FA1 are variables the range of which is the class of all finite automata. Lemma 5.6.7. If a schema P is a structural subschema of a schema R, then there is a structural VE-monomorphism of P into R. Proof is left as an exercise.
Definition 5.6.25. A schema P is a subschema of a schema R if all nodes of P belong to the set of nodes of R, all links of P belong to the set of links of R, all ports of P belong to the set of ports of R, and the internal and external port assignment functions pIP and pEP and port-link adjacency function cP of P is a restriction of the internal and external port assignment functions pIR and pER and port-link adjacency function cR of R, respectively. It is denoted by P ⊆ R. Let P be a subschema of a schema R. Proposition 5.6.19. For any concretization Con P (abstraction Abs P ) of the schema P , there is a unique minimal concretization Con R (abstraction Abs R) of the schema R such that Con P (Abs P ) is a subschema of Con R (Abs R).
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Proof . (1) The concretization Con P of the schema P is constructed by changing some variables from P to constants. Let us take be the set Vch of all changed variables in P and make the same changes in R as those that are performed in P . This will give us the concretization Con R of the schema R, in which Con P is a subschema and which is minimal as the least number of variables from R necessary for making Con P a subschema Con R is changed to constants. (2) The abstraction Abs P of the schema P is constructed by changing some constants from P to variables. Let us take be the set Cch of all changed constants in P and make the same changes in R as those that are performed in P . This will give us the abstraction Abs R of the schema R, in which Abs P is a subschema and which is minimal as the least number of constants from R necessary for making Abs P a subschema Abs R is changed to variables. The Proposition is proved. Let P be a subschema of a schema R. Proposition 5.6.20. For any concretization Con R (abstraction Abs R) of the schema R, there is a unique concretization Con P (abstraction Abs P ) of the schema P such that Con P (Abs P ) is a subschema of Con R (Abs R). Indeed, to build the necessary subschema Con P (Abs P ) of the schema P , we take the minimal subschema of Con R (Abs R) that contains all nodes, links and, if necessary, ports from P . Remark 5.6.12. The concept of a subschema of a schema is defined to both informal and formal schemas, as well as for basic and port schemas. Proposition 5.6.21. Any port schema potentially is a subschema of a closed port schema. Indeed, a port schema is open when there are free ports, i.e., ports that are assigned to any link. Thus, it is possible to make any port schema close if we assign links to free ports so, that the new schema will not have ports for external interaction and thus, it will be closed.
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In the neurophysiological schema theory (Arbib, 1995; 2005), it is important to be able to elaborate different schemas into networks of interacting subschemas until finally it becomes possible to realize these constructs in terms of neural networks or other appropriate circuitry. To provide means for construction of new schemas from given schemas, it is possible to use operations introduced in the mathematical schema theory (Burgin, 2010a). Example 5.6.15. The (informal) schema of grasping given in Figure 5.32 is a subschema of the schema from Figure 5.23. Example 5.6.16. The (formal) schema given in Figure 5.33 is a subschema of the schema from Figure 5.26. This schema has connections/links of two types. Here is the list of variables in this schema: X4 is a variable (algorithms or X5 is a variable (algorithms or
the range of which is the class of all schemas neural assemblages) for fast phase movement; the range of which is the class of all schemas neural assemblages) for hand preshape; kinesthetic
visual and
and tactile
kinesthetic input
input
Hand
Hand
Preshape
Rotation
Actual Grasp Grasping Figure 5.32. Dashed lines denote activation of signals and solid lines denote transfer of data
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X4
X7
X5
X6
X8
Figure 5.33. Dashed lines represent activation of signals, while solid lines represent transfer of data
X6 is a variable (algorithms or X7 is a variable (algorithms or X8 is a variable (algorithms or
the range of which is the class of all schemas neural assemblages) for hand rotation; the range of which is the class of all schemas neural assemblages) for slow phase movement; the range of which is the class of all schemas neural assemblages) for actual grasp.
Proposition 5.6.22. If a schema P is a (structural) subschema of a schema R, and the schema R is a (structural) subschema of a schema Q, then the schema P is a (structural) subschema of the schema Q. Proof . Let us assume that a schema P is a structural subschema of a schema R, and the schema R is a structural subschema of a schema Q. It means that the grid G(P ) of P is a generalized oriented submultigraph of the grid G(R) of R and the grid G(R) of R is a generalized oriented submultigraph of the grid G(Q) of QR. Consequently, the grid G(P ) of P is a generalized oriented submultigraph of the grid G(Q) of Q. By definition, it means that the schema P is a structural subschema of the schema Q. Now let us assume that a schema P is a subschema of a schema R, and the schema R is a subschema of a schema Q. The relation “to be a subschema” is defined by inclusion of sets (of nodes, links and ports) and by restrictions of functions. As inclusion of sets and restriction of functions are transitive relations, the schema P is a subschema of the schema Q.
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Proposition 5.6.23. If f : R → P is a (structural) homomorphism [V-monomorphism, E-monomorphism, VE-monomorphism] of schemas and Q is a subschema of the schema R, then f defines a restriction fQ of f on Q, which is a (structural) homomorphism [V-monomorphism, E-monomorphism, VE-monomorphism, respectively] of schemas Definition 5.6.26. A subschema Q of a schema R is called V-complete in R if Q contains all nodes from R. The concept of a V-complete subschema is especially useful when connections (links) are variable while all nodes are constants. In such a way, it is possible to investigate how connections (links) between a set of given nodes are changing with time. Definition 5.6.27. A subschema Q of a schema R is called E-complete in R if Q contains all links from R that are connected in R to some node of Q. The concept of a E-complete subschema is especially useful when nodes are variable while all connections (links) are constants. In such a way, it is possible to explore how nodes are changing with time while connections (links) remain stable. Definition 5.6.28. A subschema Q of a schema R is called P-complete in R if Q contains all ports from R. Definitions imply the following result. Proposition 5.6.24. A subschema Q of a schema R is E-complete, P-complete, and V-complete at the same time in R if and only if it coincides with R. Proposition 5.6.25. If a schema R does not have nodes without ports, then P-completeness of a subschema Q implies V-completeness of Q. Indeed, ports cannot be separate from nodes. Therefore, if all ports are present in a subschema, then all nodes are also present in this subschema.
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Proposition 5.6.26. If a schema R does not have links without ports and open ports, then P-completeness of a subschema Q without open ports implies E-completeness of Q. Indeed, if there no open ports in R, then any of its ports is connected by a link to another port. The subschema Q contains all ports from the schema R. Not to be open, these ports need all links from the schema R. Therefore, if all ports are present in the subschema Q, then all nodes are also present in Q and it is E-complete. Proposition 5.6.27. If f : R → P is a structural V-epimorphism (E-epimorphism) of schemas, then the inverse image f −1 (Q) of a V-complete (E-complete) subschema Q of a schema P is V-complete (E-complete). Proof . (a) Let Q be a V-complete subschema of a schema P . Then f −1 (Q) contains all nodes from R as f : R → P is a structural V-epimorphism, i.e., f −1 (Q) is V-complete in R. (b) Let Q be a E-complete subschema of a schema P and r be an link from R, at least, one end of which is connected to a node A such that f (A) is a node from Q. Then by the definition of a structural homomorphism, f (r) is connected to the node f (A), at least, one end. Thus, f (r) is r be a link from Q as Q is E-complete in R. Consequently, r belongs to f −1 (Q). As r is an arbitrary link from R connected to a node from f −1 (Q), the schema f −1 (Q) is E-complete in R. Proposition is proved. Definition 5.6.29. A schema P is open if it is connected to some other systems. Otherwise, P is closed. Multigraphs of the schemas allow one to characterize definite classes of schemas. Proposition 5.6.28. A schema R is closed if its multigraph G(R) is not generalized. Proof is left as an exercise.
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Corollary 5.6.4. A schema R is a transducer only if its multigraph G(R) has edges connected by their beginning and edges connected by their end. Proposition 5.6.29. If f : R → P is a (structural) homomorphism of schemas and R is a closed schema, then the image f (R) is closed. Proof is left as an exercise. Corollary 5.6.5. An epimorphic image of a closed schema is closed. Proposition 5.6.30. If f : R → P is a (structural) homomorphism of schemas and R is a connected schema, then the image f (R) is connected. Proof is left as an exercise. Corollary 5.6.6. An epimorphic image of a connected schema is connected. An important part of the mathematical schema theory is concerned with operations with schemas. There are operations with schemas aimed at the creation and development of computing and communication networks, some of these operations are constructed and studied in (Burgin, 2010a). Another class of schema operations is related to concepts because schemas provide efficient representation of concepts. That is why an important operation with conceptual schemas is conceptual blending. It is a process that operates below the level of consciousness and involves connecting two concepts to create new meaning (Fauconnier and Turner, 2002; Guhe et al., 2011). Researchers use this operation to explain abstract thought, creativity, and language. For instance, Fauconnier and Turner (2002) argue that all learning and all thinking consist of blends of concepts and metaphors based on various physical experiences. Blending is a cyclic operation, in which the results of previous blendings are then themselves blended together into an increasingly rich structure that makes up people’s mental functioning in modern society.
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Chapter 6
Knowledge Structure and Functioning: Megalevel or Global Theory of Knowledge Knowledge is not a series of self consistent theories that converges towards an ideal view; it is rather an ever increasing ocean of mutually incompatible (and perhaps even incommensurable) alternatives, each single theory, each fairy tale, each myth. Paul Feyerabend
In the context of global knowledge, that is, on the knowledge megalevel, there are two approaches to knowledge structuration: — System structuration organizes knowledge in the form of interacting knowledge systems. — Typological structuration groups knowledge with respect to different types. For instance, scientific and mathematical knowledge are structured as systems of interacting theories. At the same time, typological structuration organizes knowledge systems according to their types. For instance, as it is done in Chapter 2, three basic types of knowledge are discerned: descriptive, representational, and operational knowledge. Each type forms a subsystem or component of an advanced scientific or mathematical theory.
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A global knowledge system encompasses knowledge about a domain that consists of immense amount of objects, relations, processes, transformations, and interactions. Any wide-ranging and advanced scientific or mathematical theory is an example of global knowledge systems and consequently, models of scientific and mathematical theories are models of global knowledge systems of the definite type. Namely, scientific and mathematical theories, in general, are more organized and more advanced than other types of knowledge. As a rule, these theories provide more complete representation of global knowledge in comparison with other kinds of knowledge such as engineering knowledge or medical knowledge. One more distinction is made between comprehensive and partial knowledge systems. Definition 6.1. A comprehensive knowledge system has all types of knowledge and knowledge of each type is usually organized as the corresponding subsystem of the whole knowledge system (cf., Table 4.1). Advanced scientific theories, such as relativity theory, quantum mechanics or genetics, are comprehensive knowledge systems. Definition 6.2. A partial knowledge system does not have all types of knowledge. Formal theories, such as axiomatic set theory or Peano arithmetic, are examples of partial knowledge systems. Structural analysis of big knowledge systems in general and scientific theories, in particular, has been, as a rule, concerned only with the inner structure of knowledge systems. The reason for this was the limited understanding of the concept of structure that had existed for a long time. In the general theory of structures developed in (Burgin, 2012), this limitation was eliminated by demonstration that any system has five types of structures — internal structure, inner structure, intermediate structure, external structure, and outer structure. In addition, the new theory demonstrated that the traditional understanding of the concept of structure, as well as its mathematical formalizations are incomplete and the complete concept of structure and its mathematical formalization was created. As a result, in the
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knowledge realm, the theory of mathematical structures developed by Bourbaki (1960) became a special formalized subtheory of the general theory of structures (Burgin, 2012). 6.1. A typology of structures and scientific knowledge Every task involves constraint, Solve the thing without complaint; There are magic links and chains Forged to loose our rigid brains. Structures, structures, though they bind, Strangely liberate the mind. James Falen
While the traditional approach takes into account only one structure of a system, the general theory of structures postulates that any system has different structures, which belong to five basic types: inner, internal, intermediate, outer, and external structures. Definition 6.1.1. An internal structure Q of a system R contains only inner structural parts, components and elements, i.e., parts, components, and elements of R; relations between these parts, components and elements; relations between these parts, components, elements and relations from Q; and relations between relations from Q. A complex system can have different internal structures because it is possible to consider this system selecting different parts, components and elements. For instance, it is possible to treat a text as a system consisting of words and relations between them or as a system consisting of letters and relations between them. It will give us different internal structures of the same text. Moreover, if we select definite parts, components and elements, it is possible to have different internal structures taking dissimilar relations between them. For instance, taking words as elements of a text, we obtain one internal structure when we choose syntactic relations between these words and another internal structure when we choose semantic relations between these words. Note that by this definition, it is possible to treat the system R as a part of itself. In this case, relations between elements and the whole system R, as well as relations between relations in R and the
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whole system R are also included in the internal structure. However, it is also useful to exclude such relations from consideration. It brings us to the special case of internal structures — inner structure. Definition 6.1.2. An inner structure Q of a system R is an internal structure when the whole system R is not a part, component or element of itself. For instance, the inner structure of an abstract set does not include any relations between elements of this set, while the internal structure of an abstract set is based on the membership relation between elements of this set and the set itself, i.e., if X is an abstract set and x is an element from X, then we have the relation x ∈ X. As in the case of internal structures, a complex system can have different inner structures. Definition 6.1.3. An external structure T of a system R is an extension of the internal structure, in which other systems, their parts, components and elements are included, as well as relations between all these included parts, components and elements, relations between these parts, components, elements and relations from T and relations between relations from T . For instance, considering an organization A, we take members and divisions of this organization, other organizations, divisions and people with which members and divisions of this organization have relations as elements of the external structure of A, and take relations between these elements and relations as relations of the external structure of A. In the external structure of a scientific theory T , we include other theories related to T , scientists who have been developing this theory, applications of T , and relations between all these systems. As in the case of internal and inner structures, a complex system can have different external structures. Definition 6.1.4. An intermediate structure T of a system R is an external structure when the whole system R is not a part, component, or element of itself and other systems are also excluded from T , as well as relations that include these systems.
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For instance, the intermediate structure of a mathematical theory M includes axioms, statements, and theorems of other mathematical theories related to axioms, statements, and theorems of M , relations between all these objects and relations between these relations. If we take, for example, the theory of groups, then its axioms and many of its theorems are also axioms and theorems of the theory of abelian groups, as well as of the theory of free groups. As in the case of internal and inner structures, a complex system can have different intermediate structures. Definition 6.1.5. An outer structure of a system R is an inner structure of a system U in which R is only one of the inner elements of the internal structure Q of the system U . For instance, the internal structure of the organization where an individual works is an outer structure of this individual. The internal structure of person’s family also is an outer structure of this person. The outer structure of a group is the category of all groups, while the outer structure of a set is the category of all sets. As in the case of internal and inner structures, a complex system can have different outer structures. Besides, when models of knowledge systems are developed, it is reasonable to make a difference between knowledge systems and systems of other nature. It brings us to two types of intermediate structures, two types of external structures, and two types of outer structures of a knowledge system: — — — — — —
knowledge-bounded intermediate structure; beyond-knowledge intermediate structure; knowledge-bounded external structure; beyond-knowledge external structure; knowledge-bounded outer structure; beyond-knowledge outer structure.
Advanced scientific theories are model examples of comprehensive and partial global knowledge systems. However, scientific theories represent only a specific type of global knowledge systems. Gopnik and Meltzoff (1997) give what is probably the most comprehensive set
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of conditions that scientific theories have to satisfy. These conditions fall into three categories: structural, functional, and dynamic. It is interesting that these three categories exactly correspond to the three types of complexity measures of algorithms and computations (Burgin, 2005). Structurally, scientific theories are abstract, coherent, causally organized, and ontologically committed bodies of knowledge. They are abstract in that they posit entities and laws using a vocabulary that differs from the vocabulary used to state the evidence that supports them. They are coherent in that there are systematic relations between the entities posited by the theory and the experimental and observational data. Scientific theories are causal insofar as the structure that they posit in the world to explain observable regularities is ordinarily a causal one. Keil (1989) suggests that causal relations are central to scientific theories; especially, in those components that are homeostatic and hierarchically organized. Finally, scientific theories are ontologically committed as the entities that they hypothesize correspond to real kinds supporting counterfactuals about how things would be under various non-actual circumstances. Functionally, scientific theories must make predictions, interpret evidence in new ways, and impart explanations of phenomena in their domain. The predictions of scientific theories often go beyond simple generalizations of the evidence, and include ranges of phenomena that the theory was not initially developed to cover. Scientific theories interpret experimental and observational data by providing new descriptions that influence what is seen as pertinent or salient and what is not. Associative relations produced by interpretations supply the raw data for theoretical development, as well as counterexamples for discarding the theory. In addition, relevant scientific theories provide explanations of phenomena in their domain. Dynamically, scientific theories are not static representations of their domains but are transformed and developed by scientists. Dynamic processes that involve scientific theories include: — an initial period involving preliminary hypothesis formation and the accumulation of evidence via processes of experimentation and observation,
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— — — —
discovery of counterexamples, possible discounting of such examples as noise, generation of hypotheses to modify a theory, production of a new theory when an old one has accumulated too many counterexamples or repulsive and complicated auxiliary amendments, — applications of a theory. Applying scientific approach, many researchers studied inner structures of scientific theories building their models, which are often called reconstructions, and testing their validity by application to existing scientific theories. The most popular is the standard (positivist) model (reconstruction) of a scientific theory, which utilizes means of logic representing a scientific theory as a system of propositions (cf., for example, (Carnap, 1934/1937; Suppe, 1974; 1979; 1999)). Namely, according to Carnap, a scientific theory is an interpreted axiomatic formal system, which consists of: • a formal language, including logical and non-logical terms; • a set of logical axioms and corresponding rules of construction and inference; • a set of mathematical axioms and corresponding rules of construction and inference; • a set of non-logical axioms, which express the empirical content of the theory; • a set of semantic postulates defining the meaning of non-logical terms, which formalize the analytic truths of the theory; • a set of rules of correspondence, which give an empirical interpretation of the theory. Another popular approach to description of the scientific theory structure is the structuralist model (reconstruction) of a scientific theory (cf., for example, (Sneed, 1971; Balzer et al., 1987)), which utilizes means of set theory representing a scientific theory as a system of models of the theory domain. Some researchers treat scientific theories as devices for the formulating and resolving scientific problems. In this context, they model
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scientific theories by systems of statements and questions (problems) including (in some models) various forms of problem representation, rules and heuristics for resolving problems and utilizing erotetic logic for rigorous analysis of problems and problem-solving (cf., for example, (Lobovikov, 1984; Garrison, 1988)). In his model, Thagard (1988) represented a scientific theory as a highly organized package of rules, concepts, and problem solutions. Some researchers treat scientific theories as devices for the formulating and resolving scientific problems. In this context, they model scientific theories by systems of statements and questions (problems) including (in some models) various forms of problem representation, rules and heuristics for resolving problems and utilizing erotetic logic for rigorous analysis of problems and problem-solving (cf., for example, (Lobovikov, 1984; Garrison, 1988)). In his model, Thagard (1988) represented a scientific theory as a highly organized package of rules, concepts, and problem solutions. All these and some other approaches were unified in the structurenominative model or reconstruction (SNR) of a scientific theory (Burgin and Kuznetsov, 1989; 1989a; 1991; 1992; 1993; 1994; Burgin et al., 1989; Balzer et al., 1991). As a result, other models of theoretical knowledge that describe inner structure of big knowledge systems, such a scientific theories, became subsystems of the structure-nominative model of scientific knowledge (a scientific theory) and all structures used in those models are either named sets or systems of named sets. For instance, the structuralist model of a scientific theory (cf., for example, (Sneed, 1971; Balzer et al., 1987)) is represented as the model-representing subsystem of a scientific theory, while the standard (positivist model) of a scientific theory (cf., for example, (Suppe, 1974; 1979; 1999)) is represented as the logic-linguistic subsystem (LLS) of a scientific theory. The structure-nominative model has been applied to the analysis of laws and models of physical theories (Burgin and Kuznetsov, 1993), to formal and informal concept formation from a general point of view (Burgin and Kuznetsov, 1988; 1990), to the analysis
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of the structure and development of mathematical theories (Burgin and Kuznetsov, 1991a), to problems of intellectual activity and cognition (Burgin and Kuznetsov, 1988a; 1988b), to problems of pedagogy (Burgin et al., 1989), to esthetic features of scientific theories (Burgin and Kuznetsov, 1993a) and to knowledge representation in AI systems (Burgin and Kuznetsov, 1987; 1988a). The structure-nominative model has provided efficient means for studies of scientific reduction (Balzer et al., 1991) demonstrating that the structure-nominative general schema for reduction entails as special cases Schr¨oder-Heister–Schafer’s weak and strong representation and reduction concepts (Schr¨oder-Heister and Schafer, 1989), as well as the structuralist concept of reduction (Balzer et al., 1987). Kuokkanen (1993) suggested one more application of the structurenominative model, namely, to study Rantala’s concept of correspondence (Rantala, 1989). A new development of the structure-nominative model of scientific knowledge is elaborated in (Burgin, 2011) where the complete beyond-knowledge external structure of a scientific/mathematical theory is constructed. The complete beyond-knowledge external structure of a scientific/mathematical theory (comprehensive knowledge system) developed in the extended structure-nominative model or reconstruction (ESNR) is given in Figure 6.1. The model of global knowledge systems presented in this chapter is the further development of the SNR of comprehensive scientific knowledge system (Burgin and Kuznetsov, 1989; 1989a; 1991; 1992; 1993; 1994; Burgin et al., 1989; Balzer et al., 1991), as well as of the ESNR of comprehensive scientific knowledge system elaborated in (Burgin, 2011). Scientific and mathematical theories represent a transition form, from the macrolevel to the megalevel of knowledge. When a mathematical or scientific theory appears, it is, as a rule, small and is situated on the macrolevel of knowledge. This was the situation with the calculus at the end of 17th century, with non-Euclidean geometries in the middle of the 19th century or with non-Diophantine arithmetics now. At the same time, mature theories, such as geometry, algebra, genetics, or quantum physics, are on the megalevel.
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Affirmative Thinking
Problem Thinking
Thinking
Logical Part
Heuristic Part
Logic-Linguistic Subsystem
Problem-Heuristic Subsystem
Linguistic Part
Problem Part
Subsystem of Ties Axiological Part
Nomological Part
Pragmatic-Procedural Subsystem Procedural Part
Model-representing Subsystem Model Part
Object Domain
Action
World
Figure 6.1. The complete beyond-knowledge external structure of a scientific/ mathematical theory (comprehensive knowledge system), which includes the inner structure (the middle layer of the diagram), beyond-knowledge intermediate structure, and is developed in the ESNR of comprehensive knowledge systems
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6.2. Nuclear and comprehensive knowledge systems Integrity without knowledge is weak and useless, and knowledge without integrity is dangerous and dreadful. Samuel Johnson
A nuclear knowledge system performs a specific function in knowledge functioning. There are several types of nuclear knowledge systems: descriptive knowledge systems, representational knowledge systems, operational knowledge systems, assertoric knowledge systems, erotetic knowledge systems, and heuristic knowledge systems. A comprehensive knowledge system performs all these functions combining all nuclear knowledge systems in a comprehensive totality of knowledge. There are different models of knowledge on the megalevel. The most popular models, such as positivists logical representation of knowledge where scientific theory is represented by logical propositions and/or predicates (cf., (Carnap, 1934/1937; Popper, 1965; 1979; Suppe, 1999)) or model-oriented representation of knowledge, which assumes that a theory is a collection of models (cf., (Suppes, 1967; Suppe, 1979; Sneed, 1979; van Frassen, 2000)), represent only nuclear knowledge systems working well on the macrolevel but being essentially incomplete on the megalevel. The structure-nominative model created by Burgin and Kuznetsov (Burgin and Kuznetsov, 1991; 1992; 1993; 1994; Burgin et al., 1989; Balzer et al., 1991) was the first conceptual and mathematical model of comprehensive knowledge systems. This model encompasses all other models of theoretical and practical knowledge that existed before giving an advanced structural representation of scientific knowledge in general and scientific theories in particular. Providing means for exploration of various traits and regularities of scientific knowledge structure and functioning, the structure-nominative model forms the base of the structure-nominative direction in the methodology of science, and it is demonstrated that named sets form the structural base of the SNR reconstruction of scientific knowledge
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(Burgin and Kuznetsov, 1989; 1989a; 1991; 1992; 1993; 1994; Balzer et al., 1991). Here we further develop the SNR model building the modal stratified bond model (MSB model) of comprehensive knowledge systems in general, as well as advanced scientific and mathematical theories, in particular, describing their structure and functioning. To build the MSB model, we configure the global knowledge in tree directions — systemic, modal and hierarchical. Namely, we take into account three modalities of knowledge considered in Chapter 2 to construct the modal direction: ∗ Assertoric knowledge consists of epistemic structures with implicit or explicit affirmation of being knowledge. ∗ Hypothetic or heuristic knowledge consists of epistemic structures with implicit or explicit supposition that they may be knowledge. ∗ Erotetic knowledge consists of epistemic structures that express lack of knowledge. Logical propositions or statements, such as “The Sun is a star”, are examples of assertoric units of knowledge. Beliefs with low extent of certainty, i.e., when they are not sufficiently grounded, are examples of hypothetic knowledge. Questions and problems are examples of erotetic knowledge. Knowledge with different modalities forms strata in knowledge systems determining the horizontal structure of comprehensive knowledge systems. In the hierarchical direction, we separate three levels of global knowledge systems: 1. The componential level of global knowledge. 2. The attributed level of global knowledge. 3. The productive level of global knowledge. The componential level consists of elements, parts and blocks from which systems from the attributive level are built. In some sense, the componential level is the substructural level of a global knowledge system.
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The attributed level reflects the static structure of global knowledge as a system constructed from elements, parts and blocks from the componential level. The productive level reflects the cognitive (dynamic) structure of global knowledge, containing means for knowledge acquisition, production, and transmission. Note that each of these three levels has its strata and sublevels. Levels in global knowledge systems determine the vertical structure of this system. As it is explained in Chapter 2, it is natural to distinguish three categories of knowledge, which form the systemic direction of knowledge structuration: ∗ Descriptive knowledge. ∗ Representational knowledge. ∗ Operational knowledge. Each of these categories is subdivided into three groups. Descriptive knowledge has three types: ◦ Informal knowledge based on natural languages. ◦ Semiformal knowledge based on logic. ◦ Formal knowledge. Let us consider definitions of a limit as examples of three types of descriptive knowledge. An informal definition: A number a is a limit of a sequence of numbers ai (i = 1, 2, 3, . . .) if the distance between a and all but a finite number of elements from the sequence is smaller than any arbitrarily small positive number. A semiformal definition: A number a is called a limit of a sequence l if for any ε ∈ R++ the inequality |a − ai | < ε is valid for almost all ai , i.e., there is such n that for any i > n, we have |a − ai | < ε. A formal definition: a = limi→∞ ai if for any ∀ε ∈ R++ ∃n ∈ N ∀i > n(|a − ai | < ε). Note that the formal definition is much shorter than both informal and semiformal definitions.
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Representational knowledge has three modes: ◦ Knowledge representing statics. ◦ Knowledge representing dynamics, which expresses two distinctive modes: • Knowledge representing functions; • Knowledge representing processes. The statement “The Sun is a star” is an example of static representational knowledge. The statement “The Sun gives light” is an example of static representational knowledge representing functions. The statement “The Earth rotates about the Sun” is an example of representational knowledge representing processes. Operational knowledge has three types: ◦ Procedural knowledge. ◦ Instrumental knowledge. ◦ Axiological knowledge. An algorithm, e.g., rules for adding decimal numbers, is an example of procedural knowledge. An abstract automaton, e.g., a Turing machine (Burgin, 2005), is an example of instrumental knowledge. The statement “A theory has to be able to explain the results of experiments” is an example of axiological knowledge. Thus, in the systemic context, we discern five knowledge subsystems: the logic-linguistic subsystem (LSS), the model-representation subsystem (MSS), the procedural subsystem (PSS), the axiological subsystem (ASS), and the instrumental subsystem (ISS) of a complete knowledge system. Together, the PSS, ISS, and ASS form the operational subsystem (OS) of a complete knowledge system but taking into account the intermediate structure (cf., (Burgin, 2012)) of knowledge, we consider these three subsystems separately. All these subsystems are explicated and studied based on functions and functioning of real comprehensive knowledge systems. As a theory contains knowledge, it possesses a variety of linguistic tools,
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the main of which are different languages. From this perspective, a theory is regarded as a sophisticated system of statements about its domain, e.g., quantum mechanics describes the microlevel of the physical world. In addition, theoretical knowledge has developed logical structures, while logical apparatus, such as inference, is used for scientific and mathematical cognition, as well as for grounding and testing acquired knowledge. To perform these functions, a mature theory employs specific means from linguistic and logic, which are amalgamated in the LSS of the theory. Any comprehensive knowledge system models some domain and results in the necessity of the MSS. Operational functions of a comprehensive knowledge system are performed by the PSS, the ASS, and the ISS. Each subsystem has the same horizontal and vertical structures as the whole comprehensive knowledge system. All subsystems of global (comprehensive) knowledge systems have several levels, which are presented in Table 6.1 in the unified form and considered in more detail in Sections 6.2–6.6. In addition to types of knowledge, we have three knowledge modalities: — Assertoric knowledge; — Erotetic knowledge; — Hypothetic or heuristic knowledge. An assertoric knowledge item asserts that something is or is not the case. For instance, the statement “The Sun is a star” is an assertoric proposition. Assertoric propositions and assertoric predicates are studied by assertoric logics such as the Aristotelian syllogistics. An erotetic (from Greek, er¯ot¯esis — questioning) knowledge item expresses an inquiry or a question. For instance, the statement “Is the Sun a star?” is an erotetic proposition (a question). Erotetic knowledge is studied by erotetic logics (Belnap and Steel, 1976). Hypothetic (heuristic) knowledge is knowledge that is not sufficiently grounded, e.g., a hypothesis or a conjecture. For instance, the statement “The age of the Sun is nine billion years” is a hypothetic proposition (a hypothesis). Hypothetic logic studies hypothetic propositions and hypothetic predicates (Baldoni et al., 1998).
Representational knowledge
Operational knowledge
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Subsystem (Part)
Componential Nominalistic part: level concepts, lexicon, vocabularies, alphabets Linguistic part: grammars, linguistic relations, languages
Productive level
Logical part: logical calculi, logics, deduction/ inference rules, logical varieties, prevarieties and quasivarieties
Aspect part: properties
PSS Operating part: primitive operations, data
ASS Scaling part: relation, scales and their systems
ISS
Fragment part: parts and components of automata, devices, machines Modeling part: Operator part: Evaluation part: Performer part: parametric, instructions, estimates, automata, devices, attributive and operators judgments, norms, machines, relational models goals, measures, instruments criteria, and values Nomological part: Algorithmic part: Combination part: System part: systems and algorithms, algebras and systems and algebras of models procedures, calculi of networks of operational properties, automata, devices, schemas, scenarios estimates, machines judgments, norms, goals, measures, criteria, and values
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MSS
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Table 6.1. Parts and levels of global knowledge systems.
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Modalities of knowledge determine three knowledge strata: — The assertoric stratum; — The erotetic stratum; — The hypothetic, or heuristic, stratum. Each of these strata comprises all types of knowledge — descriptive, representational, and operational — of the global knowledge system, e.g., of a scientific theory. Namely, the erotetic stratum has logical, model, procedural, axiological, and instrumental components. Problems and questions related to the knowledge in the LSS are united in the logical component of the erotetic stratum. Problems and questions related to the theory models are combined in the model component of the erotetic stratum. Problems and questions related to algorithms, procedures and methods are amalgamated in the procedural component of the erotetic stratum. Problems and questions related to properties, estimates, and parameters are collected in the axiological component of the erotetic stratum. Problems and questions related to instrumental issues are integrated in the instrumental component of the erotetic stratum. In a similar way, the heuristic stratum has logical, model, procedural, axiological, and instrumental components. Conjectures, heuristics, and hypotheses in the linguistic form are united in the logical component of the heuristic stratum. Heuristic and hypothetic models, i.e., models that are not sufficiently validated, are integrated in the model component of the heuristic stratum. Heuristic and hypothetic properties, estimates, and parameters are unified in the axiological component of the heuristic stratum. Heuristic and hypothetic algorithms, procedures and methods are collected in the procedural component of the heuristic stratum. Descriptions of heuristic instruments and devices, as well as their properties are integrated in the instrumental component of the erotetic stratum. The five subsystems of a theory (knowledge system) and theory strata are not disconnected — they function together, have common elements, and there are various ties between all subsystems. For instance, statements from LSS are interpreted in models from MSS, while problems (questions) from the erotetic stratum
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can have solutions (answers) in the assertoric stratum. All these ties are collected in the subsystem of bonds or ties of the theory. Inference rules and algorithms in general belong both to the PSS and to the LSS, while the PHS shares construction/generation rules from erotetic and heuristic languages and inference rules from erotetic and heuristic logics with the PPS. We remind that erotetic languages describe and erotetic logics study the logic and pragmatics of questions and answers, heuristic languages describe and heuristic logics study the logic and pragmatics of hypotheses and conjectures, while assertoric languages represent and assertoric logics study the logic and pragmatics of assertions (statements or propositions). That is why in any developed knowledge system, there is the subsystem of bonds or ties (BSS), which unites all other subsystems and their strata, comprising connections and ties between their elements and components. Examples of such connections are: — the interpretations of languages and calculi (from LSS) in models (from MSS); — the procedural interpretations (in PSS) of languages and calculi (from LSS); — the correspondence between algorithms and procedures of inference (from PSS) and logics (from LSS) where they are used; — the relations between conjectures (from the hypothetic stratum) and their properties (from the assertoric stratum); — the relations between problems (the erotetic stratum) and algorithms (from the assertoric stratum) that solve these problems; — the correspondence between generative grammars (from PSS) and languages (from LSS) generated by these grammars; — the relations between hypotheses (from the hypothetic stratum) and corresponding problems (from the erotetic stratum); — the relations between heuristics (from the hypothetic stratum) and their properties (from the assertoric stratum). All subsystems, their levels and strata form the inner structure of a comprehensive knowledge system. This structure with
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the beyond-knowledge intermediate structure and beyond-knowledge outer structure of a comprehensive knowledge system in general and of a scientific or mathematical theory, in particular, is presented in Figure 6.2.
Mental World Hypothetic Thinking Affirmative Thinking
Attitude evaluation
Thinking
Inquisitive Thinking
Structural World
Hypothetic stratum
Axiological Subsystem
Logic-Linguistic Subsystem
Assertoric stratum
Subsystem Subsystem of Bonds of Bonds
Procedural Subsystem
Instrumental Subsystem
Model-representation Subsystem
Physical World
Erotetic stratum
Object Domain
Action
Figure 6.2. The complete beyond-knowledge external structure of a comprehensive knowledge system, e.g., a scientific/mathematical theory, which includes the inner structure (the middle layer of the diagram), and beyond-knowledge intermediate structure
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6.3. Logic-linguistic knowledge system and descriptive knowledge Five paths to a single destination. What a waste. Better a labyrinth that leads everywhere and nowhere. Umberto Eco
A great bulk of knowledge and many think all knowledge is represented in a linguistic form, that is, using language, or actually, a variety of languages. There are many natural languages, which are used for information exchange and knowledge preservation. There is a diversity of artificial languages created for specific purposes. Scientific languages, such as languages of physics or biology, have been created for scientific cognition and accumulating its results. Some artificial languages, such as Esperanto, were created providing better tools for communication in society. Programming languages have been continuously created for controlling computers. Mathematical languages, such as the language of arithmetic with its numbers or the language of geometry with its figures, have been created for mathematical cognition and accumulating its results. Creation of logical languages have been aimed at formalization of reasoning. In any case, any global system of knowledge has a component or a subsystem, in which knowledge is represented by languages and has the logical form, i.e., according to the rules of logic used to organized knowledge in a reliable and efficient system. Many philosophers and logicians even implied that any scientific or mathematical theory as a kind of global knowledge is a system of statements (propositions). This is the standard model of a scientific theory introduced by positivists (cf., (Carnap, 1934/1937; Popper, 1965; 1979; Suppe, 1999)). The comprehensive models of global knowledge in general and scientific theories, in particular, such as the structure-nominative model (Burgin and Kuznetsov, 1991; 1992; 1993; 1994; Burgin et al., 1989; Balzer et al., 1991), the extended structure-nominative model (Burgin, 2011) and the MSB model presented in this book, do not reject importance of logic and linguistic for knowledge systems but
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LSS Logical part Calculi
Linguistic part
Varieties
Productive level
Languages Grammars Conceptual part
Attributed level
Concepts Lexicon
Componential level
Figure 6.3. The vertical structure of the LSS
reflect their presence in separation of the specific LSS in comprehensive knowledge systems. The LSS has three basic levels and three corresponding parts (cf., Figure 6.3): • The componential level, which gives elements and components, such as concepts, terms, names, and lexicons, for building next levels; • The attributed level, where languages and their constituents, such as grammars and interpreters, are situated; • The productive level, where tools, such as logical calculi, varieties, quasi-varieties and prevarieties, for knowledge production and organization belong.
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Each level exists as the corresponding part of a global knowledge system (scientific theory): • The conceptual part, • The linguistic part, • The logical part. Each basic level is, in turn, divided into sublevels, which structurally form a named set sequence in the sense of (Burgin, 2011). Let us consider all these levels. The first level of the LSS consists of the symbols used in the languages of this subsystem. For instance, mathematical theories use such symbols as: decimal digits, Latin letters, Greekletters, Hebrew letters, additional mathematical symbols, e.g., ∅ or , and letters of the natural language used by mathematicians, e.g., French, German, or Russian. On the second level, symbols are organized into alphabets. For instance, decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 form the alphabet of the decimal positional numerical system. If a knowledge system uses several languages, e.g., a physical theory, as a rule, uses natural languages, mathematical languages and the language of the area of this theory, then it is possible to take the union of all its alphabets as the alphabet of this theory. The third level of the LSS consists of the rules for building words, expressions and well-formed formulas from the symbols. On the fourth level, some words and expressions are chosen as names and terms. For instance, 123 is a name of natural number in the decimal positional numerical system. In the binary positional numerical system, the same number has the name 1111011. Concepts used in the considered knowledge system or connected to this system form the fifth level of the LSS. Concept structure and properties are modeled and studied by means of various named sets (Burgin and Kuznetsov, 1988; 1990). Symbolic forms of concept expressions (terms) are used for the construction of the lexicon comprising alphabets and vocabularies of languages from the whole knowledge system, which are elements of the sixth level of LLS.
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The seventh level consists of the construction rules for expressions, e.g., phrases, sentences, and texts, from the languages used in the whole knowledge system in general and in the LSS, in particular. For instance, in the propositional calculus, logicians use the construction rule: If a and b are propositions, then a&b is also a proposition. The eighth level of the LSS contains various languages treated as systems of expressions built from symbols of alphabets in accordance with construction rules. All these languages as systems of expression are elements of the LSS. However, their semantics are defined by means of other subsystems, while in some cases the languages also belong to other subsystems. For instance, semantics of model languages is defined in the MSS and they also belong to this subsystem. In a similar way, semantics of algorithmic languages is defined in the procedural subsystem (PSS), to which these languages also belong. The axiological subsystem (ASS) shares languages and logics of norms and values with the logic-linguistic subsystem. The ninth level of the LSS contains rules for transforming expressions from the theory’s languages. In accordance with the classification of languages, there are many kinds of transformation rules, for example, deduction and substitution are the most frequently utilized and the best known of them. The tenth level, which is sometimes divided into three sublevels, contains formal calculi. Any calculus is a named set C = (A, R, T ) where A is an axiom system, T is the set of theorems of the given calculus, and R consists of the rules used in the process of deducing these theorems, i.e., rules from the ninth level (cf., Section 3.8). The structure of a logical calculus brings on three sublevels of the tenth level: — Systems of axioms, i.e., of expressions accepted without proofs; — Systems of theorems, i.e., of expressions that are proved; — Logical calculi, which combine axioms, deduction rules and theorems into unified systems. The eleventh level of the LSS contains towers of calculi introduced for representation of dynamic aspects of formal theories (Maslov,
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1983) and other logical varieties, prevarieties, and quasi-varieties (Burgin, 1991d; 1997f; 2004c; Burgin and de Vey Mestdagh, 2011; 2015). Additionally, a comprehensive knowledge system, e.g., an advanced scientific theory, usually is a source and repository of problems, conjectures, heuristics, and hypotheses. For instance, theories are, as a rule, estimated by problems they allow scientists/ mathematicians to solve. Thus, the theory of non-Euclidean geometries has been considered one of the most important mathematical theories because it solved the long-standing problem about the fifth postulate of Euclid. In addition to parts and levels, the LSS has three basic strata (cf., Figure 6.4): — The assertoric stratum, which contains knowledge about the knowledge system, e.g., theory, domain, e.g., propositions describing the knowledge domain properties.
Hypothetic stratum Assertoric stratum Subsystem of BondsLSS
Erotetic stratum
Figure 6.4. The horizontal structure of the LSS
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— The erotetic stratum which contains knowledge on problems, tasks, and questions about the knowledge system, e.g., theory, domain. — The hypothetic stratum, which contains hypothetic, i.e., not sufficiently validated, knowledge about the knowledge system, e.g., theory, domain, e.g., conjectures about the knowledge domain properties. The hypothetic and erotetic strata of the logic-linguistic subsystem also has several levels inheriting them from the described above assertoric stratum of the knowledge system. However, while the assertoric stratum employs assertoric languages, the erotetic stratum of the LSS uses translational, heuristic and other languages for representing problems, questions, tasks, hypotheses, and heuristic methods of construction and search. It also contains calculi, logics, and algebras of problems and questions, hypotheses, etc. The hypothetic stratum of the LSS uses translational, heuristic and other languages for representing conjectures, hypotheses, and heuristic methods of construction and search. It also contains calculi, logics, and algebras of conjectures and hypotheses, etc. It is necessary to stress that the erotetic and hypothetic strata of the majority of mathematical and scientific theories are less developed and formalized than their assertoric hypothetic stratum. 6.4. Model-representation knowledge system and representational knowledge The beginning of knowledge is the discovery of something we do not understand. Frank Herbert
Any knowledge system, e.g., a scientific or mathematical theory, consists of knowledge about some domain, called the knowledge domain, e.g., the theory domain. To perform this function, a comprehensive knowledge system, e.g., an advanced theory, uses an assortment of models of the entities from the theory domain and in some cases,
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MSS Nomological part Systems of Modeling models part
Aspect part
Models
Productive level
Attributed level
Properties Names
Componential level
Figure 6.5. The vertical structure of the MSS
of the domain as a whole. These models, e.g., mathematical models, form the MSS of the knowledge system. The MSS also has several levels, which form a named set sequence (cf., Figure 6.5). Its first level consists of various names for objects from the knowledge domain, e.g., the object field of a theory. Names play a very important role for any kind of knowledge in general and for representational knowledge, in particular. For instance, to define a process calculus, one starts with a set of names of channels for providing means of communication (Lanese et al., 2011). Note that the names of the objects are not only separate words but also formulas, expressions, or texts. For instance, an equation for electron is a name of an electron. The second level includes systems of names of objects with relations between the names. These systems are represented by semantic networks or dependency tables. There are two types of relations
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between the names of objects: linguistic relations, in which names are connected as linguistic entities, and reflection relations, which represent relations between objects. The third level contains names of properties and relations between the objects from the knowledge domain. Note that the names of the properties and relations are not only separate words but also formulas, expressions or texts. The fourth level of the MSS consists of constructions describing these properties and relations. The most important of such constructions is an abstract property P represented by a named set P = (U, p, L), where U is the universe of entities considered, p is a partial mapping, and a partially ordered set L is the scale of the property P (cf., Section 5.3). The fifth level includes systems of names of properties and relations with relations between the names. These systems are represented by semantic networks or dependency tables. There are two types of relations between the names of properties and relations: linguistic relations, in which names are connected as linguistic entities, and reflection relations, which represent relations between properties and relations. The sixth level of the MSS consists of structural hierarchies built of different sets, multisets and named sets that are used for building models and properties in the considered system of knowledge, e.g., in a scientific or mathematical theory. For instance, Bourbaki’s set scale (Bourbaki, 1960) and of the concept of universes used in non-standard analysis (Cutland, 1988), which were considered before as entities comprising the sixth level, are special cases of structural hierarchies. The seventh level of the MSS contains basic abstract entities defined on the supports of the theory models. The choice of basic properties depends on axiological estimates and judgments. The eighth level consists of models, formal and informal, mathematical, and conceptual, which are used in the theory and called theory models. In the MSB model, they have the general form of the named set M = (R(D), f, L). Here D is the set that consists of the names of the objects studied, the names of their properties
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and relations between them, the names of names, etc.; the names of abstract properties and relations corresponding to properties and relations of objects, to properties of their properties, etc.; and names of ideal entities like truth values; while R(D) is a subset of S(D) and S(D) is the set scale in the sense of (Bourbaki, 1960) with basis D. This set scale includes D and all its elements, functions from D into D, functions defined on these functions; the set of all subsets of D; and so on. In addition, L is the scale of the properties of the elements from R(D) and f is the (partial) function that assigns values of their properties to the elements from R(D). The ninth level contains relations and ties between theory models, as well as their properties and parameters. For instance, an important relation between theory models is “to be a submodel of”. Examples of properties are “standard” and “non-standard”, namely, there are standard and non-standard models. The tenth level of MSS is comprised of relations and ties between relations and ties between theory models, as well as their properties and parameters. Using the structural hierarchy described in Section 6.2, it is possible to come to higher levels of the MSS. The eleventh level contains algebras and calculi of models, as well as their properties and parameters. Examples of such algebras and calculi are: • Process algebras (Hennessy, 1988; Burgin and Smith, 2010). • Process calculi (Hoare, 1985; Milner, 1989; 1999; Moller and Tofts, 1990). • Algebras of abstract automata (Burgin, 2010d). These algebras and calculi contain models of physical (usually, computational) processes. Moreover, now there is a tendency to treat physical, biological, psychological and economical processes as computational processes. In addition to parts and levels, the MSS has three basic strata (cf., Figure 6.6): — The assertoric stratum, which contains knowledge about the knowledge system, e.g., theory, domain, e.g., propositions describing the knowledge domain properties.
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Hypothetic stratum Assertoric stratum Subsystem of BondsMSS
Erotetic stratum
Figure 6.6. The horizontal structure of the MSS
— The erotetic stratum which contains knowledge on problems, tasks, and questions about the knowledge system, e.g., theory, domain. — The hypothetic stratum, which contains hypothetic, i.e., not sufficiently validated, knowledge about the knowledge system, e.g., theory, domain, e.g., conjectures about the knowledge domain properties. The hypothetic and erotetic strata of the MSS also has several levels inheriting them from the described above assertoric stratum of the knowledge system. However, while the assertoric stratum employs validated models of knowledge objects and their properties and relations, the hypothetic stratum of the MSS uses hypothetic and heuristic models of knowledge objects and their properties and relations. It also contains calculi and algebras of such models. The erotetic stratum of the MSS contains problems and questions concerning models of knowledge objects, as well as their properties and relations.
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6.5. Procedural, axiological and instrumental knowledge systems, and operational knowledge If knowledge can create problems, it is not through ignorance that we can solve them. Isaac Asimov
Any comprehensive knowledge system contains operational knowledge, which is organized in three subsystems: • the PSS, which encompasses diverse operations, procedures, algorithms, instructions, rules for operation, tasks, action and interaction schemas, scenarios, processes and methods. • the ASS, which comprises property representations, estimates, norms, judgments, standards, indices, indicators, criteria, benchmarks, measures, qualities, characteristics, attributes, quantities, judgments, norms, goals, criteria, and values. • the ISS, which contains descriptions of various instruments, abstract automata and machines, mechanisms and devices. For instance, any theory is a tool for scientific inquiry and knowledge integration. Thus, a theory has different operations, procedures, algorithms, rules, action schemas, and methods for performing these functions, as well as for estimation obtained results and representation of properties of studied objects. All these tools are comprised by the PSS of the theory. The PSS of the theory has three components: inner, internal, and external. The inner component includes operations, procedures, algorithms, scenarios, rules, and action schemas that are used inside the PSS. Algorithms that build other algorithms or enumerate other algorithms are examples of elements from the interior component of the PSS. The internal component includes operations, procedures, algorithms, scenarios, rules, and action schemas that are used inside the comprehensive knowledge system, e.g., a theory, but outside its PSS. Inference rules are examples of elements from the internal component of the PSS. The external component includes operations, procedures, algorithms, scenarios, rules, and action schemas that are used outside the comprehensive knowledge system, e.g., a theory. Measurement and approximation
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algorithms and procedures are examples of elements from the external component of the PSS of physical theories. Similar to other subsystems of comprehensive knowledge systems, the PSS is also organized hierarchically. The first (basic) level of the PSS consists of elementary actions and operations. Their structure and properties are modeled and studied by means of various named sets or fundamental triads (cf., (Burgin, 2011)). The second level of the PSS contains relations between and properties of elementary actions and operations. For instance, arithmetical operations of addition and multiplication are commutative and associative. An example of relations between arithmetical operations is reduction, for instance, multiplication is reduced to addition. Instructions and rules for performing elementary actions and operations, such as transition rules in abstract automata, instructions of Turing machines or instructions in programming languages, are elements of the third level of the PSS. The fourth level consists of construction rules for integrating elementary actions and operations into algorithms, programs, procedures, scenarios, processes and methods as well as construction rules for systems that perform these actions and operations. Examples of construction rules are sequential composition and parallel composition of automata and algorithms, as well as many operations described in (Burgin, 2010d). The fifth level of the PSS contains various procedures, algorithms, tasks, programs, schemas, scenarios, processes, and methods built from elementary actions and operations by construction rules and instructions. As it is demonstrated in (Burgin, 2011), algorithms and procedures are represented by named sets and systems of named sets, such as named set chains. The sixth level of the PSS contains relations between and properties of procedures, algorithms, tasks, schemas, programs, scenarios, processes, and methods from the fourth level. Examples of such properties are time complexity, space complexity, computational complexity, communication complexity, power and reliability
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of algorithms, procedures, and programs. Examples of such relations are linguistic equivalence, functional equivalence, and reducibility of algorithms, procedures, and programs. The seventh level of the PSS contains systems of procedures, algorithms, tasks, schemas, scenarios, processes, and methods from the fourth level. Examples of such systems are software systems, classes of algorithms of the same type, such as the class of all finite automata or the class of all Turing machines (cf., Appendix B), and classes of algorithms that solve the same problem, algorithms of data compression. The eighth level contains composition operations for algorithms, automata, procedures, and other models of processes (cf., for example, (Burgin, 2005b; 2010d)). Composition operations allow building new algorithms, processes and procedures from other algorithms, processes and procedures. As algorithms, processes and procedures are represented by named sets and systems of named sets, compositions of algorithms and procedures are represented by operations with named sets and systems of named sets, such as operations with named set chains. The ninth level of the PSS contains algebras and calculi of algorithms, processes and procedures (cf., for example, (Baeten, 1990; Baeten and Bergstra, 1991; 1996; Bergstra and Klop, 1984; Burgin, 1997j; Burgin and Smith, 2010)), as well as algebras of operations with algorithms, automata, procedures, and processes. Note that such algebras and calculi also belong to the MSS. The tenth level consists of construction rules and metarules (rules for operation) for combining conventional algorithms, automata, and procedures into algorithms, automata, and procedures of the second level (Burgin and Gupta, 2012; Burgin and Debnath, 2010). The eleventh level of the PSS consists of algorithms, automata and procedures of the second level (Burgin and Gupta, 2012; Burgin and Debnath, 2010). Building algorithms of higher and higher levels it is possible to go to higher levels of the PSS. Levels and parts of the PSS form its vertical structure (cf., Figure 6.7).
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PSS Algorithmic part Algorithms, procedures, Operator scenarios part
Operating part
Operators
Productive level
Attributed level
Operations Data
Componential level
Figure 6.7. The vertical structure of the PSS
In addition to levels, the PSS has three basic strata (cf., Figure 6.8): — The assertoric stratum contains operational knowledge in the form of operations, procedures, algorithms, instructions, rules for operation, tasks, action and interaction schemas, scenarios, processes and methods. — The erotetic stratum contains knowledge on problems, tasks and questions about operations, procedures, algorithms, instructions, rules for operation, tasks, action and interaction schemas, scenarios, processes and methods. — The hypothetic, or heuristic, stratum contains heuristic, i.e., not sufficiently validated, and empirical, i.e., not theoretically based, operations, procedures, algorithms, instructions, rules for operation, tasks, action and interaction schemas, scenarios, processes, and methods.
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Hypothetic stratum Assertoric
stratum
BondsPSS
Erotetic stratum
Figure 6.8. The horizontal structure of the PSS
The PSS is closely connected to other two subsystems containing operational knowledge. The closest to it is the ISS, which contains descriptions of various instruments, abstract automata and machines, mechanisms, and devices for performing operations, procedures, algorithms, rules for operation and tasks from the PSS, e.g., measuring or experimental devices. Similar to other subsystems of comprehensive knowledge systems, the ISS is also organized hierarchically. The first (basic) level of the ISS consists of descriptions of elements, components and parts of machines, devices, abstract automata, and instruments that are related to the comprehensive knowledge system in question. For instance, alphabets of abstract automata or descriptions of the parts of experimental devices belong to the first level. Taking such an instrument as a single-beam absorption spectrometer, we see that it has such parts as the amplifier and the detector (cf., for example, (Rothbart, 1997)). The second level consists of properties of and relations between the elements, components, and parts described on the first level. For instance, rules of automata can be deterministic or non-deterministic.
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An example of a relation is to “be after”, e.g., the amplifier is after the detector in a single-beam absorption spectrometer. The third level consists of construction rules for constructing machines, devices, and instruments from their elements, components and parts described on the first level. The fourth third level of the ISS contains descriptions of various machines, devices, abstract automata and instruments built by construction rules from their elements, components and parts. The fifth level consists of properties of and relations between the machines, devices, abstract automata and instruments constructed on the fourth level. The sixth level of the ISS contains systems of abstract automata, such as finite automata; abstract machines, such as Turing machines, random access machines, cellular automata, and inductive Turing machines; descriptions of devices and instruments, such as computers or networks. The seventh level contains composition operations for abstract automata, abstract machines, devices and instruments (cf., for example, (Burgin, 2005b; 2010d) for operations with abstract automata and abstract machines). Composition operations allow building new abstract automata, machines, devices and instruments from other abstract automata, machines, devices, and instruments. The eighth level of the ISS contains algebras and calculi of abstract automata and abstract machines (cf., for example, (Burgin, 2010d)). Note that such algebras and calculi also belong to the MSS. Levels and parts of the ISS form its vertical structure (cf., Figure 6.9). In addition to levels, the instrumental subsystem has three basic strata (cf., Figure 6.10): — The assertoric stratum contains operational knowledge in the form of abstract automata and descriptions of instruments, devices and machines; — The erotetic stratum contains knowledge on problems, tasks and questions about abstract automata and descriptions of instruments, devices and machines;
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ISS Fragment part Parts and components of automata Performer part
Productive level
automata machines devices System part
Attributed level
Systems and networks of automata
Componential level
Figure 6.9. The vertical structure of the ISS
— The hypothetic, or heuristic, stratum contains descriptions of heuristic, i.e., not sufficiently validated, and empirical instruments, devices and machines. Finally, the ASS of a comprehensive knowledge system encompasses property representations, estimates, judgments, norms, goals, measures, indices, indicators, attributes, criteria, and values used in this system, e.g., in a scientific theory. Some of these objects are applied inside the knowledge system they belong to, while others are used for the knowledge domain. For instance, the truth value (from the ASS) is applied to the assertions (from the LSS of a scientific/mathematical theory. There are also other judgments and estimates, such as the adequacy of a model, the time complexity or space complexity of an algorithm, and so forth, which are used in the knowledge system itself. Other objects from the axiological
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Hypothetic stratum Assertoric stratum
BondsISS
Erotetic stratum
Figure 6.10. The horizontal structure of the ISS
subsystem, such as measures, criteria and goals, are used in the knowledge domain. For instance, in the theory of algorithms, such measures as time complexity, space complexity, and Kolmogorov complexity, are used for estimation of programs, software systems and concrete algorithms, such as approximation algorithms or search programs. The ASS is also organized hierarchically containing the following levels. Its first (basic) level contains names of properties, indices, indicators, attributes, estimates, judgments, norms, goals, measures, criteria, and values. The second level of the ASS contains scales of properties, attributes, measures and norms. The third level consists of abstract properties and quantities, which are a specific kind of named sets (cf., Section 5.3). The fourth level of the ASS consists of natural properties, measures, and characteristics.
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The fifth level contains various criteria, indices, indicators, and attributes. Their structure and properties are modeled and studied by means of an assortment of named sets. The sixth level of the ASS contains a diversity of estimates, norms, judgments, norms, goals, standards, values, and benchmarks. The seventh level contains operations with abstract properties, quantities, real properties, measures, characteristics, criteria, estimates, norms, standards, values, and benchmarks. The eighth level of the ASS contains algebras of abstract properties and quantities, which are systems of named sets. Their structure and properties are also modeled and studied by means of various named sets (cf., Section 3.6). The ninth level of the ASS contains algebras of real properties, measures, and characteristics, which are also represented by systems of named sets. The tenth level contains algebras of criteria. The eleventh level of the ASS contains algebras of indices, indicators, attributes, estimates, norms, standards, values, and benchmarks. Levels and parts of the ASS form its vertical structure (cf., Figure 6.11). In addition to levels, the instrumental subsystem has three basic strata (cf., Figure 6.12): — The assertoric stratum contains operational knowledge in the form of abstract automata and descriptions of instruments, devices and machines; — The erotetic stratum contains knowledge on problems, tasks and questions about abstract automata and descriptions of instruments, devices and machines; — The hypothetic, or heuristic, stratum contains descriptions of heuristic, i.e., not sufficiently validated, and empirical instruments, devices and machines. To conclude, it is necessary to remark that the described structuration of subsystems efficiently works in the studies of the statics and dynamics of scientific knowledge in general and of mathematical
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ASS Combination part Algebras & calculi of estimates, norms, Evaluation values part
Productive level
Estimates Scaling part
Norms Values
Attributed level
Scales
Componential level
Figure 6.11. The vertical structure of the ASS
and scientific theories, in particular (cf., (Burgin and Kuznetsov, 1989; 1989a; 1991; 1992; 1993; 1994; Balzer et al., 1991)). However, it is not unique because it is possible to separate other levels, parts, and strata. The choice of structuration depends on the studied knowledge system and on the problems being solved. The BSS inherits structuration into different levels, parts, and strata from other subsystems of the knowledge system. 6.6. Relations between and operations with global knowledge systems The fact that scientists do not consciously practice a formal methodology is very poor evidence that no such methodology exists. It could be said–has been said–that there is a distinctive methodology of science which scientists practice unwittingly, like the chap in Moliere who found that all his life, unknowingly, he had been speaking prose. Peter B. Medawar
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Hypothetic stratum Assertoric stratum
BondsASS
Erotetic stratum
Figure 6.12. The horizontal structure of the ASS
It is natural to define set-theoretical relations between and operations with knowledge systems treating these systems as sets of knowledge items or knowledge units, e.g., sets of propositions. Here are some of set-theoretical relations. 1. Set-theoretical inclusion of knowledge systems: A knowledge system A is a subsystem of a knowledge system B if A is a subset of B. 2. The binary relation to be disjoint: Knowledge systems A and B are disjoint if A does not intersect with B. Here are some of set-theoretical operations. 1. Set-theoretical union of knowledge systems: A knowledge system C is the union of knowledge systems A and B if C consists of knowledge items from A and from B. 2. Set-theoretical intersection of knowledge systems: A knowledge system D is the intersection of knowledge systems A and B if D consists of all knowledge items that belong both to A and to B.
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3. Set-theoretical difference of knowledge systems: A knowledge system E is the union of knowledge systems A and B if E consists of all knowledge items that belong to A but not to B. For instance, knowledge about cats and dogs is the union of knowledge about cats and knowledge about dogs. It is also possible to define structural relations between and structural operations with knowledge systems that have some structure, e.g., a propositional algebra, i.e., a set of propositions closed with respect to logical operations of disjunction, conjunction, implication and negation, or a logical calculus (cf., Section 5.2). Here are some of these relations. 1. The binary relation to be a structural subsystem: A knowledge system A is a structural subsystem of a knowledge system B if A is a subset of B inheriting its structure from B. For instance, a subcalculus A of a calculus B is a structural subsystem of B. A subtheory R of a formal theory T is a structural subsystem of T . 2. The binary relation to be a structural extension: A knowledge system B is a structural extension of a knowledge system A if B is a structured system that contains A. For instance, any calculus B that contains a set of formulas A is a structural extension of A. 3. The binary relation to be a strict structural extension: A knowledge system B is a strict structural extension of a knowledge system A if B is a minimal structured system that contains A. For instance, a minimal calculus B that contains a set of formulas A is a strict structural extension of A. Here are examples of structural operations. 1. The logical intersection of axiomatic theories (calculi) T1 and T2 is built by taking the intersection A of the axioms of T1 and T2 and generating a theory from the axioms in A. 2. The logical union of axiomatic theories (calculi) T1 and T2 is built by taking the union D of the axioms of T1 and T2 and generating a theory from the axioms in D.
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To organize theory-elements on the second level of the theory hierarchy of the knowledge-bounded outer structure, structuralists use the specialization relation σ between theory-elements, which means that one theory is a specialization of another theory (Sneed, 1971). Namely, we have the relation σ
T1 → T2 , when the theory T2 are constructed from the theory T1 by adding additional conditions (laws or axioms). For instance, the theory of differentiable manifolds is a specialization of the theory of topological manifolds. Functioning of science and development of scientific knowledge often involves the construction relation β between theory-elements. Namely, we have the relation β
T1 → T2 , when the basic concepts of the theory (knowledge system) T2 are constructed from the basic concepts of the theory (knowledge system) T1 . There are three basic operations with formal theories, which are nuclear (in this case logical) global knowledge systems: expansion, contraction, and revision. The first two of them are binary operations, in which the first argument is a formal theory and the second argument is an arbitrary set of formulas, while the third one is a ternary operation, in which the first argument is a formal theory and the second and the third arguments are arbitrary sets of formulas. To define these operations in a formal way, we select a system of consistency conditions C and assume that all considered theories are consistent with respect to C (cf., Section 2.3.2). The model example of consistency for formal theories is the classical consistency when a theory T is consistent if it does not contain a formula and its negation. Definition 6.6.1. Given a formal theory T and a set of formulas F , the expansion Exp(T , F ) of T by F is a minimal consistent formal theory Q that contains both T and F .
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Note that if there is no consistent formal theory Q that contains both T and F , the expansion Exp(T , F ) is not defined. This happens, for example, when T contains a formula A and T contains its negation. Proposition 6.6.1. If the intersection of consistent with respect to C formal theories is a consistent with respect to C formal theory, then the expansion Exp(T , F ) is defined in a unique way. Otherwise, expansion of formal theories is a multivalued operation. Definition 6.6.2. Given a formal theory T and a set of formulas F , the contraction Cnt(T , F ) of T by F is a maximal consistent formal subtheory Q of T that does not contain F . Note that to build the contraction Exp(T , F ) of T by F , we need to exclude from T not only formulas from F but also such formulas that allow deduction of formulas from F . Proposition 6.6.2. If the logical closure of consistent with respect to C formal theories without formulas from F is a consistent with respect to C formal theory does not contain formulas from F , then the contraction Cnt(T , F ) is defined in a unique way. Note that contraction of formal theories is a multivalued operation in the general case. Definition 6.6.3. (a) Given a formal theory T and two sets of formulas F and G, the positive revision Rvnp(T , F , G) of T by F is a minimal consistent formal theory Q that contains both T and the set F \G. (b) Given a formal theory T and two sets of formulas F and G, the negative revision Rvnn(T , F , G) of T by F and G is the largest consistent formal subtheory Q of T that does not contain the set G\F . Proposition 6.6.3. If sets of formulas F and G do not intersect, then the negative revision Rvnn(T, F, G) coincides with the contraction Cnt(T, G) and the positive revision Rvnp(T, F, G) coincides with the expansion Exp(T, F ).
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Indeed, if sets of formulas F and G do not intersect, then F \G = F and G\F = G. An important operation is structuration of knowledge systems. For instance, structuration of quantum knowledge is described in Chapter 4. Structuration of full-size (global) knowledge systems is described in Section 6.2. A special case of structuration is stratification of a knowledge system. For instance, stratification of full-size (global) knowledge systems into assertoric, erotetic, and hypothetic strata is described in Section 6.2. Abstraction of knowledge items also induces stratification of knowledge into levels of abstractions (Section 2.3.8). In Section 2.4, metaknowledge is stratified. 6.7. Hierarchies of knowledge systems The only good is knowledge and the only evil is ignorance. Socrates
Knowledge in science is not always well organized on the global plane. That is why, we consider here only the hierarchy of scientific knowledge, which is properly structured by various relations. This hierarchical structure was elaborated in the structuralist direction of the methodology of science in the form of a theory-net, theory-evolution, and theory-holon, representing the structural hierarchy of scientific knowledge in the knowledge-bounded outer structure of a scientific theory T (Sneed, 1971; Balzer et al., 1987). Here we further develop this approach. At the first (lowest) level of the theory hierarchy, structuralists logically position theory-elements. Definition 6.7.1. A theory-element is the smallest unit regarded as a theory. For instance, propositional calculus and number theory are theory-elements, while theory of groups and topology are theorynets. The knowledge-bounded outer structure of a scientific theory T is represented by theory-nets, theory-evolutions and theory-holons
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that contain T . Concepts of a theory-net, theory-evolution and theoryholon were introduced in the structuralist model of scientific knowledge (Sneed, 1979; Balzer et al., 1987). To organize theory-elements on the second level of the theory hierarchy of the knowledge-bounded outer structure, structuralists use the specialization relation σ between theory-elements, which means that one theory is a specialization of another theory. Namely, we have the relation σ
T1 → T2 , when the theory T2 are constructed from the theory T1 by adding additional conditions (laws or axioms). For instance, the theory of differentiable manifolds is a specialization of the theory of topological manifolds. At the second level of the theory hierarchy, structuralists place theory-nets. Definition 6.7.2. A theory-net is a net of theory-elements connected by the specialization relation σ. An example of a structuralist theory-net in algebra is given in Figure 6.13. An important relation between theory-nets is inclusion when one theory-net is a part of another theory-net. For instance, the theorynet “Topological algebra” is included (as a subnet) into the theorynet “Algebra” and the theory-net “Topology.” Algebra theory of universal algebras σ
σ
semigroup theory
σ
ring theory
module theory
σ group theory
σ σ
field theory
σ
σ
theory of vector spaces `
theory of associative rings
σ theory of abelian groups
theory of finite groups
Figure 6.13. A part of the theory-net “Algebra”
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Scientific theories are not static systems. They are born, developing and often going to another world — from science to the history of science. To represent theory dynamics, structuralists introduced the concept theory-evolution. Definition 6.7.3. A theory-evolution is a theory-net “moving” through historical time. It is possible to find examples of theory-evolutions in (Sneed, 1971; Balzer et al., 1987). Besides, theory-nets form a hierarchical structure. To reflect this structure, the concept theory-holon was introduced. Definition 6.7.4. A theory-holon is a complex of theory-nets tied by “essential” links. Any big field in mathematics or in physics has to be represented by a theory-holon. In essence, the whole mathematics or theoretical physics are theory-holons. The structuralist model gives a realistic picture of theoretical knowledge, when it includes several theories. However, to have a more exact and complete model of theoretical knowledge, the structurenominative model was elaborated. Here we go to a higher level building the MSB model of advanced knowledge systems. In this model, not only the specialization relation σ is used to form a mathematical description of a theory-net but also other inter-theoretical relations are employed. For instance, functioning of science and development of scientific knowledge often involves the construction relation β between theory-elements. Namely, we have the relation: β
T1 → T2 , when the basic concepts of the theory (knowledge system) T2 are constructed from the basic concepts of the theory (knowledge system) T1 . For instance, in algebra, a field F is an abelian group with respect to addition and the set F \{0}, i.e., all elements from F but 0, is an abelian group with respect to multiplication, the concept field is based on the concept abelian group. As a result, the structure nominative model of the knowledge-bounded outer structure of a
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scientific theory T , i.e., a theory-net containing T , can be different from the corresponding structuralist model (theory-net). There are also other inter-theoretical relations, i.e., relations between theories, such as “to be a logical extension”, (Shoenfield, 2001) “to be a functional extension” (Burgin and Kuznetsov, 1985). Besides, levels of theoreticity, abstraction, constructivity, and structuration also define specific relations between theory-elements in a theory-net (Burgin and Kuznetsov, 1994). Based on these considerations, here we build the MSB hierarchy of advanced theoretical knowledge systems. Definition 6.7.5. A theory-component is a system of knowledge that plays the role of a component of a theory. In the SNR, five theory-components are discerned on the upper level of a theoretical knowledge system: the LSS, the MSS, the PPS, the PHS, and the subsystem of ties (ST). On the lower level of the theory inner hierarchy, the structure-nominative model has such components as the ASS, the procedural (operational) subsystem, logical subsystem, and linguistic subsystem. In the MSB model, different five theory-components are discerned on the upper level of a theoretical knowledge system: the LSS, the MSS, the PSS, the ASS, ISS and the BS or BT of a complete knowledge system. In addition, each level of each subsystem also is a theory-component (cf., Section 6.1). Definition 6.7.6. A theory-unit is a system of knowledge treated as a theory. Analytical mechanics, quantum mechanics, group theory, number theory, theory of manifolds, and probability theory are examples of theory-units. As in the structuralist model, the knowledge-bounded outer structure of a scientific theory T is represented by theory-nets, theory-evolutions and theory-holons that contain T . In addition, theory-net-evolutions and theory-holon-evolutions are also introduced to represent dynamics not only of theory-units but also of theory-nets, theory-evolutions, and theory-holons.
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σ
general topology
σ
algebraic topology
σ
differential topology σ
theory of topological groups
topology of surfaces
σ topology of manifolds
σ σ theory of Lie groups
theory of Lie groups
Figure 6.14. A structure-nominative theory-net in topology
Definition 6.7.7. A theory-net is a net of theory-units connected by inter-theoretical relations. An example of a structure-nominative theory-net in topology is given in Figure 6.14. It is useful to understand that it is possible to treat some theories as theory-units in some situations and as theory-nets in other situations. Analytical mechanics, quantum mechanics, group theory, the calculus and probability theory are examples of such theories. At the same time, there are theoretical knowledge systems, such as quantum theory or algebra, that are definitely theory-nets. Although theory-nets can be very big, on some level of complexity, the philosophy, and methodology of science come to a qualitative alteration in complexity, which demands another concept for its representation. Definition 6.7.8. A theory-holon is a complex of theory-nets tied by knowledge-related links. Global knowledge systems in general and scientific theories are not static systems. They are going over various transformations after emergence. To represent global knowledge dynamics, we introduce the following concepts. Definition 6.7.9. A theory-state is a theory-unit taken at definite interval of time. For instance, it is possible to consider quantum mechanics in 1930 or number theory in 2000 as examples of theory-states.
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Definition 6.7.10. A theory-evolution is a sequence of theory-states taken at successive intervals of time. For instance, it is possible to consider the sequence of the states of quantum mechanics in 1930, 1940, 1950, and 1960 as a theoryevolution of this discipline. We have another theory-evolution taking the sequence of the states of number theory in 1900, 1925, 1950, 1975, and 2000. Not only scientific theories are changing with time. Theory-nets and theory-holons are also changing. To represent their dynamics, we introduce the following concepts. Definition 6.7.11. A theory-net-state is a theory-net taken at definite interval of time. For instance, it is possible to consider quantum mechanics in 1930 or number theory in 2000 as examples of theory-net-states. Definition 6.7.12. A theory-net-evolution is a sequence of theorynet-states taken at successive intervals of time. For instance, it is possible to consider the sequence of the states of quantum mechanics in 1930, 1940, 1950, and 1960 as a theory-netevolution. We have another theory-net-evolution taking the sequence of the states of topology in 1920, 1940, 1960, 1980, and 2000. Definition 6.7.13. A theory-holon-state is a theory-holon taken at definite interval of time. For instance, it is possible to consider quantum theory in 1930 or mathematics in 2000 as examples of theory-holon-states. Definition 6.7.14. A theory-holon-evolution is a sequence of theoryholon-states taken at successive intervals of time. For instance, it is possible to consider the sequence of the states of physics in 1930, 1940, 1950 and 1960 as a theory-holon-evolution. We have another theory-holon-evolution taking the sequence of the states of mathematics in 1900, 1925, 1950, 1975, and 2000. Named set chains give a natural mathematical representation for theoryevolutions, theory-net-evolutions and theory-holon-evolutions (Burgin, 2008a; 2011).
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Chapter 7
Knowledge Production, Acquisition, Engineering, and Application It is better to learn late than never. Publilius Syrus
In the previous chapters, we studied knowledge properties, their evaluation and verification, as well as knowledge structures on different levels. The goal of this chapter is to describe activities and processes going on in the knowledge universe, which consists of a diversity of knowledge systems and items together with their representations and carriers. We consider such processes and activities as cognition, knowledge production, learning, knowledge acquisition, knowledge discovery, reasoning, knowledge management and application. It is necessary to understand that these processes and activities utilize data and knowledge representations, e.g., texts, schemas, formulas, etc., which, as we have found before, are not knowledge itself as the same knowledge can have different representations. However, working with representations, we process knowledge, which is a very important kind of structures (cf., Section 8.2).
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7.1. Knowledge production, learning, and acquisition as basic cognitive processes Knowledge has to be improved, challenged, and increased constantly, or it vanishes. Peter Drucker
There are many processes in which people obtain knowledge, as well as many names for these processes — cognition, knowledge production, knowledge acquisition, knowledge creation, knowledge capture, learning, knowledge reception, experience, observation, experimentation, thinking, reasoning, perception, knowledge discernment, knowledge apprehension, understanding, judgment, knowledge comprehension, knowledge grasp, insight, knowledge purchase, and knowledge discovery. There are different interpretations of and opinions on the meaning of these terms. Here we take cognition as the most generic term for obtaining knowledge and contemplate other processes in this area as specific kinds and types of cognition. In addition, we ascribe cognitive abilities not only to people but also to other cognitive or epistemic systems such as intelligent technical systems, e.g., computers with the corresponding software, organizations, social groups, and communities. Cognitive studies explicated three basic ways of cognition: 1. Knowledge creation or production when new knowledge is produced. 2. Knowledge acquisition/capture, which may be treated as proactive learning where the learner (cognizer) actively seeks existing knowledge in some area. 3. Knowledge reception or reflexive learning when knowledge carriers, such as instructions or statements, are given (are sent) to the learner (cognizer). In what follows, we study these ways of cognition in more detail. Creation (production) of knowledge goes on three major levels — personal, group, and social. In this process, personal, group, and social intelligence and creativity applied to personal/group/social knowledge often produces (creates) new knowledge.
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On the personal level, according to (Polanyi, 1966), knowledge creation involves an ongoing process of transformation and integration of existing explicit and tacit items of knowledge. It is a highly personal process, which depends on many factors: on the abilities and knowledge of the individual, motivation, particular situation and person’s perception of the situation (Polanyi, 1958). Although some researchers think that knowledge is created only by individuals, history of the humankind shows that knowledge is also created by groups, communities and society as a whole but different individuals play different (if any) roles in this creation. For instance, the initial knowledge in the mathematical calculus was created by the group of two people — Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716). It is assumed that both mathematicians did this independently. At the same time, some groups work together in knowledge creation. For instance, the group of two mathematicians Kenneth Appel and Wolfgang Haken created knowledge that the Four-Color Conjecture is true proving the Four-Color Theorem, which states that any map drawn on a plane can use only four colors without any two adjacent countries having the same color (Appel and Haken, 1977). In contrast to Newton and Leibniz, they worked together proving the Four-Color Theorem. Mathematical knowledge as a whole is created by the mathematical community, in which mathematicians interact, discuss their results and use knowledge obtained by other mathematicians. It is necessary to explain that although knowledge creation and knowledge production are placed in the same category of processes, they are different when treated on a deeper level. Here we employ the following definitions. Definition 7.1.1. Creation in general and knowledge creation, in particular, is an individual, often unique action. Definition 7.1.2. Production in general and knowledge production, in particular, is a process that consists of separate actions of creation. It is possible to find formalization of the concepts action and process in (Burgin and Smith, 2010).
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Knowledge is acquired by an epistemic system E under the action of information, i.e., knowledge is the result of information impact. Note that it is possible to treat knowledge production as a kind of acquisition of knowledge from the system that (who) produces this knowledge. However, this will be ungrounded if we presuppose that acquisition of some knowledge means that this knowledge already exists when acquisition starts. Procedural/instructional knowledge/information production and creation has three forms: — Generation of new knowledge/information; — Transformation of existing knowledge/information into new information/knowledge; it is a kind of knowledge/information evolution; — Reorganization/restructuring of existing information/knowledge with the goal to obtain new knowledge/information. This classification essentially depends of the utilized classifications of instructions/rules. For instance, if deduction rules, such as modus ponens, are treated as generation instructions, then deduction is knowledge generation. However, if deduction rules are interpreted as transformation instructions, then deduction is knowledge transformation. The difference between knowledge transformation and knowledge restructuring is also relative. When we consider knowledge as a whole, then knowledge restructuring is a kind of knowledge transformation. However, when we deal with knowledge items as separate objects, then knowledge transformation changes these items, while knowledge restructuring changes only relations between these items. Tuomi (1999) separates three modes of knowledge generation: — Anticipation when the model of a phenomenon we have changes (breaks down), producing new knowledge from the tension between anticipation and what is observed. — Appropriation of knowledge existing in another system.
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— Articulation means explication and reconfiguration of epistemic relationships within the meaning system available for the cognizer. Three basic ways of cognition considered above correspond to three main sources of knowledge/information: 1. Personal and social intelligence and creativity applied to personal/social knowledge and data can give new knowledge. 2. Material activity, such as observation, experimentation, request for information, work, search in books, databases and on the Internet, communication, etc., is a resourceful source of knowledge and experience. In turn, experience can give new knowledge to those who can learn from experience. Note that people can learn from their own experience and from experience of others. 3. Carriers of knowledge, for example, other people, books, knowledge bases, computers, and networks, such as the Internet, can provide knowledge on their own, e.g., people give lectures or companies disseminate information about their products and services. Note that there are people who are creative but lack intelligence and there are intelligent people who are deficient in creative skills. There are different forms of material activity that involve knowledge processes: 1. 2. 3. 4. 5. 6. 7.
Observation, which is typically reflexive learning. Experimentation, which is frequently proactive learning. Practical activity, which can include all kinds of learning. Interaction and its special kind communication. Search as intellectual activity. Selection as intellectual activity. Games, which can include other material activities.
Note that observation, experimentation and practical activity can be material, e.g., observation of planet movements, as well as mental, e.g., experimentation in mathematics (cf., for example, (Poincar´e,
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1902; 1905; 1908; Burgin, 1998) or computational, e.g., computer simulation of driving a car. Three basic stages of knowledge acquisition by a cognitive (intelligent) system are: — Information search and selection, — Information extraction, acquisition and accumulation, — Transformation of information into knowledge. All these stages are based on knowledge of the cognizing system. Therefore, it is possible to say that knowledge generates (increases) knowledge. Although in some cases, it may be false knowledge. Perception as the process of creation of perceptual knowledge depends on the use of knowledge stored in the system and not available in the immediate sensory input. This peculiarity makes possible recognizing and remembering objects and scenes that make sense, as opposed to those that do not (Biederman et al., 1974; Potter, 1975). Although in this case, the implicit knowledge is used unconsciously, the perceiver would normally be able to identify and describe its source in other cases when the knowledge source is still not apparent to the observer. For instance, some researchers suggest that motion perception appears to involve implicit familiarity with the physics of transparency (Stoner et al., 1990). It is interesting that there are situations when false knowledge can help to acquire true knowledge. Usually it happens when an individual, e.g., a scientist, take some existing knowledge, makes observations and experiments and then based on these observations and experiments obtains true (correct) knowledge. For instance, taking false knowledge that the Sun rotates around the Earth, Copernicus came to the conclusion that this was not true and the true knowledge was that the Earth rotated around the Sun. In a similar situation, Galileo had false knowledge that heavy objects fell faster than lighter ones because Aristotle had written so. In spite of this, Galileo decided to test this knowledge dropping two different weights from the Leaning Tower in Pisa. He found that they landed at the same time. After repeating this experiment many
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times, Galileo correctly concluded that the velocity of a falling body did not depend on its weight. This was true knowledge. In these cases, researchers abandoned false knowledge after finding true knowledge. However, there are situations when false knowledge is not discarded by acquisition of true knowledge. The most famous example is Maxwell’s inference of the formula of gas pressure. According to Fabrikant (1985), deriving this formula, Maxwell, at first, did one mistake, obtaining incorrect knowledge, but then continuing his derivation, he did another mistake, which compensated the first mistake. As the result, Maxwell obtained the correct formula, which represented true knowledge. There are three basic sources in knowledge acquisition: from practice/experience, from reasoning/thinking and from authority/opinion. Namely, we have the following classification: 1. Knowledge acquisition by practice/experience means that the cognitive agent gets knowledge from practical activity, e.g., it allows better achieving some goals, and our experience gives evidence for this. 2. Knowledge acquisition from authority/opinion means that an epistemic structure (knowledge) is acquired from opinion, which is usually held as an authoritative one. Note that it may be an opinion of an individual or of a social group taken from some source, such as a book, magazine, or the Internet. 3. Knowledge acquisition by reasoning/thinking is performed in the mentality of the cognitive agent and is explicit justification. However, this traditional classification is incomplete. To show this, we use the Componential Triune Brain model (CTBM) developed in (Burgin, 2010) and utilized in Section 7.2 for understanding intuition. According to this model, the brain has three basic components — the System of Rational Intelligence — SRI (also called the System of Reasoning), the System of Emotions (the System of Affective States — SAS) and the System of Will and Instinct — SWI. Each of these systems works as a source in knowledge acquisition but only the SRI is taken into account in the above classification. Consequently, there are two other kinds of mental-based knowledge
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acquisition when the cognitive agent gets knowledge by emotions and by instructions/assertions. Therefore, the third source has to be interpreted in the following way: 3a. Knowledge acquisition by mental information processing is performed in the mentality of the cognitive agent and can be either explicit justification when it is reasoning or instruction, or implicit justification when it comes from emotions or will. There are three types of knowledge/information search: — internal when a system searches in its own knowledge/information storages such as memory, database or knowledge base; — external when a system searches in its environment; — mixed when a system searches everywhere. Search and selection of relevant knowledge/information are cognitive processes based on knowledge that the system already has. Indeed, it is necessary, at least, to know what to search, where to search, and what tools are suitable for the search. For instance, perception, recognition and recollection of objects and scenes depends on the use of knowledge stored in the system and demands selection of relevant schemas (Biederman et al., 1974; Potter, 1975). There are three basic types of knowledge/information production/creation: — Knowledge/information production by reasoning, e.g., by logical inference, which includes deduction, induction, and abduction; — Procedural/instructional knowledge/information production, e.g., by construction, by experimentation or by transformation; — Intuitive knowledge/information production by subconscious contemplation, by guessing or by emotions, e.g., the so-called, gut feeling. According to Popper (1979), the aptitude for solving problems, is “a creative ability to produce new guesses, and more new guesses” and growth of knowledge is due to a procedure of “trial and error”.
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There are three basic types of knowledge capture or proactive learning: • knowledge capture by search, • knowledge capture by selection, • knowledge capture by explication or extraction. Often these processes (actions) go one after another forming the proactive learning cycle. There are also three basic types of knowledge reception or reflexive learning: ◦ knowledge reception by exchange; ◦ knowledge reception by inquiry; ◦ knowledge reception by acceptance of what is coming. The main components of cognitive proficiency are intelligence, persistence, and creativity. There are various theories of human intelligence, as well as of artificial intelligence (AI), for which human intelligence is taken as the model. Usually it is assumed that intelligence is based on the intellect of a person, which is a subsystem of personality, i.e., some structure. Another approach interpreted intelligence in the same way as the intellect treating it as a static property of an individual (such as gifts, will, or moral beliefs). Thus, intellect and intelligence have been considered as a static entity related to a human being, while intellect and intelligence are always displayed in the behavior and more exactly, in intellectual activity of a person. Consequently, many proponents of behavior-based AI (cf., for example, (Pfeifer and Scheier, 1999; Markman, 2000)) argue that the traditional methods for studying intelligence have failed to provide sufficient insight into what intelligence is and how it works. The root of this failure, they contend, is that cognitive science has focused on complex internal mental processes to the exclusion of the factors that permit intelligent agents to interact with their environment. However, psychology has its own approach to behavioral aspects of intelligence. It is based on intellectual activity. The term intellectual activity has been extensively used by different authors for
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a long time, but it has not been sufficiently exact lacking an adequate and efficient definition (Bogoyavlenskaya, 1983). Consequently, this absence has implied many difficulties for understanding and investigation of this important phenomenon. Such an exact definition was constructed by Burgin (1995d; 1996a; 1998a). According to this definition, Intellectual activity is a meaningful functioning of mind (intelligent thinking). This definition provides for the dynamic expression of human intelligence as well as for elaboration of efficient means for its study. That is why, an investigation of various properties of intellectual activity is of the greatest interest to psychology and pedagogy because intelligence has been always considered as main characteristic of a human being. The main assumption of this approach is that intelligence is always displayed in different kinds of behavior and, in particular, in cognition. Thus, knowing very little about inner structure of intelligence, it is more efficient to consider intelligent behavior, or more exactly, intellectual activity of a person. Taking essential components of human activity as the base, different types and grades of intellectual activity are explicated and explored. With respect to the result, there are three types of intellectual activity: — The reproductive intellectual activity is selection in and reproduction of the given knowledge. — The bounded productive intellectual activity is search for necessary knowledge. — The productive intellectual activity is creation/production of new knowledge. We can observe the reproductive intellectual activity when a student learns some material given by the teacher or by reading the textbook. We can observe the bounded productive intellectual activity when a person searches the Internet or an encyclopedia for necessary information.
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We can observe the productive intellectual activity when a physicist discovers a new law of physics or when a mathematician proves a new theorem. It is necessary to understand that in some cases, selection and/or reproduction can be more complicated actions than search and even creation of new knowledge. That is why selection and reproduction also demand a definite level of intelligence. With respect to the means used in achieving the result, there are also three types of intellectual activity: — The reproductive instrumental intellectual activity, — The extended instrumental intellectual activity, — The creative intellectual activity. In the reproductive instrumental intellectual activity, an individual or a group uses known means (tools) in the conventional way. For instance, when a student solves a problem by the technique taught by the teacher, it is the reproductive instrumental intellectual activity. In the extended instrumental intellectual activity, an individual or a group uses known means (tools) in a new way. For instance, when a student solves a problem by the technique usually used in a different area, it is the extended instrumental intellectual activity. Another example of the extended instrumental intellectual activity comes from computer programming when programmers and computer scientists started to use logic for building programming languages and writing programs. In the creative intellectual activity, an individual or a group invents new means (tools) for their activity, e.g., solving the problem. For instance, the famous physicists Richard Feynman (1918– 1988) invented path integral for giving a new model of quantum mechanics. Classification of intellectual activity is useful in several aspects. First, it helps to study cognition and creativity from different perspectives. Second, it allows developing person oriented cognitive methodology and technology. Third, it provides a theoretical base for
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the development of cognitive and creative skills. In particular, intellectual activity has been applied to problems of education (Burgin, 1995d). Intellectual activity is an expression of intelligence in its operational/behavioral form. This feature allows using intellectual activity as base for studying intelligence, while measures and methods of evaluation of intellectual activity provide tools for estimation active intelligence of people. In psychological studies, the intellectual activity approach is orthogonal to the classical psychological approach to intelligence where intelligence is considered as a trait or faculty of people. For instance, intellectual activity is complementary to the well-known triarchic model of human intelligence of Robert Sternberg. The triarchic model asserts that human intelligence consists of three types of faculties: analytical giftedness, synthetic giftedness (creativity), and contextual (practical) giftedness. According to the triarchic model, analytical giftedness is dominant in being able to take apart problems and being able to see solutions not often seen. To explain these abilities, analytical giftedness is associated with three types of components in the functioning of the mind: metacomponents, performance components, and knowledgeacquisition components (Sternberg, 1985; 1997). Definition 7.1.3. The metacomponents control functioning of the mind telling what the mind has to do. These components are especially important in problem solving and decision making. Definition 7.1.4. The performance components are the processes in the mind that actually carry out the actions the metacomponents dictate allowing people perceiving problems and relations between objects, transforming images and applying relations to another set of terms. Definition 7.1.5. The knowledge-acquisition components are used in acquisition of new information, selection of the useful information from irrelevant information and construction of new knowledge combining the various pieces of information.
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Intellectually gifted individuals are proficient in using knowledgeacquisition components because they are able to learn new information at a greater rate. Synthetic giftedness is especially important for creating new ideas in the process of generating new ideas, posing and solving new problems and managing novel situations. Contextual (practical) giftedness involves the aptitude to apply synthetic and analytic skills to everyday situations. The effectiveness with which an individual matches the surroundings and contends with daily situations reflects degree of definite intelligence. Practically gifted people are able to succeed in any setting, creating a supreme fit between themselves and their environment. Contextual (practical) giftedness is expressed in the three processes: adaptation, shaping, and selection. Definition 7.1.6. Adaptation is a process of making changes within oneself to better adjust to the environment. For instance, when the weather changes and temperatures drop, people adapt by wearing extra layers of clothing to remain warm outside and heat up their homes to remain warm inside. Definition 7.1.7. Shaping is a process of making changes in the environment to better suit one’s needs. For instance, people grow trees and bushes to have better climate, more oxygen and enhanced surroundings. The suggested terminology can be extended based on the classification of developmental processes suggested by Jean Piaget (1896– 1980). Namely, Piaget suggested that development consists of accommodation and assimilation (Piaget, 1964). Definition 7.1.8. Accommodation includes reinterpretation of the current situation and changing the cognitive model and behavioral schemas that are used. Thus, accommodation is inner shaping of the situation. Definition 7.1.9. Assimilation is the process of adjustment to the current situation without changing its interpretation, e.g., by changing the behavioral schemas that are used.
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We see that in his model, Piaget describes behavioral learning and adjustment means primarily cognitive adaptation described in Definition 7.1.6. Definition 7.1.10. Selection is a process of changing the place of living to find the environment that better meets the individual’s goals. For instance, immigrants leave their native countries where they endure economical, religious, or/and social oppression and go to other countries in search of a better and less strained life. With respect to knowledge, analytical giftedness expresses itself in analyzing knowledge and solving cognitive problems. Synthetic giftedness ameliorates creation of new knowledge, while contextual (practical) giftedness facilitates applying knowledge to practical problems. There are three basic sources of knowledge for an epistemic system E, i.e., a system, which has or/and produces knowledge and other epistemic structures: 1. Another epistemic system R, e.g., a book, database, individual, organization or movie. 2. Another physical system R, e.g., a molecule, atom, star, car, tree, or the Earth. 3. The system E itself, for example, when E extracts (recollects) its own knowledge or produces new knowledge. There are three types of knowledge sources: — Proactive sources send information themselves and the cognizer only accepts (or does not accept) what was sent. — Reactive sources send information in response to an inquiry or request. — Passive sources do not send information and so, the cognizer has to extract information from passive sources. First two types are active sources. There are three ways of knowledge (information) extraction: — Extraction by non-intrusive interaction when it is possible to disregard the impact on the source, e.g., by observation.
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— Extraction by intrusive interaction when it is necessary to take into account the impact on the source, e.g., by experiment. — Extraction by inquiry when a message is send to source, which responds sending a reply to this message. Important processes going in society are knowledge translation and knowledge translation Integration. Definition 7.1.11. Knowledge translation is transmission of knowledge from one area to another and its adaptation, e.g., reinterpretation, to the new area. Definition 7.1.12. Integration of knowledge from two areas is its transmission to the common area and its mutual adaptation. An important way of knowledge acquisition is interpretation, which is considered as a procedure or function that corresponds knowledge to a given system, e.g., to a text, facilitating its understanding. There is a special theoretical area of interpretation called hermeneutics. In other words, hermeneutics is a methodology of obtaining knowledge of a text by extracting it from the text. The Greek word ‘hermenuein’ means to express, explain, translate or interpret (Thiselton, 1998). Hermeneutics appeared in 1654 as an art of interpretation of biblical texts. Schleiermacher (1819) welded different approaches into this field making the main input in its development and applying it not to the Bible but also to other texts. As a result, hermeneutics became a theory of understanding in the broadest sense, and Dilthey (1981) applied it to all human acts and products, including history and human life. He asked the question, “How do the social or human sciences differ from the natural sciences?” On his opinion, while the natural sciences explain, the social and human sciences understand. In contrast to this, modern methodology of science persuasively demonstrated (Agazzi, 1992; Burgin, 1995) that structurally explanation and understanding are similar processes. The main difference is in methodology they use. Natural science is based on direct
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observations and experiments, while social science develops from indirect evidence and opinions. Contemporary hermeneutics, according to Dilthey (1981), considers two levels of understanding. The lower, immediate understanding of simple expressions and higher understanding that involves comprehension of individuals. In comparison with this, hermeneutics developed in Jewish tradition is a system called PARDES, which comprises four levels of understanding and is considered in more detail in Section 7.1.1. Besides, achievements of modern science, advanced epistemology, and innovative hermeneutics made it possible to build up more than ten levels of cognition and understanding (Burgin, 1996c; 1996d). Now let us consider some important categories of cognition in more detail. 7.1.1. Scientific cognition We’ve arranged a civilization in which most crucial elements profoundly depend on science and technology. Carl Sagan
Science emerged as an efficient tool for exploration and understanding, at first, of nature and later of the whole world in which people live. Scientists created powerful means of cognition. Scientific cognition typically utilizes three basic processes: • Theoretical reasoning; • Observation and experiment; • Intuitive insight. As a rule, these processes are separate but depend on and support one another expanding in the concurrent mode with multiple cycles and iterations. The formalized mode of theoretical reasoning is logical inference (information/knowledge production), which has three main forms: — Deduction; — Induction; — Abduction.
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While philosophers have studied deduction and induction for a long time, the term abduction was first introduced by the American philosopher and semiotician Charles Sanders Peirce (1839–1914) only in the 19th century. These three basic methods of reasoning are applied to declarative knowledge in general and as formalized techniques, to knowledge in the form of expressions (formulas) from the utilized scientific or logical language in particular. Deduction, induction and abduction are also used to obtain properties of operational and representational knowledge. Deduction is a type of logical inference of knowledge performed by application of specific deduction rules. which, in general, have the form: A→B
(7.1)
A B.
(7.2)
or
Here A is called the assumption of the rule, B is called the conclusion of the rule and each of them is a finite number of expressions or formulas. For instance, taking the expression “X is in Y ”, we can build the deduction rule: “U is in V ”, “V is in W ” “U is in W ”. Applying this rule to two propositions “We live in the USA” and “The USA is situated in America” as A, we deduce the proposition “We live in America” as B. A valid deduction guarantees the truth of the conclusion given the truth of the assumptions. In a general case, it is necessary to apply deduction rules several times to obtain the necessary conclusion. The most utilized deduction rule is modus ponens, which has the form: ϕ, ϕ → ψ ψ, where ϕ and ψ are statements or propositions. This rule has the following meaning: If ϕ is true and ϕ implies ψ, then ψ is true.
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Mathematical and scientific practice shows that deduction is used not so much for knowledge production as for knowledge justification. Recursive algorithms (cf., (Burgin, 2005)) and the majority of logical systems formalize deduction (cf., Chapter 5). However, as Aristotle observed, scientific discovery by deduction is impossible, except one knows the “first” primary premises . . . and it is necessary to obtain these premises by induction. Induction, in its main interpretation, is a form of logical inference that allows inferring a general statement from a sufficient number of particular cases, which provide evidence for this general statement (the conclusion). However, if the evidence is not complete, the conclusion may be incorrect. For instance, Aristotle saw that all of the swans at the places where he lived were white, so he induced that all swans are white in general. However, much later European came to Australia and discovered black swans. This shows that in contrast to deduction, induction does not always give correct results because it works with incomplete information, while the number of initial cases is not bounded and the researcher does not know for sure when to stop. However, the whole science is actually built on induction because scientific laws have to be in agreement with nature for natural sciences and with social systems for social sciences, while it is possible to make only a finite number of experiments. Induction has a long history. For instance, it is possible to find elements of inductive reasoning in Plato’s Parmenides (ca. 370 B.C.E.). Later mathematicians started using inductive proofs, which afterward were shaped in the Principle of Induction. We can see (cf., for example, (Katz, 1996; Rashed, 1994)) indication of inductive inference in works of the famous Euclid (ca. 300 B.C.E.), Islamic mathematician al-Karaji (ca. 1000) and prominent Indian mathematician Bhaskara (1114 — ca. 1185). In the explicit form, mathematical induction was used (cf., (Rabinovitch, 1970)) by Jewish mathematician Levi ben Gerson (1288–1344), while great French mathematician Blaise Pascal (1623– 1662) unambiguously formulated the Principle of Induction in his Trait´e du triangle arithm´etique. Later mathematical induction has become a popular tool in mathematics.
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In addition to inductive reasoning, there is also inductive learning, which involves making (often uncertain) inferences that go beyond direct experience and are based on intuition and insight. This is the main procedure in learning from experience. Analysis of computational processes allows discovering one more kind of induction — constructive mathematical induction. In the theory of computation, it is called recursion. Indeed, informally recursion is a technique such that given the value f (n) of some function defined for natural numbers, it allows computing the value f (n + 1) (Shoenfield, 2001; Burgin, 2005). When this technique is given, it is assumed that it makes possible to compute all values of the function f . Similar to the conventional mathematical induction, in constructive mathematical induction, description of a general step of computation presupposes the possibility to perform computation for an infinite quantity of inputs. Note that inductive reasoning is computation of the truth function. Thus, it is possible to discern two types of induction as knowledge production: the observational induction and abstract induction. The observational induction, or more generally, empirical induction, is a conclusion made for a large collection of objects, e.g., events, systems or processes, based on observation of (and experiments with) some (usually small) part of this collection. The abstract induction is a conclusion made for a large (often infinite) collection of objects, e.g., numbers, structures or procedures, based on reasoning about some finite (usually small) part of this collection. In its essence, induction is reasoning based on a simple rule: If it was so, then it will be so. We call it the straightforward induction. For instance, when people see that the Sun rises every morning, they suppose (induce) that it will be always. However, such induction is not always reliable. For instance, when people go into the space in a spaceship, they find that their previous supposition is violated. Correct application of the straightforward induction demands highly developed intuition and can be invalidated by some new
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observations or experiments. The straightforward induction is the cornerstone cognitive technique in physics and other sciences. When scientists declare that an experiment validates a scientific theory, they implicitly apply the straightforward induction any experiment can tell only that the theory is correct in one particular case. Mathematicians, being largely dissatisfied by absence of absolute reliability in the straightforward induction, invented rules for making induction (totally, or at least, sufficiently) trustworthy. This brought forward the imperative induction, which is based on definite rules for testing a part of a big collection of objects and making conclusions about the whole collection. Note that the straightforward induction involves a divergent process when it is impossible or unreasonable to get direct knowledge about each object from the whole collection. Introducing rules for making the process finite and tractable is renormalization of the reasoning process similar to renormalization used in physics. The most popular kinds of the imperative induction are mathematical induction and statistical inference. Statistical inference allows obtaining properties of a large set of objects called population in statistics, using data about these properties drawn from some (usually small) parts (subsets) of the whole population. These parts (subsets) are called samples, which are usually selected according to definite rules. Validity of statistical inference strongly depends on the relevance of the statistical model used for inference to the modeled problem (Cox, 2006). A statistical model consists of three parts: • A set of assumptions related to the whole population • Rules for sample selection • Rules for estimation of properties of the selected samples and for making inference from obtained data. Statisticians use three types of modeling assumptions; • Fully parametric when it is assumed that the probability distributions describing the population and process generating the
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statistical data are fully described by a family of probability distributions involving only a finite number of unknown parameters. • Non-parametric when the assumptions made about the population and the process generating the statistical data are minimal. • Semi-parametric when the assumptions are not as complete as in the fully parametric case but are not so little as in the nonparametric case. Mathematical induction, in essence, reduces induction to deduction by the Principle of Induction formalized in the axiom of induction, which according to (Poincar´e, 1905), has the following form. Axiom of induction (numerical form). If a theorem is proved for 1 and it is also proved that it is true for n + 1 whenever it is true for n, then it is true for all natural numbers. However, mathematical induction is applied not only to numbers but also to a variety of mathematical and other objects. These applications are grounded by the following axiom of induction, which is also called the principle of mathematical induction. Assume that an infinite sequence of statements (propositions) A1 , A2 , A3 , . . . , An , . . . is given and A is the statement (proposition) that all are true. Axiom of induction (logical form). If the statement (proposition) A1 is proved and it is also proved that An+1 is true whenever An is true for an arbitrary n = 1, 2, 3, . . . , then all statements (propositions) An are true. However, a more general form of the axiom of induction is useful in many situations. Assume that an infinite sequence of objects O1 , O2 , O3 , . . . , On , . . . is given and P is a property. Axiom of induction (general form). If a theorem is proved that the object O1 has the property P and it is also proved that the object On+1 has the property P whenever the object On has the property P , then all objects O1 , O2 , O3 , . . . , On , . . . have the property P . Note that that the general and logical forms of the axiom of induction are equivalent when all properties are binary and are defined by
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predicates, while having a property is represented by a logical statement (proposition). However, axioms of induction represent only the simplest form of mathematical induction. Mathematicians have used many other kinds of mathematical induction. Principle of extended induction. If the statement (proposition) A1 is proved and it is also proved that An+1 is true whenever A1 , A2 , . . . , An are true for an arbitrary n = 1, 2, 3, . . . , then all statements (propositions) An are true. One more kind of mathematical induction, which starts not from the first statement, shifted mathematical induction. Principle of shifted induction. If the statement (proposition) Ak is proved and for an arbitrary n ≥ k, it is also proved that An+1 is true whenever An are true, then all statements (propositions) An are true (n = 1, 2, 3, . . .). In some cases, it is necessary to extend our assumptions. Principle of extended shifted induction. If the statement (proposition) Ak is proved and for an arbitrary n ≥ k, it is also proved that An+1 is true whenever Ak , Ak+1 , . . . , An are true, then all statements (propositions) An are true (n = 1, 2, 3, . . .). There are also: • • • • •
Odd-even mathematical induction Backward mathematical induction Spiral mathematical induction Double mathematical induction Mathematical induction with parameter
While mathematical induction eliminates necessity of intuition in making the conclusion, acceptance of the axiom of induction and its application still demand intuition (Poincar´e, 1905). There are special logical systems that formalize induction and inductive reasoning (Kyburg, 1969; H´ ajek and Havranek, 1978). Naturally, they develop logical representations only for some kind of mathematical induction imbedding mathematical induction, which is an inference procedure (deduction rule), into a logical calculus.
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At the same time, empirical induction, which is prevalent in science, is modeled and explored in the theory of abstract automata, algorithms and computation. There are three main directions in mathematical modeling of empirical induction: the, so-called, Solomonoff’s theory of universal inductive inference, inductive inference based on learning in the limit and inductive inference based on inductive Turing machines. Solomonoff’s approach interprets knowledge acquisition as gaining an ability to predict a symbol in a sequence based upon the knowledge of previous symbols from this sequence (Solomonoff, 1964). The basic assumption made in this theory is that symbols in the sequence follow some unknown but computable probability distribution. Inductive inference based on learning in the limit stems from the innovative paper of Mark Gold (1965), where limiting recursion was introduced in the form of limiting recursive and limiting partial recursive functions. The goal of inductive inference is to recognize a function given some of its values. In this context, only recursively computable functions are considered as descriptions of scientific laws and epistemic systems are represented by inductive inference machines (Angluin and Smith, 1983; Osherson et al., 1991). In the third mathematically based direction, the process of inductive inference is performed by an abstract automaton called an inductive Turing machine, which represents the next step in the development of computer science providing more adequate and efficient models for contemporary computers and computer networks, constituting an important class of super-recursive algorithms and satisfying all conditions in the informal definition of algorithm (Burgin, 2005). In this context, cognition is represented by computations of inductive Turing machines or inductive cellular automata (Burgin, 2005; 2015). Abduction, according to Peirce, is a form of logical inference aimed at finding a hypothetical explanation E for an observed phenomenon P (Peirce, 1931–1935). In contrast to induction, abduction does not demand many observations. In contrast to deduction, there are no exact infallible rules for abduction.
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Peirce assumed that abductive reasoning constitutes the first stage of a scientific research, as well as of any interpretive processes (Peirce, 1931–1935). In this role, abduction utilizes two operations — formation of plausible hypotheses or premises by the procedure of reconstructing plausible causes or intentions, and selection some of them by rational principles and “guessing instinct”. However, it is reasonable to add three more operations to the process of abduction: — Grounding the acceptance of the selected hypothesis; — Experimentation of the selected hypothesis; — Application of the selected hypothesis. The first of these operations is aimed at maintenance of logical coherence, while the second and the third operations pursue the pragmatic relevance. Peirce compares the process of abduction to the Darwinian model of evolution where formation of hypotheses corresponds to the birth of living being and selection by the “guessing instinct” and the rational “principle of economy” matches natural selection by a fitness rule. Bonfantini and Proni (1983) suggest to understand the abductive “guessing instinct” not only as a natural insight, which is inborn, but also as a cultural insight, which is learned and rooted in the person’s background. This connects the abductive reasoning to hermeneutic processes (Eco, 1990). As a rule, abduction also utilizes intuitive solutions for finding a hypothetical explanation for an observed phenomenon because there are no deterministic rules for this process (cf., Section 7.1.1). Even deduction and induction demand intuition — deduction when axioms for theories are selected and induction when the decision is made that number of particular cases is sufficient to make the conclusion (cf., Section 7.1.1). An important and popular kind of abduction is analogy. Analogy as a method of reasoning (inference) is based on a similarity relation (Osuga and Saeki, 1990; Burgin, 1993). To define analogical reasoning in a formal way, we consider a similarity relation σ between statements or propositions.
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Mathematically similarity relation is represented by tolerance relation, which is a binary relation that is reflexive and symmetric (cf., Appendix A). For instance, letters “a” and “a” are similar, while letters “a” and “b” are not similar. Taking some set of objects, it is possible to define that two objects are similar when they have, at least, one common property. In this case, any two letters are similar but a mountain and a letter are not similar. The procedure of analogical reasoning is described by the following operation: An inference system, e.g., a person or a machine, considers a system of conditions a1 , a2 , a3 , . . . , an that imply a fact, e.g., a statement or effect, C as a consequence, and another system of conditions b1 , b2 , b3 , . . . , bn such that each ai is similar to the corresponding bi ., i.e., ai σbi is true. Then the inference system finds a fact B similar to C and assumes that B is a consequence of conditions b1 , b2 , b3 , . . . , bn . Nevertheless, in spite of serious studies of abductive reasoning in epistemology and its increasing impact on linguistics, hermeneutics and AI, the status of abductive reasoning remains a controversial issue in philosophy of science and epistemology. Sowa (1984) suggested describing deduction, induction, and abduction in the following way: • Deduction is described by the following formal rule: If A and A → B are true, then conclude B. • Abduction is described by the following informal rule: If B and A → B are true, then assume A, or when several statements A imply B, then use some selection rule or preference relation to select the best A. • Induction is described by the following informal rule: If B is true for every case when A has been observed and there was a sufficient number of observations, then assume A → B.
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This description explicates differences between three approaches. Abduction and induction both use the tentative verb assume instead of the secure and comfortable verb conclude. Induction demands some quantity of observations as the premise and has an implicit quantifier every in the conclusion. This makes induction much more complex than deduction, which generates a conclusion in a single step. In comparison with abduction, induction is simpler because finding a cause for something is usually more complicated than observation and experiment. It is interesting to know that the basic scientific methods of cognition — deduction, induction and abduction — intrinsically correspond to three ways of cognition and understanding crystallized in Jewish tradition. According to it, there is a system called PARDES, which literally means “orchard” or “garden” and comprises four manners or levels of cognition and understanding: Peshat or Pshat, Remez, Derush or Drash and Sod. An expert in Judaism Gershom Scholem (1897–1982) defines these levels in the following way (Scholem, 1995): • Peshat (Pshat) is the literal perception of what is given, • Remez is the allegoric cognition and understanding, • Derush (Drash) is the hermeneutical or logical cognition and understanding, • Sod is the mystical insight or esoteric cognition. In this system, it is possible to correspond Drash to logical deduction, Remez to abduction and Pshat to induction. Esoteric cognition (Sod) and mystics lies outside science and scientific cognition. The development of science and philosophy allows extending traditional and inventing new methods of cognition. As a result, achievements of modern science, advanced epistemology, and hermeneutics make possible separation and development of more than ten levels of cognition and understanding in comparison with four classical methods of PARDES (Burgin, 1996c; 1996d). Usually reasoning is performed with expressions in natural languages or formulas in logical languages. However, as we have seen in Chapter 5 some popular forms of knowledge representations
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use geometrical images such as graphs, diagrams, networks, and schemas. As a result, people created another type of reasoning with visual images. For instance, using diagrams, which are usually twodimensional and only sometimes three-dimensional. Diagrammatic reasoning is reasoning using diagrams as visual representations of knowledge and information. According to psychological studies, 80% of all information that people get through their senses is visual information (Atkinson et al., 1990). Thus, diagrammatic reasoning helps understanding of concepts and ideas, facilitates reasoning and comprehension. Different knowledge representations bring us to separation of three forms of reasoning: 1. Symbolic reasoning is performed with symbolic expressions such as texts or formulas. 2. Diagrammatic reasoning is performed with firmly structured images such as graphs, diagrams, networks, and schemas. 3. Picture reasoning is performed with weakly structured and unstructured images such as pictures or paintings. To conclude, it is necessary to remark that scientific reasoning in general and logical inference, in particular, may include all three basic forms of theoretical analysis — deduction, induction, and abduction. For instance, at first, a scientific law is obtained by inductive inference based on experimental results or by abduction based on intuition. Then different consequences of this law are acquired by deduction. One of the most prominent examples of this situation is obtaining the Newton’s law of gravitation and deducing Kepler’s laws of motion from the Newton’s law. 7.1.2. Intuition as a cognitive instrument Man knows more than he understands. Alfred Adler
Although general public assumes that scientific cognition is based on exact methods of reasoning, intuition plays an important role in science. For instance, abduction cannot be efficient without qualified
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intuitive insights. Mathematics also relies on intuition. For instance, to be able to select a relevant set of axioms for a mathematical theory or to suggest a reasonable hypothesis, it is necessary to have a developed mathematical intuition. According to Hintikka (2003), the Latin term intuitio was introduced into the mainstream philosophical usage by the scholastics, who gave a variety of interpretations for the term intuitive cognition. However, the broad scholastic conception of intuition was not long-lived. The very general understanding (1) of the term intuition relates this phenomenon to any source of ostensible knowledge not obtained by conscious inference from the possessed knowledge. Note that as sense-perception satisfies explanation (1), we come to a specific kind of intuition called sensory intuition. At the same time, Kant regarded intuition as a phenomenon strictly distinct from any sensation. In spite of many attempts to uniquely characterize intuition, no efforts of researchers have resulted in the definition acceptable by the majority of philosophers, psychologists, scientists, and mathematicians. Therefore, there are many competing definitions and descriptions of intuition, and we consider some of them. Intuition is a deeper perception of inherent possibilities and inner meanings. Intuition is the ability to understand something instinctively, without the need for conscious reasoning. Intuitions are simply opinions . . . (Lewis, 1983). Intuition is also characterized as the vehicle for knowing. Intuitions are the tendencies that make certain beliefs attractive to us, that ‘move’ us in the direction of accepting certain propositions without taking us all the way to acceptance (van Inwagen, 1997). According to Locke, intuitive knowledge is the perception of the certain agreement or disagreement of two directly compared ideas, i.e., intuition is restricted to knowledge of ideas. An intuition is a propositional attitude that either seems to be true (Bealer, 1998; Pust, 2000; Huemer, 2001; 2005), or it is presented
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to the subject as true (Chudnoff, 2011) or it pushes the subject to believe a proposition in it (Koksvik, 2011). Some researchers even regard intuition as a constituent of knowledge. For instance, Charles Oppenheim, contends that “knowledge is a combination of information and a person’s experience, intuition and expertise.” (cf., (Zins, 2007)). The phenomenon of intuition has been explored for a long time. Many philosophers and some mathematicians, such as Descartes, Kant, Bergson, Poincar´e, G¨odel, Bunge and Kuhn to name but a few of many, considered intuition as one of the main cognitive tools. It is possible to find allusion to intuition in the work of the great Aristotle, who declared that intuition would be the initial source of knowledge (Aristotle, 1984). Ren´e Descartes (1596–1650) regarded intuition as a clear and attentive mind giving birth to truth in a way more reliable than deduction. As a result, intuition provided propositions that had to be chosen as axioms and postulates. As an example of an absolutely clear, intuitive statement, Descartes suggested the equality 2 + 2 = 4. However, as it happened with the statement of Kant that the Euclidean geometry is intuitively unique, this statement of Descartes was invalidated with the discovery of non-Diophantine arithmetics (Burgin, 1977; 1997g; 2007; 2010c). Baruch Spinoza (1632–1677) elaborated the threefold division of knowledge: • Knowledge from imagination; • Knowledge from intuition; • Knowledge from the intellect. We see that for Spinoza, intuition is one of the basic sources and engines of knowledge. Bunge (1962) suggested that intuition in the sense of Spinoza consisted of the fast inference of conclusions. However, Spinoza also treats intuitive knowledge as transition from an adequate idea of certain attributes of God to the adequate knowledge of the essence of things (Parkinson, 1954). Similar to Spinoza, Immanuel Kant (1724–1804) envisaged a triadic structure of knowledge creation and acquisition coming from
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three engines of knowledge: • Empirical intuition; • Reasoning; • Pure intuition. According to Kant, intuition is a conscious, objective representation, which is truly different from any sensation, while the latter is not a representation of something but only a state of the subject. In addition, intuitions are passive representation, by means of which sensibility enables sensations. Concepts are also representations, which are general and mediate, while intuitions are singular, immediate representations. For Kant, experience is the mixture of an intuition with a concept in the form of a judgment. Thus, we see that that in the epistemology of Kant, intuition plays a fundamentally important role in knowledge creation. For Henri-Louis Bergson (1859–1941), the doctrine of intuition is of the innermost importance. For instance, he contends that philosophical intuition is the instrument of metaphysical knowledge and is the center of all philosophical work, or at least, of all philosophy which deserves this name. It is intuition, not reasoning, which brings people into sympathetic acquaintance with mental reality. In contrast to the misleading shell of common knowledge, which is framed for action, intuition enables people to understand the intrinsic properties of the world, as well as the external and relative character of the impressions and thoughts about the world. Bergson describes intuition as a simple, indivisible experience through which people transcend into the inner nature of an object to comprehend what is unique and indescribable within it. The absolute essence that is grasped always perfectly correlates with the object and is infinite because being grasped as a whole through a simple, indivisible act of intuition, it encloses boundless representations when analyzed (Bergson, 1923; 2007). Intuition is especially important for metaphysics, which is, according to Bergson, the science that dispenses with symbols to grasp the
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absolute because metaphysics involves an inversion of the habitual modes of thinking and needs its own method (Bergson, 1923; 2007). Edmund Gustav Albrecht Husserl (1859–1938), who was also intensely interested in intuition, differentiated between different kinds of intuition. The way we know sensible objects he called sensible intuition. At the same time, immediately valid judgments, which serve as the foundation of all proofs required in genuine science, derive their validity from originally presentive intuitions. Intuition displays aspects of reality and not only impressions of these aspects. It is adequate because it provides the interface of consciousness and of the object of consciousness. At the same time, intuition does not presuppose for Husserl any special source of insight or knowledge or any particular human capacity. The famous psychologist Carl Gustav Jung (1875–1961) considered four way of knowledge creation and reality comprehension: — — — —
Sensation Intuition Thinking Feeling
Jung understood intuition as a deeper perception of inherent possibilities and inner meanings when intuitive perception ignores the details and focuses instead upon the general context or environment adding meaning by encapsulating things into the situation that are not immediately apparent (Jung, 1969). In his philosophical and methodological writings, the great mathematician Jules Henri Poincar´e (1854–1912) explained the important role played by intuition in creation of mathematical and physical knowledge. He assumes that there is a specific mathematical intuition, which allows finding hidden harmony and relations (Poincar´e, 1908). Giving many examples, Poincar´e affirm that it impossible to create new mathematical knowledge without this mathematical intuition.
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In addition, Poincar´e specifies several sorts of the mathematical intuition (Poincar´e, 1905): — Intuition based on feelings and imagination; — Intuition of generalization by induction, which similar to intuition in natural sciences; — Intuition of pure number, which serves as a base for mathematical induction. In a similar way, the prominent French mathematician Jean Alexandre Eug`ene Dieudonn´e (1906–1992) described the unifying role of the mathematical intuition: “Mathematics has less than ever been reduced to a purely mechanical game of isolated formulas; more than ever does intuition dominate in the genesis of discoveries”. (Dieudonn´e, 1975)
Other prominent mathematicians Hilbert and Cohn-Vossen (1952) write that the mathematical knowledge progresses is understood and learned through a common, simple, highly integrated creative process involving both intuitive insight and intellectual analysis. G¨ odel (1906–1978) discussed problems of intuition with a considerable attention promoting the view that there is a faculty called rational intuition, which resembles sense-perception but is directed towards abstract ideas rather than concrete bodies (G¨ odel, 1947). Rational intuition as applied specifically to mathematical concepts is usually called mathematical intuition. G¨ odel called mathematical intuition applied specifically to set-theoretic concepts by the name set-theoretic intuition. The notable philosopher Thomas Samuel Kuhn (1922–1996) assumes that paradigm change is a value change and while normal science can articulate a paradigm but cannot change it, only intuition can change paradigms: “Paradigms are not corrigible by normal science at all ... normal science ultimately leads only to the recognition of anomalies and to crises. And these are terminated, not by deliberations and interpretation, but by a relatively sudden and unstructured event like the gestalt switch. Scientists then often speak of the “scales falling from the eyes” or of the “lightning flash” that “inundates” a previously obscure puzzle. On other
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occasions, the relevant illumination comes in sleep. No ordinary sense of the term “interpretation” fits these flashes of intuition through which a new paradigm is borne.” (Kuhn, 1962).
It is also necessary to remark that intuition is the cornerstone of a notable direction in mathematics, which is called intuitionism. According to intuitionism, mathematics is a free creation of the human mind based on intuition, and a mathematical object or property exists if and only if it can be physically or mentally constructed. The basic principle of intuitionism is the primordial intuition of natural numbers and mathematical induction (cf., (Fraenkel and Bar-Hillel, 1958)). That is why some researchers and other people discard intuition arguing that it is unreliable and lacks scientific explanation. Discussing the prejudice many have against intuition and in favor of perception, Sosa states, “opposition to the reliability of intuition appears to involve a self-defeating appeal to intuition” (Sosa, 2006). Besides, as we have seen, perception also involves intuition — the, so-called, perceptual intuition. In addition, Sosa suggests an analogy between intuition and eye-witness testimony, which shows that observers in general and witness, in particular, are frequently mistaken about their perceptions because witnesses to the same event can give radically varying accounts of it. This analogy shows that intuitions is as subjective as eye-witness accounts, and the difference in intuitions among persons happens as often as divergence of eyewitness testimonies. In spite of this, intuitions can be and are relevant in many cases and the epistemic role of intuition is not easily filled by other cognitive abilities and sources of knowledge (Sosa, 2006; Gilbert, 2008). Philosophers and mathematicians considered different types of intuition. Parsons (2008) distinguishes intuition of from intuition that. For instance, it is possible to have an intuition of a straight line in the Euclidean plane without having an intuition that given any straight line and any point beyond this line, there is one and only one straight line parallel to the given straight line.
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Burgess (2014) contemplates mathematical intuition as a kind of rational intuition with three forms: — The set-theoretic intuition is intuition related to set-theoretic concepts. — The geometric in the sense of G¨odel, intuition supports the belief that the three-dimensional Euclidean space correctly represents a certain structure existing in the realm of mathematical objects. — The chronometric intuition is intuition related to time. In addition, Burgess separates two forms of empirical geometric intuition: — The spatial intuition supports beliefs about the physical space. — The temporal intuition is intuition related to physical time. Fraenkel and Bar-Hillel (1958) consider two types of intuition: the primordial intuition of natural numbers and mathematical induction and the global intuition, which allows one to determine when two symbols coincide or have the same type. Bunge (1962) suggests much more types of intuition, which are named and listed below: 1. Perceptual intuition is immediate identification of a thing, phenomenon or symbol. 2. Comprehension intuition is clear understanding of the meaning and/or interrelations of system of symbols such as a text or a diagram. 3. Interpretation intuition is easiness of interpretation of conventional signs and symbols. 4. Geometrical intuition is the ability to envisage absent things and construct visual models and schemas. 5. Metaphoric imagination is the ability to apprehend and develop metaphors. 6. Creativity intuition is interpreted as creative imagination. 7. Reasoning intuition is interpreted as high-speed reasoning. 8. Synthetic intuition is the ability to easily synthesize different elements, systems, and objects in a unified system.
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9. Common sense intuition is the ability to make decision without utilization of scientific knowledge or sophisticated reasoning. 10. Practical intuition is the ability to make sound judgments and estimates. It is possible to consider three types of intuitive knowledge/ information production: — Analogy; — Extension or generalization; — Guessing. In this context, extension is a form of logical inference from less general to more general. To understand neurophysiologic foundation of intuition, we utilize the Componential Triune Brain model (CTBM) developed in (Burgin, 2010), which is the further development of the Triune Brain model (TBM) introduced and studied by MacLean (1913–2007). The main conception of the TBM is existence of three levels of perception and action that are controlled by three corresponding centers of perception in the human brain (MacLean, 1973; 1982). These three centers together form the Triune Brain with the structure of a triad constituting the neural basis, or framework, of the brain consists of three parts: the spinal cord, hindbrain, and midbrain. Besides, centuries of evolution have endowed people with three distinct cerebral systems (cf., Figure 7.1). The oldest of these is called the reptilian brain or R-complex. It programs behavior that is primarily related to instinctual actions based on ancestral learning and memories, satisfying basic needs such as self-defense, reproduction, and digestion. The reptilian brain is fundamental in acts such as primary motor functions, primitive sensations, dominance, establishing territory, hunting, breeding, and mating. Through evolution, people have developed a second cerebral system, the limbic system, which MacLean refers to as the paleomammalian brain and which contains hippocampus, amygdala, hypothalamus, pituitary gland, and thalamus. This system is situated
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Neomammalian Paleomammalian
Reptilian
Figure 7.1. The triune brain
around the R-complex, is shared by humans with other mammals, and plays an important role in human emotional behavior. The most recent addition to the cerebral hierarchy is called the neomammalian brain, or the neocortex. It constitutes 85% of the whole human brain mass and receives its information from the external environment through the eyes, ears, and other organs of senses. This brain component (neocortex) contains cerebrum, corpus callosum, and cerebral cortex. The cerebrum and cerebral cortex are divided into two hemispheres, while the corpus callosum connects these hemispheres. The neocortex deals with information in a logical and algorithmic way. It governs creative and intellectual functions of people, such as social interaction and advance planning. The left hemisphere works with symbolic information, applying step-by-step reasoning, while the right hemisphere handles images processed by massively parallel (gestalt) algorithms. Even psychologists who have objections to the Anatomic Triune Brain model admit that it is a useful, although oversimplified, metaphor, as the structure presented as the triune brain is based on a sound idea of three functional subsystems of the brain. In the development of neurophysiology and neuropsychology, MacLean’s theory was used as a base for the Whole Brain model, developed
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by Herrmann (1990). The main idea of this development is a synthesis of the Anatomic Triune Brain model with the two-hemisphere approach to the brain functioning. The theory of the triune brain (reptilian, old mammalian and new mammalian) is used as a metaphor and a model of the interplay between instinct, emotion, and rationality in humans. Cory (1999) applied this model to economic and political structures. In Cory’s schema, the reptilian brain mediates the claims of self-interest, whereas the old mammalian brain mediates the claims of empathy. If selfish interests of an individual are denied for too long, there is discontent due to a feeling of being unjustly treated. If empathic interests are denied for too long, there is discontent due to guilt. In either case, the center of intelligence at the prefrontal cortex plays the role of a mediator. Its executive function is required to restore balance, generating the reciprocity required for effective social and economic structures. The TBM is used to explain hyperactivity of youngsters studied by Zametkin (1990) and other researchers. Peter Levine base his approach to trauma treatment on the TBM (Levine, 1990). According to Levine, there are three types of the uniform stress and relaxation responses to a threatening situation that are active in all animal species through the autonomic nervous system. In the everyday language, these responses are metaphorically called fight, flight, or freeze. The first two of them are well known, while the third one was introduced by Levine. The freezing or immobility response has evolved over millions of years and it has served an adaptive purpose well for all species — except humans. In an individual, it can lead to trauma. Many physical ailments are actually residues of thwarted trauma reactions incurred during stressful events. What usually happens to non-human species is that after the threatening situation resolves itself, the animal forgets the stress and goes on its way without being traumatized. In contrast to this people can get stuck in the freezing response, while the reasoning mind resists or blocks the natural bodily sensations and fine motor movements needed to come out of the freeze response. The contemporary rationalistic culture is not helpful in
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supporting people in such a traumatizing situation. The feelings that people go through after experiencing a traumatic event are outside of their voluntary control, often being frightening and even potentially re-traumatizing. Levine (1999) postulates that trauma exists not in the event or in the story of the event, but is stored within the nervous system and thus, is expressed in definite reactions and behaviors. The main principle of the Levine’s treatment approach is that the body has a natural, innate, and miraculous capacity to heal once these reactions and behaviors are understood and guided. Although the Triune Brain has become a well-known model in contemporary psychology, it caused several objections on the ground of the development and structure of the triune brain system. First, there is evidence that the, so-called, paleomammalian and neomammalian brains appeared, although in an undeveloped form, on much earlier stages of evolution than it is assumed by MacLean. Second, there are experimental data that in the neocortex, regions that are homological to the, so-called, paleomammalian and reptilian brains exist and perform similar kinds of information processing. For instance, neuropsychological data give evidence that amygdala, which is a part of a limbic system, performs the low-level emotion processing, while the ventromedial cortex performs the high-level emotion processing. This shows that emotions exist, at least, on three levels: on the subconscious level of limbic system, on the conscious intuitive level, and on the conscious rational level in the cortex. The first level utilizes direct affective information, while the second and, to some extent, the third levels make use of cognitive emotional information (Burgin, 2010). At the same time, the development of the system of Will demands inclusion of some regions that are not included into the R-complex (the reptilian brain in MacLean’s theory) into this system. It means that the centers of rational intelligence, emotion and will are not concentrated in three separate regions of the brain but are highly distributed among several components of the brain. Thus, it might be better to call them not centers but systems of intelligence, emotion, and will. This extension of functional characteristics results in
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System of rational intelligence (SRI)
System of emotions/affective states (SAS)
System of will and instinct (SWI) Figure 7.2. Three basic systems of the brain
the necessity to change the TBM making it more adequate to experimental data. All this brings us to the CTBM described in (Burgin, 2010), which portrays the brain as a structure consisting of the three basic systems (cf., Figure 7.2): • the System (or Component) of Rational Intelligence (also called the System of Reasoning) (SRI) • the System (or Component) of Emotions (or more generally, of Affective States) (SAS) • the System (or Component) of Will and Instinct (SWI). Note that in contrast to MacLean’s TBM, the Componential Triune Brain model does not assume that each basic system belongs only to one anatomic part of the brain. These systems can cover regions of different anatomic parts of the brain, e.g., system of emotions includes amygdala, which is a part of a limbic system, and the ventromedial cortex, and even have common fragments. All three systems of the brain are structural schemas in the sense of the schema theory, which is developed as a specific direction of the brain theory (Anderson, 1977; Arbib, 1992; 2005; Armbruster, 1996; Burgin, 2005a; 2006; 2010a). According to this theory, brain schemes interact in a way of concurrent competition and coordination. All
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these interactions are based on physical processes but have an inherent informational essence related to a specific type of information. Information processes in the brain are more exactly reflected by the theory of the triadic mental information than by the conventional information theory that deals only with cognitive information. At the same time, dynamic brain schemas are non-phenomenal representations of mental knowledge. It is natural to call the initial MacLean’s structure by the name the Anatomic Triune Brain model because it is based on the anatomy of the brain where three indispensable parts are distinguished: the neocortex, limbic system and R-complex. In standard structuring of the brain, we also find these three systems. In the conventional setting (cf., for example, (DeArmond et al., 1989; Russell, 1992)), the brain includes three components: the forebrain, midbrain, and hindbrain. The forebrain is the largest division of the brain involved in a wide range of activities that make people human. The forebrain has a developed inner structure. It includes the cerebrum, which consists of two cerebral hemispheres. The cerebrum is the nucleus of the system (center) of rational intelligence. Under the cerebrum is the diencephalon, which contains the thalamus and hypothalamus. The thalamus is the main relay center between the medulla and the cerebrum. The hypothalamus is an important control center for sex drive, pleasure, pain, hunger, thirst, blood pressure, body temperature, and other visceral functions. The forebrain also contains the limbic system, which is directly linked to the experience of emotion. The limbic system is the nucleus of the system (center) of emotions (or more generally, of affective states). The midbrain is the smallest division and it makes connections with the other two divisions — forebrain and hindbrain — and alerts the forebrain to incoming sensations. The hindbrain is involved in sleeping, waking, body movements, and the control of vital reflexes such as heart rate, blood pressure. The structures of the hindbrain include the pons, medulla, and cerebellum. The hindbrain is the nucleus of the system (center) of will and instinct.
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The SRI realizes rational thinking. It includes both symbol and image processing, which go on in different hemispheres of the brain. The SAS governs sensibility and the emotional sphere of personality. The SWI directs behavior and thinking. Two other systems influence behavior only through the will. For instance, a person can know that it is necessary to help others, especially, those who are in need and deserve helping. However, in many cases, this person does nothing without a will to help. In a similar way, we know situations when an individual loves somebody but neither tells this nor explicitly shows this due to an absence of a sufficient will. It is necessary to remark that discussing the will of an individual we distinguish conscious will, unconscious will, and instinct. All of them are controlled by the SWI. In addition, it is necessary to make distinctions between thoughts about intentions to do something and the actual will to do this. Thoughts are generated in the SRI, while the will dwells in the SWI. In other words, thoughts and words about wills, wishes, and intentions may be deceptive if they are not based on a will. Will is a direct internal injunction, as well as any kind of motivation. That is, the forces that act on or within an organism to initiate and direct behavior, has to be transformed into a will in order to cause the corresponding action. The will is considered as a process that deliberates on what is to be done (Spence, 2000). The CTBM is not only a necessary extension of the TBM but it also continues a long standing approach to the brain stratification. As Smith (2010) demonstrates, triadic models of the brain and psyche have featured through two and half millennia of Western thought, starting with works of Pythagoras, Plato, and Aristotle and receiving a modern airing in Paul MacLean’s the TBM. A generation later after Pythagoras, Plato and Aristotle, Herophilus and Erasistratus from Alexandria put together a more anatomically informed triadic theory, which was modified by Galen in the 2nd century and remained the prevailing paradigm for nearly 1,500 years until it was overturned by the great thinkers of the Renaissance. Nonetheless, the notion that the human neuropsychological system is somehow best thought of as having a triadic (tripartite) structure has remained
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remarkably resilient and has reappeared time and again in modern and early modern times. For instance, the TBM well correlates with the Freud’s model of personality, which has the structure of the triad (Id, Ego, Super-ego). In the correspondence between the TBM and the Freud’s model, the reptilian complex corresponds to Id, the limbic system corresponds to Ego, and the neocortical complex corresponds to Super-ego. In this venue, it is also possible to consider the triarchic theory of intelligence developed by Robert Sternberg (1985). Taking each of these the centers, SRI, SAS, and SWI, as a specific infological system, we find three types of information. One is the conventional information that acts on the center of reasoning and of other higher brain functions (SRI), which is situated in the neocortex. This information gives knowledge, changes beliefs and generates ideas. Thus, it is natural to call it cognitive information. Information of the second type acts on the SAS, which includes the paleomammalian brain. It is natural to call this information by the name direct emotional information, or direct affective information or emotive information. Information of the third type acts on the SWI, which contains the reptilian brain. It is natural to call this information by the name direct regulative or direct effective information. Thus, anthropic information has three dimensions: Cognitive information changes the content of the SRI, which includes the knowledge system (thesaurus) and neocortex (neomammalian brain) as its carrier. Direct emotional/affective information changes the content of the SAS, which includes the paleomammalian brain (limbic system). Direct regulative/effective information changes the content of the SWI, which includes the reptilian brain or R-complex. However, in general, emotions constitute only one part of affective states, which also include moods, feelings, etc. That is why in general, direct affective information is more general than direct emotional information. However, as there is no consensus on the differences between emotions and affective states, these two types of information are used without differentiation. Interactions between the basic brain systems imply dependencies between thinking, emotions, and actions of people. Emphasizing
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some of these relations, psychologists build their theories and psychotherapists develop their therapeutic approaches. Giving priority to the SRI, the so-called “cognitive revolution” has taken hold around the world. It influenced both psychology, resulting in the emergence of cognitive psychology (Neisser, 1967; 1976), and cognitive psychotherapy, inspiring creation of cognitive therapy (Beck, 1995). The main postulate in these areas is that all processes in the human psyche are information processes. In psychology, the word cognitive often means thinking in many contexts of contemporary life (Freeman and DeWolf, 1992). The cognitive therapeutic approach begins by using the extremely powerful reasoning abilities of the human brain. This is important because our emotions and our actions are not separate from our thoughts. They are all interrelated. Thinking (SRI) is the gateway to our emotions (SAS) — and our emotions are the gateway to our actions through motivation and will (SWI). This is only another way of saying that information from the SRI goes to the SAS — and from it to the SWI that controls our actions. Consequently, the cognitive psychotherapeutic approach, which has been successfully utilized for treating many mental disorders, gives additional supportive evidence for the theory of the triune brain and behavior, as well as for the theory of the triadic mental information. The latter explains that while going from the SRI to the SAS and to the SWI, information is transformed from cognitive information, to direct emotional/affective information to direct effective/regulative information. As a result, we have interaction processes of the personality components of the Componential Triune Brain presented in Figure 7.3. An interesting connection between the TBMs and the Stoic philosophy have been found recently. The prominent Roman–Greek Stoic philosopher Epictetus (ca. 55–135) suggested that the apprentice philosopher should be trained in three distinct areas or topoi (Epictetus, 2008): 1. Desires (orexeis) and aversions (ekkliseis); 2. Impulse to act (hormas) and not to act (aphormas); 3. Freedom from deception, hasty judgment, and anything else related to assents (sunkatatheseis).
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Reasoning
Emotions/Affective states
Will/instinct
Behavior Figure 7.3. Interaction between components of personality
These three areas of training exactly correspond to the three types of philosophical discourse referred to by earlier Stoics (cf., (Diogenes Laertius, 1991)): 1. The logical is the topos, concerning assent to an impression or the value-judgment. 2. The ethical is the topos, concerning impulse (hormˆe). 3. The physical is the topos, concerning desire (orexis). Naturally, the logical is directly related to the SRI, the ethical is associated with the SAS, and interpreting desire as will or instinct, the physical is linked to the SWI. The structure of Stoic philosophy also correlates with the Existential Triad of the world (Section 2.2). Indeed, the logical belongs to the World of Structures, the physical naturally belongs to the Physical World and the ethical, as you would expect, belongs to the Mental World. Analyzing three ways of intuitive knowledge/information production considered above, i.e., by subconscious contemplation, by guessing or by emotions, e.g., the so-called, gut feeling, we see that the Componential Triune Brain allows us to explicate three mental capacities, which form the operational base of intuition: • Unconscious deliberation in the system of mental reasoning, i.e., in the SRI. • Emotional intelligence in the sense of (Burgin, 2010), which functions in the SAS.
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• Embodied instincts and instructions for control guessing based on the SWI. These types constitute only inner intuition, while many people speak and write about intuition from external sources — from God, from the Cosmos/Universe or the Earth, and from extraterrestrials. Intuition is also extremely important in everyday life because it allows utilization of partial knowledge without inference or the use of reason. In this context, intuition is associated with instinct combined with innate knowledge, i.e., it is related to the SWI, which is linked to ethics and knowing instinctively the ‘right’ and the ‘wrong’ way to behave. There are approaches to formalization of intuitive information/ knowledge production. An attempt to make guessing grounded gave birth to probability theory created in the 17th century by Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665). The goal of probability introduction was changing some types of intuitive guessing to reasoning with probabilities (cf., for example, (Burton, 1997)). For a couple of centuries this reasoning was not formalized and only in the 19th century, the first attempts to synthesize logic with probability started. The notable logician George Boole aimed to reconcile classical logic (which tends to express complete knowledge or complete ignorance) and probability theory (which tends to express partial or/and imprecise knowledge or ignorance), introducing imprecise probability (Boole, 1854). His approach represented subjective interpretation of probabilities because people often do not have enough information to assign definite numbers to probabilities of given events. In the 20th century, various probabilistic logics also called probability logics were introduced with the goal to combine the ability of probability theory to work with uncertainty with the capacity of deductive logic for formal inference — deduction. In particular, in probabilistic logics, truth values are not confined only to two values — true and False — as in traditional logics but are probabilities of being a true expression. In comparison with conventional logics, probabilistic logics provide richer and more expressive formalisms with a broad range of possible application areas. However, a richer formalism brings difficulties with probabilistic logics. First,
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they bring higher computational complexity of processing of their probabilistic and logical components. Second, there is a possibility of counter-intuitive results. Another approach to formalization of intuitive information/ knowledge production is based on fuzzy logics and linguistic variables (Bandemer and Gottwald, 1996), as well as fuzzy logical varieties and prevarieties (Burgin and Rybalov, 2003). Note that, in essence, all scientific methods of cognition demand intuition. Indeed, experimentation need intuition what experiments it would be useful to conduct and how to organize these experiments. In many cases, existing rules are insufficient. Observation also needs intuition for making right selection from all observed events — not all of them are equally important for the definite case. Reasoning also employs intuition. It is evident for abduction where there no formal rules for generation and selection of possible explanations and rational intuition can direct the researcher in the right way, or mislead her or him. What concerns induction, we have already indicated that, as Poincar´e (1905) stressed, induction always involves a specific kind of intuition. Even deduction engages intuition in many cases. For instance, in non-monotonic logics, knowledge exchange and selection, which demand intuition, precede deduction (cf., Section 3.3). In a similar way, performing deduction in logical varieties and prevarieties, intuition is helpful for choosing the right component for this operation (cf., Section 3.3). To conclude, it is necessary to remark that although psychologists, mathematicians, and scientists studied different forms of intuition, they have not achieved a sufficiently full understanding of this phenomenon. 7.1.3. Computers and networks as cognitive tools The good news: Computers allow us to work 100% faster. The bad news: They generate 300% more work. Unknown
One of the most popular directions in application of computers to obtaining knowledge is knowledge discovery in databases (KDD).
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It is not coincidental. As Wright (1998) explains, the amount of data being collected in databases today far exceeds our ability to reduce and analyze data without the use of automated analysis techniques. Many scientific and transactional business databases grow at a phenomenal rate. Databases are increasing in size in two ways: (1) the number N of records or objects in the database grows and (2) the number d of fields or attributes to an object increases (Fayyad et al., 1996). Databases where the number of objects, the order is more than 100 are becoming increasingly common, for example, in the astronomical sciences. At the same time, the number of fields has the order more than 100 occur in many databases, for example, in medical diagnostic applications. That is why KDD emerged as the field that is evolving to provide people an efficient approach to extract knowledge from these volumes and volumes of data. KDD is aimed at an essential growth of the efficiency of information processing in comparison with direct data mining. KDD is a growing field and there are many knowledge discovery methodologies in use and under development. However, the majority of these methodologies are empirical and deficiency of sound theoretical foundations for KDD prevents from achieving even sufficient efficiency in work with knowledge by means of computers. Let us analyze the process of data transformation into knowledge. Knowledge is achieved through information retrieval, which in its turn is based on data collection, mining, and analysis. The amount of data being collected in databases today far exceeds our ability to reduce and analyze data without the use of automated analysis techniques. Many scientific and transactional business databases grow at a phenomenal rate. KDD is the new field that is evolving to provide automated analysis solutions. The term knowledge discovery in databases was coined at the first KDD workshop in 1989 to denote processes in which knowledge is the end-product, while data are the raw material (Piatetsky-Shapiro, 1991). A particular step in this process is called data mining, which is the application of specific algorithms for extracting patterns from data (Fayyad et al., 1996; Wright, 1998) although in many cases, KDD is not distinguished from data mining.
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Different methods of data mining have been developed to deal with such amounts of data. However, data mining, while giving more generalized and/or adequate data, does not provide knowledge per se. As an example we can consider data mining on the Internet. This process provides large amounts of data to a user (some relevant, some not), but the user has himself/herself to convert these data into knowledge. Knowledge is formed only inside some knowledge system. It may be the mind of a user or an automated knowledge system on a computer. In addition, before data can become knowledge, it is necessary to extract from these data appropriate information. KDD provides the capability to discover new and meaningful information by using existing data. KDD quickly exceeds the human capacity to analyze large data sets. The amount of data that requires processing and analysis in a large database exceeds human capabilities, and the difficulty of accurately transforming raw data into knowledge surpasses the limits of traditional databases. Therefore, the full utilization of stored data depends on the use of knowledge discovery techniques. As the first approximation, we develop a model of knowledge discovery, which consists of three levels: data mining, information retrieval, and knowledge formation (cf., Figure 7.4). Each of these three levels may be divided into several sublevels. For instance, data mining is performed on the level of raw data, level of interpreted data, and level of attributed data. Separation of data mining, information retrieval, and knowledge formation, as well as their sublevels, is based on the general theory of information (Burgin, 2010) and Data input
Data mining
Information retrieval
Knowledge formation
Knowledge output
Figure 7.4. The triadic model of KDD
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system theory of knowledge presented in this book. Mathematical techniques from the theory of named sets, category theory, and general theory of properties are utilized. Each of these processes, data mining, information retrieval, and knowledge formation, is an essential component of information processing, both by people and computers. Data mining provides a base for information retrieval, which, in its turn, allows knowledge formation and production. Under their conventions, the information retrieval process takes the raw results from data mining (the process of extracting trends or patterns from data) and carefully and accurately transforms them into useful and understandable information. This information is not typically retrievable by standard techniques but is uncovered through the use of AI techniques. In its turn, the retrieved information is transformed into knowledge through the knowledge discovery process, which places information into a definite knowledge system, which is coordinated with the knowledge system of a user. Thus, we have the triadic dynamic model of knowledge discovery presented in the following diagram (Burgin and Gantenbein, 2002): This stratification of knowledge discovery is based on principal distinctions between data, information and knowledge. Consequently, there is a difference in working with data, processing information, and acquiring or producing knowledge. This shows that the definition of knowledge discovery as “the non-trivial extraction of implicit, unknown, and potentially useful information from data” given in (Frawley et al., 1991) is essentially incomplete. A more detailed stratification of KDD is developed in (Fayyad et al., 1996). In their model, data are treated as a set of facts (e.g., cases in a database), while a pattern is an expression in some language describing a subset of the data or a model applicable to the subset. The KDD process is interactive and iterative, involving numerous steps, iterations, and loops demanding many decisions from the user.
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Knowledge Stages 8 and 9: Interpretation/evaluation Patterns
Stage 7: Data mining Algorithms, models and methods Stages 5 and 6: Selection of algorithms, models and methods
The transformed data Stage 4: Data transformation, which includes reduction The preprocessed data Stage 3: Preprocessing, which includes cleaning The target data Stage 2: Data selection
Tools
The initial data
Goals
Stage 1: Identification of the application domain, goals, tools and initial database (or databases) for KDD The initial problem
Figure 7.5. The expanded model of KDD
The basic stages of the KDD process are (cf., Figure 7.5): The first stage is developing an understanding of the application domain, finding the relevant prior knowledge and identifying the goals of the KDD process from the customer’s viewpoint.
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The second stage is formation of a target entity in the database, i.e., either selection of a data set, or focusing on a subset of variables or data samples, on which knowledge discovery has to be performed. The third stage is data cleaning and preprocessing including such basic operations as (1) removing noise if appropriate; (2) collecting the necessary information to model or account for noise; (3) deciding on strategies for handling missing data and/or data fields; and (4) accounting for time-sequence information and known changes. The fourth stage is data and variables reduction and projection with the goal of finding useful features and invariant representations for the chosen data depending on the goal of the task. The fifth stage is matching the goals of the KDD process obtained on the first stage to a particular data-mining method, such as summarization, classification, regression, clustering, and so on. The sixth stage is an exploratory analysis with model and hypothesis selection, choice of the data-mining algorithms and selecting methods for data pattern searching. This process includes deciding which models and parameters might be appropriate, e.g., models of categorical data are different than models of vectors over the real numbers, and matching a particular data-mining method with the overall criteria of the KDD process, e.g., the end user might be more interested in understanding the model than its predictive capabilities. The seventh stage is data mining, which is searching for patterns of interest in a particular representational form or a set of such representations, including classification rules or trees, regression, and clustering. The user can significantly optimize the data-mining method by correctly performing the preceding operations. The eighth stage is interpreting mined patterns, visualizing the extracted patterns, models and/or data, obtaining knowledge, and possibly returning to any of the preceding stages for further iteration. The ninth stage is acting on the discovered knowledge by using the knowledge directly, incorporating the knowledge into another system for further action, or simply documenting it and reporting it to interested parties. This process also includes checking for and resolving potential conflicts with previously possessed or extracted knowledge.
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There are many different methods classified as KDD techniques and used for information retrieval and knowledge discovery. There are quantitative methods, such as the probabilistic and statistical approaches. There are KDD schemas that utilize visualization techniques (cf., for example, (Gantenbein and Sung, 2001)). There are classification approaches to KDD such as Bayesian classification, inductive logic, data cleaning/pattern discovery, and decision tree analysis. Other approaches include deviation and trend analysis, genetic algorithms, neural networks, and hybrid methodologies, which combine two or more techniques. Many of them are described and classified in (Wright, 1998). Other popular techniques of knowledge discovery and acquisition based on computers are pattern recognition, computer simulation, learning algorithms, and expert system technology. For instance, computer simulations are an essential part of all experiments in some fields, such as high-energy physics. It is possible to assert that without computer simulations many experiments in these fields would be impossible. In addition, computer simulations can essentially decrease the cost of experimentation and save environment from the negative impact of some physical experiments. Mathematicians also started to use computers in proving new theorems and finding properties of mathematical objects. The most famous result obtained with the help of computers was the Four-Color Theorem, according to which any map drawn on a plane can use only four colors without any two adjacent countries having the same color (Appel and Haken, 1977). In the process of proving this theorem, a computer checked through a large number of particular maps confirming that all of them satisfied the condition of the theorem. Another prominent mathematical result obtained with the help of computers is a proof of the famous Kepler conjecture. In 1611, the great astronomer and mathematician Johann Kepler (1571–1630) studied the problem of how to stack spherical balls, e.g., apples or oranges, in the best way, i.e., so that they fill space as densely as possible. He found several ways to do this and conjectured that one of them is the best possible.
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In 1997, Thomas Hales of the University of Michigan announced a proof of this conjecture, which involved, like the famous computer proof of the 4-color theorem, computer checking of thousands of separate cases, many of them individually very laborious. The proof contained over 3 gigabytes of computer code and data. To justify this result, it was necessary to check it (cf., Chapter 3). However, the manuscript with the proof contained around 250 pages. With all codes and formulas, it was hard to do for one mathematician. So, a committee of 12 experts was appointed to verify the result. They started to work but after 4 years, the committee announced that they had found no errors, but still could not confirm the correctness and gave up. This well demonstrates existing problems with computerized proofs of complex mathematical results. In addition to using computers as assistants in their work, mathematicians and computer scientists has been developing automated systems for theorem proving, deduction, and general reasoning. It is necessary to remark that automated theorem provers also found applications beyond mathematics, for example, in validation of software and hardware correctness. However, inside mathematics, achievements of automated theorem provers have been very moderate. At the same, mathematicians, computer scientists and psychologists started developing computational cognition as the computational basis of learning and inference trying to find foundations of cognitive processes and utilizing mathematical modeling, computer simulation, and behavioral experiments. It is also necessary to emphasize that the Internet has become a powerful depository of information and thus, a useful cognitive tool. Access to immense amounts of information allows the Internet users to successfully create new knowledge. One more direction in application of computers to cognition is scientific and cognitive visualization (cf., for example, (Butler and Bryson, 1992; Hartley and Barnden, 1997; Burgin, 1997f; Burgin et al., 2001; 2001a; 2001b; Gantenbein and Sung, 2001; Liu et al., 2001; Zellweger, 2011)).
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7.1.4. Learning “The most erroneous stories are those we think we know best — and therefore never scrutinize or question.” Stephen Jay Gould
Traditionally learning is understood as the process of knowledge acquisition or knowledge transfer from one individual to another or from a book to an individual. There is also a much broader interpretation of learning. For instance, Tuomi (1999) suggests that learning is synonymous to internalization of new knowledge, creation of knowledge or/and the development of new skills. However, this broad approach makes learning equivalent to cognition in contrast to the customary utilization of these two terms. To avoid blending of these terms, we use two definitions. Definition 7.1.13. Learning in the exact sense is internalization of knowledge transferred from another carrier of knowledge, e.g., another person, organization, textbook, or the Internet. Learning of students at a college or university is learning in the exact sense. Note that this definition includes the development of new skills because it is natural to treat skills as embodied operational knowledge. This definition also includes learning not only of individuals but also learning of groups, organizations, and societies. Definition 7.1.14. Learning in the broad sense is internalization of knowledge created, discovered or obtained by the knower. According to this definition, it is possible to learn from experience or from nature. Bearing in mind learning in the broad sense, Tuomi (1999) develops a classification of modes, sources and processes of ontogenetic learning, which is presented in Table 7.1 with some modifications. The given definition, allows us to discern different types and levels of learning: • The Xerox learning is simple acquisition of knowledge from another system.
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Knowledge Production, Acquisition, Engineering, and Application Table 7.1. Modes, sources, and processes of ontogenetic learning. A source of the behavioral change
Environment
Society
Self
Language Verbal mode
Generation of spontaneous empirical concepts and ideas
Training Generation of scientific concepts and ideas Participation in thought communities
Conceptual thinking
Knowledge processing Self-referential non-linguistic mode
Experience Empirical experiment
Reflective socialization
Imagination
Organism Non-referential non-linguistic mode
Habit formation Skill acquisition
Tacit socialization
Intuition
• The comprehensive learning is acquisition and understanding of knowledge from another system. • The functional learning is acquisition of knowledge from another system and developing skills for its application. • The articulate learning is acquisition of knowledge from another system and being able to explain (transmit) it to somebody else. • The behavioral learning is acquisition of knowledge from another system and expressing this knowledge in ones behavior/ functioning. For instance, we can observe the Xerox learning when somebody learns by heart a definite text. A teacher needs the articulate learning to be able to teach what she learned before. Professional learning has to be the functional learning because it is necessary to use what is learned in one’s professional activity.
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When students are taught in schools, colleges and universities, it is the behavioral learning because they have to express their knowledge in tests and quizzes. Differentiation between learners gives us the following classification: ◦ The individual learning is learning of an individual. ◦ The collective learning is learning of more than one individual. In turn, collective learning can be: ◦ The group learning is learning of a group where group can any assembly of people. ◦ The organizational learning is learning of an organization, i.e., of a structured group. ◦ The social learning is learning of society. Based on his types of knowledge (cf., Section 2.1), Tuomi (1999) introduces several types of learning: — The inter-generational learning is learning inside a generation of the learner. — The intra-generational learning is learning across different generations, e.g., formation of instincts (instinctive knowledge). — The cognitive or conceptual learning is internalization of cognitive knowledge. — The structural learning is acquisition of habits on the individual level and practices/routines on the organizational and social levels, as well as the development of instincts. — The ontogenetic learning is learning that takes place in the ontogenetic development. — The phylogenetic learning is learning that takes place in the phylogenetic development. Note that although traditionally instincts are linked only to individuals, it is possible to speak about instincts of crowds, organization and societies in the same way as we can speak about collective intelligence (Burgin, 2012).
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Learning is a kind of intellectual activity. That is why, three basic types of intellectual activity induces the corresponding typology of learning in the exact sense: — The reactive learning is selection of the given knowledge items and reproduction of the selected items in the memory of the learner. — The active learning intellectual activity is search for and extraction of the necessary knowledge. — The passive learning is reproduction of the given knowledge in the memory of the learner. In the case of learning in the broad sense, we have one more type: — The proactive learning, which involves creation of new knowledge. Gregory Bateson (1904–1980) introduced several levels of learning (Bateson, 1973): • Learning 0 means that there is no change, while the system is involved in repetitive behaviors of individuals, groups or organizations ‘inside the box’. It is characterized by specificity of response, which — right or wrong — is not corrected. • Learning I is gradual, incremental change involving corrections and adaptations through behavioral flexibility and extensions. While these modifications may help to increase the capabilities of individuals, groups or organizations, they are still ‘inside the box’. It is characterized by the modification of specificity of response by correction of errors in the choice of alternatives. • Learning II is a change in the process of Learning I, e.g., a corrective change in the set of alternatives from which the choice on the level I is made or a change in how the set of alternatives is organized. It is a fast and often discontinuous change involving instantaneous shifts of responses to an entirely different category or class of behavior and performing switches from one type of “box” to another such as changes in policies, values, or priorities. • Learning III is a change in the process of Learning II. It is characterized by momentous revisions, which go beyond the boundaries
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of the current identity of the individual, group, or organization. It is possible to say that the change is “outside a collection of boxes’. • Learning IV is a change in the process of Learning III, and in its essence, it is revolutionary change involving beginning of something completely new, unique and transformative. Researchers also study learning strategies. For instance, in (Molnar, 1997) six learning strategies are described: — memory learning strategy includes grouping and structured reviewing of information, as well as applying images and sound and employing action — compensation learning strategy involves guessing and inferring — cognitive learning strategy entails analyzing, reasoning, transferring information, taking notes, practicing, highlighting, and summarizing — meta-cognitive learning strategy includes organization, evaluation, and planning of learning by setting goals and objectives — affective learning strategy is based on emotions when learners regulate their emotional state — social learning strategy implies working with others and asking questions There are different models of the learning process. For instance, John Dewey (1859–1952), who by the opinion of many educators, was the most significant educational thinker of his time, suggested a model that is consistent with all types of learning but applies only to reactive learning. It consists of the following steps (Dewey, 1938): 1. 2. 3. 4. 5. 6. 7.
Interruption of the routine functioning. Problem definition and conceptualization. Working hypothesis formation. Inference and thought experiment for testing the hypothesis. Experimental action for testing the hypothesis. Problem solving as the confirmation of the hypothesis. Returning to the new routine functioning.
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We can see that for Dewey learning is not only knowledge acquisition but mostly solving a problem, while the solution may be not only new knowledge but also different behavior or new way of functioning. It is possible to ask a question how this schema works when the routine functioning is learning. For instance, a scientist has to learn all the time and not only when a new problem emerges. Besides, in the case of Xerox learning, there is no hypothesis formation because knowledge representation is routinely stored in the memory of the learner. Several factors influence the utilization of strategies by learners: — — — — — —
learner ability gender culture attitude motivation learner ability to manage the learning process
All considered above learning processes are based on a simple schema when the learner learns by receiving information from some source, e.g., from a teacher, instructor or textbook. In reality, learning can include more stages and involve more participants. To reflect these peculiarities, researchers introduced and studied the concept of iterated learning (cf., for example, (Kalish et al., 2007)). Iterated learning is intergenerational knowledge and information transmission in the social context of the cultural evolution. Cultural information transmission by iterated learning may explain why language is structured explicating key sources of its evolution. There is a mathematical model of iterated learning based on the constructive hierarchy of inductive Turing machines (Burgin and Klinger, 2004). This model demonstrates that deductive iterated learning strategies do not increase learning possibilities in comparison with individual learning. At the same time, inductive iterated learning strategies essentially increase learning abilities in comparison with individual learning.
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Learning is intrinsically connected to teaching. Researchers distinguish three basic approaches to teaching: — The teacher-centered approach is realized when the teacher is the main authority figure, while the primary role of students is passive information reception via lectures and direct instruction having the end goal of testing and assessment through objectively graded tests and assessments. — The student-centered approach is realized when the teacher and students play an equally active role in the learning process, while student achievements are measured through both formal and informal forms of assessment, including group projects, student portfolios, and class participation. — In the content-centered approach, the teacher makes an effort to give more information to the students making them to acquire (learn) more knowledge. In the teacher-centered approach, the teacher the sole supplier of knowledge and information, who can play the following roles of: — A formal authority who is in position of power focusing on rules and expectations — An expert who possesses all knowledge and expertise within the classroom — A role model who is leading by example demonstrating how to access information and obtain knowledge In the student-centered approach focuses on student investigation and hands-on learning when the teacher can play the following roles of: — A facilitator who place a strong emphasis on teacher–student relationships, — A delegator who plays a passive role in the student learning, — An organizer who organizes the process of the student learning. Discussing learning of individuals and organizations, it is necessary not to forget that contemporary learning technologies often
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involve computers and networks, e.g., the Internet (Keenan, 1964; Molnar, 1997). As Molnar (1997) states, “research shows that educational technology, when properly applied, can provide an effective means for learning.” There are different terms that denote utilization of computers in education: • • • • • • •
Computer Computer Computer Computer Computer Computer Computer
Assisted Instruction (CAI) Aided Instruction (CAI) Assisted Learning (CAL) Based Education (CBE) Based Instruction (CBI) Enriched Instruction (CEI) Managed Instruction (CMI)
When educators started using the Internet for teaching/learning/ training, new terms appeared: • Web Based Training • Web Based Learning • Web Based Instruction It is possible to discern three directions in utilization of computers for the purpose of learning/teaching/training: • The control technology when the computer plays the role of an instructor telling students what to do. • The teacher instrumental technology when the computer is used as a tool of an instructor. • The student instrumental technology when the computer is used as a helping device of a student, who is telling computer what she/he needs. These first two directions correlate with two teaching strategies: • The control technology corresponds to the teacher as an instructor. • The instrumental technology corresponds to the teacher as a facilitator.
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For instance, in programmed instruction, which is a kind of the control technology, the computer presents learning material to the student, often employing text, graphics, sound and video. In the process of presentation, questions are posed to the student, and based on her answers, the computer, or more exactly, the education software (program) makes a choice how to continue its presentation. In some cases, new topics are chosen for exposition, while in other situations, the computer repeats previous topics if they were not properly learned. The control technology is also used in the simulation assisted training when the student works with a simulation of the real world systems, processes, and situations. Simulation assisted training is used when it is not practical or feasible to provide the learning with real-life systems, for example, in preparatory pilot training. At the same time, the teacher can use computers as a tool for enhancing her teaching, for example, suggesting the students to participate in simulating experiments, when students explore simulations of the real experiments on the computer screen instead of doing this either in a laboratory or in the field. Educationally-oriented computer games are also used for enhancing teaching. Games generally include a competitive element reinforcing motivation of learning and knowledge that the student is assumed to have. Each of the two computerized teaching/training technologies have three specific forms. The control technology can have the following forms: • A collaborative instruction allows the student has a possibility to change the process of instruction/teaching. • A feedback-based instruction is organized taking into account the feedback the student gives in response to questions and problems posed by the computer. • A commanding instruction simply gives an exposition of the necessary material. The teacher instrumental technology corresponds to the teacher as a facilitator
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The teacher instrumental technology can have the form of: • Proactive interaction with the student • Active interaction with the student • Reactive interaction with the student In addition, there is an active research in the area of automated learning, or machine learning, when computers are themselves learning (cf., for example, (Weiss and Kulikowski, 1991)). To conclude, it is necessary to note that according to the general theory of information (Burgin, 2010), a learner cannot get knowledge directly, e.g., a teacher cannot give knowledge directly to her students. The learner, e.g., a student, obtains only information, which can be converted into knowledge or not. Only the learner can perform conversion (transformation) of information into knowledge and the teacher or a textbook is able to help or hinder the student in this process. 7.1.5. Knowledge creation in organizations Advances are made by answering questions. Discoveries are made by questioning answers. Bernhard Haisch
Nonaka et al. (2000) suggested their model of knowledge creation in organizations. It consists of three components: 1. The socialization, externalization, combination, internalization (SECI) process of knowledge creation through the conversion of tacit and explicit knowledge. 2. The shared context for knowledge creation called ‘ba’ in Japanese. 3. The utilized knowledge assets, such as the inputs, outputs, and moderators of the knowledge-creating process. According to this model, the knowledge creation process has the form of a spiral, which develops based on these three components and dialectical thinking as the leading mechanism of the process. The top and middle management of the organization has to generate
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ba efficient for knowledge creation. In such a way, using existing knowledge assets, an organization creates new knowledge through the SECI process that goes on in ba, while new knowledge, once created, becomes, in turn, the basis for the next stage of the knowledge creation process. The main steps in the SECI process are socialization, externalization, combination, and internalization. Socialization is the process of collecting, accumulating, and disseminating tacit knowledge. Members of the organization collect and accumulate knowledge from their professional activity, from daily social life, interaction with external experts and informal meetings with competitors outside the organization (external search), from their colleagues inside the organization (internal search) and grow craftsmanship and expertise through practice and demonstrations by a master. This process is called socialization because it goes on in social interactions. Externalization is the process of converting tacit knowledge into explicit knowledge by means of creative and essential dialogues, utilization of metaphors, abductive thinking of the members of the organization and knowledge sessions of experts (knowledge engineers) with the knowers, i.e., with the members of the organization who successfully perform their functions. Combination is the process of acquisition, integration, synthesis, processing, and disseminating explicit knowledge by planning strategies and operations, assembling internal, and external data by using published literature, computer simulation and forecasting, writing manuals, documents, and building databases on products and services. Internalization is the process of converting explicit knowledge into tacit knowledge and embodying it into the members of the organization in the form of shared mental models or technical know-how. In such a way, explicit knowledge created is shared throughout an organization becoming a valuable asset. For instance, documents and manuals contain explicit knowledge but without the necessary application skills of this knowledge, it usually remains useless. Skills are based on tacit operational knowledge. Therefore, only conversion of
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Tacit
T Socialization a c Empathizing i t
Internalization
T a Embodying c i t
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E x p l i c i t
Externalization Articulating
E x p l i c i t
Combination Connecting
Explicit
Explicit Figure 7.6. The SECI process
explicit descriptions from documents and manuals to tacit proficiency allows organization to make these documents and manuals truly useful. The whole SECI process is presented in Figure 7.6, which illustrates the four ways of knowledge conversion and the evolving spiral movement of knowledge through the SECI process. An important peculiarity of the SECI process is that it moves through the four modes of knowledge conversion not in a circle but in a spiral, which becomes larger in scale and scope as it moves up through the ontological levels. On any stage, created knowledge can set off a new spiral of knowledge creation, expanding horizontally and vertically inside and beyond the organization, starting at the individual level and growing as it moves through communities of interaction that transcend individual, sectional, departmental, divisional, and even organizational boundaries. Being properly organized, organizational knowledge creation is a never-ending process that accelerates and enhances itself continuously.
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However, this schema is incomplete because organizations and individuals also use two other knowledge creation processes: — Formation is the process of amalgamating explicit and tacit knowledge into tacit knowledge by means of learning from books, manuals and instructions to enhance individual and organizational skills. — Construction is the process of synthesizing explicit and tacit knowledge into explicit knowledge by means of creativity techniques and procedures, such as brainstorming, thinking outside the box, USIT, TRIZ, and morphological analysis. Let us look at definitions of these creativity techniques. Definition 7.1.15. Brainstorming is a group creativity technique by which efforts are made to find a conclusion for a specific problem by gathering a list of ideas spontaneously contributed by the members of the participating group, discussion of these ideas and selection of the most efficient ones. Definition 7.1.16. Thinking outside the box (also called thinking out of the box or thinking beyond the box) is a creativity technique, both group and individual, in which people try solving a problem in a more general context than it is originally given, thinking differently, unconventionally, or from a new perspective. Definition 7.1.17. USIT is an individual or group creativity technique, which is based on three fundamental components: objects, attributes, and the effects they support. There are two groups of effects: beneficial effects, called functions, or not beneficial effects, called unwanted effects. USIT consists of three common phases: — Problem definition in terms of objects, attributes, and effects. — Problem analysis, which has two forms: (1) the “closed-world” analysis of the problem to understand functional connectivity of objects, attributes and effects; and (2) the “particles method”, which starts from a possible solution and works back to the problem situation.
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— Application of solution strategies: utilization, nullification, and elimination of the unwanted effect. It is assumed that each phase takes equal time when the problem is solved. Definition 7.1.18. TRIZ (literally meaning a theory of the resolution of invention-related tasks), which also called the theory of inventive problem solving, and occasionally goes by the English acronym TIPS, is a problem-solving, analysis and forecasting tool developed by the Soviet inventor and science-fiction author Genrich Altshuller (1926–1998) and his colleagues, beginning in 1946. However, TRIZ is not only a theory but also includes a practical methodology, creative tools, a knowledge base, and model-based technology for innovative problem solving. Definition 7.1.19. Morphological analysis is a method for systematically structuring and investigating the total set of relationships contained in given problem complexes with the goal of solving a definite problem. It was developed by Fritz Zwicky (1898–1974) for exploring all possible solutions to a multidimensional, non-quantified complex problem (Zwicky, 1967, 1969). Researchers have also developed different mathematical tools for modeling, study, and improvement of knowledge processes in organizations. For instance, Heather and Rossiter (2009) apply category theory to problems of interoperability of knowledge systems, providing a powerful abstraction of the business process. To conclude, it is necessary to clarify a controversial issue of who creates knowledge. Some researchers insist that only individuals create new knowledge, while it cannot be created by groups, organizations and society. For instance, Nonaka and Takeuchi (1995) write: “In a strict sense, knowledge is created only by individuals. Organizational knowledge creation, therefore, should be understood as a process that ‘organizationally’ amplifies the knowledge created by individuals and crystallizes it as a part of the knowledge network of the organization.”
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However, this is a misconception. For instance, such a knowledge creation technique as brainstorming implies that the participating group solves the problem creating new knowledge. Besides, there are examples from the history of science and mathematics when a group solves a definite problem although the majority of problems have been solved by individual researchers. In many cases, the groups of scientists (mathematicians) who together solve some problem work also together. For instance, working together Marie Sklodowska Curie (1867–1934) and Pierre Curie (1859–1906) discovered two previously unknown elements — polonium and radium — creating new physical and chemical knowledge. However, there are situations when a group of people working separately creates the new knowledge. The famous example of such a situation is the discovery of the subatomic particle called positron by a group of two. History tells us that in 1928, the famous physicist Paul Dirac (1902–1984) introduced what was later called the Dirac equation. It unified quantum mechanics and special relativity describing the properties of the electron. However, calculations from this equation indicated two particles: one was the ordinary negatively charged electron and another was a positively charged particle with the mass as that of the negatively charged electron. Dirac did not neglect the negative answer suggesting existence of an unknown particle but nobody believed him and many physicists even mocked at him. However, in 1932, another physicist Carl Anderson (1905–1991) discovered a particles with the parameters predicted by the theory of Dirac. This shows that the new particle called positron and knowledge about it was discovered by the group of two physicists — Dirac and Anderson. At the same time, there is a class of situations when groups and, in some sense, organizations create new knowledge. This happened in the case of geographical discoveries because they were made when groups of people reached new lands. For instance, when Columbus came to America, new knowledge about this continent was created not only by Columbus but by all members of the ship crews that reached America, while any crew of a ship is an organization.
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7.2. Knowledge organization and engineering Real knowledge is to know the extent of one’s ignorance. Confucius
Engineering, in general, is defined as application of scientific principles to practical ends, such as the design, manufacture, and operation of structures and mechanisms. Consequently, knowledge engineering is application of scientific principles to practical ends associated with knowledge, such as the design, manufacture, maintenance, and operation of knowledge systems. In a similar way, Feigenbaum and McCorduck (1983) define knowledge engineering as an engineering discipline that involves integrating knowledge into computer systems in order to solve complex problems normally requiring a high level of human expertise. However, there other definitions of knowledge engineering, which belong either to the transfer approach or to the modeling approach. The definition of Feigenbaum and McCorduck is the objective for the Transfer Approach, which treats knowledge engineering as application of engineering techniques to transfer human knowledge into AI systems. It usually means that the human knowledge for solving a problem is transferred to the knowledge base assuming that experts already have this knowledge in the explicit form. Thus, the transfer approach disregards the tacit knowledge an individual possesses for solving the problem. This deficiency caused the paradigm shift towards the modeling approach, which is more adequate to reality describing problem-solving as a dynamic, cyclic, incessant process dependent on the knowledge acquired and the interpretations made by the system. Therefore, in the Modeling Approach, the goal of knowledge engineering is to include problem solving techniques and knowledge of the domain expert into AI systems modeling how an expert solves problems in real life (Schreiber et al., 2000). In the contemporary practice, knowledge engineering consists of construction, maintenance, and development of knowledge-based
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systems and is related to many computer science domains such as AI, databases, data mining, expert systems, decision support systems, and geographic information systems. Besides, knowledge engineering is also linked to mathematical logic, as well as to cognitive science and socio-cognitive engineering. Knowledge engineering includes various activities specific for the development of a knowledge-based system: • Assessment of the problem; • Development of a knowledge-based system shell/structure; • Acquisition of the relevant information, knowledge and specific preferences (IPK model) is the principal stage of the process; • Knowledge representation for which available representation schemas e.g., rules or semantic networks, are used; • Knowledge structuring; • Translation of the structured knowledge into knowledge bases; • Testing and validation of the inserted knowledge; • Integration and maintenance of the system; • Revision and evaluation of the system; • Inference of new knowledge; • Explanation and justification of new knowledge. Being still more art than engineering, knowledge engineering does not exactly follow the above list in practice. The phases overlap, the process might be iterative, and contains many cycles. One of the central goals of knowledge engineering is to identify an appropriate conceptual lexicon building an ontology. Ontologies allow building models of the knowledge domain and defining terms used inside the domain and the relationships between them. There are different types of ontologies such as domain ontologies, generic ontologies, application ontologies, and representational ontologies. Ontology engineering, or ontology building, is a subfield of knowledge engineering that studies the methods and methodologies for working with ontologies including ontology development processes, the ontology life cycle, the methods and methodologies for building ontologies, and the tool suites, and languages that support them. The
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goal of ontology engineering is extraction of knowledge from software applications, enterprise and business procedures, and organizational practice. For instance, ontology engineering develops approaches to solving the interoperability problems brought about by semantic obstacles, such as the obstacles related to the definitions of business terms and software classes. Another goal of knowledge engineering is building, maintaining, and using knowledge-based systems, such as expert systems, by extracting knowledge from a human expert and then translating this knowledge into a knowledge base and using various technical, scientific and social tools. Knowledge engineers have developed a number of principles, methods and tools to improve the knowledge acquisition and structuring. Some of the key principles are: • There are different: ◦ types of knowledge each requiring its own approach and technique. ◦ types of experts and expertise, such that methods should be chosen appropriately. ◦ ways of representing knowledge, which can aid the acquisition, validation and re-use of knowledge. ◦ ways of using knowledge, so that the acquisition process can be guided by the project aims (goal-oriented). • Structured methods allow increasing efficiency of knowledge acquisition and structuring processes. • Knowledge engineering is aimed at efficient structuring knowledge and organization of knowledge processes. Organization of knowledge processes involves the following activities: — — — — —
Process Process Process Process Process
identification; design; implementation; facilitation; monitoring;
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— Process analysis; — Process mending. Thus, we can see that knowledge engineering is a field of engineering where the knowledge is the object of operations and is a sphere of activity of knowledge engineers. Knowledge engineering is closely related to software engineering. Successful knowledge engineering demands various skills including: computer and network skills, effective communication skills, logical thinking, understanding of organizations and individuals, self-confidence, and patience. 7.3. Knowledge management and application A little knowledge that acts is worth infinitely more than much knowledge that is idle. Kahlil Gibran
Knowledge management (KM) has become such a hot topic that it has been dubbed the business mantra of 1990s (Hallal, 1998). To explain this situation, new terms such as knowledge-oriented organization or knowledge-generating organization were introduced. In the contemporary economical and social environment, efficient KM is necessary to enable an organization to function resourcefully in the long run. At the same time, while KM has grown to be a highly prominent topic of research and practice, the term remains rather ambiguous and controversial, what impedes progress in articulating what KM entails and what knowledge-based organizations will look like (Baker and Badamshina, 2002). The most popular approach to KM describes it as a collection of processes directed at creation, capturing, accumulating, sharing, applying and reusing knowledge (Sydanmaanlakka, 2000). Another approach interprets KM as delivering the right knowledge to the right persons at the right time. Baker and Badamshina (2002) advocate a very different perspective by comprehending KM as developing and managing integrated, well-configured knowledge systems
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and increasingly embedding work systems within these knowledge systems, rather than managing something as nebulous as knowledge per se. Here we suggest looking at KM from a more general point of view within the business model or organization. Management has always been responsible for effective acquisition and allocation of people, resources and tools at all levels in an organization, as well as for the planning of processes and the evaluation of the results. In a similar way, it is possible to define KM as effective acquisition and allocation of knowledge at all levels in an organization, as well as for the planning of knowledge processes and the evaluation of the results. Naturally, this includes building and enhancing knowledge systems although not embedding work systems within these knowledge systems but efficiently embedding knowledge systems within work systems. Whereas there is no general agreement upon definition of KM, there is near consensus that it constitutes the combination of all the actions necessary for ensuring that organizations learn from past practice and make effective use of all skills and knowledge that their staff possesses (Powell, 2003). In many organizations ‘managing’ knowledge has come to occupy a central place in a manager’s work although knowledge engineers ought to play the leading role in this process. Knowledge management makes great demands on the strategic insight, problem-solving ability, and tact of the person involved in this activity. To be efficient, KM has to include the following activities: 1. Determination and identification of needs for knowledge (information). 2. Location of needs in knowledge (information). 3. Search for knowledge (information). It is necessary to understand that such a search can unsuccessful, i.e., without finding/discovery of the necessary knowledge. 4. Knowledge discovery and collection consists of finding the necessary knowledge and bringing it to the organization. According to the current terminology, knowledge discovery is a process of
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5.
6.
7.
8.
9.
10.
11.
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transformation of data into knowledge using data mining and information extraction. Knowledge creation/production is a process of generating (constructing) of overall new knowledge or of reconstruction of locally new knowledge, i.e., knowledge that is new for those who reconstruct it but it existed somewhere before this reconstruction. Knowledge reception is a process of receiving information (data) sent by another system, e.g., an agent or organization, and converting it into knowledge. Knowledge acquisition has several forms: (1) it is a process of acceptance of the created, received or found knowledge into the system, e.g., into the organization or the knowledge base; (2) it is a process of capturing existing implicit knowledge and transforming it into explicit knowledge, which is performed mainly through knowledge sessions of experts (knowledge engineers) with the knowers, i.e., people who have this implicit knowledge. Knowledge appropriation and representation is a process of making knowledge suitable for definite people and/or definite tasks and/or definite organizations. It is performed by transformation and transmutation of knowledge, knowledge representations, and knowledge carriers. Knowledge codification is a process of changing knowledge representation aimed at providing a possibility for placing knowledge in a structured repository using a knowledge models. Knowledge storing is a process of accumulation of knowledge in a physical repository such as a database, knowledge base, library or archive. Knowledge integration has several forms: (1) it is a process of integration of one knowledge system into another knowledge system; (2) it is a process of transformation of several knowledge systems into one knowledge system; and (3) it is a process of integration of a knowledge system into some area of activity. Evaluation of knowledge assets is a process of finding essential properties, parameters, characteristics and attributes knowledge in the organization.
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13. Knowledge sharing and dissemination is a process of transmission and distribution of knowledge inside an organization, which is often done through person-to-person contacts, lectures, workshops, seminars, webinars, and e-mails. 14. Knowledge hiding is a process of protecting knowledge assets from unsanctioned access, which is often missed in descriptions of knowledge management in organizations. 15. Knowledge translation is a process of transmitting knowledge from one are to another, e.g., from creation and discovery to applications. 16. Knowledge maintenance consists of actions aimed at modifying, updating, and correcting organizational knowledge so that to keep it operational and acceptable to its users increasing its usability. 17. Knowledge application,implementation, and utilization are processes in which knowledge existing in an organization is used for supporting various activities, e.g., for problem solving or decision-making. 18. Knowledge monitoring is a process of monitoring and evaluating of knowledge usage. 19. Knowledge exchange and trade are performed in interaction with other organizations and individuals. 20. Knowledge revision is a process of evaluation the situation and changing knowledge when it is reasonable or necessary. 21. Knowledge retirement is a process of deleting and excluding some knowledge items. Each of the activities from the knowledge management process includes the following stages:
➢ ➢ ➢ ➢ ➢
Goal determination; Means determination; Activity organization; Activity realization; Result evaluation.
All these activities go on concurrently shaping different cycles. For instance, search for knowledge may be repeated several times before
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the result will be obtained or the cycle knowledge creation — knowledge appropriation — knowledge storing is performed many times through the whole process of KM. An important component of knowledge management is the Organizational Knowledge cycle: 1. Knowledge acquisition; 2. Knowledge dissemination; 3. Knowledge utilization. Some of these activities, e.g., knowledge codification or knowledge storing, involve only explicit knowledge, while others, e.g., knowledge creation or knowledge appropriation, involve both explicit and tacit knowledge. Knowledge acquisition includes extraction, collection, analysis, modeling and validation of knowledge by means of knowledge engineering for knowledge management projects. Usually it is possible to access explicit knowledge in an organization at three stages: before, during, or after KM-related activities. However, access to tacit knowledge is rather restricted. Note that all research and application of knowledge management concentrate their attention on knowledge discovery, creation, acquisition, and storing. At the same time, an important type of processes in knowledge management has been overlooked for a long time, namely, knowledge protection by means of knowledge hiding. In contrast to knowledge sharing, hiding is an intentional concealment of knowledge requested or supposedly looked for by another person or organization. In the perfect world, all knowledge would be open for everybody. In contrast to this ideal picture, there are many restrictions on the access to different knowledge items in the world we live. This is caused by various reasons. An example of such restrictions is paid access versus open access to scientific and other publications. The reason for this is that publishers have to spend money on publication and they do not want to do this without compensation of the expenses and even some profit. Another situation emerges due to the competition between companies, which necessitates protection of their knowledge from competitors.
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Only recently, researchers started studies of knowledge hiding in organizations (cf., for example, (Connelly et al., 2011)). Researchers found that there are three ways of knowledge hiding: evasive hiding, rationalized hiding, and playing dumb. Another activity missed by knowledge management studies is knowledge retirement. Remember to forget is an essential principle for efficient knowledge management. Forgetting, i.e., eliminating or deleting knowledge, is important in the following situations: • When the knowledge base has a limited space, it is necessary to make space for new knowledge. In this case, it is necessary to correctly select what is possible to delete without impairing functioning of the person or organization. • When the new knowledge contradicts to some stored knowledge and preservation of the old knowledge may decrease efficiency of the system, it is necessary to delete old knowledge. This is a general situation in expert systems and non-monotonic and default logics when old knowledge is excluded to evade contradictions and inconsistency (cf., Section 3.3). • When some knowledge may be detrimental to the system, its elimination is vital. Operational knowledge in the form of computer viruses gives examples of such knowledge. Now knowledge maintenance is mostly developed for operational knowledge in the form of computer and network software. For instance, software developers often release upgrades in the form of a “patch” in order to correct program bugs that are an inevitable part of software development (Burgin and Debnath, 2006). Many software companies also suggest updates for solving different problems with their software. At the same time, it is also necessary maintain descriptive and representational knowledge. For instance, it is important permanently to update information in databases and knowledge bases because outdated information can be very detrimental. Such processes are continuously going on in web search systems, which constantly update the stored search data by means of the system called the Index.
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PEOPLE Define the roles of, and knowledge needed by Help design and then use
Provides support for
Help design and then operate
PROCESSES
Makes possible new kinds of Determine the need for
TECHNOLOGY Figure 7.7. Internal structure of knowledge management
Knowledge management goes in the system, which has three basic components: people, technology and processes, which are connected in the following way (Edwards, 2009; 2011): Arrows in Figure 7.7 show what one component is doing with another component. For instance, processes define the roles of and knowledge needed by people, while technology provides support for people. Processes determine needs for technology. In turn, technology makes possible new kinds of processes. To conclude, it is necessary to make two remarks. First, knowledge management is a very popular and flourishing area of research and here we presented only some basic elements from its theory and methodology. It is possible to find much more in the literature on this topic. Second, it is vital to understand that knowledge itself does not solve our problems or problems of the organization. Even having the best knowledge what to do, people often do not or cannot do what it is necessary to achieve their goals. Thus, it is extremely important how we utilize knowledge we have. It is related to individuals, organizations, and societies.
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Chapter 8
Knowledge, Data, and Information The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe. Karl Pearson
Knowledge is intrinsically related to data and information. This association became evident with the advent of computers. However, people still struggle to achieve unambiguous understanding of relations between these concepts. There is a huge diversity of judgments on this topic. Many papers and books present views of their authors on such relations. Some works reflect points of view of different authors. For instance, 130 definitions of data, information, and knowledge formulated by 45 scholars are collected in (Zins, 2007). A variety of approaches to this problem is also described in the book (Burgin, 2010). Here based on the general theory of information (Burgin, 2010), we further develop the understanding according to which knowledge, data, and information do not belong to the same plane of reality having, as a result, dissimilar functions. Knowledge and data are of the same ilk but situated on different levels of structural reality playing the role of substance in the world of structures, while information functions in the world of structures as energy works in physical world. Consequently, information is complementary to knowledge and data constituting orthogonal dimensions in the world of structures.
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8.1. Epistemic structures and cognitive information The greatest obstacle to discovering the shape of the earth, the continents, and the oceans was not ignorance but the illusion of knowledge. Daniel J. Boorstin
As it is demonstrated in Section 3.1, epistemic structures provide a sound foundation for knowledge studies. At the same time, they are also crucial for understanding the information phenomenon and building information theories. When people speak about information, they mean, as a rule, cognitive information. Indeed, since the 16th century, we find the word information in ordinary French, English, Spanish, and Italian in the sense we use it today: to instruct, to furnish with knowledge (Capurro, 1991). However, scientific usage of the notion of information (cf., for example, (Loewenstein, 1999)) implies a necessity to have a more general definition. For instance, Heidegger pointed to the naturalization of the concept of information in biology in the form of genetic information (Heidegger and Fink, 1970). An example of such a situation is when biologists discuss information in DNA in general or in the human genome, in particular. As a result, we come to the world of structures. Modification of structural features of a system, and transformation of other system characteristics through these changes, is the essence of information in the strict sense. This correlates with the von Weizs¨aker’s remark that information being neither matter nor energy, according to (Wiener, 1961), has a similar status as the “platonic eidos” and the “Aristotelian form” (Weizs¨ acker, 1974). In the context of the general theory of information, these ideas are crystallized in the following principle (Burgin, 2010). Ontological Principle O2a (the Special Transformation Principle). Information in the strict sense or proper information or, simply, information for a system R, is a capacity to change structural infological elements from an infological system IF(R) of the system R.
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The infological system plays the role of a free parameter in the definition of information. One of the ways to vary this parameter is to choose a definite kind of infological elements. Additional conditions on infological elements imply a more restricted concept of information. Note that it is possible to comprehend the concept of information in a much broader sense, which includes energy as a particular form of information (Burgin, 2010). There are researchers whose interpretations of information come close to the definition given in the Special Transformation Principle. One of them is developed by Karpatschof (2000) based on the concept of release mechanism and activity theory. He postulates existence of systems with stored potential energy, which is released by specific release mechanisms triggered by some signals with a low energy. Then information is defined as a property of a signal to trigger the release mechanism of a system (Karpatschof, 2000). If we take the release mechanism of a system R as its infological system IF(R) and take this definition, we come to the definition from the Special Transformation Principle. Discussing definitions of data, knowledge and information, Capurro explains that he prefers the everyday meaning (since Modernity) of the term information, namely, information is “the act of communicating knowledge” (cf., (Zins, 2007)). In the same way as energy is a source of physical system transformation and emergence, information is a source of transformation and emergence in epistemic systems in general and in knowledge systems, in particular. Madden (2000) defines information as “a stimulus originating in one system that affects the interpretation by another system of either the second system’s relationship to the first, or of the relationship the two systems share with a given environment . . .”. Thus, if we take interpretations or relationships of a system R as its infological system IF(R) and take this definition, we come to the definition from the Special Transformation Principle. In another work (Madden, 2004), information is defined as “a stimulus which expands or amends the World View of the
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informed”. Thus, if we take the World View of a system R as its infological system IF(R) and take the new definition, we once more come to the definition from the Transformation Principle formulated above or the Cognitive presented below. However, here we are mostly interested in the particular case of proper information — in cognitive information. It is identified by the Cognitive Transformation Principle. To better understand how infological system can help to explicate the concept of information in the strict sense, we consider cognitive infological systems. Definition 8.1.1. An infological system IF(R) of the system R is called cognitive if it contains (stores) such epistemic structures (elements) such as knowledge, data, images, ideas, fancies, abstractions, beliefs, etc. Cognitive infological systems are standard examples of infological systems, while their elements, such as knowledge, data, images, ideas, fantasies, abstractions, and beliefs, are standard examples of infological elements. Cognitive infological systems are very important, especially, for intelligent systems as the majority of researchers believe that information is intrinsically connected to knowledge (cf., (Fl¨ uckiger, 1995)). The system of knowledge KIF(R) of a system R is a kind of infological systems of an intelligent system R. In cybernetics, the knowledge system of R is called the thesaurus Th(R) of the system R. A thesaurus is a part of larger cognitive infological systems. Another example of an infological system is the memory of a computer. Such a memory is a place in which data and programs are stored. A cognitive infological system of a system R is denoted by CIF(R) and is related to cognitive information. This relation is described in the following ontological principle. Ontological Principle O2c (the Cognitive Transformation Principle). Cognitive information for a system R is a (potential) capacity to cause changes in the cognitive infological system CIF(R) of the system R. In this context, a cognitive infological system CIF(R) contains, acquires, stores, and processes epistemic structures, such as
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knowledge, data, ideas, beliefs, images, algorithms, tasks, procedures, problems, schemas, scenarios, values, measures, opinions, goals, ideals, fantasies, abstractions, etc. A cognitive structure (element) E is any system (element) used for cognition. It represents or reflects, i.e., contains information about, some domain or system, which is called the domain of the epistemic structure E. We discern pure epistemic structures and weighted epistemic structure. A pure epistemic structure p contains information only about its domain. Example 8.1.1. A proposition is a pure epistemic structure. Example 8.1.2. A semantic network is a pure epistemic structure. Example 8.1.3. A frame is a pure epistemic structure. A weighted epistemic structure consists of a pure epistemic structure p and its weights. Namely, a weighted epistemic structure has the form (p; w1 , w2 , w3 , . . . , wk ) where wi is the weight of p in the dimension i. Examples of weighted epistemic structure are considered in Section 3.1. Some researchers also relate information to structures. For instance, information is characterized as a property of how entities are organized and arranged, but not the property of entities themselves (Reading, 2006). Other researchers have related information to form, while form is an explicit structure of an object (Burgin, 2012). For instance, Dretske (2000) characterizes information as an attribute of the form (in-form-ation) that matter and energy have but not as a feature of the matter and energy themselves. Cognitive infological systems are very important, especially, for intelligent systems. Indeed, the majority of researchers believe that information in general is intrinsically connected to cognition, while cognitive information is one of the three basic types of anthropic information studied in (Burgin, 2011a). Moreover, some researchers believe that people’s knowledge about physical reality is the result of information they obtain from external sources (von Weizs¨ acker, 1958;
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von Weizs¨acker et al., 1958; Wheeler, 1990; Frieden, 1998). Understanding that physicists study physical systems not directly but only through information they get from these systems has created a school of thought about the role of information processing in physical processes and its influence on physical theories. According to one of the leading physicists of the 20th century John Archibald Wheeler (1911–2008), it means that every physical quantity derives its ultimate significance from information. He called this idea “It from Bit” where “It” stands for things, while “Bit” impersonates information as the most popular information unit (Wheeler, 1990). For Wheeler and his followers, space–time itself must be understood and described in terms of a more fundamental pregeometry without dimensions and classical causality. These features of the physical world only appear as emergent properties in the ideal modeling the physical reality based on information about complex interactions of very simple basic elements, such as subatomic particles. Not all proper information is cognitive. For instance, there are also such types of information as emotional information and effective information, which are proper but not cognitive as they impact the emotional and effective subsystems of the brain (Burgin, 2010). In the mathematical representation of information, a cognitive infological system is modeled by a mathematical structure, for example, a space of knowledge, beliefs, and fantasies. Information, or more exactly, a unit of information, is an operator acting on this structure. A global unit of information is also an operator. Its place is in a higher level of hierarchy as it acts on the space of all (or some) cognitive infological systems. Knowledge constitutes a substantial part of the cognitive infological system. Some researchers even equate cognition with knowledge acquisition and consider the system of knowledge often called thesaurus as the whole cognitive infological system. In a general case, knowledge, as a whole, constitutes a huge system, which is organized hierarchically and has many levels as it is demonstrated in Chapters 4–6. However, two issues, absence of the exact concept of structure and lack of understanding that structures can objectively exist, result
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in contradictions and misconceptions related to information. For instance, one author writes that information is simply a construct used to explain causal interaction, and in the next sentence, the same author asserts that information is a fundamental source of change in the natural world. Constructs cannot be sources of change — they can only explain change. The Ontological Principle O2c implies that information is not of the same kind as knowledge and data, which are structures. Actually, if we take that matter is the name for all substances as opposed to energy and the vacuum, we have the relation that is represented by the diagram called the Structure-Information-Matter-Energy (SIME) Square considered in the next section. 8.2. Structural aspects of knowledge–information duality Information is not knowledge. Albert Einstein
In the traditional approach, relations between knowledge and information are usually analyzed in the context of the triad Data– Information–Knowledge, which attracted attention of researchers only in the 20th century. In contrast to this, the best minds in society were concerned with the problem of knowledge from ancient times. Concepts of information and data in contemporary understanding were introduced much later. Consequently, information attracted interests of researchers millennia after knowledge had become one of the primary concerns of philosophers. Only in 1920s, the first publications appeared that later formed the core of two principal directions of information theory. In 1922, Fisher introduced an important information measure now called Fisher information and used in a variety of areas — from physics to information geometry and from biology to sociology. Two other papers (Nyquist, 1924) and (Hartley, 1928), which started the development of the most popular direction in information theory, appeared a little bit later in the communication theory. Later many other directions, such as algorithmic information
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theory, qualitative information theory, semantic information theory, pragmatic information theory, economic information theory, dynamic information theory, quantum information theory, and operator information theory, have been developed by different researchers (Burgin, 2010). Interest to the third component, data, from the triad Data– Information–Knowledge emerged in the research community only after computers became the foremost information processing devices. As a result, researchers started to study the whole triad only in 80s of the 20th century. It is possible to approach relations between data, information and knowledge from many directions. Now the most popular approach to the triad Data–Information–Knowledge represents hierarchical relations between them in a form of a pyramid with data at the bottom, knowledge at the top and information in the middle (cf., for example, (Landauer, 1998; Boisot and Canals, 2004; Sharma, 2005; Rowley, 2007)). In knowledge management literature, this hierarchy is mostly referred to as the Knowledge Hierarchy or the Knowledge Pyramid, while in the information science domain, the same hierarchy is called Information Hierarchy or Information Pyramid for obvious reasons. Often the choice between using the label “Information” or “Knowledge” is based on what the particular profession believes to fit best into the scope of this profession. Here we use the term Data– Information–Knowledge Hierarchy or Data–Information–Knowledge
Knowledge
Information
Data
Figure 8.1. The Data–Information–Knowledge pyramid
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Pyramid because the researchers prefer the image of a pyramid as it conveys more implicit information than has an arbitrary hierarchy. Besides, researchers extended the Data–Information– Knowledge Pyramid by one more level — wisdom. The extended form of this pyramid is called the Data–Information–Knowledge–Wisdom Hierarchy (Pyramid) (Sharma, 2005). The form of the pyramid implies that the bulk of data in the world is much larger than the volume of information, which in turn, is smaller than the amount of knowledge in the world. Another metaphor conveyed by the form of a pyramid implies that data are compressed into information, while information, which in turn, is condensed into knowledge. In other words, the concepts go from the least to most processed or integrated, with data the rawest, and knowledge the most rarefied. The message contained in the form of the pyramid, as well as in the concept of a hierarchy, is that information is situated on the higher level than data, while knowledge is situated on the higher level than information. Here higher level can mean “more valuable” or “better organized”. As Sharma writes (2005), there are two separate threads that lead to the origin of the Data–Information–Knowledge Hierarchy. In knowledge management, Ackoff is often cited as the originator of the Data–Information–Knowledge Hierarchy. His 1988 Presidential Address at International Symposium in Geotechnical Safety and Risk (ISGSR) (Ackoff, 1989) is considered by many to be the earliest mention of the hierarchy as he gives its detailed description and does not cite any earlier sources on this topic. However, Zeleny (1987) described this hierarchy in the field of knowledge management two years earlier than Ackoff. In his paper, Zeleny defined data as “know-nothing”, information as “know-what”, knowledge as “knowhow”, and added wisdom as “know-why” (cf., Figure 8.2). Almost at the same time as Zeleny’s article appeared, Cooley (1987) built the Data–Information–Knowledge Hierarchy in his discussion of tacit knowledge and common sense. In all of these papers, no earlier work is cited or referred to. Nevertheless, much earlier in information science, Cleveland (1982) described the Data–Information–Knowledge Hierarchy in
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Wisdom
Knowledge Information
Data Figure 8.2. The extended Data–Information–Knowledge Pyramid or Data– Information–Knowledge–Wisdom Hierarchy
detail. Besides, Cleveland pointed that the surprising origin of the hierarchy itself is due to the poet T.S. Eliot who wrote in “The Rock” (1934): Where is the Life we have lost in living? Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information? This is the first vague mention of the Data–Information– Knowledge Hierarchy, which was expanded by Cleveland and others by adding the layer of “Data” or in terms of Cleveland, of “facts and ideas”. Cleveland (1982) concedes that information scientists are “still struggling with the definitions of basic terms” of the hierarchy. He uses Elliot’s metaphor as a starting point to explain the basic terms. Cleveland also agrees that there are many ways in which the elements of the hierarchy may be defined, yet universal agreement on them need not be a goal in itself. At the same time, the majority of researchers assume that the difference between data, information, and knowledge has pivotal importance in our information age with its information technology and knowledge-based economy. For instance, Landauer (1998) writes, “the repeated failure of natural language models to succeed beyond the most superficial level
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is due to their not taking into account these different levels (of the Data–Information–Knowledge Pyramid) and the fundamentally different processing required within and between them”. The most popular approach to the relation between data and information implies that information is an organized collection of facts and data. In this context, the process of transition from data to knowledge goes in two steps: at first, data are transformed into information and then information is converted into knowledge by structuring processes. Thus, information comes into view as an intermediate level of similar phenomena situated between data and knowledge forming the triad Data–Information–Knowledge. However, the triad Data–Information–Knowledge is not always visually represented by a three-level pyramid. Another structure (cf., Figure 8.3) of the system Data–Information–Knowledge — the chain — is considered by Liew (2007). In this structure, not only
Internalized: Absorbed & understood by the human mind
Processed & Analyzed: A reconstructed picture of historical events &/or projection of possible future events
Information
Knowledge
Externalized: Verbalized &/or illustrated
Captured & stored
Data
Figure 8.3. The Data–Information–Knowledge chain
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levels of the system Data–Information–Knowledge are explicated but also connections between these levels are described. In this Data–Information–Knowledge Chain, data are interpreted as recorded (captured and stored) symbols and signal readings, while symbols are interpreted in a broad sense and include words (texts and/or verbal expressions), numbers, diagrams, and images, and signals also interpreted in a broad sense include sensor and/or sensory readings of light, sound, smell, taste, and touch. Information, according to Liew (2007), is a message that contains relevant meaning, implication, or input for decision and/or action, while knowledge is the (1) cognition or recognition (know-what), (2) capacity to act (know-how), and (3) understanding (know-why) that resides or is contained within the mind or in the brain. Some researchers extend the Data–Information–Knowledge Pyramid by inclusion of additional level called wisdom labeling it by the name the Data–Information–Knowledge–Wisdom Hierarchy (cf., Figure 8.2). Other researchers include one more level, “understanding”, in the Data–Information–Knowledge Pyramid. As Ackoff writes (1989), descending from wisdom there are understanding, knowledge, information, and, at the bottom, data (cf., Figure 8.4). Each of the levels includes the categories that fall below it. However, the majority of researchers do not add understanding to their studies of the Data– Information–Knowledge Pyramid. The Data–Information–Knowledge Pyramid is a very simple schema. That is why it is so appealing to researchers. However, as we know from physics and other sciences, reality does not always comply with the simplest schemes. Therefore, to find relevance of this pyramid to the objective reality, we need to understand all its levels — data, information, and knowledge. According to Davis, “data is the plural of datum, although the singular form is rarely used. Purists who remember their first-year Latin may insist on using a plural verb with data, but they forget that English grammar permits collective nouns. Depending on the context, data can be used in the plural or as a singular word meaning a set or collection of facts. Etymologically, data, as noted, is the plural
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Wisdom
Understanding
Knowledge Information
Data Figure 8.4. The Data–Information–Knowledge–Understanding–Wisdom pyramid or hierarchy
of datum, a noun formed from the past participle of the Latin verb dare — to give. Originally, data were things that were given (accepted as “true”). A data element, d , “is the smallest thing which can be recognized as a discrete element of that class of things named by a specific attribute, for a given unit of measure with a given precision of measurement”. (cf., (Zins, 2007)) Data has experienced a variety of definitions, largely depending on the context of its use. With the advent of information technology the word data became very popular and is used in a diversity of ways. For instance, information science defines data as unprocessed information, while in other domains data are treated as a representation of objective facts. In computer science, expressions such as a data stream and packets of data are commonly used. The conceptualizations of data as a flow in both a data stream and drowning in data occur due to our common experience of conflating a multiplicity of moving objects with a flowing substance. Data can travel down a communication channel. Other commonly encountered ways of talking about data include having sources of data or working with raw data. We can place data in storage, e.g., in files or in databases, or fill a repository with data. Data are viewed as discrete entities.
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They can pile-up, be recorded or stored and manipulated, or captured and retrieved. Data can be mined for useful information or we can extract knowledge from data. Databases contain data. We can look at the data, process data, or experience the tedium of data-entry. It is possible to separate different classes of data, such as operational data, product data, account data, planning data, input data, output data, and so on. All these expressions reflecting usage of the term data assign some meaning to this term. In particular, this gives us two important conceptualizations of data: data are a resource, and data are manipulable objects. In turn, this may implicitly imply that data are solid, physical, things with an objective existence that allow manipulation and transformation of data, such as rearrangement of data, conversion to a different form or sending data from one system to another. Ackoff defines data as symbols that represent properties of objects, events and their environments (Ackoff, 1989). In this context, data are products of observation by people or automatic instrument systems. According to Hu and Feng (2006), data is a set of values recorded in an information system, which are collected from the real world, generated from some pre-defined procedures, indicating the nature of stored values, or regarding usage of stored values themselves. Data are also considered as discernible differences between states of some systems in the world (Lloyd, 2000; 2002), i.e., light versus dark, present versus absent, hot versus cold, 1 versus 0, + versus −, etc. In many cases, binary digits, or bits, represent such differences and thus, carry information about these differences. However, according to Boisot (2002), information itself is a relation between these discernible states and an observer. A given state may be informative for someone. Information is then what an observer will extract from data as a function of his/her expectations or prior knowledge (Boisot, 1998). Many understand data as discrete atomistic tiny packets with no inherent structure or necessary relationship between them. However, this is not true. Besides, there are different kinds and types of data. In addition, data as an abundant resource can pile-up to such an
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extent that many people find themselves drowning in data. As a substance, data can be measured. The most popular way to measure data is in bits where bit is derived from the term a binary digit, 0 or 1. The reason for this is very simple. Namely, the vast majority of information processing and storage systems, such as computers, calculators, embedded devices, CD, DVD, flash memory storage devices, magneto-optical drives, and other electronic storage devices, represent data in the form of sequences of binary digits. It would be useful to better clarify our understanding of data. At first glance, data look like some material things in contrast to knowledge, which is structural in essence. However, when we start analyzing what data are, we see that this is not so. Indeed, numbers are a kind of data. Let us take a number, say 10. In mathematics, it is called a natural number. From mathematics, we also know that natural numbers are equivalent collections of sets that have the same number of elements (Bourbaki, 1960). This is a structure. Another way to define natural numbers is to axiomatically define the set N of all natural structures and to call its elements by the name natural number (Kuratowski and Mostowski, 1967). Abstract sets and their elements definitely are structures. One more way to represent natural numbers is to use abstract properties (Burgin, 1989). This approach develops the idea of Cantor, who defined the cardinal number of a set A as a property of A that remains after abstraction from qualities of elements from A and their order (cf., for example, (Kuratowski and Mostowski, 1967)). Properties, abstract and real, are also structures. Thus, in any mathematical sense, a natural number, in particular, number 10, is a structure. Moreover, the number 10 has different concrete representations in mathematical structures. In the decimal numerical system, it is represented as 10. In the binary numerical system, it is represented as 1010. In the ternary numerical system, it is represented as 101. In English, it is represented by the word ten, and so on. In the material world, it is possible to represent number 10 as a written word on paper, as a written word on a board, as a said word, as a geometrical shape on the computer or TV screen, as signals, as a state of some system and so on. All these representations form a structure that we call number 10.
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We come to similar situations analyzing letters and words, which are also data. Let us consider the letter a. It also has different physical representations: a, a, a, a, a, a, a , A, A, A, a, a, a, a, a, A, A, A, A, A, and many others. In addition, it is possible to represent the letter a as a written symbol on paper, as a said word, as a written symbol on a board, as a geometrical shape on the computer or TV screen, as signals, as a state of some system, and so on. Thus, the letter a, as well as any other letter or word, is a structure. It is known that words are represented by written and printed symbols, pixels on the screen, electrical charges, and brain-waves (Suppes and Han, 2000). Let us look how the upper level of the Data–Information– Knowledge pyramid — knowledge — is defined and treated in contemporary studies. Logical analysis shows that knowledge is difficult to define. Taking knowledge as the third and upper component of the Data–Information–Knowledge pyramid, we see that in spite of the long history of knowledge studies, there is no consensus on what knowledge is. As we have seen, over the millennia, the philosophers of each age and epoch have added their own ideas on the essence and nature of knowledge to the list. Science has extended this list as well. As a result, there is a lot of confusion in this area. As Land et al. (2007) write, knowledge itself is understood to be a slippery concept, which has many definitions. In any case, our civilization is based on knowledge and information processing. That is why it is so important to know what knowledge is. For instance, the principal problem for computer science as well as for computer technology is to process not only data but also knowledge. Knowledge processing and management make problem solving much more efficient and are crucial (if not vital) for big companies and institutions (Ueno et al., 1987; Osuga, 1989; Dalkir, 2005). To achieve this goal, it is necessary to distinct knowledge and knowledge representation, to know regularities of knowledge structure, functioning and representation, and to develop software (and in some cases, hardware) that is based on theses regularities. Many intelligent systems search concept spaces that are explicitly or implicitly predefined by the choice of knowledge representation. In effect, the knowledge representation serves as a strong bias.
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In contrast to this, some researchers conceive information as a process, whereas knowledge is perceived as a state (Machlup et al., 1983). According to Sholle (1999), information and knowledge are distinguished along three axes: multiplicity, temporality, and spatiality. Multiplicity means that information is piecemeal, fragmented, and particular, while knowledge is structured, coherent, and universal. Temporality means that information is timely, transitory, and even ephemeral, while knowledge is enduring and temporally expansive. Spatiality means that information is a flow across spaces, while knowledge is a stock, specifically located, yet spatially expansive. If we take formal definitions of knowledge, we see that they determine only some specific knowledge representation. For instance, in logic, knowledge is represented by logical propositions and predicates. On one hand, informal definitions of knowledge provide little opportunities for computer processing of knowledge because computers can process only formalized information. On the other hand, there is a great variety of formalized knowledge representation schemes and techniques: semantic and functional networks, frames, productions, formal scenarios, relational, and logical structures. However, without explicit knowledge about knowledge structures per se, these means of representation are used inefficiently. The middle level in the Data–Information–Knowledge pyramid is information. We can see the lack of agreement about the definition of this term although a quantity of definitions have been suggested in information sciences, knowledge management, and database theory. Moreover, the effort to define information has been active in other disciplines such as epistemology, cognitive sciences, computer science, electrical engineering, and systems theory, among others. Now let us look in more detail structural relations between information, knowledge, and data. A systemic approach demands not to consider components from the Data–Information–Knowledge pyramid as given but to define these concepts in a more consistent way according to the understanding of the majority of researchers: Information is structuring of data (or structured data); Knowledge is structuring of information (or structured information).
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However, another approach (cf., for example, (Meadow and Yuan, 1997)) suggests a different picture: Data usually means a set of symbols with little or no meaning to a recipient. Information is a set of symbols that does have meaning or significance to their recipient. Knowledge is the accumulation and integration of information received and processed by a recipient. From the perspective of knowledge management, information is used to designate isolated pieces of meaningful data. These data integrated within a context constitute knowledge (Gundry, 2001; Probst, Raub, and Romhard, 1999). As a result, it is often assumed that data themselves are of no value until they are transformed into a relevant form. This implies that the difference between data and information is functional, not structural. Stenmark (2002) collected definitions of the components from the Data–Information–Knowledge pyramid from seven sources. Wiig (1993) considers information as facts organized to describe a situation or condition, while knowledge consists of truths and beliefs, perspectives and concepts, judgments and expectations, methodologies and “know how”. Nonaka and Takeuchi (1995) consider information as a flow of meaningful messages, while knowledge consists of commitments and beliefs created from these messages. According to Spek and Spijkervet (1997) data are not yet interpreted symbols, information consists of data with meaning, while knowledge is the ability to assign meaning. According to Davenport (1997) data are simple observations, information consists of data with relevance and purpose, while knowledge is valuable information from the human mind. According to Davenport and Prusak (1998) data are discrete facts, information consists of messages meant to change the receiver’s
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perception, while knowledge is Experiences, values, insights, and contextual information. Quigley and Debons (1999) treat data as a text that does not answer questions to a particular problem. They look upon information as a text that answers the questions who, when, what, or where. In addition, they deal with knowledge as a text that answers the questions why and how. Choo et al. (2000) understand data as facts and messages, assume that information is data vested with meaning, and perceive knowledge as justified, true beliefs. Dalkir (2005) treats data as “content that is directly observable or verifiable,” information as “content that represents analyzed data” or as “analyzed data — facts that have been organized in order to impart meaning”, while knowledge is defined in his book as subjective and valuable information. One more view of the difference between data and information is that data is potential information (Meadow, 1996). A message as set of data may potentially be information but the potential is not always realized. This distinction is similar to the distinction between potential energy and kinetic energy. In mechanics, kinetic energy is that associated with movement. Potential energy is associated with the position of a thing, e.g., a weight raised to a height. If the weight falls, its energy becomes kinetic. Impact is what happens if the weight, with its kinetic energy, hits something. Meadow and Yuan (1997) suggest that in the information world, impact is what happens after a recipient receives and in some manner acts upon information. This perfectly correlates with Ontological Principles O1 and O2 from the general theory of information (Burgin, 2010). The most popular in computer science approach to information is expressed by Rochester (1996) who defines information as an organized collection of facts and data. Rochester develops this definition through building a hierarchy in which data are transformed into information into knowledge into wisdom. Thus, information appears
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as an intermediate level of similar phenomena leading from data to knowledge. An interesting approach to understanding data, information, and knowledge in a unified context of semiotics is developed by Lenski (2004). He suggests that data denote the syntactical dimension of a sign, knowledge denotes the semantic dimension of a sign, and information denotes the pragmatic dimension of a sign. Data, in the sense of Lenski, are a system that is organized by structural and grammatical rules of sign. In contrast to this, knowledge results from neglecting the amount of individual contribution to the semantic abstraction process. However, such an abstraction process may be only subjectively acknowledged resulting in personal knowledge. Comprising pragmatic dimension, information is bound to a (cognitive) system that processes the possible contributions provided by signs that constitute data for a possible action. Moreover, information inherits the same interpretation relation as knowledge with the difference that the latter is abstracted from any reference to the actual performance whereas information, in contrast, emphasizes reaction, and performance. As a result, knowledge and information are closely tied together. To elaborate his own definition of information, Lenski (2004) uses seven principles as a system of prerequisites for any subsequent theory of information that claims to capture the essentials of a publication-related concept of information. These principles were introduced by other researchers as determining features of information. Principle 1. According to Bateson (1973), information is a difference that makes a difference. Principle 2. According to Losee (1997), information is the value of characteristics in the processes’ output. Principle 3. According to Belkin and Robertson (1976), information is that which is capable of transforming structure. Principle 4. According to Brookes (1977), information is that which modifies . . . a knowledge structure.
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Principle 5. According to Brookes (1980), knowledge is a linked structure of concepts. Principle 6. According to Brookes (1980), information is a small part of such a structure. Principle 7. According to Mason (1978), information can be viewed as a collection of symbols. After formulating these principles, Lenski gives his own interpretation, explicating their relation to the “fundamental equation” (8.1) of Brookes (1980). K(S) + ∆I = K(S + ∆S).
(8.1)
This equation reflects the situation when a portion of information ∆I acts on a knowledge structure K(S), transforming it into the structure K(S + ∆S) with ∆S as the effect of this change. In this context, principles of Lenski acquire the following meaning. Principle 1 reflects the overall characterization of information along with its functional behavior and refers to the ∆-operator in the “fundamental equation”. Principle 2 expresses a process involved with results ∆I that are constituents of information. Principle 3 specifies the concept of difference as a transformation process resulting in K[S + ∆S]. Principle 4 shows on what information acts, namely, on the knowledge (or structure) system K[S]. Principle 5 explains what is knowledge (or knowledge structure). Principle 6 relates information to knowledge structures. Principle 7 specifies carriers of information. To achieve his goal, Lenski (2004) formulates one more principle. Principle 8. The emergence of information is problem-driven. Based on these principles, Lenski (2004) presents a working definition for a publication-related concept of information. Definition 8.2.1. Information is the result of a problem-driven differentiation process in a structured knowledge base.
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Thus, very often, it is assumed that being different, knowledge and information nevertheless have the same nature. For instance, the sociologist Merton (1968) writes that knowledge implies a body of facts or ideas, whereas information carries no such implication of systematically connected facts or ideas. In many books and papers, the terms knowledge and information are used interchangeably, even though the two entities, being intertwined and interrelated concepts, are far from identical. Moreover, some researchers define information in terms of data and knowledge. At the same time, other researchers define knowledge and/or data in terms of information. For instance, Kogut and Zander (1992) conceive information as “knowledge which can be transmitted without loss of integrity”, while Meadow and Yuan (1997) write that knowledge is the accumulation and integration of information. MacKay (1969) also assumes that knowledge itself turns out to be a special kind of information. In a similar way, Davenport (1997) treats information as data with relevance and purpose, while Tuomi (1999) argues that data emerge as a result of adding value to information. In a similar way, commonplace usage of words data and information blurs differences between these concepts. For instance, Machlup and Mansfield write (1980): “Data are the things given to the analyst, investigator, or problem-solver; they may be numbers, words, sentences, records, assumptions — just anything given, no matter in what form and of what origin . . . Many writers prefer to see data themselves as a type of information, while others want information to be a type of data.”
At the same time, according to Frost (1986), “Knowledge is the symbolic representation of aspects of some named universe of discourse”, while data are the symbolic representation of simple aspects of some named universe of discourse. The universe of discourse may be the actual universe or a fictional one, one in the future, or in some belief. In any case, this means that data are a special case of knowledge. All these and many other inconsistencies related to the Data– Information–Knowledge pyramid cause grounded critic of this
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approach to understanding data, knowledge, and information. For instance, Capurro and Hjorland (2003) write that the semantic concept of information, located between data and knowledge is not consistent with the view that equates information management with information technology. Boisot and Canals (2004) criticize distinctions that have been drawn between data, information, and knowledge by those who analyzed the Data–Information–Knowledge pyramid. Fricke (2008) also offers valid arguments that this pyramid is unsound and methodologically undesirable explaining that it has no foundation. In addition, researchers criticized various implications of the Data–Information–Knowledge pyramid. For instance, challenging the traditional approach, Tuomi (1999) argues that in contrast to the conventional estimation that knowledge is more valuable than information, while information is superior to data, these relations have to be reversed. Thus, data emerge as a result of adding value to information, which in turn is knowledge that has been structured and verbalized. Moreover, there are no “raw” data as any observable, measurable, and collectible fact has been affected by the very knowledge that made this fact observable, measurable, and collectible. According to Tuomi, knowledge, embedded in minds of people, is a prerequisite for getting information. Some researchers treat knowledge in the mind as information, which, in turn, is explicit and appropriate for processing when it is codified into data. Since only data can effectively be processed by computers, Tuomi (1999) explains, data is from the technological perspective the most valuable of the three components of the Data–Information–Knowledge pyramid, which consequently should be turned upside-down. In a similar way, Childers reasons “. . . knowledge is that which is known, and it exists in the mind of the knower in electrical pulses. Alternatively, it can be disembodied into symbolic representations of that knowledge (at this point becoming a particular kind of information, not knowledge). Strictly speaking, represented knowledge is information. Knowledge — that which is known — is by definition subjective, even when aggregated to the level of social, or public, knowledge — which is the sum, in a sense, of individual “knowings”.
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Data and information can be studied as perceived by and “embodied” (known) by the person or as found in the world outside the person” (cf., (Zins, 2007)). Le Coadic also explains “datum (in our sector mainly electronic) is the conventional representation, after coding (using ASCII, for example), of information. Information is knowledge recorded on a spatio-temporal support. Knowledge is the result of forming in mind an idea of something” (cf., (Zins, 2007)). It means that information is a kind of knowledge, while data are obtained from information. In opposition to the conventional approach, according to which knowledge is a kind of information, Zins (2007) believes that in the subjective domain, information is empirical knowledge. In turn, Poli (cf., (Zins, 2007)) does not agree that data, information, knowledge, and message are placed on the same level of analysis. On his opinion, message is the “vehicle” carrying either data or information (which can be taken as synonymous). At the same time, knowledge hints to either a systematic framework (e.g., laws, rules or regularities, that is higher-order “abstractions” from data) or what somebody or some community knows. Stewart (2002) advocates subjective approach to relations between data and knowledge writing “. . . one man’s data can be another man’s knowledge, and vice versa, depending on context”. All these considerations show that it is necessary to make a distinction between data, knowledge, beliefs, ideas and their representations. For knowledge, it is a well-known fact and knowledge representation is an active research area (cf., for example, (Ueno et al., 1987)). Change of representation is called codification. Knowledge codification serves the pivotal role of allowing what is collectively known to be shared and used (Dalkir, 2005). At the same time, differences between data and data representations are often ignored. A variety of perspectives on the triad Data–Information– Knowledge is collected in (Zins, 2007), where 45 scholars formulated 130 definitions of data, information, and knowledge mapping the major conceptual approaches for defining these three key concepts.
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Here we acquaint the reader with some of these approaches from (Zins, 2007) in cases that are different from the conventional approach to the Data–Information–Knowledge pyramid. In contrast to the widespread opinion, Rafael Capurro suggests that data are (or datum is) an abstraction. Indeed, while the concept of data or datum suggests that there is something there that is purely given and that can be known as such, this understanding contradicts to the grounded opinion that there is nothing like ‘the given’ or ‘naked facts’ but that every (human) experience/knowledge is biased. In contrast to data, Capurro comprehends knowledge as the event of meaning selection of a (psychic/social) system from its “world” on the basis of communication. To know is then to understand on the grounds of making a difference between message as a meaning offer and information as meaning selection. According to Capurro, information is a multi-layered concept with Latin roots (where informatio means “to give a form”) going back to Greek ontology and epistemology. The use of this concept in information science is, at the first sight, highly controversial when it refers to the everyday meaning (since Modernity), which explains that information is “the act of communicating knowledge”. In addition, Capurro suggests using this definition as far as it points to the phenomenon of message that he treats as the basic one in information science. It brings us to the Capurro communication triad (cf., Figure 8.5)). Following the main principles of systems theory and second-order cybernetics, Capurro advocates that a message is a meaning offer, while information refers to the meaning selection within a system and understanding expresses the possibility that the receiver integrates the selection within his/her pre-knowledge. Thus, Capurro concludes, “Putting the three concepts (“data”, “information”, and “knowledge”) gives the impression of a logical Message
Information
Understanding
Figure 8.5. The Capurro communication triad
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hierarchy: information is set together out of data and knowledge comes out from putting together information. This is a fairytale”. Summing up all discussions related to the Data–Information– Knowledge pyramid and the Data–Information–Knowledge–Wisdom pyramid, Bates (2010) writes that “it is difficult to take it from its popular meaning and develop it into something sufficiently refined to be useful for research”. In comparison with knowledge, information is an active structure. As some researchers have observed (cf., for example, (Hodgson and Knudsen, 2007)), information causes some action. This well correlates with the approach elaborated in the general theory of information (Burgin, 2010) where in contrast to the Data– Information–Knowledge Pyramid (Figure 8.1), Data–Information– Knowledge Chain (Figure 8.3) and Lenski’s approach, a different schema called the Knowledge–Information–Matter–Energy (KIME) Square is elaborated. It is based on the Ontological Principle O2a (the Special Transformation Principle) of the general theory of information discussed in the previous section. In essence, the Ontological Principle O2a implies that information is not of the same kind as knowledge and data, which are structures (Burgin, 2010). Taking matter as the name for all physical substances as opposed to energy and the vacuum, we have the system of relations represented by the diagram in Figure 8.6. The SIME Square visualizes and embodies the following principle: Information is related to structures as energy is related to matter similar Energy
≈
contains
Information
contain similar
Matter
≈
Structures
Figure 8.6. The SIME Square
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Some researchers also related information to structure of an object. For instance, information is characterized as a property of how entities are organized and arranged, but not the property of entities themselves (Reading, 2006). Other researchers have related information to form, while form is an explicit structure of an object. For instance, information is characterized as an attribute of the form (in-form-ation) that matter and energy have and not of the matter and energy themselves (Dretske, 2000). However, two issues, absence of the exact concept of structure and lack of understanding that structures can objectively exist, result in contradictions and misconceptions related to information. For instance, one author writes that information is simply a construct used to explain causal interaction, and in the next sentence, the same author asserts that information is a fundamental source of change in the natural world. Constructs cannot be sources of change, they can only explain change. Assigning the place for information in the world, some researchers also contrasted it to matter and energy. For instance, Wiener (1961) wrote, “Information is information, not matter or energy.” Bates (2010) also argues that information is not identical to the physical material that composes it; rather information is the pattern of organization of that material, not the material itself. However, pattern of organization is a pattern of structure, and according to the comprising definition of structure (cf., Sections 5.1.2 and 6.1), a pattern of structure is itself a structure. This is the main distinction of her definition from the definition in the general theory of information. Here we are interested not in all structures but only in structures called knowledge. That is why we build a special form of the SIME Square, which is called the KIME Square (cf., Figure 8.7). The KIME Square visualizes and embodies the following principle: Information is related to knowledge and data as energy is related to matter This schema means that information has essentially different nature than knowledge and data, which are of the same kind.
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similar Energy
≈
contains
Information
contains similar
Matter
≈
Knowledge/Data
Figure 8.7. The KIME square
Knowledge and data are structures, while information is only represented and can be carried by structures. This approach to knowledge is to a great extent similar to the theory of MacKay (1969), who asserted that knowledge must be understood as a coherent representation. Here a representation is any structure (a pattern, picture, or model) whether abstract or concrete which, by virtue of its features, could symbolize some other structure. Such representation structures contain information elements. Information elements in the sense of MacKay are not shapeless isolated things, but are imbedded in a structure, which helps the receiver to infer the meaning of the information intended by the sender. MacKay illustrates this thought with the following example. A single word rarely makes sense. Usually it takes a group of words or even several sentences acting together to clarify the role of a single word within a sentence. In her view the receiver will find it all the easier to integrate a piece of information in what she knows, the more familiar she is with all words in the group or, at least, with some parts of it. In particular, when people receive the same message several times from different sources, usually it will each time become more familiar and seem more plausible to them. This effect is by no means limited to verbal communication. For instance, the result of a scientific experiment gains in evidence the more often it is reproduced under similar conditions. However, the KIME representation of relations between knowledge and other basic phenomena (the KIME Square) is different from the approach of MacKay in several essential issues. First, MacKay
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does not specify representation of what knowledge is, while the KIME model does it (Burgin, 2010). Second, MacKay does not specify what kind of representation knowledge is, while the KIME model does it, assuming that knowledge is a structure or a system of structures. Third, the KIME model provides an explication of the quantum knowledge structure (cf., Section 4.1). The KIME Square well correlates with the distinction some researchers in information sciences have continually made between information and knowledge. For instance, Machlup (1983) and Sholle (1999) distinguish information and knowledge along three axes: 1. Multiplicity: Information is piecemeal, fragmented, particular, while knowledge is structured, coherent and universal. 2. Time: Information is timely, transitory, even ephemeral, while knowledge is enduring and temporally expansive. 3. Space: Information is a flow across spaces, while knowledge is a stock, specifically located, yet spatially expansive. These distinctions show that information is conceived of as a process, whereas knowledge is a specific kind of substance. This understanding well correlates with the information concept determined by the Ontological Principle 2c from the general theory of information (Burgin, 2010), which is incorporated in the KIME Square. Indeed, cognitive information is conceived as a kind of energy in the World of Structures, which under definite conditions induces structural work on knowledge structures. At the same time, knowledge is a substance in the World of Structures. Mey (1986) considered matter, energy, and information as a triad elements of which in their interaction constitute those features of (an) object that can be perceived by people. The SIME Square shows that it is necessary to add structure to this triad. Some researchers, such as von Weizs¨acker or Mattessich, wrote about similarities between energy and information. For instance, Mattessich (1993), assuming a holistic point of view and tracing information back to the physical level, concluded that every manifestation of active or potential energy is connected to some kind
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of information, just as the transmission of any kind of information requires active or potential energy. It is possible to ask why data and knowledge occupy the same place in the KIME Square (cf., Figure 8.6). It is possible to explain similarities and distinctions between data and knowledge with the following two metaphors. Metaphor A. Data and knowledge are like molecules, but data are like molecules of water, which has two atoms, while knowledge is like molecules of DNA, which unites billions of atoms. Metaphor B. Data and knowledge are like living beings, but data are like bacteria, while knowledge is like a human being. It is possible to base our understanding of how knowledge is different from data on the following descriptions of knowledge and data: • Knowledge is a compressed, goal oriented structural depository (carrier) of refined information. • Data is a raw structural depository (carrier) of information. It is possible to treat conversion of data into knowledge as refinement of information. An interesting question is which of these three essences, data, information, and knowledge, are external phenomena, i.e., they belong to the physical world, and which are internal phenomena, they belong to the mental world. Zins (2007) specifies five models of these relations: 1. Data and information are external phenomena, while knowledge is an internal phenomenon. 2. Data constitute an external phenomenon, while knowledge and information are internal phenomena. 3. Data constitute an external phenomenon, while knowledge and information can be both external and internal. 4. Data and information can be both external and internal, while knowledge is an internal phenomenon. 5. Data, knowledge, and information can be both external and internal.
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According to the general theory of information, data, knowledge, and information all belong to the world of structures having carriers in both mental and physical worlds (Burgin, 2010). New understanding that comes from the general theory of information necessitates changes of the definition of the term information in the American Heritage Dictionary (1996). The main implication is that it is more adequate to say and write that information gives knowledge of a specific event or situation. When people say and write that information is a collection of facts or data (The American Heritage Dictionary, 1996), the general theory of information suggests that it is more adequate to say and write that a collection of facts or data contains information. According to the American Heritage Dictionary, Information is: 1. 2. 3. 4.
Knowledge derived from study, experience, or instruction. Knowledge of a specific event or situation; intelligence. A collection of facts or data: “statistical information”. The act of informing or the condition of being informed; communication of knowledge: “Safety instructions are provided for the information of our passengers”. 5. A non-accidental signal or character used as an input to a computer or communications system (in Computer Science). 6. A numerical measure of the uncertainty of an experimental outcome. 7. A formal accusation of a crime made by a public officer rather than by grand jury indictment (in Law). According to the general theory of information, more adequate expressions for the above definitions are: 1. Knowledge is derived from information obtained from study, experience, or instruction. 2. Information gives knowledge of a specific event or situation; or information provides intelligence. 3. A collection of facts or data contains (statistical) information.
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4. The act of informing and the condition of being informed are components of information transmission; communication is an exchange of information. 5. A non-accidental signal or character used as an input to a computer or communications system contains information. 6. A numerical measure of the uncertainty of an experimental outcome is a measure of information. 7. A formal accusation of a crime made by a public officer contains information on who committed the crime. These expressions give some properties of information. It is possible to do similar explication of the correct meaning for the following definitions of information given in the Roget’s New Thesaurus (1995): 1. That which is known about a specific subject or situation: data, fact (used in plural), intelligence, knowledge, lore. 2. That which is known; the sum of what has been perceived, discovered, or inferred: knowledge, lore, wisdom. According to the general theory of information, more adequate expressions for the above definitions are: 1. That which is known about a specific subject or situation, i.e., data, facts, intelligence, knowledge, and lore, contains information. 2. That which is known, i.e., knowledge, contains information; the sum of what has been perceived, discovered, or inferred contains information, i.e., knowledge and lore contain information, while wisdom assumes possession of big quantity of information, i.e., a wise person has a lot of information and what is even more important, this person can properly use that information. This analysis of the common usage of the word information shows that the general theory of information does not essentially reverse the conventional meaning. The theory makes this meaning more precise by separating information from its carriers and representations. In our everyday speech, we do not differentiate between information
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and its representation. As a rule, it does not matter. However, this differentiation can be important in science and even in the everyday communication, for example, when we need to find the intended meaning of a message, going beyond its literal understanding. An important component of information processes is their context. According to the existential triad, it has three components: the structural information context, physical information context, and mental information context. These contexts are analyzed based on the general theory of communication and interaction, as well as the general theory of information. Thus, the physical information context consists of: ♦ ♦ ♦ ♦ ♦ ♦ ♦
A physical system that is the source of information. A physical system that is the receiver/recipient of information. A physical system that is the object of information or represents this object, e.g., brain processes represent thoughts. A physical system that is the carrier of information. A physical system that is the channel of information transmission or/and communication space. Information transmission as a physical process, which includes interactions of the source and recipient with the carrier. Physical environment in which information transmission goes on. Thus, the mental information context consists of:
♦ ♦ ♦
♦
An infological system that is the source of information, e.g., mind of a person. An infological system that is the recipient of information. Mental representations of the object of information, e.g., if the object of information is a book, then its mental representation is a reflection of the book content in the mind. Mental environment in which information transmission goes on. The structural information context consists of:
♦
A structural infological system of the source of information, e.g., knowledge of a person.
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A structural infological system of the receiver/recipient of information. The structure of the object of information. The structure of the carrier of information. The structure of the channel of information transmission or/and communication space. The structure of information transmission, which includes interactions of the source and recipient with the carrier. The structure of the environment in which information transmission goes on.
While the first two components of the structural information context are well understood in the majority of cases, the object of information is very often neglected. Given a portion of information I, we define information object as the object to which changes in the infological system caused by the portion of information I are related (Burgin, 2010). For instance, the object of a portion of cognitive information is the object about which this information gives knowledge to the receiver/recipient. The object of a portion of emotional information is the object to which emotions caused by this information in the receiver/recipient are related. Note that an object can be a complex system or even a collection of some objects. In the algorithmic information theory (Burgin, 2010), objects are words or texts of some languages. Knowledge structures explicated in previous chapters allow us to discern several kinds of data situated between raw data and knowledge providing a mathematical perspective on transformation of data into knowledge. According to the quantum theory of knowledge, the surface outer structure of knowledge has the following form: Knowledge domain (object) D
knowledge K .
(8.2)
As it was demonstrated, any knowledge object (domain) D has some indicative knowledge (a name of this object or domain) N
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and the aspect A reflected by the knowledge K. This gives us Diagram (8.3) representing the first-order knowledge structure. g A
K
q
p D
(8.3)
N f
It means that in Diagram (8.3), we have the knowledge domain (knowledge object) D, an aspect of the domain (object) D, the symbol N , which denotes the name of D or a class of names of the objects from D (a name of the object D), and K is a knowledge item (unit). The symbolic form of the first-order knowledge structure presented by Diagram (8.3) is [(D, f, N ), (q, p), (A, g, K)]. Knowledge quanta studied in Section 4.1 have the same structure, i.e., it is possible to describe them by Diagram (8.3). When Diagram (8.3) represents the structure of descriptive (quantum) knowledge, the aspect A of the domain (object) D is a property (feature) of D (the objects from D) and knowledge K is the attribute (abstract property) that corresponds to A (cf., Section 4.1). When Diagram (8.3) represents the structure of representational (quantum) knowledge, the aspect A of the domain (object) D is an intrinsic structure of D (of the objects from D) and knowledge K is a model (ascribed structure) of D (of the objects from D) (cf., Section 4.1). When Diagram (8.3) represents the structure of operational (quantum) knowledge, the knowledge domain (knowledge object) D consists of actions, operations or processes (is an action, operation or process), the aspect A is the class of structures of the actions, operations and processes from the domain D (the structure of the action, operation or process D), the symbol N denotes the class of names of these objects from D (the name of the object D), and K is the class of symbolic representations of objects from D, such as systems of instructions, algorithms or procedures (cf., Section 4.1).
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Giving an exact definition of the first-order knowledge structure by building its mathematical model in the form [(D, f, N ), (q, p), (A, g, K)], allows us to study data in a more exact form than before, discern data from knowledge, and specify several types of data. The first type is raw or uninterpreted data. Their first-order structure is represented by Diagram (8.4). q U
A .
(8.4)
In this diagram, U consists of some objects, q is a relation between U and A, while A denotes: (1) attributes of these objects in the case of descriptive data, (2) representations of these objects in the case of representational data, and (3) operations, action and processes related to these objects in the case of operational data. For instance, all three types of data are used in the object-oriented programming (OOP) for object description. OOP is a programming paradigm based on the concept of abstract objects represented by data structures that include attributes in the form of descriptive data, characteristics in the form of representational data and methods in the form of operational data. Raw data correspond to the substantial component of knowledge (cf., Section 4.1). The difference is that in contrast to the substantial component of knowledge, raw data are not related to any definite property. Having raw data, a person or a computer system can transform them into knowledge by means of additional knowledge that this person or computer system already has. The second type of data is formally interpreted data. They are related to the abstract property P , forming a named subset of the information component of a knowledge unit represented by Diagram (8.3), and the first-order structure of this component is represented by Diagram (8.5). p N
L .
(8.5)
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Here N consists of the names of objects and L is the set of values (the scale) of the property P on the names of objects that tentatively have these properties, while p is the relation that connects names of considered objects with values of the ascribed properties of these objects, i.e., p is the functional component (evaluation function) of the property P . Example 8.2.1. Automatic instrument systems perform measurement, which is a process of data acquisition, producing formally interpreted data. An unmanned weather station, for example, may record daily maximum and minimum temperatures. Such recordings are formally interpreted data because numbers (the set L) are corresponded to names “maximal temperature” and “minimal temperature” (the set N ). Formally interpreted data correspond to the symbolic component of knowledge (cf., Section 4.1). The difference is that formally interpreted data are not necessarily included in a knowledge system. The third type is attributed data. They are related not to the abstract property P but to the values of an intrinsic property, i.e., to the attribute A. The first-order structure of attributed data is represented by Diagram (8.6). g A
L
.
(8.6)
The fourth type of data is naming data, the first-order structure of which is represented by Diagram (8.7). f U
N
.
(8.7)
There are two more types of data, which are more enhanced and are closer to knowledge. The fourth type is object interpreted data. Their first-order structure is represented by Diagram (8.8). L p f U
N
.
(8.8)
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Taking Example 8.2.1 and adding information that maximal and minimal temperatures are temperatures of air (or water), we obtain object interpreted data. In this process, Diagram (8.5) is extended to Diagram (8.8). The fifth type of data is object attributed data. Their first-order structure is represented by Diagram (8.9). g L
A q
.
(8.9)
N
It is interesting to remark that the statement about the correspondence between linguistic constructions representing knowledge and things in the external world as a necessary component of knowledge, which makes it different from data, was discovered in (Burgin, 1989a) and then reiterated in (Davis et al., 1993) and (Burgin, 1995a; 2004; 2010). Remark 8.2.1. It is possible to treat all types of data as incomplete knowledge. Formally naming data are names of considered objects and correspond to the naming component of knowledge (cf., Section 4.1). The difference is that naming data are not necessarily included in a knowledge system. It is important to understand that data and knowledge can themselves be objects, which have names, as well as intrinsic and ascribed properties. In particular, when the domain U consists of (some kind of) data, then in this case, we come to named data, which play an important role in the recent ideas for the development of the Internet (Jacobson et al., 2012; Ntuli and Han, 2012). To understand what named data are and why they are so popular, we consider the schema of the data transfer on the Internet. The contemporary Internet is based on the TCP/IP communication protocol. In it, the transmission control protocol (TCP) part is performs separation of the file/message into packets on the source computer and reassembling the received packets at the destination,
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e.g., at the recipient computer. The internet protocol (IP) part handles the address of the destination computer so that each packet is routed (sent) to its proper destination. In named data networking (NDN) architecture for the future Internet, the transmitted packets of data carry data names rather than source or destination addresses. The developers of this architecture believe that this conceptually simple shift will have far-reaching implications for how people design, develop, deploy, and use networks and applications. The named data principle implies that a communication network should allow a user to focus on the data identified by their names he or she needs, rather than having to reference a specific, physical location where that data would be retrieved. Actually, Internet packets of data are already named by destination addresses and the new approach suggests changing these names to the original data names (identifiers). It is assumed that such a renaming brings potential for a wide range of benefits such as simpler configuration of network devices, building security into the network at the data level and content caching to reduce congestion and improve delivery speed. In addition, sustained growth in e-commerce, digital media, social networking, and smartphone applications has led to prevailing use of the Internet in the role of a distribution network. Utilization of a point-to-point communication protocol in distribution networks is complex and error-prone, while NDN better suits distribution environment. In this context, named sets give a natural mathematical model for named data, which form the naming component of the knowledge quanta with data as their object or domain (Burgin and Tandon, 2006). Consequently, named set theory provides powerful means for network algorithms and procedures in the form of various operations and correspondences (Burgin, 2011). Another example of naming data is named graphs, which are a key structure of the Semantic Web architecture. In it, a set of resource description framework (RDF) statements (a graph) are identified using a universal resource identifier (URI), allowing derivation of descriptions of context, provenance information or other metadata. This shows that named graphs form an extension of the RDF data
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model giving additional evidence for importance of naming data in contemporary information technology. 8.3. Information as a source of knowledge Some people drink deeply from the fountain of knowledge. Others just gargle. Grant M. Bright
As the cognitive infological system contains knowledge of the system it belongs, cognitive information is the source of knowledge changes. This perfectly correlates with the approach of Dretske (1983) and Goldman (1967) who defined knowledge as information-caused belief, i.e., information produces beliefs that are called knowledge. Moreover, it is impossible to obtain knowledge without information. Dretske (1983) develops this idea, implying that information produces beliefs, which, according to our definition, are also elements of the cognitive infological system. Moreover, many researchers relate all information exclusively to knowledge. For instance, Mackay (1969) writes: “Suppose we begin by asking ourselves what we mean by information. Roughly speaking, we say that we have gained information when we know something now that we didn’t know before; when ‘what we know’ has changed.”
However, information comes to the knower not by itself but in some carrier. Often this carrier is called data. Therefore, the new understanding of relations between information and knowledge suggests that a specific fundamental triad called the Data–Knowledge Triad provides a more relevant theoretical explication for Elliot’s metaphor than the Data–Information–knowledge Pyramid. Namely, the Data–Knowledge Triad has the following form: information Data
Knowledge
.
(8.10)
Synthesizing data, information, and knowledge into a conceptual structure, the triadic representation (8.10) has the following interpretation, which reveals data–knowledge dynamics:
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Knowledge, Data, and Information
Data, under the influence (action) of additional information, become (are transformed into) knowledge That is, information is the active essence that transforms data into knowledge. It is similar to the situation in the physical world, where energy is used to perform work, which changes material things, their positions and dynamics. Diagram (8.10) can be also interpreted in the following way:
➢ Knowledge is a compressed, goal oriented structural depository of refined information. ➢ Data is a raw symbolic depository of information. ➢ Conversion of data into knowledge is a refinement of information. According to Bates (2010), Dretske conceives information as commodity that capable of yielding knowledge (Dretske, 1981). Although Dretske assumes that knowledge and hence, information are always true, his definition well correlates with the Data–Knowledge triad. Barwise and Seligman (1997) write “information is closely tied to knowledge”. Meadow and Yuan suggest (1997) that the recipient’s knowledge increases or changes as a result of receipt and processing of new information. Cognitive information related to knowledge was also studied by Shreider (1967). At the same time, other researchers connect cognitive information to experience. For instance, Boulding (1956) calls a collection of experiences by the name image and explains that messages consist of information as they are structured experiences, while the meaning of a message is the change that it produces in the image. However, experience as trait of a personality is, in essence, explicit and implicit knowledge obtained from experience as a process. So, this understanding also connects information to knowledge. Diagram (8.10) is a sub-diagram (a part) of Diagram (8.11). effective information Cognitive Information
effective information Data
Knowledge
(8.11) This relation between information and knowledge is well understood in the area of information management. For instance, Kirk
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prior expectations
the World
data
data filters
information
the agents knowledge base
Figure 8.8. Transformation of data into knowledge
(1999) writes, “the effectiveness of information management can be measured by the extent of knowledge creation or innovation in organizations”. The triadic representation also well correlates with opinions of Bar-Hillel (1964), Heyderhoff and Hildebrand (1973), Brookes (1980), and Mizzaro (1996; 1998; 2001), who describe an information process as a process of knowledge acquisition, which is often understood as reduction of uncertainty. The information triad (8.10) also well correlates with the approach of Boisot (1998) where information is what an observer extracts from data as a function of his/her expectations or prior knowledge. Boisot illustrates the point by means of the diagram given in Figure 8.8. The diagram from Figure 8.8 indicates that data, or more exactly, data representations, are physical, but not information. Therefore, data are rooted in the world’s physical properties. Knowledge, by contrast, is rooted in the comprehensions, interpretations, estimates, and expectations of individuals. Information is what a knowing individual can use for creation and extraction of knowledge from data, given the capacity of his/her knowledge. Physical extraction essentially depends on used types of knowledge, i.e., whether it is represented by (contained in) algorithms and logic, models and language, operations and procedures, goals, and problems. The situation is similar to the situation with (physical) energy when energy is extracted from different substances and processes: from petroleum, natural gas, coal, wood, sunlight, wind, ocean tides, etc. This extraction essentially depends on used devices, technologies, and techniques. For instance, solar cells are used to convert sunlight into electricity. Wind
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and water turbines rotate magnets and in such a way create electric current. Petroleum-powered engines make cars ride and planes fly. The KIME Square shows essential distinction between knowledge and information in general, as well as between knowledge and cognitive information, in particular. This distinction has important implications for education. For instance, transaction of information (for example, in a teaching process) does not give knowledge itself. It only causes such changes that may result in the growth of knowledge. This correlates with the approaches of Dretske (1981) and MacKay (1969), who declare that information increases knowledge and knowledge is considered as a completed act of information. However, the general theory of information differs from Dretske’s and MacKay’s conceptions of information because the general theory of information demonstrates that information transaction may result not only in the growth of knowledge but also in the decrease of knowledge (Burgin, 1994). An obvious case for the decrease of knowledge is misinformation and disinformation. For instance, people know about the tragedy of the Holocaust during the World War II. However, when articles and books denying the Holocaust appear, some people believe this and lose their knowledge about the Holocaust. Disinformation, or false information, is even used to corrupt opponent’s knowledge in information warfare. However, the general theory of information differs from this conception of information because it demonstrates that information transaction may result also in the decrease of knowledge (Burgin, 1994). Namely, even genuine information can decrease knowledge. For instance, some outstanding thinkers in ancient time, e.g., Greek philosophers Leucippus (5th century B.C.E.) and Democritus (ca. 460–370 B.C.E.), knew, in some sense, that all physical things consisted of atoms. Having no proofs of this feature of nature and lacking any other information supporting it, people lost this knowledge. Although some sources preserved these ideas, they were considered as false beliefs. So, information about absence of supporting evidence often decreases knowledge in society. Nevertheless later when physics and chemistry matured, they found experimental evidence for atomic
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structure of physical objects, and knowledge about these structures was not only restored but also essentially expanded. In a similar way, knowledge about America was periodically lost in Europe and America was rediscovered several times. A thousand years ago, nearly half a millennium before Columbus, the Norse extended their explorations from Iceland and Greenland to the shores of Northeastern North America, and, possibly, beyond. According to the Olaf saga, the glory of having discovered America belongs to Bjarni, son of Herjulf, who was believed to have discovered Vinland, Markland, and Helluland as early as 985 or 986 on a voyage from Iceland to Greenland (Reeves et al., 1906). There are also stories of other pre-Columbian discoveries of America by Phoenician, Irish, and Welsh, but all accounts of such discoveries rest on insufficiently vague or unreliable testimony. Sometimes people lose some knowledge and achieve other knowledge. For instance, for centuries mathematicians knew that there was only one geometry. The famous German philosopher Kant (1724– 1804) wrote that knowledge about the Euclidean geometry is given to people a priory, i.e., without special learning. However, when mathematicians accepted the discovery of non-Euclidean geometries, they lost the knowledge about uniqueness of geometry. This knowledge was substituted by a more exact knowledge about a diversity of geometries. Some may argue that it was only a belief in uniqueness but not a real knowledge. However, in the 18th century, for example, the statement There is only one geometry (8.12) was sufficiently validated (for that time). In addition, it well correlated with reality known at that time as the Euclidean geometry was successfully applied in physics. Thus, there all grounds to assume that the statement (8.12) represented knowledge of the 18th century in Europe. Likewise, now the vast majority of people know that two plus two is equal to four although existence of non-Diophantine arithmetics shows that there are situations when two plus two is not equal to four (Burgin, 1997c; 2007; 2010c). Distinction between knowledge and cognitive information implies that transaction of information (for example, in a teaching process)
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does not give knowledge itself. It only causes changes that may result in the growth of knowledge. In other words, it is possible to transmit only information from one system to another, allowing a corresponding infological system to transform data into knowledge. In microphysics, the main objects are subatomic particles and quantum fields of interaction. In this context, knowledge and data play role of particles, while information realizes interaction. As we have seen, usually people assume that information creates knowledge. For instance, when an individual sends some text by e-mail, this text, which is usually a knowledge representation, is converted to data packages, which are then transmitted to the recipient. However, utilizing information, it is possible to create data from knowledge. This shows that the triad (8.13) inverse to the triad (8.10) is also meaningful and reflects a definite type of information processes. Knowledge
Data .
(8.13)
At the same time, data are used not only knowledge generation but also for deriving other epistemic structures such as beliefs and fantasies. It gives us two more information diagrams. information Data
(8.14)
Beliefs
and information Data
Fantasy
.
(8.15)
To explain why it is more efficient, i.e., more adequate to reality and more productive as cognitive hypothesis, etc., to consider information as the knowledge content than to treat information in the same category as data and knowledge, let us consider people’s beliefs. Can we say that belief is structuring of information or structured information? No. However, we understand that people’s beliefs are formed by the impact some information has on people’s mind. People process and refine information and form beliefs, knowledge, ideas, hypotheses, etc. Beliefs are structured in the same way as
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knowledge, but in contrast to knowledge, they are not sufficiently justified. Thus, the concept of information in the sense of Stonier (1991; 1992) does not allow fitting beliefs into the general schema, while the concept of information in the sense of the general theory of information naturally integrates beliefs into the system data, information, and knowledge. There are other examples that give evidence to support the statement that Diagrams (8.10) and (8.11) correctly explains relation between information, data and knowledge. In spite of this, people are very conservative in their beliefs and do not want to change what they “know” to a more grounded knowledge mostly because it demands an intellectual effort (action) and even people are systems that comply with the principle of minimal action. 8.4. Dynamic aspects of knowledge, data, and information interaction Knowledge must come through action; you can have no test which is not fanciful, save by trial. Sophocles
As it is demonstrated in the previous section, information changes epistemic structures in general and knowledge, in particular. Using epistemic spaces (cf., Section 3.1) as theoretical patterns of knowledge systems, we model information by epistemic information operators. They act either in pure or in weighted epistemic spaces, transforming these spaces and describing dynamics of infological systems and epistemic spaces, which consists of epistemic structures modeling infological systems. Let us consider two (weighted) epistemic spaces E and H. Definition 8.4.1. (a) A (partial) mapping A : E → H is called an epistemic information operator. (b) If both epistemic spaces E and H have the same structure and an information operator A : E → H preserves this structure, then A is called a structured information operator or an information homomorphism.
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(c) When E = H, the operator A is called an inner epistemic information operator. An inner epistemic information operator changes elements or states or elements of (weighted) epistemic spaces and multispaces. There are three pure basic types of inner epistemic information operators: content, bond, and weight operators. Definition 8.4.2. A content epistemic information operator acts on symbolic epistemic items (structures) in an epistemic space changing its state. For instance, all information operators studied in (Mizzaro, 2001; Burgin, 2010; 2011a) are content epistemic information operators. Definition 8.4.3. A bond epistemic information operator acts on connections (bonds or relations) between symbolic epistemic items (structures) in an epistemic space changing its state. Such operators as interpretation and reinterpretation of information/knowledge items (Burgin, 2011) are bond epistemic information operators. In weighted epistemic spaces, we have one more type of inner epistemic information operators. Definition 8.4.4. A weight epistemic information operator acts on weights of symbolic epistemic items (structures) in an epistemic space changing its state. In addition, there are mixed epistemic information operators. Definition 8.4.5. A mixed epistemic information operator acts on symbolic epistemic items (structures) and on connections (bonds or relations) between symbolic epistemic items (structures) and/or weights of symbolic epistemic items (structures) in an epistemic space changing its state. Operators of logical inference, such as rules of deduction, are mixed epistemic information operators act because they add new knowledge items in the form of propositions or/and predicates and establish relations of inferrability/deducibility between propositions or/and predicates.
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There are three key content inner epistemic information operators: • An addition/deletion content operator AD (DL) adds a symbolic epistemic item (knowledge item) to the state of the (weighted) epistemic space. • A transformation/substitution content operator TR (ST) transforms or substitutes a symbolic epistemic item in the state of the (weighted) epistemic space. • A substantiation content operator switches on or off an existing symbolic epistemic item in the state of the (weighted) epistemic space. Thesaurus information operators are a special type of epistemic information operators. Definition 8.4.6. A content thesaurus information operator acts on knowledge items in a knowledge space. For instance, all information operators studied in (Mizzaro, 1996; 1998; 2001; Burgin, 2010a) are content thesaurus information operators. Definition 8.4.7. A bond thesaurus information operator acts on connections (bonds or relations) between knowledge items in a knowledge state. Such operators as interpretation and reinterpretation of information/knowledge items are bond thesaurus information operators. Example 8.4.1. When an intelligent agent learns, it usually adds knowledge items to its knowledge base changing in such a way the state of this base. The knowledge base is naturally represented by an appropriate epistemic space. Thus, growth of knowledge in the process of learning is modeled by application of addition content information operators. Definition 8.4.8. (a) A replica of an epistemic (knowledge) item is another knowledge item equivalent to the initial one.
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(b) A replication epistemic information operator REPL makes a replica of an epistemic (knowledge) item and adds it to the current epistemic (knowledge) state. Note that copies of an epistemic (knowledge) item always are its replicas but a replica of an epistemic (knowledge) item is not always its copy. Example 8.4.2. Let us consider logical knowledge representation in which knowledge items are propositions. Then according to laws of logic there are equivalent propositions. For instance, taking the proposition (1) “B implies A”, we have equivalent propositions (2) “A follows from B”, (3) “If B, then A”, and (4) “A is a consequence of B”. All of them are replicas of one another although they are not copies. Addition of symbolic epistemic items can be performed by five operations: • by generation of a new item inside the current state of the (weighted) epistemic space; • by generation of a new item outside (the current state of) the epistemic space and its transition into the current state of the (weighted) epistemic space; • by transition of an existing item from the epistemic space into the current state of the (weighted) epistemic space; • by replication of an item from the current state of the (weighted) epistemic space; • by replication of an item outside (the current state of) the epistemic space and transition of this replica into the current state of the (weighted) epistemic space. Consequently, substitution of symbolic epistemic items can be performed by five operations because substitution is the sequential composition of elimination and addition. In the case of stratified epistemic spaces, there is one more type of key content epistemic operators, namely, a moving operators MV, which moves epistemic items from one strata to another (Burgin, 2011a).
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A transformation epistemic information operator TR takes a group of epistemic (knowledge) items (may be, one item) from the current epistemic (knowledge) state and transforms it into another group of epistemic (knowledge) items (may be, into one item). A generation epistemic information operator GR takes a group of epistemic (knowledge) items (may be, one item) from the current epistemic (knowledge) state and generates another group of epistemic (knowledge) items (may be, one item). The difference between transformation and generation is that in generation, the initial group of epistemic items is preserved, while in transformation, it is not preserved. There are three key bond epistemic information operators: (1) An addition/deletion bond operator adds or deletes a connection (bond or relation) between symbolic epistemic items in the current epistemic (knowledge) state of the (weighted) epistemic space. (2) A substitution bond operator changes a connection (bond or relation) between symbolic epistemic items in the current epistemic (knowledge) state of the (weighted) epistemic space to another connection (bond or relation). (3) A substantiation bond operator switches on or off an existing connection (bond or relation) between symbolic epistemic items in the current epistemic (knowledge) state of the (weighted) epistemic space. There are three basic weight epistemic information operators: (1) An addition/deletion weight operator adds or deletes a weight to symbolic epistemic items in the weighted epistemic space. For instance, epistemic items had one weight constructible, which indicates whether the structure is constructible or not. Then the weight complexity was added by the addition operator. The new weight reflects complexity of the structure construction. (2) A transformation/substitution weight operator substitutes one weight of symbolic epistemic items in the weighted epistemic space by another weight.
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For instance, let us consider the situation when weighted epistemic items (structures) had the weight justification, which is then substituted by the weight provability. In different circumstances, a transformation operator changes the weight complexity for the weight hardship. Such situation happens in software engineering (Burgin and Debnath, 2003). (3) A value-changing weight operator changes weights of symbolic epistemic items in the weighted epistemic space. For instance, let us assume the value of the weight complexity for texts was estimated based on recursive algorithms, such as Turing machines. Later the complexity estimate was obtained by means of super- recursive algorithms, such as inductive Turing machines. As it is proved that super-recursive algorithms decrease algorithmic complexity (Burgin, 2005), the value-changing operator has to be applied to give correct complexity of the texts processed by inductive Turing machines. There are also mixed epistemic information operators. A mixed epistemic information operator acts on symbolic epistemic items (structures) in some epistemic state, on their weights and on their connections (bonds or relations). For instance, a mixed epistemic information operator can act on knowledge items in a knowledge state, their weights and their connections (bonds or relations). Operators of logical inference, such as rules of deduction, are mixed epistemic information operators act because they add new knowledge items in the form of propositions or/and predicates and establish relations of provability/deducibility between propositions or/and predicates. Subspaces of knowledge spaces represent subsystems of knowledge systems. For instance, in large knowledge systems, such as a scientific theory, it is possible to separate the subsystem of denotational knowledge and the subsystem of operational knowledge. It looks like it might be sufficient to consider only finite or at least, locally finite agents. However, if knowledge is represented by logical statements and it is assumed (as it is done, for example, in the theory
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of semantic information developed by Bar-Hillel and Carnap (1958) that any knowledge system contains all logical consequences of all its elements, then an agent with such knowledge system is infinite. In information algebras, portions of information are represented by close subsets of sentences from a logical language L (Kohlas and St¨ ark, 2007). However, in conventional logics closed with respect to such information operators as deduction, sets are infinite because any sentence p implies p ∨ q for any sentence q from L, which is, as a rule, infinite (cf., for example, (Shoenfield, 1967)). Thus, in the context of classical logic and information algebras any portion of information has infinitely many representations. Consequently, such a portion generates a system with the infinite number of knowledge items. Let us consider epistemic information operators from a weighted epistemic space E into a weighted epistemic space H. Definition 8.4.9. An epistemic information operator A : E → H is called: (a) stationary if for any epistemic structures e and l, the equality A(e; w1 , . . . , wk ) = (l; v1 , . . . , vh ) implies the equality A(e; u1 , . . . , uk ) = (l; q1 , . . . , qh ) for any (e; u1 , . . . , uk ). (b) permanent if for any weighted epistemic structure (e; w1 , . . . , wk ), we have A(e; w1 , . . . , wk ) = (e; v1 , . . . , vk ). (c) semi-permanent if for any epistemic structure e and any number, k, there is a number h such that for any system of weights (w1 , . . . , wk ) of e, we have A(e; w1 , . . . , wk ) = (e; v1 , . . . , vh ). Definitions imply the following result. Lemma 8.4.1. Any permanent epistemic information operator A is semi-permanent, while any semi-permanent epistemic information operator B is stationary. Lemma 8.4.2. (a) Any weight epistemic information operator is semi-permanent.
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(b) Operators of adding weights and of deleting weights are semipermanent but not permanent. (c) Operators of substituting weights and of changing values of weights are permanent. Stationary epistemic information operator are related to morphisms of epistemic vector bundles. We remind (Le Potier, 1997) that a morphism of a vector bundle E = (E, p, B) into a vector bundle H = (H, r, D) is a pair of continuous mappings f : E → H and g : B → D such that Diagram (8.16) is commutative. f E
H
p
r B
.
(8.16)
D g
Note that morphisms of a vector bundles are mappings (morphisms) of named sets, which are studied in the theory of named sets (Burgin, 2011). Thus, weighted epistemic spaces form a category with vector bundle morphisms as its morphisms. It makes possible application of results for categorical information modeling (Burgin, 2010b; 2011b) to epistemic information operators in weighted epistemic spaces. Let us consider two weighted epistemic spacesE and H. Proposition 8.4.1. An epistemic information operator A : E → H is stationary if and only if it induces a morphism of the vector bundle E = (E, pE , Ee ) into the vector bundle H = (H, pH , He ). Proof . Necessity. Let us consider a stationary epistemic information operator A : E → H. Then by Definition 8.4.9, each fiber Fa of the vector bundle E = (E, pE , Ee ) is mapped by A into a single fiber Gb of the vector bundle H = (H, pH , He ). That is why, we can build the mapping C : Ee → He defining C(a) = b. By construction, we have
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C(pE (x)) = pH (A(x)) for all elements x from E. This gives us the commutative diagram (8.17). A E
H pH
pE EE
C
.
(8.17)
HE
It means that the pair (A, C) is a morphism of the vector bundle E = (E, pE , Ee ) into the vector bundle H = (H, pH , He ). Sufficiency. If the pair (A, C) is a morphism of the vector bundle E = (E, pE , Ee ) into the vector bundle H = (H, pH , He ), i.e., the Diagram (4) is commutative. Consequently, each fiber Fa of the vector bundle E = (E, pE , Ee ) is mapped by A into a single fiber Gb of the vector bundle H = (H, pH , He ), i.e., A is a stationary epistemic information operator. Definition 8.4.10. A stationary epistemic information operator A : E → H is called: (a) uniform if for any real number a and any weighted epistemic structures (e; w1 , . . . , wk ) and (l; v1 , . . . , vh ), the equality A(e; w1 , . . . , wk ) = (l; v1 , . . . , vh ) implies the equality A(e; au 1 , . . . , au k ) = (l; av 1 , . . . , av h ). (b) additive if A(e; w1 + u1 , . . . , wk + uk ) = A(e; w1 , . . . , wk ) + A(e; u1 , . . . , uk ). (c) linear if it is uniform and additive. Example 8.4.3. Let us consider a weighted epistemic space E, in which there are n epistemic structures e1 , e2 , e3 , . . . , en , the distance between any two of them is 1 and each of them has one weight w the range of which is the real line R. Taking a real number t, we define the epistemic information operator A by the following rule: A(ek , w) = (ek , tw ). By definition, this operator is linear and thus, uniform and additive.
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At the same time, the epistemic information operator B with B(ek , w) = (ek , k) is neither uniform nor additive nor linear. To study linear epistemic information operators, we need some topological constructions. In (Burgin, 2004b), the concept of path Q-connectedness is introduced and studied. To find relations between linearity and boundedness of information operators, we need to further develop this concept for metric spaces. Path Q-connectedness is important for epistemic spaces because in many cases, epistemic spaces have the structure of a graph or network of epistemic items connected by various relations. For instance, Motter et al. (2002) build the conceptual network of a language using exactly this approach. Let C be a subspace of a metric space U with the distance function d. Definition 8.4.11. The space C is called path (q, r)-connected in U if for any two points a and b in C, there exists a sequence a1 , a2 , a3 , . . . , an of points in C such that d(a, a1 ) ≤ r, d(an , b) ≤ r, d(ai , ai+1 ) ≤ r for all i = 1, 2, 3, . . . , n − 1 and [d(a, a1 ) + d(a1 , a2 ) + d(a2 , a3 ) + · · · + d(an−1 , an ) + d(an , b)] < q · d(a, b). Example 8.4.4. Taking a square in which the length of the side is equal and which is situated in an Euclidean plane (cf., Figure 8.9) and the space of its vertices C = {A, B, C, D}, we see that it is path (2, 1)-connected but it is not path (1, 1)-connected and not path A
B
C
D
Figure 8.9. A topological space that is path (2, 12 )-connected but it is not path (1, 1)-connected
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√ (2, 12 )-connected. Indeed, d(A, C) = 2, the lengths of both paths (A, B, C) and (A, D, C) is equal to 2 with d(A, B) = d(B, C) = d(A, D) = d(D, C) = 1. Consequently, d(A, B) + d(B, C) < 2d(A, C) but d(A, B) + d(B, C) > 1 · d(A, C). Thus, the space C is path (2, 1)-connected but it is not path (1, 1)-connected. In addition, it is not path (q, p)-connected when p < 1 because there no paths between A and C such that the distance between two consecutive points is less than 1. However, not all sets in metric spaces are path (q, r)-connected. Example 8.4.5. The parabola y = x2 as the space C, we see that it is not path (q, r)-connected for any numbers q and r. Indeed, taking points u = (x, x2 ) and w = (−x, x2 ), we see that the distance between these points d(u, w) = 2x. Now let us suppose that this parabola C is path (q, r)-connected for some numbers q and r. It means that there is a sequence a1 , a2 , a3 . . . , an of points in C such that d(u, a1 ) ≤ r, d(an , w) ≤ r, d(ai , ai+1 ) ≤ r for all i = 1, 2, 3, . . . , n − 1. As all points a1 , a2 , a3 , . . . , an belong to C, the sum [d(u, a1 ) + d(a1 , a2 ) + d(a2 , a3 ) + · · · + d(an−1 , an ) + d(an , w)] is larger than x2 − ( 12 r)2 . At the same time, we have [d(u, a1 ) + d(a1 , a2 ) + d(a2 , a3 ) + · · · + d(an−1 , an ) + d(an , w)] < q · d(u, w). Thus, x2 − ( 12 r)2 < 2qx because d(u, w) = 2x. Transforming this inequality, we obtain 2 1 2 r , x − 2qx < 2 2 1 2 r . x(x − 2q) < 2 For sufficiently big x, this inequality cannot be valid because r is a fixed number. Thus, our assumption is not true and the parabola y = x2 is not path (q, r)-connected for any numbers q and r. Here we are mostly interested in content epistemic information operators, which we simply call epistemic information operators in what follows. Using classical concepts of continuity and boundedness (Kuratowski, 1966; Alexandroff, 1961), as well as the concept of (p, q)-continuity from neoclassical analysis (Burgin, 2008), we determine important classes of epistemic information operators.
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Let us consider epistemic information operators from a weighted epistemic space E with a metric d into a weighted epistemic space H with a metric d. In this case, it is possible to define the diameter d of sets in E and in H. Namely, if X ⊆ E, then d(X) = sup{d(x, z); x, z ∈ X} when this supremum exists and undefined otherwise. Definition 8.4.12. An epistemic information operator A : E → H is called: (a) bounded if given a state X of E, for any number r, there is a number t such that the condition d(X) ≤ r, implies the condition d(A(X)) ≤ t. (b) uniformly bounded if for any number r, there is a number t such that for any state X of E, the condition d(X) ≤ r, implies the condition d(A(X)) ≤ t. (c) continuous if A is a continuous mapping. (d) (p, q)-continuous if A is a (p, q)-continuous mapping. Boundedness of an epistemic information operator A means that when the distances between epistemic structures in a set X are bounded, then the distances between epistemic structures in the image A(X) of the set X are bounded. (p, q)-continuity of an epistemic information operator A informally means that when the distances between epistemic structures in a set X are not larger than p, then the distances between epistemic structures in the image A(X) of the set X are not larger than q. Lemma 8.4.3. Any uniformly bounded epistemic information operator A: E → H is bounded. In a discrete metric space, any point is an open and a closed set (Kuratowski, 1966). Thus, any epistemic information operator in an epistemic space E is continuous, open and closed. Results from (Burgin, 2008) give us the following property of epistemic information operators. Lemma 8.4.4. An epistemic information operator A : E → H continuous if and only if it is (0, 0)-continuous.
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In addition, we need some constructions from neoclassical analysis, such as fuzzy limits and fuzzy continuity (Burgin, 2008). Let r ∈ R + . Definition 8.4.13. (a) An element a from E is called an r-limit of a sequence l (it is denoted by a = r − limi→∞ ai or a = r − lim l) if for any ε ∈ R++ the inequality d(a, ai ) < r + ε is valid for almost all ai , i.e., there is such n that for any i > n, we have d(a, ai ) < r + ε. (b) A sequence l that has an r-limit is called r-convergent and it is said that l r-converges to its r-limit a. Informally, a is an r-limit of a sequence l if for an arbitrarily small ε, the distance between a and all but a finite number of elements from l is smaller than r + ε. In other words, an element a is an r-limit of a sequence l if for any ε ∈ R++ almost all ai belong to the interval (a − r − ε, a + r + ε). It is a natural generalization of the classical concept of a limit as the following result demonstrates. Lemma 8.4.5. A point a limit of a sequence l if and only if it is a 0-limit of the sequence l. We also need fuzzy continuity. Definition 8.4.14. (a) A partial function f : R → R is called (q, r)-continuous at a point a ∈ R if for any sequence l = {ai ∈ R; i = 1, 2, 3, . . .}, for which a is an q-limit, the point f (a) is an r-limit of the sequence {f (ai ) ∈ R; i = 1, 2, 3, . . .}. (b) A function f : R → R is called (q, r)-continuous in (inside) set X ⊆ R if f (x) (the restriction of f (x) on X) is (q, r)-continuous at each point a from X ∩ Domf . Fuzzy continuity is a natural generalization of the classical concept of continuity as the following result demonstrates. Lemma 8.4.6. A function f (x) is continuous at a point a ∈ R if and only if it is (0, 0)-continuous at the point a. These results show that the concept of (q, r)-continuity is a natural extension of the concept of conventional continuity. Lemma 8.4.7. If t > r, and p < q, then any (q, r)-continuous at a function f (x) is also (p, t)-continuous at a.
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Note that if q < p, then it is possible that a (q, r)-continuous at a function is not (p, r)-continuous at a. For instance, the function f (x) = x is (0, 0)-continuous at the point 0, but for any p > 0, it is not (p, 0)-continuous at 0. Let us consider two (weighted) epistemic spaces E and H, assuming that the space E is a path (q, 1)-connected metric space with the metric d and the space H is a metric space with the metric d. Denoting metrics in different spaces by the same letter d follows the mathematical tradition and does not cause confusion. Theorem 8.4.1. A epistemic information operator A : E → H is (1, k)-continuous for some positive number k if and only if it is uniformly bounded. Proof . Necessity. Let us consider a (1, k)-continuous epistemic information operator A : E → H and two points a and b from a set X in E. By Definition 8.4.13, there exists a path (sequence of points) l = {a1 , a2 , a3 , . . . , an } in E such that d(a, a1 ) ≤ 1,
d(an , b) ≤ 1,
d(ai , ai+1 ) ≤ 1
for all i = 1, 2, 3, . . . , n − 1 and [d(a, a1 ) + d(a1 , a2 ) + d(a2 , a3 ) + · · · + d(an−1 , an ) + d(an , b)] < q · d(a, b). If for some i, d(ai , ai+1 ) < 12 and d(ai+1 , ai+2 ) < 12 , then it is possible to eliminate the point ai+1 , from the path because by properties of metric, d(ai , ai+2 ) ≤ d(ai+1 , ai+2 ) + d(ai , ai+1 ) < 12 + 12 = 1 and the new path is not longer than the previous one. We can reduce the initial path in such a way and assume that we have an irreducible path l = {a1 , a2 , a3 , . . . , an } between a and b. As A is a (1, k)-continuous epistemic information operator, d(A(a), A(a1 )) < k, d(A(an ), A(b)) < k, d(A(ai ), A(ai+1 )) < k for all i = 1, 2, 3, . . . , n − 1. Thus, d(A(a), A(b)) ≤ d(A(a), A(a1 )) + d(A(a1 ), A(a2 )) + · · · + d(A(ai−1 ), A(an )) + d(A(an ), A(b)) < k(n + 1).
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Now let us estimate the number n. Taking an irreducible path l = {a1 , a2 , a3 , . . . , an }, we know that between any two pairs (ai , ai+1 ) and (ai+2 , ai+3 ) with the distance less than 12 , there is at least, one pair (ai+1 , ai+2 ) with the distance larger than 12 . Thus, the number of pairs (ai , ai+1 ) the distance between which larger than 12 is more than ( 14 )n. Consequently, the length of the path l = {a1 , a2 , a3 , . . . , an } is larger than 12 · ( 14 )n = ( 18 )n, i.e., ( 18 )n < [d(a, a1 ) + d(a1 , a2 ) + d(a2 , a3 ) + · · · + d(an−1 , an ) + d(an , b)] < q · d(a, b) = q · d where the distance d(a, b) is equal to d. Thus, n < 8q · d. Let us assume that X is a bounded set. It means that there is a positive number h such that for two points a and b from X, the distance d(a, b) is less than h. It is possible to assume that h > 1. Then n < 8q · h and d(A(a), A(b)) < k(n + 1) < k(8q · h + 1) = t < (8qk + k) · h. It means that the operator A is uniformly bounded because numbers k and q are constants and t depends only on one variable h. Necessity is proved. Sufficiency. Let us consider a uniformly bounded epistemic information operator A: E → H. Then (cf., Definition 8.4.14) for any number r, there is a number t such that for any state X of E, the condition d(X) ≤ r, implies the condition d(A(X)) ≤ t. In particular, for the number 1, there is a number k such that for any points a and b from E, the condition d(a, b) ≤ 1, implies the condition d(A(a), A(b)) ≤ k. It means that the operator A is (1, k)continuous. Remark 8.4.1. The proof of Theorem 8.4.1 is sufficiently general. Therefore, this result remains true for general metric spaces that satisfy the necessary conditions. One of the basic results of functional analysis is the theorem stating that a linear operator in Banach space is continuous if and only if it is uniformly bounded (Dunford and Schwartz, 1958; Rudin, 1991). It is demonstrated that (1, 0)-continuity is stronger than continuity in metric spaces (Burgin, 2008). Thus, it is possible to ask a question whether it would be possible to change the condition of (1, k)-continuity to the condition of continuity in Theorem 8.4.1.
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The following example shows that it is impossible as there are linear epistemic information operators that are continuous but not uniformly bounded. Example 8.4.6. Let us consider a weighted epistemic space E, in which there is a countable number of epistemic structures e1 , e2 , e3 , . . . , en , . . ., the distance between any two of them is 1 and each of them has one weight w the range of which is the real line R. We define the epistemic information operator A by the following rule: A(en , w) = (en , nw ). By definitions, this operator is linear and continuous but it is not uniformly bounded. Indeed, d((e1 , 1), (en , 1)) ≤ 2, while d(A(e1 , 1)A(en , 1)) > n for any n. Even more, there are linear epistemic information operators that are continuous but not bounded. Example 8.4.7. Let us consider a weighted epistemic space E, in which there is a countable number of epistemic items e1 , e2 , e3 , . . . , en , . . . such that d(en , en+1 ) = (1/2n ) and d(en , en+k ) = n+k−1 (1/2i ). Each of these epistemic items has one weight w the Σi=n range of which is the real line R. We define the epistemic information operator A by the following rule A(en , w) = (en , nw ). By definitions, the set U = {(e1 , 1), (e2 , 1), (e3 , 1), . . . , (en , 1), . . .} is bounded as d(U ) = sup d(en , en+k ) < 3. The operator A is linear and continuous. However, it is not bounded because in the set A(U ), there are pairs of points with the arbitrary big distance between them, e.g., d(A(e1 , 1), A(en , 1)) > n. Although continuity is insufficient for boundedness of a linear epistemic information operator in a general case, there are situations when boundedness is still equivalent to continuity. Let us consider two weighted epistemic spaces E and H such that both spaces Ee and He are metric spaces with the metric d, the base epistemic space Ee is finite and all fibers Fa of the vector bundle E = (E, p, Ee ) and fibers Ga of the vector bundle H = (H, p, He ) are hyperseminormed vector spaces (Burgin, 2013).
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Theorem 8.4.2. A linear epistemic information operator A : E → H is continuous in each fiber Fa of the vector bundle E = (E, p, Ee ) if and only if A is bounded. Proof . Necessity. Let us consider an epistemic information operator A : E → H continuous in each fiber Fa of the vector bundle E = (E, p, Ee ) and a bounded set X in E. Then each intersection Xa = X ∩ Fa is a bounded set because a subset of a bounded set is also bounded. As the operator A is continuous in the fiber Fa , the image A(Xa ) of Xa is bounded (Burgin, 2013). As the union of a finite number of bounded sets is bounded, X = ∪a∈Ee Xa is a bounded set. Sufficiency. Let us consider a bounded epistemic information operator A : E → H. Then it is bounded on any subset of E, in particular, on each fiber Fa of the vector bundle E = (E, p, Ee ). As each fiber Fa of the vector bundle E = (E, p, Ee ) is a hyperseminormed vector space, the results from (Burgin, 2013) show that the epistemic information operator A is continuous in each fiber Fa of the vector bundle E = (E, p, Ee ). In many applications, the base epistemic space Ee is finite. For instance, a popular type of the base epistemic space Ee is a semantic network (cf., Chapter 5) and all known semantic networks are finite. Thus, the conditions from Theorem 8.4.2 are almost always satisfied. As normed vector spaces are an important special case of hyperseminormed vector spaces (Burgin, 2013) the following result is implied by Theorem 8.4.2. Corollary 8.4.1. If all fibers Fa and Ga are normed vector spaces, then a linear epistemic information operator A : E → H is continuous in each fiber Fa of the vector bundle E = (E, p, Ee ) if and only if A is bounded. For linear operators in vector spaces, uniform boundedness coincides with boundedness (Dunford and Schwartz, 1958; Rudin, 1991). This gives us the following result. Corollary 8.4.2. If all fibers Fa and Ga are seminormed and, in particular, normed, vector spaces, then a linear epistemic information
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operator A : E → H is continuous in each fiber Fa of the vector bundle E = (E, p, Ee ) if and only if A is uniformly bounded. When Ee consists of single element, Theorem 8.4.2 gives us the classical result of functional analysis. Corollary 8.4.3. (Dunford and Schwartz, 1958; Rudin, 1991). A linear operator in Banach space is continuous if and only if it is uniformly bounded. Many epistemic spaces in general and M-spaces in particular are stratified (cf., Chapter 3). For instance, an important technique is consistent stratification of inconsistent knowledge using logical varieties, prevarieties and quasi-varieties (Burgin, 1991d; Burgin and de Vey Mestdagh, 2015). Stratification of the knowledge system and the corresponding M-space allows defining specific classes of epistemic information operators. Definition 8.4.15. An epistemic information operator A is called stratified if for any j ∈ J, there is k ∈ J such that for any Ki from KSM , we have A(Kij ) ⊆ Kik . Stratified information operators preserve the structure, i.e., stratification, of knowledge states. Note that addition and deletion operators are intrinsically stratified. Definition 8.4.16. (a) An epistemic information operator A is called closed if for any j ∈ J and for any Ki from KSM , we have A(Kij ) ⊆ Kij . (b) An epistemic information operator A is called closed in a Mizzaro space Ki from KSM if A(Ki ) ⊆ Ki . Lemma 8.4.8. Any closed epistemic information operator A is stratified. Definition 8.4.17. (a) A stratified epistemic information operator A in a linearly stratified M-space M is called monotone (antitone) if for any n ∈ N , there is k ∈ N such that k ≥ n(k ≤ n) and for any Ki from KSM , we have A(Kij ) ⊆ Kik .
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(b) A stratified epistemic information operator A in a linearly stratified M-space M is called strictly monotone (strictly antitone) if for any n ∈ N , there is k ∈ N such that k > n(k < n) and for any Ki from KSM , we have A(Kij ) ⊆ Kik . Definitions imply the following property of epistemic information operators. Lemma 8.4.9. Any strictly monotone (strictly antitone) epistemic information operator A is monotone (antitone). Lemma 8.4.10. In a finite linearly stratified M-space M, there are no strictly monotone and strictly antitone information operators. Definition 8.4.18. An epistemic information operator A is called contracting if there is k ∈ J such that for any Ki from KSM , we have A(Kij ) ⊆ Kik . Definitions imply the following result. Lemma 8.4.11. Any contracting epistemic information operator A is stratified. There are five types of basic epistemic operations: adding, deleting, moving, replicating, and transforming knowledge, and five types of corresponding basic epistemic information operators: addition AD, deletion DL, moving MV, replication REPL, generation GR and transformation TR epistemic information operators. Definition 8.4.19. A transformation epistemic information operator TR takes a group of knowledge items (may be, one item) from the current knowledge state and transforms it into another group of knowledge items (may be, into one item). An example of a transformation epistemic information operator is operation of substitution in logic described in Section 5.2.3. Definition 8.4.20. A generation epistemic information operator GR takes a group of knowledge items (may be, one item) from the current knowledge state and generates another group of knowledge items (may be, one item).
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The difference between transformation and generation is that in generation the initial group of knowledge items is preserved, while in transformation it is not preserved. Lemma 8.4.12. AD is equal to GEN with the empty set of the initial knowledge items. Definition 8.4.21. A moving epistemic information operator MV moves a knowledge item from one stratum into another one. For instance, an operator that moves a knowledge item from the short-term memory to the long-term memory is an example of a moving operator. Definition 8.4.22. (a) A replica of a knowledge item is another knowledge item equivalent to the initial one. (b) A replication epistemic information operator REPL makes a replica of a knowledge item and adds it to the current knowledge state. Example 8.4.8. Let us consider logical knowledge representation in which knowledge items are propositions. Then according to laws of logic there are equivalent propositions. For instance, taking the proposition (1) “B implies A”, we have equivalent propositions (2) “A follows from B”, (3) “If B, then A”, and (4) “A is a consequence of B”. All of them are replicas of one another although they are not copies. If the proposition (1) belongs to the stratum K1 , then its replication to the stratum K2 can introduce either proposition (1) or proposition (2) or proposition (3) to the stratum K2 , while its copying to the stratum K2 can introduce only proposition (1) to the stratum K2 . An important special case of a replication epistemic information operator is a copying epistemic information operator COPY, which makes a copy of a knowledge item and adds it to the current knowledge state. Another important special case of a replication epistemic information operator is a restricted replication epistemic information operator REPL0 , which replicates a knowledge item and adds it only
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to a stratum of the current knowledge state that does not have the same replica. One of its special cases is a restricted copying epistemic information operator COPY0 , which makes a copy of a knowledge item and adds it only to a stratum of the current knowledge state that does not have the same replica. Operators REPL0 , and COPY0 are used in stratified M-spaces not to make these spaces stratified M-multispaces. Proposition 8.4.2. The operator COPY0 can copy a knowledge item only to a different stratum, i.e., if a ∈ Ki and COPY0 a ∈ Kj , then i = j. Indeed, if this condition is violated, then the initial M-space is converted to an M-multispace. Complex information operations and operators are studied in (Burgin, 1997e). Definition 8.4.23. An epistemic information operator C is called the sequential composition of an epistemic information operator A with an epistemic information operator B if C(x) is defined and equal to B(A(x)) when: 1) A(x) is defined and belongs to the domain of B; 2) B(A(x)) is defined. Otherwise, C gives no result being applied to x, i.e., C(x) = ∗ . It is denoted by B ◦ A. Taking sequential composition of an epistemic information operator A with itself, we obtain sequential powers An of the operator A. In the general case, the sequential composition of epistemic information operators is not commutative in M-spaces as the following example demonstrates. Example 8.4.9. Let us consider a structured M-space M = {KS M ; OSM } where KS M = ∪i∈I KS M i . In this space, the operator MVaij moves an element a from the stratum KS M i to the stratum KS M j , and does not change other elements from KS M . Taking the sequential composition of such operators, we have MV aij ◦ MV aik = MV aij = MV aik ◦ MV aij = MV aik if i = j, k = j, and i = k. Thus, operators MV aij do not commute with one another.
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At the same time, all these operators are idempotents, i.e., MV aij ◦ MV aij = MV aij . It is necessary to remark that in a structured M-multispace M with an infinite number of elements a in each stratum KS M i , operators MV aij and MV aik commute with one another. This demonstrates difference between M-spaces and M-multispaces. Proposition 8.4.3. If A and B are closed (closed in a Mizzaro space Ki ) operators, then their sequential composition A◦B is also a closed (closed in a Mizzaro space Ki ) operator. Indeed, if A and B are closed epistemic information operators in a structured M-multispace M , then for any j ∈ J and for any Ki from KSM , we have A(Kij ) ⊆ Kij and B(Kij ) ⊆ Kij . Thus, (A ◦ B)(Kij ) = B(A(Kij ) ⊆ B(Kij ) ⊆ Kij . For closed in a Mizzaro space Ki operators, the proof is similar. Proposition 8.4.4. If A and B are contracting operators, then their sequential composition A ◦ B is also a contracting operator. Proof is similar to the proof of Proposition 8.4.3. Proposition 8.4.5. If A and B are stratified operators, then their sequential composition A ◦ B is also a stratified operator. Proof is similar to the proof of Proposition 8.4.3. Proposition 8.4.6. If A and B are (strictly) monotone [antitone] operators, then their sequential composition A ◦ B is also a (strictly) monotone [antitone] operator. Indeed, if A and B are monotone epistemic information operators in a structured M-space M , then for any Ki from KSM , we have A(Kij ) ⊆ Kik with k ≥ j and B(Kik ) ⊆ Kih with h ≥ k. Thus, (A ◦ B)(Kij ) = B(A(Kij ) ⊆ B(Kik ) ⊆ Kih with h ≥ j. Considerations for strictly monotone, antitone and strictly antitone epistemic information operators are similar. Let us consider an M-space M with a finite linear stratification.
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Proposition 8.4.7. For any monotone and any antitone epistemic information operator A, there is a number n such that the sequential power An is also a closed epistemic information operator. Indeed, if A is a monotone epistemic information operator in a structured M-space M , then making each step, it either increases the number of the stratum or the image of a stratum remains in the same stratum. If the second case is true for all strata of M , then A itself is a closed epistemic information operator. Otherwise, A can increase the number of a stratum only for a finite number of steps because there are only a finite number of strata in M . Thus, after some number of repetitions, the image of a stratum remains in the same stratum. Taking the largest number of such steps, we obtain the necessary number n. Note that n cannot be larger than the number of strata in M . Let us explore relations between basic epistemic information operators. Definition 8.4.24. (Burgin, 2010d). Two operators A and B are functionally equivalent if they have the same definability domain D and A(x) = B(x) for any element x from D. For instance, two algorithms that compute the same function on natural numbers are functionally equivalent. Proposition 8.4.8. A transformation epistemic information operator TR is functionally equivalent to the sequential composition of a deletion epistemic information operator DEL and addition epistemic information operator AD that act in the same stratum of the M-space. Indeed, if TR takes items a1 , a2 , . . . , an , from KSM and transforms them into b1 , b2 , . . . , bm , then it is possible to achieve the same result by deleting a1 , a2 , . . . , an , and adding b1 , b2 , . . . , bm , to the corresponding stratum of KSM . Proposition 8.4.9. A moving epistemic information operator MV is functionally equivalent to deletion of a knowledge item in one stratum and adding the same knowledge item to another stratum.
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For instance, it is possible to describe the process of transition of a knowledge item X from the short-term memory in the brain to the long-term memory as deletion of X from the short-term memory and addition of X to the long-term memory. Proposition 8.4.10. A replication epistemic information operator REPL is functionally equivalent to adding an equivalent knowledge item to the corresponding stratum. For instance, it is possible to describe the process of rewriting a word X that represents a knowledge item from one tape t1 of a Turing machine (cf., Appendix B) to another tape t2 as addition of X to the tape t2 without changing the tape t1 . Proposition 8.4.11. For any M-space M, there is a superspace H, in which all deletion and addition epistemic information operators DEL and AD in M are functionally equivalent to moving epistemic information operators MV in H. Proof . To build a superspace H with the necessary properties, we add one more stratum E called the external stratum to the initial M-space M . In addition, we assume that E contains all elements from the universal set (multiset) W and each element has infinitely many copies in E. In this case, any deletion of an element a from a state K from M is equivalent to moving the same element a to the stratum E. In a similar way, any addition of an element a to a state K from M is equivalent to moving the same element a from the stratum E to the state K. Proposition is proved. Proposition 8.4.12. A moving epistemic information operator MV can be (functionally) simulated by copy COPY and deletion DEL epistemic information operators. Indeed, instead of moving a knowledge item a from a stratum Ki of a state K to a stratum Kj of a state K , it is possible to copy a from Ki to Ki and then to delete this element from Ki .
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Proposition 8.4.13. A generation epistemic information operator GEN is functionally equivalent to the sequential composition of a transformation epistemic information operator TR and addition epistemic information operator AD that act in the same stratum of the M-space. Proof is similar to the proof of Proposition 8.4.12. Proposition 8.4.14. A transformation epistemic information operator TR is functionally equivalent to the sequential composition of a generation epistemic information operator GEN and deletion epistemic information operator DEL that act in the same stratum of the M-space. Proof is similar to the proof of Proposition 8.4.12. Definition 8.4.25. A system B of epistemic information operators is an operator basis of an M-space M if any A from OSM is a composition of elements from B. Operator bases can be useful in many situations. For instance, knowing properties of operators from such a base and properties of compositions, we can find properties of other operators. Assuming that all operators in an M-space M are compositions of basic epistemic information operators, we have the following results. Proposition 8.4.15. (a) {AD , DEL} is an operator basis of an arbitrary (stratified) M-space M. (b) {TR, MV } is an operator basis of an arbitrary (stratified) M-space M. (c) {TR} is an operator basis of an arbitrary (i.e., non-stratified) M-space M. Proof is based on Propositions 8.4.9–8.4.14. Proposition 8.4.16. (a) In a stratified M-space M with the external stratum, {REPL0 , DEL} is an operator basis. (b) In a stratified M-space M with the external stratum, {MV } is an operator basis.
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(c) In a stratified M-multispace M with the external stratum, {REPL, DEL} is an operator basis. Proof is based on Propositions 8.4.9–8.4.14. Different classes of epistemic information operator mathematically represent properties of information and its action on knowledge spaces, which are mathematically modeled by stratified M-spaces and stratified M-multispaces. This is the functional mathematical model of dynamic features of knowledge and information interaction. Another mathematical model of dynamic features of knowledge and information interaction uses category theory and is studied in (Burgin, 2010b; 2011b). 8.5. Knowledge as a measure of information He who knows, does not speak. He who speaks, does not know. Lao-Tzu
The methodological principles of the anthropic information explication and classification bring us to cognitive information and its important subclass called epistemic information. Portions of epistemic information are modeled/represented by epistemic information operators acting in spaces of knowledge, which are represented by a formal construction called a Mizzaro space. These spaces consist of knowledge items often unified by structural relations. In a general setting, epistemic information has been studied by different authors. Bar-Hillel and Carnap (1958), Hintikka (1968; 1970; 1971) and Israel and Perry (1990) explored information in knowledge represented by means of mathematical logic. Shreider (1965), Mackay (1969), Brookes (1980), Mizzaro (1996; 1998; 2001) and Gackowski (2004) base their theories on the following assumption: Information is a change in a knowledge system. Later this principle has been made more exact (Mizzaro, 2001) and formulated as Epistemic information is a change in a knowledge system.
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The general theory of information (Burgin, 2010) makes the next step to the better understanding of epistemic information. Namely, it is explained that Epistemic information is a capacity to cause changes in a knowledge system and it is possible to measure this capacity by changes in the knowledge system impacted by information. Such changes in the knowledge system of a system R reflect accepted information. Note that information can be accepted not only as true information but also as false information. In this case, changes in the knowledge system can result in exclusion of some knowledge or in labeling this knowledge as false, e.g., treating it as a misconception or blunder. To reflect interactions between information and knowledge in the mathematical form, we assume existence of a universal knowledge set (knowledge multiset) W of knowledge units (knowledge items) and consider a class A of dynamic systems, e.g., intelligent or cognitive agents, that have knowledge. This assumption is based on a traditional approach to knowledge, which is called the representational atomism (Campbell, 1998), according to which, knowledge is built from some basic or primitive or elementary units by combining them into a more complex structure. It is possible to take the set W C of elementary knowledge units mathematically modeled in Chapter 4 as the universal set (multiset) W . Another possibility for W is realized by the set (multiset) W L of propositions and/or predicates from a logical language L. Propositions and predicates are symbolic knowledge units in the logical approach developed in works of Bar-Hillel and Carnap (1958), Hintikka (1968; 1970) and some other authors. Shreider (1965) interpreted symbolic knowledge units as texts in a thesaurus. Many researchers employ mental schemas as cognitive/symbolic knowledge units in the brain (cf., for example, (Anderson, 1977; Arbib, 1992; Armbruster, 1996; Burgin, 2006)). Taking descriptions of these situations possible in a world U , which can be a physical world, a mental world or a world of some organization, as knowledge items or knowledge units, we obtain one more universal knowledge set (multiset) W S (cf., for example, (Barwise and Perry, 1983)).
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As it is explained in Section 3.1, the set (multiset) W is called universal because we assume that the following axiom is true. UKEFA 1 (the Internal Representation Axiom). Knowledge of any system A from the class A is organized in the form of subsets (submultisets) of the set (multiset) W . Note that UKEFA 1 is a particular case of the Internal Cognitive Representation Axiom (UCEF 1) introduced in Section 3.1. It is possible to interpret W as the base of all knowledge that systems from the class A are able to have about their environment. If A is a system from the class A, then knowledge of A at some moment is called a knowledge state (KS) of A or of the knowledge space of A. By Axiom UKEFA 1, any KS is a subset (submultiset) of the set (multiset) W . It means that knowledge of any system A from the class A consists of atomic components called knowledge items or knowledge units (KI) Additionally, we assume that knowledge of A is stratified, which means that elements from a KS are situated in several strata. According to our model (cf., Section 3.1), the set (multiset) W and all stratified knowledge state (KS) are knowledge spaces. This model further develops the approach of Mizzaro (1996; 1998; 2001) and Burgin (2010; 2011a) to epistemic information. The set of all actual knowledge states of a system A is denoted by KSA and the set of all possible knowledge states of the system A is denoted by PKSA. By Axiom UKEFA 1, both of these sets are subsets of the power-set 2W . Thus, any knowledge of a system A state consists of two parts — the actual knowledge KA of A and the potential knowledge KP of A. Note that the potential knowledge KP of A., i.e., knowledge accessible by A from this state without incoming information, depends on the actual knowledge KA of A in this state. As it is explained in Section 3.1, information is modeled by epistemic information operators, which change knowledge states (KS) of systems from the class A. Namely, each portion (piece) of information is represented by an epistemic information operator.
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Definition 8.5.1. (a) A (partial) mapping R : KSA → KSA is called an actual epistemic information operator in the actual knowledge space of the system A. (b) A pair R, P of (partial) mappings R : KSA → KSA and P : PKSA → PKSA is called an extended epistemic information operator in the extended knowledge space of the system A. If the knowledge state of the system A is KI , then the action of the operator R changes it to the state KRF . To define change caused by the operator R, we make a distinction between transition changes and multistep changes. There are three basic kinds of such changes (operations): knowledge items are added to the initial knowledge state, knowledge items are removed (deleted) from the initial knowledge state and knowledge items from the initial knowledge state are moved from one stratum to another. In transition changes, each operation is performed at most one time with one knowledge item. In multistep changes, each operation is performed at most one time with one knowledge item on each step of the whole process. For instance, when changes are multistep, it is possible to remove a knowledge item k at the first step and then to add k at the step three. The same knowledge item can be moved several times inside the knowledge system and/or can be moved several times into and out from the knowledge system. We use the following notation: • KRO is the set of all knowledge items added in the process of information acceptation, i.e., in the process caused in the knowledge space of A by the actual epistemic information operator R. • KRD is the set of all knowledge items removed (deleted) in the process of information acceptation. • KRM is the set of all knowledge items moved between strata in the process of information acceptation. Note that it is possible to treat removed or deleted knowledge as knowledge moved outside the system A. Proposition 8.5.1. In the context of transition (one-step) changes, the following equalities are true for any actual epistemic information
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operator R KRO = KRF \KI , and KRD = KI \KRF . Proof . Knowledge items that belong to KRF but does not belong to KI are definitely added when the operator R is applied. This gives us the inclusion KRO ⊇ KRF \KI . Besides, knowledge items that belong to KI cannot be added in the transition process — it is only possible to move or to delete them. This gives us the equality KRO = KRF \KI . Knowledge items that belong to KI but does not belong to KRF are definitely deleted when the operator R is applied. This gives us the inclusion KRD ⊇ KI \KRF . Besides, knowledge items that belong to KRF cannot be deleted in the transition process — it is only possible to move or to add them. This gives us the equality KRD = KI \KRF . Proposition is proved. Introduced components of knowledge states allow us to define measures of information. At first, we define measures of received information. Let us consider an actual epistemic information operator R and a system A from K in the knowledge state KI . Definition 8.5.2. The transitional measure |IRA | of information IRA transmitted by R to A represents changes of knowledge in A with the knowledge state KI under the impact of R and is defined by the following formula |IRA | = ∆K = |KRF | − |KI | where |X| is the number of elements in the set X. This shows that transitional measure of transmitted information can be positive when more knowledge items are added than deleted or negative when less knowledge items are added than deleted. As in the transition process, each operation is performed at most one time with each of knowledge item from KI , Proposition 8.5.1 gives us the following result.
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Proposition 8.5.2. In the context of transition (one-step) changes, the following equalities are true for any actual epistemic information operator R |IRA | = |KRF \KI | − |KI \KRF | = |KRO | − |KRD | Usually, acceptation of information is a complex process that involves many steps of operation. Consequently, we need a more exact measure taking into account different operations performed at all steps. Definition 8.5.3. The operational measure IRA of information IRA transmitted by R to A represents changes of knowledge in A with the knowledge state KI under the impact of R and is defined by the following formula IRA = ∆K = |KRO | + |KRD | + |KRM |, where |X| is the number of elements in the set X. Corollary 8.5.1. When the knowledge system of A is not stratified, we have IRA = ∆K = |KRO | + |KRD |. The operational measure of transmitted information is always non-negative and can be applied both to transitional and multistep changes. Definitions imply the following result. Proposition 8.5.3. For any actual epistemic information operator R and any system A from the class A, the following inequality is true |IRA | ≤ IRA . Proof is left as an exercise. Proposition 8.5.3 gives us the following result. Proposition 8.5.4. In the context of transition (one-step) changes, the following equalities are true for any actual epistemic information operator R IRA = |KRF \KI | + |KI \KRF | + |KRM |.
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Corollary 8.5.2. When the knowledge system of A is not stratified, we have IRA = |KRF \KI | + |KI \KRF |. Measures of transmitted information are used to define measures of a portion of information represented by an actual epistemic information operator R. At first, we define measures of R for an object A. Definition 8.5.4. The upper transitional measure |IR A | of information of R for A represents maximal changes of knowledge in A under the impact of R and is defined by the following formula U |IR A | = max{|IRA |; for all KI ∈ KSA}.
Similar to the transitional measure of transmitted information, the upper transitional measure can be positive or negative. Note that negativity of the upper transitional measure for the object A means that the portion of information represented by the operator R always causes decrease of knowledge in A independently of its state. Proposition 8.5.2 implies the following result. Proposition 8.5.5. In the context of transition (one-step) changes, the following inequalities are true for any actual epistemic information operator R min{|KRF \KI |; for all KI ∈ KSA} − max{|KI \KRF |; for all KI ∈ KSA} U ≤ |IR A| ≤
≤ max{|KRF \KI |; for all KI ∈ KSA} − min{|KI \KRF |; for all KI ∈ KSA}. Definition 8.5.5. The lower transitional measure IR A of information of R for A represents minimal changes of knowledge in A under the impact of R and is defined by the following formula R IA = min{|IRA |; for all KI ∈ KSA}. L
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Similar to the transitional measure of transmitted information, the lower transitional measure can be positive or negative. Note that positivity of the lower transitional measure for the object A means that the portion of information represented by the operator R always causes increase of knowledge in A independently of its state. Naturally, we have the inequality R U |IR A |L ≤ |IA | . Proposition 8.5.6. In the context of transition (one-step) changes, the following inequalities are true for any actual epistemic information operator R min{|KRF \KI |; for all KI ∈ KSA} − max{|KI \KRF |; for all KI ∈ KSA} ≤ |IR A |L ≤ max{|KRF \KI |; for all KI ∈ KSA} − min{|KI \KRF |; for all KI ∈ KSA}. Usually, acceptation of information is a complex process that involves many steps of operation. Consequently, we need more exact measures taking into account different operations performed at all steps of information transmission. Definition 8.5.6. The upper operational measure IR A of information of R for A represents maximal changes of knowledge in A under the impact of R and is defined by the following formula U IR A = max{IRA ; for all KI ∈ KSA}. Similar to the operational measure of transmitted information, the upper operational measure is always non-negative. Definition 8.5.6 implies the following result. Proposition 8.5.7. In the context of multistep changes, the following inequalities are true for any actual epistemic information operator R min{|KRO |; for all KI ∈ KSA} + min{|KRD |; for all KI ∈ KSA} R U + min{|KRM |; for all KI ∈ KSA} ≤ IA
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≤ max{|KRO |; for all KI ∈ KSA} + max{|KRD |; for all KI ∈ KSA} + max{|KRM |; for all KI ∈ KSA}. Definition 8.5.7. The lower operational measure IR A of information of R for A represents minimal changes of knowledge in A under the impact of R and is defined by the following formula R L = min{IRA ; for all KI ∈ KSA}. IA
Similar to the operational measure of transmitted information, the lower operational measure is always non-negative Naturally, we have the inequality R U IR A L ≤ ||IA .
Definition 8.5.7 implies the following result. Proposition 8.5.8. In the context of multistep changes, the following inequalities are true for any actual epistemic information operator R min{|KRO |; for all KI ∈ KSA} + min{|KRD |; for all KI ∈ KSA} + min{|KRM |; for all KI ∈ KSA} ≤ IR A L ≤ ≤ max{|KRO |; for all KI ∈ KSA} + max{|KRD |; for all KI ∈ KSA} + max{|KRM |; for all KI ∈ KSA}. It is necessary to remark that measuring knowledge by the number of knowledge items and applying these measures to measuring information gives us only the first approximation to modeling information processes related to knowledge transformations. Indeed, if some knowledge item contains another knowledge item, then the second knowledge item does not add knowledge to the first knowledge item. Besides, one knowledge item can contain much more knowledge
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than another knowledge item. In addition, the number of knowledge items as a measure of knowledge reflects only the substantial dimensions of knowledge, while the relational dimensions, which describe relations between knowledge items, belong to higher-level models of knowledge transformations and operational information theory, which are studied elsewhere. Taking into account such relations allows one to achieve higher precision in measuring information and knowledge. However, there are many situations where precision provided by quantities of knowledge items is sufficient. For instance, exactly this level of precision is successfully used in software engineering and technology where program instructions, which belong to the procedural type of knowledge, are used as knowledge items that determine measures of information and knowledge transformations (Baer and Zeidman, 2009). The most popular measures of information are Shannon’s entropy and Kolmogorov (or algorithmic) complexity (Burgin, 2010). The goal of Claude Elwood Shannon (1916–2001) was to measure transmission information. To achieve this goal, he introduced the entropy of a message m that informs about happening of an event E or the outcome D of an experiment H, assuming that there are n possible alternatives E1 , E2 , E3 , . . . , En , one of which is E, to the event E or there are n possible outcomes D1 , D2 , D3 , . . . , Dn of the experiment H, one of which is D (Shannon, 1948). In this case, the entropy of the message m is defined by the following formula H(m) = H(p1 , p2 , . . . , pn ) = −Σni=1 pi · log2 pi .
(3.2.4)
Probabilities used in the formula (3.2.4) are obtained from experiments and observation, using relative frequencies. By definition, the entropy H(m) of the message m measures information by knowledge given by the message m about happening of an event E or the outcome D of an experiment H. Kolmogorov (or algorithmic) complexity was introduced by three authors — Ray Solomonoff (1926–2009), Andrey Nikolayevich Kolmogorov (1903–1987) and Gregory Chaitin. In this context, information is considered not as some intrinsic property of different objects but is related to algorithms that use, extract or produce
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this information. As a result, this approach was the first developed form of a complexity measure of objects, such as a piece of text or an arbitrary sequence of symbols, which estimates the computational resources needed to specify the object. Namely, the Kolmogorov (or algorithmic) complexity CA (x) of an object (word) x with respect to an algorithm A is defined as CA (x) = min{l(p); A(p) = x}, in the case when there is an input word (program of A)p with the length l(p) such that A(p) = x; otherwise CA (x) is not defined. This shows that the Kolmogorov complexity CA (x) of an object (word) x measures information sufficient to reconstruction of the object (word) x. With respect to the algorithm A, its program or input data playing the role of a program are representations of operational knowledge as they determine how to reconstruct the object. The length of a program (word) is its measure. Therefore, in this case, information is also measures by knowledge. In addition to Kolmogorov complexity CA (x), there also other variations of this measure — uniform complexity KR(x), prefix complexity or prefix-free complexity K(x), monotone complexity Km(x), conditional Kolmogorov complexity CD(x), time-bounded Kolmogorov complexity Ct (x), space-bounded Kolmogorov complexity Cs (x), resource-bounded Kolmogorov complexity Ct,s (x) and inductive Kolmogorov complexity (cf., (Burgin, 2004c; 2010)). All these measures estimate information by the amount of operational knowledge. Axiomatic approach to measuring information by the amount of operational knowledge is developed in (Burgin, 1982a; 1983; 1990b; Cˆampeanu, 2012). All kinds of Kolmogorov/algorithmic complexity are encompassed by this approach. Kolmogorov/algorithmic complexity as a measure of information has been applied in a variety of areas such as medicine, biology, neurophysiology, physics, economics, hardware, and software engineering.
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Chapter 9
Conclusion We now know enough to know that we will never know everything. Maria Popova “Creating a “Fourth Culture” of Knowledge”
Thus, we can see that knowledge is an extremely complex and at the same time, more than ever important phenomenon. It plays a pivotal role in the life of individuals, functioning of organizations and the whole existence of society. It is important to understand the two-sided function of knowledge for a system that possesses knowledge. On the one hand, knowledge can be and in many cases, is the source of change and development. On the level of an individual (the knower), good knowledge and its correct application can lead to a better life and higher achievements. On the level of organization, knowledge can enhance extension of activities, expansion into new domains, and improvement of organizational culture and practices. On the level of society, knowledge brings steady technological and economical progress, which accelerates all the time. In essence, it is possible to envision knowledge processes as fundamental drivers of life on all levels. On the other hand, knowledge allows a system to preserve dynamic stability and consistency on all levels. For an individual, it would be very hard (if not impossible) to preserve wellbeing and even life without proper knowledge in the permanently changing and sometimes hostile environment. Organizations always live in the state of competition and insufficient knowledge and/or its wrong 803
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application brings bad consequences, up to decline and disintegration. History shows that when cognition is not supported and/or its results are not properly used, society declines and decays. Thus, we come to the necessity to have knowledge about knowledge. Synthesizing existing and developing new approaches in knowledge studies, an exposition of the synthetic theory of knowledge is given in this book. It includes philosophical, methodological, theoretical and applied aspects of knowledge and knowledge processes. This theory is constructed on three levels — quantum, average, and global. In the context of the synthetic theory of knowledge, several mathematical techniques for modeling knowledge processes are described. Epistemic structures, epistemic spaces and epistemic information operators provide efficient means for knowledge studies. Different types of knowledge spaces presented in this book — unstructured knowledge spaces, stratified knowledge spaces, typological knowledge spaces, and structured knowledge spaces — are aimed at the development of knowledge processing in technical systems, such as computers, computer networks, robots, etc. Studied here mathematical models and constructions, such as logical varieties and prevarieties, open new possibilities for artificial intelligence. Novel theoretical and practical ideas are introduced. For instance, the conditional approach to knowledge definition is developed in Chapter 2. This approach better fits the practice of utilization of the term knowledge allowing efficient differentiation and integration of knowledge. At the same time, a more complete description of knowledge management is elaborated in the book. Namely, new aspects and stages of knowledge management, such as knowledge hiding, knowledge retirement and knowledge maintenance, are explicated and explored in this book. The necessity of strongly multidisciplinary and transdisciplinary approaches in knowledge studies has been demonstrated. That is why we have explored basic properties of knowledge, knowledge representations, knowledge processes and knowledge functions from philosophical, methodological, mathematical, scientific and practical perspectives opening new and emphasizing existing directions and areas in knowledge studies.
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For instance, an important direction is the measurement theory of knowledge and information. It is necessary to build efficient measures of knowledge providing effective tools for evaluation of knowledge assets of individuals and organizations. There are already some simple measures such as the number of propositions and predicates as the measure of descriptive knowledge in the propositional form or the length of a program as the measure of operational knowledge in the procedural form. However, it is necessary to have more measures for evaluation of different knowledge properties based on sound theoretical foundations. It is possible to essentially extend the range of practical applications for some theories presented in this book. For instance, mathematical schema theory presented in Chapter 5 can be used for studies of the brain and human intelligence (Burgin, 2005a), for building computer and communication networks (Burgin, 2006), and for organizational planning. One more prospective application of mathematical schemas is the three-schema approach, or the Three Schema Concept, in software and system engineering. According to this approach, construction of information systems such as databases and knowledge bases demands conceptual modeling as the key condition to achieving high-quality data and knowledge integration (Loomis, 1987). The three-schema model comprises three types of schemas: • External schemas are representing user; • Conceptual schemas integrate external schemas in a logical structure; • Internal schemas define physical storage structures for knowledge and data. The external schema is the highest level of abstraction, which is situated at the view level, reflecting the user vision of the system and its interface. The conceptual schema, which is situated at the logical level, defines the logical structure of the entire system and the ontology of the system data, information and/or knowledge. For instance, in the case of databases and knowledge bases, the conceptual schema describes what data and/or knowledge are stored in the system, the relationships among the data (knowledge items) and
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complete description of the user’s requirements without any concern for the physical implementation, which is relegated to the next level. The physical schema, which is situated at the internal or physical level, which is the lowest level of the system representation. It deals with the physical representation of data and/or knowledge describing how they are physically stored and organized on the storage medium. To conclude, we formulate some open problems in knowledge studies, which, in turn, open new directions for the future research. As psychologists found, people used mental schemas all the time (Neisser, 1967; Anderson, 1977; Rumelhart, 1980). For instance, image schema establish patterns of understanding and reasoning (Johnson, 1987; Lakoff, 1987; Rohrer, 2006). Interaction schemas are determined by the execution of tasks involving the physical environment (Arbib, 1992; 1995). At the same time, mathematical schema theory provides more efficient tools than empiric approaches in schema theory (Burgin, 2005; 2006; 2010a). This brings us to the following problems. Problem 1. Use mathematical schema theory for modeling and exploration of mental activity of people. Problem 2. Use mathematical schema theory for modeling and exploration of the brain. To make the theory of knowledge really scientific, it is necessary to develop experimentation techniques in this area. This bring us to the following problems. Problem 3. Create a theory of knowledge measurement and evaluation. Problem 4. Elaborate a theory of properties of and relations between knowledge items. In this book, we constructed various operations with knowledge items. The next step is to add new operations and organize a unified structure of all operations with knowledge items. This results in the following problems. Problem 5. Build algebraic systems (algebras) of knowledge.
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Problem 6. Develop a mathematical theory of knowledge algebras (algebraic systems) and study their properties. Contemporary logic is based on logical calculi, their semantics and models. Logical varieties, quasivarieties and prevarieties studied in this book (Chapter 3) are powerful generalizations of logical calculi providing many additional possibilities. This naturally brings us to the following problem. Problem 7. Develop logic as a theoretical discipline based on logical varieties, quasi-varieties and prevarieties of all three types — syntactic, semantic, and model varieties, quasi-varieties and prevarieties. Finally, some theories presented in this book, such as the mathematical theory of schemas (Chapter 5), the theory of logical varieties, quasi-varieties, and prevarieties (Chapter 3) or the theory of abstract properties (Chapter 5), allow and need further development.
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The science of Pure Mathematics, in its modern developments, may claim to be the most original creation of the human spirit. A. N. Whitehead
A. Set theoretical foundations ∅ is the empty set. If X is a set, then r ∈ X means that r belongs to X or r is a member of X. If X and Y are sets, then Y ⊆ X means that Y is a subset of X, i.e., Y is a set such that all elements of Y belong to X. The union Y ∪X of two sets Y and X is the set that consists of all elements from Y and from X. The intersection Y ∩X of two sets Y and X is the set that consists of all elements that belong both to Y and to X. The union i∈I Xi of sets Xi is the set that consists of all elements from all sets Xi , i ∈ I. The intersection i∈I Xi of sets Xi is the set that consists of all elements that belong to each set Xi , i ∈ I. The difference Y \X of two sets Y and X is the set that consists of all elements that belong to Y but does not belong to X. If X is a set, then 2X is the power set of X, which consists of all subsets of X. The power set of X is also denoted by P(X). 809
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If X and Y are sets, then X × Y = {(x, y); x ∈ X, y ∈ Y } is the direct or Cartesian product of X and Y , in other words, X × Y is the set of all pairs (x, y), in which x belongs to X and y belongs to Y . Y X is the set of all mappings from X into Y . · · × X × X. Xn = X × X × · n
Elements of the set X n have the form (x1 , x2 , . . . , xn ) with all xi ∈ X and are called n-tuples, or simply, tuples. If X is a set, then |X| is the cardinality of set X. If a set X is finite, then |X| is the number of elements in the set X. A fundamental structure of mathematics is function. However, functions are special kinds of binary relations between two sets. A binary relation T between sets X and Y is a subset of the direct product X × Y . The set X is called the domain of T (X = Dom(T )) and Y is called the codomain of T (Y = CD(T )). The range of the relation T is Rg(T ) = {y; ∃x ∈ X ((x, y) ∈ T )}. The domain of definition of the relation T is DDom(T ) = {x; ∃y ∈ Y ((x, y) ∈ T )}. If (x, y) ∈ T , then one says that the elements x and y are in relation T , and one also writes T (x, y). Binary relations are also called multivalued functions (mappings or maps). If T is a binary relation between sets X and Y , then the binary relation T −1 = {(y, x) for all (x, y) ∈ T } is called the inverse of the relations T . Taking binary two relations R ⊆ X × Y and Q ⊆ Y × Z, it is possible to build a new relation QR ⊆ X × Z also denoted by Q ◦ R that is called composition or superposition of relations R and Q and is defined by the following rule QR = {(x, z) ∈ X × Z; ∃y ∈ Y ((x, y) ∈ R and (y, z) ∈ Q)}. If T is a binary relation between sets X and Y , and R is a binary relation between sets Z and V , then the direct or Cartesian product T × R of binary relation between sets T and R is defined as T × R = {((x, z); (y, v); (x, y) ∈ T, (z, v) ∈ R}.
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A family J of subsets of X, is called a filter if it satisfies the following conditions: If P ∈ J and P ⊆ Q, then Q ∈ J. If P ,Q ∈ J, then P ∩ Q ∈ J. It is possible to read more on set theory, for example, in (Bourbaki, 1960; Kuratowski and Mostowski, 1967). A preorder (also called quasiorder) on a set X is a binary relation Q on X that satisfies the following axioms: 1. Q is reflexive, i.e., xQx for all x from X. 2. Q is transitive, i.e., xQy and yQz imply xQz for all x, y, z ∈ X. A partial order is a preorder that satisfies the following additional axiom: 3. Q is antisymmetric, i.e., xQy and yQx imply x = y for all x, y ∈ X. 4. A strict partial order is a preorder that is not reflexive, is transitive and satisfies the following additional axiom: Q is asymmetric, i.e., only one relation xQy or yQx is true for all x, y ∈ X. An equivalence on a set X is a binary relation Q on X that is reflexive, transitive and satisfies the following additional axiom: 5. Q is symmetric, i.e., xQy implies yQx for all x and y from X. A function (also called a mapping or map or total function or total mapping) f from X to Y is a binary relation between sets X and Y in which there are no elements from X which are corresponded to more than one element from Y and to any element from X, some element from Y is corresponded. Often total functions are also called everywhere defined functions. Traditionally, the element f (a) is called the image of the element a and denotes the value of f on the element a from X. At the same time, the function f is also denoted by f : X → Y or by f (x). In the latter formula, x is a variable and not a concrete element from X.
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A partial function (or partial mapping) f from X to Y is a binary relation between sets X and Y in which there are no elements from X that are corresponded to more than one element from Y . Thus, any function is also a partial function. Sometimes, when the domain of a partial function is not specified, we call it simply a function because any partial function is a total function on its domain. A multivalued function (or mapping) f from X to Y is any binary relation between sets X and Y . f (x) ≡ a means that the function f (x) is equal to a at all points where f (x) is defined. Two important concepts of mathematics are the domain and range of a function. However, there is some ambiguity for the first of them. Namely, there are two distinct meanings in current mathematical usage for this concept. In the majority of mathematical areas, including the calculus and analysis, the term “domain of f ” is used for the set of all values x such that f (x) is defined. However, some mathematicians (in particular, category theorists), consider the domain of a function f : X → Y to be X, irrespective of whether f (x) is defined for all x in X. To eliminate this ambiguity, we suggest the following terminology consistent with the current practice in mathematics. If f is a function from X into Y , then the set X is called the domain of f (it is denoted by Dom f ) and Y is called the codomain of T (it is denoted by Codom f ). The range Rg f of the function f is the set of all elements from Y assigned by f to, at least, one element from X, or formally, Rg f = {y; ∃x ∈ X(f (x) = y)}. The domain of definition DDom f of the function f is the set of all elements from X that related by f to, at least, one element from Y is or formally, DDom f = {x; ∃y ∈ Y (f (x) = y)}. Thus, for a partial function f (x), its domain of definition DDom f is the set of all elements for which f (x) is defined. Taking two mappings (functions) f : X → Y and g : Y → Z, it is possible to build a new mapping (function) gf : X → Z that is called composition or superposition of mappings (functions) f and g and defined by the rule gf(x) = g(f (x)) for all x from X.
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For any set S, χS (x) is its characteristic function, also called set indicator function, if χS (x) is equal to 1 when x ∈ S and is equal to 0 when x ∈ / S, and CS (x) is its partial characteristic function if / S. CS (x) is equal to 1 when x ∈ S and is undefined when x ∈ If f : X → Y is a function and Z ⊆ X, then the restriction f|Z of f on Z is the function defined only for elements from Z and f|Z (z) = f (z) for each element z from Z. If U is a correspondence of a set X to a set Y (a binary relation between X and Y ), i.e., U ⊆ X ×Y , then U (x) = {y ∈ Y ; (x, y) ∈ U } and U −1 (y) = {x ∈ X; (x, y) ∈ U }. An n-ary relation R in a set X is a subset of the nth power of X, i.e., R ⊆ X n . If (a1 , a2 , . . . , an ) ∈ R, then one says that the elements a1 , a2 , . . . , an from X are in relation R. Let X be a set. An integral operation W on the set X is a mapping that given a subset of X, corresponds to it an element from X, and for any x ∈ X, W ({x}) = x. Examples of integral operations are: sums, products, taking minimum, taking maximum, taking infimum, taking supremum, integration, taking the first element from a given subset, taking the sum of the first and second elements from a given subset, and so on. Examples of finite integral operations defined for numbers are: sums, products, taking minimum, taking maximum, taking average, weighted average, taking the first element from a given subset, and so on. As a rule, integral operations are partial, that is, they assign values, e.g., numbers, only to some subsets of X. Proposition A.1. Any binary operation in X generates a finite ordinal integral operation on X. It is possible to read more about integral operations and their applications in (Burgin and Karasik, 1976; Burgin, 2004a). Set theory is correctly considered the base of the major part of contemporary mathematics. For a long time, the word set was used in mathematics as an informal notion. Only at the end of the 19th century and at the beginning of the 20th century this notion was
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formalized in the process of the set theory development. In turn, set theory provided rigorous foundations for the whole mathematics. However, applications of mathematical structures to real life phenomena demonstrated limitations of sets. As a result various generalizations of sets have been suggested. The most popular of these generalizations are fuzzy sets and multisets. A multiset is similar to a set, but can contain indiscernible elements or different copies of the same elements. It is possible to read more about multisets in (Aigner, 1979; Knuth, 1997). A fuzzy set A in a set U is the triad (U, µA , [0, 1]), where [0, 1] is an interval of real numbers, µA : U → [0, 1] is a membership function of A, and µA (x) is the degree of membership in A of x ∈ U . It is possible to read more about fuzzy sets, for example, in (Klir and Folger, 1988; Zimmermann, 2001). Named sets as the most encompassing and fundamental mathematical construction encompass all generalizations of ordinary sets and provide unified foundations for the whole mathematics (Burgin, 2011). A named set (also called a fundamental triad) has the following graphic representation (Burgin, 1990; 1991; 1997; 2011): connection Entity 1
Entity 2
,
correspondence Essence 1
Essence 2
(1) .
(2)
In the fundamental triad (named set) (1) or (2), Entity 1 (Essence 1) is called the support, the Entity 2 (Essence 2) is called the reflector (also called the set or component of names) and the connection (correspondence) between Entity 1 (Essence 1) and Entity 2 (Essence 2) is called the reflection (also called the naming correspondence) of the fundamental triad (1) (respectively, (2). In the symbolic form, a named set (fundamental triad) X is a triad (X,f ,I) where X is the support of X and is denoted by S(X), I is the component of names (also called set of names or reflector) of X and is denoted by N(X), and f is the naming correspondence (also called reflection) of the named set X and is denoted by n(X). The most
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popular type of named sets is a named set X = (X, f, I) in which X and I are sets and f consists of connections between their elements. When these connections are set theoretical, i.e., each connection is represented by a pair (x, a) where x is an element from X and a is its name from I, we have a set theoretical named set, which is a binary relation. Even before the concept of a fundamental triad was introduced, Bourbaki in their fundamental monograph (1960) had also represented binary relations in a form of a triad (named set). Using the term triad, it is necessary to distinguish it from the notion of a triplet. A triad is a system that consists of three parts (elements or components), while a triplet is any three objects. Thus, any triad is a triplet, but not any triplet is a triad. In a triad, there are ties and/or relations between all three parts (objects from the triad), while for a triplet, this is not necessary. There are many named sets that are not set theoretical. For instance, an algorithmic named set A = (X, A, Y ) consists of an algorithm A, the set X of inputs and the set Y of outputs. Let us named set
I
f
X
Figure A1. A set theoretical named set X = (X, f, I)
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take as X and Y the set of all words in some alphabet Q and all Turing machines that work with words in the alphabet Q as algorithms. Then theory of algorithms tells us that, there are much more algorithmic named sets than relations (set theoretical named sets) because several different algorithms (e.g., Turing machines) can define the same function or relation. Thus, algorithmic named sets are different from set theoretical named sets. Mereological named sets are essentially different from set theoretical named sets (Le´sniewski, 1916; 1992; Leonard and Goodman, 1940; Burgin, 2011). Categorical named sets (fundamental triads) also are different from set theoretical named sets. For instance, an arrow in a category is a fundamental triad but does not include sets as components (Herrlich and Strecker, 1973). Named sets with physical components, such as (a woman and her name), (an article and its title), (a book and its title) and many others, are far from being set theoretical. People meet fundamental triads (named sets) constantly in their everyday life. People and their names constitute a named set. Cars and their owners constitute another named set. Books and their authors constitute one more named set. A different example of a named set (fundamental triad) is given by the traditional scheme of communication: Sender
Receiver .
In this case, connection may be one of the following: a channel, communication media, or a message. People can even see some fundamental triads (named sets). Here are some examples. When it is raining, we see the fundamental triad that consists of a cloud(s) (Entity 1), the Earth where we stand (Entity 2) and flows of water (the correspondence). When we see a lightning, we see another fundamental triad that consists of a cloud(s) (Entity 1), the Earth where we stand (Entity 2) and the lightning (the correspondence). There are many fundamental triads in which Entity 1 is some set, Entity 2 consists of the names of the elements from the Entity 1 and elements are connected with their names by the naming relation.
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This explains the name “named set” that has been applied to this structure (Burgin, 1990; 1991; 2011). A standard model of a named set is a set of people who constitute the carrier, their names that form the set of names, and the naming relation consists of the correspondence between people and their names. Many mathematical systems are particular cases of named sets or fundamental triads. The most important of such systems are fuzzy sets (Zadeh, 1965; 1973; Zimmermann, 1991), multisets (Knuth, 1997), graphs and hypergraphs (Berge, 1973), topological and fiber bundles (Husemoller, 1994; Herrlich and Strecker, 1973). Moreover, any ordinary set is, as a matter of fact, some named set, and namely, a singlenamed set, i.e., such a named set in which all elements have the same name (Burgin, 2011). It is interesting that utilization of singlenamed sets instead of ordinary sets allows one to solve some problems in science, e.g., Gregg’s paradox in biology (Ruse, 1973; Burgin, 1983). It is possible to find many named sets in physics. For instance, according to particle physics, any particle has a corresponding antiparticle, e.g., electron corresponds to positron, while proton corresponds to antiproton. Thus, we have a named set with particles as its support and antiparticles as its set of names. A particle and its antiparticle have identical mass and spin, but have opposite value for all other non-zero quantum number labels. These labels are electric charge, color charge, flavor, electron number, muon number, tau number, and barion number. Particles and their quantum number labels form another named set with particles as its support and quantum number labels as its set of names. When we study information and information processes, fundamental triads become extremely important. Each direction in information theory has fundamental concepts that are and models of which are either fundamental triads or systems built of fundamental triads. Indeed, the relation between information and the receiver/recipient introduced in the Ontological Principle O1 is a fundamental triad, while the Ontological Principles O4 and O4a introduce the interaction and second communication triads (Burgin, 2010). The first communication triad is basic in statistical
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information theory (Chapter 3). A special case of named sets are Chu spaces used in information studies, introduced by Barr (1979) and theoretically developed by Chu. A Chu space C over a set B is a triad (A, r, X) with r : A × X → B. The set A is called the carrier and X the cocarrier of the Chu space C. Matrices are often used to represent Chu spaces. Note that in the form (A, r, X), Chu space is a triad but not a fundamental triad. At the same time, in the more complete form (A × X, r, B), Chu space is a particular case of fundamental triads. One more example of named sets is given by operands from the multidimensional structured model of computer systems and computations (Burgin and Karasik, 1976; Burgin, 1976; 1982). An operand Q is a triad (Z, r, C) where Z = Z1 × Z2 × · · · × Zn , each Zi is a subset of the set Z of all integer numbers, r : Z × Z → C and C is a set. The set Z is called the support of the operand Q. Taking an operand with n = 2 and arbitrary sets Z1 and Z2 , we get the concept of a valued relation (Dukhovny and Ovchinnikov, 2000; Frascella and Guido, 2008). An operator A in the multidimensional structured model of computer systems and computations is a mapping A = Ql → Qh , where Qk is the set of all k-tuples (Q1 , Q2 , . . . Qk ) and Qi all are operands (i = 1, 2, 3, . . . , k; k ∈ {l, h}). Many constructions, such as valued relations, fuzzy relations (Salii, 1965), classifications in the sense of (Barwise and Seligman, 1997), and Chu spaces, are particular cases of such operands. Named sets are explicitly used in many areas: in models of computers and computation (Burgin and Karasik, 1976), artificial intelligence (Burgin and Gladun, 1989; Burgin and Gorsky, 1991; Burgin and Kuznetsov, 1992), mathematical linguistics (Burgin and Burgina, 1982), software engineering (Browne et al., 1995) and Internet technology (Balakrishnan et al., 2004; Cunnigham, 2004). Set-theoretical named sets are very popular in databases and knowledge engineering due to the fact that both binary relations and hierarchical structures are specific kinds of named sets. Even images of
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data and knowledge structures and operations in such structures are presented as named sets (cf., for example, (Elmasri and Navathe, 2000). The reason for such a popularity of named sets in these areas is the fact that it is possible to represent any data structure by named sets and their chains (Burgin, 1997; Burgin and Zellweger, 2005; Burgin, 2008a). There are different formal mathematical definitions of named sets/fundamental triads: in categories, in set theory and by axioms (Burgin, 2011). Axiomatic representation of named sets shows that named set theory, as a formalized mathematical theory, is independent from set theory and category theory. When category theory is built independently from set theory, then categorical representations of named sets are also independent from set theory. It is also necessary to emphasize that physical fundamental triads (named sets), i.e., fundamental triads that are structures of physical objects, are independent from set theory. An important category of named sets is abstract properties. An abstract property P of objects from the universe U is a named set P = (U, p, L) where L is a partially ordered set called the scale Sc(P ) of the abstract property P and p : U → L is a partial function (L-predicate) called the evaluation function Ev(P ) of P . L is called the scale Sc(P ) of the abstract property P . Abstract properties are used as mathematical models of real properties. It is possible to read more about named sets in (Burgin, 1990; 1991; 1992a; 1997; 2011). B. Elements of the theory of algorithms The theory of algorithms is the most abstract part of the theory of algorithms, computation and automata. Abstract algorithms and automata work with symbolic data, usually, in the form of words. An alphabet is a set of symbols, e.g., A = {1, 0} or B = {a, b, c}. A string is a sequence of alphabet symbols, e.g., 101000. A word is a string that belongs to some language. A formal language is a set of words in a fixed alphabet.
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The most general structure in the theory of abstract automata is state transition machine. A state transition machine (STM), also called a state transition system, A consists of three structures and can be represented as A = (L, S, δ): — The linguistic structure L = (AI , Q, AO ) where AI is an alphabet of input symbols, Q is a set of states, and AO is a set of output symbols of the STMA; — The state structure S = (Q, q0 , F ) where q0 is an element from Q called the initial or start state and F is a subset of Q called the set of final states of the STM A; — The action structure δ traditionally called the transition function, or more exactly, transition relation of the STMA; the transition relation δ determines how input and the current state determine the next state and output, i.e., δ : AI × Q → Q × AO δ is, in general, a multivalued function. It can be represented by two relations (functions): The state transition relation (function): δtr : AI × Q → Q and the output relation (function): δout : AI × Q → AO . Examples of state transition machines are finite automata, pushdown automata, Turing machines, inductive Turing machines, limit Turing machines, Petri nets, neural networks, and cellular automata (cf., (Burgin, 2005)). The structure of an inductive Turing machine or a Turing machine, as an abstract automaton, consists of three components called hardware, software, and infware. Infware is a description and specification of information that is processed by an (inductive) Turing machine. Computer infware consists of data processed by the computer. Inductive Turing machines and conventional Turing machines are abstract automata working with the same symbolic
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information in the form of words. Consequently, formal languages with which (inductive) Turing machines works constitute their infware. Computer hardware consists of all devices (the processor, system of memory, display, keyboard, etc.) that constitute the computer. In a similar way, an inductive Turing machine (a Turing machine) M has three abstract devices: a control device A, which is a finite automaton and controls performance of M ; a processor or operating device H, which corresponds to one or several heads of a conventional Turing machine; and the memory E, which corresponds to the tape or tapes of a conventional Turing machine. The memory E of the simplest inductive Turing machine consists of three linear tapes, and the operating device consists of three heads, each of which is the same as the head of a Turing machine and works with the corresponding tape. The control device A is a finite automaton that regulates: the state of the whole machine M , the processing of information by H, and the storage of information in the memory E. The memory E is divided into different but, as a rule, uniform cells. It is structured by a system of relations that organize memory as a well-structured system and provide connections or ties between cells. In particular, input registers, the working memory, and output registers of the inductive Turing machine M are separated. Connections between cells form an additional structure K of E. Each cell can contain a symbol from an alphabet of the languages of the machine M or it can be empty. In a general case, cells may be of different types. Different types of cells may be used for storing different kinds of data. For example, binary cells, which have type B, store bits of information represented by symbols 1 and 0. Byte cells (type BT) store information represented by strings of eight binary digits. Symbol cells (type SB) store symbols of the alphabet(s) of the machine M . Cells in conventional Turing machines have SB type. Natural number cells, which have type NN, are used in random access machines. Cells in the memory of quantum computers (type QB) store q-bits or quantum bits (Deutsch, 1985). Cells of the tape(s) of real-number Turing machines
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(Burgin, 2005) have type RN and store real numbers. When different kinds of devices are combined into one, this new device has several types of memory cells. In addition, different types of cells facilitate modeling the brain neuron structure by inductive Turing machines. It is possible to realize an arbitrary structured memory of an inductive Turing machine M , using only one linear one-sided tape L. To do this, the cells of L are enumerated in the natural order from the first one to infinity. Then L is decomposed into three parts according to the input and output registers and the working memory of M . After this, nonlinear connections between cells are installed. When an inductive Turing machine with this memory works, the head/processor is not moving only to the right or to the left cell from a given cell, but uses the installed nonlinear connections. Such realization of the structured memory allows us to consider an inductive Turing machine with a structured memory as an inductive Turing machine with conventional tapes in which additional connections are established. This approach has many advantages. One of them is that inductive Turing machines with a structured memory can be treated as multitape automata that have additional structure on their tapes. Then it is conceivable to study different ways to construct this structure. In addition, this representation of memory allows us to consider any configuration in the structured memory E as a word written on this unstructured tape. If we look at other devices of the inductive Turing machine M , we can see that the processor H performs information processing in M . However, in comparison to computers, this operational device performs very simple operations. When H consists of one unit, it can change a symbol in the cell that is observed by H, and go from this cell to another using a connection from K. This is exactly what the head of a Turing machine does. It is possible that the processor H consists of several processing units similar to heads of a multihead Turing machine. This allows one to model in a natural way various real and abstract computing systems by inductive Turing machines. Examples of such systems are: multiprocessor computers; Turing machines with several tapes;
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networks, grids and clusters of computers; cellular automata; neural networks; and systolic arrays. We know that programs constitute computer software and tell the system what to do (and what not to do). The software R of the inductive Turing machine (Turing machine) M is also a program in the form of simple rules: (B.1) qh ai → aj qk , qh ai → cq k ,
(B.2)
(B.3) qh ai → aj qk c. Here qh and qk are states of A, ai and aj are symbols of the alphabet of M , and c is a type of connection in the memory E. Each rule directs one step of computation of the inductive Turing machine M . The rule (1) means that if the state of the control device A of M is qh and the processor H observes in the cell the symbol ai , then the state of A becomes qk and the processor H writes the symbol aj in the cell where it is situated. The rule (2) means that the processor H then moves to the next cell by a connection of the type c. In Turing machines with linear tapes, there are only two types of connections: R is the connection to the right neighbor and L is the connection to the left neighbor of the cell. The rule (3) is a combination of rules (1) and (2). Like Turing machines, inductive Turing machines can be deterministic and non-deterministic. For a deterministic inductive Turing machine, there is at most one connection of any type from any cell. In a non-deterministic inductive Turing machine, several connections of the same type may go from some cells, connecting them with (different) other cells. If there is no connection of the prescribed by an instruction type that goes from the cell that is observed by H, then H stays in the same cell. There may be connections of a cell with itself. Then H also stays in the same cell. It is possible that H observes an empty cell. To represent this situation, we use the symbol Λ. Thus, it is possible that some elements ai and/or aj in the rules from R are equal to ε in the rules of all types. Such rules describe situations when H observes an empty cell and/or when H simply erases the symbol from some cell, writing nothing in it.
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The rules of the type (3) allow an inductive Turing machine to rewrite a symbol in a cell and to make a move in one step. Other rules (1) and (2) separate these operations. Rules of the inductive Turing machine M define the transition function of M and describe changes of A, H, and E. Consequently, these rules also determine the transition functions of A, H, and E. A general step of the machine M has the following form. At the beginning of any step, the processor H observes some cell with a symbol ai (for an empty cell the symbol is Λ) and the control device A is in some state qh . Then the control device A (and/or the processor H) chooses from the system R of rules a rule r with the left part equal to qh ai and performs the operation prescribed by this rule. If there is no rule in R with such a left part, the machine M stops functioning. If there are several rules with the same left part, M works as a nondeterministic Turing machine, performing all possible operations. When A comes to one of the final states from F , the machine M also stops functioning. In all other cases, it continues operation without stopping. For an abstract automaton, as well as for a computer, three things are important: how it receives data, process data and obtains its results. In contrast to Turing machines, inductive Turing machines obtain results even in the case when their operation is not terminated. This results in essential increase of performance abilities of systems of algorithms. The computational result of the inductive Turing machine M is the word that is written in the output register of M : when M halts while its control device A is in some final state from F , or when M never stops but at some step of computation the content of the output register becomes fixed and does not change although the machine M continues to function. In all other cases, M gives no result. The memory E is called recursive if all relations that define its structure are recursive. Here recursive means that there are some Turing machines that decide/build all naming mappings and relations in the structured memory.
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Inductive Turing machines with recursive memory are called inductive Turing machines of the first order. The memory E is called n-inductive if all relations that define its structure are constructed by an inductive Turing machine of order n. Inductive Turing machines with n-inductive memory are called inductive Turing machines of the order n + 1. Limit Turing machines have the same structure (hardware) as inductive Turing machines. The difference is in a more general way of obtaining the result of computation. To obtain their result, limit Turing machines need some topology in the set of all words that are processed by these machines. Let a limit Turing machine L works with words in an alphabet A and in the set A* of all such words, a topology T is defined. While the machine L works, it produces words w1 , w2 , . . . , wn , . . . in the output tape (output memory). Then the result of computation of the limit Turing machine L is the limit of this sequence of words in the topology T . When the set A* has the discrete topology, limit Turing machines coincide with inductive Turing machines. Turing machines are special cases of inductive Turing machines. The difference is in a more general way of obtaining the result of computation. To obtain their result, Turing machine gives the result if and only if its control device A comes to a final state from F . Besides, the memory of a Turing machine consists of one (or several) potentially infinite one-dimensional (or multidimensional) tapes. It is possible to read more on algorithms in (Sipser, 1997; Hopcroft et al., 2007; Burgin, 2005). C. Elements of algebra and category theory An algebraic system is a structure A = (X, Ω, R) that consists of: a non-empty set X called the carrier or the underlying set of A and elements of which are called the elements of A; a family Ω of algebraic operations, which are mappings ωi : X ni → X(i ∈ I); and a family R of relations rj ⊆ X mj (j ∈ J) defined on X.
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The non-negative integers ni and mj are called the arities of the respective operations ωi and relations rj . When ω is an operation from Ω with arity n, then the image ω(a1 , a2 , . . . , an ) of the element (a1 , a2 , . . . , an ) from X n under the mapping ω : X n → X is called the value of the operation ω for elements a1 , a2 , . . . , an . In a general case, some operations from Ω may be partial mappings. For instance, division in such a universal algebra (algebraic system) as a field is not defined for 0, i.e., it is impossible to divide by 0. The operations from Ω and the relations from R are called basic or primitive. Using basic operations, it is possible to build many different derivative operations in the algebra A. The pair of families ({ni ; i ∈ I}; {mj ; j ∈ J}) is called the type of the algebraic system A. Two algebraic systems A and A have the same type if I = I , J = J , and ni = ni and mj = mj for all i ∈ I, j ∈ J. An algebraic system A is called finite if the set X is finite and it is called of finite type if the set I ∪ J is finite. An algebraic system A = (X, Ω, R) is called a universal algebra or simply, an algebra if the set R of its basic relations is empty. An algebraic system A = (X, Ω, R) is called a model (in logic) or a relational system if the set Ω of basic operations is empty. A heterogeneous or many-sorted or multibase universal algebra A is a set A with a system of operations Σ where A is the union {Ai ; i ∈ I} of the indexed system (family) {Ai ; i ∈ I} of sets and each operation is a mapping of the form f : Ai1 ×Ai2 ×· · ·×Aik → Ai . The system A is called the carrier or support of the multibase algebra A and the system I is called the collection of sorts of the multibase algebra A. For instance, a deterministic finite automaton A with the input alphabet Σ, the input alphabet Ω and the set of states Q is a many-sorted algebra, which has the support {Σ, Q, Ω}, two binary operations δ : Σ × Q → Q, and σ : Σ × Q → Ω, and several unary operations σ0 , σ1 , . . . , σk on the set Q : σ0 = q0 , σ1 = q1 , . . . , σk = qk with q1 , . . . , qk ∈ F . Classical algebraic systems are groups, rings, linear (vector) spaces, linear algebras, lattices, ordered sets, ordered groups, etc.
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Groups, rings, modules, fields, linear spaces, linear algebras, lattices, and semigroups are classical universal algebras. For instance, a linear space or a vector space L over the field F, e.g., the field R of real numbers, elements of which are often called vectors, has two operations: addition: L × L → L denoted by x + y where x and y belong to L; scalar multiplication: F × L → L denoted by ax where a ∈ F and x ∈ L. These operations satisfy the following axioms: Addition is associative: For all x, y, z from L, we have x + (y + z) = (x + y) + z. Addition is commutative: For all x, y from L, we have x + y = y + x. Addition has an identity element: There exists an element 0 from L, called the zero vector, such that x + 0 = x for all x from L. Addition has an inverse element: For any x from L, there exists an element z from L, called the additive inverse of x, such that x + z = 0. Scalar multiplication is distributive over addition in L: For all elements a from F and vectors y, w from L, we have a (y + w) = a y + a w. Scalar multiplication is distributive over addition in F : For all element elements a, b from F and any vector y from L, we have (a + b)y = ay + by. Scalar multiplication is compatible with multiplication in F :
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For all elements a, b from F and any vector y from L, we have a(by) = (ab)y. The identity element 1 from the field F also is an identity element for scalar multiplication: For all vectors x from L, we have 1x = x. Vectors x1 , x2 , . . . , xn from L are called linearly dependent in L if there is an equality Σni=1 ai xi = 0 where ai are elements from R (from F , in a general case) and not all of them are equal to 0. When there are no such an equality, vectors x1 , x2 , . . . , xn are called linearly independent. A system B of linearly independent vectors from L is called a basis of L if any element x from L is equal to a sum Σni=1 ai xi where n is some natural number, xi are elements from B and ai are elements from R (from F , in a general case). The number of elements in a basis is called the dimension of the space L. It is proved that all bases of the same space have the same number of elements. The space R is a one-dimensional vector (linear) space over itself. The space Rn is an n-dimensional vector (linear) space over R. An n-ary operation of a universal algebra A is called commutative if for any permutation i1 , i2 , i3 , . . . , in of numbers 1, 2, 3, . . . , n, we have ω(x1 , x2 , x3 , . . . , xn ) = ω(xi1 , xi2 , xi3 , . . . , xin ) for any elements x1 , x2 , x3 , . . . , xn from A. A subsystem of a universal algebra A closed with respect to basic operations is called a sub-algebra of A. A subsystem of a model is called a sub-model. The concept of a sub-algebra essentially depends on the set of operations of the algebra under consideration. In contrast to this, any non-empty subset of a model is a sub-model. A set V of universal algebras of a type Ω is called a variety if there is a system of identities such that V consists of all universal algebras of a type Ω that satisfy these identities. A variety of universal algebras may be characterized as a non-empty class of algebras closed under taking quotient algebras, sub-algebras and direct products.
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It is possible to find other structures from algebra and their properties, for example, in (Kurosh, 1963; Van der Varden, 1971). There are two approaches to the mathematical structure called a category. One approach treats categories in the framework of the general set-theoretical mathematics. Another approach establishes categories independently of sets and uses them as a foundation of mathematics different from set theory. It is possible to build the whole mathematics in the framework of categories. For instance, such a basic concept as a binary relation is frequently studied in categories. Toposes allow one to reconstruct set theory as a subtheory of category theory (cf., for example, (Goldblatt, 1979)). According to the first approach, we have the following definition of a category. A category C consists of two collections Ob C, the objects of C, and Mor C, the morphisms of C that satisfy the following three axioms: A1. For every pair A, B of objects, there is a set MorC (A, B), also denoted by HC (A, B) or HomC (A, B), elements of which are called morphisms from A to B in C. When f is a morphism from A to B, it is denoted by f : A → B. A2. For every three objects A, B and C from Ob C, there is a binary partial operation, which is a partial function from pairs of morphisms that belong to the direct product MorC (A, B) × MorC (B, C) to morphisms in MorC (A, C). In other words, when f : A → B and g : B → C, there is a morphism g ◦ f : A → C called the composition of morphisms g and f in C. This composition is associative, that is, if f : A → B, g : B → C and h : C → D, then h ◦ (g ◦ f ) = (h ◦ g) ◦ f . A3. For every object A, there is a morphism 1A in MorC (A, A), called the identity on A, for which if f : A → B, then 1B ◦f = f and f ◦ 1A = f . Examples of categories: The category of sets SET: objects are arbitrary sets and morphisms are mappings of these sets. The category of groups GRP: objects are arbitrary groups and morphisms are homomorphisms of these groups.
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The category of topological spaces TOP: objects are arbitrary topological spaces and morphisms are continuous mappings of these topological spaces. Mapping of categories that preserve their structure are called functors. There are functors of two types: covariant functors and contravariant functors. A covariant functor F : C → K, also called a functor, from a category C to a category K is a mapping that is stratified into two related mappings FObC : Ob C → Ob K and FMor C : Mor C → Mor K, i.e., FObC associates an object F (A) from the category K to each object A from the category C and FMorC associates a morphism F (f ) : F (A) → F (B) from the category K to each morphism f : A → B from the category C. In addition, F satisfies the following two conditions: F (1A ) = 1F (A) for every object A from the category C; F (f ◦ g) = F (f ) ◦ F (g) for all morphisms f and g from the category C when their composition f ◦ g exists. That is, functors preserve identity morphisms and composition of morphisms. A contravariant functor F : C → K from a category C to a category K consists of two mappings FObC : Ob C → Ob K and FMorC : Mor C → Mor K, i.e., FObC associates an object F (A) from the category K to each object A from the category C and FMorC associates a morphism F (f ) : F (A) → F (B) from the category K to each morphism f : A → B from the category C, that satisfy the following two conditions: F (1A ) = 1F (A) for every object A from the category C; F (f ◦ g) = F (g) ◦ F (f ) for all morphisms f and g from the category C when their composition f ◦ g exists. It is possible to define a contravariant functor as a covariant functor on the dual category Cop . A functor from a category to itself is called an endofunctor. There is an approach to the definition of a category in which a category C consists only of the collection Mor C of the morphisms
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(also called arrows) of C with corresponding axioms. Objects of C are associated with identity morphisms 1A . It is possible to do this because 1A is unique in each set MorC (A, A), and uniquely identifies the object A. In any case, morphism is the central concept in a category. But what is a morphism? If f : A → B is a morphism, then it is a one-to-one named set ({A}, f, {B}). Thus, the main object of a category is a named set (fundamental triad), and categories are built from these named sets (fundamental triad). Besides, a construction or separation of a category begins with separation of all elements into two sets and calling all elements from one of these sets by the name “objects” and elements from the other sets by the name “morphisms”. In such a way, two named sets appear. In addition, composition of morphisms, as any algebraic operation, is also represented by a named set (MorC (A, B)×MorC (B, C), ◦, MorC (A, C)). This shows that the informal notion of a named set is prior both to categories and sets. As a result, we come to the conclusion that any category is built of different named sets. Moreover, functors between categories, which are structured mappings of categories (Herrlich and Strecker, 1973), are morphisms of those named sets. It is possible to read more about categories, functors and their properties, for example, in (Goldblatt, 1979; Herrlich and Strecker, 1973). D. Numbers and numerical functions N is the set of all natural numbers 1, 2, . . . , n, . . . . ω is the sequence of all natural numbers. N0 is the set of all whole numbers 0, 1, 2, . . . , n, . . . . Z is the set of all integer numbers or of integers. Q is the set of all rational numbers. R is the set of all real numbers or of reals. The geometric form of the set R is called the real line. C is the set of all complex numbers. ∞ is the positive infinity, −∞ is the negative infinity. Usually, these elements are added to the set R. The new set is denoted by R∞ = R ∪ {∞, −∞}.
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If a is a real number, then |a| or a denotes its absolute value or modulus. Thus, |a| = a when a is non-negative and |a| = −a when a is negative. Two more important constructions are the integral part or integral value [a] (also denoted by a ) of a, which is equal to the largest integer number that is less than a, and ]a[ (also denoted by
a),which is equal to the least integer number that is larger than a. For instance, [1.2] = 1, [7.99] = 7 and [−1.2] = −2. The difference a — [a], also denoted by a mod 1, or {a}, is called the fractional part of a. All these construction define real-valued functions |x|, [x], {x}, and ]x[ where ]x[ is sometimes called the ceiling function and [x] is sometimes called the floor function. The symbol ≈ denotes the relation “approximately equal”. For instance, we have 5 ≈ 5.001. Axioms for operations with real and complex numbers: Commutativity of addition: a + b = b + a; Associativity of addition: (a + b) + c = a+ (b + c); Commutativity of multiplication: a · b = b · a; Associativity of multiplication: (a · b) · c = a · (b · c); Distributivity of multiplication with respect to addition: a · (b + c) = a · b + a · c. Zero is a neutral element with respect to addition: a + 0 = 0 + a = a. One is a neutral element with respect to multiplication: a · 1 = 1 · a = a. A function with R as its range is called a real function or a realvalued function. A function with C as its range is called a complex function or a complex-valued function. Operations with numbers induce similar operations with functions: Addition of functions (f + g)(x) = f (x) + g(x). Subtraction of functions (f − g)(x) = f (x) − g(x). Multiplication of functions (f · g)(x) = f (x) · g(x). Scalar multiplication of functions (k ·f )(x) = k ·g(x) by a number k.
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E. Topological, metric and normed spaces A topology in a set X is a system O(X) of subsets of X that are called open subsets and satisfy the following axioms: T1. X ∈ O(X) and ∅ ∈ O(X). T2. For all A, B, if A, B ∈ O(X), then A ∩ B ∈ O(X). T3. For all Ai , i ∈ I, if all Ai ∈ O(X), then i∈I Ai ∈ O(X). A set X with a topology in it is called a topological space. Topology in a set can be also defined by a system of neighborhoods of points from this set. In this case, a set is open in this topology if it contains a standard neighborhood of each of its points. For instance, if a is a real number and t ∈ R++ , then an open interval Ot a = {x ∈ R; a − t < x < a + t} is a standard neighborhood of a. If X is a subset of a topological space, then Cl(X) denotes the closure of the set X. In many interesting cases, topology is defined by a metric. A metric in a set X is a mapping d : X × X → R+ that defines distances between points of X and satisfies the following axioms: M1. d(x, y) = 0 if and only if x = y, i.e., the distance between an element and itself is equal to zero, while the distance between any different elements is equal to a positive number. M2. d(x, y) = d(y, x) for all x, y ∈ X, i.e., the distance between x and y is equal to the distance between x and y. M3. The triangle inequality: d(x, y) ≤ d(x, z) + d(z, y)
for all x, y, z ∈ X.
That is, the distance from x through z to y is never less than the distance directly from x to y, or the shortest distance between any two points is a straight line. A set X with a metric d is called a metric space. The number d(x, y) is called the distance between x and y in the metric space X. For instance, in the set R of all real numbers, the distance d(x, y) between numbers x and y is the absolute value |x − y| , i.e., d(x, y) = |x−y|. This metric defines the following topology in R. If a is a point
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from R, then a standard neighborhood of a has the form Or (a) = {y; |a − y| < r with r ∈ R++ }. A set is open in this topology if it contains a standard neighborhood of each of its points. Mappings of metric (topological) spaces that preserve the metric (topological) structure are called continuous. It is possible to find other structures from topology and their properties, for example, in (Kuratowski, 1966). n i=1 ci denotes the sum c1 + c2 + c3 + · · · + cn . If U is a subset of a metric space ⊆ R, then the diameter D(U ) of the set U is equal to sup{d(x, y); x, y ∈ U }. If l = {ai ∈ M ; i = 1, 2, 3, . . .} is a sequence, and f : M → L is a mapping, then f (l) = {f (ai ); i = 1, 2, 3, . . .}. a = lim l means that a number a is a limit of a sequence l. It is possible to introduce a natural metric in the space Rn : if x, y ∈ Rn , x = ni=1 ai xi , and y = ni=1 bi xi where xi are elements from a basis B of the space Rn , then d(x, y) = (a1 − b1 )2 + (a2 − b2 )2 + · · · + (an − bn )2 . It is called the Euclidean metric in the space Rn . The space Rn with the Euclidean metric is called an Euclidean space and often denoted by E n . Another natural metric in Rn is the Manhattan distance, where the distance between any two points, or vectors, is the sum of the distances between corresponding coordinates, i.e., d(x, y) = |a1 − b1 | + |a2 − b2 | + · · · + |an − bn |. It is possible to consider any non-empty set X as a metric space with the distance d(x, y) = 1 for all x not equal to y and d(x, y) = 0 otherwise. It is a discrete metric. A norm in a linear space L over the field R is a mapping : L → R+ that satisfies the following axioms: N1. x = 0 if and only if x = 0, i.e., the zero vector has zero length, while any other vector has a positive length. N2. For any positive number a from R, we have ax = a x , i.e., multiplying a vector by a positive number has the same effect on the length.
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N3. The triangle inequality:
x + y ≤ x + y
i.e., the norm of a sum of vectors is never larger than the sum of their norms. This implies:
x − y ≤ x + y . A vector space L with a norm is called a normed vector space or simply, a normed space. The space Rn is a normed vector space with the following norm: where xi are elements from a basis B if x ∈ Rn and x = ni=1 ai xi of the space Rn , then x = a21 + a22 + · · · + a2n . Proposition D.1. Any normed vector space is a metric space. Indeed, we can define d(x, y) = x−y and check that all axioms of metric are valid for this distance. Another natural metric in a normed vector space is the British Rail metric (also called the Post Office metric or the SNCF metric) on a normed vector space, given by d(x, y) = x + y for distinct vectors x and y, and d(x, x) = 0. A Hilbert space is an abstract linear (vector) space over the field R of real numbers or the field C of complex numbers with an inner product and complete as a metric space. An inner product in a vector space V is a function ·, · : V × V → R (or in the complex case ·, · : V × V → C) that satisfies the following properties for all vectors x, y, and x from V and number a: Conjugate symmetry: x, y = y, x. Linearity in the first argument: ax , y = ax, y, x + z, y = x, y + z, y. Positive-definiteness: x, y > 0 for all x = 0 from V .
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A fiber bundle B (also called fibre bundle) is a triad (E, p, B) where the topological space E is called the total space or simply, space of the fiber bundle B, the topological space B the base space or simply, base of the fiber bundle B, and p is a topological projection of E onto B such that every point in the base space has a neighborhood U such that p−1 (b) = F for all points b from B and p−1 (U ) is homeomorphic to the direct product U × F . The topological space F is called the fiber of the fiber bundle B. Informally, a fiber bundle is a topological space which looks locally like a product space U × F . Fiber bundles are special cases of topological bundles. A topological bundle D is a triad (named set) (E, p, B) where E and B are topological spaces and p is a topological (i.e., continuous) projection of E onto B. It is possible to find introductory information on topological manifolds, fiber and topological bundles, and other topological structures in (Gauld, 1974; Husemoller, 1994; Lee, 2000).
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b2334-index
Index
universal, 192 Algorithm, 49–50, 157, 424–425 representation, 160 constructive, 487 data-mining, 693 deduction, 279 first-level, 160 generating, 403 genetic, 694 learning, 163 local search, 163 measuring, 220 network, 759 nondeterministic, 63 optimization, 163 partial search, 163 power of, 304 probabilistic, 63 quantum, 77 randomized, 64 recursive, 74 second-level, 160–162 super-recursive, 220 symbolic, 54 wired, 54 Algorithmic information theory, 131 complexity, 96, 131 ladder, 304
A Abstract, 28, 30, 41, 52, 138 Abstraction, 28, 65, 138, 175 ladder, 138 Abstractness, 95, 137 level of, 137 Action, 14, 21, 49 structure, 820 Acquisition, x, 6 Activity, 15, 17 economic, 7 intellectual, xv mental, 55 practical, 55 Adaptation, 537 Adequate, 561 Aggregate, 82 Algebra, 422, 601 of sets, 236 Boolean, 439 epistemic quantum, 369 information, 192 knowledge, 358 Lie, 420 linear, 827 multibase, 359 process, 420 relational, 153 927
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928 language, 404, 423, 425 problem, 218 process, 125 representation, 160, 402 space, 179 verification, 258 Alphabet, 100 Analogy, 677 Analysis axiological, 3 functional, 3 methodological, 48 sociological, 48 structural, 2 Analytic representation, 34 Anti-knowledge, 76 Architecture concept, 454 named data networking, 759 network, 155 networking, 759 semantic web, 759 three-schema, 553 Arithmetic, 24, 33, 35–36, 76, 764 Array, 153, 410 systolic, 823 Arrow, 329 Artificial intelligence, 2–3, 79 neural network, 54 Aspect, 75 Assertion, 260 Automaton abstract, 157 accepting, 159 cellular, 820 deterministic, 100, 159 finite, 157, 159 grid, 161, 571 non-deterministic, 361 Axiom, 62, 108, 239 Axiomatic approach, 131 mathematical system, 26
b2334-index
Subject Index semantics, 145 set theory, 279 theory of algorithms, 248 B Base, 56, 59 Belief, 169, 175 excessive, 122 false, 763 justified, 23 true, 23 Binary code, 328 information operator, 192 numerical system, 349 operation, 360 property, 105 relation, 188, 280, 375 representation, 132 Biology, 403 Block-schema, 551 Bond, 201, 340 Boolean algebra, 439 grammar, 305 Bounded intellectual activity, 652 rationality, 140 set, 211 uniformly, 214 Bundle, 782 C Calculation, 36 Calculus basic, 75 classical, 57 logical, 216 process, 420 propositional, 97 Cardinality, 293, 810 Category, 417 abstract, 417 algebraic, 417 Cell, 821
page 928
September 27, 2016
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Theory of Knowledge: Structures and Processes - 9in x 6in
Subject Index Century, 8, 11 Certainty, 24, 27–28 Clarity, 130 Classification bidirectional, 66 confidence, 66 domain-oriented, 64 eight-fold bidirectional, 67–68 epistemic, 244 hierarchical, 66–67 ontological, 66 problem-oriented, 61 social, 66 three-dimensional, 65 Creation, 87, 152 Creativity, 87 Culture, 206, 350 Code, 328 Codification, 744 Codomain, 810 Cognition, 3, 9, 15, 20, 28, 78–79, 122, 124, 128–129, 171, 175, 196, 232, 239, 344, 451, 537, 540, 644, 647, 653, 658, 665, 688, 696, 725, 732 Cognitive classification, 55 information, 42 process, 29, 42 semantics, 144 skills, 49 Cognitology, 3 Communication, 117 Completeness, 139 Complex, 124 mathematical object, 473 metaphor, 422 operational knowledge, 123 phenomenon, 123 process, 796 system, 123 Complexity, 53, 125–127 algorithmic, 96, 131 average, 131 average space, 131 average time, 131
b2334-index
page 929
929
axiomatic complexity measure, 127 cognitive, 127 computational, 301 measure, 131 direct complexity measure, 131 dual complexity measure, 131 dynamic complexity measure, 131 inductive Kolmogorov, 801 Kolmogorov, 131, 629, 800–801 of acquisition, 53 of integration, 53 of knowledge, 53 of learning, 53 operational knowledge complexity measure, 131 prefix, 801 problem, 128 space, 131 static complexity measure, 131 time, 131 transformation, 128 uniform, 801 utilization, 128 Component, 20, 47 Composition, 153 Comprehensibility, 130 Computability, 280 Computation, 304, 402 concurren, 280 cooperative, 543 distributed, 556 sequential, 545 symbol-based, 545 Computational complexity, 301 neuroscience, 548 parallel, 550 practice, 425 process, 320 Computer graphics, 556 hardware, 560 program, 562
September 27, 2016
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Theory of Knowledge: Structures and Processes - 9in x 6in
930 programming, 653 science, 556 Computing cloud, 161 metrics, 220 power, 304 practice, 424 Concept, 6, 47 analysis, 184 behavioral, 173 formation, 205 illusive, 124 informal, 185 knowledge, 65 learning, 205 mathematical, 72 meaning, 143 model of, 447, 452–454 predicate, 34 pure, 34 subject, 34 Confidence level, 121 interval, 121 coefficient, 121 bound, 121 Configuration, 71, 346, 546, 822 Connectedness, 207 Consistency, 96 local, 263 global, 263 Consistent, 97, 107, 276, 296 Constructive algorithm, 487 definition, 487, 496 logic, 442 representation, 441, 487, 496 Constructivity, 639 Context, 148 abstract, 103 algebraic, 103 formal, 284 Control, 21, 49 device, 821 Contraction, 634
b2334-index
Subject Index Core, 452–454, 535 Correct descriptively, 256 functionally, 256 operationally, 257 Correctness, 96 consumer-oriented, 259 designer-oriented, 259 programmer-oriented, 259 software, 97, 257 user-oriented, 259 Correlation, 96 Creativity, 399 Criterion pragmatic, 98 semantic, 98 syntactic, 98 Cycle, 19, 249 D Data, 735–740 Data-mining, 690 algorithm, 693 method, 693 Decidability, 301, 512 Decidable set, 512 Decision algorithm, 694 Decidable problem, 512 Deduction, 37, 232 algorithm, 279 rule, 156 Definability, 788 Description of algorithm, 161 Descriptive programming, 401 Design, 8, 126, 711 Device, 3, 424, 478 Differentiation, 497, 684 Dimension abstraction, 65 abstractness, 95 certainty, 95 codification, 65 completeness, 95
page 930
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Theory of Knowledge: Structures and Processes - 9in x 6in
page 931
931
Subject Index complexity, 95 confidence, 95 correctness, 95 diffusion, 66 dynamic, 65 exactness, 95 feature, 95 integration, 95 meaning, 95 possession, 65 separation, 95 typological, 65 validation, 95 Direct product, 208 Direction Buddhist religious, 17 in Chinese philosophy, 431 classical, 431 in epistemology, 139 in Indian philosophy, 10 of information theory, 727 in linguistic semantics, 144 in logic, 269 methodological, x in methodology, 46 notable, 675 philosophical, x, 477 popular, 269, 727 of research, 415 scientific, 395 specific, 681 structuralist, 46 structure-nominative, 47, 603 Discreteness, 95 Disjunctive normal form, 373–374 Dissemination, 6 Domain, 61 Doxastology, 45 Dyadic model, 348, 350 Dynamical system, 240 fuzzy, 236 hybrid discrete–continuous, 286 Dynamics
b2334-index
external, 568 internal, 568 E Efficiency, 135 Efficient, 135 Effective procedure, 424 Element chemical, 523 cognitive, 725 competitive, 704 construction, 158 data, 733 discrete, 733 environment, 525 explicit, 56 grammatical, 427 infological, 722–724 information, 748 inner, 597 knowledge, 54, 93–94 minimal, 481 quantum, 307 rare earth, 523 schema, 556–557 semiotic, 351 structural, 590 tacit, 56 traditional, 346 triad, 749 unknown, 710 XML, 552 Elementary data, 328 image, 351 knowledge unit, 179, 189, 307, 792 particle, 104, 315, 325, 524 phenomenon, 652 script, 533 unit, 328–330, 792 Emergence, 640, 685, 723, 741 Emotion, 15, 56, 82, 141, 406–407, 650 Enumeration, 511
September 27, 2016
19:41
Theory of Knowledge: Structures and Processes - 9in x 6in
932
b2334-index
Subject Index
Energy, 7, 127 Epistemic, 170, 178 Epistemology, 9 Equivalence, 186, 341, 361, 366 Euclidean geometry, 671 metrics, 473–474 plane, 675, 775 space, 211, 676, 834 Evaluation, 169 Evidence, 267 Existence material, 84 mental, 84 physical, 84 structural, 84 Expansion, 634 Experience, 56 Expression, 278 Extension, 677
characteristic, 813 elementary, 87 computable, 665 fuzzy, 558 general recursive, 305 non-deterministic, 558 partial, 338, 503, 778, 812 partial recursive, 341, 665 probabilistic, 558 recursive, 280 set-valued, 558 total, 811 transition, 160 variable, 558 Functional complexity measure, 127 programming language, 426 Functioning, 701 Fuzzy, 103
F
Generality, 137 Generating rules, 403 Goal, 32, 39, 104–105, 155, 164, 171, 173, 197, 492, 665, 709, 736 Government, 4, 8, 21 Grammar context-free, 304–305 context-sensitive, 305 phrase-structure, 305 regular, 305 Graph, 518 Grid automaton, 561–562, 571–572 Guessing, 677
False, 52 absolutely, 52 Fantasy, 225–226, 765 Feeling, 673 Finite sequence, 784 Finite word, 782 Flow-chart, 561 Form analytic, 411 dynamic, 286 static, 521 Formal grammar, 487 language, 487 Formula Boolean, 501 open, 565 closed, 565 Frame, 536 Function adjacency, 564 algorithmic, 567
G
H Hardware abstract, 400 Head, 431 Hierarchy, 152, 197, 568, 570, 636, 729 High-performance computing, 130 History, 227
page 932
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Theory of Knowledge: Structures and Processes - 9in x 6in
Subject Index I Icon, 355 Idea, 175 Ideal, 175 Illogical, 429 Image, 113 conceptual, 113 Inconsistency, 108, 472–473, 719 Inconsistent, 41, 57, 107–108, 218, 270 Index, 355 Indian philosophy, 9 Induction, 37 Inductive computation, 320 hierarchy, 414 inference, 665, 669 process, 660 Turing machine, 157 Industry, 8, 125 Inference, 37, 107 Infinity, 822 actual, 822 negative 831 positive 831 Infological system, 118, 172, 175, 180, 203, 723–724, 754, 765 cognitive, 175 Information cognitive, 42 concept of, 722 essence of, 722 genetic, 722 in the strict sense, 722, 724 operation, 786 phenomenon, 722 processing, 4, 175, 264–265, 307, 541, 545, 680, 689, 691 proper, 722, 724, 726 science, 5, 8, 154, 526, 728–729 storage, 195–196, 650 theory, 131, 170, 179, 328, 347, 727 transmission, 404–405, 407, 409, 532, 752–754 triadic mental, 86, 682, 685
b2334-index
page 933
933
Information processing system (see also IPS) abstract, 566 autonomous, 567 real, 566 Information theory algorithmic, 131, 727, 754 dynamic, 728 economic, 728 general, 728 operator, 728 pragmatic, 728 qualitative, 728 quantum, 728 semantic, 728 Inference, 9 Infware abstract, 820 Input, 72 alphabet, 826 condition, 253 data, 251 kinesthetic, 547 port, 556 register, 821 relevant, 424 rule, 161 sensory, 648 symbol, 159 symbolic, 424 tactile, 547 test, 260 variable, 260 visual, 547 word, 160 Insight rational, 55 Instruction of a Turing machine, 565 Integration, 12, 155, 367, 497, 706, 712, 804 Intellectual activity bounded productive, 652 productive, 652 reproductive, 652
September 27, 2016
19:41
Theory of Knowledge: Structures and Processes - 9in x 6in
934
Subject Index
Intelligence artificial, 2–3, 79, 151, 264, 295, 395, 526, 651, 804, 818 collective, 85, 698 Interaction, 42 Interactive process, 201, 301 Turing machine, 157 Interface, 673 Internet, 690 Interpretation conceptual, 187 erotetic, 187 operational, 187 Intuition, 676 chronometric, 676 common sense, 676 comprehension, 676 creativity, 676 geometrical, 676 global, 676 interpretation, 676 mathematical, 670, 673–674 metaphoric, 676 perceptual, 676 practical, 676 primordial, 675–676 reasoning, 676 set-theoretic, 676 spatial, 676 synthetic, 676 temporal, 676 J Journal, 60, 90, 120, 154, 165 Justification, 23 by authority/opinion, 244 by practice/experience, 244 by reasoning/thinking, 244 coherence, 242 doxatic, 242 epistemological, 215 existential, 244 external, 242 faith-based, 245
b2334-index
foundational, 242 internal, 242 knowledge, 242 probabilistic, 242 procedural, 245 process, 243 rehabilistic, 243 Justified by observation, 23 by experience, 33 by probable arguments, 33 by weather forecast, 222 by the recent inspection, 266 K Key levels of knowledge, 41 Knowledge abstract, 28, 66 acquisition, 2, 9, 17, 605 active, 65 agentive, 61 amount of, 96 analytic, 34 analytical, 35 ancillary, 65 a posteriori, 55 application, 3 a priori, 55 articulated, 56–57 assertoric, 607 auditory, 68 axiological, 49 base, 56 biological, 68 breadth of, 138 carrier, 80 case-specific, 65 certainty of, 119 characteristics of, 36, 52, 96 clarity of, 122 classification of, 44, 64–65, 75 codified, 65 cognitive, 67–68 coherent, 64 collective, 322
page 934
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Theory of Knowledge: Structures and Processes - 9in x 6in
Subject Index completeness of, 139 complexity of, 53, 125, 129, 303–304 communal, 55 compiled, 64 concealed, 168 concept, 65 conceptual, 67 conscious, 168 conditional, 70 confidence in, 119 consistency of, 96, 107 correctness of, 97, 112, 117, 231, 242 creation, 2, 644–645, 671–673, 705–710, 716, 718, 762 cultural, 66 declarative, 48, 69 deep, 67 deep-level, 67 definitional, 65 depth of, 138 descriptive, 48, 211, 605 descriptive properties of, 91, 149 differentiation of, 50 diffused, 66 digital, 68 dissemination, 717–718 domain, 96 domain-specific, 64 domain-oriented, 64 economic, 69 educational, 69 efficiency of, 134 embedded, 53 embodied, 53 embrained, 53 economy, 7 efficiency of, 134, 147, 521, 713 elementary knowledge unit, 179, 189, 307, 792 empirical, 55, 744 encultured, 53, 65 engineering, 1 entailed, 65
b2334-index
page 935
935 erotetic, 607 esoteric, 61 evaluation, 215 exact, 61 exactness of, 102 existential, 49 existential characteristics of, 40, 77, 81 exoteric, 61 expectational, 65 explicit, 91, 705–706, 716, 718, 737 explicitly absent, 168 external, 59 external characteristics of, 149 externally explicit, 57–58 factual, 66 false, 118, 648 fictitious, 66 focal, 59 formal, 49 function, 2–3, 804 future, 66 fuzzy, 61 general, 65 generality of, 137 geological, 68 global theory of, 593 graph, 554 group, 58 habitual, 67 higher, 9 hypothetic, 607 implicit, 56–57, 59, 648, 716, 761 incommunicable, 57 incomplete, 265, 272, 758 indeterminate, 17, 61 inductive, 64 informal, 49 individual, 58, 69 instinctive, 67, 698 instrumental, 49 intellective, 61 intellectual, 69 internally explicit, 57
September 27, 2016
936
19:41
Theory of Knowledge: Structures and Processes - 9in x 6in
b2334-index
Subject Index intuitive, 31, 650, 670–671, 677, 686 justification, 215 learned, 67 link, 330 logico-mathematical, 60 lower, 9 management, x, 1 mathematical, 24, 124, 158, 256, 479, 593, 645, 673–674 meaning of, 140 measure, 111 memorized, 64 methodological, 65 modality of, 100 molecular, 68 moral, 64 network, 55, 709 node, 330 non-referential, 55 objectified, 58 of ideas, 670 olfactory, 68 ontogenetic, 67 operational, 49 organization, 92–93, 301, 711 organizational, 58 partial, 71, 139–140, 594, 687 passive, 65, 67 perceptual, 64 personal, 65 phylogenetic, 67 physical, 60, 68 political, 68 practical, 60, 69 pragmatic, 69–70 precision of, 139 printed, 68 probabilistic, 61 procedural, 49, 69 processing, 4, 147, 239, 538, 697, 736, 804 production, 1, 4, 7–8, 643–644 productive, 60 professional, 53
procedural, 67, 69 propositional, 48, 98 public, 58, 65 quantum, 68, 93–94 quantum theory of, 41, 309–310, 341, 362, 754 quantum unit of, 309 real, 66 referential, 55 regularity of, 405 relational, 70 relevance of, 104 reliability of, 32, 95, 136 religious, 69 representation, 5, 41, 80, 97, 131, 185, 263, 280, 341, 398–400, 449 representation of, 141, 177, 400, 526, 603 representational, 49, 605 role of, 3, 7, 40, 179 scientific, 46–47, 124, 158, 401, 600, 634, 638 sediment, 67 self-referential, 67 semiformal, 49 shared, 65 significance of, 131 situational, 67 sociological, 53 social, 60 society, 6 social, 60, 69–70, 131, 548, 644, 647 socio-cultural knowledge, 67 source of, 32, 42, 136, 176, 433, 760 space, 185 specific, 66 spiritual, 69 state, 189 state-referential, 55 stockpiled, 67 structure, 547 strategic, 67
page 936
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Theory of Knowledge: Structures and Processes - 9in x 6in
b2334-index
937
Subject Index studies, 1–2, 9, 39–40, 117, 151, 722, 804 structural, 49 structure of, 92 subsymbolic, 54 superficial, 67 subconscious, 59 supplementary, 65 surface, 67 symbol-type, 53 symbolic, 54, 68 synthetic, 35 system, 98 tacit, 55 technology, 1 testing, 215 theoretical, 60 theory, 142 theory of, 9 transgenerational, 67 true, 648 truthfulness, 112 unarticulated, 57 undiffused, 66 unit of, 98, 311 utilization, 1 validation, 253 verification, 253 visual, 68 wired, 54 written, 68 Knowledge system axiological, 606 comprehensive, 594, 601, 603, 611, 626 instrumental, 606, 622 logic-linguistic, 612 model-representation, 617 nuclear, 603 procedural, 67, 69, 142, 606 Knowledge systems hierarchies of, 636 operations with, 359 relations between, 72 Knowledge unit
page 937
descriptive, 311 operations with, 386 L Language abstract, 380 algorithmic, 425, 427, 496, 615 block-scheme, 539 context-free, 304 empty, 119 formal, 403 functional programming, 426 imperative programming, 426 input, 72 natural, 93, 145, 316, 343, 403–404, 427, 441, 466, 519, 605, 730 of mathematics, 403 of science, 411 object-oriented programming, 401, 426, 470 output, 705 procedural programming, 426 programming, 50 regular, 304 representation of a, 452 working, 129 Learning automated, 705 machine, 705 Length, 204 Level attributive, 604 componential, 604 of the world, 87 productive, 604 Limit, 605 Limit partial recursive function, 665 Limit recursive function, 665 Limited recursion, 665 Link, 329–340, 557 arrow semantic, 330 complete semantic, 330 inner semantic, 330
September 27, 2016
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Theory of Knowledge: Structures and Processes - 9in x 6in
938 knowledge, 330 semantic, 330 Linguistic representation, 148, 177, 401, 446 structure, 820 Logic algebraic, 441 Aristotelian, 441 autoepistemic, 442 Avicennian, 432 belief, 442 Buddhist, 430 business, 442 categorical, 442 classical, 441 combinatory, 442 complex, 442 computability, 442 conclusion, 442 conditional, 442 conservative, 442 constructive, 442 contemporary, 440 continuous, 442 cumulative, 442 default, 442 deontic, 442 dependence, 442 deviant, 442 dialectic, 440 discussive, 442 discursive, 442 dynamic, 442 epistemic, 442 equational, 442 erotetic, 442 European, 430, 432 fibring, 442 first-order, 442 formal, 440 free, 442 functor, 442 fuzzy, 442 higher-order, 442
b2334-index
Subject Index hybrid, 442 hypermodal, 443 illocutionary, 443 inclusive, 443 Indian, 430 inductive, 443 infinitary, 443 informal, 443 information-theoretic, 443 intensional, 443 intermediate, 443 interpretability, 443 intuitionistic, 443 Jain, 429 Judaic, 433 labeled, 443 linear, 443 local, 443 many-sorted, 443 many-valued, 443 market, 440 mathematical, 441, 463, 485, 493–494, 555 modal, 443 nominal, 443 non-symbolic, 443 of computation, 426 of decision, 443 of diagnosis, 440 of discovery, 443 of epistemology, 440 of formal inconsistency, 443 of names, 443 of physics, 440 of science, 440 operational, 443 paraconsistent, 443 polar, 443 polyadic, 443 polymodal, 443 Port-Royal, 358, 438 possibilistic, 443 predicate, 441 probabilistic, 443 process, 443
page 938
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Theory of Knowledge: Structures and Processes - 9in x 6in
Subject Index prohairetic, 443 propositional, 439, 441 provability, 443 quantum, 443 relevant logic, 443 resource, 443 scholastic, 440 second-order, 443 slash, 443 spatial, 443 stationary, 443 substructural, 443 superintuitionistic, 443 symbolic, 440 temporal l, 443 tense, 443 triadic, 443 type-free, 443 universal, 440 Logical calculus, 281 implication, 438 language, 97, 119, 141, 167, 179, 181, 189–190, 290, 400, 496, 659 positivism, 46, 218 prevariety, 281 quasivariety, 281 reasoning, 429, 487 semantics, 144, 181, 429, 441 theory, 280 variety,186, 191–192, 270–274, 281, 294, 305 Logic-based programming languages, 401, 426 M Machine finite state, 568 inductive Turing, 822 limit Turing, 825 Turing, 825 Macrolevel, 41 Macrosystem, 307 Maintenance, 6
b2334-index
page 939
939
Manifold, 278 Many-worlds theory, 291 Mathematics, 35, 62, 86, 130, 156 Mathematical, 41 Meaning, 145 contextual, 147 denotation, 147 estimate, 147 implicational, 147 relation, 147 sender, 145 sentence, 145 speaker, 145 textual, 147 Measure direct complexity, 131 dual complexity, 131 dynamic complexity, 131 functional complexity, 131 of information, 118, 802 integral complexity, 129 semantic, 118 static complexity, 131 Memory bubble, 200 collective, 85 computer, 90, 178, 191, 197, 199, 310, 400, 465 core, 199 declarative, 51 echoic, 195–196 eidetic, 52, 195 electro-acoustic, 200 episodic, 51 explicit, 51 flash, 199 haptic, 196 human, 195–197 iconic, 196 implicit, 51 internal, 197 long-term, 51, 195–196, 785, 789 main, 197–198 molecular, 200 n-inductive, 825
September 27, 2016
940
19:41
Theory of Knowledge: Structures and Processes - 9in x 6in
b2334-index
Subject Index
perceptual, 197 personal, 197 phase-change, 200 primary, 195 procedural, 51 secondary, 195 semantic, 51–52 semiconductor, 199 sensory, 195–196 short-term, 195–196, 785, 789 skill, 197 social, 85 static, 199 structured, 822 system, 197 thin-film, 199 twistor, 199 vacuum tube, 200 working, 542, 821–822 Megalevel, 41–42, 328, 593, 601, 603 Mental world, 84–87, 554, 792 Mentality, 84–85, 87, 89–91, 244, 477–479, 540, 545, 650 Metadata, 40, 151–155, 551, 553, 759 Meta-epistemology, 164 Meta-ethics, 164 Metaheuristic, 163 Metaknowledge, 40, 151–153, 158, 162, 165–167, 353, 636 Metalanguage, 167 Metalogic, 164 Metaphilosophy, 164 Metarule, 159 Methodology, 158 Metric, 833 Microlevel, 41, 307 Microsystem, 307 Minimization, 197 Misconception, 83 Modality anticipation, 76 bygone, 76 confidential, 76 current, 76 of knowledge, 100
temporal, 228 Mode accepting, 567 computing, 598 concurrent, 599 deciding, 587 of activity, 571 Model constructive, 205 many-world, 291 mathematical, 287 multidimensional structured, 265 structure-nominative, 600 Modeling, 6 Multigraph, 575 Multiplicity, 178 Multiset, 178 N Name, 345 Named data, 759 set, 487, 814 Natural number, 35 Network abstract neural,425 artificial neural, 558 architecture, 155 assertional, 520 definitional, 520 executable, 521 hybrid, 521 implicational, 520–521 learning, 521 linguistic, 521 neural, 694 semantic, 32, 520 statement, 521 theory, 520 unsupervised neural, 544 Neuron, 419 Node accepting, 568 generating, 568 transducing, 568
page 940
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Theory of Knowledge: Structures and Processes - 9in x 6in
Subject Index Norm, 834–835 Novelty, 96, 150 O ∆-operator, 741 Object abstract, 83, 312 constructive, 96 natural, 312 Object-oriented programming (see also OOP), 401, 426, 470, 756 Observers abstract, 83 external, 83 internal, 83 real, 83 Ontology, 42, 84 Operation arithmetical, 516 effective, 424 functional, 492 information, 786 integral, 813 order, 305 topological, 412, 415 Operational decomposition, 548 Operational device, 822 Operational programming language, 160 Operator, 49 addition bond, 770 addition content, 768 addition epistemic information, 788–790 addition weight, 770 additive epistemic information, 788 antitone epistemic information, 784 binary, 360 bond, 767 bond epistemic information, 767 bond thesaurus information, 768 bounded epistemic information, 777, 780
b2334-index
page 941
941 closed epistemic information, 783 content, 767 content epistemic information, 767 content thesaurus information, 768 continuous epistemic information, 777 contracting epistemic information, 784 copying epistemic information, 785–786 database, 484 deletion bond, 770 deletion content, 768 deletion weight, 770 emotional information, 171 epistemic information, 170, 193, 766 generation epistemic information, 770 information, 191–193 inner epistemic information, 767 instructional information, 171 knowledge information, 171 linear, 780 linear epistemic information, 775, 781–782 logical, 489 mixed epistemic information, 767 modal, 489 monotone epistemic information, 783 moving, 769, 785 moving epistemic information, 785, 788–789 information theory, 328 of adding weights, 773 of deleting weights, 773 of substituting weights, 773 permanent epistemic information, 772 projection, 487 (p, q)-continuous epistemic
September 27, 2016
942
19:41
Theory of Knowledge: Structures and Processes - 9in x 6in
b2334-index
Subject Index
information, 777 replication epistemic information, 768 schema, 550 selection, 487 semipermanent epistemic information, 772 stationary epistemic information, 772 stratified epistemic information, 787 strictly antitone epistemic information, 784, 787 strictly monotone epistemic information, 784 structured information, 766 substantiation bond, 770 substantiation content, 768 substitution bond, 770 substitution content, 768 substitution weight, 770 tense, 432 transformation content, 768 transformation weight, 770 transformation epistemic information, 770 uniform epistemic information, 777 uniformly bounded epistemic information, 780 value-changing weight, 771 weight, 767 weight epistemic information, 767 Operationalism, 46 Operationism, 46 Opinion, 244 Order, 165 Ordinal, 234, 813 Organization, 1, 7–8, 149, 656, 700, 719 Output alphabet, 826 condition, 253 data, 260
port, 557 variable, 260 P Part
algorithmic, 608, 625 aspect, 608, 618 combination, 608, 631 evaluation, 608, 631 fragment, 608, 628 linguistic, 608, 613–614 logical, 602, 608, 613 nominalistic, 608 nomological, 602, 608, 618 operating, 608, 625 operator, 608, 625 scaling, 608, 631 system, 608, 628 Partial order, 811 Partial projection, 208 Partial recursive function, 341 Perception, 9 Philosophy, 158 Physics, 361, 403, 416 Physical, 423 Potential infinity, 825 Potential process, 109 Port, 559 Power accepting, 670 computing, 304 decision, 309 set, 237, 809 Pragmatics, 145 Precision, 139 Predicate, 308 Prefix function, 801 Preprocessing, 692–693 Principle methodological, 791 of minimal action, 766 named data, 329, 758 ontological, 817 special transformation, 722–723, 746
page 942
September 27, 2016
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Theory of Knowledge: Structures and Processes - 9in x 6in
b2334-index
Subject Index Problem Decidable, 512 halting, 128, 565 undecidable, 301 Procedural programming, 426 Procedure, 461 Process, 11, 48 algorithmic, 125 business, 56 cognitive, 29 complex, 796 dynamic, 78 integration, 125 problem-solving, 67 thermonuclear, 68 Processor, 821 Product, 4, 363 Production, 427–428 Program, 126–127, 131 Property abstract, 41, 91 ascribed, 113, 311–312 contextual, 92 descriptive, 91 existential, 40 intellectual, 4 intrinsic, 113, 312 of knowledge, 74, 101, 133 natural, 311 relational, 92 Proposition, 176 Psychology social, 85 Q Quality primary, 312 secondary, 312 Quantifier existential, 486 universal, 486 Quantum, 316 Query, 303, 481 Question, 481, 600
R Range, 558 Real number, 132, 214, 315, 774, 832–833 Realizability, 265 Recursion, 172, 661 Recursive relation, 351 Reduction, 301, 367 Relation, 147 Relationship, 275 Relevance, 96 Reliability, 95 Representation of a language, 141 operational, 187 Representational type, 217 Resource, 4, 127 Restriction, 392 Result final, 530 of acceptation, 796 of computation, 825 Revision, 634 Robustness, 124 Rule construction, 160 data transformation, 159 deduction, 555 derivation, 465 execution, 159 formation, 487 logical, 493, 495 midot, 433 of a Turing machine, 158 of correspondence, 422, 599 of inference, 265, 284, 402 of interpretation, 429 of propagation, 465 production, 487 syntactic, 258, 427 S Scale, 170 Scenario, 624
page 943
943
September 27, 2016
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Theory of Knowledge: Structures and Processes - 9in x 6in
944 Schema, 539 basic, 559 closed, 567 conceptual, 553 content, 549 database, 553 dataflow, 551 deterministic, 567 external, 553 formal, 548 function, 549 ideological, 548 image, 546 interaction, 541 internal, 553 linguistic, 549 mental, 541, 547 motor, 541 network, 543 non-deterministic, 567 open, 567 perceptual, 541 port, 559 potentially open, 568 process, 549 role, 549 theory, 544 social, 548 specification, 552 star, 553 static, 549 theory, 555 XML, 551–552 Script, 527 Search, 163 Semantic link, 329 node, 329 network, 538–539, 725, 782 Semantic link network, 330 Semantics axiomatic, 145 cognitive, 144 conceptual, 144 denotational, 145
b2334-index
Subject Index formal, 144 frame, 144 lexical, 144 linguistic, 144 logical, 144 operational, 145 structural, 144 Semiotic triangle, 356 Sequential composition, 153, 512–513, 623, 769, 786 Set acceptable, 72 computable, 364 empty, 187 enumerable, 511 fuzzy, 103 named, 330, 814 Sign, 345 conceptual, 343 material, 343 Size, 542–543 Social context, 3, 463, 701 influence, 701 intelligence, 85, 644, 647 mentality, 85 memory, 85 psychology, 85 recognition, 4 Society, 80 Sociology, 347, 415, 556 Software, 97 Software metric, 220 Solution, 709 Source, 154 Space abstract epistemic, 178–182 conceptual, 204 conceptual epistemic, 206 epistemic, 178–182 Euclidean, 676, 834 knowledge, 187 linear, 827 metric, 833–835
page 944
September 27, 2016
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Theory of Knowledge: Structures and Processes - 9in x 6in
b2334-index
Subject Index organization, 566 structure, 566 symbolic epistemic, 178–182 topological, 189, 833 topological vector, 189 vector, 208 weighted epistemic, 206 weighted propositional epistemic, 207 Spatial, 676 Spatiality, 737 Specification, 26, 258, 274, 457 Square Structure-Information-MatterEnergy (see also SIME), 727, 746, 749 Specialized concurrent processing, 426 State accepting, 794 configuration, 346 final, 820 initial, 174 inner, 247 space, 192, 201 start, 159, 380, 820 structure, 820 Statement, 19, 24 Stoic philosophy, 686 Stratification, 84, 284 Stratum assertoric, 609 erothetic, 609 heuristic, 609 hypothetic, 609 String, 819 Structuralism, 415 Structuration, 84, 576, 593 Structure, action, 820 algebraic, 412, 415 cognitive, 725 concept, 614 dynamic, 605 dynamic space, 566 epistemic, 170
external, 415, 569 global space, 566 inner, 415 intermediate, 415 internal, 415, 569 knowledge, 603 mathematical, 157 of a system, 368 order, 412 outer, 415 pure epistemic, 725 region space, 542 space, 566 state, 820 static, 605 static space, 606 static synthetic, 804 static systemic, 604 topological, 412 weighted epistemic, 725 Structured memory, 822 memory of order n, 305 programming, 305 proposition, 483 Subconcept, 185 Subset, 362 Substance, 476 Subsystem axiological, 606 of bonds, 610 instrumental, 606 logic-linguistic, 606 model-representing, 606 of ties, 639 operational, 639 pragmatic-procedural, 602 problem-heuristical, 602 procedural, 602 structural, 603 Syllogism, 16, 25 Symbol, 355 binary, 360 empty, 360 of the alphabet, 380
page 945
945
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Theory of Knowledge: Structures and Processes - 9in x 6in
946 Symbolic, 441 Synchronization, 421 System abstract, 83 AI, 42 algebraic, 807 artificial, 41 assertoric knowledge, 603 automated, 695 axiom, 497 brain, 684 cerebral, 677 cognitive, 177–178 cognizing, 648 complex, 123 comprehensive knowledge, 77 computer, 395, 468 computational, 46 conceptual, 170 decision support, 712 descriptive knowledge, 805 epistemic, 178 expert, 237, 265 formal, 164, 404 functioning, 49 geographic information, 712 global knowledge, 594 heuristic knowledge, 603–604 holistic, 13 inference, 277 infological, 684 information processing, 566 informational, 56 intelligent, 175 knowledge, 178 knowledge-based, 712 limbic, 677 logical, 745 mathematical, 754 nervous, 85 neuropsychological, 683 nuclear knowledge, 603 numerical, 614 of Affective States (see also SAS), 649, 681–686
b2334-index
Subject Index of of of of of of of of of of
Emotions, 649 conditions, 667 equalities, 328 instructions, 320 named sets, 323 names, 323 operations, 329 philosophy, 12 properties, 324 Rational Intelligence (see also SRI), 649, 681–686 of Reasoning, 244, 649, 681 of symbols, 676 of Will and Instinct (see also SWI), 681–686, 770 operational knowledge, 603 partial knowledge, 594 physical, 656, 753 processing, 566 religious, 11 representation, 806 representational knowledge, Samkhya, 589 search, 279 sign, 298 software, 264 Solar, 71 static, 168 storage, 148 structuration, 658 structured, 159 symbolic, 25, 369 technical, 354 theory, 258 theoretical, 369 threefold, 458 transition, 125, 326 Vedanta, 11–12 Systemic context, 125 T Table, 528 Table form, 725 Tape, 159
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Theory of Knowledge: Structures and Processes - 9in x 6in
Subject Index Task, 45, 259 Technology computer, 698 information, 687 software, 254 theory of, 248 Temporal, 51, 76 Temporality, 737 Testing action, 253 ad hoc, 255 alpha, 255 auditing, 252 beta, 255 black-box, 254 decision, 253 diagnostic, 253 exploitation, 251–252 exploratory, 255 functional, 252–253 load, 254 logical, 252 performance, 251–252 processual, 252–253 recovery, 255 simulation, 251 smoke, 255 static, 252 stress, 254 structural, 252 verification, 253 volume, 255 white-box, 254 Theorem, 62 Theory of average knowledge, 346 fuzzy set, 103, 558 global, 647 many-worlds, 489 mathematical, 247 mathematical schema, 555 of information, 365 of knowledge, 3 physical, 270 quantum, 208
b2334-index
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scientific, 325 Theory-element, 368 Time physical, 676 system, 127 Tool, 487 Topology, 184, 567, 833 discrete, 825 Total function, 478 Transmission, 6 Triad, 208, 281, 329 attributive, 219 Bacon/Augustine Sign, 345 balanced sign, 350 communication, 745 Data-Knowledge, 760 Data-Information-Knowledge, 42 dyadic sign, 347 dynamic sign, 357 epistemic, 172 epistemological, 142 evaluation, 220–221 existential, 84–89, 554 functional, 261 fundamental, 341, 448, 487, 496, 814 general Popper, 89 information, 407 language, 405 Locke triad of the world, 89 Locke triad of science, 89 process, 262 reflection epistemic, 173 sign, 345 substantiation epistemic, 173 symbolic, 330 triangular, 587 Triadic approach, 261 concept, 357 context, 456 dynamic model, 417 logic, 444 mental information, 257 model, 457
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Theory of Knowledge: Structures and Processes - 9in x 6in
948 relation, 357 representation, 367 sign model, 348 stratification, 387 structure, 405, 467 typology, 247 Triune Brain, 677 Componential Triune Brain, 677 True, 52 absolutely, 52 Turing machine, 157 Typology acquisition, 67 dynamic, 67 triadic, 67 U Unconscious collective, 89 Understanding, 57 Universal, 324 University, 696 Utilization, 598 V Validation, 169 Value, 328 Variable, 558 Vector bundle, 211, 780 Verification, 369 W Weight, 208 Wisdom, 598
b2334-index
Subject Index Word empty, 367 infinite, 498 finite, 657 length of a, 217 Working tape, 647 World actual, 398 business, 720 conceptual, 658 existing, 347 extended mental, 87 external, 89 information, 739 material, 84–88 mental, 84–88 mind-independent, 96 mystical, 147 natural, 348 objective, 647 outside, 702 physical, 84–88 of ideas, 324 of forms, 301 of structures, 84–88 perfect, 654 possible, 478 real, 415 structural, 84–88 view, 650
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