Handbook of mereology 9783884050903, 3884050907

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Handbook of Mereology

Analytica Investigations in Logic, Ontology and the Philosophy of Language Editors: Hans Burkhardt (Founding Editor †) Ignacio Angelelli, Joseph M. Bocheński (†) Christian Thiel Managing Editor: Stamatios Gerogiorgakis

_____ Handbook of Mereology Editorial Board Peter Forrest (Armidale, AU) Thomas Mormann (San Sebastian, E) Peter M. Simons (Dublin, IRL) Barry Smith (Buffalo, NY, US) Achille Varzi (Columbia, NY, US) Roberto Casati (Paris, F) Herbert Hochberg (Austin, Texas, US) Kevin Mulligan (Geneva, CH) Burkhard Schäfer (Edinburgh, UK)

Handbook of Mereology Editors Hans Burkhardt (Founding Editor †) Johanna Seibt Guido Imaguire Stamatios Gerogiorgakis

Philosophia

ISBN 978-3-88405-090-3 © 2017 by Philosophia Verlag GmbH München Printed in Germany 2017

Preface

The present volume is the first comprehensive reference work for research on partwhole relations – or better, and quite appropriately, a substantive part thereof. According to our guiding conception, developed by Burkhardt and Seibt more than a decade ago, the Handbook of Mereology was to offer a wide scope, inclusive presentation of contemporary research on part-whole relations that would draw out systematic, historical, and interdisciplinary trajectories, show the subject’s fertility, and inspire future explorations. In particular, we wanted to impress that mereology is much more than the study of axiomatised systems. The relationship between part and whole is a basic schema of cognitive organisation that operates not only at the level of language and propositional thought, but also at the level of sensory input processing, especially visual and auditory. In the natural, social, and human sciences, as well as in the Humanities, part-whole relations organize all three: data domains, methods, and theories. In short, part-whole relations play a fundamental role in how we perceive and interact with nature, how we speak and think about the world and ourselves, as societies and as individuals. For this reason the study of part-whole relations, both within and across domains, begins long before the metamathematically motivated inquiries of logicians at the beginning of the 20th century, and goes far beyond it. That this first edition of the Handbook of Mereology had to remain a part of the envisaged whole is to some extent a reflection of the current research landscape in mereology, the peculiarities of which led to an unusually protracted production process. The international research community in mereology is still so small that contributors and reviewers are not easily come by and often the only option was to wait rather than to replace. Moreover, since European scholars in the history of philosophy operate predominantly within their native language and abide by different national stylistic and expository conventions, extended review and revision periods were necessary, and translations and major editing tasks presented timeconsuming obstacles. But the delay of the production process also had quite contingent reasons. Due to illness, Hans Burkhardt soon had to decrease his involvement, and the second editor, with increasing support by the third editor, had to take over most of the handbook’s scientific and practical organisation. From 2012 onwards the second and third editor, who originally had committed themselves to short-term assignments, struggled with the unexpected task of having to accommo-

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PREFACE

date over 3000 additional work hours within already busy work schedules. In 2013 Stamatios Gerogiorgakis joined the editorial team, lending additional research competences and linguistic editing support. When Hans Burkhardt died in May 2015, the three remaining editors decided to consolidate the results achieved so far and to present, in honor of his name and his sustained efforts for research on the history of mereology, a first edition of the Handbook that can qualify as a sufficiently substantive partial realisation of the project idea. We trust that the unique scope of the 131 contributions collected here will bring into view the fundamental significance of mereology in the wide sense of the term – not only for logic or ontology, but also for philosophy of biology, chemistry, and quantum physics; for a philosophy of nature that can accommodate chaos and emergence; for philosophy of art, for ethics; and for philosophy of cognition. The included domain-specific investigations of part-whole relations might even provide useful heuristics for researchers in various scientific disciplines (e.g., cognitive science, psychology, biology, or chemistry). There are of course a good many entries explaining and exploring the meta-mathematical interest of formal mereology and its use for the development of nominalist descriptions of the domain of mathematical science – the reader will find historically and systematically informative presentations of the relationship between mereology, Boolean algebra, and topology, as well as explorations in the intersection of metamathematics and metaphysics, such as on how to ground differential equations in a region-based ontology. But the Handbook also features several contributions that convey the richness and complexity of linguistic encodings of part-whole relations. Importantly, many entries suggest that and how our common sense reasoning departs from the classical formalisations of part-whole relationships that have influenced much of contemporary ontology. A considerable portion of the historical dimension of mereology could be documented, especially with respect to the medieval period and the phenomenological tradition. The Handbook’s entries are written by internationally renowned specialists but with an interdisciplinary readership in view. Most of the entries not only report the state of the art but present new original research published here for the first time. The protracted production process of this first edition has been unusually trying for all contributors, some of whom submitted their entries many years ago. We would like to thank all authors for their contributions, but we also want to express our particular gratitude to our authors for their sustained commitment and extraordinary patience. We gratefully acknowledge the support of many colleagues who served as external reviewers. Our special thanks go to Andrew Arlig, who took over several translations and language revisions. We also thank Ulrich Staudinger, the owner of the Philosophia Verlag, for staying with the project throughout the contingent difficulties we encountered. Perhaps it is quite fitting that this first edition of the Handbook of Mereology remains a part of the envisaged whole – it may serve as a useful reminder that some

PREFACE

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ambitions can only be discharged piecemeal, with each new attempt garnering energies from the shortcomings of predecessors. We hope that this portion of a comprehensive reference work for research on part-whole relations will stimulate the preparation of a second edition to fill in what had to be omitted here. In other words, we hope that the mereological supplementation principle not only holds in theory but also in praxis. Johanna Seibt, Guido Imaguire, Stamatios Gerogiorgakis

ENTRIES

9

Table of Entries

Abelard

21

Abstract

25

Act and Action

28

Activity

31

Albert of Saxony

39

Animal

41

Anselm

45

Aristotle’s Theory of Parts

49

Aristotle’s Theory of Wholes

56

Art

62

Artifact

68

Atomism in Ancient Greek Philosophy

70

Atomism, Logical

73

Atomism, Medieval

81

Atomism, Metaphysical Axiomatic Method

Comments on “The Calculus of Individuals and Its Uses”

127

Carnap, Rudolf

128

Category

131

Causation

133

Chaos

137

Chemistry

141

Coincidence

147

Collectives and Compounds

150

Common Sense Reasoning About Parts and Wholes 152 Conscious Experience

160

Continuants and Occurrents

168

Cosmology

171

85

Deontic modalities

179

89

Descartes

182

Dispositions

190

Dynamical Systems

193

Bergmann, Gustav

93

Biological Parts

96

Body

99

Elements

197

Boethius

103

Emergence

200

Bolzano, Bernard

106

Ethics

206

Boolean Algebras

114

Experience

209

Brentano, Franz

119

10

ENTRIES

Linguistic Structures

293

Locke

297

224

Mally, Ernst

307

227

Material Constitution

311

Medicine

315

Facts

213

Fiction

218

Fractals

223

Frege, Gottlob Fusion

Gestalt

231

God

235

Good Life, The

237

Goodman, Nelson

240

Grammar

242

Granularity

245

Grossmann, Reinhardt S.

249

Holes

253

Homeomerous and Automerous 255 Husserl, Edmund

262

Medieval Discussions of Temporal Parts and Wholes 331 Medieval Mereology

338

Meinong, Alexius

345

Mereological Essentialism

349

Mereological Triangle

352

Mereotopology

354

Metamathematics of Mereology 361

Naïve Mereology

369

Natural Science

371

Nominalism

379

Ingarden, Roman

271

Non-literal Language Use and PartWhole Relations 382

Intentionality

274

Non-Wellfounded Mereology

383

Jurisprudence

277

Ontological Dependence

389

Order

393

Paradoxes

397

Parmenides

399

Language

287

Leśniewski, Stanisław and Polish Mereology 287

ENTRIES

Perceptual Whole

401

Persistence

409

Philosophy of Mathematics

412

Phonology

424

Piece

428

Plato

432

Points

435

Possession and Partitives

438

Powers

442

Praedicabilia

444

Privation

447

Propositions

455

Quantum Mechanics

461

Quantum Mereology

476

11

Society, Individualism and Holism (Collectivism) in the Study of, 508 Stoics

510

Structure

512

Structure of Appearance, Goodman’s 519 Stumpf, Carl

522

Subject, Person, Self

525

Substance

529

Substrate

534

Sum

538

Syllogism

540

Syntax

544

Tarski, Alfred

549

Temporal Parts

551

Theoretical Mereology

554

Radulphus Brito

481

Thomas Aquinas

562

Raimundus Lullus and Lullism

482

Topology

565

Reduplication

487

Totum Potentiale

568

Reinach, Adolf

489

Transitivity

570

Rhetoric

492

Tropes

579

Russell, Bertrand

496

Twardowski, Kazimierz

585

Scherzer, Johann Adam

499

Universal

589

Segelberg, Ivar

502

Shadows

505

Whitehead, Alfred North

593

12

ENTRIES

Whitehead’s Metaphysics

598

World, Actual

602

Index

607

CONTRIBUTORS

13

Contributors Albertazzi, Liliana Department of Humanities, University of Trento Perceptual Whole Andrae, Benjamin Munich School of Philosophy, Ludwig-Maximilians University Experience, Whitehead’s Metaphysics Arlig, Andrew Department of Philosophy, Brooklyn College, CUNY Abelard, Boethius, Medieval Mereology Bäck, Allan Department of Philosophy, Kutztown University Aristotle’s Theory of Parts, Reduplication, Syllogism Beck, Hans Institute for Physics, University of Neuchâtel Chaos Boucher, Pol Institut de l’Ouest: Droit et Europe, University of Rennes Jurisprudence Bränmark, Johan Department of Global Political Studies, Malmö University Ethics, The Good Life Bromand, Joachim Institute of Philosophy, University of Bonn/Institute of Philosophy, RWTH Aachen University Paradoxes Burkhardt, Hans Department of Philosophy, Ludwig-Maximilians University at Munich Albert of Saxony, Medicine, Mereological Triangle (with Stamatios Gerogiorgakis), Radulphus Brito, Raimundus Lullus and Lullism, Scherzer (with Stamatios Gerogiorgakis) Cameron, Ross P. Department of Philosophy, University of Virginia Mereological Essentialism

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CONTRIBUTORS

Cohnitz, Daniel Department of Philosophy and Religious Studies, Utrecht University Goodman, Nelson (with Marcus Rossberg), Structure of Appearance, Goodman’s (with Marcus Rossberg) Correia, Fabrice Department of Philosophy, University of Geneva Ontological Dependence Cosans, Christopher Department of Philosophy, University of Marylands Animals Cotnoir, A. J. Department of Philosophy, University of St Andrews Non-Wellfounded Mereology Donelly, Maureen Department of Philosophy, University at Buffalo Granularity Eberle, Rolf Faculty of Philosophy, University of Rochester Order Enfield, Nicholas J. Max Planck Institute for Psycholinguistics, Nijmegen Body (with Asifa Majid) Forrest, Peter School of Humanities, University of New England in Armidale, NSW, Australia Theoretical Mereology Gerogiorgakis, Stamatios Philosophy Department, University of Erfurt Medieval Discussions of Temporal Parts and Wholes, Mereological Triangle (with Hans Burkhardt), Privation, Scherzer (with Hans Burkhardt), Individualism and Holism (Collectivism) in the Study of Society, Thomas Aquinas Hammond, Michael Department of Linguistics, University of Arizona Phonology

CONTRIBUTORS

Harte, Verity Department of Philosophy, Yale University Plato Hawley, Katherine Department of Philosophy, University of St Andrews Fusion, Temporal Parts Hellman, Geoffrey Department of Philosophy, University of Minnesota Philosophy of Mathematics Herre, Heinrich Institute for Informatics, University of Leipzig Boolean Algebras Herstein, Gary Independent Scholar Cosmology Hochberg, Herbert Department of Philosophy, University of Texas at Austin Logical Atomism, Facts, Segelberg, Tropes Huebner, Johannes Department of Philosophy, Martin-Luther University at Halle-Wittenberg Act, Action Imaguire, Guido Department of Philosophy, Federal University of Rio de Janeiro Frege, Nominalism, Propositions, Russell Jacquette, Dale Philosophy Institute, University of Bern Fractals, Tarski Jansen, Ludger Institute of Philosophy, University of Rostock Collectives and Compounds, Dispositions, Substance, Substrate Johansson, Ingvar Department of Historical, Philosophical and Religious Studies, Umeå University Natural Science

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16

CONTRIBUTORS

Kaiser, Marie I. Philosophy Department, University of Bielefeld Biological Parts Kanzian, Christian Department of Philosophy, Theological Faculty of the University Innsbruck Persistence Koons, Robert Department of Philosophy, University of Texas at Austin Causation Koptjevskaja-Tamm, Maria Department of Linguistics, Stockholm University Possession and Partitives Koslicki, Kathrin Department of Philosophy, University of Alberta Structure Krause, Décio Department of Philosophy, Federal University of Santa Catarina Quantum Mereology Krecz, Charles A. Department of Philosophy, University of Texas at Austin Piece Leonard, Henry S., Sr. Philosophy Department, Michigan State University Comments on “Calculus of Individuals and Its Uses” (prefaced by Henry S. Leonard, Jr.) Lowe, Jonathan E. Department of Philosophy, Durham University Coincidence Majid, Asifa Max Planck Institute for Psycholinguistics, Nijmegen Body (with Nicholas Enfield) McGivern, Patrick Department of Philosophy, University of Wollongong Dynamical Systems (with Alex Rueger)

CONTRIBUTORS

17

McGregor, William B. Linguistics, Cognitive Science and Semiotics, School of Communication and Culture, Aarhus University Grammar Meixner, Uwe Institute of Philosophy, University of Augsburg Abstract, Axiomatic Method, Universal Michael, Emily Department of Philosophy, City University of New York Medieval Atomism Miller, Kristie Department of Philosophy, University of Sydney Holes, Actual World Mittmann, Rainer Independent Scholar Descartes Moravcsik, Edith Department of Linguistics, University of Wisconsin-Milwaukee Syntax Mormann, Thomas Department of Logic and Philosophy of Science, University of the Basque Country Carnap, Points, Topology Müller, Thomas Department of Philosophy, University of Konstanz Deontic modalities Mumford, Stephen Department of Philosophy, Durham University/School of Economics and Business, Norwegian University of Life Sciences Powers Needham, Paul Department of Philosophy, Stockholm University Chemistry, Elements

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CONTRIBUTORS

Oderberg, David S. Department of Philosophy, University of Reading Shadows Pietruszczak, Andrzej Department of Logic, Nicolaus Copernicus University Metamathematics of Mereology Poli, Roberto Department of Philosophy, University of Trento Aristotle’s Theory of Wholes Potter, Luke Department of Philosophy, University of Notre Dame Material constitution (with Michael Rea) Rea, Michael Department of Philosophy, University of Notre Dame Material constitution (with Luke Potter) Reicher, Maria E. Department of Philosophy, RWTH Aachen University Meinong Rijkhoff, Jan Department of Linguistics, Aarhus University Linguistic Structures, Non-literal Language Use and Part-Whole Relations Rosiak, Marek Department of Logic, University of Lodz Husserl, Ingarden, Twardowski Rossberg, Marcus Philosophy Department, University of Connecticut Goodman, Nelson (with Daniel Cohnitz), Structure of Appearance, Goodman’s (with Daniel Cohnitz) Rueger, Alex Faculty of Arts, University of Alberta Dynamical Systems (with Patrick McGivern) Salice, Alessandro Philosophy Department, University College Cork Brentano, Intentionality, Mally, Reinach, Stumpf

CONTRIBUTORS

19

Sanford, David H. Department of Philosophy, Duke University Naïve Mereology, Sum Schaffer, Jonathan Department of Philosophy, Rutgers University Metaphysical Atomism Schalley, Andrea C. Department of Language, Literature and Intercultural Studies, Karlstad University Common Sense Reasoning About Parts and Wholes Schantz, Richard Department of Philosophy, University of Siegen Gestalt Scherb, Jürgen L. Department of Philosophy, Ludwig-Maximilians University at Munich Anselm Schnieder, Benjamin Department of Philosophy, University of Hamburg Bolzano Seibt, Johanna Department for Philosophy and the History of Ideas, University Aarhus Activity, Homeomerous and Automerous, Subject (with Bartłomiej Skowron), Transitivity Sedley, David Christ’s College, University of Cambridge Atomism in Ancient Greek Philosophy Silberstein, Michael Department of Philosophy, Elizabethtown College Conscious Experience, Quantum Mechanics Simons, Peter Department of Philosophy, Trinity College Dublin Artifact, Category, Whitehead

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CONTRIBUTORS

Skowron, Bartłomiej International Center for Formal Ontology, Faculty of Administration and Social Sciences, Warsaw University of Technology Mereotopology, Subject (with Johanna Seibt) Stephan, Achim Institute of Cognitive Science, Osnabrück University Emergence Storm-Henningsen, Peter Lillebælt Academy, University of Applied Sciences Art Tegtmeier, Erwin Department of Philosophy, University of Mannheim Bergmann, Continuants and Occurrents, Grossman Thom, Paul Department of Philosophy, University of Sydney Parmenides, Stoics van Zantwijk, Temilo Institut of Philosophy, Friedrich-Schiller-University Jena Rhetoric von Wachter, Daniel International Academy of Philosophy in the Principality of Liechtentein God Woleński, Jan Department of Social Sciences, University of Information, Technology and Management at Rzeszow Leśniewski and Polish mereology Woods, John Department of Philosophy, University of British Columbia Fiction Wyllie, Guilherme Department of Philosophy, Fluminense Federal University Praedicabilia, Totum potentiale

ABELARD

A Abelard There are several sophisticated twelfth-century inquiries into the nature of parts and wholes (Henry 1991, ch. 2; Arlig 2013; King 2015). Peter Abelard (1079-1142 CE) is one of the foremost contributors to this high point in the history of medieval mereology. Abelard's mereological ideas are developed from within a classificatory system that he adapts from Boethius’s On Division (see the entry on Boethius). Abelard first observes that, when we use the term ‘whole’, sometimes we are referring to substance, sometimes to form, and sometimes to form and substance taken together (1970: 546). ‘Whole’ can be said with respect to substance in two ways: first, it can make reference to a universal whole; second, there is the integral whole, which is something in so far as it has or is delimited by a quantity. ‘Whole’ said with respect to form makes reference to the soul, which in some sense consists of its powers. ‘Whole’ said with respect to substance and form taken together refers to a combination of something that plays a material role and something that plays a formal role. This can be either a species in so far as it is analysable into a genus and a differentiae, or a thing that is a combi-

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nation of a ‘substance’, which plays the material role, and a form. (Abelard’s account of material substances and their hylomorphic composition has some idiosyncratic features. For a helpful overview, see Marenbon 2013, pp.168-74). Abelard is famous for his defense of the claim that universal wholes are not things, but rather only words (1919: 10-16; 1933: 515-522). Abelard allows logicians to say that universals are a kind of whole, but this effectively means that these words have the logical properties of classes whose elements (viz. the ‘parts’) all have the same nature. Abelard's discussions of the soul and substanceform composition, while interesting in a number of respects, do not always easily compare with contemporary mereological inquiries. Those aspects of Abelard’s investigations of parts and wholes that most readily map onto later mereological discussions appear in his treatment of integral wholes. Hence integral wholes will be the focus of the remainder of this entry. Abelard asserts that there are many kinds of integral whole. The first broad division is between discrete and continuous integral wholes. Abelard’s understanding of this division diverges from the traditional Aristotelian formulation, since he holds that a continuous whole is merely one whose parts are placed together without any gaps between them (1970: 73). Nevertheless, like many Aristotelians, Abelard holds that only natural composite entities (e.g. gold, cacti, cats, and humans) are continuous

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ABELARD

wholes. All other items, including most notably all artifacts, are discrete wholes, for even the parts of a human-made object, no matter how well made, always have gaps between them. Abelard insists that only God can create continuous wholes. Thus, in a case like that of a glass sphere, Abelard insists that God is the agent who creates the glass from the ingredients placed in the furnace. Humans merely provide the conditions for the generation of the glass and impose a spherical arrangement upon God’s creation (1970: 419-20). Discrete wholes come in three varieties: some are structured wholes, others are aggregations, and still others are mere pluralities (1921: 170-1). A house is an example of a structured discrete whole since its parts have not only been aggregated but also given an arrangement. Crowds and flocks are examples of aggregates. Most of Abelard’s contemporaries acknowledge the existence of aggregates and structured wholes. However, Abelard seems to be one of the few medieval thinkers who accepts the existence of wholes whose parts are arbitrarily taken together. In the extreme case, these ‘pluralities’ can consist of parts that belong to different categories. For example, the collection of this whiteness (a quality) and this finger (a substance) counts as a whole, albeit a whole that has a very weak degree of unity (1970: 548). Abelard however does not believe that composition is utterly unrestricted, since the parts, no matter how loosely connected, must all exist at the same time. Indeed, Abelard argues that items like hours, days,

and years are not strictly speaking wholes, although it is sometimes useful to treat them as if they are (1921: 186-7; 1970: 554). Abelard also makes a number of interesting remarks about what it is to be an integral part. Two will be noted. First, Abelard insists that there can be no whole that consists of one part, since if that were true, the quantity of the whole would fail to exceed that of the part (1970: 554). On this point, Abelard is in agreement with virtually all medieval thinkers. Second, Abelard distinguishes between the ingredients (literally, ‘those things out of which the whole is’) of a whole and the parts of a whole. All parts are ingredients, but not all ingredients are parts, since in order for x to be a part of y, x must continue to persist in y after the ingredients have been combined or collected together (1970: 575). Therefore, the flour, water and the eggs are ingredients of the bread, but they are not its parts. Abelard seems to be one of the first medieval thinkers to appreciate the phenomenon of mereological overlap. Abelard makes use of this notion to answer several sophisms employed by an unnamed opponent which purport to demonstrate that ‘whole’ is merely a word (1958: 119-120). Abelard’s appreciation of the notion of overlap furthermore allows him to develop an original and noteworthy account of numerical identity (1969: 247-55). According to Abelard, A is the same in essentia as B only if A and B share all and only the same parts. He also asserts that if A is the same in essentia as B, then A and B

ABELARD

are numerically the same. But even if A differs in essentia from B, it does not follow that A is numerically different from B since non-coincident, overlapping objects (such as a house and its wall) are not numerically different from one another. Abelard’s theory allows him to assert that Socrates’s body and Socrates’s body less one hand were not numerically distinct bodies. This would give him the conceptual resources to resist a version of Geach’s puzzle about many nearly coincident cats (Geach 1969: 252; Normore 2006: 749). Perhaps Abelard’s most controversial mereological thesis is that “no thing possesses more parts at one time than at another” (1921: 300; 1970: 423). This thesis leads him to assert explicitly that discrete integral wholes depend upon their parts for their persistence. For instance, if this house loses a brick, it ceases to exist – even though another house will exist (in most cases) after the removal of the brick (1970: 551). Unfortunately, despite the seemingly general character of his proposition, it is unclear whether Abelard really intends to extend this thesis about persistence to all things. He does not explicitly discuss the persistence conditions of plants and brutes, and his remarks about personal identity are terse and hard to interpret. He concedes that the act of clipping Socrates’s fingernail would not amount to a case of homicide (1970: 552). But this seems to be due to a technicality: homicide only occurs when the human soul has been dislodged from a body. Even though he perhaps has the conceptual resources at hand (recall his distinc-

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tion between integral wholes and substance-form wholes), and unlike later medieval thinkers who seem to share many of his convictions, such as John Buridan and Albert of Saxony (Normore 2006; Pasnau 2011, ch. 29), Abelard does not appear to be interested in developing a complete and general account of persistence through mereological changes. However, some of Abelard’s followers, namely the so-called Nominales, may have done so. The evidence is fragmentary, but the Nominales appear to hold that no thing can survive a change of parts, including the thing that is Socrates, even though they concede that the person who is Socrates can survive a change in parts (Martin 1998: 6-12). See also > Aristotle, Boethius, Material Constitution, Mereological Essentialism, Medieval Mereology, Persistence, Universals. Bibliographical remarks

Brower, J., 2004. Includes a helpful discussion of Abelard’s theory of identity and its application to the problem of material constitution; supplement with Arlig 2012, Marenbon 2007, and Marenbon 2013 (pp. 195-198). Henry, D. P., 1990. Consists of a translation and close study of Abelard 1958; the discussion in Henry 1991, section 2.7, is substantially the same but somewhat less accessible. Henry, D. P., 1991. Sections 2.1-2.7 (pp. 64-180) contain an extensive and

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ABELARD

very useful study of Abelard’s mereological writings. King, P., 2004. Contains a helpful overview of some of Abelard’s metaphysical theses concerning mereology and identity; it should be read in conjunction with Marenbon 2013 (ch. 6) which criticises King and provides an alternative interpretation of some of Abelard's general ontological commitments. Martin, C. J., 1998. Includes a very important discussion of Abelard’s views on persistence through mereological change as well as an inquiry into the striking positions of the Nominales; it should be studied in conjunction with Arlig 2007, Arlig 2013, and Henry 1991. References and further readings

Abelard, P., 1919, Logica ‘ingredientibus’, pt. 1: Glossae super Porphyrium, in Peter Abealards Philosophische Schriften. I. Die Logica ‘Ingredientibus. 1: die Glossen zu Porphyrius, Bernhard Geyer, ed., Beiträge zur Geschichte der Philosophie des Mittelalters, no. 21, pt. 1. Münster: Aschendorffshen Buchhandlung. Abelard, P., 1921, Logica ‘ingredientibus’, pt. 2: Glossae super Praedicamenta Aristotelis, in Peter Abealards Philosophische Schriften. I. Die Logica ‘Ingredientibus’. 2: die Glossen zu den Kategorien, Bernhard Geyer, ed., Beiträge zur Geschichte der Philosophie des Mittelalters, no. 21, pt. 2. Münster: Aschendorffshen Buchhandlung.

Abelard, P., 1933, Logica ‘nostrorum petitioni sociorum’: glossula super Porphyrium, in Peter Abealards Philosophische Schriften. II. Die Logica ‘nostrorum petitioni sociorum: die Glossen zu Porphyrius, Bernhard Geyer, ed., Beiträge zur Geschichte der Philosophie und Theologie des Mittelalters, no. 21, no. 4. Münster: Aschendorffshen Buchhandlung. Abelard, P., 1958, Secundum Magistrum Petrum sententie, in L. MinioPaluello, ed., Twelfth Century Logic, Texts and Studies, II: Abaelardiana inedita, Rome: Edizioni di Storia e Letteratura, 109-21. Abelard, P., 1969, Theologia Christiana, Buytaert E. M. (ed.), Corpus Christianorum Continuatio Mediaevalis, no. 12, Turnhout: Brepols. Abelard, P., 1970, Dialectica, 2nd ed., L. M. de Rijk, ed., Assen: Van Gorcum. Arlig, A., 2007, “Abelard’s Assault on Everyday Objects”, American Catholic Philosophical Quarterly 81: 209-227. Arlig, A., 2012, “Peter Abelard on Material Constitution”, Archiv für Geschichte der Philosophie 94: 119146. Arlig, A., 2013, “Some TwelfthCentury Reflections on Mereological Essentialism”, Oxford Studies in Medieval Philosophy 1: 83-112. Brower, J., 2004, “Trinity”, in Brower, J.; Guilfoy, K. (eds.), The Cambridge Companion to Abelard, C

ABSTRACT

ambridge: Cambridge Press, 223-57.

University

Freddoso, A., 1978, “Abailard on Collective Realism”, Journal of Philosophy 75: 527-38. Henry, D. P., 1985, “Abelard’s Mereological Terminology”, in Medieval semantics and metaphysics: Studies dedicated to L. M. De Rijk, E. P. Bos, ed., Nijmegen: Ingenium Publishers, 65-92. Henry, D. P., 1990, “Master Peter’s Mereology”, in De ortu grammaticae: Studies in medieval grammar and linguistic theory in memory of Jan Pinborg, Bursill-Hall, G. L.; Ebbesen, S.; Korner, K. (eds.), Philadelphia: John Benjamins, 99-115. Henry, D. P., 1991, Medieval Mereology, Amsterdam: B. R. Grüner. King, P., 2004, “Metaphysics”, in J. Brower and K. Guilfoy, eds., The Cambridge Companion to Abelard, Cambridge: Cambridge University Press, 65-125. King, P., 2015, “Pseudo-Joscelin: The Treatise on Genera and Species”, Oxford Studies in Medieval Philosophy 2: 104-211. Marenbon, J., 2007, “Abelard’s Changing Thoughts on Sameness and Difference in Logic and Theology”, American Catholic Philosophical Quarterly 81: 229-50. Marenbon, J., 2013, Abelard in Four Dimensions: A Twelfth-century Philosopher in his Context and ours, Notre Dame: University of Notre Dame Press.

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Martin, C. J., 1998, “The Logic of Growth: Twelfth-Century Nominalists and the Development of Theories of the Incarnation”, Medieval Philosophy and Theology 7: 1-15. Normore, C., 2006, “Ockham's Metaphysics of Parts”, Journal of Philosophy 103: 737-54. Pasnau, R., 2011, Metaphysical Themes, 1274-1671, Oxford: Clarendon Press. Andrew Arlig

Abstract All abstract individuals have neither spatial nor temporal parts, or in other words: they have neither a spatial nor a temporal localisation in a literal sense (though some of them – for example, certain sets – can be assigned a spatial or temporal localisation derivatively, and do have that location analogically). Perhaps all individuals that have neither spatial nor temporal parts are abstract, perhaps not: God, angels, and souls would be individuals that have neither spatial nor temporal parts, yet one would not call any one of them ‘abstract’. Not only certain individuals but also certain non-individuals are abstract. In fact, there is a long-standing tendency in ontology to consider all non-individuals to be abstract entities, the rationale for this being that all non-individuals have neither spatial nor temporal parts. But, as in the case of individuals, so also in the case of non-individuals (for example,

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ABSTRACT

universals and states of affairs): the absence of spatial and temporal parts does not appear to be a sufficient condition for abstractness (though it is a necessary condition in both cases). Prima facie it does not appear appropriate to call, say, the state of affairs of the earth’s revolving around the sun an abstract entity. It has been suggested that what makes an entity abstract is its lack of causal powers. But the absence of causal powers, too, is not a sufficient condition for abstractness (although, again, necessary for it), because not all causal epiphenomena are bound to be abstract. If some conscious experiences had no causal powers, it would certainly not make them abstract. Nor would I be abstract if I – in contrast to my brain – had no causal powers. Whatever may be the precise meaning of abstractness, propositions and concepts are abstract throughout: very plausibly, all propositions and all concepts are abstract. This implies that propositions and concepts have neither spatial nor temporal parts. Their lack of spatial and temporal parts, however, does not prevent propositions and concepts from having, and being, parts in a certain clear sense, and hence it does not prevent them from having their own abstract mereologies. Their mereologies are abstract in a much more radical sense than the mereology of, say, abstract geometrical shapes, or of Meinongian objects. Moreover, the beginnings of the mereology of concepts go back at least to Plato’s dialogue Parmenides; it has the longest

history of all mereologies. For these reasons, I will here concentrate on the abstract mereologies of concepts and propositions. The mereology of propositions can be built on the notion of logical part, where proposition p is a logical part of proposition q if, and only if, q logically entails p. It should be noted that propositions are here taken to be non-linguistic entities. The invoked relation of logical parthood is, therefore, not the relation of logical implication, which I take to hold between sentences, not propositions; rather, logical parthood is the ontological basis of logical implication. The intended part-relation is, moreover, logical entailment broadly conceived; it is not logical entailment as codified in some logical system, say, firstorder predicate logic. But logical entailment broadly conceived extensionally encompasses the entailment relation of first-order predicate logic – if this latter entailment relation is referred to propositions instead of sentences. The mereology of concepts (also taken to be non-linguistic entities) can be built on top of the mereology of propositions by making use of the following definition, which extends the notion of logical part from propositions to concepts: The concept F is a logical part of the concept G if, and only if, (1) F and G are meaningfully applicable to exactly the same entities, and (2) for all x to which F is meaningfully applicable: the proposition that F truthfully applies to x is a logical part of the

ABSTRACT

proposition that G truthfully applies to x. Thus, for example, the concept of extendedness is a logical part of the concept of coloredness, because (1) both concepts are meaningfully applicable to exactly the same entities and because (2) for all x to which extendedness is meaningfully applicable: the proposition that extendedness truthfully applies to x is a logical part of the proposition that coloredness truthfully applies to x. A mereology with primitive partrelation, and not with some other mereological relation as primitive, is either a proper-parts or a proper-orimproper-parts mereology. Clearly, the mereology of propositions, as based on the notion of logical part taken in the sense specified above, is a proper-or-improper-parts mereology. The part-relation on propositions is transitive and reflexive, which is familiar from other proper-orimproper parts mereologies. But it must be noted that the abstract mereology of propositions differs significantly from classical extensional mereology. In contrast to classical extensional mereology, the principle of mereological extensionality – according to which entities that are parts of each other are identical to each other – fails to hold in the case of the mereology of propositions: Even though the proposition that Jack is older than Mack and the proposition that Mack is younger than Jack are logical parts of each other, the two propositions are two propositions, and not one and the same. Moreover, it is not a feature of clas-

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sical mereologies that some entity that belongs to the field of their partrelation is a part of every entity in that field. For example, there certainly is no volume of space which is a subvolume of every volume of space. In contrast, there are many – indeed, infinitely many – propositions which are a logical part of every proposition: the proposition that 1=1, the proposition that what is not extended is not colored, the proposition that not every proposition is false, etc. Especially the feature of the mereology of propositions that has just been described may suggest to some that the so-called mereology of propositions is not really – but only analogically – a mereology. But what is an intuitively satisfactory criterion according to which one is to decide whether a theory is really (literally, genuinely) a mereology? Here is such a criterion: A mereology-like theory is a genuine mereology if, and only if, its partrelation satisfies the following schema: For all x and y: x is a part of y only if the conjunction (or sum) of x and y is identical to y. And, as a matter of fact, we find that the part-relation of the mereology of propositions does not satisfy that schema: The proposition that Mack is male is a logical part of the proposition that Mack is the son of Jack. But it is not the case that the conjunction of the two propositions is identical to the proposition that Mack is the son of Jack: the proposition that Mack is male and the son of Jack is obviously different from the proposition that Mack is the son of Jack. Hence one is

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quite justified in concluding that the mereology of propositions is not a genuine mereology, but a mereology only in an analogical sense. However, the situation changes fundamentally if propositions are identified with coarsely individuated states of affairs, as some authors have suggested. Then the propositions that Jack is older than Mack and that Mack is younger than Jack are identical (since they are the same state of affairs) and the proposition that Mack is male and the son of Jack is identical to the proposition that Mack is the son of Jack. If propositions are identified with states of affairs, the mereology of propositions becomes a classical extensional mereology – except for the fact that there still is a proposition that is a part of every proposition. But from the point of view of abstract mereology, that there is such a proposition (only one if propositions are identified with coarsely individuated states of affairs) is nothing to balk at. See also > Boolean Algebra, Propositions, Universal. References and further readings

Giraud, T., 2013, “An Abstract Mereology for Meinongian Objects”, Humana Mente 25: 177-210. Künne, W., 1982, “Criteria of Abstractness”, in: Smith, B., ed., Parts and Moments. Studies in Logic and Formal Ontology, Munich: Philosophia, 401-437.

Kutschera, F. v., 1995, Platons “Parmenides”, Berlin: De Gruyter. Lewis, D., 1986, On the Plurality of Worlds, Oxford: Blackwell. Meixner, U., 1997, Axiomatic Formal Ontology, Dordrecht: Kluwer. Meixner, U., 2006, The Theory of Ontic Modalities, Heusenstamm: Ontos. Uwe Meixner

Act and Action Preliminaries. Actions are character-

ised by at least three contrasts. First, actions are occurrences and not dispositions. Second, an action is an active doing and not merely something that happens to us. Third, an action is a doing that is rationally explainable, in contrast to a form of behavior where it would be pointless to inquire for reasons. Some philosophers claim that to act is to cause some change and that inanimate substances act in this sense (Alvarez & Hyman 1998). Here the usual view is adopted according to which all actions are (from some point of view) intentional (Davidson 1980, essay 3). Accordingly some but not all mental episodes are actions. Some authors stipulate a difference in meaning between ‘act’ and ‘action’ but no such distinction is obviously adequate. The nature of actions. Apart from purely mental acts, actions typically are connected with sequences of causally related events. If A moves his finger and thereby turns on the

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light, there will presumably be some mental episode (e.g. the intention to turn on the light), bodily events (contractions of muscles, movement of A’s finger) and events outside A’s body (movement of the switch, flow of electrons). There are three general approaches concerning the nature of actions. Each of these approaches identifies actions with different parts of the sequences of causally related events connected with actions. 1. According to causal theories, the best known version of which is propounded by D. Davidson, an action is a bodily event caused by an appropriate mental item which is not part of the action (Davidson 1980, essays 1 & 3). Davidson regards the mental item as a reason composed of some beliefs and pro-attitudes the content of which rationally explains the action. Others claim that Davidson fails to identify the proximate causes of actions and employ intentions (Sellars 1973, sec. IV; Brand 1984, chap. 2). 2. Volitionist theories claim that certain mental episodes are either parts of actions or identical with them, the mental episodes playing a causal role and being described as acts of will, intentions, decisions, and, recently, as tryings (Hornsby 1980; Pietrowski 2000). Volitionist theories come in several varieties. a) According to John Stuart Mill an action is “the state of mind called a volition followed by an effect” (Logic, Bk. 1, chap. iii, sec. 5). So a volition and some bodily effect are parts that jointly compose an action. b) The view that volitions are both actions and causes of actions faces Ryle’s famous regress argument: If actions

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are caused by volitions and if volitions themselves are actions any volition must be caused by a further volition (Ryle 1949 chap. 3 sec. 2). c) Most volitionists hold that volitions are actions but that it is bodily movements rather than actions that are caused by them. Being part of a causal sequence is sometimes seen as essential for actions insofar as volitions are not actions unless they cause bodily movement (Hornsby 1980: 44; contrast Ginet 1990: 30). 3. Agent-causal theories claim that the agents cannot serve merely as arenas for mental and bodily events but must themselves be causes. Recent versions typically deny that the agent causes his actions but rather view an action as the causing of some (mental or bodily) event by an agent (O’Connor 2003: 266; contrast Taylor 1966: 115). Individuation of actions. If A moves

his finger and thereby flips the switch, does he perform one action or two possibly overlapping actions? More generally, if A φ’s by ψ-ing, is his φ-ing identical with his ψ-ing? Three types of answer have been given. 1. According to the unifier account, all descriptions pick out the same action. On this account, if one talks about A’s flipping the switch one does not introduce a new action but describes the same action (A’s moving his finger) in terms of one of its effects (Davidson 1980, essay 3). 2. According to the multiplier account, an action is the exemplification of an act-property (Goldman 1970: 10). Whenever A φ’s by ψ-ing, A’s φ-ing is different from A’s ψ-ing if φ and ψ introduce different act-

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properties. Support for this account partially rests upon the controversial claim that ‘by’ expresses a relation between actions (for critique cf. Bennett 1994). 3. According to middle accounts, A’s φ-ing might be identical with A’s ψ-ing, but will not invariably be so. Some adherents of this view claim that actions typically are classified with reference to their effects: shooting is causing a shot, killing is causing a death. Call the effect with reference to which an action is defined its result and further effects of it its consequences (von Wright 1963: 39). Then the criterion may be stated thus: A’s φ-ing is identical with his ψ-ing just in case they have the same result (Alvarez & Hyman 1998: 234). Accordingly, A’s flipping the switch is different from A’s moving his finger since its result is not the result of A’s moving his finger but a consequence thereof. Basic actions. Basic actions may be

regarded as the minimal (possibly improper) parts of actions which are themselves actions. Usually it is held that A’s φ-ing is basic if A does not φ by doing something else. This idea has to be refined in two respects. First, one should classify not actions but descriptions of actions as basic. Second, the ‘by’-locution does not pick out a unique standard of basicness. From the agent’s point of view it would be appropriate to say that A contracts his muscles by raising his arm whereas from a physiologist’s point of view it is the other way round. Therefore it is reasonable to introduce a causal notion of basicness (Hornsby 1980, chap. 5). Adopting the terminology from above yields: A

description d of A’s φ-ing is basic if the result introduced by d is not the consequence of any other action of A. See also > Activity, Brentano, Intentionality, Stumpf. Bibliographical remarks

Davidson, D., 1980. Propounds what still is the dominant theory of action. Hornsby, J., 1980. Careful analysis; claims that all actions are tryings and therefore occur inside the body. Mele, A.R. ed., 1997. The essays provide a useful overview. Wittgenstein, L., 1953. Wittgenstein’s questions have shaped the debate. References and further readings

Alvarez, M. & Hyman, J., 1998, “Agents and their Actions”, Philosophy 73: 219-245. Bennett, J., 1994, “The ‘Namely’Analysis of the ‘by’-Locution”, Linguistics and Philosophy 17: 29-51. Brand, M., 1984, Intending and Acting, Cambridge, MA: MIT Press. Davidson, D., 1980, Actions and Events, Oxford: Clarendon Press. Ginet, C., 1990, On Action, Cambridge: CUP. Goldman, A., 1970, A Theory of Human Action, Englewood Cliffs, NJ: Prentice-Hall.

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Hornsby, J., 1980, Actions, London: Routlegde and Kegan Paul. Mele, A.R. ed., 1997, The Philosophy of Action, Oxford: Clarendon Press. O’Connor, T., 2003, “Agent Causation”, in: G. Watson ed., Free Will, Oxford: Clarendon Press, 257-284. Pietroski, P., 2000, Causing Actions, Oxford: Clarendon Press. Ryle, G., 1949, The Concept of Mind, London: Hutchinson, Chap. 3. Sellars, W., 1973, “Actions and Events”, Nous 7: 179-202. Taylor, R., 1966, Action and Purpose, Englewood Cliffs, NJ: Prentice-Hall. von Wright, H., 1963, Norm and Action. A Logical Inquiry, London: Routlegde and Kegan Paul. Wittgenstein, L., 1953, Philosophical Investigations, Oxford: Blackwell, §§ 611-660. Johannes Huebner

Activity In contrast to action or event, the notion of activity so far has received comparatively little attention in contemporary analytical ontology. However, detailed investigations of partwhole relations on activities may have considerable import for a number of systematic ontological problems, including the question of whether the part-whole relation is transitive.

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Pre-philosophical usage. In everyday

reasoning and informal glosses in science, the term ‘activity’ is used in a narrow and a wide sense. According to the narrow sense, an activity is an occurrence that is (i) an intentional doing performed by one or more subjects capable of intentions (humans and perhaps also higher animals), and (ii) in some fashion repetitive or ‘uniform’. For example, the sentences ‘Kim is reading’, ‘today’s choir singing was beautiful’, or ‘wolves howled throughout the night’ are about activities in this narrow sense of the term. By contrast, according to the wider notion, activities do not need to be intentional doings – all that matters is condition (ii). Activities in this wider sense include afflictions of human subjects (e.g. the denotation of ‘Kim was coughing all night long’) but also the doings and afflictions of many other kinds of ‘subjects’: things (cf. ‘the wheel was spinning’), pluralities of things or pieces of matter (cf. ‘thousands of snowflakes hurtled through the valley’), stuffs (cf. ‘the surf was pounding the coast’, ‘water evaporates at 100° C’), occurrences (cf. ‘the magnet’s movement creates a current’, ‘meteor strikes were pounding the earth’) and others (‘news will be spreading fast’). In fact, as C. D. Broad (1933, Vol. 1, 141ff.) pointed out, some activities are even ‘subjectless’ occurrences in the sense that there is no object or medium that can be said literally to bring about or undergo the occurrence. For example, sentences with impersonal pronouns such as ‘it is raining’ or ‘it is itching’ seem to denote subjectless activities

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in this sense, but also sentences with less obvious ‘dummy subjects’ (Sellars 1981), e.g., ‘volcanic activity has been shaping the region for centuries’, ‘the radiation has not been contained’, or ‘the fire continued to burn’. In sum, as we prephilosophically use the term ‘activity’, it serves to characterise repetitive or otherwise uniform occurrences, including those that are brought about by intentions, but also goingson that do not seem to involve any thing or material medium at all. Tasks for an ontology of activities. Going by the standard (Carnap-Quine) conception of ontology as the domain theory of a natural or scientific language L (theory), an ontological investigation of activities has (at least) three tasks, and in all of them partwhole relations play an important role. First, an ontology of activities should describe which types of entities speakers of L are referring to when they speak about activities. Thus the pre-philosophical data need to be clarified – precisely which sentences should be targeted to get at the inferential meaning of our concept of an activity, in contrast to other occurrences? Second, an ontology of activities should devise a categorytheoretical description of the type of entities that sentences about activities are about – are these entities, e.g., particular or general, concrete or abstract, simple or complex, etc? Third, an ontology of activities should explore how the ontological counterparts that make our sentences about activities true relate to other categories needed to describe the domain of the relevant language.

(1) Inferential data for an ontology of activities. In Metaphysics Theta.6

(1048b 18-b35) Aristotle distinguishes two sorts of actions (praxeis) – some, like seeing and thinking, are ‘complete’ (teleia) while others, such as learning or building a house, are ‘incomplete’ (atelēs). Whether or not an action is complete depends on whether it ‘contains’ that which it is directed at, its completion or telos, and that in turn, Aristotle suggests by illustration, can depend on the following condition, later called the ‘Aristotle’s completeness test’: “in the same moment (hama) one is seeing and has seen, is understanding and has understood, is thinking and has thought”. Aristotle says that incomplete and complete actions should be called kinēseis and energeiai, respectively, and in later translations these terms were translated – guided by Aristotle’s illustrations here and in other passages – as movements and activities, respectively. In the 20th century, the philosopher G. Ryle took Aristotle’s ‘completeness test’ to mark “differences in the logical behavior of verbs” (1949: 149; my emphasis), and thereby formulated a research project that later was worked out by Z. Vendler (1957) and A. Kenny (1963). Both Vendler and Kenny aimed to classify ‘action verbs’ according to systematic differences in inferences licensed by these verbs, and thereby to define different concepts of action or ‘action types’. While Kenny distinguished between three classes of action types: ‘activities’, ‘performances’, and ‘states’, Vendler presented a fourfold division that was to become the more influen-

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tial, distinguishing between ‘activities’, ‘accomplishments’, ‘achievements’, and ‘states’. The inferential hallmarks of ‘activity verbs’ – such as ‘run’ or ‘read’ – of Vendler’s classification can be summarised and labeled as follows (cf. Seibt 2004). Verb V of language L, V is an ‘activity verb’ if (A1) through (A4) hold: (V1) Dynamic duration: ‘[Noun phrase] is V-ing’ is a grammatical sentence of L. (V2) Unboundedness: the predicate ‘[noun phrase] finished V-ing’ is false in L for any instantiation of the noun phrase. (V3) Distributivity: For every temporal interval t, if ‘[noun phrase] Ved during t’ is true then ‘[noun phrase] V-ed during t*’ is true for every t* that is part of t. (V4) Homeomereity: V denotes an occurrence all temporal parts of which are of the “same nature” as the whole occurrence. By contrast, ‘accomplishment verbs’ like ‘build a house’ or ‘grow up’ fail with respect to (A2) through (A4) – they denote an occurrence that we conceptualize as bounded by a completion point, thus they are incomplete before the completion point is reached and not distributive, as well as continuously changing towards the completion point and thus not ‘homeomerous’ or ‘likeparted’. (In On Plants, 818a 21, Aristotle characterises stuffs as ‘ὁµοιοµερῆ’, lit. ‘of like parts’, which should be transliterated as ‘homoeomerous’ but is often, especially in contemporary ontological

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and linguistic discussion, stated as ‘homeomerous’ or ‘homomerous’). While providing important insights into the inferential contrast between our notions of activities and accomplishments, Vendler’s and Kenny’s classifications proved unsuccessful, however. Both philosophers falsely assume that the inferentially relevant meaning that characterises the common-sense concept of an activity hinges on the lexical meaning of certain ‘verbs’. As pointed out by Verkuyl (1972), Dowty (1977) and, in a particularly seminal paper, by Mourelatos (1978), the classifications overlook the fact, well-known in linguistic semantics, that the shifts in the verbal aspect of a predication can change the occurrence type (in linguistic terminology: ‘Aktionsart’) that the lexical meaning of the verb expresses as a default. For example, the lexical meaning of the verbal predicate ‘build a house’ includes that the occurrence denoted is by default to be conceptualised as an ‘accomplishment’, and in the sentential context of ‘Max built a house two years ago’ the verbal predicate functions according to its default meaning, denoting an accomplishment. However, the same verbal predicate behaves like an ‘activity verb’ in the context of ‘Max was building houses for many years’. In fact, whether an occurrence is conceptualised as, e.g., activity or accomplishment, depends not only on tense, aspect, and the lexical meaning of the verbal predication but on all components of the sentence, such as the number value (singular or plural)

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of the subject of the verbal predication (cf. Rijksbaron 1989). These relationships have been explored in great detail and across languages in linguistic ‘aspectology’ (Sasse 2002), but so far there is little crossfertilising discourse between aspectology and ontology. Some analytical ontologists introduce the common sense notion of an activity or process as expressed by a sentence in the progressive (Steward 2015, Stout 2016). But this does not capture the full inferential meaning of the notion of an activity. Using linguistic research as heuristics for ontology, Seibt (2004; 2005) proposes that our common sense conception of an activity, as documented in inferential meaning, is constituted by the conjunction of six conditions. An activity is an occurrence that (A1) has temporal duration, (A2) is unbounded, (A3) is at any time completed (in the sense of (V3) but also in that it occurs for the sake of nothing else but its own occurrence), (A4) can be suspended and resumed. With these four features one can reformulate a fourfold classification of occurrence types (activities, developments, results, and states) that is expressed in terms of sets of ‘aspectual inferences’ (i.e., inference from one type of aspectual information to another, as in: ‘S is an activity sentence if S with progressive aspect entails S with perfective aspect’). Aspectual inferences do not make reference to specific linguistic material (unlike e.g., (V2) above), and to the extent to which the linguistic definitions of aspects hold cross-linguistically, the classification of occurrence types in

terms of aspectual inference networks can be used across languages (Seibt 2015). Condition (A3), that an activity A is ‘self-completing’ or at any time complete, implies that A is ‘sameparted’ in the following sense: (A4) Homeomereity: any part of the temporal interval during which A occurs is an interval where an occurrence of the same (proximate) kind as A occurs. However, (A3) also implies that activities fulfill the stronger condition of being ‘self-parted’: (A5) Automereity: any part of the temporal interval during which A occurs is an interval where A occurs. Additional elements of the inferential meaning of activities (versus accomplishment) are worked out by Kühl (2008), who considers also starts and endings of these action types. Condition (A5) has gone virtually unnoticed (cf. Zemach 1970, Roberts 1979, Seibt 1997). Most authors read (A5) as the postulate of homeomereity and reformulate the latter using Goodman’s (1951: 53) notion of ‘dissectivity’– a dissective predicate applies to all parts of anything to which it applies. This reading immediately turns activities into filled spacetime regions satisfying certain predicates (see e.g. Simons 1987: 139). But (A5) says something more extraordinary – it states that activities literally recur and that is of crucial relevance for the following two tasks for an ontology of activities.

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(2) Categorisations of ontological truth-makers for activity sentences.

Once the concept of activity has been clarified – that is, once we have clarified what is (not) entailed when we conceptualize an occurrence as an activity, we can turn to the task of defining truth-makers for sentences about activities. Aristotle proposed that statements about activities (e.g., about energeiai such as living, Aristotle’s prime illustration, but also knowing, seeing, or walking-about) denote the interaction of an entity’s active capacity (form) and passive capacity (‘functional matter’) for manifesting the functional features that characterise the occurrence (e.g., as an occurrence of living, knowing, seeing etc.). Within the context of Aristotle’s system, this ontological interpretation, and the associated interpretation for kinēseis or developments (movements, changes), are explanatorily quite powerful. For they may be used to account for the generation, persistence, and destruction (deterioration) of biological individuals and artifacts (cf. Gill 1989). However, Aristotle’s core distinctions – active versus passive, first-level versus second-level potentiality, and, in particular, potentiality versus actuality – are not strictly axiomatically defined but derive a good part of their explanatory meaning from the common-sense concepts (e.g., of the division between activities and goal-oriented developments) they are supposed to explain, creating a hermeneutic circle that may be somewhat too tight. Most ontologists currently assume that we do not need a new ontologi-

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cal category to describe the denotations of our discourse about activities. They tend to lump together sentences about activities with sentences about developments/accomplishments and postulate as their truth-makers either substances, or four-dimensional regions, or (collections of) stages of such regions. Setting aside the fact that only the last two approaches can accommodate ‘subjectless’ (agentless) activities, the envisaged truth-makers are in all three cases categorised as concrete particular individuals that are ‘static’. A few analytical ontologists, however, postulate entities that are – in a sense that so far has not been defined – ‘dynamic’, and distinguish between two sorts of dynamic particulars, namely, the category of ‘events’ as truth-makers for our talk about accomplishments or developments, and the category of ‘processes’ as truth-makers for our talk about activities (cf. e.g., Needham 2004, Galton & Mizoguchi 2008, Steward 1997; 2012; Stout 1997; 2016). However, if we take the distributivity condition (A3) to imply automereity (A5) then the truthmakers of activity sentences cannot be particulars; rather, they are non-particulars since they are multiply occurrent in time. Rescher (1996) concludes from this that ‘processes’ are concrete universals. By contrast, Seibt (1995; 2005; 2015) argues that the truth-makers of our talk about activities must be conceived of as a new category of concrete non-particular individuals, since they are neither dependent nor predicable in the sense in which this holds for the category of universals (that activities are non-particular individu-

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als is also considered by Hornsby (2013)).

aspect’ to a ‘count aspect’ (Bunt 1985).

Whether we need a new category of non-particular individuals or not thus hinges on the notion of automereity, and some philosophers (e.g., Gill 1997) have objected that activities are homeomerous and automerous only down to a certain ‘grain size’ – while running occurs during every 10 second interval of 10 minutes of running, during an interval of 50 milliseconds only a constitutive movement of the running occurs, e.g., the pushing off from the left foot. However, this objection operates with a clandestine conceptual shift. We can conceive of the 10 minute interval of running either as an activity or as a sequence of short-term accomplishments/development (pushing off from the left foot, bringing the right arm forward, etc.). If we conceptualize it as an activity, then our common sense concept of an activity prescribes that running occurs in every temporal part of a 10 minute interval of running (Seibt 2004).

Task 3: The explanatory role of truthmakers for activity sentences. The in-

The point may be thrown into better relief by a comparison with the spatial homeomereity and automereity of stuffs (masses) like wood or water. Every spatial region that is a spatial part of a region R in which water exists is a region in which water exists. If we object that the spatial region occupied by an oxygen atom is not a region in which water exists, we clandestinely reconceptualize that which occupies R as a collection of ‘things’ (atoms) and view it no longer as stuff – we shift from a ‘mass

ferential symmetries between our concepts of activities and stuffs on the one hand, and things and developments (or events, or accomplishments) on the other hand (see in particular Mourelatos 1978, but also Mayo 1961, Zemach 1970, Roberts 1979, Crowther 2011) give rise to the question of how the truth-makers of activity sentences – call them processes* – relate to other categories. For example, given the inferential symmetries between the concept of activity and the concept of stuff, can we treat processes* and the truthmakers of sentences about stuffs subcategories of a more basic category of concrete non-particular individuals? But there also seem to be important inferential symmetries between the concept of activity and the concept of a thing – if activities are temporally automerous, and thus multi-located in time in the way in which this holds for things, should we then conclude that processes* can also be the truthmakers for sentences about things or even persons, i.e., items that we take to be numerically identical throughout time and change? Recently some authors have begun to discuss whether the standard division between the categories of ‘occurrent’ and ‘continuant’ needs to be revised to form the category of an “occurrent continuant” (Stout 2016; Steward 2013). In General Process Theory a much more radical approach to the reconfigura-

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tion of the traditional categorial landscape is suggested; here it is shown that we can define truth-makers for sentences about activities, stuffs, developments, things, collections and several other common sense types of entities in terms of one basic category (‘general process’), occurring in different ‘modes’ that are characterised by different combinations of spatial and temporal homeomereity and automereity (Seibt 2005; 2015a). Another line of inquiry investigates whether processes*, the truth-makers of activity sentences, offer us new paths to the explanation of common sense relational concepts, such as emergence (Bickhard & Campbell 2012), causation (Ingthorsson 2002), or agency (Hornsby 2013), or the integration of sensory qualities into a naturalistic metaphysics (Sellars 1981). One of the core tasks for this direction of research is to clarify which formal mereological framework can be used to state part-whole relations on processes*. Those who postulate filled four-dimensional regions as truth-makers of activities, i.e., those who work with weak identity conditions for processes*, can avail themselves of classical extensional mereology (Simons 1987: 127). In this case the relation ‘x is part of y’ on a process* is identical to the relation of ‘x is spatiotemporal part of y’. In this setting, the condition of automereity cannot be maintained – the whole spatio-temporal region would need to occur in one of its spatio-temporal parts, which is absurd. Here it is relevant to remind ourselves that in many

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contexts spatio-temporally coincident activities count as different occurrences both in common sense and in scientific reasoning (e.g., running in spatiotemporal region R and exercising in R; a metal disc’s rotating in R and its warming up in R; cellmetabolism in R and the sum of cooccurrent bio-chemical processes constituting cell metabolism). Thus it seems preferable not to identify the truth-makers for sentences about activities with filled spatiotemporal regions, i.e., operate with stricter identity conditions for processes*. If processes* are individuated by ‘what they do’, i.e., by their – in the widest sense of the term – ‘functional’ relationships, they can occur multiply in space and time and the automereity condition can be respected. The relation ‘is part of’ on processes* is then no longer the relation ‘is a spatiotemporal part of’ as modelled by classical extensional mereology, but rather a conceptually more basic relationship that has functional connotations. Whether and how one can formally capture the inferential meaning of this more basic meaning of ‘is part of’ by a non-transitive (irreflexive, asymmetric) relation is little explored so far; in Seibt (2005; 2015b) transitive extensions of the non-transitive ‘is part of’ are explicitly defined across different ‘levels of depth’ in partitions that represent processes*. In such a ‘Leveled Mereology’ – which operates with an acyclic partrelation yet also allows for a formal expression of mutual dependencies of processes* – the ‘proper parts principle’ is retained, but is re-defined as

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holding up to, and relative to, certain partition levels. See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Homeomereity and Automereity, Persistence, Transitivity. References and further readings

Bari, C., 2009, Aspect in Ancient Greek, PhD Thesis University of Nijmegen. Broad, C. D., 1933, Examination of McTaggart's Philosophy, Cambridge: Cambridge University Press. Bunt, H., 1985, Mass Terms and Model Theoretic Semantics, Cambridge: Cambridge University Press. Dowty, D., 1977, “Toward a Semantic Analysis of Verb Aspect and the English ‘Imperfective’ Progressive”, Linguistics and Philosophy 1: 45-78. Crowther, T., 2011, “The Matter of Events”, The Review of Metaphysics 65: 3-39 Galton, A., & Mizoguchi, R., 2009, “The Water Falls but the Waterfall Does not Fall: New Perspectives on Objects, Processes and Events”, Applied Ontology 4: 71-107.

Gill, M.-L., 2004, “Aristotle's Distinction between Change and Activity”, Axiomathes 14: 3-22. Goodman, N., 1951, The Structure of Appearance, Cambridge, MA: Harvard University Press. Hornsby, J., 2012, “Actions and Activity”, Philosophical Issues 22: 233245. Ingthorsson, R. D., 2002, “Causal Production as Interaction”, Metaphysica 3: 87-119. Kenny, A., 1963 Actions, Emotions, and Will, London: Routledge. Kühl, C.-E., 2008, “Kinesis, Energeia, and what Follows – Outline of a Typology of Human Actions”, Axiomathes 18: 303-338. Mayo, B., 1961, “Objects, Events, and Complementarity”, Philosophical Review 70: 340-361. Mourelatos, A., 1978, “Events, Processes, and States”, Linguistics and Philosophy 2, 415-434. Reprinted with corrections in P. J. Tedeschi and A. Zaenen, (eds.), Tense and Aspect, New York, Academic Press, 1981: 191-212. Rescher, N., 1996, Introduction to Process Metaphysics, Albany: SUNY Press.

Gill, K., 1993, “On the Metaphysical Distinction Between Processes and Events”, Canadian Journal of Philosophy 23: 365-384.

Rijksbaron, A., 1989, Aristotle, Verb Meaning and Functional Grammar. Towards a New Typology of States of Affairs, Amsterdam: J. C. Gieben.

Gill, M.-L., 1989, Aristotle on Substance — The Paradox of Unity, Princeton, NJ: Princeton Univ. Press.

Roberts, J. H., 1979, “Actions and Performances Considered as Objects and Events”, Philosophical Studies 35: 171-185.

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Sasse, H.-J., 2002, “Recent Activity in the Theory of Aspect: Accomplishments, Achievements, or just Non-Progressive State?”, Linguistic Typology 6: 199-273. Seibt, J., 1995, “Individuen als Prozesse: Zur prozess-ontologischen Revision des Substanzparadigmas”, Logos 2: 352-384. Seibt, J., 1997, “Existence in Time: From Substance to Process”, in: J. Faye, U. Scheffler, M. Urs, (eds.), Perspectives on Time. Boston Studies in Philosophy of Science, Dordrecht: Kluwer, 143-182. Seibt, J., 2004, “Free Process Theory: Towards a Typology of Processes”, Axiomathes 14: 23-57. Seibt, J., 2005, General Processes – A Study in Ontological Category Construction, Habilitationsschrift at the University of Konstanz. Partly republished (forthcoming 2018) as Activities, Berlin: DeGruyter. Seibt, J., 2015a, “Ontological Scope and Linguistic Diversity: Are There Universal Categories?”, The Monist 98: 318-343. Seibt, J., 2015b, “Non-Transitive Parthood, Leveled Mereology, and the Representation of Emergent Parts of Processes”, Grazer Philosophische Studien 91: 165-190. Sellars, W., 1981, “Foundations for a Metaphysics of Pure Process”, Monist 64: 3-90. Stewart, H., 2013, “Processes, Continuants, and Individuals”, Mind 122: 781-812.

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Steward, H., 2015, “What is a Continuant?”, Aristotelian Society Supplementary Volume 89: 109-123. Stout, R., 2016, “The Category of Occurrent Continuants”, Mind 125: 41-62. Vendler, Z., 1957, “Verbs and Times”, The Philosophical Review 66: 143-160. Verkuyl, H., 1972, On the Compositional Nature of the Aspect, Dordrecht: Reidel. Zemach, E., 1970, “Four Ontologies”, Journal of Philosophy 23: 231247. Johanna Seibt

Albert of Saxony Albert of Saxony (c 1320-1390), German philosopher and theologian, was the rector of the University of Paris, the first rector of the University of Vienna, and the bishop of Halberstadt. In his Sophismata 45, 46 and 49 he discusses several different kinds of mereological problems. Sophism 45 is the first sophism on the part-whole relation that Albert considers; it runs: Totus Sortes est minor Sorte – the whole Sortes is less than Sortes. This statement seems false, but it also could appear true if one reasons as follows: any one of Sortes’ body parts is less than Sortes, and Sortes is the sum (whole) of his body parts. When analysing this sophism, Albert distinguishes between a syncategorematic whole and a categorematic whole. As an exam-

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ple Albert uses the Latin proposition: In oculo meo est totum quod est in mundo. This proposition is true, if ‘whole’ (totum) is understood in its categorematic sense, so that the proposition in understood in this way: ‘There is a whole in my eye which exists in this world’. For example, if Albert sees his pupil, who is a whole that exists in the world, then the pupil (in a sense) is in Albert`s eye. The proposition is false, if we understand ‘totum’ in is syncategorematic sense, so that the sentence means: ‘Any part of the world is in my eye’. If ‘the whole Sortes’ is taken in the categorematic sense, as the ‘complete Sortes’, the statement becomes false, while it is true for the syncategorematic reading of ‘the whole Sortes’ as ‘all parts of Sortes.’

there would be a whole class of Socrateses, and ‘Socrates’ would not be an individual name, but a species name of a mereological homo Socraticum.

A related set of problems of mereological concepts is discussed in Sophism 46 Totus Sortes est pars Sortis – The whole Sortes is a part of Sortes. Albert distinguishes in this context between: a: Sortes minus one finger, b: the finger, and c: composite of a and b. In this case we get the following sophism: a is Sortes, c is Sortes, and a is a proper part of c, ergo: Sortes is a part of Sortes.

A qualitative whole is a whole composed of parts that are (i) not outside of each other; (ii) that can be made more perfect by another part; and (iii) condition (ii) holds both in an accidental and in an essential way.

One of the solutions of this problem leads to the discussion of mereological essentialism pursued in our days by Roderick M. Chisholm and others. The problem can be formulated in the following way. If there are several accidental parts, then there also have to be several substantial wholes, for example: Socrates minus one finger, Socrates minus two fingers, Socrates minus one toe, and so on. In this case

In the sophism 49: Omne totum est maius suum parte (“Every whole is greater than its part”), Albert introduces the important distinction between a totum quantitativum (quantitative whole) and a totum qualitativum (qualitative whole). He defines the two notions as follows: A quantitative whole has parts that are (i) outside of each other; (ii) that are not potential parts of one another; and (iii) that could not be ‘made more perfect’ by another part – i.e., the parts do not overlap and have full actuality.

Albert draws in this sophism two distinctions. On the one hand, he contrasts totum quantitativum finitum (finite quantitative whole) and a totum quantitativum infinitum (infinite quantitative whole); on the other hand, he differentiates between a totum qualitativum essentiale (qualitative essential whole) and totum qualitativum accidentale (qualitative accidental whole). In discussing a quantitative infinite whole, Albert shows, by appealing to the case of an infinite time, that one infinite whole is neither more or less nor equal in relation to another infinite whole, because he

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holds a part of an infinite whole to be always an infinite whole itself.

gic and Grammar, DordrechtBoston-London: Kluwer, 288-303.

In the case of a qualitative whole, for example in the case of a beautiful face, a part, which is not beautiful (for example the nose), can be made more beautiful by the other parts and the whole face remains beautiful. There are parts in a beautiful face, which are not beautiful. But, according to Albert, there is a potentiality in the relation between the parts, which can turn a less beautiful part of a face into a part that not only contributes to the beauty of the whole, but is itself more beautiful. Thus qualitative parts, in contradistinction to quantitative parts, can exhibit a greater or lesser degree of some property. But in a qualitative whole a property of the whole, such as being beautiful or being the best, is not communicable (i.e., does not transfer) from the whole to all the parts nor are the properties of the parts communicable (transferable) to the whole.

Fitzgerald, M., 2009, “Time as a Part of Physical Objects: The Modern 'Descartes-minus' Argument and an Analogous Argument from Fourteenth-Century Logic (William Heytesbury and Albert of Saxony)”, Vivarium 47: 54-73.

See also > Abelard, Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Boethius, Medieval Mereology, Medieval Discussions of Temporal Parts and Wholes, Radolphus Brito, Totum Potentiale. References and further readings

Albertus de Saxonia, 1502, Sophismata. Nachdruck: 1975, Hildesheim: Georg Olms Verlag. Biard, J., 1993, “Albert de Saxe et les sophismes de l'infini”, in Stephen Read, ed., Sophisms in Medieval Lo-

Hans Burkhardt

Animal In our time animals are considered one of the five kingdoms of life (the others being that of plants, fungi, protests, and monera). Traditionally, animal referred to a living thing that was able to move around, in contrast to plants. As a category, the concept of animal can be traced back to the beginnings of Western thought. Aristotle (384-324 BC) did much to forge an idea of animals as substances with a special way of existence. In On the Soul, he argued that all living things can be regarded as existing at one of three levels. Plants only have the powers of digesting nutrients and reproducing, while animals have the additional powers of sensation and locomotion. Aristotle viewed humans as a subset of animals in virtue of their rational abilities. In his anatomical works, such as Parts of Animals, Aristotle argued that animals comprised parts that worked together for the sake of the whole. He claimed that parts could be understood as organs, or tools, that were used by the whole animal for its activities. In

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History of Animals, he reports that, in some type of animals, groups work together for common ends, so that individual animals can are parts of their larger group. In his discussion of human politics, Aristotle likened the way the citizen is a part of his state to the way that the hand is part of the body (Politics, I.2, 1253 a 20 29). In late antiquity, the physician Galen (129 - about 199 AD) advanced the concept of part-whole relationship in animals. His work, Usefulness of the Parts, systematically goes through every part of the human body to explain what it does for the sake of the whole, while his Anatomical Procedures gives directions for directly observing the nature of the parts by dissections and vivisection experiments. Galen worked out in detail how such structures as nerves, muscles, and bones are articulated so that given motions are produced. With his experimental procedures, he was able to show mechanisms in detail. Galen argues that the extent to which the parts work together indicates that nature manifests a kind of intelligence. He claimed throughout the work that nature generates and adapts the parts so that they work together as well as possible. In the Renaissance, anatomists built upon the anatomical and physiological accomplishments of Aristotle and Galen. In 1543, Andreas Vesalius (1514-1564) published his work on the Fabric of the Human Body, which contained systematic illustrations of the human body and corrected some of Galen’s anatomical descriptions,

which had been based on animal not human dissections. In his Motion of the Heart and Blood, William Harvey (1578-1657) produced a new theory that held the animal heart pumps blood which circulated through the arteries back to the heart through the veins. Harvey made many positive references to the approach towards understanding animal life taken by Aristotle and Galen, and continued to embrace the teleological notion that nature has fashioned the parts to work together for the whole. In his Treatise on Man, which was published in 1664, Rene Descartes (1596-1650) argued for a new conception of animals. He claimed that animals lack anything like what Aristotle had called a soul and rejected the teleological approach to studying an animal’s anatomy. Instead, he claimed that animals were machines – something like clocks – and that all their motions could be understood as arising from the shape and motions of their parts. The only activity in any animal that he attributed to a nonmechanical cause was thinking, which could only be found in humans. He located thought in the soul, which was an unextended substance. A different approach to both animal and plant life was taken by Immanuel Kant (1724-1804) in his Critique of Judgment. Kant argued that organisms had to be understood as more than mere mechanisms. They exhibited teleological properties, that is, the arrangement and function of the parts could only be understood as being caused by the design of the

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whole. Hence he claimed each part can be seen as both a means and ends, since it contributes to the life of the animal as well as receives its sustenance from it. At the end of the 18th century, the notion arose that animals can change over time in such a way that one species can evolve from another. In Germany, Johann von Goethe (1749 1832) developed an approach to studying morphology that built on the analysis of Kant and suggested the possibility of change. In France, Jean-Baptiste de Lamarck (17741829) and Etienne Geoffroy SaintHilaire (1772-1844) both advanced the theory that, during the course of long periods of time, the parts of one organism can change so that one species evolves into another. In England, the theory of evolution gained popular acceptance with the publication of Robert Chamber’s (18021871) Vestiges of the Natural History of Creation in 1844, but was rejected by most of the scientific community. In 1859, however, Charles Darwin’s (1809 - 1882) Origin of Species convinced many scientists that evolution occurs by a process he called natural selection. According to the theory of natural selection, animals both inherit parental traits but also display differences. Any animals with parts that have varied in a useful fashion would be more likely to live and leave progeny themselves, with the result that parts would be altered over many generations by the accumulation of variations to the point that a new kind of animal would arise.

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Since Darwin’s time, the development of increasingly sophisticated theories and laboratory methods have led to further modifications in our understanding of animals. Studies on breeding and how organisms differ from their parents led to the discovery of genes – parts of organisms that determine which traits they share with their parents. The progress of microbiology has resulted in a displacement of the traditional division of life into plants and animals. Instead, life forms are divided between prokaryotes, which are microscopic life forms whose cells lack nuclei, and the category of eukaryotes, which encompasses all life forms whose cells have their genes enclosed in a nucleus. The prokaryotes constitute the kingdom of monera. The eukaryotes are further divided into the kingdoms of animals, plants, fungi, and protists, which consist of either a single cell or a group of similarly structured cells. Research into genetics, developmental biology and the potential origins of life has led to continued inquiry by biologists on whether life can be reduced to a mechanical analysis of the parts as Descartes had suggested, or whether each organism manifests an interrelated structure that can only be understood in a consideration of the whole as Kant had suggested. See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Biological Parts, Medicine, Structure.

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Amundson, R., 2005, The Changing Role of the Embryo in Evolutionary Thought: Roots of Evo-Devo, Cambridge: Cambridge University Press.

Galen, 1968, Galen on the Usefulness of the Parts of the Body: Peri chreias morion De usu partium, translated by M. T. May, Ithaca: Cornell University Press.

Aristotle, 1983, Parts of Animals, translated and edited by A. L. Peck, Cambridge: Harvard University Press.

Hall, T., 1972, Rene Descartes: Treatise of Man, Cambridge: Harvard University Press.

Brogaard, B., 2004, “Species as Individuals”, Biology and Philosophy 19: 223-224.

Hankinson, R., 1989, “Galen and the Best of All Possible Worlds”, Classical Quarterly 39: 206-227.

Buss, L., 1987, The Evolution of Individuality, Princeton: Princeton University Press.

Harvey, W., 1995, The Anatomical Exercises: De Motu Cordis and De Circulatione Sanguinis, translated by G. Keynes, Mineola, New York: Dover Publications

References and further readings

Clarke, E., 2010, “The Problem of Biological Individuality”, Biological Theory 5: 312-325. Clarke, E., 2012, “Plant Individuality: A Solution to the Demographer's Dilemma”, Biology and Philosophy 27: 321-361. Cornell, J., 1986, “Newton of the Grassblade? Darwin and the Problem of Organic Teleology”, ISIS 77: 405421. Cosans, C., 1998, “Aristotle’s Anatomical Philosophy of Nature”, Biology and Philosophy 13: 311-339. Darwin, C., 1859, On the Origin of Species, a facsimile of the first edition, Cambridge: Harvard University Press, 1964. Dupré, J., 2012, Processes of Life, Essays in the Philosophy of Biology, New York: Oxford University Press. Ereshefsky, M. and Makmiller P., 2013, “Biological Individuality: The Case of Biofilms”, Biology & Philosophy 28: 331-349.

Kant, I., 1790, Critique of Judgment, translated by W. S. Pluhar, Indianapolis: Hackett, 1987. Kauffman, S., 1993, The Origins of Order: Self-Organization and Selection in Evolution, New York: Oxford University Press. Pepper, J. W. and Matthew D. Herron, 2008, “Does Biology Need an Organism Concept”, Biological Reviews 83: 621-627. Pradeu, T., 2012, Limits of the Self, New York: Oxford University Press. Raff, R., and Kaufman, T., 1983, Embryos, Genes, and Evolution: The Developmental – Genetic Basis of Evolutionary Change, Bloomington and Indianapolis: Indiana University Press. Richards, R., 1992, The Meaning of Evolution: The Morphological Construction and Ideological Reconstruction of Darwin’s Theory, Chicago: University of Chicago Press.

ANSELM

Turner, S. J., 2000, The Extended Organism, Cambridge, MA: Harvard University Press. van Inwagen, P., 1990, Material Beings, Ithaca, NY: Cornell University Press. Webster, G., and Goodwin, B., 1982, “The Origin of Species: A Structuralist Approach”, Journal of Social and Biological Structure 5: 15-47. Wilson, J., 1999, Biological Individuality: The Identity and Persistence of Living Entities, New York: Cambridge University Press. Wilson, R. A. and Barker, M., “The Biological Notion of Individual”, in The Stanford Encyclopedia of Philosophy, Spring 2013 Edition, Zalta, E. N. (ed.), http://plato.stan ford.edu/archives/spr2013/entries/bio logy-individual/. Wilson, R. A., 2005, Genes and the Agents of Life: The Individual in the Fragile Sciences: Biology, New York: Cambridge University Press. Zimmerman, D., 2002, “Persons and Bodies: Constitution Without Mereology?”, Philosophy and Phenomenological Research 64: 599-606. Christopher Cosans

Anselm Saint Anselm was born in 1033 in Aosta, which at that time was part of Burgundy. In 1056, after the death of his mother and quarrels with his father, he left home and wandered around in Burgundy and France until

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1059, when he came to Le Bec in Normandy. A year later he joined the Benedictine order and became the assistant of Lanfranc of Pavia. After Lanfranc had been appointed abbot of Caen, Anselm became prior at Le Bec. His literary career probably started with the Monologion (M) in 1076. Dissatisfied with this concatenatio argumentorum he published the more concise Proslogion (P) in 1078. Probably due to his so-called ontological arguments in chapters 2-4 of P and the appended disputation with a certain Gaunilo of Marmoutiers, Anselm’s achievements were increasingly and critically acknowledged. In the same year he became abbot of Le Bec. Later in the 1080s he wrote De Veritate (DV), De Libertate Arbitrii (DLA), De Casu Diaboli (DCD) and probably – according to Franciscus Salesius Schmitt – De Grammatico (DG). Sir Richard Southern suggests an earlier date between 1060 and 1063 for its first draft. In 1093 Anselm became Archbishop of Canterbury. Soon afterwards – in 1094 – he finished his Epistola de Incarnatione Verbi (EDIV), a demanding refutation of an attack launched by Roscelin of Compiègne on the exclusive incarnation of the second person of the Trinity in Jesus Christ. In 1098 he completed Cur Deus Homo (CDH) in exile. In 1099 he wrote De Conceptu Virginali (DCV). In 1101/2 he finished De Processione Spiritus Sancti (DPSS). Due to disagreements on investitures with Henry I, Anselm had to leave England for his second exile 1103-1106. The Epistola de Sacrificio Azymi et Fermentati (EDSA) and De Sacramentis Ecclesiae

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(DSE) were written between 1106 and 1107. From 1107-8 he wrote De Concordia (DC). His Philosophical Fragments (PF) also date from his late period. Anselm died on the 21st of April 1109. Anselm’s programme and method can be characterised as follows. Roughly following St Augustine and Boethius, Anselm modified his approach in the course of his career, turning from the method of sola ratione (“by reason alone”) in M to fides quaerens intellectum (“faith seeking understanding”) in P and later. For example, when developing his theory of atonement in CDH, he holds to high logical standards. He proceeds via necessary arguments, without relying on the authority of the Bible or the revelation in Christ. Responding to the demands of his fellow monks, Anselm’s works show an uncompromising search for a clear language, as well as simple and sound arguments. Critical analyses of religious and non-religious language play an important role in his effort to establish the Ratio fidei. His rigorous logical implementation of these methodological requirements throughout his writings brought Anselm – even among philosophers (e.g. Kurt Flasch and Desmond Paul Henry) – the honorary title Father of Scholasticism. Anselm’s arguments in Prologion (P) 2-4 can be regarded as a paradigm case for the effort to present his ideas as clearly as possible. Therefore, it is no wonder that these arguments and the appended dispute with Gaunilo recommend themselves for modern

reconstructions, be they formal or informal, sympathetic (cf. e.g., Richard Campbell, Peter Hinst, Gyula Klima, Ed Zalta) or critical (cf. e.g. Peter Millican, Edgar Morscher, Hermann Weidemann). There is much disagreement about how to reconstruct Anselm’s arguments. Among the sympathetic readings there are roughly two directions of interpretation. (i) The orthodox interpretation takes the second sentence of P 2 as a definition in a modern sense that can be used to prove indirectly that God exists in intellectu et in re. (ii) The non-orthodox interpretation sees P 2-4 as a three-stage-argument from belief to understanding. The latter holds that the argument aims at identifying God as that-than-whichnothing-greater-can-be-thought with that-which-cannot-be-thought-not-tobe (cf. Campbell 1979). Both readings (i) and (ii) are possible. Whether Anselm’s arguments are sound or – as is frequently maintained – contains some sleight of hand, is still controversial. However, recent results seem to confirm Richard Campbell’s and Ian Logan’s conclusion that the often heard prejudice that there must be a trickery in Anselm’s arguments can now be seen to be itself the product of philosophical developments from whose influence we are just beginning to free ourselves. Anselm’s philosophical theology requires a reconstructive hermeneutics. In his early writings M and P St Anselm struggles to clarify the concept of perfection and equivalent terms like ‘good’ (bonum). In M 15 he offers a series of examples, but no clear definition. It seems that he is trying

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to introduce the concept of perfection and/or good by means of the twoplace relational concept better than (melius). Furthermore, Anselm seems to presuppose such a definition in places where he talks about a good that contains the delightfulness of all goods (iucunditatem omnium bonorum), as in P 24. There is no doubt that he regards God as a something (aliquid) that does not lack any of the perfections so that his goodness is immense (immensitas bonitatis Dei; cf. P 13), which means that his goodness is infinite. This line of thought is convincingly confirmed in the debate with Gaunilo, his first critic. In his reply (R 10) to Gaunilo Anselm gives the following definition of the divine substance, which is embedded in a belief sentence: And therefore we believe of the divine substance everything which absolutely can be thought better to be than not to be (Credimus namque de divina substantia quidquid absolute cogitari potest melius esse quam non esse). If we read the part after “everything...” (quidquid absolute cogitari potest melius esse quam non esse) as a generally quantified definiens for perfection then we have an alternative definition of God as an omni-perfect entity. Altogether it seems that Anselm intended the identity-formula of P 4, where he characterises God as “id quo maius cogitari non potest”, and P 24 (as well as P 5, 11 + R 10) as equivalent. As a final observation, every time Anselm tries to define perfection via examples, he uses quasi-ethical words like ‘good’ (bonum), ‘goods’ (bona) and ‘better’ (melius). When he speaks about the Ultimate

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Reality in a more direct way he uses “greater” (maius). It remains unclear, however, how we are to understand ‘better’ and ‘greater’ in these contexts. Other questions raised by Anselm’s philosophy of perfect-being (cf. Brian Leftow 2004) concern the ontological status of his Famosa descriptio, the meaning of “exist in the understanding” (esse in intellectu) and “exist in reality” (esse in re sive actu). The main problem for a contemporary reconstruction in formal logic is to determine whether Anselm’s use of the key concept ‘perfection’ suggests that ‘perfections’ in general can be formulated in a firstorder language with definite descriptions, or whether a second-order language with quantifications over predicate variables is required. Until the second half of the 20th century Anselm was widely regarded as a gifted theologian, but as a poor logician and philosopher. Mainly due to the corrective investigations of Desmond Paul Henry from the 1950s to the 1990s, we now have a far clearer view of Anselm’s logic in general and especially of his theory of paronymy and his philosophy of language which – according to Henry – requires an ontologically neutral reconstruction language in order to be understood correctly. Henry also clarifies Anselm’s logic of identity, his treatment of empty names, his theory of action, and his manifold use of modal terms. Henry’s results have been a source of inspiration for Douglas Walton (FS XIV 1976: 298312), Simo Knuuttila (1993; 2004) and others over the last five decades. Any further investigation will have to

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take their subsequent results into account. While the reception of Anselm’s logic and ontology is already fairly advanced, his numerous applications of mereological concepts in M 20-4, P 18, R 1, 4+9 EDIV 13 and DPSS 9+13 and elsewhere, have not yet been studied properly. They are primarily motivated by theological concerns. In M, P and R Anselm argues that God exists without any parts, which means that He is simple. In EDIV and DPSS he uses partwhole terminology in a negative way. He struggles to show that in God there is nothing but unity, except where there is no relational opposition (ubi non obviat relationis oppositio) in the Trinitarian sense. These descriptions, informal as they are, still await scholarly reconstructions. A creative reconstruction of Anselm’s Nile-analogy in EDIV 13 has been suggested by Christopher Hughes (1989). It is based on mereological co-composition and therefore seems to conflict with divine simplicity. Bibliographical remarks. For an over-

view of the current state of scholarship (especially concerning truth, freedom, justice etc.) consult The Cambridge Companion to Anselm edited by Brian Davies and Brian Leftow (2004) and Sola ratione edited by Stephan Ernst and Thomas Franz (2009). With Dom. Schmitt’s critical edition of Anselm’s Opera Omnia (1968), A Concordance to the Works of St Anselm edited by Gillian Evans (NY 1984) and a historical Portrait in a Landscape by Sir Richard Southern (1990) we are well equipped for the next steps. Also

helpful and therefore worth mentioning are translations by Jasper Hopkins and Herbert Richardson (I-IV, 1974-76), by Hopkins (1986), by Davies and Evans (1998), by Ian Logan (2009) and Thomas Williams (Indianapolis 2007). Last but not least the translations of Anselm’s letters by Walter Fröhlich (1990 and 1993) should be mentioned. See also > Accidents, Action, Boethius, Causation, God, Medieval Mereology, Syntax, Leśniewski, Substance, Syntax. References and further readings

Campbell, R., 1976, From Belief to Understanding, Canberra: ANU. Knuuttilla, S., 1993, Modalities in Medieval Philosophy, London: Routledge. Knuuttilla, S., 2004, “Anselm on Modality”, in: Davies, B. and Leftow, B. (eds), The Cambridge Companion to Anselm, Cambridge UP, 111-131. Leftow, B., 2004, “Introduction” (with Brian Davies), in Davies, B. and Leftow, B. (eds), The Cambridge Companion to Anselm, Cambridge UP, 1-4. Reinmuth, F.; Siegwart, G.; Tapp, C. (eds.), 2014, Theory and Practice of Logical Reconstruction: Anselm as a Model Case, Logical Analysis and History of Philosophy 17, Muenster: Mentis. Jürgen Ludwig Scherb

ARISTOTLE’S THEORY OF PARTS

Aristotle’s Theory of Parts At first glance Aristotle does not make parts and wholes fundamental in his theory: we think rather of substance and accident, matter and form, potentiality and actuality. However, in the course of investigating the structure of substances, he came to make some distinctions about types of parts. These distinctions, coupled with his remarks on parts elsewhere came to form the basis of standard medieval doctrine of parts and wholes and his own account of the structure of substance. In general Aristotle views parts as respects or aspects of objects. In the Sophistical Refutations, he discusses fallacious inferences involving parts and wholes chiefly when he discusses the fallacy of secundum quid ad simpliciter. (On the origin of this fallacy, cf. Plato, Sophist 256 a 11-b 4; 257 a 4-5, and Simplicius, in Phys. 238,22-239,7) Here there is an inference from asserting a predicate of a subject in a certain respect to asserting it without qualification. He gives the examples: ‘Αn Ethiopian is white with respect to his teeth; therefore an Ethiopian is white’ (167 a 11-4); ‘This shield is half white and half black; therefore it is white and black, and hence white and not white’ (167 a l9-20). Aristotle says that this fallacy concerns not the language but the things being talked about (166 b 20-4). Thus the fallacy arises from improperly taking a property of the part to be a property of its whole. Half of the surface of the shield is white, and half black, but the whole surface is not both white and black.

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Aristotle himself accepts that a man, the whole, has a certain color, say, blackness, because a part of him, his skin, is black (Physics 210 a 29-30). So such paralogisms arise not from the mere inference that a property of a part is a property of its whole, as some such inferences are valid, but from an improper inference. However, most instances of the fallacy of secundum quid ad simpliciter do not obviously concern parts and wholes. For Aristotle takes as his cardinal examples of this fallacy cases like ‘This is not a man; therefore this is not’ (167 a 1-3; 180 a 32-4). Indeed he names the fallacy the one that arises from “saying simply or without qualification what is said in a respect or at some place or time or relative to something” (166 b 23-3; 166 b 37-8; 180 a 23-4). Aristotle describes the entire class of propositions containing qualifications in some respect or another in these very terms (Prior Analytics, 49 a 27-8; see Reduplication entry). Accordingly Aristotle employs a wide sense of ‘part’, following ordinary language use, where any qualification may be said to speak of “a part” of what is being talked about. Aristotle often uses the expression ‘qua’ or ‘in virtue of’ (κατά) to mark such qualifications. Aristotle admits that in a general sense accidents might be said to be parts of their substances (Physics 210 a 34b 5; Simplicius, in Phys. 552,18 ff.). However, he holds that in a more narrow sense an accident is not a part of its substance. Aristotle says that an accident “…is in something not as a

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part, and cannot exist separately from what it is in” (Categories 1 a 24-5). How then do accidents differ from parts, strictly speaking? Accidents cannot exist apart from their substances. Perhaps parts can exist apart from their wholes, even if they no longer will be called “the parts of” that substance. Thus a jar of wine is a whole, having the jar and its liquid contents as its parts. Aristotle remarks that when the wine is separated from the jar, they are no longer parts of the whole. They still exist of course as a jar and as wine. This reading agrees with what Aristotle says about the parts of animals being substances in their own right, as discussed further below (Categories 8 a 15). Aristotle continues to connect his doctrine of parts to the ‘qua’ locution in his philosophical lexicon where he distinguishes stricter senses of ‘part’ (Metaphysics V.25). (As is common in this lexicon, the senses often overlap in extension so that different senses can apply to the same example). In one sense, (1) a part is what is taken away or abstracted from a quantum qua quantum, as two is a part of three (1023 b 12-5; 1052 b 17-22). Again, (2) a part is a quantum measuring the whole quantum. In this sense of ‘part’, two is not a part of three; but presumably three would be a part of three. Aristotle considers numbers to be discrete quanta with no common boundary (Categories 4 b 2531). Thus, while two is a part (or “portion”, µόριον) of three in the sense of being able to be subtracted from it, two and one, being discontinuous, do not make up a continuous

whole. On the other hand, in the second sense, three could be part of a stick being three cubits in length (Metaphysics 1052 b 31-5). Or, perhaps, as Aristotle goes on to distinguish wholes composed of potential parts from those composed of actual parts, the difference between the first two senses is that an actual two is a part of three in the first sense while only a potential two is a part of three in the second (1023 b 32-4). Alexander explains the point of the ‘qua’ phrase in ‘quantum qua quantum’ thus: “Now he has added ‘qua quantum’ because a quality also can be abstracted from the quantum, like heat or whiteness or sweetness, but these are not parts, because they are not abstracted from it as a quantum: for not by the abstraction of these is it lessened in virtue of quantum” (In Metaph. 423,26-9). Again, (3) Aristotle recognises nonquantitative parts (or “portions”) such that the various species are parts of their common genus. Also (4) those things into which a whole is divided or from which it is composed are its parts. Aristotle allows the thing that is composed or is to be divided to be either the form or what has the form. A case of the latter is the bronze sphere, which has the bronze, its matter, and the sphere, its form, as parts. Aristotle does not give here an example of the parts of a form, but presumably these appear in its definition. E.g., ‘snubness’ is defined as ‘concavity in a nose’; here the concavity and nose would be parts of snubness (Cf. 1034 b 32-1035 b 1). This example seems much like the last sense that Aristotle distinguishes:

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(5) the constituents in the account (λόγος) of what a thing is are its parts. Here the genus will be a part of its species, as animal is part of the species goat. (Perhaps the difference with the prior sense comes from ‘nose’ not being strictly a constituent, the genus or differentia, of the definition of ‘snubness’. Cf. 1023 b 17-25). Aristotle likewise distinguishes senses of ‘whole’ in Metaphysics V.26 in terms of the kinds of parts constituting it or the sort of relation holding among the parts or their mode of existence (he has more extensive discussions of these distinctions of wholes also when he discusses ‘one’ in V.6 and X.1.). (1) An organic unity is “that from which none of its natural parts is absent”. Thus a dog or a tree is a whole having its limbs and organs as parts. All these parts naturally move together when the dog moves (1023 b 26-7; 1016 a 5-6; 1052 a 22-8). (2) A container or unifier is either (a) as a genus contains its species ready to be differentiated, as the genus animal contains the species dog, goat, and fox (1023 b 29-32; 1016 a 24-32), or (b) the unity given by a vessel containing a quantity of stuff so as to give it definite limits, as a band unifies a bundle of sticks or as a jar contains the wine in it (1016 a 1; Physics 210 a 30-2). Aristotle says that such containers are wholes “especially when (the parts) are contained in potency but, if not, in act; more so by nature than by art” (1023 b 33-4). The portions of wine in the jar are different usually only in potency. Such wholes are continuous and have continuous parts with a common boundary; the portions of wine in the

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jar become identifiable only when they are taken from the jar. In contrast the bundle of sticks is a whole having its parts, the sticks, naturally distinct in actuality and continuous only artificially via the band tying them together (1052 a 19-22). Other wholes are discrete and have parts not having a common limit: olives scattered at the bottom of a jar or numbers unified by addition. Again a jar of wine or at any rate of water is a natural whole, whereas a bundle of sticks is an artificial whole. Aristotle takes the natural as more genuine wholes than the artificial, and prefers the continuous wholes having only potential parts to the discrete wholes having actual ones. (3) Some quantitative wholes have their parts in a definite order. Some of these are continuous, like lines, planes, and solids, which have positional order. Others are discrete, like numbers, verbal utterances (λόγοι), and temporal sequences (Categories 5 a 15-37). (4) Other quantitative wholes have their parts in no definite order. Such things are totalities or sums (πάντα). Here when the parts are transposed the whole remains the same: thus for pieces of wax in a pile or a coat however it be folded. (5) Water and numbers can be called totalities but not wholes, except metaphorically. Aristotle claims this perhaps because such totalities have parts only potentially, via an arbitrary division. These things are one qua being indivisible in their species or form (1016 a 21). Totalities are composed of homogeneous parts of similar material and

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have names that are mass terms, like ‘water’, ‘flesh’ and ‘rice’. Here the inference holds that the whole is made of certain material from which the parts are being made of . If a portion of this liquid is wine, then this liquid is wine. (Simplicius, In Phys. 551,323: “Also every genus is predicated synonymously of all the species, but the whole only of the homoimeres, and of those not in virtue of being a whole.”). This does not hold for wholes typically signified by count nouns and having not homogeneous but rather heterogeneous continuous parts with a definite structure: if a portion of this goat is flesh, it does not follow that this goat is flesh; a part of a syllable is not a syllable (Metaphysics 1014 a 26-31). Again only wholes and not totalities can suffer mutilation. Here a part is removed while the substance is preserved. Such a part must not be merely accidental but a functional part of the whole, and typically an extremity that cannot grow back. So a man is mutilated if he loses his hand, but not if he loses his spleen or if he has his head shaved (Metaphysics V.27). In distinguishing wholes from totalities or sums, Aristotle is working out puzzles raised by Plato (Theaetetus 204 a 11; 204 e 8-10). Socrates gets Theaetetus to say that a whole arises out of the parts and is different from the parts. Yet wholes are not sums. The parts of a sum are still present as its elements, whereas the parts of a whole no longer are distinct: an animal is a whole composed of parts but is not merely the sum of its parts (204 a 7-9). But then Socrates gets Theaetetus to agree that the whole

and the sum are the same, since the parts are parts of the whole as well as of the sum (205 a 1-7). Socrates has Theaetetus conclude that the letters are the only parts of the syllable. But then, given that there is no account of the letters since they are simple, there will be no account of the syllable (205 a 11-c 2). Still we ought to have clearer knowledge of the elements than of their compound (206 b 6-9). Again, in the dialectical exercise of the Parmenides, it is argued both that the one is not a whole and does not have parts, for then it would be many, and that the one has parts since it is one being, i.e., it is both being and one, which are parts of a whole (137 d; 142 d 1-5; cf. Physics185 b 11-6). To allow for compound unities that are both one and many in different ways, Aristotle recognises both sorts of parts: his totalities are compounds that are just sums of their parts; his wholes proper are organic unities that are more than the sums of their parts. Wholes have a definite structure; totalities do not (Metaphysics 1044 a 45). The structure of a substance is its form; what instantiates that structure is its matter. Words have letters as their material parts but are not merely heaps, not merely the sum of their parts. The structure of a word causes the letters to have a certain order. This structure is not a part in the way that the letters are parts, but still is a constituent or causal principle of the word (Physics 195 a 16; Metaphysics 1013 b 17). For if that which gives the letters their structure were a part of the same type, then yet another structure would be needed in order to

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give a structure to that first structure and the letters: an infinite regress, like the Third Man argument. To avoid the regress, Aristotle appeals to his doctrine of wholes (Metaphysics 1041 b 19-22). Now Aristotle admits that perceptible substances have their matter and form as their parts (1034 b 34-1035 a 27). Thus the bronze statue is composed of the bronze and its shape, and a woman of body and soul. At least at times Aristotle recognises intelligible matter, and then would apply the same analysis for the circles and numbers of mathematics (1036 a 2-5; 1036 b 33; 1038 a 3-9; 1035 b 1; Frede and Patzig 1988, Vol. 2, 195-6). However Simplicius suggests that the whole is the totality of its parts for intelligible substances having no partitioning while it is not that totality for corporeal things (In Phys. 560,32-561,10). Aristotle takes the form also to have parts, as presented in its definition. These are not material parts but elements of the formula of the essence (1030 a 7-20). Here the genus and differentia are part of the species: rational and animal are parts of the species man. Such parts are prior to the definition, while the material parts are not prior but posterior to it (1035 b 3ff). It is not by coincidence that Aristotle brings up the paradigm of letters and words at the end of his account of substance (Metaphysics 1041 b 1133). Aristotle is applying his account of parts and wholes to substances. He runs the discussion of universals and the parts of a definition in tandem

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with the discussion of material parts perhaps because in both cases the parts as such, sc., under that actual description qua parts, cannot exist separately and independently, but only potentially. When they do exist independently, they do so only by becoming the constituents of some other actual thing, itself real only as an individual substance. So universals constitute the reality of substances, too, but do not exist independently as separate Platonic substances (cf. Physics 184 a 25-6; 187 b 15-6). In this way Aristotle takes individual substances as the wholes of their parts, matter and form. Both the matter and form of an individual substance can be analysed themselves as wholes in different ways. The form is a whole with a certain structure from the components of its definition (Metaphysics 1030 b 28-1031 a 1; On Interpretation 21 a 15-7). Again, the matter of an individual substance like a goat has material parts in a certain order. These parts are wholes in their own right: its head, legs, heart and lungs. Those parts themselves may be structured wholes: in this way the head has eyes, horns and nose as its parts. Yet ultimately we get to more basic parts of the goat that are totalities: its flesh and bones (1070 a 1920; 1035 b 26-7; 1036 b 3-4). Likewise the formal parts may themselves be wholes: an animal is a perceptible mobile substance, and we continue to get parts of these parts until we reach simple elements (Posterior Analytics I.19-21). The formal parts of an animal exist in re only as actualities of the substance and cannot exist separately in their own right. In contrast,

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the material parts of animals can exist as substances in their own right, while existing as parts of the animal only in their fully actualised state. Thus ends Aristotle’s account of universals. We can see an application of this solution to the problem of universals if we work out what Aristotle says about the parts of animals. Aristotle says that the (material) parts of the substance must be stated with reference to their functions. Thus the definition of a hand must include its functioning as a hand. For this the hand must be part of a living body. Hence Aristotle says that a hand severed from the body is not a real hand but a hand in name only (Parts of Animals 640 b 35-641 a 5; On the Soul 412 b 19-22). In the Metaphysics he insists upon what Ackrill has called the homonymy principle: a functioning part and a nonfunctioning part of an animal are homonyms (1035 b 23-3; Ackrill 1973: 125-7; Williams 1982: 113). On the other hand, Aristotle states in the Categories 8 a 15 that the parts of animals are substances. Moreover, in Metaphysics 1028 b 8-12 Aristotle considers whether parts of substances are indeed substances, referring to parts of animals and parts of the heavens, like the stars, and points out that they are commonly agreed to be substances (1042 a 6; 24). Moreover, in Metaphysics VIII Aristotle distinguishes the matter of a substance from its form. The matter of the house consists in the boards and bricks, as the hand and the foot are parts of an animal (1043 a 8-9; 1043

a 14-6). He says that their being will be defined by many qualities of the mixing of matter like hardness and softness (1042 b 15-31), and it seems clear that Aristotle does not describe here heads and hands as part of animals but as substances having a certain form and matter, just like the parts of houses. (Bostock 1994: 257) Thus Aristotle’s position on the parts of animals is quite puzzling. On the one hand, he states clearly that they are substances. On the other hand, he states that a part of an animal, like a hand, that does not function as a hand is a hand in name only. We have here two conceptions of parts: one material, one functional. (1) A part of an animal is composed of matter, like its flesh and bones, and persists through time. Sometimes it happens to function as a hand; sometimes it does not; sometimes it cannot; at some time it perishes. In his biological works Aristotle goes into some detail about the embryological development of parts of animals like hands and eyes, and talks of them as hands and as eyes before they can function. (2) Yet a part is a part of a whole, and so its existence as a part seems relative to that whole. In the Categories Aristotle brings up this point when discussing relations. He seems to deny that parts of animals are relatives but does not offer a clear solution. (8 a 13-29; 7 a 17-25) The names for parts of animals thus have a certain ambiguity. They can be taken to signify fully actualised, functioning parts of the whole organisms, or substances that have the potential for such functioning and at

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times come to actualize that potential. Aristotle recognises such ambiguities. Indeed, he criticises his predecessors for not taking explicit notice of them (On the Soul 425 b 26426 a 6). In his biological works, he regularly describes as “parts” and as “eyes”, “hands” etc. those portions of the embryo and child that have the capacity to develop the actual functions of those parts but do not have them yet. Puppies are born blind and unable to reproduce, and yet have “eyes” and “genitals” (Generation of Animals 779 a 27-8; 744 b 11-27). Still, even though such parts are not yet functioning, they develop for the sake of what they would become, and thus their being and becoming is tied to the whole animal – but as their final cause, not because they are relata (Lennox 1997). In short, these developing organs are the substances and potential parts of the animal; these substances are its actual parts when fully functioning (Simplicius, in Phys. 810,25-8). To the extent that these substances are fully actualised, they constitute parts of the animal body and are not called substances in their own right. Thus qua part, the hand is not a substance in its own right, although qua itself and per se it is a substance. Thus, Michael Frede and Günther Patzig 1988, Vol. 2, pp. 169-70, describe parts and wholes as relata. As Aristotle himself says: “For it is not a hand in any state that is a part of man, but the hand which can fulfill its work, which must therefore be alive; if it is not alive it is not a part” (Metaphysics 1036 b 30-2). That is, ordinarily we call something a hand only if it is

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a functioning hand: “…it is called its part when it fulfills its work. It will do its work and be called a part of a man when it is disposed by nature: when it is not but is e.g. inanimate it neither does its work nor is it called a part” (Ps.-Alexander, In Metaph. 514,36-39; 534,27-33). A material part of an animal is then a substance itself with a career independent of the whole animal, although the form of the substance causes it to develop towards the goal of becoming an actual part. That cause, the final cause, comes from the whole animal itself. Hence Aristotle says in this case the whole comes before the part, as it causes certain materials to start forming a hand. The chunk of material is a potential part, a potentially functioning hand, before becoming an actual one. In contrast, a formal part of a substance – like matter and form, or the constituents of a definition – exist prior to the whole, since they are the cause of the whole. Yet formal parts never exist as substances in their own right, as Plato or the atomists would have it. Rather, they exist actually as constituents of other substances and only potentially as constituents and causes of the substance that is under development. See also > Activity, Homeomerous, Plato, Reduplication, Syntax, Substance, Universal. References and further readings

Ackrill, J. L., 1973, “Aristotle’s Definitions of Psyche”, Proceedings of the Aristotelian Society 73.

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Alexander of Aphrodisias, 1891, In Aristotelis Metaphysica Commentaria, ed. M. Hayduck, Berlin: Felix Meiner Verlag. Alexander of Aphrodisias, 1993, On Aristotle’s Metaphysics 5, trans. & comm. W. E. Dooley, Ithaca: Cornell University Press. Aristotle, 1986, The Complete Works, ed. J. Barnes, Princeton, N.J.: Princeton University Press. Bäck, A., 2014, Aristotle’s Theory of Abstraction, Dordrecht: Synthèse Library, Springer Verlag. Bostock, D., 1994, trans. & comm, Aristotle: Metaphysics Books Z and H, Oxford: Oxford University Press. Burkhardt, H., 1991, Handbook of Metaphysics and Ontology, Vol. 2, Munich: Philosophia Verlag. Caujolle-Zaslaavsky, F., 1980, “Les relatifs dans les Catégoires,” in Concepts et catégoires dans la penseé antique, ed. P. Aubenque, Paris: J. Vrin. Cohen, S. M., 1992, “Hylomorphism and Functionalism,” in Essays on Aristotle’s De Anima, ed. M. Nussbaum & A. Rorty, Oxford: Oxford University Press. Frede, M. and Patzig, G., 1988, ed., trans. & comm., Aristoteles Metaphysik, 2 Vols., Munich: Akademie Verlag. Kirwan, C., 1971, trans. & comm., Aristotle’s Metaphysics Books ∆ and E, Oxford: Clarendon Press. Koslicki, K., 2007, “Towards a NeoAristotelian Mereology”, Dialectica 61: 127-159.

Lennox, J., 1997, “Material and Formal Natures in Aristotle's De partibus animalium”, in Aristotelische Biologie. Intentionen, Methoden, Ergebnisse, eds. Sabine Föllinger and Wolfgang Kullmann, Stuttgart: Franz Steiner Verlag. Mayhew, R., 1997, “Parts and Wholes in Aristotle’s Political Philosophy”, The Journal of Ethics 1. Morales, F., 1994, “Relational Attributes in Aristotle”, Phronesis 39 (3): 255-274. Simplicius, l882, In Aristotelis Physicos Commentaria, Berlin: F. Meiner Verlag. Wedin, M., 2000, Aristotle’s Theory of Substance, Oxford: Oxford University Press. Whiting, J., 1992, “Living Bodies”, in Essays on Aristotle’s De Anima, ed. M. Nussbaum & A. Rorty, Oxford: Oxford University Press. Williams, C. J. F, 1982, trans. & comm., Aristotle’s De Generatione, Oxford: Oxford University Press. Allan Bäck

Aristotle’s Theory of Wholes Introduction. Aristotle wrote about

parts and wholes interspersed in most of his works, which shows that the problem was a source of constant concern to him. The resulting theory was historically successful: indeed, its influence lasted for more than twenty centuries (Henry 1991). However, none of the best-established

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reconstructions of Aristotle’s thought published in the last 200 years has contained a chapter explicitly dealing with his theory of parts and wholes. This would seem to suggest that the reconstruction of Aristotle’s theory of wholes and parts leads to some deep underlying problems in Aristotle’s framework. Indeed it appears that Aristotle failed to clarify the connections among the theory of categories, the role of the principles (the tension between the dynamics of actuality and potency and the principle of the one being the main sources of difficulty), the theory of whole and parts, and the theory of the continuum. In the end, Aristotle decided to subordinate both wholes and continua to the dialectics of actuality and potentiality. He therefore asserted that whenever the whole is actual, its parts must be potential. Similarly, whenever the continuum is actual, its points are potential. And vice versa in both cases. The main reason for these claims was that “no substance is composed of substances” (Metaphysics 1041 a 5). Similarly, no whole is composed of other wholes. Therefore, parts of wholes are not wholes themselves. “Evidently even of the things that are thought to be substances, most are only potencies – both the parts of animals (for none of them exists separately; and when they are separated, then too they exist, all of them, merely as matter) and earth and fire and air; for none of them is a unity, but as it were a mere heap” (Metaphysics, 1040 b 5-9). The resulting picture has an apparent coherence to it, but the overall structure is highly unstable: as soon as the

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slightest change is made to the theory of substance, the theory of wholes and parts or the theory of the continua, the entire framework totters. (But reconfigurations of the elements of the framework are possible – Brentano, one of the most committed Aristotelians of the nineteenth century, relied explicitly on the theory of parts and wholes as the framework for both ontology and epistemology, but dramatically modified the categorial frameworks of these disciplines (Poli 2004, Albertazzi 2006); similarly, Brentano’s (and for that matter Leibniz’s) innovations of the dialectics between continua and their elements on the one hand, and between wholes and their parts on the other, gave rise to genuinely new visions of both, continua and wholes). In the following it will be worked out how Aristotle understands whole in the context of his framework. Aggregates and wholes. Aristotle distinguishes between pan (aggregate) and holon (whole) to mark a fundamental ontological difference. Both belong to the category of quantity, and both are distinguished by the position (thesis) of the parts. Pan is a quantity in which the positions of the parts (within the whole) do not produce difference (Metaphysics, 1024 a 1). In the case of pan, that is to say, the positions of the parts can be modified without changing the ontological nature of the aggregate. Holon, by contrast, is a quantity in which the positions of the parts help characterise the whole: if the positions of the parts are altered, the ontological nature of the whole changes.

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Two cases can be distinguished: in the first, only the position of some parts is important; in the second, the positions of all the parts are important for the whole. In later scholastic terminology, these two cases were called the ‘integral whole’ and the ‘essential whole’. The latter was exemplified at the formal level by definitions and at the material level by the second substances or natural kinds. In the case of homogeneous bodies, aggregates and wholes are ruled by different relations to their parts. The main distinction is between continuous bodies and contiguous bodies. Contiguous bodies contain parts whose boundaries are distinct from each other, while continuous bodies contain parts whose boundaries are fused together (continuous). This means that all the parts making up an aggregate are in actuality, while those making up a whole are only in potency. “If a mass [= body] is continuous and homogeneous, its parts are only potentially in places-proper, but if they were so divided as to be in mutual contact (as if in a heap), they would have actualised places-proper” (Physics, 212 b 4-6). Whole, one, and nature. Aristotle de-

fines a whole (holon) as something that does not lack any of the parts that by nature it should possess and which contains things in a manner such that they constitute a unity (Metaphysics, 1023 b 26-28). Clarification is therefore required of the concepts of by nature and unity. The former concept refers primarily to the living world: for Aristotle, the ob-

jects that are by nature are the organisms of the biological world. What is meant by saying that wholes are unities, or that they contain things in a manner such that they constitute a unity? In order to grasp the sense of unity relative to the concept of integral whole, it is helpful to start from the concept of one per se, and in particular from the thesis that something is one per se when it is a continuum (Metaphysics, 1015 b 37; 1052 a 18 ff.). The Aristotelian concept of continuum is the result of a process of construction which moves through various stages, beginning with the concept of the consecutive, passing to the contiguous, and only in the third stage arriving at continuous. These are the definitions: consecutive is whatever does not display any intermediate of the same kind between itself and that of which it is consecutive; contiguous is the consecutive in contact; one finally arrives at the continuous “when the limits of two things, whereby they touch each other, become one alone” (Physics 227 a 11-12). Note that for Aristotle, in the case of two contiguous objects which become a continuous object, the boundary between the given objects belongs to both of them. A further note is helpful: the construction of the continuum in stages shows that aggregates as contiguous objects are not proper wholes. Besides the three stages just discussed (of consecutiveness, contiguity, and continuity), Aristotle adds a further characteristic, that of solidarity: an object has solidarity when the

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parts move in the same instant and in the same direction as the whole. From this it follows that, for Aristotle, a body whose parts have perfect solidarity is more continuous than a body whose parts do not have (perfect) solidarity. Consequently, he conceives the continuum and the whole as one because their movements are indivisible (Metaphysics 1052 a 35). Composing, dividing, and being in wholes. The problem of parts and

wholes is but one aspect of the more general problem of composition (and of the dual problem of division). Aristotle distinguishes at least five different types of composition, viz. the composition of (1) substance and accident (inherence), (2) the composition of genus and species (the type-of hierarchy required by definitions and essences), (3) the composition of matter and form, (4) the composition of act and power, and (5) the composition of part and whole. The first two types of composition pertain to the theory of categories; the third and fourth types of composition concern the category of substance and pertain to first philosophy. The last type of composition is apparently generic and may deal with any of the former, with a slight bent in favor of the analysis of substance. Analysis of the dual procedure of dividing yields a slightly different

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picture. The procedure by composition goes from parts to wholes; the dual procedure by division goes from wholes to parts. We thus obtain six different types of parts: (A) what can be obtained by dividing a quantity; (B) what can be obtained by dividing a matter; (C) what can be obtained by dividing a form; (D) what can be obtained by dividing the synholon of matter and form; (E) what can be obtained by dividing a whole; (F) what can be obtained by dividing a definition. The connection between the two procedures is far from smooth. (2) and (F) can be taken as referring to properly dual procedures, and likewise (3) and (D), and (5) and (E). The other cases are less straightforward. Some light may be shed on the matter if we recall that, according to Aristotle, the theory of parts is but a chapter of a wider theory dealing with the different ways in which “one thing is said to be ‘in’ another” (Physics, 210 a 15 ff). Here Aristotle distinguishes eight different cases: (1) “as the finger may be said to be ‘included in’ the hand or, to put it more generally, the part ‘in’ the whole”; (2) “the whole is said to ‘consist in’ the full tale of its parts; for there is none of the whole ‘outside’ of the collective parts”; (3) as man is included in animal, “or more generally the species ‘in’ the genus”; (4) as the genus “is ‘included in’ the definition of the species”; (5) as the form in the material;

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(6) as in “‘wherever’ or ‘in what place soever’ the prime initiative (or efficient cause) resides; (7) as ‘in the contemplated end’ (or final cause); (8) “But the primary sense, from which all these are derived, is that in which we say that a thing is ‘in’ a vessel, or more generally ‘in a place’” (Physics 210 a 15-24; see also Metaphysics 1023 a 23-25). The first four cases come in couples: (1) is dual to (2), and (3) to (4). The other four cases do not present dual versions. (5), (6) and (7) refer to the causes: (5) to formal and material causes, (6) to efficient causation and (7) to final causation. One may therefore claim that the theory of the causes can be framed by a properly enlarged theory of parts and wholes. Aristotle explicitly states that all the cases so far seen can be uniformly understood as “parts are causes of the whole, and the premises are parts of the conclusion, in the sense of that out of which these respectively are made; but of these things some are causes in the sense of the substratum (e.g. the parts stand in the relation to the whole), others in the sense of the essence—the whole or the synthesis or the form” (Physics, 195a 17-21). Finally, (8) manifests the focal meaning of ‘being in’ as being in a locus. This basic meaning has sometimes been called locative copula. Proper understanding of the Aristotelian theory of parts and wholes therefore requires finding a way to coordinate the three contexts we have mentioned, namely those of composition, division and location.

Temporal relations of wholes. All of

the cases mentioned so far can be further diversified by determining whether the parts are simultaneous with, or antecedent, or posterior to their wholes, as specified in Metaphysics Z 10. Basically, conceptual parts (those composing the definition (i.e. the essence) of the whole are prior to the whole: “All constituents which are parts of the formula, and into which the formula can be divided, are prior to their wholes” (1035 b 3). On the other hand, material parts are either posterior to or simultaneous with their whole; those “into which the whole is resolved as into matter, are posterior to the whole” (1035 b 12), while indispensable parts connect properly material and formal parts and therefore are neither anterior nor posterior: “some parts are contemporary with the whole: such as are indispensable and in which the formula and the essence are primarily present; e.g. the heart or perhaps the brain” (1035 b 25-26). “A part, then, may be a part of the form (by form I mean essence), or of the concrete whole composed of form and matter, or of the matter itself” (1035 b 32-33). One may also add that “matter and form are intrinsic to the thing that is in the place” (Physics, 210 b 31, tr. amended with the content of the note) thereby connecting the principles of the substance with the locative copula mentioned above (i.e., the 8th meaning of being in). To prevent misunderstandings, it should be stressed that for Aristotle substance does not mean matter. In

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Metaphysics Z 10 he tells us that “the parts of soul are prior, either all or some of them, to the concrete ‘animal’ … and the body and its parts are posterior to this … and it is not the substance but the concrete thing that is divided into these parts as its matter” (Metaphysics 1035 a 18-22). Wholes with homeomerous parts. The

case of homogeneous bodies has been already mentioned. The theory of simultaneous parts articulates the difference between homogeneous and non-homogeneous parts. The former are usually called homeomerous parts and essentially correspond to tissues, whereas the latter are termed anhomeomerous and essentially correspond to organs. Their differences are analysed in Parts of animals, where Aristotle claims that “three sorts of composition can be distinguished. (1) First of all we may put composition out of the elements … (2) The second sort of composition is the composition of the ‘uniform’ substances found in animals (such as bone, flesh, etc) … (3) The third and last is the composition of the ‘non-uniform’ parts of the body. Such as face, hand and the like” (646 a, 12-24). These three forms of composition come in a row: uniform parts “are for the sake of” non-uniform parts (646 b 11-12) while elements “must exist for the sake of the uniform substances” (646 b 5-6). The two main axioms for wholes. Two main axioms rule material parts and

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wholes. The underlying intuition is that the destruction of (antecedent and simultaneous) parts implies the destruction of the whole, while the destruction of the whole does not imply the destruction of its parts: “the whole perishes when the parts do so, and it does not necessarily follow that the parts also have perished when the whole has perished” (Topica, 150 a 32-35). See also > Aristotle’s Theory of Parts, Biological Parts, Brentano, Collectives and Compounds, Homeomerous and Automerous, Piece, Substance, Totum Potentiale. References and further readings

Albertazzi, L., 2006, Immanent Realism. Introduction to Franz Brentano, Dordrecht: Springer. Aristotle, Metaphysics, Physics, Topica, Parts of Animals from Loeb Classical Library. Henry, D.P., 1991, Medieval Mereology, Grüner: AmsterdamPhiladelphia. Poli, R., 1998, “Qua-theories”, in L. Albertazzi (ed), Shapes of Forms. Kluwer: Dordrecht, 245-256. Poli, R., 2004, “Approaching Brentano’s Theory of Categories”, in A. Chrudzimski and W. Huemer, eds., Phenomenology and Analysis. Essays in Central European Philosophy, Ontos Verlag: Frankfurt, 285-321. Roberto Poli

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ART

Art The modern concept of art was originally was derived from the latin ars and the greek techné, which can both roughly be translated as skill or ability; in the current understanding of the term, ‘art’ denotes both the activities involved in the production of the art work, as well as – prima facie at least – a distinctive (possibly contextual) feature of these activities and their products that demarcates art from other domains such as the crafts, science, or life. Art comes in different ‘genres’ which are determined by the type of product generated in artistic activity – such as painting, music, poetry, sculpture, performance and more recently happenings, ready-mades as well as hypertext based works. Though artworks are generally considered as composite units consisting of several parts, it is not clear what should be considered as the parts of an art work, how these parts are organised in the composition of the whole, and which parts, if any, contribute to the item’s status as work of art. According to an ancient yet longstanding conception, the part of an artwork that renders it art is its structure and certain properties of that structure: art works are imitations of the harmony and beauty found in nature. This idea of ‘mimesis’ can be traced back to Plato and Aristotle and persisted for many hundred years. For instance, it still figures in the ‘modern system’ art in the eighteenth century, a classical attempt to systematically distinguish

between types of art and artworks and their properties, and more specifically in Charles Batteux’ distinction from 1747 between the “fine arts” as “imitation of beautiful nature” and the “mechanical arts” (Batteux 1970). The idea of art as mimesis is today generally overtaken by representational theories of art (for discussion and outline of the main positions, see Carroll 1999: 19-57). For example, a graphic work represents an object, if its proper parts are ordered in such a way that it depicts that particular object, i.e. if an “outline shape” (Hopkins 2006; 1998) or “visible figure” (Reid 1997) can be seen in the picture. The idea of seeing-in stems from Richard Wollheim (1980 & 1987) and it has been debated since whether such depiction is really founded in a representational function of the Gestalt properties of certain parts of the artwork, and, if so, whether the representational function presupposes resemblance relations between represented and representing items (see e.g. Bantinaki 2009 and Stecker 2010: 189-196, for discussion). A somewhat different explanation of the status of art works in terms of structural aspects was proposed by A. G Baumgarten, in the middle of the 18th century. Baumgarten famously included ‘Art’ under the broader heading of ‘Aesthetics’ (Baumgarten 1983; 2007), thereby stressing the importance of perception and aesthetic properties like beauty to be central for an analysis of the experience and knowledge of art. It is with Baumgarten that aesthetics discovers itself

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as an independent academic field, though his approach is consistent with the aspects of the mimesisapproach inherited by the Greeks. For example, Baumgarten also holds that aesthetic properties of artworks emerge from an organisation of parts, if this organisation displays certain regularities such as symmetry or the ‘golden section’ (‘golden ratio’). Such measures have been used extensively in attempts to arrive at formulas governing the appearance of aesthetic qualities, but in the contemporary debate it is controversial whether symmetry principles are central to the aesthetic perception of artworks. For example, Harold Osborne (1986) holds that the use of the golden section is hardly essential to artists in creating their works, while H. J. McWhinnie (1989) argues for the opposite view. A prominent example of a modern version of the symmetry conception is Jay Hambridge’s “dynamic symmetry” (Hambridge 1968), where the Golden Ratio is supplemented with other ratios, like the Silver Ratio. In Hambridge’s arithmeticgeometrical approach, ratios are often formulated as squares and roots, from which ‘dynamic rectangles’ can be developed, delineating symmetric proportions relating to the ratios. Hambridge’s approach could be confronted with a classical objection from Edmund Burke (Burke 1958: 94, quoted in Hanfling 1992: 43), disputing the existence of regularities that could justify aesthetic judgments. While it is commonly held that most (if not all) artworks do possess some kind of structure or form, there is not

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only disagreement about the relevant aspects and functions of this structure but also about which whole the structure is a part of and what are the parts of the structure. Again inspired by Plato and Aristotle, Clive Bell (1915) developed a theory of “significant form,” a holistic approach to artworks, which states that the parts of a work of art (e.g. a painting) are the parts of the aesthetical object, which are perceived as forms and structures, like harmony or symmetry or ugliness, and must not be confused with the physical parts (e.g., the paint or canvas). These ideas were more thoroughly analysed and worked out in detail by Roman Ingarden (Ingarden 1962). Ingarden distinguishes between ‘paintings’ (physical objects) and the corresponding ‘pictures’ (aesthetical objects), arguing that the picture is not a part of the painting. Drawing on Husserl, Ingarden advances a holistic “layer-view” of artworks, arguing that artworks embody a hierarchy of “essences” or “strata”. Ingarden’s line of thought can easily be applied to other fields such as literature or music. One might question, however, whether the relationship between, e.g., poem and text, and melody and sound, respectively, can be construed analogously to Ingarden’s analysis of the relationship between picture and painting. That is, one might question whether it is reasonable to search for one mereological theory of the artwork that could capture all the different genres (poetry, music, painting) in the light of these differences.

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ART

A further development of the distinction between aesthetic and material whole introduces the idea that aesthetic aspects are emergent parts of the art work. For example, Monroe C. Beardsley identifies aesthetic qualities of artworks as “emergent perceptual regional qualities” (Beardsley 1958, pp.82-88 & 1970) and contrasts these with corresponding “local” qualities, like a section of white in a painting, which form the base of the regional qualities. That these qualities are emergent means that they should be distinguished from merely “summative qualities”, which are mereological sums of various local qualities. The point is that a change of parts (local qualities) might change some regional qualities while leaving other regional qualities intact. An example could be a tune transposed from one key to another, or a portrait painted using watercolours, and later as oil on canvas. In response to this proposal Frank Sibley (1965) argued that only a subset of Beardsley’s emergent regional qualities constitutes aesthetic properties. Sibley (1974) introduces the notion of “determinate meritconstituting properties”, which he takes to be highly specific features that emerge from the physical parts of the artwork. According to Sibley we cannot describe the aesthetically significant features of an artwork by using general or determinable descriptions. A line is not elegant, just because it is curved (determinable). It is elegant because it is curved in exactly that particular way (determinate). Following Peter Strawson (1966), Sibley claims that it is rarely

possible to provide informative descriptions of the physical base of the emergent aesthetic properties of an artwork exactly because such properties are individual or ‘peculiar’ to the aesthetic object. Also Arthur Danto holds that aesthetic qualities can be viewed as emergent parts of a special kind (Danto 1993: 199-200). Just as a person differs from the sum of body parts, the sum of the parts of an artwork is different from the components and aspects constituting it, in situations when it is not considered a work of art. A much discussed example is Duchamp’s 1917 exposition of a urinal, entitled Fountain. Danto points out that Fountain has many properties that the urinal lacks: “It is daring, impudent, irreverent, witty, and clever” (ibid: 93-94). A mereological analysis of Danto’s view and an alternative proposal can be found in Null 1995, where it is suggested to supplement the extensional concepts of proper and improper parthood with a notion of irregular parts While some theories of art thus explicitly or implicitly make reference to the parts of artworks, according to other current theories of art the mereology of art is irrelevant to the status of art. This view is implicit in George Dickie’s Institutional Theory of Art (Dickie 1974), according to which it is the members of the “artworld” who decide which objects should be regarded as artworks, and hence the art-making properties of objects are not intrinsic. But as Danto has pointed out, Dickie’s – quite popular – version of the institutional theory

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cannot explain what exactly motivates a member of the artworld to declare an object as an artwork (Danto 1981: 5-6).

(for further discussion of aesthetic contemplation see e.g. Rorty 1992 who also provides an overview of Aristotle’s views).

Another variety of the nonmereological, ‘relational approach’ to art claims that proper experience of artworks is contemplative, i.e. that it involves a focus on the intrinsic qualities or properties of the work, setting aside any ethical or practical relations or aspects. Aesthetic contemplation of artworks is often characterised as involving an attitude of “disinterestedness” (Kant 2006) or “psychical distance” (Bullough 1957) which arises in tandem with perceiving the artwork as something of intrinsic value, i.e., something whose properties are independent of the perceiver’s reactions. To the contrary, it is the artwork that is often said to force contemplation of beauty upon the beholder, a point that led Aristotle to compare the inevitability of the experience of beauty with the inescapable charms of the sirens (Eudemian Ethics, 1230 b 21-1231 a 7). A modern proponent of this view of contemplation is for instance C.S. Lewis who emphasises that beholders of art need to “surrender” to a work of art in order to contemplate it properly (1961, p.19). As Aristotle points out in the Poetics (1449 b 24-1450 a 24), the impression of the tragedy’s independence is supported by the fact that we perceive it as closed or “complete” (1449 b 25). Even when we experience a play or read a novel for the first time and thus do not know exactly what will happen next, we do know that the narrative will progress in the order determined by the author

A recent version of the relational approach to art replaces ‘contemplation’ with ‘interaction’ or ‘immersion,’ especially in the context of discussions on “emergent art” (e.g., internet-based works). In this way partwhole relations re-enter the discussion from within the relational approach, since emergent artworks are interactively created or co-created by the audience, which is then transformed from an audience of spectators to a community of users. Some authors have begun to develop a corresponding ‘digital aesthetics’ (se e.g. Cubitt 1998). However, so far the development of ‘digital aesthetics’ has not yet received much attention by scholars in contemporary philosophy of art. Bibliographical remarks. An impres-

sive and classic analysis of the mereological constitution of artworks can be found in Ingarden 1962. As general introduction to the central discussions see Hanfling 1992, Carroll 1999 and Kieran 2006. In Hargittai 1986 & 1989 there is several interesting papers on symmetry and the golden section, which in combination makes a fine starting-point. Concerning theories of Seeing-in and depiction, Abell & Bantinaki 2010 contains a good collection of readings. Concerning “digital aesthetics”, Cubitt 1998 is a good place to start, and a comparison with Eco 1962 is illuminating.

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ART

See also > Collectives and Compounds, Emergence, Fiction, Gestalt, Husserl, Ingarden, Perceptual Whole. References and further readings

Abell & Bantinaki, ed., 2010, Philosphical Perspectives on Depiction, Oxford : Oxford University Press. Bantinaki, K., 2009, “Depiction”, in Davies et al, eds., A Companion to Aesthetics, 2.ed, Wiley Blackwell. Batteux, C., 1970, Les beaux arts réduits à un même principe, New York, Johnson Reprint. Baumgarten, A. G., 2007, Ästethik 12, Hamburg: Felix Meiner. Baumgarten, A. G., 1983, Meditationes Philosophicae de Nonullis ad Poema Pertinentibus, Hamburg: Felix Meiner. Beardsley, M. C., 1982, The Aesthetic Point of View. Selected Essays (ed. by Wreen et al.), Ithaca: Cornell University Press. Beardsley, M. C., 1970, “The Aesthetic Point of View”, in Kiefer et al., eds., Perspectives in Education, Religion and the Arts, 219-237. Reprinted in Beardsley: The Aesthetic Point of View, 15-34. Beardsley, M. C., 1958, Aesthetics – Problems in the Philosophy of Criticism, New York: Harcourt. Bell, C., 1915, Art (2.ed), London: Chatto and Windus. Bullough, E., 1957, Æsthetics – Lectures and Essays (Wilkinson ed.), London: Bowes and Bowes.

Burke, E., 1958, A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and the Beautiful, ed. by J.T. Boulton, London: Routledge and Kegan Paul. Carroll, N., 1999, Philosophy of Art – A Contemporary Introduction, London and New York: Routledge. Collinson, D., 1992, “Aesthetic Experience”, in Hanfling, ed., Philosophical Aesthetics, 111-178. Crowell, S. G., ed., 1995, The Prism of the Self, Dordrecht: Kluwer Academic Publishers. Cubitt, S., 1998, Digital Aesthetics, London: Sage Publ. Danto, A., 1981, The Transfiguration of the Commonplace, Cambridge Mass.: Harvard University Press. Dickie, G., 1974, Art and the Aesthetic – An Institutional Analysis, Ithaca: Cornell. Eco, U., 1962, Opera aperta. Forma e indeterminazione nelle poetiche contemporanee, Milano, Bompiani. English translation of the main parts in Eco, Umberto 1989, The Open Work (Transl. by A. Cancogni), Hutchinson Radius. Hambridge, J., 1968, Principles of Dynamic Symmetry, New York: Dover. Hanfling, Oswald, ed., 1992, Philosophical Aesthetics – An Introduction, Oxford: Blackwell. Hargittai, I., ed., 1986, Symmetry. Unifying Human Understanding, Oxford: Pergamon Press.

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Hargittai, I., ed., 1989, Symmetry 2. Unifying Human Understanding, Oxford: Pergamon Press.

Osborne, H., 1986, “Symmetry as an Aesthetic Factor”, in Hargittai 1986: 77-82.

Hopkins, R., 2006, “The Speaking Image: Visual Communication and the Nature of Depiction”, in Kieran 2006: 145-159.

Reid, T., 1997, An Inquiry into the Human Mind, on the Principles of Common Sense, (edited by D.R. Brookes), Edinburg: Edinburg University Press.

Hopkins, R., 1998, Picture, Image and Experience, Cambridge: Cambridge University Press. Ingarden, R., 1962, Untersuchungen zur Ontologie der Kunst, Tübingen, Max Niemeyer. English translation: Ingarden, Roman, 1989, Ontology of the Work of Art (transl. by R. Meyer), Ohio University Press. Kant, I., 2006, Kritik der Urteilskraft, Hamburg: Felix Meiner. English translation: Kant, Immanuel, 2000, Critique of the Power of Judgement (Transl. by E. Matthews), Cambridge: Cambridge University Press. Kiefer et al., eds., 1970, Perspectives in Education, Religion and the Arts, Albany: Suny Press. Kieran, M., ed., 2006, Contemporary Debates in Aesthetics and the Philosophy of Art, Malden: Blackwell. Lewis, C.S., 1961, An Experiment in Criticism, Cambridge: Cambridge University Press. Lopes, D.M., 2006, “The Domain of Depiction”, in Kieran 2006: 160-174. McWhinnie, H. J., 1989, “Influences of the Ideas of Jay Hambridge on Art and Design” in Hargittai 1989: 10011008. Null, G.T., 1995, “Art and Part: Mereology and the Ontology of Art”, in Crowell 1995: 255-275.

Rorty, A.O., ed., 1992, Essays on Aristotle's Poetics, Princeton: Princeton University Press. Sibley, F., 1965, “Aesthetic and NonAesthetic”, Philosophical Review 74: 135-59. Reprinted in Sibley 2001: 33-51. Sibley, F., 1974, “ Particularity, Art and Evaluation”, Proceedings of the Aristotelean Society, Supp. Vol. 48, 1-21. Reprinted in Sibley 2001: 88103. Sibley, F., 2001, Approach to Aesthetics, (edited by Benson et al.), Oxford: Clarendon Press. Stecker, R., 2010, Aesthetics and the Philosophy of Art, 2., ed., Rowman & Littlefield. Strawson, P.F., 1974, Freedom and Resentment and Other Essays, London: Methuen. Strawson, P.F., 1966, “Aesthetic Appraisal and Works of Art”, The Oxford Review 3. Reprinted in Strawson 1974: 178-188. Wollheim, R., 1987, Painting as an Art, London: Thames and Hudson.

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ARTIFACT

Wollheim, R., 1980, Art and its Objects, (2. ed), Cambridge: Cambridge University Press. Peter Storm-Henningsen

Artifact An artifact is an enduring object deliberately made by one or more agents in order to perform some function or fulfill some purpose. Almost all artifacts relate to mereology in several ways: artifacts are in many respects the domain of applied mereology. The making of artifacts typically involves some of the following mereological operations: (1) separation for use of a part from a larger object; (2) removal of parts from a larger object to make it suitable; (3) formation of an individual object from a mass of material; (4) joining together two or more objects to make a larger one; (5) reshaping an object, changing the relative spatial positions of its parts. Simple examples are: (1) cutting a branch from a tree to make a walking stick; (2) stripping its leaves and twigs; (3) casting molten metal to make a ferrule; (4) fitting the ferrule to the stick; (5) bending the other end to serve as a handle. These types of operation, repeated in numberless variants, are employed in making anything from chipped hand axes to aircraft. An artifact in use has parts which play different roles in its functioning. Some are essential to the use, others may be ornamental. It is the proper-

ties of the parts and their interrelations that render them fit for their roles. A knife blade should be sharp and strong and firmly fixed to its handle, a table top should be flat, supportive and stably mounted, a car’s wheels should be mounted on axles and so distributed as to ensure stability and load-bearing. Some artifacts are fixed and static: a road sign, a dam. Many are dynamic, whether internally, as a motor, or designed to move, as a door, or a canoe. Internally dynamic artifacts have parts designed to move or be moved in relation to one another: a book is to be opened and its pages turned; the pistons, cranks and wheels of an engine are to move in specifically foreseen ways. The internal configuration of a dynamic artifact’s parts may repeat cyclically like a clock or an engine, or it may follow an asymmetric course, as the expenditure of propellant in a rocket. Materials necessary to operate an artifact but not considered parts of it are consumables: they may include fuel, lubricant, air and water. Likewise not parts but often necessary to an artifact’s operation are: a medium, such as water for ships; external energy sources, such as wind for windmills, horses for carriages, an electricity supply for a building; and operators. The rowers of a galley, an aircraft’s pilot, the wielder of a screwdriver are all operators, as distinct from other involved parties, such as the passengers in a bus or the inmates of a prison. A significant change in the production of artifacts was the development of mass production. The key factor

ARTIFACT

here, despite the term, is not quantitative but mereological. In mass production, analogous component parts are made so similar to one another that they are individually interchangeable without detriment to the integrity and functioning of the whole. Artifacts come in a huge range of complexity, from dugouts to spacecraft, from tents to skyscrapers, from flags to the internet. The more complex artifacts such as modern buildings and modes of transport may consist of millions of parts, where ‘part’ here is used not in its broad mereological sense but in the sense of unified individuals contributing to the functioning of the whole. Complex artifacts may have many layers of complexity, with the simplest parts composing subassemblies or intermediate parts, these being parts of yet larger assemblies, and so on over several iterations. Some functional parts of artifacts may come together only as the whole is assembled rather than being inserted as units, such as a car’s braking or electrical systems. The design, manufacture, operation, maintenance, repair and retirement of complex artifacts requires not only technological skills but also a complex regime for managing the mereological complexity, from builders’ plans and engineers’ blueprints to computerised bills of materials. Typically, large numbers of organisations and people are involved at all stages of such an artifact’s life-cycle, and the management and co-ordination of their co-operation is also highly complex.

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Artifacts are designed to function, to work and be used in certain ways, whether statically or dynamically. Making, operating, and maintaining artifacts are all actions or sequences of actions. The agents need not be human: a bird’s nest, a spider’s web, a beaver’s dam are artifacts, though the most advanced human artifacts are orders of magnitude more complex than these. The processes involved in making and operating an artifact, and the artifact’s own functioning, are themselves mereologically complex, though their mereology has been far less investigated than that of their products. The variety of manufacturing and operating processes is considerable, and they too may have their salient parts, for example the different phases of the Otto cycle in a piston engine. Advanced manufacturing employs several mereological stratagems for reducing costs and maintaining a variety of options. Standardised parts can be employed in many situations: screws, bolts, switches, panels and other components allow interchangeability, generic tooling and large, cost-saving production runs. Whole artifacts as well as components may be modified from generic parts needing minor adaptation, such as cutting optional holes or adding optional extras. A further aid to manufacturing is the use of geometric rigs or other positioning systems, allowing costeffective repetitive actions by stationary tools and robots. These ensure that similar parts of repeat tokens of an object type are positioned in the same place relative to the tools and operators acting on them.

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See also > Activities, Aristotle's Theory of Parts, Aristotle's Theory of Wholes, Continuants and Occurrents, Common Sense Reasoning About Parts and Wholes. References and further readings

Dipert, R. R., 1993, Artifacts, Art Works, and Agency, Philadelphia: Temple University Press. Hilpinen, R., 1993, “Authors and Artifacts”, Proceedings of the Aristotelian Society 93: 155-178. Dement, C.W. and Simons, P., 1996, “Aspects of the Mereology of Artifacts” in Poli, R. and Simons, P. eds., Formal Ontology, Dordrecht: Kluwer, 255-276. Simons, P., 2013, “Varieties of Parthood: Ontology Learns from Engineering”, in D. Michelfelder, N. McCarthy and D. Goldberg, eds., Philosophy and Engineering: Reflections on Practice, Principles and Process, Dordrecht: Springer, 151163. Peter Simons

Atomism in Ancient Greek Philosophy Parmenides (early to mid 5th century BC) argued that belief in a physical world which varies over space and time is untenable: since not-being is a self-eliminating concept, that only leaves being, which without the addition of not-being can admit of neither differentiation nor division. “‘Nor is

it divisible, since it is all alike”, Parmenides wrote (B8.22). Atomism arose in the mid to late 5th century largely as a response to this, its proponents being the obscure Leucippus and his pupil, the celebrated polymath Democritus. Not-being, these atomists maintained against Parmenides, is a coherent notion, and the familiar physical world is precisely a combination of being and not-being, or – translated from metaphysical into physical terms – a combination of body and void. Parmenides, they implicitly acknowledged, was in a way right about the indivisibility of being, because division occurs not within being, but at the interstices, which consist of not-being. That is, a pure portion of body cannot be divided, but compound bodies, consisting of body plus void, can be divided at the void gaps. The portions of pure body are “atoms”, ‘atoms’, literally ‘uncuttables’ or ‘indivisibles’, and an infinite number of them distributed across an infinite universe have combined to produce infinitely many worlds. Their indivisibility makes them unchangeable and indestructible, and hence suitable as permanent primary elements underlying all phenomenal change. What makes body as such indivisible? Since Democritus insisted that atoms come in an infinite variety of shapes and sizes, he cannot have meant that they do not have proper parts (clearly the larger of two atoms has at least two proper parts: the portion which is equal in size to the smaller atom, and the portion by which it exceeds it). Therefore it seems unlikely – although this has

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been controversial among scholars (cf. Mau 1954, Furley 1967, Barnes 1979, Sorabji 1983, Makin 1989, Taylor 1999, Sedley 2008; 2007) – that the indivisibility of atoms was conceived as mathematical in character. The evidence favours physical indivisibility. According to Aristotle, On generation and corruption I 2, Democritus invoked the impossible consequences of supposing that a portion of body could be actually broken up by being cut at every point: for example, the parts thus obtained could never be reassembled into the original body, being either of zero size or altogether non-existent. It is often said that atomism was inspired by the need to answer the paradoxes of Zeno of Elea, Parmenides’ follower. Zeno had exploited the apparently absurd consequences of assuming a magnitude to be infinitely divisible: for example, it could never be traversed, since there would be infinitely many subdivisions of it to pass in sequence before arriving at the other side. Atomism would come to the rescue by denying that the number of subdivisions is infinite. But since it had to be granted that atoms have parts (see above), albeit parts into which they could never be physically separated, atomism seemed powerless to help answer Zeno, who could simply ask, in response, how even a single atom could ever be traversed, given that it must be traversed part by part. Zeno’s influence on the emergence of atomism must therefore remain questionable. What however is certain is that Zeno’s puzzles played a key part in a

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second phase of indivisibility-theory, in the mid and late 4th century BC. Xenocrates, Plato’s pupil and second successor as head of the Academy, invoked Zeno’s paradoxes of divisibility in defending his littleunderstood and idiosyncratic theory of ‘indivisible lines’. Diodorus Cronus (late 4th century), a leading member of the Dialectical School, was heavily influenced by Zeno, as well as by Aristotle’s critical discussion of the Zenonian paradoxes in Physics VI. On this basis he developed his own physical hypothesis that the ultimate particles constituting bodies are altogether ‘partless’. Finally Epicurus (341-271 BC), who inherited Democritean atomism and developed it in the light of the recent debate, constructed a two-tier system, as follows. ‘Atoms’ are indeed no more than physically indivisible bodies. However, when analysed into its ultimate constituents an atom turns out, largely for Zenonian reasons, to contain only finitely many parts. Each ultimate part is a vanishingly small dot of magnitude, such as not even to contain proper parts of its own. Each of these ‘minimal parts’ is, quite simply, the smallest magnitude that there is, and their constitutive role can be understood by recourse to an analogy with the smallest magnitude that you can see: an ordinary visible magnitude must consist of an exact number of these visible minima (since their parts are ex hypothesi invisible), and analogously any actual magnitude must consist of an exact number of minimal parts (since their parts are ex hypothesi non-existent). It follows that all magnitudes, in vir-

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tue of consisting of an exact number of these ultimate sub-multiples, are commensurable with each other. If this denial of incommensurability conflicts with Euclidean geometry, the Epicureans noted, then so much the worse for Euclidean geometry, which they indeed proceeded to reject as founded on false principles. It is possible that some later Epicureans tried to construct their own finitist geometry. Other paradoxical consequences of a theory of minima, already pointed out by Aristotle, were likewise accommodated by Diodorus and Epicurus. For example, there can be no strictly present-time motion, because there is no process of traversing a partless magnitude. Motion is therefore staccato: a moving object is successively in a series of minimally separated locations, without any smooth transition between them. As Diodorus put it, an object ‘has moved’, without its ever having been true to say of it that it ‘is moving’. Diodorus, a celebrated logician, defended the credibility of this with the parallel of Helen, who had three husbands, without its ever having been true to say that she ‘has’ three husbands. Another consequence of accepting minimal magnitudes was that there can be no differences of speed: in the time that a faster body took to traverse one minimum, a slower body would, impossibly, have to traverse a shorter distance. Epicurus accepted this consequence too. Macroscopic differences of speed are the aggregates of complex motions by the in-

numerable individual atoms constituting the moving body. Taken on its own, he maintained, each atom within a compound moves at an invariable absolute speed. Hence there are no real differences of speed. See also: Aristotle's Theory of Parts, Aristotle's Theory of Wholes, Paradoxes, Parmenides, Stoics. Bibliographical remarks

Barnes, J., 1979. Contains a rigorous discussion of Democritean atomism, with full historical contextualisation. Denyer, N., 1981. Pioneering analysis of Diodorus’ theory of partless magnitudes. Long, A.A. and D.N. Sedley, D.N., 1987. Vol. 1 translates and analyses in sections 8-12 the main texts on Epicurus’ two-tier atomic theory. Makin, S., 1989. Presents and tackles the main problems facing any interpretation of Democritean atomism. Sedley, D., 2008. Fuller exposition of the interpretation of Democritean atomism outlined above. Taylor, C. C. W., 1999. Translates and analyses the evidence for early Greek atomism. References and further readings

Barnes, J., 1979, The Presocratic Philosophers, London: Routledge and Kegan Paul.

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Denyer, N., 1981, “The Atomism of Diodorus Cronus”, Prudentia 13: 3345. Furley, D. J., 1967, Two Studies in the Greek Atomists, Princeton: Princeton University Press. Konstan, D., 1982, “Ancient Atomism and its Heritage: Minimal Parts”, Ancient Philosophy 2: 60-75. Long, A.A. and D.N. Sedley, D.N., 1987, The Hellenistic Philosophers, 2 vols., Cambridge: Cambridge University Press. Makin, S., 1989, “The Indivisibility of the Atoms”, Archiv für Geschichte der Philosophie 71: 125-49. Mau, J., 1954, Zum Problem des Infinitesimalen bei den antiken Atomisten, Berlin: Akademie-Verlag. Sedley, D., 2008, “Atomism’s Eleatic Roots”, in Curd, P. and Graham, D.W. ed., The Oxford Handbook of Presocratic Philosophy, Oxford: Oxford University Press, 305-32. Sorabji, R., 1983, Time, Creation and the Continuum, London: Duckworth. Taylor, C.C.W., 1999, The Atomists: Leucippus and Democritus, Toronto, Buffalo and London: University of Toronto Press. Verde, F., 2013, Elachista. La dottrina dei minimi dell’Epicureismo, Leuven: Leuven University Press. David Sedley

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Atomism, Logical The early years. Logical atomism has

been taken to be both a conception of philosophical analysis and a metaphysical account. Both aspects are present in B. Russell’s writings from the early 20th century that can be said to have culminated in the Lectures on Logical Atomism and On Propositions of 1918-19. Russell’s The Principles of Mathematics of 1903 and “On Denoting” of 1905, along with G. Moore’s 1903 paper The Refutation of Idealism, have long been recognised as key works in the early stages of the revival of realism that was a significant aspect of logical atomism. Two of Moore’s earlier papers, “The Nature of Judgment” (1899) and “Identity” (1900), as well as his dissertation works of 1897 and 1898 from which they are derived, have also been seen to be relevant to the emergence of logical atomism in recent years. One core aspect involved is the attempt to specify the ‘logical atoms’ that philosophical or ‘logical’ analysis purportedly uncovers in seeking the ultimate constituents of the world of our experience and in our conscious awareness of such experience. Logical atomists sometimes assumed that, as opposed to metaphysical speculation or hypothetical postulation, they employed ‘logical analysis’ of what is given in experience and thought. The tool of symbolic logic was taken as a key aid in the procedure of analysis. This latter idea led to the claim that a special symbolic framework was required – a linguistic schema along the lines of Russell and Whitehead’s epochal Principia Mathematica – as indis-

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pensable to the development of viable philosophical analyses. Russell’s 1903 The Principles of Mathematics and 1905 “On Denoting”, along with Moore’s 1903 “The Refutation of Idealism”, were key works in the early stages of the development of logical atomism. In the 1903 book Russell set out his version of logicism – that Peano arithmetic can be derived within a system of predicate logic via ‘defining’ (interpreting, modeling) of the primitive terms ‘0’, ‘successor’ and ‘number’ in the vocabulary of such a system of predicate logic The 1903 work also suggested early notions of a theory of logical types that would deal with what Russell referred to as “the contradiction” – the celebrated problem about properties and classes that would become known as Russell’s paradox – the property of all properties that do not exemplify themselves and the class of all classes that are not members of themselves. Russell was preoccupied by the paradox that bears his name from his discovery of it in 1901 until he believed he had arrived at a solution a few years later. In 1905, as he later cryptically noted, he thought his theory of descriptions was the ‘first step’ that opened the way to a solution (1967: 243). He did not explain how the theory of descriptions played the role of a ‘first step’, but he might have meant that expressions like ‘the property of not being a selfapplicable property’, ‘the class of all classes that are not members of themselves’, etc. are not to be taken as referential expressions but as con-

textually defined signs – incomplete symbols that cannot be employed in purported instantiations to yield a paradox, unless one adds an explicit claim of existence. In the 1905 paper Russell set out his theory of descriptions, which came to play a significant role in the development of analytic philosophy and metaphysics as well as in his conception of logical atomism. That role is obvious when one recalls his contrasting definite descriptions, as contextually eliminable expressions, with the indexical referring expression, like proper names or individual constants, as basic signs of a ‘logically proper’ language. Since any indexical constants of such an improved language or linguistic scheme would be referentially interpreted, all such ‘names’ or constant primitive predicates would represent, and the puzzles about non-naming names as well as supposed reference to nonexistent or merely possible entities would be avoided. Ordinary language was not only not logically sufficient for the resolution of philosophical problems; it was seen as giving rise to such problems by its overly permissive grammar and lack of logical perspicuity. Improved or ideal languages, logically proper schemata, were required. Such clarified or perspicuous schemata were taken to be necessary to reveal the logical structure of what the schema was used to represent or ‘picture’, as it was sometimes put. The idea of a perspicuous language being a ‘logical picture’ that would make various philosophical problems and purported solutions inexpressible is a theme of L. Wittgenstein’s Trac-

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tatus that lay behind some of his criticisms of Russell’s earlier writings, particularly the latter’s The Problems of Philosophy of 1912, that would be published during the time of Wittgenstein’s first visit to Cambridge that began in October of 1911. Moore’s earlier realism influenced Russell and his development of logical atomism (a phrase Russell had used in March, 1911), while posing problems for both Russell and Wittgenstein about propositional entities and the relation between thought and reality that both would subsequently address. In early works Moore held that: (a) propositions were their own grounds of truth (or falsity) by virtue of an inherent truth-making or falsemaking component or combinatorial connection; (b) the connecting ties forming propositions (facts) were independent of the mental acts that apprehended them; (c) there are independent facts, identified with true propositions, which are both bearers of and grounds of truth as nothing other than the proposition itself could be such a ground; (d) objects are identical with true existential propositions (facts), such as “This rose exists” and “Red exists now” (Moore 1899; 1900; 1900/1). These were themes in works that Russell was familiar with, referred to and was influenced by. Just as such a clarified linguistic schema would clearly display its basic or primitive terms, it would be equally clear what and how the complex (defined) terms were constructed and thus analysed. This allowed the use of a logically perspicuous

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linguistic structure to be employed to characterise and portray the ‘logical atoms’ of the world of experience, since basic terms were taken to be correlates of empirically given entities. As the defined terms were constructs from basic terms, so what they might be taken to represent were seen as reducible to what the basic signs represented and facts involving the latter. developments. While basic terms represented logical atoms in one sense, atomic sentences came to represent a further kind of ‘atom’ in another sense. The entities of different categories that were represented, particulars and properties, were dependent in the sense that both were necessarily constituents of facts. Facts, in turn, trivially required constituents as they were not simples. R. Carnap’s 1928 The Logical Construction of the World and 1934 Logical Syntax of Language are, in part, later attempts to carry out some of the themes of logical atomism while also being critical discussions of themes in Russell and Wittgenstein. One sees more recent aspects of the approach with a more pronounced awareness of ontological and metaphysical issues in G. Bergmann’s posthumously published New Foundations of Ontology (1992). The simplest sentences of a Principia type schema are subject-predicate patterns of various kinds – monadic, dyadic, triadic, and so forth. They became the ‘atomic sentences’ in the specific sense that no linguistic part of them was of a sentential form, the only constituent terms being either primitive predicates (of appropriate type) Later

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or ultimate subject terms that could not occur as predicates. The basic, or atomic, sentences were taken to represent possible situations, as Wittgenstein had put it. Situations that were actual, or realised, became the atomic facts that, for Wittgenstein, comprised the world. Atomic facts are not mereological compounds of their constituents, for the objects, in a memorable phrase of Wittgenstein’s, hang together like the “links of a chain”. Wittgenstein’s picturesque phrasing aside, the predicative linking indicates logically diverse kinds of objects – predicables and terms that connect in a unique way that is represented by the schema ‘Øx’. Consider the expressions ‘Ø’ and ‘x’. They are like links in a chain in that no further ‘thing’ (sign) connects them – it is rather their spatial relation or juxtaposition that supplies their link that represents the instantiation ‘link’ between the items they represent. But, of course, that juxtaposition is there and, though not a sign, does have a representational function. Superficially all this appears to be like mereological composition, where given two atoms, a and b, one has their fusion, ab or a + b. Yet it is nothing like that. Typically mereological systems axiomatically declare that given two atoms there is a further entity, their sum. Despite this claim, some authors, such as D. Lewis, take the sum to be “no more” than its atoms – not really some further entity. This has generated D. Armstrong’s notorious “free ontological lunches” – taking something to be such a mereological composite is not to really recognise it as an entity,

though it may be employed in one’s systematic resolution of problems. Wittgenstein, seemingly in such a reductive fashion, took the constituents to link together – though they had to be logically of the appropriate logical kinds in order to ‘fit’ – like a foot filling a shoe rather than a sum of a shoeless foot and a footless shoe – and, moreover, they had to, in fact, fit together – the foot wearing the shoe. Though Russell held in 1903 that one required instantiation or exemplification in the monadic case of attributes, he proceeded to make the same reductive move in the case of relations. Since he recognised a relation of monadic exemplification, he blocked the notorious Bradley regress by holding that one did not have to recognise a further relation connecting a relation to what it relates. This was then applied to all relations. Hence, exemplification was not required for dyadic, triadic, etc. relations. Wittgenstein, on one understanding of the Tractatus, would apparently extend this move by recognising only relational properties and hence no linking connection – between a relation and its terms. If one allows for possibilities as well as facts, such a variant of logical atomism would appear to be closer to a mereological pattern, since given any property and particular you additionally take there to be a possibility (though the Tractatus is actually more restrictive in its compartmentalisation of objects—hence comments about the logic of color, of sound, etc.). But the atomic facts of logical atomism are not mereological com-

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posites and are clearly additional entities. True atomic sentences are so since they correspond to actual atomic facts. Such facts, while taken to be complexes of their elements, are the ultimate facts that determine the world in a way that things do not. Things, of different logical kinds (particulars, properties, relations – colors, sounds) may be said to determine possibilities, the possible combinations or ‘potentialities’ in Bergmann’s later terminology, but not what is the case. Yet, it is only when you list all the actual facts and specify that they are all the atomic facts, do you determine or characterise a world. Facts are thus the crucial elements in the ontological structure of a logical atomist. As Wittgenstein would put it: the world is the totality of facts, not of things. Another core aspect of logical atomism was the logical independence of atomic facts from each other. Any such fact could exist (be actual) or not without effecting the existence or non-existence of any other. This mirrored a simple feature of the truth tables for standard two-valued propositional logic. It also revealed a further contrast with mereological systems. The stressing of the independence of atomic facts was a sign of the atomist revolt against the holistic monism of the then dominant metaphysics of absolute idealism that Russell and Moore undertook at the beginning of the 20th century. Idealists, such as F. H. Bradley and B. Bosanquet, had employed a coherence theory of truth and stressed both the interdependence of all things and the inter-connection of apparent facts

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in the all embracing monism of The Absolute or The Real. This partly resulted from their employing a mereological-like conception of objects being complexes of concepts that formed, not into independent facts, but into increasingly complex concepts that ultimately and necessarily, by way of a doctrine of internal relations, coalesced into the single unity of a monistic Absolute. The familiar declaration of the logical atomists, and later positivists, that the only necessity was logical necessity emerged in direct opposition to the idealist pattern. This, in turn, would lead to questions about the nature of logical necessity, logical systems, and the ontological implications they involved, since it is obvious, for example, that the logical distinctions between the kinds of terms employed and the rules governing them in a logical schema do not express standard logical truths of propositional and predicate logic. (Cochiarella, 1975) Eventually, in Wittgenstein’s case, such matters would lead to an expanding sense of ‘logical’ whereby one eventually took the logic of a term to involve its use and role in a linguistic framework. On the part of many of his disciples, this led into the development of what became known as ‘ordinary language’ philosophy. Compressing the ontology. The focus

on facts and truth raised serious problems concerning falsity, negations, conjunctions, generalities and other logically complex sentences (MacBride, 2013). What, if anything, did false sentences represent? How literally is one to take talk of ‘situations’

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and ‘possibility’? Are there negative facts that are the truth grounds for true negations? Are there general facts? Such questions point to major differences that emerged between Russell and Wittgenstein. Wittgenstein appears to have held that atomic sentences represent situations or possibilities that play a role somewhat like Aristotle’s prime matter – only they play such a role with respect to the characteristics (forms) of positive and negative. Hence, a possibility exists only as characterised by one and only one of the two. It thus exists as either a positive or a negative fact. Just as some Aristotelian scholars hold that prime matter doesn’t exist as formless for Aristotle, one can then say that Wittgenstein’s Tractarian situations are not taken to exist in the sense in which facts do, though they are employed in his analysis, as prime matter is in Aristotle’s discussion. In spite of the fact that without recognising properties (relations) as objects some of the discussion of possibilities and facts in the Tractatus would be incomprehensible, it has long been disputed whether properties are included as objects, even with his reported later comments to Moore and others indicating that they (or at least relations) are (Proops, 2013). Such disputes aside, possibilities played another, crucial role. Wittgenstein employed possibilities to dispense with propositional entities that played a prominent role in an important series of 1910-11 lectures that Moore had delivered about a year prior to Wittgenstein’s arrival at Cambridge (Moore, 1953). Wittgenstein’s pattern was simple. Moore

took an act of thought to be connected in some way to a proposition that gave the former its content and the linguistic expression of it meaning. Recognising propositions, however, did not allow one to avoid the problem of possibilities by recognising that propositions could be true or false. For propositions had to be connected, in turn, to (possible) facts, whose existence or non-existence would determine their truth value. This could be put in terms of the situation existing as a positive or a negative fact. Wittgenstein thus saw that propositions could be dispensed with by simply having a thought, or a linguistic expression of it, represent the situation without a mediating propositional entity. Situations sufficed to provide the conceptual content of atomic sentences. Russell, however, sought to go further and remove possibilities or situations as well as propositions as early as 1913. Prior to 1913 he had considered an atomic sentence to be true if and only if the object(s) referred to by the subject sign(s) has (have, instantiate, exemplify) the property (relation) represented by the predicate. But that is to covertly link a particular atomic sentence to a Wittgensteinian style situation – not to a fact. It is clear that if one takes a specific case of an atomic sentence pattern, ‘Fa’, one has to do two things. Link the sentence to the possibility it expresses and hold that the possibility is realised. Linking a name to an object and a predicate to a property do not suffice to supply a link for a sentence to a particular situation, as they would do if we took the ground of

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truth to be a mereological sum. One can see the two-fold linkage more clearly in the monadic case in a slightly more symbolised variant that is familiar in such contexts: ‘Fa’ is true iff ‘Fa’ represents a’s being F and a is F. When we link a name to an object or a predicate to a property, we do not have the double feature of representation and existence for a very simple reason. All logically proper names and predicates name (refer). Logically proper atomic sentences, however, were taken to express possibilities and be either true or false. This simple feature gives rise to classic problems regarding falsity, negation and speaking about what is not. Russell sought to resolve such problems from 1905 to 1913 by using his theory of definite descriptions, while Moore, in 1910-11, allowed for both non-actual facts and propositional entities. (Moore 1953; Hochberg 1978) Though Russell only set out his view in detail explicitly in the case of relations in the 1913 manuscript (Russell, 1984), it is easy to apply the radical pattern he adopted to the monadic case. Instead of taking atomic sentences to indicate possibilities, Russell employs a definite description of a purported atomic fact along the lines of: ‘Fa’ is true iff The p of the form Φx with a as x and F as Φ exists. This involves his recognition of logical forms, which he explicitly acknowledged, and his taking terms of facts to stand in relations to the facts, with quantified variables rang-

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ing over existent atomic facts. Here the view is merely indicated to note the divergence between Russell and Wittgenstein and Russell’s attempt to explicitly avoid possible, but not actual, facts by using descriptions (Hochberg, 2000). Such descriptions, with variables ranging over atomic facts, enabled him to avoid referring to non-existent facts as he had used descriptions to avoid purported reference to non-existent objects like Pegasus. In fact, he had already suggested doing that in a passage in the epochal 1905 paper. The main point of the pattern is that the above biconditional, taken as a rule for applying a truth predicate to monadic atomic sentences, trivially holds whether such a fact exists or not, and without any explicit or implicit reference to a problematic possibility. Wittgenstein’s criticism of Russell’s multiple relation analysis of intentional contexts in the manuscript unfortunately sidetracked Russell’s important suggestions and work regarding an account of truth. In the 1918 lectures, Russell argued for both negative facts and general or quantified facts. He also attacked Wittgenstein’s Tractatus view that given a list of objects there was no need to recognise a closure statement explicitly asserting that they were all the objects. Russell focused on the point that one could not derive a generality, such as that every object has the property F, from a set of atomic sentences – a is F, b is F, …. One required, in addition, the explicit or implicit claim that the objects were all the objects – every object x is such that x =a or x =b or …, and thus

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had to appeal to at least one general fact. Russell thus recognised facts that were not atomic facts. While Wittgenstein seems to have held at places in the Tractatus that the world was composed of atomic facts, he would later note that he was aware of the need for a closure clause (Proops, 2013). In the 1925 introduction to the second edition of Principia, Russell made a further reductive move by suggesting one could avoid negative facts by employing general facts. The idea was simple enough, if problematic. The proposition that ‘a is not F’ is true if and only if every atomic fact is different from the fact that a is F. This solution, appealing to the diversity of facts (or the identity of a fact with an item on a list of facts) would later be revived by others, D. M. Armstrong for example (1997), who would revert to implicitly using atomic sentences as representing possible facts in the manner of earlier logical atomism (Hochberg, 2002). See also > Bergmann, Carnap, Facts, Proposition, Russell. References and further readings

Armstrong, D. M., 1997, A World of States of Affairs, Cambridge: Cambridge University Press. Bergmann, G., 1957-8, “The Revolt Against Logical Atomism”, Philosophical Quarterly, Part I, 7 (29): 323-29; Part II, 8 (30), 1-13. Bergmann, G., 1992, New Foundations of Ontology, Madison: University of Wisconsin Press.

Carnap, R., 1928, Der logische Aufbau der Welt, Leipzig: Felix Meiner Verlag; 1967. The Logical Structure of the World. Pseudoproblems in Philosophy (trans. R. A. George), University of California Press. Carnap, R., 1937, Logical Syntax of Language, London: Kegan Paul. Carnap, R., 1947, Meaning and Necessity, Chicago: University of Chicago Press. Cochiarella, N., 1975, “Logical Atomism, Nominalism, and Modal Logic”, Synthese, 31 (1): 23-62. Cochiarella, N., 1987, Logical Studies in Early Analytic Philosophy, Columbus: Ohio State University Press. Hochberg, H., 1978, Thought, Fact and Reference: The Origins and Ontology of Logical Atomism, Minneapolis: University of Minnesota Press. Hochberg, H., 2000, “Propositions, Truth and Belief: The Wittgenstein– Russell Debate”, Theoria 66: 3–40. Hochberg, H., 2001, Russell, Moore, and Wittgenstein, New York: HanselHohenhausen. Hochberg, H., 2002, “From Logic to Ontology: Some Problems of Predication, Negation, and Possibility”, in D. Jacquette, ed., A Companion to Philosophical Logic, Oxford: Blackwell, 281-292. Klement, K., 2013, “Russell’s Logical Atomism”, Stanford Encyclopedia of Philosophy online. Lewis, D., 1991, Parts of Classes, Cambridge: Blackwell.

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MacBride, F. 2013, “Truthmakers”, Stanford Encyclopedia of Philosophy online.

Russell, B., 1954, The Problems of Philosophy, London: George Allen & Unwin Ltd.

Moore, G. E., 1899, “The Nature of Judgment”, Mind, 8 (30): 176-193.

Russell, B., 1967, The Autobiography of Bertrand Russell 1872-1914, vol. 1, New York: Little, Brown & Co.

Moore, G. E., 1900, “Necessity”, Mind, 9 (35): 289-304. Moore, G. E., 1900/1, “Identity”, Proceedings of the Aristotelian Society, 1: 103-27. Moore, G. E., 1903, “The Refutation of Idealism”, Mind 12: 433-53. Moore, G. E., 1953, Some Main Problems of Philosophy, (1910/11), George Allen & Unwin Ltd., London. Proops, I., 2013, “Wittgenstein’s Logical Atomism”, Stanford Encyclopedia of Philosophy on line. Russell, B., 1903, The Principles of Mathematics, Cambridge: Cambridge University Press. Russell, B., 1905, “On Denoting”, Mind XIV, (4): 479-493. Russell, B., 1906/7, “On the Nature of Truth”, Proceedings of the Aristotelian Society 7: 28-49. Russell, B., 1911, “Le Réalisme analytique” Bulletin de la Société francaise de philosophie, 11. March, 282291. Russell, B., 1919, Introduction to Mathematical Philosophy, London: George Allen & Unwin Ltd. Russell, B., 1952, Our Knowledge of the External World, London: George Allen & Unwin Ltd.

Russell, B., 1984, Theory of Knowledge: The 1913 Manuscript, London: George Allen & Unwin. Whitehead, A. N. and Russell, B., 1950, Principia Mathematica, vol. I, Cambridge: Cambridge University Press, Wittgenstein, L., 1963, Tractatus Logico-Philosophicus, (trans. D. Pears & B. McGuinness), London: Allen & Unwin. Herbert Hochberg

Atomism, Medieval It is a common misperception that there were no atomistic theories during the medieval period. It is true that during the medieval period, after translations of Aristotle’s works became available and were introduced in the schools in the Latin West in the thirteenth century, scholastics commonly adopted an Aristotelian framework. As is well known, Aristotle raised objections to and rejected indivisible minimal units or atoms in two contexts. First, in Physics 6, he claimed that (1) no continuum is composed of indivisibles and maintained instead that a continuum is potentially infinitely divisible. There are no actual points in a line, instants of time, or units of motion. Second,

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in Aristotle’s view, (2) no physical substance is composed of primitive indivisible and unchangeable building blocks or atoms. Instead, prime matter and substantial form are the fundamental ontological principles of natural bodies. From this hylomorphic viewpoint, the generation of natural things is the consequence of a change from potentiality to actuality by means of a change in substantial form. Nonetheless, some scholastics were atomists. First it should be noted that among Islamic thinkers, there was a long history of development of two accounts of the fundamental structure of natural things, namely, on, the one hand, hylemorphism, influenced by Aristotle’s works and adopted, in particular, by philosophers (falasifah), and, on the other hand, atomism espoused by theologians (mutakallimun) to defend the Islamic conception of God as the immediate cause of all things. Al-Ghazali (11491209), an Asharite theologian, maintained that atoms exist perpetually while all other material things are accidental and thus momentary and without causal efficacy. Only God, with atoms as his material instrument, is the cause of all contingent events. As early as the ninth century, Mu’tazilite theologians, to support occasionalism and God’s continuous creation, endorsed unextended atoms of time and space, a view available to European scholastics as presented in Latin translations of Maimonides’ Guide for the Perplexed. During the fourteenth century in the Latin West, atomistic accounts

(views that postulated indivisible minimal units) developed in response to Aristotle’s claims in both (1) and (2) above. Scholastics developed views of (1.1) mathematical or geometrical atoms that are unextended indivisible points composing a continuum and (2.1) material atoms that are the fundamental building blocks of what exists in the material world. The dominant view among Latin scholastics was that of the divisibilists, supporting (1) above, with the most notable arguments against unextended indivisible points in a continuum provided by John Duns Scotus. Among Oxford indivisibilists, supporting (1.1), Henry of Harclay (c.1270-1317) argues that there are unequal infinites (for every whole is greater than its part, so there are more parts in a whole continuum than in its half). He concludes that this entails an infinite number of indivisible points in a continuum. Walter Chatton (c.1290-1343), in the context of discussing angelic motion, maintained instead that a continuum contains a finite number of indivisible points. Scholastics supporting 2.1 accepted the mereological assumption that indivisible parts are fundamental or ontologically prior to the whole to which they belong. From this viewpoint, Oxford philosopher William Crathorn (fl.1330's) adopts the reductionist principle (P1) that a whole is nothing other than all of its parts’ (Crathorn, 1988: 217); and also the finitist principle (P2) that there are a finite number of indivisibles (or atoms) composing all the material

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things that exist. Furthermore, Crathorn postulated space as a primitive (along with atoms) and identified empty space with vacuum (Crathorn, 1988: 417); he also claims that space is dimensional, that place is the location of a body in space, and that body acquires dimension from the locus of its indivisible parts, located contiguously in space. John Wyclif (1320-1384), the most prominent fourteenth century Oxford atomist, responds directly to Aristotle’s anti-atomistic claims. Wyclif presents a series of arguments to show that points are actual parts of a line, and as such are a principle and cause ontologically prior to a line, just as instants are in regard to time and moments to motion (Wyclif, 1893, vol. 3, p. 30). Among others, Wyclif presents the following argument. Suppose that God, who has the power to make a substance the size of a point and to juxtapose points contiguously, creates such a point substance at every point in the world. Assume also that God annihilates all continuous substance, preserving point substances. Wyclif contends that this would change nothing for the world and all creatures in the world would appear the same. Since this is possible, it therefore cannot be concluded that this is not so in fact (Wyclif, 1893, vol. 3, pp.33-34). From this viewpoint, Wyclif supports a finite and fixed number of actual unextended indivisible points that are located contiguously to produce a plenum. The place of a particular point is its site relative to certain fixed points, those at the centre and poles of the universe. Though there is

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no void space in Wyclif’s world, indivisible atoms can move by one point successively replacing another. Though he was an atomist, Wyclif did not reject Aristotelian prime matter and substantial forms. In his view, indivisible points constitute prime matter and, thereby, are the material cause and building blocks of all corporeal things. Further, Wyclif adopted a hierarchical account of the parts that compose physical things. He claims that, as designed by God, different numbers of unextended indivisible points joined together and united by a substantial form, result in persistent extended (i.e., physical) elemental atoms, those of earth, air, fire, and water. These elemental atoms, when conjoined in appropriate proportional relations are the parts of minima naturalia or compound corpuscles (analogous to molecules). When joined together, like compound corpuscles, each with its inhering compound substantial form, constitute such compound substances as gold and lead, or blood, bones, and flesh. These latter compound substances, in turn, compose animal bodies that are animated by a substantial form, namely, a soul. Finally, the total set of indivisible atoms, which are the minimal units composing all individual material substances, are the fundamental indivisible parts that determine the shape and size of the world. In subsequent centuries, hylomorphic atomism of this sort was developed, for example, by J.C. Scaliger (1484 – 1558), and, in the seventeenth century, by Daniel Sennert, and his student, Johannes Sperling.

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Gerard of Odo (1290-1349), who like Crathorn accepts (P1) and (P2), was the first of the currently known Parisian atomists, a group including such atomists as Nicholas Bonetus, Marc Trevisano, John Gideonis, and Nicholas of Autrecourt. Perhaps the best known of medieval atomists, Nicholas of Autrecourt (1299-1369), rejects Aristotelian prime matter and substantial forms and maintains that change is the result of local motion and the aggregation and disaggregation of extended atomic particles. Autrecourt assumes atoms and void. Matter is identified with an atomic flow and Aristotelian substantial form is replaced by a formal atom, which is an essential part (called a ‘virtue’) that: (a) makes a compound body what it is (defining its specific and generic nature), (b) regulates, in each compound body, its natural operations and motions; and (c) gives a compound body its per se unity. In the generation of a human being, the stars serve as the efficient cause that unites a formal atom, the human soul (the formal cause), with an atomic flow, in this case of sperm, which serves as the material cause. Bibliographical remarks. Wyclif, J.

Parts and Wholes, Substance, Substrate. References and further readings

Baffioni, C., 1982, Atomismo e antiatomismo nel pensiero islamico, Naples: Gianni e Figli. Cross, R., 1998, The Physics of John Duns Scotus: The Scientific Context of a Theological Vision, Oxford: Clarendon. Dutton, B., 1996, “Nicholas of Autrecourt and William of Ockham on Atomism, Nominalism, and the Ontology of Motion”, Medieval Philosophy and Theology 5, 63-85. Crathorn, W. 1988, Quaestiones super librum sententiarum, ed. F. Hoffmann in Questionen Zum ersten Sentenzenbuch, Beiträge zur Geschichte der Philosophie und Theologie des Mittelalters, Band 29, Aschendorff, Münster. Grant, E. & J. E. Murdoch, 1987, Mathematics and its Applications in Science and Natural Philosophy in the Middle Ages, Cambridge - London - New York: Cambridge University Press.

1893. Wyclif’s discussion of indivisible atoms is raised in his discussion of propositions of place to explain his distinctive notion of place, in vol. III, Logicae Continuatio, tract 3.

Grellard, C. & Aurélien, R. (eds.), 2009, Atomism in Late Medieval Philosophy and Theology, Leiden & Boston, Brill.

See also > Ancient Atomism, Metaphysical Atomism, Collectives and Compounds, Medieval Mereology, Medieval Discussions of Temporal

Kretzman, N. (ed.), 1982, Infinity and Continuity in Ancient and Medieval Thought, Ithaca & New York: Cornell University Press.

Kenny, A. (ed.), 1986, Wyclif in his Times, Oxford: Clarendon.

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Lasswitz, K., 1890, Geschichte der Atomistik vom Mittelalter bis Newton, Hamburg: Leopold Voss. Luthy, C.; Murdoch, J.E.; Newman, W.R (eds.), 2001, Late Medieval and Early Modern Corpuscular Matter Theory, Leiden-Boston-Koln: Brill. Michael, E., 2003, “John Wyclif on Body and Mind”, Journal of the History of Ideas 64: 343-360. Murdoch, J. E., 1957, Geometry and the Continuum in the Fourteenth Century. A Philosophical Analysis of Thomas Bradwardine’s Tractatus de Continuo, Ph.D. Dissertation, University of Wisconsin. Murdoch, J. E., 1974, “Naissance et developpement de l’atomisme au bas Moyen Age latin”, in G. H. Allard et al (eds.), La science de la nature: theories et pratiques, Montreal & Paris, Institut d’etudes medievales – Vrin,11-32. Murdoch, J. E., 1981, “Henry of Harclay and the Infinite”, in A. Maieru et al (eds.), Studi sul XIV seculo in memoria di Anneliese Maier, Rome, Edizioni di Storia e Letteratura, 219261. Murdoch, J. E., 1982, “Infinity and Continuity”, in N. Kretzmann et al (eds.), The Cambridge History of Later Medieval Philosophy, Cambridge: Cambridge University Press, 564-591. Murdoch, J. E., 1984, “Atomism and Motion in the Fourteenth Century”, in E. Mendelsohn (ed.), Transformation and Tradition in the Sciences. Essays in Honor of J. Bernard Co-

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hen, Cambridge & New York: Cambridge University Press, 45-66. Murdoch, J. E. & Synan, E., 1966, “Two Questions on the Continuum: Walter Chatton, O.F.M and Adam Wodeham, O.F.M.”, Franciscan Studies 26, 212-288. Nicholas of Autrecourt, 1971, Universal Treatise, trans. I. Kennedy, R. Arnold & A. Millward, Milwaukee, Wisc: Marquette University Press. Nicholas of Autrecourt, 1939, Exigit ordo, ed. by J.R. O’Donnell, in “Nicholas of Autrecourt. Medieval Studies, I, 179-266. Pabst, B., 1994, Atomentheorien des lateinischen Mittelalters, Darmstadt, Wissenschaftliche Buchgesellschaft. Pines, S., 1997, Studies in Islamic Atomism, ed. Tzvi Langermann, Israel, Magnes Press. Originally published in German in 1936. Pyle, A., 1997, Atomism and Its Critics. From Democritus to Newton, Bristol: Toemmes Press. Wyclif, J., 1869, Trialogus, ed. G. Lechler, Oxford, Clarendon Press, esp. 87-92. (Also printed in 1525 and 1753). Wyclif, J. 1893, Tractatus de logica, ed. M. H. Dziewicki, 3 vols. London: The Wyclif Society. Emily Michael

Atomism, Metaphysical Most broadly speaking, metaphysical atomism is a view on which certain

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sorts of entities – the atoms – are granted some sort of privileged metaphysical status. Different versions of metaphysical atomism differ over what they count as an atom, and what sort of privileged metaphysical status is granted. Starting with the notion of an atom, this notion is associated with a cluster of features including: Mereological simplicity: Atoms have no proper parts. Spatial minimality: Atoms occupy points of space. Physical indivisibility: Atoms cannot be physically divided. The classical form of atomism traces back to Democritus and the Greek atomists. It resurfaces in Newton, who in the Opticks (1952: 394), hypothesises that ‘the smallest particles of matter cohere’ to ‘compose bigger particles’, which in turn compose still bigger particles, until the biggest particles ‘which by cohering compose bodies of a sensible magnitude.’ And it stands as the dominant metaphysical picture nowadays. Thus J. Kim (1993: 337) notes: “It is generally thought that there is a bottom level, one consisting of whatever microphysics is going to tell us are the most basic physical particles out of which all matter is composed (electrons, neutrons, quarks, or whatever).” Mereological simplicity, Spatial minimality, and Physical indivisibility are in principle separable. For instance, Horgan and Potrč’s (2007) existence monism holds that all that exists is

one big extended simple (the cosmos as a whole, ‘the blobject’), which is mereologically simple but not spatially minimal. One may posit spatial minima that have proper parts and are divisible (perhaps into co-located minima). And one may posit mereologically complex and spatially extended objects that are physically indivisible for purely nomic reasons (e.g., hadrons composed of confined quarks). But although simplicity, spatial minimality, and indivisibility are in principle separable, these ideas have a natural affinity, and the notion of an atom is perhaps best understood as the notion of an entity with all three features together. Turning to the idea of a metaphysically privileged status, the available statuses will depend on one’s background meta-metaphysical view. On the orthodox Quinean (1963) view on which the task of metaphysics is to say what there is, the privileged status is existence, and the metaphysical atomist then says that only atoms exist (or at least: atoms are the only concrete objects that exist – one could combine metaphysical atomism with a Platonic embrace of abstract entities). Such a view is now defended under the heading of mereological nihilism. For instance, van Inwagen, who (1990: 5) takes as an assumption “that matter is ultimately particulate” in the sense that “every material thing is composed of things that have no proper parts: ‘elementary particles’ or ‘mereological atoms’ or ‘metaphysical simples’”, defends the quasi-nihilistic view that only metaphysical atoms and biological organisms exist. And Rosen and Dorr

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(2002) explore the fully nihilistic view that only mereological atoms and no composites exist. On the neo-Aristotelian grounding view of Schaffer (2009) and other grounding-oriented perspectives, there is also the option of allowing that atoms and composites both exist, while privileging atoms as fundamental (or at least: as grounding composites – one could still allow that the atoms themselves are in turn dependent on abstracta, etc.). Such a view, which Schaffer (2010: 44) labels atomism, is now fairly orthodox. It is the thematic extension of the idea that proper parts are metaphysically prior to their wholes. For instance, Leibniz (1989a: 85) holds that “Every being derives its reality only from the reality of those beings of which it is composed,” and so (1989b: 213) ultimately maintains: “These monads are the true atoms of nature and, in brief, the elements of things”. Russell (2003: 92) echoes this perspective in saying: “The existence of the complex depends on the existence of the simple, and not vice versa”. To which he adds: “I believe that there are simple things in the universe, and that these beings have relations in virtue of which complex beings are composed”. So, on the grounding view one may distinguish: Existence atomism: Atoms are all that exists (/among concrete objects). Priority atomism: Atoms are all that is fundamental (/most fundamental among concrete objects).

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Other options may of course be possible given other background metameta-physical views. For instance, one might have a notion of a substance as a building block of nature, while allowing that non-substance objects exist, and that substantiality and fundamentality can cross-cut. Then there would be room for the view that atoms are the substances (/building blocks) of nature. Which alternatives there are to atomism will also depend on one’s background views. But on the neoAristotelian grounding view there will be an egalitarian alternative, as well as two anti-atomistic alternatives: Egalitarianism: Atoms and composites both exist, and neither grounds the other. No atoms: Atoms do not exist. Dependent atoms: Atoms exist but are not fundamental; instead they are grounded in composites. Egalitarianism is defended by Tallant (2013), and would be upheld by anyone who accepts mereological composition but rejects the ideology of grounding and fundamentality. No atoms is held by anyone who thinks that our world is gunky (limitless decomposable into proper parts, with no simples at all), such as Plotinus (1991: 97): “Atoms again (Democritus) cannot meet the need of a base. There are no atoms; all body is divisible endlessly”. No atoms is also held by existence monists like Horgan and and Potrč (2007) who only believe in the cosmos. Dependent atoms is held by anyone who accept

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the existence of atoms but thinks they are dependent on larger wholes. For instance, priority monists (such as Schaffer 2010) who also happen to believe in atoms, would believe that the atoms depend on the wholes. The extreme versions of the No atoms and Dependent atoms view invert the atomist worldview entirely to privilege the cosmos as a whole: Existence monism: Only the cosmos as a whole exists (/concretely). Priority monism: The cosmos as a whole is all that is fundamental (/most fundamental among concrete entities). If one has a separate notion of a substance (e.g. distinct from that of a fundamental object) then one might also consider the idea that the cosmos as a whole is the one and only substance. The arguments for and against metaphysical atomism will vary of course, depending on whether Existence atomism or Priority atomism is at issue, as well as which alternatives (Egalitarianism, No atoms, or Dependent atoms – perhaps even in monistic form) are under consideration. But there are some core considerations that arise through many of the different versions of metaphysical atomism. In favor of various forms of atomism – at least in comparison with views that posit atoms plus composites – are various arguments from ontological economy and causal exclusion (cf. Merricks 2001), to the effect that atoms alone can account for everything and take up all the causal work. Composite objects also

engender certain conundrums – such as Unger’s (1980) problem of the many – which atomism might be thought to resolve. Among the arguments against atomism are arguments that appeal to the possibility of gunky scenarios without atoms, and the possibility of fundamental properties borne by composites (cf. Schaffer 2007). There is still no consensus on the viability of atomism in any form. See also: Ancient Greek Atomism, Medieval Atomism, Cosmology, NonWellfounded Merology, Ontological Dependence, Point, Substance. References and further readings

Horgan, T. and Matjaž, P., 2008, Austere Realism: Contextual Semantics meets Minimal Ontology: MIT Press. Kim, J., 1993, “The Nonreductivist’s Troubles with Mental Causation” in Supervenience and Mind: Selected Philosophical Essays, Cambridge University Press: 336-57. Leibniz, G. W. F., 1989a [1687], From the Letters to Arnauld. Philosophical Essay (eds.) Ariew and Garber: 69-89. Hackett Publishing. Leibniz, G. W. F., 1989b, [1714], The Principles of Philosophy, or, the Monadology. In Philosophical Essays, eds. and trans. Ariew and Garber: 213-24. Hackett Publishing. Merricks, T., 2001, Objects and Persons. Oxford: Oxford University Press.

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Newton, I., 1952 [1704]. Opticks: Or a Treatise on the Reflections, Refractions, Inflections, and Colours of Light. Dover. Plotinus, 1991 [c. 270], The Enneads, trans. MacKenna, Penguin Books. Quine, W. V. O., 1948, “On what there is”, Review of Metaphysics 2: 21-38. Rosen, G. and Dorr, C. 2002, “Composition as a Fiction”, The Blackwell Companion to Metaphysics, ed. Gale, Basil Blackwell, 151-74. Russell, B., 2003 [1911], “Analytic Realism”, Russell on Metaphysics, ed. Mumford: Routledge, 91-6. Schaffer, J., 2007, “From Nihilism to Monism”, Australasian Journal of Philosophy 85: 175-91. Schaffer, J., 2009, “On What Grounds What”, Metametaphysics, eds. Chalmers, Manley, and Wasserman, Oxford University Press, 34783. Schaffer, J., 2010, “Monism: The Priority of the Whole”, Philosophical Review 119: 31-76. Tallant, J., 2013, “Problems of Parthood for Proponents of Priority”, Analysis 73: 429-38. Unger, P., 1980, “The Problem of the Many”, Midwest Studies in Philosophy 5: 411-68. van Inwagen, P., 1990, Material Beings, Cornell University Press. Jonathan Schaffer

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Axiomatic Method Whether in mereology or any other area, applying the axiomatic method means to specify an axiomatic system S, consisting of the axioms of S and the logic of S. The axioms of S are certain basic statements, and the logic of S is a set of basic inferencerules which can be used to generate further statements from given statements (ultimately from the axioms). The specification of S must be effective, that is, it must be in every case decidable whether or not a given statement belongs to the axioms of S, and whether or not a given inferencerule belongs to the logic of S. Relative to the axiomatic system S – the axioms plus the logic – a notion of provability is recursively defined: (1) the axioms of S are provable in S; (2) if the premise(s) of an inference-rule of the logic of S are provable in S, then also the conclusion of that inference-rule is provable in S; (3) only statements that can be obtained by (1) and (2) are provable in S. Any statement that is provable in S but is not an axiom of S is called a theorem of S. Using the term ‘valid’ in the sense of ‘legitimate’, one can state six conditions that normally hold for axiom systems. (a) The language of an axiomatic system consists of (wellformed) statements that are finitelength strings of discrete signs. (b) The inference-rules of an axiomatic system have only finitely many premises. (c) The inference-rules of an axiomatic system are validitypreserving; that is: if their premises are valid, then their conclusions must

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be valid, too. (d) The axioms of an axiomatic system are valid statements. (e) Validity entails truth. (f) The axioms and basic inference-rules of an axiomatic system are easily describable (though there may be infinitely many axioms and basic inference-rules) and their description fits on a few printed pages, perhaps on one page. If an axiomatic system is normal in the sense of displaying all six of the aspects of normality just stated, then that system fulfils the purpose of describing a certain region of beingvalidly (hence truly), rigorously, and in a compendious way. (A nonnormal axiom system would for example be a system with one or more axioms whose truth is unknown, e.g., a set-theoretic system with the continuum hypothesis, or, alternatively, with its negation.) A normal axiomatic system S is complete with regard to validity in the language of S if all the valid statements in that language are provable in S. A normal axiomatic system S is minimal if it has no proper part S´ such that the very same statements are provable in S´ that are provable in S. Completeness and minimality are more or less aesthetic virtues in a normal axiomatic system (in fact, completeness is a virtue that is realizable only within very narrow confines). Regarding the interpretation of a normal axiomatic system S, there is a fundamental alternative: either the language of S is completely interpreted, or it is not. The interesting case is the latter. In that case, S is

taken to enumerate valid (hence true) statements, but it is not entirely determined what these valid statements are about. The only thing determined is that the statements of S (its axioms and theorems) must be understood in such a way as to be valid (hence true). Such an understanding, however, can usually be achieved in various ways – there are usually different interpretations of the language of S so that all axioms and theorems of S come out as true. Moreover, if the language of S is not completely interpreted, then a multitude of axiomatic systems that are alternatives to S may suggest themselves – even systems S´ in which statements are provable the negations of which are provable in S. Nevertheless, both S and S´ can each be a normal axiomatic system. Both S and S´ can be taken to enumerate valid (and true) statements. This is made possible by the fact that the language of S – which is also the language of S´ – is incompletely interpreted, the language L – in its present incomplete state of interpretation – can be further interpreted so that the statements of S are valid; but there is also an alternative completion of L’s interpretation, according to which all the statements of S´ are valid. For illustration, consider the mereological case. We have a language L of first-order predicate logic with identity, and in that language a special predicate, P(x, y), to be read as ‘x is a part of y’. An axiomatic mereological system with respect to L is an axiomatic system, formulated in L, in whose axioms the predicate P(x, y) is the most prominent predicate.

AXIOMATIC METHOD

Consider the following three axiomatic mereological systems with respect to L, of which in each case only the first three axioms are stated: MS1 The logic of MS1: first-order predicate logic with identity. The axioms of MS1: ∀x∀y∀z (P(x, y) ∧ P(y, z) → P(x, z)) ∀x P(x, x) ∀x∀y (P(x, y) ∧ P(y, x) → x = y) etc. MS2 The logic of MS2: the logic of MS1. The axioms of MS2: ∀x∀y∀z (P(x, y) ∧ P(y, z) → P(x, z)) ∀x∀y [P(x, y) → ∃z(P(z, y) ∧ z ≠ x ∧ ¬∃u(P(u, x) ∧ P(u, z)))] ∀y∃x P(x, y) etc. MS3 The logic of MS3: the logic of MS2. The axioms of MS3: ∀x∀y∀z(P(x, y) ∧ P(y, z) → P(x, z)) ∀x∀y(P(x, y) → ¬P(y, x)) ∃y∀x¬P(x, y) etc. The three systems are pairwise contradictory to each other, in the sense that, for each pair, a statement is provable in one system the negation of which is provable in the other. Yet, each can be a normal axiomatic mereological system with respect to L, depending on how the interpretation of L is completed. The generic interpretation of ‘P(x, y)’ as ‘x is a

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part of y’ can be further specified or completed into three varieties of parthood, namely, ‘x is a proper or improper subset of y’, ‘x is a proper subvolume of y’, and ‘x is a proper subnumber of y’ (i.e., ‘x is a smaller number than y’). If the universe of discourse of L comprises precisely the subsets of the set of human beings and P(x, y) means as much as ‘x is a proper or improper subset of y’, then the three stated axioms of MS1 are valid. If, however, the universe of discourse of L comprises precisely the volumes of space and P(x, y) means as much as ‘x is a proper subvolume of y’, then the three stated axioms of MS2 are valid. If, finally, the universe of discourse of L comprises precisely the natural numbers and P(x, y) means as much as ‘x is a proper subnumber of y’, then the three stated axioms of MS3 are valid. See also > Boolean Algebra, Metamathematics of Mereology, Sum, Theoretical Mereology. References and further readings

Meixner, U., 1997, Axiomatic Formal Ontology, Dordrecht: Kluwer. Simons, P., 1987, Parts. A Study in Ontology, Oxford: Clarendon Press. Ridder, L., 2002, Mereologie, Frankfurt a. M.: Klostermann. Uwe Meixner

BERGMANN, GUSTAV

B Bergmann, Gustav Bergmann (1906-1987) argues that the problem of wholes (complexes) is the deepest problem of ontology since solutions to all other problems depend on how we approach the problem of complexes. According to Bergmann, the key element for a solution of the problem of complexes lies in the acknowledgement of a fundamental tie or ‘nexus’ which connects the constituents of a complex. Most ontologists assume a homogeneous fundamental tie, which ties entities of the same category together, while Bergman and some others operate with an inhomogeneous fundamental tie. The inhomogeneous tie is expressed by ‘exemplifies’ as, e.g., in the sentence ‘the apple exemplifies a green colour and an oval shape’. The homogeneous tie has to be expressed by ‘coincides’ as, e.g., in the sentence ‘the qualities green and oval coincide’. Ontologies with homogeneous tie were called bundle views because in them ordinary objects such apples are viewed as bundles of qualities. In Bergmann’s conception of ontological mereology one of the two fundamental ties is not really a tie but rather a function in Frege’s sense (and mistaken for a tie). The function

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which Bergmann symbolises by gamma maps n entities (its arguments) to a n+1th entity (its value) with no implication that the value consists of the arguments. Bergmann claims that in gamma-ontologies there are no genuine complexes just because gamma is not a nexus but a function and because there are no nexus in it. Bergmann argues for the need of a nexus in the following way: without the nexus there would be only a collection which would not be an entity. Then he argues that a collection of the nexus and the other members would not be an entity either and that the complex is the fact that the nexus connects the other members/parts. Bergmann symbolises the inhomogeneous fundamental tie by epsilon and calls ontologies with that nexus epsilon-ontologies. Bergmann's diagnosis then is that only in epsilon-ontologies there are proper complexes, i.e., wholes that actually consist of their parts. The alleged wholes of a gammaontology merely derive as function values from their alleged parts. A very influential variant of the gamma view tries to solve the ontological problem of wholes epistemologically. It assumes that wholes are made of their parts by mind which has the capability of synthesis. Bergmann criticises this view as a genetic fallacy and points out that explaining how a whole is made does not furnish an ontological explanation of the structure of the whole.

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Bergmann attributes to all gammaontologies a tendency to try and make do with one category only. He thinks that it is due to this tendency that gamma-ontologies use a function instead of a nexus since a nexus would bring with it the category of fact, i.e., of the complex connected by the nexus. The tendency to operate with a one-category-ontology is obvious in Platonic ontologies but also discernible in Aristotelian ontologies when one takes into account that only the substance has proper being. Bergmann discusses a contemporary gamma-ontology in which the tendency is explicit, namely, the ontology of N. Goodman who admits only the category of ‘individuals’. Goodman – who developed together with H. S. Leonard the so-called ‘calculus of individuals’, a mereological calculus – stresses that the arguments as well as the values of the sum-function (his gamma-function) in the calculus are individuals. Goodman offers his mereological calculus as an alternative to set theory and rejects sets in favour of his individuals. Bergmann also rejects sets, though for different reasons. For Goodman the core problem is that a ‘set of sets’ cannot differ from the sets, since otherwise there would be two different entities with the same content. Bergmann argues that a collection of entities is not itself an entity. Like Goodman and unlike many advocates of a mereological calculus Bergmann holds that mereological relationships are not ontologically neutral. He interprets the calculus as a gamma-ontology, i.e., a bundle view. For the gamma-ontologies the

part-whole relation is crucial since predication (the having of attributes) is viewed as such a relation. In epsilon-ontologies the fundamental tie of exemplification epsilon itself grounds the having of attributes. In contrast to Goodman and most analytical ontologists Bergman does not think that the construction of a mereological calculus provides a suitable method for research in ontology. He adopts the traditional view that the main method of scrutiny, clarification and articulation is dialectics. This procedure also aims at consistency and completeness like the construction of a calculus. However, the aims of the dialectical method are less formal and more substantial. It aims at more than formal consistency and more than formal completeness. The dialectical probing of an ontological system involves asking what the categories are and how they are connected and asking what the solutions are to the ontological problems and how the solutions depend on each other. With respect to the problem of relations Bergman points out that gamma-ontologies have to analyse all relations as internal relations, just because they do not admit genuine complexes, i.e., facts. This requires that we understand all relations either as relations of equality or else as relations of difference, which is not plausible in case of spatial and temporal relations, since gamma-ontologies have to rely on what Bergmann calls coordinate qualities, i.e., places and time points. These qualities conflict

BERGMANN, GUSTAV

with perception, however, since spatial and temporal perception is always relational in Bergmann’s view. However, coordinate qualities come in handy to solve the problem of individuation. Thus gamma-ontologists advocate individuation by a spatiotemporal differentiation. Bergmann critically remarks that this way of approaching the problem of individuation relies on laws which are not ontological but physical. The gamma-ontologist principle of individuation works, of course, only with concrete entities. Gammaontologists hold that only what is (temporally and/or spatially) localised, exists. Therefore, they have to reject universals. They solve the ontological problem of qualitative sameness and difference by quality individuals (nowadays mostly called ‘tropes’) and a relation of equality which must not be a universal. The relation of equality holds between concrete entities and is internal to them, which entails that it, too, is concrete. The epsilon-ontologists, to which Bergmann belongs, need to acknowledge universals, since in their ontology there is a category of entities without nature (so-called ‘bare particulars’). Entities without nature require that there be entities which are natures to be complete and they further require facts to connect the two sides. Bergmann thinks that the idea of a particular as individuator and bearer of properties necessarily leads to the conception of a bare particular.

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In an epsilon-ontology there are only two internal relations, namely, that of being a constituent of a fact and that of being diverse (numerically different) from. The former is a part-whole relation. The other relations are relational universals external to the relata and connected to them only in virtue of the respective relational facts. This holds also for the relation of being a spatial part. Places, which one would need to postulate if one were to conceive of spatial relations as internal relations, do not fit into epsilonontologies. Epsilon-ontologists have to operate with a relational theory of space. See also > Collectives and Compounds, Fact, Frege, Goodman, Segelberg, Sum, Universal. References and further readings

Bergmann, G., 1967, Realism, Madison: University of Madison Press. Bergmann, G., 1964, “Synthetic Apriori”, in Bergmann, G., Logic and Reality, Madison: University of Madison Press. Bergmann, G., 1964, “The Ontology of Edmund Husserl”, in Bergmann, G.: Logic and Reality, Madison. Goodman, N., 1951, The Structure of Appearance, Cambridge Mass. Erwin Tegtmeier

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Biological Parts The view that the living world is divided into part-whole hierarchies can already be found in ancient philosophy (cf. Aristoteles’ De partibus animalium) and it is deeply embedded in the biological sciences. Biologists represent objects as being constituted by a certain collection of organised parts. For example, cells are said to consist of a cell membrane that surrounds the cytoplasm that contains various organelles. Assumptions about part-whole relations are involved in classifications of biological objects into kinds (e.g., the assumption that fish have gills, whereas mammals have lungs). Furthermore, the methodological principle that one can understand the behavior of a biological object by decomposing it into its parts remains important for generating knowledge in the biological sciences (Bechtel/Richardson 2010). Despite this ubiquity and importance of part-whole relations to biology, the philosophical question of what it means for an object X to be a biological part of another object Y is still disputed. Does biological parthood require only mereological parthood? Does it require that X is a spatiotemporal part of Y (and how can this be specified)? Is one biological ‘whole’ demarcated from another by the many and intense causal interactions among its parts? Must the behavior, activity, or operation of a part be relevant to the behavior or functioning of a whole (in what sense)? An understanding of the conditions under which something is a biological part of some ‘whole’ is relevant

to the biological sciences since it might help to solve problematic cases: for example, when does a vesicle that is transported in a eukaryotic cell become a part of the Golgi apparatus? Is the case that is attached to the Caddisfly larva and that protects it against predators a part of the larva or does it belong to the larva’s environment? How can the parts of the human genome be identified? Philosophers have gone different ways in answering the question of biological parthood. Some try to develop a monistic account, that is, they seek to identify a single criterion or list of criteria that is universally applicable to all biological fields and that provides us with clear answers. Others adopt a pluralistic position and claim that in biological practice different “theoretical perspectives” (Wimsatt 2007: 182) or “partitioning frames” (Winther 2006: 475) can be found, which imply different criteria for individuating biological parts, and thus generate different decompositions. Some pluralists go even so far and argue that different decompositions of the same biological system often are not coincident and cannot be integrated into a single picture of what the system’s parts are. One way of seeking a monistic conception of biological parthood is to consult mereology. However, it quickly becomes clear that classical extensional mereology does not suffice in order to provide a criterion for biological parthood in particular (rather than for parthood in general). To see why, suppose you cut an earthworm with a scalpel into, let’s say,

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five slices of arbitrary length. According to mereology, these slices count as “proper parts” (e.g., Simons 1987) of the earthworm since the relation between slices and earthworm satisfies the mereological principles of antisymmetry, irreflexivity, and transitivity. The same holds for the body wall or the nervous system of the earthworm. But even though the slices, the body wall, and the nervous system all are mereological parts of the earthworm, only the latter two are biological parts. The reason is that the body wall and the nervous system, but not the slices, are things whose behaviors and properties biologists seek to explain and are the objects of biological reasoning, prediction, and intervention. Among those philosophers of biology who seek a monistic approach three major criteria for biological parthood are discussed: spatiotemporal inclusion, intensity of interactions, and causal relevance. Biological parts are frequently assumed to be spatiotemporal parts (e.g., Craver 2007, Leuridan 2012). But so far it has not been sufficiently analysed what the requirements of spatial and temporal inclusion amount to (cf. Kaiser 2015, Chapter V). Spatial inclusion seems to require the independent identification of a spatial boundary inside of which the parts must be located. Temporal inclusion is only possible if the relata of the parthood relation are not only continuants, such as objects, but temporally extended occurrents, such as processes, states, or events. Proponents of the interactionist account of biological parthood (e.g.

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Simon 1962; McShea 2000) argue that a set of parts can be picked out as a ‘whole’ because the parts of the whole interact more frequently and more intensively with each other than with objects in the environment. In other words, the intensity and bandwidth of causal interactions is assumed to be the criterion for identifying part-whole relations (in biology and in other areas). Gillett (2013) holds a view that seems to combine the notion of a spatiotemporal part and the interactionist approach. His main thesis is that an individual object X is a biological part of an individual object Y iff X “is a member of a spatiotemporally related team of individuals many of whose members bear powerful and/or productive relations to each other” (2013: 321). Already van Inwagen (1990: 81) has argued that biological parthood essentially involves causation. In the same spirit H. Mellor characterises biological parts as “working parts” (2008: 68) and claims that parts must have significantly large effects on the properties of the whole. This view clearly differs from the interactionist account as it requires the behavior, activities, or operations of biological parts to be causally relevant to the behavior or functioning of the ‘whole’. The holistic aspect, the reference to a whole with certain behaviour is missing in the interactionist account. The assumption that causal relevance is the proper parthood criterion need not have the controversial implication that part-whole relations are a special kind of causal relations. Additional assumptions or criteria of biological parthood, such as the spa-

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tiotemporal inclusion criterion, can prevent this implication as they violate conditions that are said to be characteristic for causal relations (e.g. asymmetry, asimultaneity, independence). Craver avoids confounding part-whole and causal relations by specifying the causal relevance criterion as “mutually manipulability” (2007: 141), which is a symmetrical relation between the parts of a mechanism and the behavior of the mechanism as a whole. See also > Activity, Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Body, Causation, Granularity, Natural Science, Processes, Temporal Parts, Transitivity, Universal. Bibliographical remarks

Craver, C. F., 2007. Chapter 4.8 contains an interesting discussion of the problems of the intensity-of-interaction approach and introduces Craver’s own account of constitutive relevance. Gillett, C., 2013. The most recent attempt to specify the causal criterion for biological parthood. Jansen, L.; Schulz, S., 2014. A very interesting attempt to bring together mereological considerations with the diversity of part-whole claims that are made in biological practice. Winther, R. G., 2006. Illustrates the importance of the concept of a part in the biological science.

References and further readings

Bechtel, W.; Richardson, R. C., 2010, Discovering Complexity: Decomposition and Localisation as Strategies in Scientific Research, Cambridge: MIT Press. Craver, C. F., 2007, Explaining the Brain. Mechanisms and the Mosaic Unity of Neuroscience, Oxford: Oxford University Press. Gillett, C., 2013, “Constitution, and Multiple Constitution, in the Sciences: Using the Neuron to Construct a Starting Framework”, Minds and Machines 23 (3): 309-337. Jansen, L.; Schulz, S., 2014, “Crisp Islands in Vague Seas: Cases of Determinate Parthood Relations in Biological Objects”, in: Calosi, C. and Graziani, P. (eds.) Mereology in the Sciences. Parts and Wholes in Contemporary Scientific Contexts, Cham: Springer, 163-188. van Inwagen, P., 1990, Material Beings, New York: Cornell University Press. Kaiser, M. I., 2015, Reductive Explanation in the Biological Sciences. An Ontic Account, Dordrecht: Springer. Kaiser, M. I., 2017, “Individuating Part-Whole in the Biological World”, in: Bueno, O.; Chen, R.-L.; Fagan, M. B. (eds.) Individuation across Experimental and Theoretical Sciences, Oxford: Oxford University Press. McShea, D. W., 2000, “Functional Complexity in Organisms: Parts as Proxies”, Biology and Philosophy 15: 641-668.

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Mellor, D. H., 2008, “MicroComposition”, Royal Institute of Philosophy Supplements 83 (62): 65-80. Simon, H. A., 1962, “The Architecture of Complexity”, Proceedings of the American Philosophical Society 106 (6): 467-482. Simons, P., 1987, Parts. A Study in Ontology, Oxford: Clarendon. Wimsatt, W. C., 2007, Re-engineering Philosophy for Limited Beings. Piecewise Approximations to Reality, Cambridge: Harvard University Press. Winther, R. G., 2006, “Parts and Theories in Compositional Biology”, Biology and Philosophy 21: 471-199. Marie I. Kaiser

Body The body is widely regarded as a template for spatial cognition, and since topology has been treated as a basis for mereological relations (Casati & Varzi 1999); the body would appear a paradigm source for deriving mereological structure. Recent linguistic work suggests, however, that the cross-cultural conception of the body (as reflected in language) does not display multi-level or otherwise rich mereological structure. The linguistic findings accord better with what is known from cognitive science about the multiple perceptual sources for segmentation of the body and the relational organisation of its segments. There are at least three distinct types of representation contributing to an

overall mental representation of the body: (1) a structural representation of the body based primarily on visual information that encodes parts and their topographic relations; (2) a dynamic representation based on sensory and motor inputs that encodes the on-line positions of body parts in relation to one another; and (3) a semantic-conceptual representation (Siri-gu, Grafman, Bressler, & Sunderland 1991). The structural and dynamic representations provide some determination of parthood, but they do not provide any uniform or consistent principles for relations between parts. Visual recognition of the body proceeds through a number of processing stages, each of which provides a distinct representation of the body. Low spatial frequency information provides a global representation. At progressively finer levels of resolution, parts can be identified, for example, the head, trunk, arms and legs, followed by hands and feet as distinct parts, then fingers, and so forth (Marr 1982; Palmer 1977). As well as being an object of external perception through vision, the body is uniquely apprehended internally through somesthetic and proprioceptive input. The primary somatosensory cortex provides a map of the body’s surface, but this is an undifferentiated, continuous representation. The primary motor cortex, however, is segmented according to the joints of the body (de Vignemont, Tsakiris & Haggard 2006). These body segmentations correspond quite well to the parts identified by vision, since the joints also provide image

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discontinuities relevant for visual segmentation. Parts of the body, then, can be identified through perception but the relations between parts are not so simply adduced. Gestalt principles of grouping – continuity, connectedness, closedness, proximity, similarity – all play a role in vision (Palmer 1977), thus are all candidate principles for determining conceptual relations in the organisation of the body part domain (cf. Palmer & Nicodemus 1977). Consistent with this, patients with autotopagnosia (an inability to localise or orient correctly to parts of the body) make errors based on these general principles, i.e., when asked to point to a specific body part these patients often mistakenly point instead to a part that is contiguous (e.g., wrist-elbow), or that shares functional similarity (e.g., kneeelbow; see Sirigu et al. 1991). These principles also operate in the conceptual organisation of body parts in normally developing children (Crowe & Prescott 2003). Turning to the linguistics of the body and its parts, cross-linguistic work by Brown (1976) and Anderson (1978) compared terms for the body and its parts in a wide range of the world’s languages and suggested an important role for mereological structure in body part nomenclature. There were three core claims: (1) the ‘body’ constitutes the ‘whole’ from which parts are recognised; (2) mereology is the core semantic principle structuring the relation between parts; and (3) there is a deep nested hierarchy, with up to 6 levels (e.g.,

‘fingernail-crescent’ is part of ‘fingernail’ is part of ‘finger’ is part of ‘hand’ is part of ‘arm’ is part of ‘body’). Subsequent work has challenged all three claims. First, linguistically, the superordinate entity for a system of body part terms need not be ‘body’, but may be ‘person’ instead. Some languages appear not to distinguish clearly between body and person (or ‘soul’; see discussion of Kuuk Thaayorre language in Majid, Enfield & van Staden 2006). The distinction between body and person has implications for other judgments about parts too. For example, ordinary language still allows meaningful talk about dismembered body parts. After van Gogh cut off his ear, the detached ear was still an ear (Cruse 1986). But the dismembered ear is no longer part of van Gogh’s body, although it may still be part of van Gogh the person. Regarding claim (2), mereology is just one of a number of possible types of conceptual relation between parts of the body, and there may be sub-types within mereology itself. Cruse (1986) distinguishes between segmental parts of the body, which have a greater degree of spatial cohesiveness (e.g., ‘head’, ‘arm’, ‘leg’) and systemic parts, which have a greater functional unity but may be spatially non-cohesive (e.g., ‘muscles’, ‘nerves’). Most cross-linguistic studies have focused on identifying segmental parts, and their relations. Swanson and Witkowski (1977) argue on the basis of data from Hopi that possession, rather than part, is the key relation between segments of

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the body (‘his arm’, ‘the hand has fingers’). Palmer and Nicodemus (1985), with data from Coeur d'Alene argue that spatial relations, such as contiguity, organize the domain (‘the hand is connected to the arm’). A collection of in-depth profiles of body part nomenclature in a range of languages (Majid et al. 2006) similarly casts doubt on the claim that body part nomenclature is organised mereologically to any significant extent. Possession plays a key role for some languages, such as Tidore, while various spatial relations hold for others, such as Punjabi. Mereological structure was found to play a role in a few languages, but it was marginal, only applying between terms referring to the limbs (which constitute only a fraction of the 100-plus inventory of terms for parts of the body). Finally, regarding claim (3) above, none of the languages investigated in Majid et al. (2006) yielded deep or systematic hierarchies of the kind predicted by Brown and Anderson. The cognitive and linguistic coding of the body and its parts sheds light on some core properties of mereology. For example, whether a part is named or not can make a difference. When English speaking children are asked to make mereological judgments between labeled and unlabeled parts (in the latter case by touching the lower arm and asking ‘Is this part of my arm?’), only parts that are not labeled are accepted as being in a part-whole relationship; labeled parts tend not to be accepted as being subparts of the whole (Johnson & Kendrick 1984).

Another problem is transitivity. If we take the three key properties of the part relation to include irreflexivity (nothing is a part of itself), asymmetry (if A is part of B, then B is not part of A) and transitivity (if A is part of B, and B is part of C, then A is part of C – see, for example, Simons (1987) then only the first two hold in the domain of the body, while transitivity appears not to. Adults experience uncertainty and a sense of absurdity when contemplating relations between parts. Brent Berlin nicely sums up the problem with making transitivity judgments: “while a fingernail is part of the finger and finger is part of the hand and hand is part of the arm, for most speakers of English it is not the case that a fingernail is part of the arm. In fact, to suggest that the finger is part of the arm is also a bit spooky” (quoted in Werner & Begishe 1970: 252). Thus, failure of transitivity is observed not only because of the shallowness of hierarchies in the domain of parts of the body, but also due to peculiarities of inference where such hierarchies do exist. From the preceding considerations one can conclude that – perhaps surprisingly – part-whole relations play a marginal role, if any, in how people conceptualize relations between parts of the body. Instead, parts of the body may be seen as related to each other in a range of ways (e.g., mereology, possession, contiguity), and languages appear to differ considerably. This is predicted by cognitive studies, which reveal multiple distinct sources of mental representation in segmentation of the body. Fitting-

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ly, there does not appear to be a single, unifying principle of relational organisation among elements of body part nomenclature across languages. If such principles exist, we suggest that they are more likely to be limited to distinct sub-systems such as the face, internal organs, or limbs. Further research aimed at extracting general principles in how parts of the body are related to one another will have to be more attentive to the large size and internal complexity of systems of body part nomenclature than studies available to date.

References and further readings

Andersen, E. S., 1978, “Lexical Universals of Body-part Terminology”, in Greenberg, J. H. (ed.) Universals of Human Language, Stanford: Stanford University Press, 335-368. Brown, C. H., 1976, “General Principles of Human Anatomical Partonomy and Speculations on the Growth of Partonomic Nomenclature”, American Ethnologist 3: 400-424. Casati, R.; Varzi, A. C., 1999, Parts and Places: The Structures of Spatial Representation, Cambridge and London: MIT Press.

See also > Animal, Biological Parts, Common Sense Reasoning About Parts and Wholes, Gestalt, Possessives and Partitives, Perceptual Wholes, Transitivity, Twardowski.

Crowe, S. J.; Prescott, T. J., 2003, “Continuity and Change in the Development of Category Structure: Insights from the Semantic Fluency Task”, International Journal of Behavioral Development 27: 467-479.

Bibliographical remarks

Cruse, D. A., 1986, Lexical Semantics, Cambridge: Cambridge University Press.

Andersen, E. S., 1978. A classic article on cross-linguistic representation of the body and mereological structure. Brown, C. H., 1976. Written independently around the same time as the Anderson article, and coming to similar conclusions about mereological structure of the body. Majid, A.; Enfield, N. J.; van Staden, M., 2006. A collection of papers describing 10 different languages, all of which show surprisingly little partonomic structure.

de Vignemont, F.; Tsakiris, M.; Haggard, P., 2006, “Body Mereology”, in Knoblich, G.; Thornton, I. M.; Grosjean, M.; Shiffrar, M. (eds.) Human Body Perception from the Inside out, Oxford: Oxford University Press, 147-170. Johnson, C. N.; Kendrick, K., 1984, “Body Partonomy: How Children Partition the Human Body”, Developmental Psychology 20: 967-974. Majid, A.; Enfield, N. J.; van Staden, M., 2006, “Parts of the Body: Crosslinguistic Categorisation”, special issue of Language Sciences 28: 137359.

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Marr, D., 1982, Vision, San Francisco: W. H. Freeman. Palmer, G. B.; Nicodemus, L., 1985, “Coeur d’Alene Exceptions to Proposed Universals of Anatomical Nomenclature”, American Ethnologist 12: 341-359. Palmer, S. E., 1977, “Hierarchical Structure in Perceptual Representation”, Cognitive Psychology 9: 441474. Simons, P., 1987, Parts: A Study in Ontology, Oxford: Oxford University Press. Sirigu, A.; Grafman, J.; Bressler, K.; Sunderland, T., 1991, “Multiple Representations Contribute to Body Knowledge Processing”, Brain 114: 629-642. Swanson, R.A.; Witkowski, S., 1977, “Hopi Ethnoanatomy: a Comparative Treatment”, Proceedings of the American Philosophical Society 121: 320-337. Werner, O.; Begishe, K.Y., 1970, “A Lexemic Typology of Navajo Anatomical Terms I: The foot”, International Journal of American Linguistics 36: 247-265. Asifa Majid Nicholas J. Enfield

Boethius In addition to his classic The Consolation of Philosophy, Anicius Manlius Severinus Boethius (born ca. 475-77 CE, died ca. 524-26) translated and commented on much of Aris-

totle’s logic. He was also the author of several important syntheses of ancient logic and of five short theological treatises, that applied Greek philosophical principles to the Incarnation and the Trinity. Given that much of Aristotle’s work was unavailable in Latin until the end of the twelfth century, Boethius’s works were indispensable to early medieval thinkers as they explored topics in logic and metaphysics. Two of Boethius’s syntheses, On Division (1998) and On Topical Differences (1990), were routinely parts of the later medieval logical curriculum (Marenbon 2003: 168-70). These two handbooks, as well as his commentary on Cicero’s Topics (1833), were major sources for medieval mereological principles. Types of wholes and parts. In On Di-

vision Boethius surveyed the many modes of division. Two of these modes are relevant for Boethius’s mereology. One is the mode of dividing a whole into its parts, the other is the mode of dividing a genus into its species. Boethius noted that there are many types of whole (1998: 38-40). Some wholes are continuous (e.g. bodies and lines); some are non-continuous (e.g. flocks and armies). Boethius also claimed that a universal, in so far as it is divisible into particulars, is a whole. And, finally, there are some wholes – most notably the soul – which consist of ‘powers’ (virtus or potentiae). In order to make an appropriate division of a whole, Boethius insisted, one should start by dividing the whole into “those parts out of which

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this very whole is perceived to consist” (1998: 38; cf. 1833: 334). In ‘manifold’ (multiplex) objects, this first division will likely be a division of the whole into heterogeneous parts. For example, a human body is divisible into the head, hands, chest, feet, and so forth. Yet, given that these wholes are manifold, they can be divided into other parts. A human body can also be divided either into homogenous parts (e.g. flesh and bones) or into matter and form. Boethius asserted that noncontinuous wholes and universal wholes are to be divided in the same manner – namely, by itemising the particulars that constitute the whole. Finally, the whole that consists of powers should be divided as follows: “Of soul one part is in plants, another in animals, and again, of that [soul] which is in animals one is rational, another sensitive” (1998: 40). Wholes versus genera. In all of his

treatments of mereology, Boethius sharply distinguished the genus from the whole. Nevertheless, he conceded that the genus is divided as if it were a kind of whole, and in definitions the genus behaves as if it were a part (1998: 36-8). The division of the genus was distinguished from the division of the whole in four ways (1998: 12-14). First, the whole is divided with respect to quantity, whereas the genus is partitioned with respect to quality. Second, every genus is ‘naturally prior’ to its species, whereas the whole is posterior to its parts (cf. 1833: 331). Third, whereas the parts

are the matter of the whole, the genus is in a sense the ‘matter’ for its species, since the genus and the differentiae (i.e. the ‘formal’ component) constitute the definition of the species. And, fourth, “the species is always the same as that which is the genus” (species idem semper quod genus est). The part is not always the same thing as its whole. In his commentary on Cicero’s Topics Boethius offered what seems to be an alternative analysis of this fourth criterion: the parts of the whole accept the name of the whole when conjoined and taken together, but not when taken individually (1833: 289 & 331). For example, the door is not a house and the roof is not a house. But when the door and the roof are attached to and taken together with the other parts of the house, they are called a house. In contrast, the name of the genus is predicable of each species of a genus taken alone and by itself. The species dog by itself is animal, and human by itself is animal. Difficulties. Boethius’s taxonomy of

wholes and his criteria for distinguishing genera from wholes left medieval interpreters with several puzzles (see, e.g., Abelard 1954: 1938; Radulphus Brito 1978: 44-6). Two of these puzzles are worth a special mention. First, it is unclear why Boethius thought that the universal, in so far as it is constituted by particulars, is a true whole, given that he could have followed other ancient logicians (see Boethius 1998: xliv-xlviii) and treated the division of the universal into

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particulars as a mode of division distinct from both the division of the genus and the division of the whole. The second and the fourth criteria that Boethius employed to distinguish the genus from the whole would seem to group the division of the universal into particulars together with the division of the genus. It is perhaps because Boethius treated the universal as if it were a whole that some early medieval thinkers considered the universal to be nothing more than a collection of particulars. Second, Boethius’s claim that a whole depends upon its parts seemed to be an endorsement of mereological essentialism. Boethius insisted that “each and every thing gets its being from those things which compose it” (On the Trinity, II, 94-95). But in this passage, Boethius might be asserting merely that the parts are the cause of the whole’s coming into existence, not that they are required in order for the whole to continue to exist. In On the Trinity, the primary things that compose a whole are its matter and form (cf. Arlig 2009: 133-37). The claim that the whole depends upon its parts in order to persist would be more plausible if it were taken to mean that the whole depends upon its matter and form. Unfortunately, in On Division Boethius did not clearly restrict the scope of his claim about dependence to matter and form, or to any other special set of parts. It was left to medieval interpreters to refine Boethius’s principle so that the whole only depends upon its essential parts. See also > Abelard, Aristotle's Theory of Parts, Aristotle's Theory of

Wholes, Medieval Mereology, Mereological Essentialism, Radulphus Brito, Totum Potentiale, Universal. Bibliographical remarks

Boethius, 1998. The definitive edition of this important text; it includes an excellent translation and very useful notes. Henry, D. P., 1991. Section 1.3 (pp. 37-45) provides a close reading of several important passages in On Division. Radulphus Brito, 1978. Radulphus’s examination of several of Boethius’s mereological principles (see especially book II, quaestio 9) is remarkably subtle. References and further readings

Abelard, P., 1954, De divisionibus incipit, in M. Dal Pra, ed., Scritti filosofici, Milan: Fratelli Bocco. Arlig, A., 2009, “The Metaphysics of Individuals in the Opuscula sacra”, in Marenbon, J. (ed.), The Cambridge Companion to Boethius, Cambridge: Cambridge University Press, 129-54. Boethius, 1833, In topica Ciceronis commentaria, ed. J. C. Orelli, in Cicero Opera Omnia, vol. 5, pt. 1, Turin: Orelli, Fuesslini & Co. Boethius, 1990, De topicis differentiis, in D. Z. Nikitas, ed., De topicis differentiis und die byzantinische Rezeption dieses Werkes, Athens and Paris: The Academy of Athens and Librairie J. Vrin.

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Boethius, 1998, De divisione liber, ed. and trans. J. Magee, Leiden: E. J. Brill Boethius, 2000, De consolatione philosophiae / Opuscula Theologica, ed. C. Moreschini, Munich and Leipzig: Saur. Henry, D. P., 1991, Medieval Mereology, Amsterdam: B. R. Grüner. Marenbon, J., 2003, Boethius, Oxford: Oxford University Press. Radulphus Brito, 1978, Commentary on Boethius’ “De differentiis topicis” in Green-Pedersen, N. J. (ed.) Cahiers du l’Insti-tut du Moyen-age Grec et Latin 26, 1-92. Stump, E. trans. (1978) Boethius’s ‘De topicis differentiis’, Ithaca, NY: Cornell University Press. Stump, E. trans. (1988) Boethius’s ‘In Ciceronis topica’, Ithaca, NY: Cornell University Press. Andrew Arlig

Bolzano, Bernard I. Life and works. Bernard Bolzano (full name: Bernardus Placidus Johann Nepomuk Bolzano; *1781 Prague; †1848 Prague) was a philosopher, mathematician, and theologian. After he was ordained to priesthood in 1805, he held a chair for philosophy of religion from 1806 until 1819, when – on a charge of heresy – he was suspended and put under police supervision. Until his death, he was forbidden to teach and to preach.

Although officially banned from publishing, Bolzano managed to get several of his writings published during his lifetime, among them his three major philosophical works: (i) The Athanasia (1827, 2ndedition 1838; henceforth: AT), a metaphysical treatment inspired by Leibniz’s Monadology. (ii) The voluminous Theory of Science (Wissenschaftslehre, 1837; henceforth: TS), in which he covers an impressively broad range of topics. In particular, he discusses numerous issues from the philosophy of language and logic (TS vols. I & II), metaphysics (TS vols. I & II), and epistemology (TS III). Finally, he carefully examines didactic issues about how to write scientific textbooks (TS IV). (iii) The Textbook of the Science of Religion (Lehrbuch der Religionswissenschaft, 1834; henceforth: TSR), a transcript of his university lectures on the philosophy of religion. Posthumously, a small treatise on the notion of infinity, Paradoxes of the Infinite (Paradoxien des Unendlichen, 1851; henceforth: PI) was published. His major work on (the philosophy of) mathematics, the Theory of Magnitudes (Größenlehre) remained unfinished; the draft is published in his Collected Works (Gesamtausgabe, for short: GA). II. Contributions to mereology. a.

Primary Sources. The concept of a collection (Inbegriff) plays a central role in Bolzano’s philosophy. In §6 of the Introduction to the Theory of Magnitudes – henceforth: ITM – he

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even calls it one of the most elementary concepts available to us (similarly: PI §3). Bolzano’s views on the topic of collections are rather complex and they are elaborated in several longer passages. The most important sources for the general conception (particularly for explications of the central concepts) are (i) TS I: §§82–88, §135; (ii) PI §§3–9; (iii) ITM §§6– 20, 88–91, 92–93, 119, 144. But there are many more occasions on which Bolzano puts the conception to use; it plays, e.g., an important role in his theory of propositions and concepts (as developed in the TS), in his conception of infinity (as developed in the PI), and it is omnipresent in ITM §§88–187, where Bolzano develops some basic foundations of mathematics. b. Bolzano’s Concept of a Collection. A collection, according to Bolzano, is any entity that is composed out of parts. Since many things we ordinarily deal with have parts, there are many collections (in Bolzano’s sense of the word) that we are quite familiar with: for instance, ordinary garden objects, such as shrubs, snails, or lawn seats are collections, because they are wholes that possess parts (e.g. leaves, feelers, and legs). Furthermore, every entity either has parts or has none – since the latter case is rare, most entities we may think of are Bolzanian collections. Some of Bolzano’s own examples of collections are: bodies, groups of people, heaps, acts of judging (their parts being acts of conceiving), the world (regarded as the collection of all fi-

nite beings), propositions (their parts being concepts), sciences (their parts being truths), arbitrary collections of any objects. Bolzano takes the concepts collection, compositeness (Zusammengesetztheit), and part (of) to be most intimately related. He surmises that the concept compositeness is primitive and not analysable, while the concept collection can be analysed as something that has compositeness (TS I: §82, ITM §6), and the concept part of x can be analysed as something that (together with something else) composes the collection x (TS I: §83.1). Note that for Bolzano, the concepts of composing and parthood must be carefully distinguished. That both x and y are parts of z does not entail that x and y together compose z. It only entails that x together with something (but not necessarily: with y) composes z, and that y together with something (but not necessarily: with x) composes z. (A consequence of this distinction will become clear below in section d.) From what has been said so far, it is clear that Bolzano’s collections must be distinguished from the sets of contemporary set theory. Firstly, every collection in Bolzano’s sense has at least two parts; neither are there any singleton collections nor is there a null collection (both is meant to follow from Bolzano’s analysis of the concept of a collection). Secondly, while sets are abstract objects, many concrete objects (shrubs and snails) are collections. (Notice that Bolzano does not himself employ the concepts concrete and abstract, but the related

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– although non-equivalent – concepts of actuality and non-actuality; on these notions cf. Schnieder 2002: 21– 27 and Schnieder 2007.) c. Designators of Collections. Bolzano recognises some canonical ways of designating collections. One is to give an explicit list of the things of which a collection is composed: COLL the collection of A1 … and An. Another is to use ‘and’ in a bare plural phrase of the form AND A1 … and An (TS I: §83, PI §3, ITM §6). Bolzano distinguishes between two readings of AND (TS I: §83): in its collective reading, such a phrase has a singular designation, namely the collection of A1 … and An (hence it is equivalent to the corresponding phrase of the form COLL). In its distributive reading, the phrase has a plural designation, namely all parts of the collection of A1 … and An. The distributive reading is forced when we insert an ‘each of’ into AND and use expressions such as EACH each of A1 … and An. So, according to Bolzano EACH designates every part of the collection of A1 … and An. A third canonical form of designating collections is by imposing a condition on its parts and using an expression of the form TOTAL the totality of Fs (das All der F). Bolzano takes such an expression to denote a collection which contains all and only the Fs as parts (TS I: §86.3).

d. Some Mereological Principles. Bolzano endorses a variety of fundamental and derivative principles about collections. For reasons of space, I present here only a selection of some noteworthy fundamental principles here: P.1 Universalism: Given two or more objects there is the collection of these objects – no matter how heterogeneous these objects may be (PI §3, ITM §6). Thus, Bolzano recognises collections of arbitrarily chosen concrete objects, collections of arbitrarily chosen abstract objects (such as numbers or propositions), and also crosscategorial collections, such as a collection composed of a number and a tornado. P.2 Objectivity: Collections are mindindependent, they are not created by thought (PI §3, ITM §§6, 7). P.3 Atomism: Every entity is either a mereological atom (i.e. it does not have any parts), or it is – at the final level of analysis – composed of mereological atoms (AT 57, TS I: §61, PI §50). P.4 Non-transitivity of parthood: Parthood is not a strictly transitive relation (TS I: §83, ITM §9). Bolzano provides some arguments for these principles: Re P.1: While Bolzano does not directly argue for P.1, one can extract some indirect arguments from his writings: Firstly, as pointed out above, Bolzano thought that a phrase of the form ‘A and B’ (in its collective reading) designates a collection. If that semantic thesis is accepted,

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one should acknowledge collections whenever such a phrase can be used (in a collective reading) as the subject term of a true statement; and given any two objects, there seem to be some truths about them (if only the simple truth that they are two objects). Secondly, Bolzano wants to reduce relations to properties, so that a relation between A and B is a property of the collection of A and B. If that reduction is accepted, one should acknowledge a collection of A and B whenever one accepts that A is somehow related to B (e.g. by distinctness). In harmony with the principle of universalism, he thinks that canonical designators for collections are seldom empty. But he holds that some constraints must be satisfied for such designators to designate a collection. First, let us consider list-like designators of collections (e.g. designators of the form COLL); at different occasions, Bolzano mentions the following constraints for their not being empty: (C1) none of the entries in the list designates an object designated by another entry in the list (PI §3, ITM §14); (C2) none of the entries designates a part of an object designated by another entry (ITM §14); (C3) all entries designate entities which are disjoint (ITM §99). (Actually, this condition is never explicitly proposed for designators of collections in general, but only for designators of what Bolzano calls Mengen; on Mengen see below, section e.).

These constraints directly carry over to the central concept involved in the semantics of the designators, i.e. the concept of composition. They do not, however, entail constraints on the concept of parthood (pace Rusnock 2013: 156). One may accept constraint (C3) and deny that two overlapping things x and y ever compose a third thing z, without holding that two overlapping things cannot both be parts of z. For illustration, consider the following rectangle:

Since the left half of the rectangle overlaps with its grey area, Bolzano would find it objectionable to say that its left half and its grey area compose the rectangle. He is not thereby committed to deny that both the left half and the grey area are parts of the rectangle. Given Bolzano’s notion of parthood, they are (which is, of course, a desirable outcome): The left half is a part of the rectangle because there is something (namely its right half) together with which it composes the rectangle, and the grey area is a part of the rectangle because there is something else (namely the black area) together with which it composes the rectangle. Let us turn to designators of the form ‘the totality of Fs’. Bolzano realised

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in his philosophical notebooks that if their semantics are understood as he proposed (see above, section c.), some such phrases cannot denote anything on pain of contradiction: ‘A remarkable example of a contradictory concept is that of ‘the totality of things’ or the sum of all things. For, this sum would have to belong to itself as a part, if it were indeed something.’ (GA 2B, 18/2, p. 29) Bolzano here presupposes a further principle, probably taking it as analytic of the concept of a part (see e.g. TS I: § 19, p. 79):

easy to follow and it cannot be discussed here (but see Krickel 1995: 187–94).

P.5 Irreflexivity of parthood: If x is a part of y, then x ≠ y.

(2) each of Gaius and Titus is a person

(NB: Despite his described worries, Bolzano sometimes talks about the totality of everything without indicating any reservations about it; see TS III: §360 or PI §11. Moreover, in the margin of the passage quoted from his notebooks, Bolzano later added a note in which he denies the contradictoriness of the totality of objects; on this matter, see Rusnock 2013: 165.)

is true, Bolzano concludes that parthood cannot be transitive. Otherwise, (1) would designate not only Gaius and Titus, but also all their parts; but then (2) would be false because the parts of Gaius and Titus are not persons.

Re P.2: Bolzano repeatedly stresses this thesis, thereby opposing a view he formerly had endorsed when he followed the Leibnizian tradition of regarding collections as unreal objects of the mind (cf. Künne 1998: 241). Bolzano argues for P.2 in PI §14, ITM §§6,7. Re P.3: In TS I (§61), Bolzano develops an argument to the effect that every composed entity must, at a final level of analysis or decomposition, be composed of atomic entities. However, Bolzano’s reasoning is not

Re P.4: In TS I (§83), Bolzano argues for P.4 based on his semantic thesis that the word ‘and’ introduces the concept of a collection. Consider an ‘and’-phrase that forces a distributive reading: (1) each of Gaius and Titus. According to Bolzano (1) designates every part of the collection of Titus and Gaius. Since the following sentence:

In ITM (§9), Bolzano justifies the claim about non-transitivity without recourse to the described semantic claim and uses a direct example: A state, he holds, is a collection of people. But while people have body parts, body parts are not parts of a state. e. Bolzano’s Distinction between Kinds of Collections. While Bolzano takes parthood not to be transitive in general, he apparently thinks that there are collections such that every part of a part of it is a part of the collection. In fact, he distinguishes between different kinds of collections, depending on whether or not

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(i) the arrangement of their parts is irrelevant to their identity, and on whether or not (ii) the parts of their parts are parts of the whole. If a collection satisfies condition (i) then Bolzano calls it a set (Menge), and if it also satisfies condition (ii) he calls it a sum (Summe) (cp. TS I: §84, PI §§4–5, ITM §§88, 92). Another sort of collection which Bolzano deems particularly important for the philosophy of mathematics he calls series (Reihen, see TS I: §85; PI §7; ITM §144): Roughly, a series is a collection governed by a general law which puts its parts into a linear order (in ITM §144, he further requires that a series is a set). An example of a finite series is the collection of monographs written by Russell, arranged in the order of their publication. An example of an infinite series is the collection of natural numbers which starts with 1 and in which the successor of a member of the series is obtained by multiplying the member by two: 1, 2, 4, 8, … What remains unclear is whether Bolzano conceived of the said distinctions as ontological ones, so that they describe collections of distinct kinds that differ with respect to their compositional natures. While some of his formulations support this interpretation, others suggest that he conceived of the distinctions as merely representational ones, so that they correspond to concepts that characterise collections in different ways. One and the same collection could then be subsumed under both a pure

collection-concept and, say, a setconcept. (On the issue, see Krickel 1995: 94–98, Behboud 1997; on Bolzano’s notions of sums, sets, series see also Lapointe 2011: ch. 9 and Rusnock 2013.) f. Applications. To conclude this survey, I will mention three philosophical issues that Bolzano brought the notion of a collection to bear upon: (i) The theory of relations: he proposes to analyse the concept of a relation holding between x and y as an attribute of the collection of x and y (TS I: §80, ITM §21). There is a noteworthy tension between this proposal and Bolzano’s views on the semantics of designators of the form of COLL. Obviously there are relations between a whole and its parts – my whole body weighs about ten times as much as my right leg. On Bolzano’s analysis of relations, the example would give him a reason to lift his constraints (C2) and (C3) on the non-emptiness of designators of the form of COLL (see above); he then must accept that ‘the collection of A and B’ can denote a collection although B is a part of A. What is worse, an entity can also stand in a relation to itself – I am identical to myself, while Narcissus admires himself. Together with Bolzano’s analysis of relations, this apparently creates a problem not only for Bolzano’s constraint (C1) on the non-emptiness of designators of the form of COLL (the examples require that some designators of the form ‘the collection of A and A’ denote a collection), but even for his tenet that every collection contains at least two

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non-identical parts (Bolzano saw this second problem; in WL I: §80, note, he tries to argue that there are no genuine cases in which an object is related to itself). (ii) Ontology: Bolzano defends cosmological arguments for the existence of God (see TSR I: §67; cf. Ganthaler & Simons 1987, Löffler 1999) and of substances (see AT 22; cf. Künne 1998, Schnieder 2002: ch. II.4.). The arguments rely on P.1, insofar as Bolzano presupposes the existence of the huge collection of all conditioned objects (in arguing for the existence of God) and the existence of the collection of all adherences (in arguing for the existence of substances) (NB: an adherence is a concrete object which is a feature of another concrete object: it is what sometimes is called an individual accident or, more recently, a trope). In presupposing that those huge collections are concrete objects, the arguments also seem to rely on a principle which has not been mentioned so far: P.6 Inheritance of ontological status: A collection inherits the ontological status of its parts, such that a collection of concrete objects is itself a concrete object, while a collection of abstract objects is itself an abstract object (cp. AT 22, TSR §67, TS I: §79.3). It is unclear, however, which status Bolzano would have ascribed to cross-categorial collections. (iii) The theory of numbers: Bolzano regards numbers as a special case of magnitudes, where a magnitude is defined in terms of collections (see

TS I: §87, Reine Zahlenlehre §1; on Bolzano’s conception of numbers cf. Behboud 2000, Krickel 1995: ch. D.I., Rusnock 2000, Simons 1999). See also > Collectives and Compounds, God, Fusion, Sum, Transitivity. Bibliographical remarks

Bolzano, B., [TS]. One of the principle sources for Bolzano’s conception of collections; contains many applications of his conception to issues in logic, metaphysics, the philosophy of language, and the philosophy of mind. Bolzano, B., [PI]. Another of the principle sources for Bolzano’s conception of collections; contains applications to the philosophy of infinity. Bolzano, B., [ITM]. Another of the principle sources for Bolzano’s conception of collections; contains important applications to the foundations of mathematics. Krickel, F., 1995. A book-length treatment of Bolzano’s conception of collections. Very detailed, thorough and helpful (unfortunately, its layout somewhat lowers its readability). Simons, P., 1997. Simons distinguishes between different notions of collective entities and argues that they are conflated in Bolzano’s theory. Behboud, A., 1997. Behboud discusses Simons (1997) and makes a tentative proposal of how to defend Bolzano from some of the charges of

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confusing different concepts of collective entities. Rusnock, P., 2013. Rusnock develops a new interpretation of Bolzano’s concept of a sum, arguing that although Bolzano’s presentation of his conception of sums is defective, the conception itself is not, and even has important applications in the foundations of arithmetics. References and further readings

Bolzano, B.: [AT] Athanasia, Sulzbach: Seidel (21838). Bolzano, B.: [GA] Gesamtausgabe 1969–…, Stuttgart: FrommannHolzboog. Bolzano, B.: [ITM] Einleitung zur Größenlehre. In: GA vol. II.2 A 7. Bolzano, B.: [PI] Paradoxien des Unendlichen (ed. by Fr. Přihonský), Leipzig: Reclam (1851). Bolzano, B.: [TS] Wissenschaftslehre, 4 vols. Sulzbach: Seidel (1837). In: GA vols. I.11–I.14. Bolzano, B.: [TSR I] Lehrbuch der Religionswissenschaft I. In: GA vol. I.6. Bolzano, B.: Reine Zahlenlehre. In: GA vol. II.2 A 8. Behboud, A., 1997, ‘Remarks on Bolzano’s Collections’, Grazer Philosophische Studien 53: 109-116. Behboud, A., 2000, Bolzanos Beiträge zur Mathematik und ihrer Philosophie, Bern: Bern Studies in the History and Philosophy of Science.

Ganthaler, H.; Simons, P., 1987, “Bernard Bolzanos kosmologischer Gottesbeweis”, Philosophia Naturalis 24: 469-475. Krickel, F., 1995, Teil und Inbegriff, Sankt Augustin: Academia Verlag. Künne, W., 1998, “Substanzen und Adhärenzen – Zur Ontologie in Bolzanos Athanasia”, Philosophiegeschichte und logische Analyse 1: 233250. Lapointe, S., 2011, Bolzano’s Theoretical Philosophy – An Introduction, Palgrave MacMillan. Löffler, W., 1999, “Bolzanos kosmologischer Gottesbeweis im historischen und systematischen Vergleich” in: Morscher 1999: 295-316. Morscher, E. (ed), 1999, Bernard Bolzanos geistiges Erbe für das 21. Jahrhundert, Sankt Augustin: Academia Verlag. Rusnock, P., 2000, Bolzano’s Philosophy and the Emergence of Modern Mathematics, Amsterdam: Rodopi. Rusnock, P., 2013, “On Bolzano’s Concept of a Sum”, History and Philsoophy of Logic 34: 155-69. Schnieder, B., 2002, Substanz und Adhärenz. Bolzanos Ontologie des Wirklichen, Sankt Augustin: Academia Verlag. Schnieder, B., 2007, “Mere Possibilities – Bolzano’s Account of NonExisting Entities”, Journal of the History of Philosophy 45: 525-50. Simons, P., 1997, “Bolzano on Collections”, Grazer Philosophische Studien 53: 87-108.

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Simons, P., 1999, “Bolzano über Zahlen” in Morscher 1999: 217-229. Benjamin Schnieder

Boolean Algebras Partial orderings and mereological Systems. The present paper focuses

on the inter-relation between mereological systems and Boolean algebras and the investigation of modeltheoretic and algorithmic properties of the corresponding theories. Some of these topics are discussed in a broader context in (Herre 2010). Throughout the paper the standard notation of first order logic and model theory is used, and familiarity with the following notions is presupposed: first order theory, model, relational structure, elementary equivalence, elementary type, and semantic deduction (for these notions see Chang 1977, Hodges 1993). A mereological system M = (E, ≤) is a relational structure which is determined by a domain E of entities and a binary relation ≤ on E, called partof relation. The theory of all mereological systems, denoted by M, is called basic mereology. M is specified by the following axioms: (M1) ∀x (x ≤ x), (reflexivity) (M2) ∀x y (x ≤ y ∧y ≤ x → x = y), (anti-symmetry)

a rather weak theory that will be extended by addition of further axioms. The formulation of the most important mereological standard theories uses the following definitions. (D1) x < y := x ≤ y ∧x ≠ y (proper part) (D2) ov(x, y) := ∃z (z ≤ x ∧ z ≤ y) (overlap), disj (x, y) := ¬ov (x, y) (x and y are disjoint) (D3) sum(x, y, z) := ∀w(ov (w, z) ↔ ov (w, x) ∨ov (w, y)) (mereological sum) (D4) intersect(x, y, z) := ∀w(w ≤ z ↔ w ≤ x ∧w ≤ y) (mereological intersection) (D5) compl(x, y) := ∀w(w ≤ x ↔ ¬ov (w, y)) (x is absolute complement of y) (D6) diff(x, y, z) := ∀w(w ≤ z ↔ w ≤ y ∧ ¬ ov (w, x)) (z = y - x is the difference between y and x)) The subsequent axioms belong to the standard setting of mereology. They are divided into axioms pertaining to several versions of supplementation and in axioms related to the fusion or mereological sum of entities. Principles of Supplementation and Extensionality (M4) ∀xy (y < x → ∃z (z < x ∧disj (z, y))) (weak supplementation principle)

(M3) ∀xyz (x ≤ y ∧y ≤ z → x ≤ z), (transitivity)

(M5) ∀xy (¬ y ≤ x → ∃z (z ≤ y ∧disj (z, x))) (strong supplementation principle)

The basic mereology M is the first order theory of partial orderings; it is

Minimal mereology, denoted by MM, is the theory containing exactly

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the axioms {M1, M2, M3, M4}. In (Simons 1987) minimal mereology is considered as constitutive for understanding the part-of relation. The theory which includes exactly the axioms M1, M2, M3, M5 is called extensional mereology and is denoted by EM. From EM follows the principle ∀ x y (∀z (ov (z, x) ↔ ov (z, y)) → x = y) (extension principle). Classical first order mereology. The

following axioms state the existence of the mereological sum and intersection. (M6) ∃z (sum(x, y, z)) (existence of the mereological sum) (M7) ov(x, y) → ∃z (intersect(x, y, z)) (existence of intersection if y and x overlap) For axiom (M8), stating the existence of difference, note that the operation of difference presupposes the strong supplementation axiom. (M8) ∀x y (¬x ≤ y → ∃z (diff(x, y, z))) (existence of difference) Classical mereology, denoted by CM, is defined by CM = EM ∪{M6, M7, M8}. Hence, the existence of the mereological sum, intersection, and difference is postulated. The following conditions follow from CM. sum(x, y, z)∧ sum(x, y, z’) → z=z’ intersect(x, y, z)∧intersect(x, y, z’) → z=z’ diff(x, y, z) ∧diff(x, y, z’) → z=z’ The uniqueness of the third argument of these relations justifies the intro-

duction of the following binary operations: sum (x, y, z) := z = x ∪ y, intersect (x, y, z) := z = x ∩ y, diff (x, y, z) := z = y - x. Furthermore, the sum of x and y is the least upper bound of {x,y}, and the intersection of x and y is the greatest lower bound of {x,y}. Additional axioms state the existence of a greatest and a least element. (M9) ∃u ∀x (x ≤ u) (There is a greatest element) (M10) ∃u ∀x (u ≤ x) (There is a least element) The least and the greatest element, respectively, are uniquely determined; they are denoted by 0 and 1, respectively. In order to combine the previous axioms with (M9) and (M1), (M6) through (M8), and associated definitions, must be modified by replacing the relation ov (x, y) by the relation ov’(x, y), defined as follows: ov’ (x,y) := ∃ z (z ≤ x ∧z ≤ y ∧ z ≠ 0), and disj’(x,y) := ¬ov’(x,y). Classical Mereology CM with a greatest element, a least element, or both can then be defined as follows. CM1 = CM ∪ {M9} CM0 = CM ∪ {M10} CM0 ,1 = CM ∪ {M9, M10}. Finally, let ϕ (x) be a formula of signature ≤ with the free variable x (where a formula (of FOL) of signature Aristotle's Theory of Parts, Aristotle's Theory of Wholes, Mereological Essentialism, Husserl, Intentionality, Meinong, Ontological Dependence, Stumpf, Substance, Twardowski. References and further readings

Antonelli, M., 2001, Seiendes, Bewußtsein, Intentionalität im Früh-

werk von Franz Brentano, Freiburg/München: Alber. Baumgartner, W.; Simons P., 1992/ 93, “Brentanos Mereologie”, Brentano Studien 4: 53-77. Brentano, F., (11862) 1960, Von der mannigfachen Bedeutung des Seienden nach Aristoteles, unver. Reimpr., Hildesheim: Olms. Eng. transl. by George, R., 1975, On the Several Senses of Being in Aristotle, Berkeley: University of California Press. Brentano, F., 11874, 2008, Psychologie vom empirischen Standpunkte. Von der Klassifikation der psychischen Phänomene, Sämtliche veröffentlichte Schriften, Band I, Binder, Th.; Chrudzimski, A. (eds.) Heusenstamm and Frankfurt: Ontos Verlag. Eng. transl. by Rancurello, A. C.; Terrell, D. B.; McAlister, L. 1973 (2nd ed., intr. by P. Simons, 1995), Psychology from an Empirical Standpoint, London: Routledge. Brentano, F., 1933, Kategorienlehre, A. Kastil, ed., Leipzig: Meiner. Eng. transl. by Chisholm, R.; Guterman, N., 1981, The Theory of Categories, The Hague: Nijhoff. Brentano, F., 1982, Deskriptive Psychologie, Chisholm, R. M.; Baumgartner, W. (eds.) Hamburg: Meiner. Eng. transl. by B. Müller, 1995, Descriptive Psychology, London: Routledge. Chisholm, R. M., 1982, “Brentano’s theory of substance and accident”, in Chisholm, R. M. (ed.) Brentano and Meinong Studies, Amsterdam: Rodopi, 3-16.

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Chisholm, R. M., 1986, Brentano and intrinsic value, Cambridge: Cambridge University Press. Chrudzimski, A.; Smith, B., 2004, “Brentano’s ontology: from conceptualism to reism”, in Jacquette, D. (ed.) The Cambridge Companion to Brentano, Cambridge: Cambridge University Press, 197-219. Mulligan, K.; Smith, B., 1985, “Franz Brentano on the ontology of mind”, Philosophy and Phenomenological Research 45-4: 627-644. Mulligan, K.; Smith, B., 1982, “Pieces of a Theory”, in Smith, B. (ed.) Parts and Moments. Studies in Logic and Formal Ontology, Munich and Vienna: Philosophia Verlag, 15-108. Simons, P., 1988, “Brentano’s Theory of Categories: A Critical Appraisal”, Brentano Studien 1: 47-61. Smith, B., 1994, Austrian Philosophy: The Legacy of Franz Brentano, Chicago and LaSalle: Open Court. Alessandro Salice

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Comments on “The Calculus of Individuals and its Uses”

C Calculus of Individuals, Comment by Henry S. Leonard. Prefatory note. Henry S. Leonard Sr.

died suddenly in 1967 of a heart attack. At that time he was working on a collection of his papers (some of which had not been published), with a view to publishing them as a book. He planned to write comments on each paper, but only three or four of the commentaries were completed. His Comments on his paper “The Calculus of Individuals and its Uses” (1940), written jointly with Nelson Goodman, were completed but have never been published before. Two remarks may be of help to the reader of the following comments. First, the Comments are written as though we can expect to find the paper itself following the Comments, but here that is not the case. Second, the page numbers provided in the Comments in the reference to Nelson Goodman’s book The Structure of Appearance are for the first edition. For the second edition of Goodman’s book, the reference would be to pages 46-56. For the third edition, the reference would be to pages 33-40. Henry S. Leonard, Jr.

Although Goodman and I published “The Calculus of Individuals” only in 1940, such a calculus had for a long time been occupying our attention, both independently and collaboratively. Concern with a part-whole relation between individuals was a major one in Goodman’s Honors Thesis, submitted to the Harvard Department of Philosophy when he was a senior in 1928. A formal development of the calculus constituted Chapter IV of my doctoral dissertation, submitted to the Harvard department in December, 1930. In the fall of that year, as I was writing the thesis, Goodman and I met together many times for exchanges of ideas. In December of 1936, we presented a collaborative paper on the calculus before a joint meeting of the Association for Symbolic Logic and the American Philosophical Association. The version of the calculus which appeared in 1940 in The Journal of Symbolic Logic, and which is reproduced below, differs only in minor details from the paper which we read in 1936. All of this work on the part of Goodman and myself had gone on before we ever heard of the analogous work of Lesniewski, referred to in Part I and footnote 8 of the paper that follows. The earliest version of the calculus, in my dissertation in 1930, differed from the later versions in certain significant respects. It was presented as an interpolation in Whitehead and

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Russell's Principia Mathematica between *14 and *20. Hence, it did not include such general notions as those of the fusion and the nucleus of a class (I.03 and I.04, below). Instead of taking the relational expression ‘x y’ (i.e., ‘x is discrete from y’) as primitive, it took the operation ‘x + y’ as primitive. It rested on more postulates than did the later versions. (In fact, the 1936 version still had five postulates). If responsibilities can be divided in a collaborative enterprise, I believe that it may be fairly stated that the major responsibility for the formal calculus (Part II, below) was mine, while the major responsibility for discussions of applications (Part III) lay with Goodman. Goodman has subsequently presented the calculus in a somewhat different form in The Structure of Appearance, pp. 42 to 51. That account takes ‘x y’ as primitive. It also omits the generic concepts of fusion and nucleus (I.03 and I.04, below) and instead of definitions like I.04 to I.08, it includes the more limited analogues to I.04' to I.08'.

References and further readings

Leonard, H.S; Goodman, N., 1940, “The Calculus of Individuals and its Uses,” The Journal of Symbolic Logic 5: 45-55. Goodman, N., 1951, The Structure of Appearance, Cambridge: Harvard University Press; 2nd ed. 1966, Indianapolis, Bobbs-Merrill; 3rd ed. 1977, Dordrecht, Holland, D. Reidel.

Whitehead, A. N.; Russell, B., 192527, Principia Mathematica, 2nd ed., 3 vols., Cambridge, The University Press. Henry S. Leonard, Sr.

Carnap, Rudolf Rudolf Carnap (1891–1970) studied philosophy, mathematics, and physics at the universities of Jena and Freiburg. In 1921 he completed his PhD, under the supervision of the neo-Kantian Bruno Bauch, with a dissertation presenting a Kantianstyle investigation of space (Der Raum, 1922). In 1925 he moved to Vienna and joined the Vienna Circle. In 1928 he obtained his Habilitation, based on a thesis entitled Der logische Aufbau der Welt (Carnap 1928; translated into English in 1967 as The Logical Structure of the World). In 1936 he emigrated to the United States, where he taught at the universities of Chicago and Los Angeles. Among his most important works are the Aufbau (Carnap 1928) and Logische Syntax der Sprache (Carnap 1934). The Aufbau may be considered as the work of Carnap that is most closely related with mereological issues, at least implicitly. The Aufbau aimed at ‘constitution theory’, i.e., a theory of constitutional systems, which could serve as a scientific successor of traditional epistemology and philosophy of science. The constitutional system mainly treated in the Aufbau was based on a set S of elementary experiences en-

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dowed with a binary similarity relation determining which elementary experiences are similar to each other. From this base all concepts of science were to be reconstructed with the aid of ‘quasi-analysis’. Quasianalysis may be described as a kind of virtual mereological representation ascribing to elementary experience quasi-parts in such a way that two elementary experiences are similar to each other iff they share at least one common quasi-part or quasicomponent (Carnap 1928, Goodman 1963). A quasi-part T is defined as a maximal set of elementary experiences all of which are similar to each other. Carnap called such a set a similarity circle. He showed that if elementary experiences are represented by the sets of similarity circles to which they belong, this representation captures the similarity relation, i.e., two elementary experience are similar iff their associated sets of similarity circles overlap. More formally, an elementary experience e is represented by the set {T; e ∈ T} of similarity circles T to which e belongs. This representation respects the similarity relation in the sense that two elementary experiences e and e* are similar iff the representing sets r(e) := {T; e ∈ T} and r(e*) := {T; e* ∈ T} have a non-empty intersection. This amounts to a settheoretical representation of the system ES of elementary experiences by a system of sets {{T; e ∈ T}, e ∈ ES}. This system gives rise to a mereological system by interpreting the settheoretical inclusion ⊆ as a parthood relation. More precisely, two elementary experiences e and e* are similar

iff their representatives r(e) and r(e*) intersect non-trivially, i.e., have a common part T ∈ r(e) ∩ r(e*) (Mormann 2009). In sum, using the suggested reconstruction we obtain a mereological representation of the similarity structure of a constitutional system. This means that Carnap’s constitution theory may be considered formally as a part of mereology. The method of quasi-analytical representation resembles Whitehead’s method of extensive abstraction (Whitehead 1929) and the method Stone used to prove his representation theorem for Boolean algebras (Mormann 2005). In the early 1930s Carnap replaced constitutional systems by (constitutional) scientific languages. This led him to consider the task of philosophy of science as the logical analysis of scientific languages (Carnap 1934). In the last 20 years of his life he mainly worked on problems of inductive logic. See also > Experience, Gestalt, Goodman, Structure of Appearance, Whitehead. Bibliographical remarks

Carnap, R., 1928. Carnap’s opus magnum. Goodman, N., 1963. Contains a thorough criticism of Carnap’s method of quasianalysis. Mormann, T., 1994. Quasianalysis is shown to be a part of a general theory of meaningful representations.

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References and further readings

Awodey, S.; Klein, C. (eds.) 2004, Carnap Brought Home. The View from Jena, Chicago and LaSalle: Open Court. Bonk, T., 2004, Language, Truth and Knowledge. Contributions to the Philosophy of Rudolf Carnap, Dordrecht: Kluwer Academic Publishers. Carnap, R., 1928, Der logische Aufbau der Welt, second edition. Hamburg: Meiner, 1961. Translated by R. George as The Logical Construction of the World, Chicago and LaSalle: Open Court, 1998. Carnap, R., 1934, Die logische Syntax der Sprache, Wien: Springer. Translated by A. Smeaton as The Logical Syntax of Language, London: Kegan Paul, 1937. Coffa, J. A., 1991, The Semantic Tradition from Kant to Carnap. To the Vienna Station, ed. by L. Wessels, Cambridge: Cambridge University Press.

Davey, B. A.; Priestley, H. A., 1990, Introduction to Lattices and Order, Cambridge: Cambridge University Press. Mormann, T., 1994, “A Representational Reconstruction of Carnap’s Quasianalysis”, in Hull, D.; Forbes, M.; Burian, R. M. (eds.) PSA 1994, volume 1, East Lansing, Michigan: Philosophy of Science Association, 96–104. Mormann, T., 2005, “Description, Construction and Representation. From Russell and Carnap to Stone”, in Imaguire, G. and Linsky, L. (eds.) On Denoting 1905 – 2005, München: Philosophia Verlag, 333-360. Mormann, T., 2009, “New Work for Carnap’s Quasi-Analysis”, Journal of Philosophical Logic 38: 249-282. Proust, J., 1989, Questions of Form, Logic and the Analytic Proposition from Kant to Carnap, Minneapolis: University of Minnesota Press.

Creath, R.; Friedman, M. (eds.), 2007, The Cambridge Companion to Carnap, Cambridge: Cambridge University Press.

Richardson, A., 1998, Carnap's Construction of the World. The Aufbau and the Emergence of Logical Empiricism, Cambridge: Cambridge University Press.

Friedman, M., 1999, Reconsidering Logical Positivism, Cambridge: Cambridge University Press.

Uebel, T. E., 1992, Overcoming Logical Positivism from Within, Atlanta: Rodopi.

Goodman, N., 1954, The Structure of Appearance, Indianapolis: BobbsMerrill.

Whitehead, A. N., 1929, Process and Reality. An Essay in Cosmology, New York: Macmillan.

Goodman, N., 1963, “The Significance of Der logische Aufbau der Welt,” in Schilpp, P. S. (ed.) The Philosophy of Rudolf Carnap, Chicago and LaSalle: Open Court, 545-558.

Thomas Mormann

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Category In this entry, ‘category’ is employed in the philosophical sense only and neither in the mathematical sense of category theory nor in the grammatical sense of syntactic category. There is also an everyday sense of ‘category’ in which it means any class in a classification; this is too unspecific to discuss here. Philosophical use of the term ‘category’ goes back to Aristotle, who in his Categories enumerates ten things “said without combination”: kinds of thing, or equivalently, ten kinds of things one may call something. They are: substance, quantity, quality, relation, where, when, posture, state, doing, and undergoing. Debates about how Aristotle arrived at his list, and what they signify, whether they are ontological (kinds of thing, or modes of being), linguistic (kinds of predicate), or cognitive (modes of thought), have persisted from then to this day. We shall concentrate on the ontological sense and focus on the interrelation between categories in this sense and mereological notions. While Aristotle’s list has dominated discussion, other philosophers have different conceptions of category. The Stoics had four categories: substance, quality, disposition and relative disposition. Unlike in Aristotle, the Stoic categories all apply to every body. Kant used the term ‘category’ in a quite different sense, for what he called “pure concepts of the understanding”, which are concepts of kinds under which anything intuited must fall in order to be a possible object of cognition. It is accordingly

an epistemological rather than an ontological notion. Prior to enumerating the categories, Aristotle makes a fourfold division based on two oppositions: things said of versus not said of a subject; and things in versus not in a subject. In later terminology, things said of are universals, those not said of are particulars, those in are accidents, those not in are substances. Of a particular accident, Aristotle says it is “in something, not as a part, and cannot exist separately from what it is in” (1 a 24-5). By ‘part’ in this passage Aristotle means something which can exist separately. It took Husserl in his third Logical Investigation to extend the notion of part to include everything going to make up an object, including its color, shape and other factors, making the distinction between separable, independent parts or pieces and inseparable, dependent parts or moments one within the broader notion of part. Aristotle’s fourfold division has been revived by Jonathan Lowe. Aristotle holds that ‘be’ or ‘is’ means something different when used in different categories – as in Caius is a man, is six feet tall, is pale, is taller than his wife, is in his courtyard, is standing, is speaking, is being warmed by the sun. The categories then are different modes of being, a conception echoed in the 20th century by Roman Ingarden, though this view is not entailed by the notion of category. For reasons to do with his theory of definition, Aristotle held that there is no universal class of objects, no overarching common concept of

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being. Hence the categories represent the most general classes of being. As such they are both mutually exclusive (no thing belongs to more than one category) and jointly exhaustive (everything belongs to a category). Mutual exclusivity rules out any category being less than maximally general, for if it were not, all objects in it would also be objects in another, more general category. Conversely, if two categories overlapped, objects in the overlap would be subclasses of the wider categories. Exhaustivity is not stated by Aristotle but is implied: in any case, subsequent theories of categories have assumed it. Provided there is more than one category of object, exclusivity and exhaustivity are non-trivial, and ensure maximal generality. If there is, contrary to Aristotle, a universal class of all beings, then the categories are at the next lower level of generality. Either way, categories mark the most fundamental divisions in being. Mereological terms such as ‘part’, ‘overlap’ stand for relations; they are not themselves categories, but are formal, applying in different categories. There is a sense in which, in the category of quality, being colored is part of being white; in quantity, weighing two pounds is part of weighing three pounds; and in action, running is part of running quickly. Brentano later called the former members of such pairs ‘metaphysical parts’ of the later items. Leibniz’s or Kant’s talk of inclusion of one concept in another is the conceptual analogue of this ontological relationship.

It is in the category of quantity in particular that mereological concepts have significant purchase, at least for the case of extensive quantities. Quantities are all subject to relations of lesser and greater, but in the case of intensive quantities such as pain or brightness the difference between the lesser and greater is not made up by adding another pain or brightness, whereas as in such cases as length and area the addition of similar parts makes sense, and therefore the mereological concept of supplementation applies. Mereological concepts afford their own local categorial distinctions: the classes of mereological minima, maxima, and intermediaries. A mereological minumum is an atom: it has no proper part. A mereological maximum is an object that is not a proper part. In classical mereology there is a unique global maximum, the universe, of which everything is part. In more liberal mereologies there can be several local maxima, which we might call mereological continents. Objects which are neither minima nor maxima are intermediaries. In some mereologies, such as that of Whitehead, all objects are intermediaries. So mereology’s categories are: atom, intermediary, and continent. Since mereological terminology can be applied to classes, subclasses being parts of more extensive classes, we can express the requirements of mutual exclusivity and joint exhaustivity in mereological terms: the extensions of categories are disjoint and their fusion is universal; or, for

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Aristotle, the categories are all the continents. See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Brentano, Husserl, Ingarden, Whitehead. References and further readings

Aristotle, 1963, Categories and De interpretatione, Oxford: Clarendon Press. Baumgartner, W. and P. Simons, 1994, “Brentano’s Mereology”, Axiomathes 5: 55–76. Husserl, E., 1970, Logical Investigations, London: Routledge & Kegan Paul. Ingarden, R., 1964, Time and Modes of Being, Springfield: Thomas. Lowe, E. J., 2006, The Four Category Ontology: A Metaphysical Foundation for Natural Science, Oxford: Oxford University Press. Peter Simons

Causation The use of mereology in the investigation of causation remains in its infancy, despite the fact that in 1843 J. S. Mill presupposed the relevance of mereological concepts to causation. Doubts about whether causation is a real relation explain this retardation. Throughout most of the twentieth century, philosophers assumed that causation should be represented as a

quasi-logical operator on pairs of statements. If causation is best represented as a statement operator, there need be no entities related by causation and thus no application for mereology. In contrast, if causation a real relation, then it has relata, and these relata may have parts. The operator view has a prima facie advantage over the relation view: ontological parsimony. However, parsimony can be outweighed if the additional relation and the related entities have explanatory value. Attempts to flesh out the operator view have run into intractable obstacles: 1. Accounting for the asymmetry of causation (Tooley 1987: 205-243) 2. The pairing problem: distinguishing real from spurious causation. (Tooley 1987: 199-202; McDermott 1989; Ehring 1997: 18-49, 61- 66) 3. The possibility of singular causation. (Anscombe 1971: 7-9) In addition, some linguistic data supports the relation view, including Davidson’s account of action sentences (Davidson 2001: 105-122) as involving implicit reference to events, the use of perfect nominals as apparent names of events (see Vendler 1967 and Bennett 1988), and Barwise and Perry’s account of naked-infinite perception sentences as involving implicit reference to “situations” (Barwise and Perry 1987). However, the relation view of causation is not without difficulties. The greatest of these is accounting for negative causation, including prevention and the causal role of absences

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(see Mellor 1995: 162-182; Molnar 2000). Defenders of the relation view can respond in any of three ways: by denying that there is negative causation, by arguing that there are two distinct kinds of causation (one involving negative causation and the other involving a real relation), or by positing the existence of negative states of affairs as real entities. Both common sense and natural language support the existence of negative causation (Schaffer 2004), while positing two distinct kind of causation is ad hoc. Critics have lodged several objections to negative states of affairs: 1. Ontological inflation. 2. The incredulous stare. 3. Alleged violations of Hume’s maxim of the modal independence of separate entities. 4. Problems involving the contingent non-existence of particulars. There are responses to each of these objections. The advantages of a unified account of positive and negative causation outweigh the disadvantages of inflation and implausibility. Hume’s maxim may be false or may not apply in this case. Finally, it could be haecceities (e.g., the property of being this particular chair) and not the particulars themselves that are constituents of the relevant states of non-existence (see Plantinga 1974). If causation involves a real relation between entities, only one more condition need be met before deploying mereology: the relata of causation must have parts. In his System of

Logic, John Stuart Mill found it quite natural to speak of the relata of causation in ways that took this for granted: “The Cause… philosophically speaking, is the sum total of the conditions positive and negative…” (Mill, 1874, Book III, chapter 5, section 3). Building on Mill’s work, J. L. Mackie proposed that a cause is an “INUS” condition of its effect: “an insufficient but non-redundant part of an unnecessary but sufficient condition.” (Mackie 1974: 62) Mill and Mackie suppose that causation relates conditions, that any proper part of a condition is itself another condition, and that several conditions can sum together into an aggregate condition. Unfortunately, as Jaegwon Kim pointed out (Kim 1971), Mackie’s conception of conditions was confused, leading to paradoxical consequences. For example, nearly any actual condition is an INUS condition of any other condition (Kim 1971: 433). What is needed is a rigorous and consistent theory of causal relata and of how we refer to them. Kim’s own proposal was a step in the right direction (Kim 1975): the individuation conditions of events are given by means of a triple consisting of an object, a property and a time. The standard name of an event is a gerund of this kind: x’s being F at t. However, Kim’s account leaves some of the most important questions unanswered. For example, when are two properties identical? Can any property be used to refer to a genuine event, or only natural or real properties? (For example, do x’s being grue at t or y’s being red-or-in-New-

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Jersey at u name events?) And, more importantly for our present purposes: when is one event a part of another? Here are two plausible principles responding to this last question: (1) If x is both F and G at t, and F and G are both natural properties, then x’s being F at t is a part of x’s being both F and G at t. (2) If G is a determinate of some determinable F, and x is G at t, then x’s being F at t is a part of x’s being G at t. Steve Yablo has used something very like principle (2) to propose a solution to the problem of mental causation in a physical world (Yablo 1992). On Yablo’s account, an event x is part of an event y just in case x and y share all of their actual categorical properties (properties that can be had without prejudice to the modal status of other properties of the same kind), and all of the essential properties of x are also essential properties of y (Yablo 1992: 276). Yablo uses the part-of relation to define two conditions of causation: a cause is both required and enough for its effect: (3) x is required for y iff for all proper parts z of x, if z had occurred without x, then y would not have occurred. (4) x is enough for y iff for all z such that x is a proper part of z, z is not required for y. Koons (2000; 2004) has used mereology to define causal priority in terms of asymmetric necessitation and to define both minimal total

cause and cause simpliciter (following the lead of Mill and Mackie, with the exception of using events or situations rather than proposition-like conditions): (5) x is causally prior to y iff the the occurrence of every part of y necessitates the occurrence of x, and the occurrence of x does not necessitate the occurrence of any part of y. (6) x is a causally quasi-sufficient condition of y iff x is causally prior to y, and x’s occurrence is a defeasibly sufficient and actually undefeated condition of the occurrence of y. (7) x is a minimal total cause of y iff x is a causally quasi-sufficient condition of y, but no proper part of x is. (8) x is a cause of y iff x is a part of a minimal total cause of y. An alternative approach might follow the lead of Hans Reichenbach (1956) and Wesley Salmon (1984), identifying causation with the occurrence of a process in which some entity (a mark, physical quantity or trope) is transferred from link to link (see Dowe 2000 or Ehring 1997). Presumably, particular quantities or quantity-tropes can stand in the partwhole relationship, and so mereology could then be applied to the relata of causation. See also > Aristotle’s Theory of Parts, Dynamical Systems, Emergence, Facts, Powers, Quantum Mechanics.

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Bibliographical remarks

Kim, J., 1975. Kim enunciates and illustrates his constitutive triple (substance, property, time) account of event individuation, compares his position with that of Davidson (1967), and discusses the problem of the essential properties of events. Mereology appears in a brief discussion of one event’s including another. Koons, R., 2000. Koons, building on the situation theory of Barwise and Perry (1987), develops a formal theory of situations as the relata of causation. Situations verify or falsify ordinary categorical sentences in accordance with the strong three-valued logic of S. Kleene. Koons makes extensive use of mereological relations among situations in defining a variety of causal relations under a range of hypotheses about the causal laws. Mackie, J. L., 1974. Mackie covers a wide range of metaphysical and epistemological aspects of causation, including a historical overview. His own account takes conditions to be the relata of causation, in a way that assumes a tight connection between actual conditions and true sentences. Mereology plays a crucial role in defining a cause as an ‘INUS’ condition: an insufficient but nonredundant part of an unnecessary but sufficient condition. Yablo, S., 1992. Yablo develops an account of the relation between determinate and determinable properties and then uses that account to define a parallel relation of one event’s determining another. This determination relation among events seems to be a kind of part-whole relation:

when one event determines another, it is something like a part of the other. Yablo uses this account of event parthood (or determination) to propose an attractive solution to the exclusion problem in the philosophy of mind: the causal completeness of the physical does not exclude mental events from being efficacious, since physical events determine (are parts of) the relevant mental events. References and further readings

Anscombe, G. E. M., 1971, Causality and Determination, Cambridge, U. K.: Cambridge University Press. Barwise, J.; Perry, J., 1983, Situations and Attitudes, Cambridge, Mass.: The MIT Press. Bennett, J., 1988, Events and their Names, Indianapolis: Hackett. Davidson, D., 1967, “Causal Relations”, Journal of Philosophy, 64: 691-703. Davidson, D., 2001, Essays on Actions and Events, 2nd edition, Oxford: Clarendon Press. Dowe, P., 2000, Physical Causation, New York: Cambridge University Press. Ehring, D., 1997, Causation and Persistence: A Theory of Causation, New York: Oxford University Press. Kim, J., 1971, “Causes and Events: Mackie on Causation”, Journal of Philosophy 68: 426-441.

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Kim, J., 1975, “Events as Property Exemplifications”, in Brand M.; Walton, D. (eds.) Action Theory Dordrecht: D. Reidel, 159-177. Koons, R., 2000, Realism Regained: An Exact Theory of Causation, Teleology, and the Mind, New York: Oxford. Koons, R., 2004, “The Logic of Causal Explanation: An Axiomatisation”, Studia Logica 77: 325-354. Mackie, J. L., 1974, The Cement of the Universe: A Study of Causation, Oxford: Clarendon Press. McDermott, M., 1995, “Redundant Causation”, British Journal for the Philosophy of Science, 46: 523-544. Mellor, D. H., 1995, The Facts of Causation, London: Routledge. Mill, J., 1868, A System of Logic, Ratiocinative and Inductive, London, Longmans, Green, Reader, and Dyer. Molnar, G., 2000, “Truthmakers for Negative Truths”, Australasian Journal of Philosophy 78: 72-86. Plantinga, A., 1974, The Nature of Necessity, Oxford: Clarendon Press. Reichenbach, H., 1956, The Direction of Time, Berkeley: University of California Press. Salmon, W., 1984, Scientific Explanation and the Causal Structure of the World, Princeton: Princeton University Press. Schaffer, J., 2004, “Causes Need not be Physically Connected to their Effects: The Case for Negative Causation”, in Hitchcock, C. (ed.) Contemporary Debates in Philosophy of Sci-

ence, Malden, Mass.: Blackwell, 197-216. Tooley, M., 1987, Causation: A Realist Approach, Oxford: Clarendon Press. Vendler, Z. 1967, “Facts and Events,” and “Causal Relations”, in Linguistics in Philosophy, Ithaca: Cornell University Press, 122-46, 167-9. Yablo, S., 1992, “Mental Causation”, Philosophical Review 101: 245-280. Robert Koons

Chaos Most people associate the term ‘chaos’ with disorder and confusion. In Greek mythology Chaos is the most ancient god existing before the universe was created. In modern science, the term chaos characterises the disordered and seemingly unpredictable motion of a system (Berge et al 1986, Gleick 1987, Ruelle 1991, Lorenz 1993). A system consists of different parts that interact with each other. For a quantitative description, appropriate variables with numerical values are attributed to the parts and the system is specified by parameters. Here are some simple examples: System I: Single pendulum Parts: Disk of the pendulum Examples of parameters: Length of the rod Variables: Position and velocity of disk On-line simulations: see references [2] below.

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System II: Driven pendulum or coupled pendula Parts: Disks Examples of parameters: Driving frequency, length of the rods Variables: Positions and velocities of disks On-line simulations: see references [1, 2] below. System III: Planet rotating around the sun Parts: Planet Examples of parameters: Mean distance from the sun Variables: Position and velocity of the planet System IV: Solar system Parts: Various planets Examples of parameters: Mean distances from the sun Variables: Positions and velocities of planets System V: Terrestrial atmosphere Parts: Volume elements of the atmosphere Examples of parameters: Heat conduction, air viscosity Variables: Density, temperature, flow velocity, humidity etc. of the air On-line simulations: see references [4, 5, 6] below. System VI: Interacting populations (animals, societies) Parts: Different species Examples of parameters: Reproduction rates Variables: Number of individuals of each species On-line simulations: see references [3] below.

Systems I through IV are mechanical systems with massive bodies as parts. The parts in V and VI, being the subject of meteorology, biology, or even economics and sociology, are rather of a more general nature. For a system to behave chaotically it must consist of at least three parts, and the more parts it has and the stronger their coupling, the more likely it is chaotic (Ruelle 1991, ch. 13). The basic objective of chaos theory consists in evaluating the system dynamics: finding out its state and properties at any time, given the corresponding information at some initial time. The interplay between different parts yields equations of motion relating the time variation of a given variable to the values of the others. These equations are deterministic: there is no random element (as in stochastic models). Thus, for given initial values, the solution is unique. Systems I and III are examples of non-chaotic dynamics: the motion is ‘ordered’ and repetitive. The gently driven pendulum of a clock is a paradigm of ‘regular’ motion. Given its evolution for some time interval, there is no room for surprise at later times. Two identical pendula will show very similar motion patterns when initially released from similar positions. Laplace said that if the universe is governed by deterministic dynamics a being who knows all the relevant variables at a given time is able to predict with certainty what the future will be. This is true, but only ‘in principle’. Around 1900 pioneers of classical dynamics – e.g. Hadamard,

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Duhem and Poincaré (Ruelle 1991, ch. 8) – found that this perfect determinism is often a ‘useless’ concept. For most dynamic systems, time evolution depends sensitively on the initial conditions that can never be determined with arbitrary precision, and the motion appears to be irregular and “practically unpredictable”. This kind of chaos is clearly exhibited in our examples II, V (e.g. the ‘Lorenz model’) and VI (e.g. the ‘logistic map’). The term ‘chaos’ has first been used by Jim Yorke (Li and Yorke 1975) who studied a model that exhibited a transition towards chaos when a control parameter was varied. Feigenbaum (Feigenbaum 1978; 1979) showed that the usual ‘route to chaos’ may start from simple periodic motion and go through a sequence of ‘period doubling’ before becoming chaotic. The availability of fast computers (see the quoted on-line simulations) has greatly improved our understanding of chaotic dynamics. It is fascinating to observe, e.g., for a strongly driven pendulum, how small oscillations of the pendulum alternate in a totally irregular fashion with full rotations around the suspension point. Two almost identical initial configurations lead to motion patterns that first look similar, but after some time interval (Lyapunov time) start to deviate from each other. Here are the main chaotic dynamics. principle precisely motion appears to never repeats. (ii)

characteristics of (i) Although in determined, the be irregular and The motion de-

pends sensitively on initial conditions, which is of significance for mereology: if the state of a single part is changed slightly, this may radically change the time evolution of the whole. (iii) With the variation of a control parameter, the transition to chaos goes through a series of ‘bifurcations’ such as period doubling (Feigenbaum 1978; 1979). (iv) After some transient time, the motion follows an ‘attractor’ with ‘fractal’ structure (Mandelbrot 1982). As may be seen from these characteristics, the term ‘chaos’ actually may not be adequate for deterministic systems, since a delicate type of ‘order’ is inherent in chaotic motion. Even complex attractors can be precisely characterised in mathematical terms. Applications of chaos theory abound. Fortunately, planetary motion in IV is only very mildly chaotic, otherwise it would be impossible to foresee the trajectories of space satellites. On the other hand, for understanding systems V and VI, the existence of chaotic regimes is crucial. Lorenz (1993) has shown that a simple set of three equations to describe the atmosphere leads to chaos. The sensitive dependence on initial values is important: since we do not know precisely the weather conditions at a given time, reliable predictions for more than a few days are impossible. A butterfly unexpectedly moving around somewhere may completely change the evolution of the weather. Models like the logistic map (VI) describing population dynamics or the evolution of economy, show the intrinsic limits for predicting the future states of these systems. The features of chaot-

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ic dynamics seems to be important even in fields like psychology or medicine (Combs and Robertson 1995; Pezard and Nandrino 2001). For example, the behaviour of a family in a relational crisis looks unpredictable, but occurs in cycles. As some authors have argued, neural networks can show chaos the attractors of which could be interpreted as ‘mental representations’ (Newman 1996), and schizophrenia may be due to a decrease of the network storage capability leading to new attractors.

the attention of the nonspecialist the ubiquity of fractals in nature. Ruelle, D., 1991. This charming little book by a pioneer in chaos describes with minimal mathematics the philosophical implications of chaos and randomness. References and further readings

Berge, P.; Pomeau, Y.; Vidal, C., 1986, Order withing Chaos: Towards a Deterministic Approach to Turbulence, John Wiley & Sons.

See also > Dynamic Systems, Emergence, Structure, Totum potentiale.

Combs A.; Robertson, R., 1995, Chaos Theory in Psychology and the Life Sciences, Lawrence Erlbaum.

Bibliographical remarks

Feigenbaum, M. J., 1978, “Quantitative Universality for a Class of Nonlinear Transformation”, J. Statistical Physics 19: 25-52.

Berge, P., Pomeau, Y., and Vidal, C., 1986. This very readable book covers the fundamentals of chaos and includes some practical applications. Combs A.; Robertson R., 1995. Interesting implications of chaotic behaviour are shown in fields that are far away from physics and mathematics. Gleick, J., 1987. Everyone should read this best-selling, historical, nontechnical account to understand why people are so excited about chaos. Lorenz, E. N., 1993. This series of popular lectures by one of the pioneers in the field, attempts to explain the mathematical ideas underlying chaos without using equations. Mandelbrot, B. B., 1982. This extended essay by the father of fractals was the seminal work that brought to

Feigenbaum, M. J., 1979, “The Universal Metric Properties of Nonlinear Transformations”, J. Statistical Physics 21, 669-706. Gleick, J., 1987, Chaos: Making a New Science, Viking Penguin. Li, T.; Yorke, J. A., 1975, “Period three Implies Chaos”, Amer. Math. Monthly 82: 985-992. Lorenz, E. N., 1993, The Essence of Chaos, Univ. of Washington Press. Mandelbrot, B. B., 1982, The Fractal Geometry of Nature, Freeman. Newman, D., 1996, “Emergence and Strange Attractors”, Philosophy of Science 63: 245-261. Pezard, L.; Nandrino, J. L., 2001,

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“Dynamic Paradigm in Psychopathology: Chaos Theory from Physics to Psychiatry”, Encephale 27(3): 260-268. Ruelle, D., 1991, Chance and Chaos, Princeton Univ. Press. On-line computer simulations

[1] http://www.dollansky.net/Chaotic _Pendulum/chaotic_pendulum.html (dynamics of double pendulum) [2] http://monet.unibas.ch/~elmer/ pendulum/vpend.htm (various types of pendula) [3] http://users.ox.ac.uk/~quee0818/ chaos/chaos.html (dynamics of logistic model, sensitivity to initial conditions) [4] http://www.geom.uiuc.edu/~wor folk /apps/Lorenz/ (Lorenz attractor) [5] http://www.cmp.caltech.edu/ ~mcc/ chaos_new/Lorenz.html (Lorenz attractor is built up, butterfly effect) [6] http://www.cmp.caltech.edu/ ~mcc/ Chaos_Course/Outline.html (Lorenz model, pendulum, various other topics with demonstrations) Hans Beck

Chemistry Fundamental to elementary modern chemistry is the distinction between, on the one hand, chemical substances and, on the other hand, the different phases exhibited by substances. The most familiar phase properties are

those of solid, liquid and gas. (Note that in scientific usage, ‘solidity’ is not a synonym for ‘impenetrability’). This distinction is muddled in Putnam’s project of delimiting the extension of the chemical kind water in terms of what bears the relation of being the same liquid to some sample. The significance of this distinction emerged with discoveries made in the latter part of the 18th century leading to a clear change in the extension of some substance terms such as ‘water’ with the development of modern chemistry, and was fully systematised in Gibbs’s chemical thermodynamics towards the end of the 19th century. For phases finer distinctions are made. ‘Ice’, for example, is a general term for the substance water in the solid phase, but investigations have led to several different solid phases of water being distinguished from ice I – common ice – and called ice II, ice III, etc. Calcium carbonate occurs as calcite and as aragonite, which are distinct solid phases of the same substance. As these examples illustrate, names of kinds of matter are not always simply names of a kind of substance, but may also include a phase determination. The term ‘water’ is ambiguous in this respect, sometimes denoting the chemical substance (when ice and steam are water) and sometimes denoting this chemical substance specifically in the liquid phase (when ice and steam are not water). But quartz is unambiguously a chemical substance, silicon dioxide, in a particular solid form, distinct from the other solid phases tridymite and cristobalite. The distinct phases of a

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single substance are related to one another in ways that depend on the prevailing conditions in accordance with general thermodynamic laws, but with many specific features characteristic of the particular substance, such as boiling points and freezing points. At normal pressure, quartz passes to tridymite at 870°C and this to cristobalite at 1470°C. Liquid sulphur, on the other hand, comprises a mixture of the clear, mobile Sλ form in equilibrium with the dark-red, viscous Sµ form, the proportion of the latter increasing with temperature. Substance and phase properties are macroscopic concepts, systematised in macroscopic theory which treats matter as continuous, and therefore closer to everyday conceptions. They are apparently expressed by mass predicates, characterised mereologically by a distributive condition (any part of whatever satisfies the predicate also satisfies the predicate) and a cumulative condition (the sum of quantities satisfying the mass predicate also satisfies the mass predicate). There is reason to think substance and phase properties are relational (see below) and that mass predicates can be polyadic. Generalising the monadic distributive condition to the polyadic case is straightforward, but an appropriate generalisation of the cumulative condition reducing to a form equivalent to that just given for 1-place predicates is not (Roeper 1983), taking the following form for the dyadic ‘Water’ predicate:

∀π´ ∀t´(π´ ⊆ π ∧ t´ ⊆ t. ⊃ ∃π´´ ∃t´´ (π´´ ⊆ π´ ∧ t´´ ⊆ t´ ∧ Water (π´´, t´´))) ⊃ Water (π, t) Here π, π´, π´´ are quantity variables, t, t,´ t´´ are time (interval) variables and ⊆ is the part relation (Needham 2007). Understanding how ‘Water’ applies to a quantity throughout an interval is discussed in Needham (2010b), together with reasons for not regarding phase properties as mass predicates, in the light of gross microscopic features. Mixture is another fundamental notion of chemistry. There are mixtures of substances and, as the above discussion illustrates, mixtures of phases. In general, a quantity of matter at equilibrium will comprise several phases over which several substances are distributed (Needham 2007). Such complex mixtures are governed by Gibbs’s phase rule, which determines the variability of conditions under which a quantity of matter with a given number of substances can sustain a given number of phases (Needham 2010b). For example, the triple point is so called because only a quantity comprising a single substance can sustain three phases with zero variability, i.e. at a unique temperature and pressure. The specific values are characteristic of the particular substance; thus whatever simultaneously sustains solid, liquid and gas phases at precisely 0.01°C and 4.58 mmHg is water. Complex mixtures exhibiting several phases are heterogeneous, each phase being homogeneous. (“By homogeneous”, Gibbs (1948: 63) says, “is meant that the part in question is uniform

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throughout, not only in chemical composition, but also in physical state”. But appearances to the contrary, “part” here cannot be understood literally if phase predicates are not mass predicates). A homogeneous mixture of several substances is called a solution. Aristotle also understood heterogeneous quantities of matter to comprise homogeneous parts, such as flesh and bone. But he seems to have thought that the phase distinction coincided with the substance distinction, speaking of flesh and bone as homoeomeries (having parts like the whole), and of water as converted to air when it evaporates, and to earth when it solidifies. Chemists were still in the grip of this conception when they interpreted Black’s discovery in 1761 of the latent heat of fusion in terms of a transformation of substances involving the combination of ice and caloric to produce water – water being understood with Aristotle as necessarily liquid – and similarly for the latent heat of vaporisation. But the old conceptions of chemical substance gave way to the modern phase-independent notion as caloric disappeared from chemical explanations and it was realised that the same substance persists through its phase transitions (unless it decomposes). Accordingly, phase properties are relational, a quantity which is liquid at one time not being so at another. And so are substance properties, or at least those falling under the general category of compound, since substances are transformed in chemical reactions. Lavoisier analysed water into hydrogen and oxy-

gen by boiling the liquid and reducing the steam over heated charcoal. The modern interpretation of the experiment is that a quantity of water first changes phase from liquid to gas and is then transformed by chemical reaction at a high temperature with carbon yielding two other substances, carbon dioxide and hydrogen. (Lavoisier himself hadn’t relinquished caloric). This is one of numerous examples where water participates in a chemical reaction as a reactant and is ‘destroyed’. It is ‘created’ in numerous others; for example, it is the product of the combustion of hydrogen in oxygen. Thus, what is water at one time often isn’t at another, and what isn’t often is. (It is certainly not the case that something is water iff it is necessarily water). Water is liquid or solid under conditions when oxygen and hydrogen are still gaseous, and it makes no sense to describe the elemental composition of a compound in terms of phase-dependent descriptions of the substances. Substance properties denote what remains the same over phase changes and what, in the case of water, is composed of hydrogen and oxygen, not of hydrogen gas and oxygen gas. Liquid water, though homogeneous like the air, is composed of fixed proportions of its constituent elements, illustrating the law of definite proportions which provided the grounds for the distinction amongst homogeneous mixtures between compounds and solutions at the beginning of the 19th century. Although gases are miscible in all proportions, liquid or solid solutions have a degree of saturation, such that when the

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concentration of one component exceeds the saturation point two phases form. Berthollet thought this fact counted against a clear compoundsolution distinction until Proust was taken to have shown that, unlike degree of saturation, combined proportions are independent of temperature and pressure. The distinction remains widely applicable, even though modern chemistry has discovered cases which muddy the dividing line. (See articles in Part 3 “Chemical Substances” of Hendry et al. 2012). Substances are thus divided into elements and compounds, and compounds at least can be transformed into other substances. Aristotle thought that the elements could be transformed, but the phaseindependent conception of substance brought with it the belief that elements persist in compounds and are indestructible. Such was Mendeleev’s adherence to this view that he resisted the claim that radioactivity involved the transformation of elements, which he thought was tantamount to a return to alchemy. But he was proved wrong, and elements are now thought to be generated from hydrogen in stars. Elemental properties too, then, are time dependent. With the development of the nonatomic conception of the atom in the 20th century, the chemical combination of atoms into larger cohesive structures (molecules, polymers, ions, ionic crystals, hydrogen-bonded structures, etc.) of greater and lesser longevity has been understood in terms of electronic structure. Concurrently, techniques of observation over

very short intervals of time make it possible to follow in detail the course of chemical reactions via the existence of short-lived intermediates, and also give insight into the dynamic processes occurring in what, from a macroscopic point of view, are stable equilibrium states. How should the interplay of theories and conceptions at the two levels be viewed – mutually complementary or the reduction of the macroscopic to the microscopic? Although popular in the past, the reductionist view has been increasingly questioned in recent years. To give some indication of what is at issue, the reduction of chemical substance can be divided into two not entirely independent questions (Hendry and Needham 2007): R1 Can chemistry’s macroscopic substances, like water or gold, be reduced to the microscopic entities (molecule, ion) of which they are composed? R2 Can chemistry’s microscopic entities, like molecules, be reduced to quantum-mechanical systems of subatomic entities like electrons and nuclei interacting according to the laws of quantum mechanics? The first is like, but more complicated than, the question whether temperature is reducible to the average molecular kinetic energy, which is doubtful (Needham 2010a). Regarding R2, quantum chemists Hans Primas and Guy Woolley have questioned whether the classical idea of molecular structure (shape) has any basis in quantum mechanics (see Sutcliffe and Woolley 2012). The application of quantum mechanics to mol-

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ecules in for example the interpretation of spectra presupposes that molecular structure is introduced, as Woolley puts it, “by hand”. Hendry (2006) argues that this raises the possibility of downwards causation. Even the so-called Aufbau principle for the periodic table is not a single general rule based on quantum mechanical principles; rather, elements are treated separately with approximation techniques which are considered adequate when the calculated electronic structure agrees with spectroscopic and chemical properties. Further, the Pauli exclusion principle on which the general idea is based, and introduced partly for that purposes, has not been reduced (Bogaard 1978, Scerri 1991; 1994). To what extent these issues are problems depends, of course, on what is to be put into the general notion of reduction, which is also a controversial issue. Many who write ‘water is H2O’ seem to understand it as illustrating an affirmative answer to R1. This presupposes some tacit articulation of the claim. The compositional formula H2O is a macroscopic concept, giving the proportions of hydrogen and oxygen in water in such a way as to say that there is as much hydrogen in water as there is in hydrogen sulphide, with the compositional formula H2S, despite the difference in gravimetric proportions. The claim must at least imply that the predicates ‘is water’ and ‘is H2O’ are coextensive. But ‘is H2O’ cannot mean ‘is a water molecule’ if it is to hold of something that is water since water, though not a water molecule, is characterised by macroscopic features of the kind

mentioned above. Presumably, the reductionist understands it to denote an appropriate group of water particles. Particles are units perhaps interacting but not cohering with their neighbours. The reductionist faces the unenviable task of making this distinction sufficiently precise. But on an informal understanding, no one with any claim to know about water thinks that the particles in water are all and only H2O molecules. A quantity of water comprises continually changing parts, with dissociation into positive and negative ions and association via hydrogen bonding into large polymeric structures. The minimal claim must be something of the kind ∀x (x is water ≡ x comprises an appropriately grouped collection of particles of certain kinds). There is also a time reference ‘throughout t’ to be added to each side of the equivalence, where the time t is an interval and the qualification can be understood along the lines of what Eisenberg and Kauzmann (1969) call a snapshot taken with a shutter-speed of t. Because of the dynamic situation, short-term changes will be blurred over longer time intervals, and different aspects of the micro-structure will emerge for different lengths of time. All this is to be packed into explicitly spelt out accounts of ‘appropriate’ and ‘certain kinds’, without any reference to the macroscopic concept water of the likes of van Inwagen’s device ‘arranged water-wise’ if the reductionist is to avoid circularity. Since explicit accounts continue to elude scientists

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investigating the structure of water, this is a tall order (Needham 2011, section 7 “Water” in Needham 2013). See also > Aristotle's Theory of Parts, Aristotle's Theory of Wholes,, Ancient Greek Atomism, Body, Collectives and Compounds, Emergence, Granularity, Homeomerity, Mereological Essentialism, Natural Science, Persistence, Quantum Mechanics, Transitivity. References and further readings

Bogaard, P. A., 1978, “The Limitations of Physics as a Chemical Reducing Agent”, Proceedings of the Philosophy of Science Association 2: 345-56. Eisenberg, D.; Kauzmann, W., 1969, The Structure and Properties of Water, Oxford: Clarendon Press. Gibbs, J. W., 1948, “On the Equilibrium of Heterogeneous Substances”, in The Collected Works of J. Willard Gibbs, Volume I, Yale University Press, New Haven. Hendry, R. F., 2006, “Is there Downwards Causation in Chemistry?” in Baird, D.; Scerri, E.; McIntyre, L. (eds.) Philosophy of Chemistry: Synthesis of a New Discipline, Dordrecht: Springer, 173-89. Hendry, R. F., The Metaphysics of Chemistry, Oxford University Press, forthcoming. Hendry, R. F.; Needham, P., 2007, “Le Poidevin on the Reduction of Chemistry”, British Journal for the Philosophy of Science, 58: 339-53.

Hendry, R. F.; Needham, P.; Woody, A. J. (eds.), 2012, Handbook of the Philosophy of Science, Vol. 6: Philosophy of Chemistry, Amsterdam: Elsevier. Needham, P., 2007, “Macroscopic Mixtures”, Journal of Philosophy 104: 26-52. Needham, P., 2010a, “Nagel’s Analysis of Reduction: Comments in Defence as Well as Critique”, Studies in History and Philosophy of Modern Physics 41: 163-170. Needham, P., 2010b, “Substance and Time”, British Journal for the Philosophy of Science 61: 485-512. Needham, P., 2011, “Microessentialism: What is the Argument?”, Noûs 45: 1-21. Needham, P., 2013, “Hydrogen Bonding: Homing in on a Tricky Chemical Concept”, Studies in History and Philosophy of Science 44: 5166. Roeper, P., 1983, “Semantics For Mass Terms with Quantifiers”, Noûs 17: 251-65. Scerri, E., 1991, “The Electronic Configuration Model, Quantum Mechanics and Reduction”, British Journal for the Philosophy of Science 42: 309-25. Scerri, E. R., 1994, “Has Chemistry Been at Least Approximately Reduced to Quantum Mechanics?” PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1994: 160-70. Sutcliffe, B.; Woolley, R. G.; 2012, “Atoms and Molecules in Classical

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Chemistry and Quantum Mechanics” in Hendry et al., 387-426. Timmermans, J., 1963, The Concept of Species in Chemistry, trans. by Ralph E. Oespar, Chemical Publishing Company, New York. van Brakel, J., 2000, Philosophy of Chemistry: Between the Manifest and the Scientific Image, Leuven University Press, Leuven. (Contains an extensive bibliography.) Weisberg, M.; Needham, P.; Hendry, R., 2011, “Philosophy of Chemistry”, in: The Stanford Encyclopedia of Philosophy, Zalta, E. N. (ed.), http://plato.stanford.edu/archives/su m2011/entries/chemistry/. Woolley, R. G., 1988, “Must a Molecule Have a Shape?”, New Scientist, 120 (22 Oct.): 53-7. Paul Needham

Coincidence Can two material objects exist in the same place at the same time? In other words, is a case of material coincidence possible? Few philosophers would want to say that two material objects of the same kind can coincide, such as two ships or two lumps of copper (Oderberg 1996 denies that they can, but Hughes 1997 takes an opposing view). However, many would say that material objects of suitably different kinds, with different persistence conditions, can coincide, such as a lump of copper and a copper statue ‘constituted’ by that lump (Wiggins 2001 defends this po-

sition). Yet this view apparently conflicts with widely-held principles of mereology – in particular, both the weak and the strong principles of extensionality. The weak principle states that composite objects with exactly the same proper parts are identical, while the strong states that composite objects with exactly the same proper parts at some level of decomposition are identical. (The strong principle entails the weak, but not vice versa). It might perhaps be denied that the copper statue and the lump of copper have exactly the same proper parts, on the grounds that it is counterintuitive to say, for example, that the head of the statue is a proper part of the lump (see Lowe 2001). However, it seems clear that they do have the same proper parts at the level of their composition by copper particles: the statue and the lump are composed by exactly the same copper particles, at any time at which the former could be said to be ‘constituted’ by the latter. So aren’t we required to say that in this case constitution is identity with the consequence that, after all, the statue and the lump must have exactly the same parts? Perhaps not, since we may prefer to see this case as a counterexample to the strong principle of extensionality. Notice that we have implicitly adopted temporally indexed versions of the extensionality principles in the foregoing discussion, talking as we are about the proper parts that certain objects have at certain times. Notice also, however, that we are not, in this context, talking about the so-called temporal parts of persisting objects.

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Now, a temporal-parts theorist might well react to the statue/lump example by invoking a time-independent principle of extensionality and claim that these objects are distinct provided that they have different temporal parts, even if they share the same such parts throughout the period of their coincidence. Such a philosopher would only have to assert an identity between a copper statue and a lump of copper which coincided throughout their entire careers. But, of course, not all philosophers are sympathetic to the doctrine of temporal parts, so let us set it aside for present purposes. Some philosophers urge that it simply makes no sense to suppose that the same copper particles, arranged in the same way, can at the same time compose two distinct material objects, such as the copper statue and the lump of copper (see Olson 2001). If it is replied that these objects are distinct because they are of different kinds, with different persistence conditions and different modal properties (such as an ability, in the case of the lump, to survive a radical change of shape, which is not possessed by the statue), such philosophers tend to retort that these supposed differences must surely arise from and depend upon facts about the material composition of the objects in question – and yet the facts in question, it seems, are just the same for ‘both’ of these objects. For example, the lump’s ability to survive a radical change of shape may be presumed to ‘supervene’ upon the properties and relations of the copper particles composing it, in much same way that its ability to

conduct electricity does. But how, then, can it be said that there is another object, the statue, which lacks the former ability despite being composed of exactly the same copper particles possessing exactly the same properties and standing in exactly the same relations? There is, however, a way for defenders of coincidence to respond to these apparent difficulties (see Lowe 2003). First of all, they can urge that not only persistence conditions, but also principles of composition, are kind-relative. Thus, they can urge that it is in virtue of different facts about some copper particles that they compose, at a certain time, a lump of copper and that they also compose, at that same time, a copper statue. Moreover, the mere facts about how the particles are ‘arranged’ at that time do not suffice to determine that they compose, at that time, either kind of object. For instance, the particles compose a lump at a certain moment just because those same particles have adhered together for a period of time including that moment. But that fact is neither necessary nor sufficient for the particles in question to compose a statue at that moment – not necessary because a statue could be composed of different particles at other times, and not sufficient because a statue, unlike a lump, needs to retain much the same shape throughout its existence. The different composition principles for statues and lumps are largely determinative of the differences between their respective persistence conditions. The implication is that the kind to which a composite object belongs and its cor-

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responding persistence conditions are not properties of that object that ‘supervene’ upon the properties and relations of the material particles composing the object at any given time, contrary to some of the opponents of coincidence. Thus, the basis of the modal facts that a copper statue cannot, whereas a lump of copper can, survive a radical change of shape is quite unlike the basis of the fact that a copper object can conduct electricity. The former facts are determined by the composition principles which tell us what it is for some copper particles to compose, on the one hand, a copper statue and, on the other, a lump of copper. The latter fact, by contrast, is a consequence of natural laws which determine how the dispositional properties of aggregated matter depend upon the properties and relations of matter at the atomic and molecular level. See also > Dispositions, Material Constitution, Persistence, Power, Substance. Bibliographical remarks

Burke, M. B., 1994a. An interesting approach not discussed in the present article. Burke, M. B., 1994b. A good discussion of an ancient puzzle of coincidence. Gallois, A., 1998. Defends a nonstandard solution to coincidence problems.

Lewis, D. K., 1983. A classic statement of the temporal-parts approach to persistence. Noonan, H. W., 1991. Defends a thesis questioned in the present article. Thomson, J. J., 1998. An important discussion of the statue/lump case. van Inwagen, P., 1981. Questions an assumption made in the present article. References and further readings

Burke, M. B., 1994a, “Preserving the Principle of One Object to a Place: A Novel Account of the Relations among Objects, Sorts, Sortals, and Persistence Conditions”, Philosophy and Phenomenological Research 54: 591-624. Burke, M. B., 1994b, “Dion and Theon: An Essentialist Solution to an Ancient Puzzle”, Journal of Philosophy 91: 129-39. Gallois, A., 1998, Occasions of Identity: The Metaphysics of Persistence, Change, and Sameness, Oxford: Clarendon Press. Hughes, C., 1997, “Same-Kind Coincidence and the Ship of Theseus”, Mind 106: 53-67. Lewis, D. K., 1983, “Survival and Identity”, in his Philosophical Papers Volume 1, New York: Oxford University Press. Lowe, sition, Self”, Body,

E. J., 2001, “Identity, Compoand the Simplicity of the in Corcoran, K. (ed.) Soul, and Survival: Essays on the

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Metaphysics of Human Persons, Ithaca, NY: Cornell University Press. Lowe, E. J., 2003, “Substantial Change and Spatiotemporal Coincidence”, Ratio 16: 140-60. Noonan, H. W., 1991, “Constitution is Identity”, Mind 102: 133-46. Oderberg, D. S., 1996, “Coincidence under a Sortal”, Philosophical Review 105: 145-71. Olson, E. T. 2001, “Material Coincidence and the Indiscernibility Problem”, Philosophical Quarterly 51: 337-55. Thomson, J. J., 1998, “The Statue and the Clay”, Noûs 32, 149-73. van Inwagen, P., 1981, “The Doctrine of Arbitrary Undetached Parts”, Pacific Philosophical Quarterly 62: 123-37. Wiggins, D., 2001, Sameness and Substance Renewed, Cambridge: Cambridge University Press. Edward Jonathan Lowe

Collectives and Compounds Wholes are often distinguished from arbitrary mereological sums. Wholes are taken to be very special mereological entities that for some are shrouded in mystery. In applied ontology, however, e.g., in the ontology of the biomedical sciences, it is useful to distinguish certain categories of entities that stand in special relations to their parts. Such categories are collectives and compounds (Schulz et al. 2006), corresponding to the relations

of granular parthood and determinate parthood (Rector et al. 2005). Collectives are thought to consist of ‘grains’ that are all of the same kind, which are normally not spatially connected and whose number can vary as long as there is at least one grain left – although some authors allow for collectives with no grain at all (Rector et al. 2005), whereas others require at least two grains to be present (Wood & Galton 2009). It is important to note that the uniformity of grains is relative to a certain level of granularity and specificity – a collection of H2O molecules is uniform at the level of molecules, but not on the level of atoms or subatomic particles, and it could also be considered to be a mixture of oxides of different hydrogen isotopes (like heavy water or super-heavy water). Similarly, a collection of jewels can be considered to be a mono-sortal collection of stones, but also a multi-sortal mixture of emeralds and diamonds. Compounds, on the other hand, typically consist of a fixed number of components that are spatially connected but may be of different kinds. This contrast between collectives and compounds uses two different pairs of opposites: mono-sortality vs. multi-sortality and numerical flexibility vs. numerical rigidity. As these are logically independent features, different intermediate cases of complex entities are conceivable (Jansen & Schulz 2011). There are numerically flexible mono-sortal complexes (e.g., the number of saccharide monomers of a polysaccharide molecule is not fixed); but multi-sortal complexes can also be flexible regarding the

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number of their parts (e.g. while a human hand typically consists of five fingers, it may have four or six fingers, and the removal of the sixth finger would not at all destroy or diminish the hand). And there are also mono-sortal complexes of a fixed number of grains (e.g., a pair of kidneys). The types of compounds discussed so far can be further refined with respect to the question whether their parts have to be connected with each other or not, and if yes, by which kind of connection. Thus further subtypes can be defined via restrictions on the relations between the parts, such as with respect to spatial connections (like in a lump of stuff), composition in certain special structures (as in a protein), or other kinds of ‘connections’ (e.g., social relations in the case of four musicians forming a string quartet). These different kinds of complex entities are interconnected in non-trivial ways, bringing with them different persistence conditions that could be applied in the same situations: If the string quartet is dissolved, the four musicians continue to form a collective of musicians (in the sense defined above); if a pair of persons is enriched by a third person, we no longer have a pair, but we still have a collective of persons; and if in a collective of uranium atoms some atoms decay and transform into atoms of other elements, then we no longer have a collective of uranium atoms, but still a collective of atoms. Mixtures (like a salt solution, bronze or a salad sauce) can be seen as com-

pounds of collectives. E.g., a salt solution is a strict compound of a collection of salt molecules and a collection of water molecules plus appropriate connections between salt and water molecules. See also > Chemistry, Coincidence, Granularity, Homeomerous, Transitivity. References and further readings

Jansen, L.; Schulz, St., 2011, “Grains, Components and Mixtures in Biomedical Ontologies”, Journal of Biomedical Semantics 2 (Suppl. 4), S2 (9. August 2011). Rector, A.; Rogers, J.; Bittner, Th., 2005, “Granularity, Scale and Collectivity: When Size Does and Does not Matter”, Journal of Biomedical Informatics 39: 333-349. Schulz, St.; Beisswanger, E.; Hahn, U.; Wermter, J.; Kumar, A.; Stenzhorn, H., 2006, “From GENIA to BioTop. Towards a top-level Ontology for Biology”, in: Proceedings of the International Conference on Formal Ontology in Information Systems (FOIS 2006). Baltimore MD, USA. Amsterdam: IOS:103-114. Wood, Z. M.; Galton, A. P., 2009, “A Taxonomy of Collective Phenomena”, Applied Ontology 4: 267-292. Ludger Jansen

152 COMMON SENSE REASONING ABOUT PARTS AND WHOLES

Common Sense Reasoning about Parts and Wholes The analysis of types of part-whole relations in common sense reasoning is focused on the question how people in their everyday reasoning conceptualize and categorize part-whole relationships. This question is related to, but as such distinct from, the diversity of part-whole relationships defined by formal mereological systems. By the same token, the question is not which or how different types of part-whole relations are explicitly coded linguistically (e.g., grammatically or lexically) across languages. Of course, language is an important part of human behaviour and hence linguistic coding does certainly provide us with information about common-sense reasoning of part-whole relations. But one should not assume a one-to-one mapping between linguistic coding and conceptualisation of part-whole relational types. In addition to linguistics, there are other sources of information about common-sense reasoning of part-whole relations. These can be found in disciplines such as psychology, knowledge representation and artificial intelligence, fields of investigation that are concerned with the behaviour of animate and artificial agents. Yet, research on part-whole relations in common-sense reasoning has primarily been based on inferential intuitions of the researchers. There is a definite desideratum of empirical investigations in this area. General characteristics of part-whole relations between objects (and their corresponding wholes and parts) that

are essential for the common-sense view include (cf. Henderson-Sellers & Barbier 1999a, 1999b; Schalley 2004): (i) The whole has more than one part at the instance level, i.e. parts are only parts if they are proper parts. (However, the parts can be of the same category or type, and parts can even be of the same category as the whole. For example, a team has several players, i.e. several parts of the same category, and a company may have companies as parts.) (ii) The whole has an independent ontological existence, which transcends its parts (e.g., a team is more than just the set of its players). This is closely related to the emergent property of the whole as indicated below. (iii) The whole conceptually incorporates its parts, i.e. the primary object of conceptualisation is the whole. This, amongst other characteristics, distinguishes part-whole relations from attachment or possession relations (such as an earring that is ‘attached’ to a person, or a house that belongs to a person). These characteristics apply to all part-whole relations and their related objects. In addition, HendersonSellers and Barbier (1999a, 1999b) posit that the whole generally has both an emergent and a resultant property – a property that is not evident in the parts (e.g. the functionality of a car is not evident in the car’s parts) and a property that can be deduced from the parts (e.g. the weight

COMMON SENSE REASONING ABOUT PARTS AND WHOLES 153

of a car can be deduced from the weights of its parts). However, there are different types of part-whole relationships: The relation between a car and its tyres is different from the relation between the car and the steel that is part of it, or even different from a car and its chassis. Although this has been noted by a number of researchers from psychology, linguistics, philosophy and artificial intelligence (Cruse 1979; 1986; Gerstl & Pribbenow 1995; 1996; Henderson-Sellers & Barbier 1999a, 1999b; Iris, Litowitz & Evens 1988; Keet & Artale 2008; Lyons 1977; Markman 1981; Nagel 1961; Schalley 2004; Sharvy 1983; Simons & Dement 1996; Winston, Chaffin & Herrmann 1987, amongst others) – and first steps have been taken towards explicating where the relations differ and what consequences this has – the study of different types of conceptualised part-whole relations is still in dire need of intensive study. Therefore, what is presented in the following should be considered a first approximation. Relational types are often described in terms of the related entities, i.e. in terms of the nature of whole and parts. Naturally, the ontological category of the whole has an effect on the relation between whole and part. If the whole is a mass (e.g. cake) or substance of some kind (e.g. White Russian), its parts are likely to be either portions (e.g. a slice) or substances that form part of the compound (e.g. vodka). Along those lines, Winston, Chaffin & Herrmann

(1987) list six types of part-whole relations: 1. component – integral object (pedal – bike) 2. member – collection (ship – fleet) 3. portion – mass (slice – pie) 4. stuff – object (steel – car) 5. feature – activity (paying – shopping) 6. place – area (Everglades – Florida)

However, a classification of partwhole relations on the basis of the nature of the related whole and part entities does not deliver a systematic and comprehensive classification, as long as the relations themselves are not classified. What is required is a classification via relational features as underlying criteria. Winston, Chaffin & Herrmann (1987) touch on this in their description of the six types, introducing the three binary features ‘FUNCTIONAL’ (parts are/are not in a specific spatial/temporal position with respect to each other which supports their functional role with respect to the whole), ‘HOMEOMEROUS’ (parts are similar/dissimilar to each other and to the whole to which they belong), and ‘SEPARABLE’ (parts can/cannot be physically disconnected, in principle, from the whole to which they are connected). The ‘component – integral object’ relation, for instance, carries the features ‘+FUNCTIONAL’, ‘–HOMEOMEROUS’, and ‘+SEPARABLE’. However, the suggested features are not sufficiently fine-grained to capture the full range of different part-whole relations and to account for important distinctions within part-whole relations. For instance, (1)–(3) below make statements about four ‘component – integral object’ relations (car –

154 COMMON SENSE REASONING ABOUT PARTS AND WHOLES

tyres, car – chassis, car – engine, and car – roof rack). Although according to Winston, Chaffin & Herrmann (1987) all four fall into the same ‘component – integral object’ category, (1)–(3) demonstrate that we are in fact dealing with different part-whole relations: (1) A car without tyres is still a car (although a defective one), but a car without a chassis is not a car any more. (2) The engine of a car is not visible and accessible from the outside, but the tyres are. (3) A car without an engine cannot be run, while a car without a roof rack can.

Hence, a classification via relational characteristics – more diversified and fine-grained than in Winston, Chaffin & Herrmann (1987) and not based on the nature of the related objects – suggests itself. A number of such nonmandatory characteristics of partwhole relations, also referred to as ‘secondary characteristics’ in the literature, have been suggested (explicitly or implicitly) in Cruse (1986), Henderson-Sellers & Barbier (1999a, 1999b), Schalley (2004), Winston, Chaffin & Herrmann (1987) and others. They are listed as Boolean features in Table 1, together with feature descriptions and examples.

ent slices of a cake are not. ENCAPSULATED

The part is internal and not visible or directly accessible from the outside.

An engine is an encapsulated part of a car, whereas the tow-bar is not.

EXCHANGE ABLE

The part can be exchanged with an equivalent one without destroying the integrity of the whole.

Tyres are exchangeable parts of a car, whereas the brain of a person is not.

FUNCTIONAL

A functional relationship exists between the part and the whole.

A car and its engine block have a functional relationship, whereas a cake and one of its slices do not.

HOMEOMEROUS

The part is similar to the whole, i.e. the part and the whole have decisive properties in common.

A slice of a cake and the cake are similar in nature, whereas the tow-bar of a car and the car are not.

HOMOGENEOUS

The part is compa- The tyres of a car rable to the other are homogenepart(s) of the ous parts of the whole in a regardcar, whereas ed aspect. The the brain of a parts are thus person is not conceptualised as homogeneous congeneric and to any other uniform. part of that person.

MANDATORY

Table 1. Secondary characteristics of part-whole relations (cf. Schalley 2004: 141–142) Secondary characteristic CONFIGURATIONAL

Description

The part cannot be removed from the whole without destroying the whole, it is not optional.

A mandatory part of a car is its chassis (without its chassis the car is not a car any more), whereas a seat cover is not.

The part is required with regard to the completeness of the whole (i.e. it is not facultative), but neither is the whole destroyed if the part is removed (it is only

The rear-view mirror is a canonically necessary part of a car, whereas the chassis as a mandatory part and seat covers as facul-

Example

A structural and/or Different parts of functional relaa car, such as tionship exists engine and between the diftank, are in a ferent parts of the structural and same whole. functional relationship, whereas differ-

CANONICALLY NECESSARY

COMMON SENSE REASONING ABOUT PARTS AND WHOLES 155 defective) nor is the part necessarily not removable.

tative parts are not.

REMOVABLE

The part can be removed from the whole.

A removable part of a car is the rear-view mirror, whereas sugar is a nonremovable part of lemonade.

SEGMENTAL

The part is the result of a partition (in a nonmathematical sense) and has a spatial cohesiveness (physical or conceptual).

A tyre is a segmental part of a car, whereas sugar is a non-segmental part of lemonade.

SEPARABLE

SHAREABLE

The part can be A sheet of paper removed from the is a separable whole and may part of a writexist independenting pad, wherely of the whole. as a finger of a person is not. The part may belong to two or more wholes at the same time.

A person can be a part of several social groups at the same time, whereas a chassis cannot be a part of several cars at the same time.

Table 1 lists characteristics which may or may not apply to different part-whole relations. Depending on the characteristics that apply with their Boolean value to the relation, part-whole relations can thus be sorted into different types and hence classified in terms of the configurations of positive and negative characteristics. For instance, the part-whole relation between a car and its engine can be characterised by the configuration: ‘+CONFIGURATIONAL’, ‘+ENCAPSULATED’, ‘+EX-CHANGEABLE’, ‘+FUNCTIONAL’, ‘– HOMEOMEROUS’, ‘–HOMOGENEOUS’, ‘– MANDATORY’, ‘+CANONICALLY NECESSARY’, ‘+REMOVABLE’, ‘+SEGMENTAL’,

‘+SEPARABLE’, ‘–SHAREABLE’. The same relation and therefore classification applies to the relation between an engine and a boat, but not to the relation between a chassis and a car, as the chassis is mandatory and not encapsulated, for instance. Not all of these characteristics are relevant to all part-whole relations. Therefore, there might be types of part-whole relations that do not have a value for each of the secondary characteristics and are consequently more coarsely conceptualised. In addition, there are inter-relationships between features. For instance, in a relation where a part is not removable from its whole, other characteristics become irrelevant as they require the part to be removable in the first place, such as EXCHANGEABLE and SEPARABLE. This in particular shows that the secondary characteristics are not necessarily orthogonal to one another. In formal approaches, orthogonality might be a desideratum, but not in common-sense reasoning, where characteristics are selected according to their assumed proximity to conceptualisation (Schalley 2004) and their coverage of conceptually salient aspects. The question if a removable part is exchangeable or separable, for instance, is indeed important from a conceptual point of view – cf. the examples in Table 2. Table 2. Configurations of the features REMOVABLE, EXCHANGEABLE, and SEPARABLE Whole

Part

REMOVABLE

EXCHANGCHANGABLE

SEPARABLE

lemonade

sugar

–

n/a

n/a

human body

brain

+

–

–

156 COMMON SENSE REASONING ABOUT PARTS AND WHOLES human body

heart

+

+

–

apple tree

apple

+

–

+

car

engine

+

+

+

The heart–human body example and its characterisation in Table 2 alludes to two additional, though not surprising aspects: (i) individual differences: characterisations of part-whole relations might be different for different people, and (ii) conceptualisation changes: characterisations of part-whole relations are subject to change and re-conceptualisation over time. Whether the heart is really exchangeable might be disputed, due to it being considered so crucial to the integrity of the human (in a specific culture or religion, for instance) that it might be conceived as not exchangeable [(i)]. Similarly, it most certainly used to be conceptualised as non-exchangeable in pre-transplantation times, with that conceptualisation having potentially changed with the introduction of heart transplantations [(ii)]. Conceptualisations of the partwhole relation between the same entities can therefore differ both synchronically and diachronically. Flowon effects to the type system of partwhole relations should, however, only very rarely surface as a result of diachronic and corresponding major conceptual changes – the underlying secondary characteristics will principally be able to capture all different conceptualisations. Whether Boolean characteristics are sufficient for a classification of partwhole relations is a question open for discussion. The Boolean secondary

characteristics listed in Table 1, for example, do not allow for the specification of the kind of functionality in a configurational and/or functional relationship, or of the kind of sharing in a shareable relationship. Although for a ‘basic-level’ common-sense classification of part-whole relations the granularity achieved without including such a level of detail is sufficient, more specific information may be conceptually prominent and hence co-conceptualised in specific contexts and situations. If the feature ‘+SHAREABLE’ applies, a resultant sharing can be either homogeneous or heterogeneous (cf. Saksena 1998): The part is either shared among different wholes of the same category or class (for example, a person can belong to two different research groups, both of which are instances of the same category or class) or among different wholes of different categories or classes (for example, a person can belong to a family and research group at the same time). Why are types of part-whole relations to be distinguished? Artale et al. (1996) list ‘built-in transitivity of parts’ as one of the minimal requirements of a conceptual model able to capture the ontological nature of both parts and wholes. Generally, formal approaches tend to assume transitivity for part-whole relationships. However, as has been demonstrated (e.g. Cruse 1979; 1986; Lyons 1977; Winston, Chaffin & Herrmann 1987), the situation is not as simple and transitivity does not apply across the board, cf. (4)–(6) where it does and (7)–(9) where it does not:

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(1) (Winston, Chaffin & Herrmann 1987) The carburetor is part of the engine. The engine is part of the car. ⇒The carburetor is part of the car. (2) Cream is part of the coffee-liquor. The coffee-liquor is part of the cocktail. ⇒ Cream is part of the cocktail. (3) Susana is part of the School of Languages and Linguistics staff. The School of Languages and Linguistics staff is part of Griffith University staff. ⇒ Susana is part of Griffith University staff. (4) The seeds are part of the apple. The apple is part of the apple tree. *⇒ The seeds are part of the apple tree. (5) (cf. Cruse 1979) The handle is part of the door. The door is part of the wall. *⇒ The handle is part of the wall. (6) (Winston, Chaffin & Herrmann 1987) Simpson’s arm is part of Simpson. Simpson is part of the Philosophy Department. *⇒ Simpson’s arm is part of the Philosophy Department.

Winston, Chaffin & Herrmann (1987) suggest that problems with transitivity arise when different partwhole relationships are combined (as in (6), where the difference lies in, e.g., the feature ENCAPSULATED, which carries the value “+” for the seeds being part of the apple, and “–” for the apple being part of the apple tree). That is, transitivity should in general apply at least in cases where the part-whole relationships are of the exact same type, i.e. have the

same secondary characteristics. (It is not clear to date though whether the exact same set of secondary characteristics is necessary for transitivity to hold, or whether some subset is sufficient). The transitivity of partwhole relations thus appears to be predictable through distinguishing different types of part-whole relations, and the explication of secondary characteristics will allow a systematic investigation of this issue. So far the discussion has revolved around objects and the types of partwhole relations they might participate in. Though these are the prototypical part-whole relations, the understudied area of types of partwhole relations of events and other types of occurrences should not remain unmentioned. Winston, Chaffin & Herrmann (1987), for instance, list the event part-whole relation ‘feature – activity’ as one of their six types, giving examples such as (i) paying as a part of shopping, (ii) bidding as part of playing bridge, or (iii) dating as part of adolescence. In addition to those broadly conceived ‘events’, other types of occurrences or ‘eventities’ (Zaefferer 2002) are also of interest, with examples such as (iv) arriving at home as part of running home, or (v) knowing x at some specified point t in time as part of knowing x (during a time interval i that comprises t). The examples demonstrate that a clear delineation is required what a part of an event or eventity is, and that it is unlikely that the classification in terms of secondary characteristics established for object part-whole relations can simply be extended to event or eventity

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part-whole relations. Additional ways of characterising different types of part-whole relations for events and eventities, including additional secondary characteristics, will have to be established. The extensive discussion about Aktionsarten in linguistics (Beavers 2008; 2012; Bertinetto et al. 1995a; 1995b; Dowty 1979; Egg 1994; Schalley 2014; Smith 1997; Vendler 1967, amongst others) as well as informal ontological research on the mereology of events, processes, activities, and acts might provide suitable starting points for an investigation of types of part-whole relations of events and eventities in common-sense reasoning. See also > Activity, Artifact, Grammar, Granularity, Homeomerous, Linguistic Structures, Non-literal Language Use and Part-Whole Relations, Naive Mereology, Transitivity. Bibliographical remarks

Artale, A., E. Franconi, N. Guarino & L. Pazzi, 1996. More formal yet still highly relevant discussion of the part-whole relationship. Cruse, D. A., 1979. First extensive discussion of the transitivity problem in part-whole relations. Henderson-Sellers, B.; Barbier, F.; 1999a. Discussion of the objectoriented Unified Modeling Language part-whole construct, aggregation, and related constructs. Henderson-Sellers, B.; Barbier, F., 1999b. Discussion of part-whole relationship as modelled in the object-

oriented guage.

Unified Modeling

Lan-

Saksena, M.; Larrondo-Petrie, M.; France, R.; Evett, M., 1998. Extension of the part-whole construct aggregation as used in the objectoriented Unified Modeling Language. Schalley, A. C., 2004. Includes a collation of secondary characteristics of part-whole relations, with suggestions for their modelling. Vendler, Z., 1967. Most influential treatment of Aktionsarten. Winston, M. E.; Chaffin, R.; Herrmann, D. J., 1987. Extensive discussion of types of part-whole relations. References and further readings

Artale, A.; Franconi, E.; Guarino, N.; Pazzi, L., 1996, “Part-Whole Relations in Object-Centered Systems: an Overview”, Data & Knowledge Engineering 20 (3), 347-383. Beavers, J., 2008. “Scalar Complexity and the Structure of Events”, in Dölling, J.; Heyde-Zybatow, T.; Schäfer, M. (eds.) Event Structures in Linguistic Form and Interpretation, Berlin: de Gruyter, 245-267. Beavers, J., 2012, “Lexical Aspect and Multiple Incremental Themes” in Demonte V.; McNally, L. (eds.) Telicity, Change, and State: A CrossCategorial View of Event Structure, Oxford: Oxford University Press, 2359. Bertinetto, P. M.; Bianchi, V.; Higginbotham, J.; Squartini, M., (eds.)

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1995a, Temporal Reference, Aspect and Actionality. I: Semantic and Syntactic Perspectives, Torino: Rosenberg & Sellier. Bertinetto, P. M.; Bianchi, V.; Dahl, Ö; Squartini, M., (eds.) 1995b, Temporal Reference, Aspect and Actionality. II: Typological Perspectives, Torino: Rosenberg & Sellier. Cruse, D. A., 1979, “On the Transitivity of the Part-Whole Relation”, Journal of Linguistics 15: 29-38. Cruse, D. A., 1986, Lexical Semantics, Cambridge: Cambridge University Press. Dowty, D. R., 1979, Word Meaning and Montague Grammar: The Semantics of Verbs and Time in Generative Semantics and Montague’s PTQ, Dordrecht/Bos-ton/London: Kluwer. Egg, M., 1994, Aktionsart und Kompositionalität. Zur kompositionellen Ableitung der Aktionsart komplexer Kategorien. Berlin: Akademie. Gerstl, P.; Pribbenow, S., 1995, “Midwinters, End Games, and Body Parts: A Classification of Part-Whole Relations”, International Journal of Human-Computer Studies 43 (5/6): 865-889. Gerstl, P. & S. Pribbenow, 1996, “A Conceptual Theory of Part-Whole Relations and its Applications”, Data & Knowledge Engineering 20 (3): 305-322. Henderson-Sellers, B.; Barbier, F., 1999a, “Black and White Diamonds” in France R.; Rumpe, B. (eds.) UML’99 – The Unified Modeling Language. Beyond the Standard.

Second International Conference, Proceedings, Berlin/Heidelberg: Springer, 550-565. Henderson-Sellers, B.; Barbier, F., 1999b, “What is this Thing Called Aggregation?” in Mitchell, R.; Wills, A. C.; Bosch, J.; Meyer, B. (eds.) Technology of Object-Oriented Languages and Systems (TOOLS 29), Proceedings, Los Alamitos: IEEE Computer Society, 236-250. Iris, M. A.; Litowitz B. E.; Evens, M. W., 1988, “Problems of the PartWhole Relation” in Evens, M. W. (ed.) Relational Models of the Lexicon. Representing Knowledge in Semantic Networks, Cambridge: Cambridge University Press, 261-288. Keet, C. M.; Artale, A., 2008, “Representing and Reasoning over a Taxonomy of Part-Whole Relations”, Applied Ontology 3 (1): 91-110. Lyons, J., 1977, Semantics 1. Cambridge: Cambridge University Press. Markman, E. M., 1981, “Two Different Principles of Conceptual Organization”, in Lamb M. E.; Brown, A. L., (eds.) Advances in Developmental Psychology, Vol. 1, Hillsdale, NJ: Erlbaum, 199-236. Nagel, N., 1961, The Structure of Science: Problems in the Logic of Scientific Explanation, New York: Harcourt, Brace & World. Saksena, M.; Larrondo-Petrie, M.; France, R.; Evett, M., 1998, “Extending Aggregation Constructs in UML”, in Bézivin, J.; Muller, P.-A. (eds.) The Unified Modeling Language, UML’98 – Beyond the No

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tation, Berlin/Heidelberg: Springer, 273-280. Schalley, A. C., 2004, Cognitive Modeling and Verbal Semantics, Berlin/New York: Mouton de Gruyter. Schalley, A. C., 2014, “Objectorientation and the Semantics of Verbs”, in Robering, K. (ed.) Events, Arguments, and Aspects – Topics in the Semantics of Verbs. Amsterdam/Philadelphia: John Benjamins, 159-186. Sharvy, R., 1983, “Aristotle on Mixtures”, The Journal of Philosophy 80 (8): 439-456. Simons, P.; Dement, C., 1996, “Aspects of the Mereology of Artifacts”, in Poli, R.; Simons, P. (eds.) Formal Ontology, Dordrecht: Kluwer, 255276. Smith, C. S., 1997, The Parameter of Aspect, 2nd ed., Dordrecht: Kluwer. Vendler, Z., 1967, “Verbs and Times” in Z. Vendler (ed.) Linguistics in Philosophy, Ithaca, NY: Cornell University Press, 97-121. Winston, M. E., Chaffin, R.; Herrmann, D. J.; 1987, “A Taxonomy of Part-Whole Relations”, Cognitive Science 11 (4): 417-444. Zaefferer, D., 2002, “Polysemy, Polyvalence, and Linking Mismatches. The Concept of RAIN and its Codings in English, German, Italian, and Spanisch”, DELTA – Documentação de Estudos em Lingüística Téorica e Aplicada 18 (spe.): 27-56. Special Issue: Polysemy. Andrea C. Schalley

Conscious Experience Discussions about mereology and conscious experience, e.g., qualia, subjectivity, etc., are difficult for a few reasons. First, unlike physical entities, there is no agreement as to whether part and whole talk even applies sensibly to conscious experience. Whether it does apply to conscious experience or not depends on which feature of conscious experience is being focused on such as its first-person nature, or more generally one’s conception of conscious experience. There are many different and incompatible conceptions of conscious experience out there, such as eliminativism about qualia versus consciousness as an entity, and they each have different mereological implications. Second, there are multiple dimensions in which one can raise questions about conscious experience and mereology. One can ask whether or not conscious experience has parts that are themselves conscious experiences such as co-conscious qualia like seeing red and feeling pain that are somehow joined to create phenomenal unity. More generally one can ask if conscious experience is divisible into numerically distinct streams, centers or subjects as for example some people claim is the case with split-brain patients (cf. Brook/Raymont 2010). This is all complicated by the fact that not everyone agrees that conscious experience is unified, and those who do agree it’s unified often disagree about how it is unified, e.g., does it have something to do with the phenomenal

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subject, binding mechanisms in the brain, etc. Nor does everyone agree that conscious experience is even made of parts. Then there is the question of how conscious experience relates mereologically to the brain or more fundamental physical entities. There is currently no consensus here either. For example one might hold that only a subset of the brain is responsible for all of conscious experience or that some global brain process is necessary. There is also no agreement on which scale of brain activity is behind conscious experience, theories range from subcellular mechanisms to neural connections to large-scale neural synchrony and beyond. Moving beyond brains to more fundamental physical entities, panpsychism and panprotopsychism hold that basic particles have some sort of (proto-) conscious experiences that somehow combine to form unified conscious subjects such as ourselves. This is known as the combination problem in the panpsychism literature and very few believe it has been answered (Silberstein 2010; Silberstein, Stuckey and McDevitt 2017 chps. 7 and 8). Allegedly the advantage of panpsychism over emergentist accounts of conscious experience is that the former does not have to explain how conscious experience comes from matter, but the combination problem puts a qualifier on any such claims. Of course if we define matter as essentially non-conscious or nonmental this is an impossible task for emergentism. Thus historically many people who advocate for emergence

have explicitly advocated for strong emergence wherein conscious experience is explained by brute bridgelaws connecting conscious states with brain states (Chalmers 1996, Silberstein 2017). Brute-bridge laws rub many people the wrong way because they are not explanatory as in the ‘and then a miracle occurred’ cartoon by Sidney Harris, and because they detract from the unity of the world and of science. There are exceptions among emergentists however, some have attempted to elaborate a type of emergence that is neither strong or weak, i.e., a type of emergence that is truly ontological and explanatory (Silberstein 1998; 1999; 2001; 2002; 2006 and 2009; Humphreys 1997). Silberstein for example discusses what he calls mereological emergence (wholes that have properties and causal capacities not determined by the properties of their basic parts) and focuses on quantum entanglement as an example (ibid; also see entry on quantum mechanics this volume). He has speculated that something of that nature could explain how conscious experience emerges from matter (ibid). It must be stressed that this is only an analogy, no claim is being made that consciousness has anything to do with quantum mechanics! As a contrast to both panpsychism and emergence, neutral monism claims that mind and matter are not distinct types at all and both co-emerge from some neutral base (Silberstein 2010, Silberstein, Stuckey and McDevitt 2017 chps. 7 and 8). If neutral monism is true there may be no mereological relations between matter and conscious

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experience at all. Therefore given all the complexities involved here we must restrict ourselves in some fashion. The central questions pertaining to conscious experience and mereology discussed herein are as follows: (1) is consciousness made of qualia-like parts, is it atomistic such that every unified conscious state with its auditory, visual, tactile, etc. aspects can be decomposed into those experiential elements and perhaps further divided or is consciousness essentially or necessarily indivisible? (2) What exactly is the relationship between conscious states, brain states and the environment? There are several dimensions to this question. Is there a part/whole relationship between brain states and conscious states? Is consciousness in some sense more than the ‘sum of its parts’? Does consciousness ‘supervene’ on or is it ‘realised’ synchronically by anything physical such as brain states and if so, exactly which physical features determine and explain conscious states? How does neural activity generate the various unities of conscious experience such as phenomenal unity: the fact that everything is experienced together as a unity in a fully unified conscious state? How does phenomenal unity relate to other forms of mental, cognitive or neurological unity? The task of clarifying the relationship between conscious experience and mereology is notoriously tricky and slippery. For example, Bennett and Hacker (2003) have been accusing reductive neuroscientists and reduc-

tive neuro-philosophers of committing the ‘mereological fallacy’: attributing properties to parts that properly only belong to wholes. Their point is that when people say things like ‘the brain thinks, feels, decides, etc.’, they are committing what Ryle would call a category mistake because it is only the whole person that has these attributes. But even if one is sympathetic to this bit of Wittgenstein inspired conceptual analysis, it does not settle any ontological or scientific questions (for those of us who have not abandoned such quests) pertaining to the mereological relationship between say conscious states and brain states. Things are made much worse by the fact that the brain includes all the domains and spatial and temporal scales of reality that consist of parts and wholes ranging from microphysics to conscious experience, and both the brain and the mind have many complex interactions with the body and the environment. For example some claim that quantum effects that occur within subcellular structures known as microtubules that are inside neurons are responsible for conscious experience while on the other end of the spectrum some integrated information theories of conscious experience just identify conscious experience with some measure or description of integrated information in the brain, thus making conscious experience an information theoretic property that need not be essentially biological at all (cf. Van Gulick 2014). Some neuroscientists are searching for the neural correlate (NCC) of specific conscious experiences such

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as specific kinds of visual perceptions like seeing colour – the NCC being a sufficient condition for the occurrence of said percept (Koch 2004). This certainly suggests an atomistic conception of consciousness. Note also that the very idea of the ‘binding problem’ in the cognitive neuroscience of consciousness presupposes the atomistic model. On this view the first goal of the cognitive neuroscience of consciousness is to identify the NCC of every experiential model. However, there are plenty of dissenters. Searle calls the atomistic view of consciousness the ‘building-block approach’ and rejects it in favour of what he calls the ‘unified-field approach’ (2004). He thinks that any changes in experiential content are just modifications in the already active conscious field. The primary question for neuroscience on this view is how does the brain produce the conscious state as such as opposed to the unconscious state – how does it produce the whole conscious field? What is the difference between the conscious brain and the unconscious brain as such? The secondary question will be about the modifications of that field. The claim is that the experience of blue for instance can only happen in a brain that is already conscious – and thus a fully unified conscious state is not merely an aggregate of such discrete percepts. Searle is clear that the field notion is meant merely as a metaphor and he is really advocating for the claim that consciousness is essentially or necessarily phenomenologically unified (ibid). Another well-known dissenter is Tye (2003). He argues

against the building-block approach in favour of what he calls the ‘one experience view’, the idea that each fully unified conscious state is just one experience followed by another one and so on. The Tye and Searle models take phenomenological unity as essential to conscious experience and they both assume if that unity itself is to be explained, the explanation must come from beyond the domain of phenomenology itself, e.g., neuroscience. Whatever their views about the unity of conscious experience, most theorists do think that an individual’s conscious states supervene on their brain states, at least nomologically in the actual world. For instance even Chalmers’ an avowed antireductionist and dualist holds this (ibid). Minimal supervenience physicalism entails that there can be no change in conscious experience without a change of brain state and that people in exactly the same brain state will be having the same experiences. Notice that the supervenience claim is not itself explanatory and is compatible with a number of different explanatory relations. In fact minimalist supervenience physicalism does not entail reduction, causal closure of the physical, etc. For example, as is well known, Mysterians believe that science may never be able to explain conscious experience even though it does supervene on the brain. Most theorists however hold that neuroscience will one day actually explain consciousness. The nature of the explanation in question looks

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very different for different theories. In Searle’s schema for example the order of explanation in the cognitive neuroscience of consciousness will be as follows: 1) collect NCCs, 2) discover efficient causal relations between global brain states and unified conscious experience and then 3) come up with a grand theory. He says we are still early in phase 1 (Searle, 2013). Accordingly, as we have already noted, different theories and models will focus on different scales and aspects of the brain such as quantum processes, sub-cellular processes, neuronal action, large-scale neural synchrony or oscillations, multiple brain regions in interaction, topological network properties and even more functional information-theoretic features such as Integrated Information type theories. Given that Searle adopts a field view of conscious experience he expects to find a global brain state or process that somehow efficiently causes the conscious field (ibid). He suggests we look for some ‘massive’ activity of the brain capable of giving rise to full phenomenological unity. Bickle (2003) on the other hand gives a defense of reducing mental states (including conscious states) and functions to molecular and genetic features of neural systems. With the building-block approach the idea is to find the NCC of a particular quale such as seeing red and then somehow go beyond correlates to more robust explanations such as causal mechanical (localisation and decomposition) or identity. However the essential unity model has suggested to some that one should reject

such ‘smallism’ and search rather for an isomorphism in the brain’s neurochemical activity to phenomenological unity. Many people believe the explanation is to be found in largescale neural dynamics that gives rise to large-scale conscious integration via phase synchrony or some other form of “generalised synchrony” across different frequencies (Thompson 2007). The reality of cognitive neuroscience is that it uses a number of explanatory modes that range from trying to discover NCCs, delimiting causal mechanisms to quantifying much more mathematically abstract causal relations and information theoretic relations between brain components or different domains. The specific tools involved here include the localisation and decomposition of mechanistic biology, various knockout methods, various forms of imaging, etc. A growing minority of cognitive scientists and systems neuroscientists have eschewed mechanistic explanations and embraced dynamical systems theory and graph theory as the primary mode of explanation (Chemero and Silberstein 2008; Silberstein and Chemero 2013), more on this shortly. Some theorists want to try and accept some aspects of both building-block models and essential unity models of consciousness experience. In the past Koch for example has made a distinction between enabling factors and specific factors of consciousness that maps directly onto the consciousness as such and content of consciousness distinction. He suggests we look for the NCC of both and he claims that the NCC of the former will likely be

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some kind of global large-scale synchrony while the NCC of the latter will be more local, short-lived and unstable (Koch 2004). Searle and Koch are perhaps in agreement that the thalamocortical system is the most likely candidate for the largescale neural dynamics underlying the unified conscious field. As with building-block theories, one can accept the essential unity thesis and disagree about the particulars of the isomorphism story. It is widely agreed that along with cortical activity, midbrain and brainstem mechanisms are essential for consciousness as such, while thalamocortical networks are the most likely place for large-scale neural integration isomorphic to or correlated with transient unified conscious states – content of consciousness (Baars 2005). But again, whatever model of conscious experience one accepts they may have very different stories about how this maps onto the brain. Van Gulick gives a good example of this in what follows: One particular recent controversy has concerned the issue of whether global or merely local recurrent activity is sufficient for phenomenal consciousness. Supporters of the global neuronal workspace model (Dehaene 2000) have argued that consciousness of any sort can occur only when contents are activated with a large scale pattern of recurrent activity involving frontal and parietal areas as well as primary sensory areas of cortex. Others in particular the psychologist Victor Lamme (2006) and the philosopher Ned Block (2007) have argued that local recurrent activity between

higher and lower areas within sensory cortex (e.g. with visual cortex) can suffice for phenomenal consciousness even in the absence of verbal reportability and other indicators of access consciousness (cf. Van Gulick 2014). While there is little consensus about any of this in cognitive neuroscience, the following is accepted by many: (1) the reticular activating system (RAS) in the brainstem seems to be the mechanism of wakefulness – transitions from unconscious sleep to waking, (2) in the cortical EEG this transition is indicated by changes from high voltage, slow, and regular activity of stage four sleep to the low voltage, fast and irregular waves of the waking and dreaming state, (3) the critical link between the brainstem reticular formation (RF) and the cortex is the thalamus, (4) the reticular-thalamic core is a necessary condition for consciousness – large areas of cortex up to an entire cerebral hemisphere can be surgically removed without the loss of consciousness as such, but very small lesions (a few cubic centimeters) to parts of the reticular-thalamic core (especially the intralaminar nuclei) cause coma, and (5) lesions to the cortex selectively impair conscious features and functions (content of consciousness) of otherwise unified states such as color perception, object perception, face recognition, etc. (Baars 2005). Finally, there are some who reject the NCC picture entirely and hold that the necessary and jointly sufficient conditions for conscious experience are brains, bodies and environments

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in complex interactions over various spatial and temporal scales (Thompson 2007; Silberstein 2006; Silberstein and Chemero 2011a; 2011b). This is a rejection of the conception of conscious experience as a virtual construct generated by neural activity. Advocates of such a view are usually working out of a handful of different but related traditions that include: enactivism, ecological psychology, radical empiricism, phenomenology, extended cognition, etc. Thompson (ibid) for example, who also accepts the essential unity of consciousness, thinks that since consciousness as such depends on midbrain and brainstem structures as well as regulatory systems of the entire body that it is wrong to say the unified conscious field ‘supervenes’ on only the thalamocortical system. On his view a transient conscious state is neurally ‘embodied’ as a large-scale dynamical state that entrains more local brain activities. This transient state itself is embedded within a more overarching state of ‘background consciousness’ (consciousness as such) configured by dynamic integration along the entire neuraxis. This is a fancy way of saying that Thompson (but not Koch or Searle) advocates for an embodied and embedded account of consciousness. Rooted in ecological psychology and dynamical cognitive science, Silberstein and Chemero (2011a, 2011b) have argued for a similar view, extending the dynamical system associated with conscious experience to include non-linear interactions between brain, body and environment over various spatial and temporal

scales (ibid). They call such conscious agents: extendedphenomenological-cognitive systems (ibid). Silberstein has further argued that the best way to characterise the idea of extended conscious experience is neutral monism (Silberstein 2010; Silberstein and Chemero 2015). Of course there are many other conceptions of conscious experience and its relation to the physical world, but hopefully the preceding is sufficient to provide a broad map of the territory. See also > Descartes, Dynamical Systems, Experience, Gestalt, Perceptual Whole, Whitehead's Metaphysics. References and further readings

Baars, B. J., 2005, “Global Workspace Theory of Consciousness: toward a Cognitive Neuroscience of Human Experience”, Progress in Brain Research, vol. 150: 45-53. Bennett, M. R., Hacker, P. M. S., 2003, Philosophical Foundations of Neuroscience, Wiley-Blackwell. Bickle, J., 2003, Philosophy and Neuroscience: A Ruthlessly Reductive Account, Norwell, MA: Kluwer Academic Press. Brook, A.; Raymont, P., 2010, “The Unity of Consciousness”, in: The Stanford Encyclopedia of Philosophy, Zalta, E. N. (ed.), http://plato.stanford.edu/entries/consc iousness-unity/ Chalmers, D., 1996, The Conscious Mind: In Search of a Fundamental

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Theory, Oxford: Oxford University Press. Chemero, A.; Silberstein, M., 2008, “After the Philosophy of Mind: Replacing Scholasticism with Science”, Philosophy of Science vol. 75, No. 1: 1-27. Humphreys, P., 1997, “How Properties Emerge”, Philosophy of Science 64: 1-17. Koch, C., 2004, The Quest for Consciousness: A Neurobiological Approach. Englewood, CO: Roberts & Company. Searle, J., 2004, Mind: A Brief Introduction, Oxford: Oxford University Press. Searle, J., 2013, Our Shared Condition-Consciousness, http://www.ted. com/talks/john_searle_our_shared_c ondition_consciousness Silberstein, M., 1998, “Emergence and the Mind-Body Problem”, Journal of Consciousness Studies vol. 5, No. 4: 464-482. Silberstein, M., 1999, “The Search for Ontological Emergence”, Philosophical Quarterly 49: 182-200. Silberstein, M., 2001, “Converging on Emergence: Consciousness, Causation and Explanation”, Journal of Consciousness Studies Vol. 8, No. 910: 61-98. Silberstein, M., 2002, “Reduction, Emergence, and Explanation”, in The Blackwell guide to the philosophy of science, Machamer. P.; M. Silberstein, eds., Malden, MA: Blackwell, 203-226.

Silberstein, M., 2006, “In Defense of Ontological Emergence and Mental Causation”, in The Re-emergence of Emergence. P. Davies, Editor. Chapter 9, Oxford: Oxford University Press. Silberstein, M., 2009, “Emergence and consciousness”, in Oxford Companion to Consciousness, Bayne, T., Cleeremans, A.; Wilken, P., eds., Oxford: Oxford University Press. Silberstein, M., 2010, “Why Neutral Monism is Superior to Panpsychism” in Mind and Matter: an International Interdisciplinary Journal of MindMatter Research. Imprint Academic Mind & Matter Vol. 7(2): 239-248. Silberstein, M.; Chemero, A., 2011a, “Complexity and Extended Phenomenological-Cognitive Systems”, Topics in Cognitive Science: Special Issue on the Role of Complex Systems in Cognitive Science. Guy Van, 2011: 1-16. Silberstein, M.; Chemero, A., 2011b, “Dynamics, Agency and Intentional Action”, Humana.Mente: Journal of Philosophical Studies. Special issue on Agency: from Embodied Cognition to Free Will. Issue 15. Silberstein, M.; Chemero, A., 2013, “Constraints on Localisation and Decomposition as Explanatory Strategies in the Biological Sciences”, Philosophy of Science, vol. 80, nr. 5: 958-970. Silberstein, M.; Chemero, A., 2015, “Extending Neutral Monism to the Hard Problem”, Journal of Consciousness Studies, special issue Consciousness Unbound: Going Be

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yond the Brain, Silberstein, M.; Chemero, A. (eds.), 22 (3-4): 181194. Silberstein, M., 2017, “Neutral Monism Reborn: Breaking the Gridlock Between Emergent versus Inherent”, Seager, W. (ed.) Routledge Companion to Panpsychism. Silberstein, M.; Stuckey; McDevitt, 2017, Beyond the Dynamical Universe: Unifying Block Universe Physics and Time as Experienced, Oxford: Oxford University Press. Thompson, E., 2007, Mind in Life: Biology, Phenomenology, and the Sciences of Mind, Harvard: Harvard University Press. Tye, M., 2003, Consciousness and Person, MIT Press. Van Gulick, R., 2014, “Consciousness”, in: The Stanford Encyclopedia of Philosophy, Zalta, E. N. (ed.), http://plato.stanford.edu/entries/consc iousness/#QuaThe. Michael Silberstein

Continuants and Occurrents The term ‘continuant’ has been introduced by W. E. Johnson to replace Aristotle's term of “substance” (ousia) as a residual term, freed from certain controversial philosophical implications. Johnson characterises the continuant as “that which continues to exist while its states and relations may be changing” (Logic I, p. 199). Thus a continuant has a relatively long duration, is a subject, i.e., a bearer of properties and stands in

relations to other items. The category of continuants is contrasted with the category of occurrents, which resemble Aristotelian accidents in that they inhere in continuants. Unlike recent uses of the term ‘occurrent’, where ‘occurrent’ is predominantly used to cover the subcategories of processes and events, respectively, in Johnson occurrents are momentary and simple. Continuants and occurrents are thus compatible categories and can be part of the same ontology, but Johnson uses the term ‘occurrent’ also to characterise the empiricist alternative to a continuant-based ontology. He takes empiricists to champion a mono-categorial approach that is based on the category of occurrents only. This fits well with Locke's mockery of the idea of a bearer of properties and with Berkeley's and Hume's claim that only entities of short duration are given to us and that ordinary objects are composed of such momentary entities. The premise in the background is that perception is always restricted to a present moment and that entities exceeding a moment cannot be perceived. Frequently, the issue discussed by empiricists is identity, for example, the question of whether various experiences can be attributed to the same self, more generally, whether various occurrents can be known to belong to the same continuant. However, as Johnson points out, the real issue is whether continuants exist or not. Hume clearly denies physical as well as mental continuants in favour of a temporal series of occurrents.

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Against this view Kant argues that continuants – though they are not given to us in immediate experience – are necessary constructs required for our understanding of change. Without a continuant as subject, so the argument goes, the change of an ordinary object would lack unity, as is the case with Berkeley’s and Hume’s temporal series. Undoubtedly, to refer to a temporal series of occurrents does not provide for an adequate ontological grounding of the phenomenon of a changing thing, since occurrents of different things can also stand in the relation of temporal precedence. In order to serve as an adequate ontological grounding the temporal series would need to be a series of occurrents that also stand in a suitably tight causal relationship. Different ontological analyses of the phenomenon of the changing thing are offered: firstly, the simple continuant analysis. The temporal series analysis comes in two varieties depending on whether a series is acknowledged as an entity or existent (hereafter: ‘strong series analysis’) or not (hereafter ‘weak series analysis’). According to the strong series analysis there are entities of longer duration, though these are complex in contrast to a continuant which normally is taken to be simple. Whether the classical empiricsts are committed to a strong or to a weak series analysis is often difficult to decide, for although complexes (including series) loom large in empiricism, their structure and status is rarely fully investigated. It seems clear, though, that classical empiri-

cists are at least committed to the weak series analysis and do not admit of anything that is simple and exists for a longer period of time. In the contemporary debate about the ontological interpretation of the phenomenon of change in things the series analysis of the empiricists has been revived using the notion of temporal parts. D. Lewis – following Mark Johnston – distinguishes between “endurance” and “perdurance” accounts of the persistence of a thing through change. The endurance account is the continuant analysis, while the perdurance account reformulates the strong series analysis (the weak series analysis is not taken into consideration). Lewis explains the distinction in terms of temporal localisation. To endure means to be localised as a whole at more than one point in time. To perdure means to exist as a series of temporal parts each of which exists at exactly one point in time (or very short period). In contrast to Lewis’ distinction between endurance and perdurance, the distinction between simple continuant analysis, strong series analysis, and weak series is at bottom mereological – the three analyses postulate different degrees of unity for the entity that is to serve as the ontological ground of a persistent changing thing. Much of the contemporary debate focuses on the clarification of the mereological structure of a perduring entity and in particular on the question of how to ground the unity of the temporal complex, i.e., of explaining the way in which the different tem-

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poral parts form together the same persistent thing. This difficulty does not arise with a view that advocates a continuant analysis of changing things that allows for continuants to have temporal parts and to be nevertheless simple (Tegtmeier 2008). On this analysis continuants both endure but they also perdure, which becomes possible if one, firstly, adopts a relational view of time, thus rejecting independently existing points in time, and, secondly, postulates a relation of temporal parthood holding between the continuant as a whole and an unchangig temporal part which bears an occurrent. The respective relational fact is a complex consisting of two particulars (the temporal part and the simple continuant) and the relation of temporal part. The contradictions of change discovered by the Ancient Greek philosopher Parmenides – nowadays often called “the problem of temporary intrinsics” (D. Lewis) – traditionally has been solved by presentism, i.e., the thesis that all and only that which is present exists. The traditional solution to the contradictions of change (expounded exemplarily by Aristotle) postulates that only contradictions between respectively present entities are problematic; given that what was the case before the change, and would be incompatible with the present state, no longer exists, no contradiction arises. A particular sophisticated version of presentism, based on a mereological theory of continua and boundaries, has been offered by F. Brentano.

The particular strength of the series analysis with momentary members (occurrents) is that it straightforwardly eliminates the contradictions of change, since the contradictory attributes before and after the change are not related to the same persistent entity (e.g., continuant). Many philosophers (e.g. D. Wiggins) think that the contradictions of change are easily solved, and thereby the traditional continuant analysis retained, if attributions are related to times. However, there are ontological difficulties with this solution – either the instantiation relation is turned into a three-place relation (holding between attributes/occurrents, continuants, and times) or properties have to be turned into relations. Moreover, this strategy of resolving the contradictions of change cannot be combined with a relational view of time. Finally, in addition to Tegtmeier’s proposal to disengage the contrast between continuants and occurrents on the one hand, from the contrast between endurance and perdurance on the other hand, recent work in analytical process ontology has challenged both of these contrasts, reconceiving of endurance as “recurrence” (Seibt 2008) or postulating outright a category of “occurrent continuants” (Stout 2016). References and further readings

Aristotle: Metaphysica Z Berkeley, G., Three Dialogues between Hylas and Philonous I Hume, D., A Treatise of Human Nature. Book I Part I

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Bergmann, G. 1964, Realism, University of Wisconsin Press, Section Six Brentano, F. 1976, Philosophische Untersuchungen zu Raum, Zeit und Kontinuum. Meiner Johnson, W.E. 1924, Logic III. Constable. Chapter 7 Lewis, D., 1986, On the Plurality of Worlds, Blackwell. Lewis, D, 1976, “Survival and Identity”, in Rorty A. O. (ed.), The Identities of Persons, University of California Press. Seibt, J., 2008, “Beyond Endurance and Perdurance: Recurrent Dynamics.” In Kanzian, C. (ed.), Persistence, Ontos Verlag, 121-153 Stout, R., 2016, “The Category of Occurrent Continuants”, Mind 125: 41-62. Tegtmeier, E. 2008, “Persistence” in Kanzian, C. (ed.), Persistence. Ontos Verlag. 185-195 Wiggins, D., 2008, Sameness and Substance Renewed. 2 ed. Blackwell Erwin Tegtmeier

Cosmology The idea of applying mereological concepts to the study of cosmology is at once very old and very new, but it is dogged by a peculiar ambiguity. In speaking of mereology (or its more refined cousin, mereotopology) within the context of cosmology, is the approach specifically philosophical

in character, or is it more physical/scientific? There are often overlaps between the two approaches, but in general they will employ differing methodological stances toward the mereological tools being employed. Yet the distinction is not one that is always respected. Very broadly, philosophical cosmology can be viewed as a general theory of explanation, whereas scientific cosmology can be viewed as a specific theory of nature, in both cases in the broadest terms possible. Where they differ, then, is over the question of whether nature is all there is in the cosmos. The original sense of the word ‘cosmos’ was of a system of order, so the question then becomes whether all order is strictly within nature, or possibly involves logical aspects that go beyond naturalism, depending on how widely or narrowly such ‘naturalism’ is conceived. For the Greeks, insofar as they can be said to have practiced science, there was no real distinction between their scientific and philosophical cosmology. Certainly it is the case that the question of the nature of space and whether it had parts or was all of a whole was of considerable interest. Questions included: How does any given place define itself as a part of the space (s) around it? Does space have a boundary? Is that boundary a part of space? Is the place of the earth within the cosmos a distinguished part of space? Are there any such distinguished parts? However, since the subject of mereology in ancient Greek philosophy is covered elsewhere in this volume, we will not spend much time on the subject here.

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The one exception will be a brief review of Michael C. Rea’s (2001) argument on the intelligibility of Eleatic monism, because it illustrates a classical position in a contemporary light. Rea’s argument centers on four principles which, he claims, are individually acceptable, and together lead to Eleatic monism. Quoting Rea, these claims are (2001: 130): EXTENSIONISM – There are no unextended material objects. EXCLUSIVISM – Not every filled region of space at every time is filled by a material object. ETERNALISM – There are some past objects, there are some future objects, and there neither were nor will be objects that do not exist. THE PLENUM PRINCIPLE – Spacetime is a connected set of points, and every region of spacetime, no matter how small, is filled by matter. Notice that material objects, in Exclusivism, and matter in the Plenum Principle, refer to different things. Material objects are just what one would expect them to be, while matter per se is akin to the material ether of 19th C. physics. Rea examines these principles in various combinations (and in more detail than can be traced out in this brief entry) and concludes that, while Eleatic monism might be counterintuitive, even false, it is a logically coherent position to take. For our purposes, the Plenum Principle is of particular interest as it applies to contemporary forms of cosmology.

Mereology makes its first appearance in contemporary Western philosophy in the work of Alfred North Whitehead (1861–1947). It is worth noting that the Polish mathematician Leśniewski began developing his mereological theories some years before Whitehead’s were ever published. But Leśniewski's work was in Polish, and did not become available to Western European readers until much later (see Simons 1987, ch. 1). In addition, Leśniewski's work was on the foundations of mathematics, whereas Whitehead was interested in generalising concepts of space used in empirical science so as not to rely on infinitesimal points (see Herstein 2006, ch. 3 and 4). However, relating Whitehead’s work to cosmology requires a bit of care. When Whitehead was developing his mereology, his focus was on natural science and the philosophy of nature. When, on the other hand, he turned his attention explicitly to cosmology, that philosophy of nature became a proper part (as it were) of his broader, cosmological program, and the tool employed evolved into mereotopology. Whitehead’s mereological works appeared in the first two of three works that presented his philosophy of nature/science (see Whitehead 1919; 1920; Whitehead 1922 did not contribute to the development of his mereology, but rather took those works for granted in developing his criticisms of Einstein’s general theory of relativity, and what will be referred to below as the ‘measurement problem of cosmology’). Note that at this stage of his career, Whitehead was not doing cosmology in either

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the scientific or philosophical sense. The mereological concepts that Whitehead was developing were aimed at an epistemological goal, that of ‘natural knowledge’. To this end, he proposed his theory of ‘extensive abstraction’. Not trusting the infinitesimal points of geometry (as this was naively construed) as properly representative of either nature or our ability to investigate it, Whitehead’s mereological technique involved a system of idealised abstractions from the given whole of nature, to smaller and smaller parts. For Whitehead, there is no ultimate bottom, no ‘atom’ or ‘point’ that can ever be reached, either in thought or in nature. This merits comparison to the Plenum Principle mentioned above, because if there are such atoms, then one is in a Democritean universe of atoms and void (contra the Plenum Principle), and the world is not fully filled and connected with ‘matter’. For Whitehead in these books, there are no mereological atoms: all parts have yet further parts. This notion shows up in the more recent mereological literature as ‘gunk’, of which more will be said presently. However, Whitehead did argue that the particular character of the world was such that extensive abstraction could ‘zero in’ on increasingly simple parts until a dominant character of nature could be taken as being the case at the idealised (but not actual) ‘point’ which the process of extensive abstraction enclosed in ever smaller (but never final) parts. Responses to his philosophy of nature led Whitehead to broaden the

nature of his task to include speculative philosophy and, specifically, philosophical cosmology, which resulted in his magnum opus, Process and Reality (Whitehead 1929; referred to as ‘PR’ hereafter), tellingly subtitled “An Essay in Cosmology”. One of the most unfortunate errors of interpretation that has continuously dogged this work is the belief that it is about scientific, rather than philosophical, cosmology; many to this day believe Whitehead was proposing an interpretation of the then still nascent quantum mechanics. There is a compelling, if not yet widely read or accepted, argument that such a ‘physicalist’ reading is mistaken, and it is that reading which will govern the remainder of this discussion of Whitehead (Auxier & Herstein 2017). The theory of extension that Whitehead brought forward into PR was significantly influenced by a series of articles by Theodore de Laguna (especially Laguna 1922). Rather than an endless series of ever shrinking parts within parts, de Laguna argued for incorporating the topological ideas of neighborhood and contact to achieve Whitehead's ultimate goal of characterising an idealised point. Whitehead adopted de Laguna’s suggestion, and it became his mature theory of extension in PR. As a philosopher, Whitehead did not spend much effort in formalising these ideas, so it fell do others to do so. Alfred Tarski picked up Whitehead’s notions from Concept of Nature and axiomatised those with various contributions of his own (Tarski 1929). Bowman Clarke, in a pair of articles,

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formalised Whitehead's mereotopology, but Clarke's work depended on quantifying over second order entities, so it fell to others to render Clarke’s work in first order logic (see Clarke 1981; 1985; for a first order representation see, e.g., Pratt & Schoop 1998). Even though Whitehead always endorsed his previous works on the philosophy of nature, some confusion can be found in the literature over whether he rejected the ‘no smallest part’ thesis found in his earlier works when he wrote PR. This is because Whitehead describes the world as being ‘atomic’ at numerous points PR. However, caution needs to be exercised here. It is very clear from Whitehead's text that these ‘atoms’ are not micro-physical corpuscles; indeed, the considerations in Whitehead’s cosmology are taking place at a logically more basic level than metrical space, so characterising anything as ‘large’ or ‘small’ would be meaningless. Rather, Whitehead's ‘atoms’ owe their sense to the original Greek word meaning ‘uncut’ or ‘undivided’. They are ‘atomic’ because they are undivided wholes (Auxier and Herstein 2017). This brings us to the topic of ‘gunk’. The term ‘gunk’ was originally coined by David Lewis (1991). Gunk has been characterised as, “matter every part of which has proper parts, so that there are no ultimate parts to form an atomistic base” (Schaffer 2010: 61). One can see right away the relationship with the Plenum Principle, and to Whitehead’s infinitely divisible “ether of events” (Whitehead 1919).

The word itself was coined so as to offer a minimal number of ontological prejudices as to what sort of ‘stuff’ gunk might happen to be. One can go no further than say a ‘minimal number’ of such prejudices, because the idea of gunk is not metaphysically neutral. One of the issues that wracks the idea of gunk is that denying the ultimacy of atoms does not suffice to explain gunk's relationship(s) to the various concepts of the continuum. Suppose the universe is ‘gunky’, that is, it possesses an underlying mereology of gunk. What, then, is the status of infinitesimal points? On the one hand, orthodox mathematical thinking tells us that a continuum is composed of points, and isn't gunk supposed to be a continuum? But on the other hand, a point is “that which has no parts,” (Euclid, Elements, book 1, definition 1), which looks very much like a mereological atom. Thus, the point on the real number line corresponding to the number pi is not further divisible into smaller units. As we saw, Whitehead rejected points as having any ontological status, but was he correct in doing so? In Whitehead's day, it certainly seemed as if puzzles about the continuum – tracing back to the days of Zeno – had been solved (Russell 1903: 346ff). But these solutions were settheoretic in nature, and it is arguably unwise to attempt to collapse mereology into set theory. (Recall that Leśniewski's work was intended to find a non-set-theoretic foundation for mathematics in mereology; this issue is made even worse recently by Cohen’s (2008) model theoretic tool

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called ‘forcing’ which can demonstrate ‘levels of infinity’ between that of the integers and the real numbers, a notion which set-theoretic concepts of the continuum have no place for – where, then, lies the ‘genuine’ continuum?). On the surface, such questions might seem of purely mathematical interest, but they bear directly on questions of whether a cosmos governed by the rules of general relativity can or cannot be ‘gunky’. If, as is the case with gunk, for every object x there is some y which is a proper part of x (that is, y is a part of x, but it is not the case that y = x) then what does that mean about the structure of fourdimensional space time (e.g., Amtzenius 2008; Russell 2008)? Infinitesimal points and mereological ‘atoms’ are often collected under the common heading of ‘simples’, allowing more generalised questions to be asked regarding space time (Simons 2004). The issues are trickier than one might initially suppose because the topological structure of 4-manifolds (which relativistic spacetime is) turns out to be a more difficult problem to solve than for either higher or lower dimensions (Freedmann and Quinn 1990; see in particular the introduction). But the discussion of simples can go well beyond our scientifically informed notions. Recalling Whitehead’s use of ‘atom’ to mean an undivided whole, the question can be broached whether a person qualifies as an extended simple (Lowe 1996). Finally, it should be added that it is at least arguably the case that the mereological choice between gunk and atoms (in the non-Whiteheadian sense) may

not cover all of the possibilities (Simons 2004). In addition to playing a role in discussions of general relativity, it has also been argued that mereology can play a key part in the understanding and interpretation of quantum mechanics (see e.g., Calosi et al 2011). This leads us to finally ask about the role of mereology in a unified scientific cosmology, and to turn to the subject of String Theory. ‘Elementary’ texts on String Theory are readily available (see e.g., Tong 2009, Siegel 2001), but these are ‘elementary’ only in comparison to the far more difficult advanced texts. Nevertheless, a few intuitive remarks can be offered here. String theory came into vogue in an attempt to bring the non-linear, macroscopic theory of relativity into a coherent unity with the linear, microphysical theory of quantum mechanics, wherein higher dimensional ‘filamental’ elements that bore a formal analogy to mundane strings, could have those higher dimensions ‘packed’ into a single point within a lower dimensional space. These ‘strings’ have the interesting mereological property of being both extended and simple – that is, they do have extension, but they don't have parts (McDaniel 2007). There are many who would argue that such a concept is nonsense. But this arguably takes the relationship of parthood to be itself strictly a matter of arbitrary divisibility within an extended region (see eg. Simons 2006). Further, qualitatively heterogenous extensions may have their own, inde-

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pendent justification for qualifying as simple (McDaniel 2000). Which is to say, an extended simple need not be qualitatively uniform throughout the entirety of its extension. So the mereological interest of Strings is manifest. However, there are far more questions than answers here. For one thing, String Theory as applied to physical problems turns out to result in multiple, independent ‘theories’ without any physical test to distinguish between them. These multiple solutions form what is known as ‘M-theory’, and divide themselves out into families that are known as ‘branes’ (see e.g. Ohta 2002). The mereological puzzles here are beyond number. Quite aside from the arguments over extended simples, of a (comparatively!) simple string, part/whole relationships now come into view across families of strings, across branes, and throughout all of M-theory! There is yet another salient point that needs to be made here. While it is not of specifically mereological significance, it must also be noted that these brane extravaganzas in physics are empirically vacuous, and consequently have little or no claim to being called scientific. This is not merely a philosophical cavil; more than a few well-respected physicists have argued this point as well (see e.g. Smolin 2007 and Woit 2006). One last piece of cosmological speculation will bring us back where we started, with Whitehead's cosmology. Leemon McHenry has recently conjectured about possible parallels between Whitehead's con-

cept of a ‘cosmic epoch’ in PR, and contemporary speculation in physics about multiple and multiplying universes – the ‘multiverse’ theory (McHenry 2011). This latter has been the source of much rollicking good fun in popular science fiction and fantasy for many decades now. But as physical science, the multiverse theory has two problematic aspects with regard to McHenry's conjecture. The first is that, like M-theory, it is empirically vacuous: there is no possible test or observation that could give the multiverse ‘theory’ any particle of scientific content. The second point, however, is more relevant to cosmology in the mereological context. The multiverse ‘theory’ (and it is essential to scare quote the word ‘theory’ in this situation) is about the structure of possibility within the framework of quantum mechanics. (Note that in order for a notion to qualify as a scientific theory, not only must it be testable in the abstract – which is the minimal qualification of an hypothesis – it must have also survived such tests on numerous occasions, so much so that reasoned doubt regarding the notion would face significant obstacles before it could gain any scientific legitimacy). How is one to interpret the quantum mechanical wave function, which never gives us a definite answer about what is actual, only a probabilistic spread about what might occur? There are mereological issues here about the actual and the possible – is the actual only a part of the possible? And there are ways of engaging this question that do not require the acceptance of empirically empty con-

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jectures (Auxier & Herstein 2017). But McHenry’s speculative connection of Whitehead’s ‘cosmic epochs’ with the multiverse ‘theory’ is arguably misplaced. Whitehead’s cosmic epochs are not about the structure of possibilities, but rather about the limits of rationally justifiable speculation. For Whitehead, the structure of possibility is already embedded within any cosmic epoch. The mereological interest of a cosmic epoch lies not in the structure of possibility, but rather in the question of whether the extensive continuum (as Whitehead conceived it) can be rationally extended (an unavoidable pun) to epochs beyond our own. Does our logic of part and whole go beyond the cosmos as we find it, to any possible cosmos? There is no empirical evidence that can answer this question, and the logical evidence can only go so far. This becomes a purely speculative matter of philosophical cosmology. See also > Ancient Greek Atomism, Mereotopology, Nature, Philosophy of Mathematics, Quantum Mereology, Whitehead. References and further readings

Arntzenius, F., 2008, “Gunk, Topology and Measure”, in Dean Zimmerman, ed., Oxford Studies in Metaphysics vol. 4, Oxford: Oxford University Press: 225-247. Auxier, R.; Herstein G. L., The Quantum of Explanation: Whitehead’s Radical Empiricism, New

York: Routledge Studies in American Philosophy. Calosi, C.; Fano, V.; Tarozzi, G., 2011, “Quantum Ontology and Extensional Mereology”, Foundations of Physics, 41, 1740. doi:10.1007/ s10701-011-9590-z. Clarke, B., 1981, “A Calculus of Individuals Based on ‘Connection”, Notre Dame Journal of Formal Logic, 22 (3): 204-218. Clarke, B., 1985, “Individuals and Points”, Notre Dame Journal of Formal Logic, 26 (1): 61-75. Cohen, P. J., 2008, Set Theory and the Continuum Hypothesis, Mineola: Dover Publications. De Laguna, T., 1922, “Point, Line, and Surface, as Sets of Solids,” The Journal of Philosophy 19 (17): 449461. Freedman, M. F.; Quinn, F., 1990, Topology of 4-Manifolds, Princeton: Princeton University Press. Herstein, G. L., 2006, Whitehead and the Measurement Problem of Cosmology, Frankfurt am Main: Ontos Verlag, 2006. McDaniel, K., 2007, “Extended simples”, Philosophical Studies 133: 131-141. McDaniel, K., 2000, “Extended Simples and Qualitative Heterogeneity”, The Philosophical Quarterly Vol. 59, No. 235: 325-331. McHenry, L., 2011, “The Multiverse Conjecture: Whitehead’s Cosmic Epochs and Contemporary Cosmology”, Process Studies 40 (1): 5-24.

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Lewis, D., 1991, Parts of Classes, Cambridge: Wiley-Blackwell. Lowe, E. J., 1996, Subjects of Experience, Cambridge: Cambridge University Press. Ohta, N., 2002, “Introduction to Branes and M-Theory for Relativists and Cosmologists”, Progress of Theoretical Physics Supplement No. 148: 1-28. Pratt, I.; Schoop, D., 1998, “A Complete Axiom System for Polygonal Mereotopology of the Real Plane”, Journal of Philosophical Logic 27: 621-658. Rea, M. C., 2001, “How to Be an Eleatic Monist”, Philosophical Perspectives, vol. 15, Metaphysics, 129151. Russell, B., 1903, Principles of Mathematics, Cambridge: Cambridge University Press. Russell, J. S., 2008, “The Structure of Gunk: Adventures in the Ontology of Space,” in Zimmerman, D., ed., Oxford Studies in Metaphysics vol. 4, Oxford: Oxford University Press: 248-274. Schaffer, J., 2010, “The Priority of the Whole”, The Philosophical Review 119 (1): 31-76. Siegel, W., 2001, Introduction to String Field Theory, https://arxiv. org/pdf/hep-th/0107094v1.pdf, verified January 27, 2017. Simons, P., Parts: A Study in Ontology, Oxford: Clarendon Press, 1987. Simons, P., 2004, “Extended Simples: A Third Way Between Atoms and Gunk”, The Monist, 87: 371-84.

Simons, P., 2006, “Real Wholes, Real Parts: Mereology without Algebra”, The Journal of Philosophy 103 (12) Special Issue: Parts and Wholes, 597-613. Smolin, L., 2007., The Trouble with Physics, Boston: Mariner Books. Tarski, A., 1983 [1929], “Foundations of the Geometry of Solids,” in Logic, Semantics, Metamathematics, trans. J. H. Woodger, 2nd ed., Indianapolis: Hackett, 24-29. Tong, D., 2009, String Theory: University of Cambridge Part III Mathematical Tripos, https://arxiv.org/abs/ 0908.0333, verified January 27, 2017. Whitehead, A. N., 1919, Enquiry into the Principles of Natural Knowledge, Cambridge: Cambridge University Press. Whitehead, A. N., 1920, The Concept of Nature, Cambridge: Cambridge University Press. Whitehead, A. N., 1922, The Principle of Relativity, Cambridge: Cambridge University Press. Whitehead, A. N., 1929, Process and Reality: An Essay in Cosmology, corrected edition, edited by David Griffin and Donald W. Sherburne, New York: The Free Press, 2nd edition 1978. Woit, P., 2006, Not even Wrong: The Failure of String Theory and the Search for Unity in Physical Law, New York: Basic Books. Gary Herstein

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D Deontic Modalities Modalities, or modal operators, modify propositions to form new propositions, thus constituting a formallanguage analogue of adverbial modification in natural language. In many instances there is a pair of dual oneplace modalities, such as the alethic modalities of ‘it is possible that’ and ‘it is necessary that’. Such is also the case for the deontic modalities ‘it is permitted that’ (often symbolised as ‘P’) and ‘it is obligatory that’ (often symbolised as ‘O’). These modalities are dual in the sense that ‘non-P-non’ corresponds to ‘O’: if it is obligatory that p, then it is not permitted that non-p. (Consequently, ‘non-O-non’ corresponds to ‘P’). In formal contexts it is customary to take the strong ‘O’ modality as basic and to treat the weak ‘P’ as an abbreviation; sometimes ‘F’ (‘it is forbidden that’) is used as an abbreviation for ‘Onon’ (equivalently, for ‘non-P’). Like in many other areas of modal logic, there are medieval precursors of a study of deontic modalities (cf. Knuuttila 1981). Mally (1926) provided the first modern attempt at establishing a formal logic including deontic modalities, which however was flawed (cf. Lokhorst 2013). The 1950s saw the establishment of dif-

ferent formal systems of modalities, not all of them based on propositional modification; von Wright (1951) is the locus classicus of this stage in the development of deontic logic. Some years later the relational (Kripke-) semantics for modal logics were discovered, and the system of standard deontic logic (SDL) was established. Implicitly, this meant adopting a certain stance on the relation between ‘ought to do’ and ‘ought to be’, viz., the so-called Meinong-Chisholm reduction (cf. Chisholm 1964): It is assumed that ‘A is obligated to bring it about that p’ is equivalent to ‘it is obligatory that A brings it about that p’, thus transforming the logically distinct category of agent-directed obligations (‘ought to do’) to the category of ‘ought to be’ that is amenable to an analysis via sentential modification. SDL is derived from propositional logic by adding the propositional ‘O’ operator with the two axioms (K)

O(p → q) → (Op → Oq)

(D)

Op → ¬O¬p

and the standard modal logic rule of necessitation, from ⊢ p to derive ⊢ Op (i.e., if it is a theorem that p, then it is also a theorem that Op). Semantically, SDL is thus the normal modal logic KD, the logic of serial frames, in which each point, or possible world, has a successor with respect to the accessibility relation, which is interpreted as relative deontic perfection. Anderson (1958) explores the idea of deriving a similar deontic logic from alethic modal log-

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ic by adding a single ‘sanction’ constant S. One main advantage of SDL is its formal transparency. Building upon SDL and its extensions, many applications in computer science have been developed (cf. the conference series DEON, e.g., v.d. Meyden and v.d. Torre 2008 and Cariani et al. 2014), capturing important aspects of deontic modalities based upon a distinction of obligatory, permitted and forbidden states. SDL is however subject to a large number of interpretational difficulties, also known (somewhat misleadingly) as ‘paradoxes of deontic logic’. E.g., Ross’s paradox (Ross 1941) consists in the observation that in SDL, from Op one can derive O(p ∨ q). Taking p to mean ‘the letter is mailed’ and q to mean ‘the letter is burned’, one can thus derive an obligation to either mail or burn a letter from an obligation to mail a letter, which seems odd. SDL also excludes the occurrence of conflicting obligations, while such dilemmatic situations seem to occur in real life. McNamara (2014) discusses many further problematic aspects of SDL, including issues of conditional obligations and secondary obligations (so-called ‘contrary-to-duties’, cf. Carmo and Jones 2002), i.e., obligations following the violation of an obligation. Quantification and the iteration of deontic modalities pose further difficulties. Many of the interpretational problems of SDL have to do with the fact that deontic concepts that play a role in human interaction, such as A’s

having a right against B that she do ϕ, are relational, or bipolar, and resist analysis in terms of the monadic ‘it is obligatory that p’ (cf. Thompson 2004). Geach (1982) furthermore argues that a useful deontic logic will have to be built upon deontic modalities that modify action terms, not propositions, thus questioning the Meinong-Chisholm reduction. Steps towards integrating deontic logic with a logic of agency have been made (Horty 2001), but there is still no generally accepted framework that captures all of the deontic concepts playing a role in human affairs. A further desideratum concerning extant formal approaches to deontic modalities is connected with the relation between deontic logic and mereology, which appears to be severely understudied. Clearly, most actions have parts, and there are interesting connections between the means-ends relationship between actions, practical reasoning, and the part-whole relationship between actions. For example, an action aiming at the means will usually be a part of an action aiming at the end (breaking the eggs is a means to, and part of, making an omelette), and if the end is obligatory, so (one may think) should be the means. Anscombe (1963) is the locus classicus for renewed interest in practical reasoning and practical knowledge; she highlights the importance of considering chains of actions linked by ‘by’ (‘I am replenishing the water supply by pumping’). Thompson (2008) takes up Anscombe’s framework and gives what may be viewed as a mereological template for action explanation

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under the heading of ‘naïve action theory’, arguing that the basic structure of action explanation has the form ‘I am doing A because I am doing B’. These authors do not, however, consider relations to deontic logic. It would appear that this bridge has yet to be built. In the current research situation Thomason (2014) may be the best starting point for such a project since he gives a rich survey of examples and challenges for deontic logic linked with the formalisation of practical reasoning and, specifically, means-ends reasoning. It seems almost certain that such research should benefit from the use of the formal tools available in mereology. See also > Act and Action, Activity, Ernst Mally, Paradoxes, Propositions. Bibliographical remarks

Åqvist, L., 2002. In-depth overview article with a focus on the development of deontic logic until the 1980s. Carmo, J. and Jones, A., 2002. Indepth overview article focussing on more recent developments, especially on conditional obligations. McNamara, P., 2014. Provides a good overview of standard deontic logic and its development, including a gentle introduction to its semantics and a discussion of many of its known problems.

References and further readings

Anscombe, G. E. M., 1963, Intention, 2nd edition, Oxford: Blackwell. Anderson, A. R., 1958, “A Reduction of Deontic Logic to Alethic Modal Logic”, Mind 67: 100-103. Åqvist, L., 2002, “Deontic Logic” in D. M. Gabbay; Guenthner, F., eds., Handbook of Philosophical Logic, 2nd ed., Vol. 8, 147-264. Carmo, J.; Jones, A., 2002, “Deontic Logic and Contrary-to-Duties”, in Gabay D. M; F. Guenthner, eds., Handbook of Philosophical Logic, 2nd ed., Vol. 8, 265-343. Cariani, F.; Grossi, D.; Meheus, J.; Parent, X., 2014, Deontic Logic and Normative Systems. 12th International Conference, DEON 2014 = Springer Lecture Notes in Computer Science, Vol. 8554. Berlin: Springer. Chisholm, R., 1964, “The Ethics of Requirement”, American Philosophical Quarterly 1: 147-153. Geach, P., 1982, “Whatever happened to deontic logic?”, Philosophia 11: 1-12. Horty, J. F., 2001, Agency and Deontic Logic, Oxford: Oxford University Press. Knuuttila, S., 1981, “The Emergence of Deontic Logic in the Fourteenth Century”, in Hilpinen, R., ed., New Studies in Deontic Logic. Dordrecht: Reidel, 225-248. Lokhorst, G.-J., 2013, “Mally’s Deontic Logic”, in The Stanford Encyclopedia of Philosophy, Zalta, E. (ed.),

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http://plato.stanford.edu/archives/win 2013/ent ries/mally-deontic/

von Wright, G. H., 1951, “Deontic Logic”, Mind 60: 1-15.

McNamara, P., 2014, “Deontic Logic”, in The Stanford Encyclopedia of Philosophy, Zalta, E. N. (ed.), http://plato.stanford.edu/archives/win 2014/entries/logic-deontic/.

Thomas Müller

Mally, E., 1926, Grundgesetze des Sollens: Elemente der Logik des Willens, Graz: Leuschner und Lubensky. Reprint in Mally, E., 1971, Logische Schriften: Großes Logikfragment, Grundgesetze des Sollens, Wolf, K.; Weingartner, P., eds., Dordrecht: D. Reidel, 227-324. Meyden, R. van der; Torre, L. van der, eds., 2008, Deontic Logic in Computer Science.9th International Conference, DEON 2008 = Springer Lecture Notes in Artificial Intelligence, Vol. 5076. Berlin: Springer. Ross, A., 1941, “Imperatives and Logic”, Theoria 7: 53-71. Thomason, R. H., 2014, “The Formalisation of Practical Reasoning: Problems and Prospects”, The IfCoLog Journal of Logics and Their Applications 1: 47–76. Thompson, M., 2004, “What is it to wrong someone? A puzzle about justice,” in R. J. Wallace; Pettit, P., Scheffler, S.; Smith, M., eds., Reason and Value. Themes from the Moral Philosophy of Joseph Raz, Oxford: Oxford University Press, 333-384. Thompson, M., 2008, Life and Action. Elementary Structures of Practice and Practical Thought, Cambridge MA: Harvard University Press.

Descartes Two ultimate classes of things. Des-

cartes’ doctrine about anything that exists, whether it is a whole or a part or an essence of an actual or potential being, departs from his doctrine of strict substance dualism. Descartes’ notion of substance is deeply affected by the established Aristotelianscholastic tradition, which recognised by the term ‘substance’ an individual object, composed of material and form. (For a discussion of the Aristotelian heritage and its influence to the Cartesian notion of substance and its properties see Burkhardt 2007). Nevertheless, Descartes breaks from the Aristotelian tradition in a decisive, epoch-making manner. Taking ‘I am’ (sum) as his ‘unshakeable foundation’ he concludes the necessary actual existence of God, and from these two starting points, he then deduces his other metaphysical theses. One fundamental conclusion he reaches is that there are two kinds of substances, the corporeal substance and thinking or intellectual substances, and that the corporeal substance and the thinking substances are really distinct and separable. He gives, as he views it, deductions and proofs for these theses in the 2nd and 6th of his Meditationes de Prima Philosophia (AT VII, 23ff./ PWD II, 16ff.; AT VII, 71ff./PWD II, 50ff.). The totality of the actual corporeal

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and all thinking substances just is the totality of all that exists. As he maintains in his Principia Philosophiae, there are no other kinds of independent or distinct existing entities (cf. Kenny 1970: 697, where Kenny calls Descartes “the father of modern Platonism”; cf. Perler 1996: 81ff.): But I recognise only two ultimate classes (summa genera) of things: first intellectual or thinking things, i.e. those which pertain to mind or thinking substance; and secondly, material things, i.e. those which pertain to extended substance or body. (Principia I, 48, AT IX-2, 45/PWD I, 208).

Descartes defines substances as selfsubsistent existing beings and as he explains, “by substance we can understand nothing other than a thing which exists in such a way as to depend on no other thing for its existence”. He however continues, however, precisely speaking “there is only one substance, which can be understood to depend on no other thing whatsoever, namely God”, for all other beings lack perfection and depend for their existence on His power. Indeed, all created beings cannot exist for a moment without His concurrence. Employing the medieval theory of Concursus Dei (the concurrence of God) in a strong way. Descartes holds that all beings once created have to be preserved by God’s continuous creative power and that the divine acts of creation and preservation are the same: … It is quite clear … that the same power and action are needed to preserve anything … as would be required to create that thing anew. … Hence the distinction between preservation and creation is only

a conceptual one… (3rd Med., AT VII, 49/PWD II, 33; for a deeper discussion of the ‘Concursus Dei’ theory see Dufour 1991; for ‘concurrence’ in Descartes see also Cottingham 1993: 40).

Yet in a weaker sense, created beings are to be called substances, even though they are brought into existence and kept existent by God in creatio continua (continuous creation) and thus are causally and existentially dependent on Him (for ontological dependence relations on Descartes see Burkhardt 2007). “Hence the term ‘substance’ does not apply univocally … to God and to other things” (Principia I, 51., AT IX-2, 57/PWD I, 210). In his correspondence Descartes gives a more detailed description of what he understands by created substances, given that they are not self-subsistent without divine concurrence: We mean only that it is the kind of thing that can exist without any created thing; and this is something that cannot be said about the modes of things, like shape or number. (to Hyperaspistes, August 1641, AT III, 429/PWD III, 193f.). Attributes and modes. Substances are

known clearly and distinctly through their immutable properties or attributes, “but each substance has one principal property which constitutes its nature and essence, and to which all its other properties are referred. Thus extension in length, breadth and depth constitutes the nature of corporeal substance; and thought constitutes the nature of thinking substance”. Modes of an individual substance show its actual contingent state within the scope of its associated attributes. These modes are real

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substantial properties and must not to be mixed up with secondary qualities like color, heat or material texture. Secondary qualities are not actual, contingent states of the substance, but rather effects which are caused within the perceiving subject. Shape and motion, for example, are intelligible only as modes of extension, and imagination, sensation, and will are intelligible only as modes of a thinking thing. Yet, as Descartes maintains, “it is possible to understand extension without shape or movement, and thought without imagination or sensation, and so on” (Principia I, 53., AT IX-2, 48/PWD I, 210f., cf. also Principia II, 4., AT IX-2, 65/PWD 224). A mode doesn’t possess independent existence, but belongs inherently to the substance, to which it corresponds and on which it depends logically. Modal distinctness (distinctio modalis) of a particular mode of substance represents everything that differentiates it from any other particular mode of the same substance and also conceptually from the substance per se. “A modal distinction can be taken in two ways: firstly as a distinction between a mode, properly so called, and the substance of which it is a mode; and secondly, as a distinction between two modes of the same substance” (Principia I, 61., AT IX-2, 52/PWD I, 213f.). Because we can “easily come to know a substance by one of its attributes”, we also can conclude “in virtue of the common notion that nothingness possesses no attributes, that is to say, no properties or qualities”, that everything which holds attributes, must exist necessarily: “Thus

if we perceive the presence of some attribute, we can infer that there must also be present an existing thing or substance to which it may be attributed. (Principia I, 52, AT IX-2, 47/PWD I, 210). Formal and objective being. Putting to

one side the dependency of all created corporeal and intellectual creatures on God, the inherence of a mode in its substance represents the only kind of dependency within or between distinct entities (Principia I, 61, AT IX-2, 52/ PWD I, 214). Appealing to these two forms of dependence, Descartes constructs an ascending sequence of different grades of reality. The infinite substance holds the highest grade of reality and the finite substance a higher one than a mode. Because nothing can result from nothingness, an effect always requires a cause with at least the same grade of reality. This also applies for the containment of objects in ideas. Provided that what we perceive corresponds as a cause exactly with that which is then objectively contained (or imagined) in the objects of our ideas, the perceived actual thing is said to exist in formal reality (esse formaliter). If the actual perceived thing does not exactly correspond with the contents of our ideas, but its reality is at least of such (or of even higher) greatness that it can act in place of something corresponding exactly, it is said to exist ‘eminently’ (esse eminenter, 2nd Med. Resp., AT VII, 161/PWD II, 114). Objective existence in ideas (realitas obiectiva) compared to formal existence does not constitute a selfsubsistent ontological kind of being. (For further information on objective

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being in Descartes see Cronin 1966). Both kinds of being – a thing’s actual existence (i.e. its formal or eminent being) and the thing’s objective being in an idea – are just two different modes of being belonging to the same entity. Objective being as contents of an idea is materially always a mode of the thinking substance and thus as such fully dependent on that substance. So it has to be considered as less perfect and of a lower grade of reality than the thinking substance itself. Given that the required formal reality of a thing corresponding to the objective reality of an idea must be of such greatness, one can trust this reality does not reside in the subject formally or eminently, and the cause of that idea must exist outside the subject. This is the sole point of departure in the Cartesian system by which actual reality of beings outside the subject can be demonstrated (3rd Med., AT VII, 42/PWD II, 29, 6th Med., At VII, 71ff/PWD II, 50ff). And for the same reason we can derive from our idea of God that the sole cause for this idea of an infinite being must be a real existing God (3rd Med., AT VII, 41ff./PWD II, 29; 2nd Med. Resp., AT VII, 165ff./PWD II, 116ff.). Thinking substance. The type of

thinking or intellectual substance (res cogitans) includes both arbitrarily many individual finite substance tokens and one infinite substance, namely God. Both the finite thinking substances and the infinite thinking substance are by nature indivisible. Thinking, being finite, and being created by God are the attributes of finite thinking substances. Because no

kind of variation can occur in the divine substance, for God no properties nor modifications, but only attributes are intelligible (Principia I, 56., AT IX-2, 49/PWD I, 211). In his 3rd Meditation Descartes characterises God as “a substance that is infinite, independent, supremely intelligent, supremely powerful, and which created both myself and everything else … that exists” (AT VII, 45/PWD II, 31). An important claim, which is prevalent throughout Descartes´ writings, is that God’s knowing and acting are necessarily simple. Even though human perception, which acts independently on the intellect, represents perfection, all perceptions embody some passion, and passivity means some kind of dependency on something. This of course is out of the question for God. So God always acts in a perfect, simple way, simultaneously understanding, willing and doing (Principia I, 23, AT IX-2, 35/PWD I, 201). Simplicity without any temporal or natural priority, constitutes a basic foundation for Descartes’ doctrine on the eternal necessary truths created by God. Finite intellectual substances are characterised by their power to effect consciousness and thereby mental incidents (cogitationes). The latter result into two fundamental faculties, the intellect, which produces different kinds of ideas (perceptions, imaginations, pure thinking, summarised as perceptiones), and the will, which produces any kind of volitive or emotive thinking, wishes or affects (desires, declines, assertions, doubts,

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summarised as volitiones) (Principia I, 32, AT IX-2, 39/PWD I, 204). A combination of these two faculties enables a human to form judgments. All mental incidents constitute as ideas in a broader sense intentional modes of the thinking substance. In his late Passions de l’âme however Descartes admits also particular emotional states as mental incidents, even though they don‘t exhibit intentionality. Both faculties of the mind are indeed functionally disjoined analysable, but they don’t represent discrete ontological parts. In his Discours de la Méthode Descartes also holds, that knowing and beliefs constitute different mental acts (AT VI, 23/PWD I, 23f.). But these separations must also be viewed as purely distinctional: As for the faculties of willing, of understanding, of sensory perception and so on, these cannot be termed parts of the mind, since it is one and the same mind that wills, and understands and has sensory perceptions. (6th Med., AT VII, 86/PWD II, 59, see also Wagner 1984). Corporeal or extended substance. The characterisation of corporeal substance as an essentially extended thing (res extensa) leads to the elaboration of its other primary properties like figure, divisibility and motion. Descartes´ conception of corporeal substance is shaped by his violent refusal of scholastic hylemorphism as well by his general realism of universals. Cartesian matter is always actual, it is never about potentiality. Forms like triangularity don’t exist as such, but merely in a body as consequences arising out of accidental con-

figurations of bodily parts (Principia I, 58.f., AT IX-2, 50/PWD I, 212f.). Following Descartes, the term ‘extension’ – something which has length, width and height – needs no further explication, because it is about a simple and pure nature (natura simplex et pura) and thus a basic notion common to all humans. Being corporeal and extended are one and the same. The extended substance exhibits a space continuum of arbitrary extension, entirely filled with matter. Even though this extended substance can be divided into indefinite many parts, those parts are not substances themselves. There is only one instance for the substance type res extensa, which means there is no substantial corporeal plurality. Because parts of the corporeal substance are endless repeatedly divisible into corporeal parts the Cartesian universe does not allow for atoms (e. g. see Principia II, 20, AT IX-2, 74/PWD I, 231). Unfortunately, Descartes uses the term ‘body’ (corpus) for both the extended substance (e. g. in Principia II, 4) and its parts (especially in the Meditationes and the Principia). When he refers to the human body even a third kind of meaning is in use. Often it is to be derived from the context whether the ‘body in general’ is meant, or an ‘individual body’, or the ‘human body’. In some paragraphs Descartes switches from one signification to the other, e.g. when he describes in the Principia the nature of a particular individual body, a stone, by means of the nature of bodies in general (Principia II, 11., AT

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IX-2, 69/PWD 227). Particularly in the 6th Meditation Descartes often uses the word corpus to refer to the human body. But even here, where Descartes shows the real distinction between mind and body, one must determine carefully whether in the particular passage the human body or the body in general as substance is meant (for a more detailed discussion of the term ‘body’ see Cottingham 1993: 22f.). That one must exercise proper care when interpreting Descartes’ usage of the words corpus and (corporeal) substantia can be illustrated by a reply to an objection of Arnauld’s, when he discusses the necessary conditions for a thing’s complete understanding. Descartes here employs the term corpus as designation for ‘human body’ as well for ‘individual part’ and in the plural also as synonym for the (corporeal) substance. In this section he explains that whether a substance is ‘complete’ and ‘incomplete’ depends on the point of view how a particular thing is perceived, the same ‘substance’; e.g., a hand might be called either complete or incomplete. The context of these prima facie puzzling statements shows, however, that Descartes‘ concern merely is to explain what is necessary in his view for a complete understanding of something. Complete understanding means to have well enough information to perceive the thing in a clear and distinct way, in contrast to having adequate understanding, which would require full understanding of the thing in every detail. In Descartes’s view two things can be understood as ‘complete’ if

‘each of one can be understood apart from the other’. He borrows here the denotation of ‘incomplete substances’, which stems from the Aristotelian-scholastic tradition, and allows it as a manner of speaking, as long as the reason for calling substances incomplete is not that they are unable to exist on their own. However as long as ‘incomplete substances’ just are considered as functional breakdown of a common greater whole, it is …possible to call a substance incomplete in the sense that, although it has nothing incomplete about it qua substance, it is incomplete in so far as it is referred to some other substance in conjunction with which it forms something which is a unity in its own right. Thus a hand is an incomplete substance when it is referred to the whole body of which is it a part; but it is a complete substance, when it is considered on its own. (4th Resp., AT VII, 222/PWD II, 156f.).

Descartes continues to point out that “in just the same way the mind and the body are also incomplete substances, when they are referred to a human being which together they make up. But if they are considered on their own, they are complete” (ibid). As Descartes holds, a particular location of an individual body is merely a matter of thinking, because it depends on the choice of a reference point. (Principia II, 13., AT IX-2, 77/ PWD I, 228). Thus, in the Cartesian view of the world (in contrast to the view of Isaac Newton) there is no absolute location where a bodily part can be found, but only a location relative to a chosen point of reference.

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Similarly, any distinction as to whether a body is in motion or not looses significance in the absolute sense. Motion can only be understood as a relative change of distance between the observed body and the point of observation. So whether a body is in motion or at rest is a pure rational distinction (distinctio rationis): … the same thing can be said to be changing and not changing its place at the same time; and similarly the same thing can be said to be moving or not moving. For example, a man sitting on board a ship … considers himself to be moving relative to the shore, which he regards as fixed; but he does not think of himself as moving relative to the ship, since his position is unchanged relative to its parts. (Principia II, 24., AT IX-2, 76f./PWD I, 233)

Descartes holds that any conceptual mereological diversity of individual bodies which occurs by “division into parts … simply in our thought” resulting into “all the variety in matter, all the diversity in its forms, depends on motion” (Principia II, 21ff., AT IX-2, 74ff./PWD I, 232). Matter is always in local motion, where motion here means transfer of a body, or of adjacent parts of the substance grouped together, “from the vicinity of other bodies which are in immediate contact with it, and which are regarded as being in rest, to the vicinity of other bodies” (Principia II, 25., AT IX-2, 76/PWD I, 233). Thus individual bodies result from specific arrangements of pieces of matter and their relative movement to other pieces of matter or bodies.

Motion is governed by three basic Cartesian mechanical laws (Le Monde ou Traité de la lumière, AT XI, 38, 41, 43f./PWD I, 93ff), which lead Descartes to his basic understanding of the world’s complexity. The smallest parts of matter unite themselves into three elements, the finest ones into fire, bigger ones into water and finally into earth (Le Monde, AT XI, 23ff/PWD I, 88f.). The system of planets he compares with a vortex of celestial matter (Principia III, 30., AT IX-2, 115f./PWD 253f.). Celestial matter and its attributes, together with its general tendency to undergo corporeal modifications, equals terrestrial matter. Even if there were countless worlds, this would not change anything concerning the uniqueness of matter. Actually this is a reason for Descartes to assume just one world. Given that corporeal substances are essentially extended and extended things are divisible, corporeal things are imperfect. Because “the nature of body includes divisibility along with extension in space, and since being divisible is an imperfection, it is certain that God is not a body” (Principia I, 23., AT IX-2, 35/PWD I, 201). Moreover, since divisibility implies imperfection, God cannot be composed of two or more substances (Discours V, 4., AT VI, 35f./PWD I, 128f.). The fact that every human being requires for its life on earth by necessity two substances Descartes interprets as a sign of deficiency and dependency. The causal relationship between subrd stances. In his 3 Meditation Des-

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cartes claims that a cause „gives“ reality to the effect (for a comprehensive discussion see Perler 1996: 143ff.). How this „giving“ comes to pass, he doesn’t say (3rd Med., AT VII, 40/PWD II, 28). In his August 1649 letter to Henry More, Descartes even holds that the transference of modifications from one bodily part to another was impossible (AT V, 404/PWD III, 381f.). Neither a substance nor a part of it could transmit something to another. But even if it seems impossible to give a causal account in its strong sense, Descartes thinks that it is still possible to bring about a coordinated course of actions between substances. In various passages Descartes maintains that the soul by virtue of particular bodily based opportunities, has a chance to ‘induce’ particular associated ideas. So causality in a stronger sense is replaced by more or less likely state correlations by pairs in mutual causal dependency. As he states in a February 1647 letter to Chanut: The soul’s natural capacity for union with a body brings with it the possibility of an association between each of its thoughts and certain motions or conditions of this body so that when the same conditions recur in the body, they induce the soul to have the same thought; and conversely when the same thought recurs, it disposes the body to return to the same condition. (to Chanut, 1 February 1647, AT IV, 603f./PWD III, 307, see also Traité de l’Homme, AT XI, 148f.; for a deeper discussion of Descartes´ dualistic mind-body approach and its reception by modern analytic philosophers see Owens 1991; see also Perler 1996: 149ff.).

As a matter of principle we know our mind better than our body, because everything what improves our knowledge of bodies improves also our knowledge of minds (2nd Med., AT VII, 33f./PWD II, 22f.; Principia I, 11., AT IX-2, 30/PWD I, 196). The thinking substance is not necessitated to perish together with its associated body, because it is able to exist without the body. Thus the mind is immortal by its nature while the body decays easily (Syn. Med., AT VII, 14/PWD II, 10). Nevertheless, in the end, the problem of body-mind relationship and mutual dependency remains unresolved in Descartes, and subsequent attempts to overcome it, from Malebranche’s ‘occasionalism’ to Leibniz’ ‘monadism’ and Spinoza’s ‘monism’, to later versions of materialism and idealism cannot not really fill the gap (see Hunter 1991). See also > Conscious Experience, Substance, Whitehead’s Metaphysics. References and further readings

Burkhardt, H., 2007, “Substances, Attributes, and Modes – Substantial Structures in Descartes, Spinoza and Leibniz”, in Kanzian, C. and Legenhausen, M., eds. Substance and Attribute, Heusenstamm: Ontos Verlag, 7-23. Cottingham, 1993, A Descartes Dictionary, Oxford: The Blackwell Philosopher Dictionaries.

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Cronin, T., 1966, Objective Being in Descartes and in Suarez, Rom: Gregorian University Press. Descartes, R., Adam C., and Tannery P. eds., 12 Vols., 1897-1912, Oeuvres de Descartes, Paris: Librairie Philosophique J. Vrin, [=AT]. Descartes, R. transl. by Cottingham J., Stoothoff R., Murdoch D. and Kenny A. (Vol. 3 only), 3 Vols., 1984-1991, The Philosophical Writings of Descartes, Cambridge/New York: Cambridge University Press [=PWD]. Dufour, C. A., 1991, “Concursus Dei”, in: Burkhardt, H.; Smith, B. (eds.) Handbook of Metaphysics and Ontology, Munich / Vienna: Analytica, 174-176. Hunter, G. J., 1991, “Descartes, René”, in: Burkhardt, H.; Smith, B. (eds.) Handbook of Metaphysics and Ontology, Munich / Vienna: Analytica, 210-212. Kenny, A., 1970, “The Cartesian Circle and the Eternal Truths”, Journal of Philosophy 67, 685-700. Owens, J., 1991, “Mind-Body”, in: Burkhardt, H.; Smith, B. (eds.) Handbook of Metaphysics and Ontology, Munich Vienna: Analytica. Perler, D., 1996, Repräsentationen bei Descartes, Frankfurt am Main: Klostermann. Wagner, S., 1984, “Descartes on the Parts of the Soul”, Philosophy and Phenomenological Research 45: 5170. Rainer Mittmann

Dispositions Adjectives like ‘water-soluble’ or ‘conductive’ are typical ‘dispositional’ predicates. Such ‘dispositional’ predicates tell us something about the causal roles played by the items these predicates apply to. Often such predicates are said to ascribe dispositional properties or, in short, dispositions, to certain items. In order for an item to have a certain disposition, it is not necessary that it actually plays that causal role at any point in time, i.e. that its disposition is ever realised or manifested; it is sufficient that it is disposed to do so, were appropriate circumstances to obtain. Dispositional properties are normally contrasted with ‘categorical’ properties. Different Types of dispositions correspond to different types of events – namely, the events that realize the dispositions (and may or may not happen). For categorical properties, however, there is no such correspondence with events. The word ‘disposition’ is often taken in a narrow sense to refer to ‘sure-fire dispositions’ only, i.e. to dispositions that will invariably realize given appropriate realizability conditions. In a wider sense, it can also be used to include other kinds of realizable entities, like abilities, virtues, capabilities or probabilistic dispositions. Probabilistic dispositions (also called ‘propensities’ or ‘tendencies’; cf. Popper 1990, Jansen 2007) are realised with a certain probability only (e.g., the tendency of an atom to decay). Abilities require an agent’s decision to be realised (e.g., the ability of a doctor to heal her or his patients), and as we

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normally cannot guarantee the success of our actions, many abilities can also be considered as being probabilistic. Dispositions often come in pairs of active powers and passive potencies – for example, the active power of the key to unlock the lock and the passive potency of the lock to be unlocked by the key; or the active power of water to dissolve sugar and the passive potency of sugar to dissolve in water. As we can never observe dispositions directly but only via their realisations or manifestations, many philosophers with verificationist leanings tend to prefer categorical properties over dispositions. Other arguments against dispositions allege that dispositions have a dubious ontic status between the existing and the non-existing, and that they are explanatorily idle. A standard argument for the explanatorily idleness of disposition is the socalled virtus dormitiva objection that can be traced back to Molière’s play “The Imaginary Invalid”. In this play, a medical student is asked in an examination why opium makes someone sleepy. In his answer, the student refers to opium’s dormitive virtue (sleep-inducing power). The objection of some philosophers is that linking effects to dispositions in this way does not amount to an explanation but rather rephrases the phenomenon. Because of such difficulties, some philosophers deny that dispositions exist and propose other truthmakers for the predication of dispositional predicates (Ryle 1963, Lowe 2006). In contrast, others claim that all

properties are dispositional in nature (like Popper 1957: 70 and Bird 2007). Still others argue that properties as such are neither dispositional nor categorical in nature, but defer this distinction to the linguistic level (Mumford 1998). Many, however, accept the existence of dispositions, but try to reduce them to what goes by the name of their ‘categorical basis’ (e.g., Prior 1985). The categorical basis of a disposition is meant to be the complex (micro-)structure of the parts of the bearer of a disposition and their categorical (structural) properties that account for the disposition in question. Often, a type-type reduction is not possible because instances of the very same type of dispositions (e.g., the disposition to trigger a sensation of the color red in humans) can have categorical bases of quite different types (e.g., different patterns of reflected light). But a token-token reduction is not possible either, as the categorical properties of the parts alone can never account for the dispositions of the whole. For example, opium can be said to have a disposition to reduce pain because it contains morphine – but this explanation is complete only once we have added that it is morphine that has a disposition to reduce pain. Morphine, in turn, can be said to have a disposition to reduce pain because its molecules have a certain structure. Again, this explanation requires in addition that this structure carries with it the disposition to combine with, e.g., opioid receptors of human cells. Dispositions of wholes, that is, can only be reduced to the categorical plus the

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dispositional properties of the parts of the wholes in question. The opium example also illustrates that we have to distinguish between two different types of cases. In the case of the morphine and the opioid structure, the whole (morphine) and the part (the opioid structure) have different types of dispositions. By contrast, in the case of opium and morphine the disposition of the whole and the disposition of the part are of the same type. This shows that in some cases the dispositions of parts are also attributed to the respective wholes. In other cases, the direction of attribution is inverse – dispositions of the whole are attributed to crucial parts. In this vein, the eye is often said to be able to see or the brain is said to be able to think, while it would in fact be more appropriate to point out the instrumental role of these parts for the respective realisation – that is, it is a human being that is able to see with her or his eyes (among other organic parts), and it is a human being that is able to think with her or his brain (cf. Aristotle, De anima I 4, 408b 13-15 and Aquinas, Summa theologica I q. 75, a. 2, ad 2). See also > Category, Causation, Power, Structure, Transitivity. References and further readings

Armstrong, D. M.; Martin, C. B.; Place, U. T., 1996, Dispositions: A Debate, London: Routledge.

Bird, A., 2007, Nature’s Metaphysics. Laws and Properties, Oxford: Clarendon Press. Damschen, G.; Schnepf, R.; Stüber, K., eds., 2009, Debating Dispositions. Issues in Metaphysics, Epistemology and Philosophy of Mind, Berlin/New York: de Gruyter. Harré, R.; Madden, E. H., 1975, Causal Powers: A Theory of Natural Necessity, Oxford: Basil Blackwell. Jansen, L., 2007, “Tendencies and Other Realizables in Medical Information Sciences”, The Monist 90: 534-554. Kistler, M.; Gnassounou, B., eds., 2007, Dispositions and Causal Powers, Aldershot: Ashgate. Loewe, J. E., 2006, The FourCategory Ontology, Oxford: Oxford University Press. Marmodoro, A., ed., 2010, The Metaphysics of Powers: Their Grounding and Their Manifestations, London: Routledge. Molnar, G., 2003, Powers. A Study in Metaphysics, Oxford: Oxford University Press. Mumford, S., 1998, Dispositions, Oxford: Oxford University Press. Popper, K. R., 1957, “The Propensity Theory of Probability, and the Quantum Theory”, in: S. Körner (ed.), Observations and Interpretation, London: Thoemmes Press. Popper, K. R., 1990, A World of Propensities, London: Butterworth. Prior, E., 1985, Dispositions, Aberdeen: Aberdeen University Press.

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Ryle, G., 1963, The Concept of Mind, London: Penguin. Tuomela, R., ed., 1978, Dispositions, Dordrecht: Reidel. Ludger Jansen

Dynamical Systems The term dynamical system is used in a loose sense and in a precise sense which is associated with dynamical systems theory. Any object or group of objects subject to a development in time can loosely be called a dynamical system; in this sense dynamical systems are closely associated with processes. For the more precise meaning of the term one has to distinguish – as in all cases of mathematical modeling – between (i) the (concrete) system being modeled and (ii) the (abstract) mathematical structure used to model that concrete system. Either system can be referred to as a dynamical system. A dynamical system in sense (ii) is represented by its states at given times and a rule that evolves the states to states at other times. More technically, a dynamical system (in sense (ii)) is a structure , where M is the multi-dimensional state or phase space of the system, ft is the evolution rule that relates the state at each time to states at other times, and T is the one-dimensional space of points in time (so that t ∈ T). In this framework the state of a system (usually consisting of a group of entities) is represented as a single entity (a point) in phase space. Phase

spaces typically have many dimensions, each corresponding to a particular property of interest in the system; the system’s evolution in time, its change of state, corresponds to the ‘motion’ of a point through phase space (extended by a time axis). A particular system’s development thus is represented by a trajectory in phase space, and the collection of trajectories originating from different initial conditions forms the phase space portrait of that system. Properties of a system’s behavior in the limit of long time can be studied by studying geometrical features of its phase space portrait. Various restrictions can be imposed on the ingredients of a structure , leading to different sorts of dynamical systems. Time, for instance, can be chosen as continuous or discrete; in the former case, ft is usually given by a differential equation, in the latter case it is a map. Although differential equations of different kinds can be used, in many applications ordinary differential equations, associated with finitedimensional state spaces, are common. The restrictions imposed in such contexts usually ensure that the evolution of the systems is deterministic: for a given state at any time, there is a unique state associated with any other time. If the evolution rule is confined to ordinary differential equations, dynamical systems do not include paradigmatic processes like the propagation of waves, since these must be described by partial differential equations and require infinitedimensional state spaces. Because a system has to satisfy relatively few

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conditions in order to be described as a dynamical system, dynamical systems theory is thought to have very wide applicability; even the human mind has been conjectured to be a dynamical system (Port/van Gelder 1995). One noteworthy limitation to the reach of the theory is the microworld: quantum systems are not dynamical systems in the precise sense of the term because quantum systems cannot be described in a phase space. One property of dynamical systems of particular interest is the stability of their behaviour, that is, how the systems respond to perturbations of various sorts. In one sense, a dynamical system is stable if its trajectories, associated with slightly different initial states, eventually are all confined to a particular region of phase space. Such a system’s behaviour is regular: it settles on a particular final state or a final sort of behaviour, regardless of the initial state. By contrast, the behaviour is chaotic (unstable) if slightly different initial states lead to widely different later states. A second sense of stability is associated with variations in the parameters characterising a system. Formally, parameters are distinct from the variables characterising the state of the system, since variables are subject to evolution in time while parameters are not. Therefore, a variation in a parameter, strictly speaking, means a switch to a different system. We can then study the way in which a system’s behaviour changes under variations of a parameter (rather than initial conditions). If the dynamics of the system changes qualitatively un-

der such perturbations (e.g., a switch from regular to chaotic behaviour), the system is said to be structurally unstable. Both concepts of stability have attracted the attention of philosophers who are interested in finding scientifically respectable examples of emergent properties (Newman 1996, Rueger 2000). Another area of philosophical interest concerns the type of explanation dynamical systems theory can provide. Does it give us causal explanations, or a different type of explanation, perhaps a type sui generis? (For affirmative answers see Kellert 1993, Berger 1998). Does the theory involve a scientific methodology that is not ‘micro-reductive’? Some have suggested that the study of chaotic dynamical systems in particular is not compatible with a conventional micro-reductive methodology, where the behavior of a complex system is explained in terms of the interactions of its parts. The argument is that in such cases, the components of the abstract system cannot be separated out and analysed independently of each other, and thus that interesting features of the system’s behavior cannot be predicted based on a knowledge of the behavior of its parts. Even as a purely methodological claim without any suggestion of ontological holism or emergence, this is a matter of controversy (Smith 1998, Silberstein/McGeever 1999). One of the difficulties in assessing such claims lies in the unclarified notion of parthood in dynamical systems. There are two directions of interest here. The first concerns how

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the ‘familiar’ parts of a concrete system, such as the individual oscillators in a coupled pair, are represented in the abstract representation of a dynamical system. Should we expect such concrete parts to have independently identifiable representations in the abstract system? The second interest goes the other way: does the mathematical treatment of abstract systems reveal interesting new kinds of parts of concrete systems? For example, a standard mathematical technique used in dynamical systems theory involves decomposing a system’s behavior into component systems operating on different temporal scales (see, e.g., Holmes 1995). A realistic understanding of such decompositions would be interesting since unlike ordinary ‘spatial’ parts, these components are not smaller than or spatially contained within the systems they compose. Understanding these ‘non-spatial’ parts is an intriguing problem for mereology. See also > Causation, Chaos, Collectives and Compounds, Emergence, Structure, Transitivity. Bibliographical remarks

Abraham, R. H., Shaw, C. D., 1992. A reliable introduction that focuses on phase space pictures rather than on equations. Auyang, S., 1998 (Ch. 8). An informal presentation with philosophical perspectives. Smith, P., 1998. A philosophically sophisticated introduction.

References and further readings

Abraham, R. H.; Shaw, C. D., 1992, Dynamics: The Geometry of Behaviour, Redwood City: Addison-Wesley. Auyang, S., 1998, Foundations of Complex System Theories, Cambridge: Cambridge University Press. Berger, R., 1998, “Understanding Science: Why Causes Are Not Enough”, Philosophy of Science 65: 306-332. Hasselblatt, B. et al. (eds.), 2002, Handbook of Dynamical Systems. 2 vols, Amsterdam: Elsevier. Holmes, M., 1995, Introduction to Perturbation Methods, New York: Springer Kellert, S., 1993, In the Wake of Chaos, Chicago: University of Chicago Press. Newman, D., 1996, “Emergence and Strange Attractors”, Philosophy of Science 63: 246-261. Port, R. F.; van Gelder, T. (eds.), 1995, Mind as Motion, Cambridge Mass.: MIT Press. Rueger, A., 2000, “Physical Emergence, Diachronic and Synchronic”, Synthese 124: 297-322. Smith, P., 1998, Explaining Chaos, Cambridge: Cambridge University Press. Silverstein, M.; McGeever, J., 1999, “The Search for Ontological Emergence”, Philosophical Quarterly 49: 182-200. Patrick McGivern Alex Ruege

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E Elements Aristotle agreed with his predecessor Empedocles that Fire, Water, Air, and Earth are the four elements, distinguished from compounds like flesh and bone but like them in being, contrary to the atomists’ view, homoeomerous: “any part of such a compound is the same as the whole, just as any part of water is water” (De generatione et corruptione (DG) I.10, 328 a 10f.). Rather than simply assuming the elements were four in number, however, he claimed to prove this from more fundamental principles in connection with a new theory of mixture, involving combination by the interaction of the original substances and developed from the same fundamental principles. These interactions arise in virtue of the mutual powers and susceptibilities conferred by properties which can be reduced to grades of warmth and humidity (DG II.2). There are maximal and minimal “contrary extremes” (DG II.8, 335 a 8) of each of these two fundamental scales, hot and cold being the extremes of warmth, moist and dry those of humidity, and elements are substances with these extremal properties. Since “it is impossible for the same thing to be hot and cold, or moist and dry … Fire is hot and dry, whereas Air is hot

and moist …; and Water is cold and moist, while Earth is cold and dry” (DG II.3, 330 a 30-330 b 5). But combination will occur on mixing any two substances with different degrees of these two fundamental scales, and original substances are only distinguished as elements by virtue of their extreme degrees of warmth and humidity. The theory of mixing contains no restriction confining the original ingredients to elements as just defined. So when he goes on to agree with “all who make the simple bodies elements” (330 b 7), it is not clear what the simplicity of the elements amounts to since the notion of being an original ingredient is not characteristic of elements and atomic conceptions have been rejected. A definition in terms of simplicity is given in De caelo III.3, however, where Aristotle says “An element, we take it, is a body into which other bodies may be analysed, present in them potentially or in actuality (which of these is still disputable), and not itself divisible into bodies different in form. That, or something like it, is what all men in every case mean by element” (302 a 15ff.). An account of simplicity in terms of possessing “a principle of movement in their own nature” (268 b 28) is elaborated earlier. There are many apparent conflicts in the texts which are not easily resolved and questions arising which are not addressed. Amongst the latter, whilst it is understandable how bodies with different degrees of warmth or humidity should mix on contact, it is difficult to see what on this theory drives the separation of original in-

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gredients from a mixture which Aristotle thinks possible (the elements are potentially present in a mixt). This may have been one reason why the Stoics thought elements were actually present in separable mixtures. Again, Aristotle distinguishes mixing resulting in intermediate degrees of warmth and humidity from ‘overwhelming’, when the proportion of one ingredient to the other is so great that the primary properties of the former persist without augmentation or diminution throughout the body emerging from the interaction. This is the mechanism by which elements are transmuted (DG II.4). When is the proportion sufficiently great? What does ‘proportion’ (which Aristotle uses in many contexts) mean (how is it measured) since he had no concept of mass? Echoing the De caelo definition, Lavoisier followed the analytic course and defined an element as the “last point which analysis is capable of reaching” (1965: xxiv). He stressed that there can be no further restrictions on the number and character of the elements, and continues, “we must admit, as elements, all the substances into which we are capable, by any means, to reduce bodies by decomposition”. The Aristotelian doctrine of the four elements is criticised accordingly. To appreciate how difficult this definition was to apply, note that whether a reaction is a decomposition, releasing a product combined in (one of) the original reactant(s), could only be determined by an independent criterion. The relative simplicity of substances is what was at issue in the dispute over phlo-

giston. That Lavoisier’s criterion of reduction in mass was by no means the obvious solution is apparent from the fact that Lavoisier himself didn’t always adhere to it when independent principles dictated otherwise. (Chlorine wasn’t counted an element because then muriatic (hydrochloric) acid would not contain oxygen, and caloric is counted an (imponderable) element in order to account for latent heat of fusion and boiling). Dalton gave Lavoisier’s idea an atomic interpretation some 25 years later, introducing into atomic doctrine for the first time the thesis that each kind of atom corresponds to a distinct macroscopic elemental substance. Some chemists still clung to this idea when controversy was at its height over the nature of radioactivity some 100 years later, maintaining that newly discovered atoms such as the electron were distinct chemical elements (Hendry 2005; 2012). In the immediate prehistory to the modern non-atomic theory of the atom, elements were characterised by atomic weights, for example by Mendeleev in the course of building his periodic table. Elements’ distinctive chemical properties in their formation of compounds display similarities allowing them to be grouped in such a way that as atomic weight increases, the groups are traversed in a fixed order, returning to the first and then traversing the groups again, and so on up to the element with greatest atomic weight. Identification of elements by their position in the periodic groups by virtue of their chemical properties ceased to be uniquely correlated with atomic

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weight as the new properties of radioactivity were discovered and measurement and separation techniques improved. With the development of atomic theory, distinct isotopes correlated with the same position in the periodic table (same chemical properties) came to be understood as differing in the number of neutrons in their atomic nuclei, the number of protons (atomic number) being the same. Since controversy centred on how to distinguish between physical and chemical properties, the question arose of exactly what criterion determines sameness of position in the periodic table (sameness of chemical kind). One idea was suggested by the definite rates of chemical reactions, which are determined by concentration (in moles) of the chemical element. Replacement of any portion of an element by an isotope, whilst maintaining the molar concentration, seemed (within the limits of experimental error) to maintain the rate of chemical reaction. On the basis of such considerations, the IUPAC (International Union of Pure and Applied Chemistry) in 1923 adopted atomic number (number of protons in the nucleus) as defining a chemical element. Modern techniques of measurement show that rates of reaction are not, in fact, independent of isotopic composition. In the extreme case of hydrogen, replacement of protium oxide by deuterium oxide slows down biochemical processes in the body so much as to make the latter poisonous. D2O has pH of 7.41, compared with 7.0 for H2O. But this is exceptional. Usually, rates and other properties differ very little and

atomic number is still taken to define a notion of chemical element, even though it groups together substances distinguishable by thermodynamic and kinetic criteria under the same heading, because for most purposes these differences are negligible (Needham 2008). The physical chemist F. A. Paneth (1931) evidently thought that this didn’t clear up all the issues and distinguished between ‘simple’ and ‘basic’ elemental substances, elements in the isolated state being simple and basic when either isolated or combined. This recalls Aristotle, whose definitions of the elements in terms of extreme contraries characterise what we would call elements in isolation. His theory of mixing allowed only that the elements were potentially, not actually, present in a compound. One of the principal motivations of the theory is the idea that combination results from mutual interaction changing the character of the original ingredients. Modern theory seeks to understand chemical reactivity of the elements in terms of electronic structure. Atomic number is correlated with electronic structure in the isolated atom, where the number of electrons is fixed by the number of protons in order to maintain electrical neutrality. But electronic structure is changed by combination, resulting on the simple Lewis model in shared electrons (covalent bonding) or transferred electrons (ionic bonding) or more thoroughly integrated structures on modern molecular orbital accounts of bonding. Although atomic nuclei are preserved in compounds, atoms themselves, which

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include an electronic structure, are not, or at least not intact. The question therefore remains of what, exactly, the basic elemental stuff is if compounds are considered to be mereologically partitioned into their elements. Otherwise, however the claim that basic elements are actually present in their compounds is construed, it is not a mereological claim about parts (Needham 2006). See also > Aristotle’s Theory of Parts, Ancient Greek Atomism, Body, Chemistry, Collectives and Compounds, Emergence, Homeomerous and Automerous, Mereological Essentialism, Natural Science, Quantum Mechanics.

References and further readings

Aristotle, 1984, The Complete Works of Aristotle, ed. Jonathan Barnes, Vol. 1, Princeton University Press, Princeton. Hendry, R. F., 2005, “Lavoisier and Mendeleev on the Elements”, Foundations of Chemistry 7: 31-48. Hendry, R. F., 2012, “Elements”, in Hendry, R. F.; Needham P.; Woody, A. J. (eds.), Handbook of the Philosophy of Science, vol. 6: Philosophy of Chemistry, Elsevier: Amsterdam. Kragh, H., 2000, “Conceptual Changes in Chemistry: The Notion of a Chemical Element, ca. 1900-1925”, Studies in History and Philosophy of Modern Physics 31B: 435-50. Lavoisier, A., 1965, Elements of Chemistry, trans. by Robert Kerr

(1790) of Traité élémentaire de Chimie, Paris 1789. Dover reprint, New York. Needham, P., 2002, “Duhem’s Theory of Mixture in the Light of the Stoic Challenge to the Aristotelian Conception”, Studies in History and Philosophy of Science 33: 685-708. Needham, P., 2006, “Substance and Modality”, Philosophy of Science, 73: 829-40. Needham, P., 2008, “Is Water a Mixture? – Bridging the Distinction Between Physical and Chemical Properties”, Studies in History and Philosophy of Science 39: 66-77. Needham, P., 2009, “An Aristotelian Theory of Chemical Substance”, Logical Analysis and History of Philosophy 12: 149-64. Paneth, F. A., [1931] 1962, “Über die erkenntnistheoretische Stellung des chemischen Elementbegriffs”, Schriften der Königsberger Gelehrten Gesellschaft, Naturwissenschaftliche Klasse 8 (Heft 4): 101-25. Translated by Heinz Post as “The Epistemological Status of the Chemical Concept of Element”, British Journal for the Philosophy of Science 13: 1-14 and 144-60. Paul Needham

Emergence During the last two decades the idea of emergence has seen a strong revival in many different fields and disciplines such as philosophy of mind, physics, biology, the social

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sciences, artificial life, connectionism, dynamical systems theory, theories of self-organisation, and theories of creativity. However, in these fields the notion of emergence is far from being used in a uniform way. While in ordinary language ‘emergence’ means something like the ‘coming into view or existence’, ‘rising above a surrounding medium’, or ‘occurring unexpectedly’, its technical use is associated with features such as ‘being novel, unpredictable, or irreducible’, ‘showing interesting nonprogrammed patterns’, or ‘having novel causal powers’. Hence, it is highly controversial what the criteria are by which ‘genuine’ emergent phenomena should be distinguished from non-emergent phenomena. Some of these criteria are very strong, so that few, if any, properties count as emergent. Other criteria are inflationary in that they count many, if not all, system properties as emergent. One of the difficulties of this debate is that it is not based on an antecedent clarification of the meanings of the terms ‘emergence’ and ‘emergent.’ The first step towards such a clarification is to distinguish between different kinds of emergence. The various kinds of emergence can be structured relative to three different dimensions. The first dimension determines whether the notion of emergence refers to an epistemological or a metaphysical relationship, the second dimension whether it refers to a synchronic or diachronic relationship, and the third dimension whether it is introduced as a weak or

rather a strong notion. For expositional reasons emphasis is here put on the contrast in the second dimension. Varieties of synchronic emergence.

Most of the more ambitious theories of emergence and, in fact, all classical approaches are based on a common notion of emergence from which they can be developed by adding further features (cf. Stephan 2007: 6672). This weak, synchronic and metaphysical notion of emergence specifies the minimal criteria for emergent properties. It comprises three features: the thesis of physical monism, the thesis of systemic properties, and the thesis of synchronic determination. The thesis of physical monism is about the nature of systems that have emergent properties or structures. It claims that the bearers of emergent features consist of physical entities only and rejects all substancedualistic positions, which, for example, attribute properties such as being alive or having cognitive states to supernatural bearers (e.g., an entelechy or a res cogitans, respectively). According to the second thesis, only systemic properties are candidates for emergent properties. These are system properties that are different in kind from those had by the system’s proper parts. It is uncontroversial that many systems exhibit systemic properties (a bacterium, e.g., is alive, but none of its proper parts). The thesis of synchronic determination specifies the relationship that holds between the systemic properties of a system and its microstruc-

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ture (i.e., the specific arrangement of the system’s parts together with their properties): Systemic properties and dispositions depend nomologically on the microstructure of their bearers. There can be no difference in a system’s systemic properties without some difference in the properties or arrangement of its parts. Anyone who denies the thesis of synchronic determination either has to admit systemic properties that are not dependent on the microstructures of their bearers, or she has to suppose that some additional factors, e.g. nonnatural entities, are responsible for differences in systemic properties and dispositions of otherwise physically identical systems. Weak (synchronic and metaphysical) emergence is equivalent to what van Gulick has called modest kind emergence (van Gulick 2001: 17). This type of emergence is compatible with contemporary reductionist approaches. Particularly for this reason, philosophers of science and cognitive scientists take this position seriously or endorse it explicitly, see e.g. Bunge (2003: 9-25) and Varela, Thompson, and Rosch (1991: 85103). Van Gulick considers an even weaker type of synchronic metaphysical emergence, namely specific value emergence. Specific value emergence obtains whenever a whole and its parts have features of the same kind, but have different sub-types or values of that kind (the mass of a whole, e.g., is different in value from the mass of any of its proper parts). Nobody, however, has ever thought

to characterise specific values as emergent properties. On the contrary, from the very beginning of British Emergentism, features of wholes that differ only in value but not in kind from features of their parts have served as a contrast class for emergent properties (cf. McLaughlin 1992). This is already present in both Mill’s distinction between heteropathic and homopathic laws (Mill 1843, bk 3, ch 6, § 2) and in Lewes’s distinction between resultant and emergent effects (Lewes 1875: 412; cf. also Stephan 2007: 78-87). Hence, there is no reason to treat specific values as emergent properties, if their bearers share them in kind with their parts. Van Gulick also considers a much stronger type of synchronic metaphysical emergence, namely radical kind emergence (van Gulick 2001: 17). In this case a whole has features that are not necessitated by the features of its parts, their mode of combination and the law-like regularities governing the features of its parts (i.e., the system’s microstructure). Radical emergence thus rejects one of the core elements of traditional theories of emergence, which is a center piece of C.D. Broad’s classical definition (Broad 1925: 61), and which is the third feature of weak synchronic emergence as introduced above: synchronic determination (sometimes also called mereological supervenience, cf. Kim 1999: 7). Kim, therefore, doubts whether radical emergence gives us a form of emergence at all (Kim 2006: 192).

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Among the more ambitious epistemological notions of emergence both a synchronic and a diachronic version deserve closer attention. While both share the features of weak emergence they differ with respect to the features of irreducibility (synchronic emergentism) and unpredictability (diachronic emergentism). The idea of strong synchronic (epistemic) emergence has its origin in Broad’s theory of emergence (cf. Broad 1925: 43-94) and involves the irreducibility of certain systemic properties as its core feature, i.e., the principled failure of reductive explanations for these properties. For a reductive explanation to be successful, several conditions must be met: first, the property to be reduced must be functionally specifiable; second, it must be shown that the specified functional role is filled by the behavior of the system’s parts; and third, the behavior of the system’s parts must follow from the behavior they show in simpler systems than the system in question. If all conditions are met, the behavior of the system’s parts in other contexts reveals the systemic properties of the actual system. Since the three conditions are independent of each other, we have to distinguish three different types of strong synchronic emergence, which have three different implications (for further details, see Boogerd et al. 2005). The first type of strong synchronic emergence obtains when a systemic property is irreducible due to the fact that the behavior of the components of the system in their

current arrangement is not reducible to the behaviors these components show in simpler systems. In this case the system itself (or its specific structure) seems to exert some ‘downward causal influence’ on its parts. However, this sort of downward causation would not violate the principle of the causal closure of the physical domain. We would just have to accept additional types of causal influences within the physical domain besides the known types of mutual interactions. The second type of strong synchronic emergence obtains if it is impossible to show that the behaviors of the system’s parts can constitute the causal role adequately attributed to the functionally specified systemic property. In this case the systemic property itself seems to have causal powers different from those of the system’s microstructure. If in addition, the systemic property were a nonphysical (mental) property with a causal influence on the physical world, we would have to admit a violation of the principle of the causal closure of the physical realm. The third sort of strong synchronic emergence obtains when the system has properties that are not functionally specifiable at all. This case is neutral concerning downward causation. It neither implies nor excludes downward causation. Systems with properties that admit of no functional analysis need not be constituted by components whose own behavior is irreducible. Nor is it implied that the system’s structure has a downward causal influence on the system’s

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parts. It may just be that some of the system’s properties are nonstructural properties, i.e. basic properties whose instantiation does not consist in the instantiation of distinct properties by the system or its parts (see O’Connor/Wong 2005: 663). O’Connor and Wong, however, introduce nonstructurality as a metaphysical (and not as an epistemological) feature of strong emergent properties. In addition, they think that novel causal powers might be conferred on the system by its own emergent features (O’Connor/Wong 2005: 665-667). Varieties of diachronic emergence. All

diachronic theories of emergence are based on a thesis about the occurrence of novelties in evolution. According to this thesis, in the course of evolution exemplifications of genuine novelties occur again and again. Existing entities combine into new configurations and structures that constitute new entities with new properties and behaviors. However, the thesis of novelty does not by itself turn a weak theory of emergence into a strong one, since reductive physicalism remains compatible with it. Only the addition of the thesis of unpredictability, in principle, will lead to stronger forms of diachronic emergentism. The structure of an arising new system can be unpredictable, in principle, for two reasons. The system’s arrangement may either be a result of indeterministic processes, or it may be the result of deterministic but chaotic processes. Within emergentism, only the second option is discussed.

It is captured by the thesis of structure unpredictability, which claims that the rise of a novel structure is unpredictable, in principle, if its formation is governed by laws of deterministic chaos. Likewise, any property that is instantiated by such a novel structure is unpredictable, in principle. Emergence as structure unpredictability has a great deal in common with an approach that views emergence as “uncompressible unfolding” (Clark 2001: 116-117). This expression refers to those patterns or macro-states of a system that can only be derived by complete simulations of all interactions at the component’s level and the external influences on the system (cf. Bedau 2010: 52). Such complete simulations of the underlying micro-dynamics would also be necessary for long-term predictions of structure formations governed by deterministic chaos. Since these simulations are not available, given that information compressing short cuts are not adequate for longer intervals, structure unpredictability as introduced above is unpredictability, in principle. Bedau who once introduced the simulation-based notion of emergence in the context of game-oflife environments (Bedau 1997: 311) unfortunately dubbed it weak emergence, too, to distinguish it from the strong (synchronic) type, which comes with irreducibility. Recently, Humphreys has argued that the criteria for synchronic emergence are not sufficient for a state or property instance to count as emergent because the historical development of

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a system’s dynamic is often crucial for them being emergent (Humphreys 2008: 432). This criticism overlooks, however, that both synchronic and diachronic as well as both epistemological and metaphysical notions of emergence have fruitful applications. What matters is to keep them apart. See also >, Causation, Chaos, Dispositios, Dynamical Systems, Power, Quantum Mechanics, Quantum Mereology. Bibliographical remarks

Beckermann, A.; Flohr, H.; Kim, J. (eds.), 1992. The first of the more recent and important collections of papers dedicated to the ideas of emergence and reduction. Bedau, M. A.; Humphreys, P. (eds.), 2007. The best collection of canonical essays to the topic. O’Connor, T.; Wong, H. Y., 2002. Gives a first good detailed overview. Stephan, A., ³2007. Historically and systematically comprehensive. Three more recent collections of articles to the notion of emergence in science and philosophy: Clayton, P.; Davies, P. (eds.), 2006, The Re-Emergence of Emergence. The Emergentist Hypothesis from Science to Religion, Oxford: Oxford University Press. Corradini, A.; O’Connor, T. (eds.), 2010, Emergence in Science and Philosophy, New York/London: Routledge.

Macdonald, C.; Macdonald, G. (eds.), 2010, Emergence in Mind, Oxford: Oxford University Press. References and further readings

Beckermann, A.; Flohr, H.; Kim, J. (eds.), 1992, Emergence or Reduction? Essays on the Prospects of Nonreductive Physicalism. Berlin/New York: Walter de Gruyter. Bedau, M., 1997, “Weak Emergence”, Philosophical Perspectives 11: 375-399. Bedau, M., 2010, “Weak Emergence and Context-Sensitive Reductionism” in Corradini, A.; O’Connor, T. (eds.), Emergence in Science and Philosophy, New York / London: Routledge, 46-63. Bedau, M. A.; Humphreys, P. (eds.), 2007, Emergence: Contemporary Readings in Philosophy and Science, London: MIT Press. Boogerd, F. C.; Bruggeman F. J.; Richardson, R. C.; Stephan, A.; Westerhoff, H. V., 2005, “Emergence and its Place in Nature: A Case Study of Biochemical Networks”, Synthese 145: 131-164. Broad, C. D., 1925, The Mind and its Place in Nature, London: Kegan Paul. Bunge, M., 2003, Emergence and Convergence. Qualitative Novelty and the Unity of Knowledge, Toronto: University of Toronto Press. Clark, A., 2001, Mindware. An Introduction to the Philosophy of Cognitive Science, Oxford: Oxford University Press

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Clayton, P.; Davies, P. (eds.), 2006, The Re-Emergence of Emergence. The Emergentist Hypothesis from Science to Religion, Oxford: Oxford University Press. Corradini, A.; O’Connor, T. (eds.), 2010, Emergence in Science and Philosophy, New York / London: Routledge. Humphreys, P., 2008, “Synchronic and Diachronic Emergence”, Minds & Machines 18: 431-442. Kim, J., 1999, “Making Sense of Emergence”, Philosophical Studies 95: 3-36. Kim, J., 2006, “Being Realistic about Emergence”, in Clayton, P.; Davies, P. (eds.), The Re-Emergence of Emergence. The Emergentist Hypothesis from Science to Religion, Oxford: Oxford University Press, 189-202. Lewes, G. H., 1875, Problems of Life and Mind. Volume 2. London: Kegan Paul, Trench, Turbner, & Co. Macdonald, C.; Macdonald, G. (eds.), 2010, Emergence in Mind, Oxford: Oxford University Press. McLaughlin, B. P., 1992, “The Rise and Fall of British Emergentism”, in: Beckermann, A.; Flohr, H.; Kim, J. (eds.), Emergence or Reduction? Essays on the Prospects of Nonreductive Physicalism. Berlin, New York: de Gruyter, 49-93. Mill, J. St., 1843, A System of Logic. Ratiocinative and Inductive. Collected Works, Vol. VII und VIII. Toronto, Buffalo: University of Toronto Press, 1974.

O’Connor, T.; Wong, H. Y., 2002 (substantive revision 2006), “Emergent Properties”, in The Stanford Encyclopedia of Philosophy, Zalta, E. N. (ed.), http://plato.stanford. edu/entries/properties-emergent/ O’Connor, T.; Wong, H. Y., 2005, “The Metaphysics of Emergence”, Nous 39: 659-679. Stephan, A., 2007, Emergenz. Von der Unvorhersagbarkeit zur Selbstorganisation, Paderborn: Mentis Verlag. Van Gulick, R., 2001, “Reduction, Emergence and Other Recent Options on the Mind-Body Problem: A Philosophical Overview”, Journal of Consciousness Studies 8 No. 9-10: 134. Varela, F. J.; Thompson, E.; Rosch, E., 1991, The Embodied Mind. Cognitive Science and Human Experience, Cambridge, MA: MIT Press. Achim Stephan

Ethics Ethical theorising has largely centred on the right and the good and the relation between the two. Consequentialists predominantly understand rightness as a matter of maximising the good, and accordingly focus on understanding which things are good; deontologists tend to see acting rightly as a matter of acting in accordance with a number of rules or principles. Some consequentialists emphasize the role of rules in ethical life and some deontologists put great weight

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on our obligation to do good, but generally writers in the consequentialist tradition have discussed mereological issues concerning how good wholes are made good by their parts, while deontologists argue about the way in which aspects of situations interact to constitute what we ought to do on the whole. With respect to good wholes and their parts, the locus classicus in terms of mereology is G. E. Moore’s Principia Ethica, where he defends what he calls the principle of organic unities, which says: “The value of a whole must not be assumed to be the same as the sum of the value of its parts” (1903: 28). By parts, Moore here means proper parts; he in turn distinguishes between the value that a whole has as a whole, and the value it has on the whole. The latter is a sum of the value of the whole considered as a part of itself and the value of its proper parts. Moore’s principle constitutes a break with a common ambition among consequentialists, namely to identify a number of things that are basic values, value-atoms so to speak, and to calculate the value of the outcome of any action as a simple sum of these. There are however examples where the way basic valueunits are combined seem to matter greatly. Take punishment: a committed crime has negative value and locking a person up has a negative value; but on the whole, locking the criminal up still makes the situation slightly better. So while the criminal’s punishment in one sense just adds another bad thing to the situation, in another sense it makes it better (by completing a certain type of

whole). Or take the case of sadistic pleasure: pleasure as such seems to be a good, but the situational whole consisting in pleasure being taken in someone else’s suffering is bad not just because it has suffering as a proper part, but also because the situation as a whole is bad. It should be pointed out that even if there are organic unities in the Moorean sense it might still be possible to list all types of them and, in the end, provide an exhaustive axiomatisation of normative ethics; still, if there are organic unities this will be a much more complex project. One possible response to Moore is simply that organic unity-phenomena are just symptoms of us not having a fine-grained enough list of the basic goods, for instance the relevant basic good might not be pleasure but nonsadistic pleasure. The problem with this response is that it might seem that even if non-sadistic pleasure is valuable, that which has value in the given situation is just the pleasure. This point leads naturally to another response to Moore, namely that he does not go far enough in recognising the importance of how wholes are composed since he stills maintains that the value of the proper parts of a relevant whole is invariable: no matter what else is included in the whole, these basics goods retain their value. Some writers, like Thomas Hurka (1998), have argued that there is another type of organic unity which has to do with the fact that many values are conditional on being located in the right circumstances. We might still be able to list all things that are of final value, but for each value we

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will also have to give the enabling conditions that need to obtain in order for the aspect in question, e.g. pleasure, to be of value in a concrete situation. Others, like Noah Lemos (1994), have defended Moore, maintaining that at least in this case, the badness of non-sadistic pleasure on the whole is not properly understood if one does not recognise that it is a matter of something inherently good, pleasure, being perverted by something inherently bad, sadism. Just as in the consequentialist tradition, deontologists have been divided between those, such as Immanuel Kant, who believe that there is a single supreme moral principle and those, such as W. D. Ross (1930), who believe that there is an irreducible plurality of moral principles which we have to abide by in our actions. Mereological issues arise mainly for this latter deontological position. Once one admits that there are several duties or moral reasons which might all be relevant in a single situation, one might wonder to what extent and how these combine. The answer suggested by Ross is that our actual duty is like a resultant determined by the vectors constituted by what he calls our prima facie duties. For instance, if I can spare someone enough pain by telling him a lie, the resultant duty might be precisely to tell the lie. On Ross’ account the strength and valence of duties or reasons are however invariable and this aspect has been criticised as untenable by particularist writers, the leading one being Jonathan Dancy (2004), who ad-

vocates a holism about reasons, i.e., that a feature can be a reason of a certain strength and valence in one situation without being so in another. For instance, that I have promised is often a reason to do something, but in certain circumstances it might not just be outweighed by other concerns, but not count as a reason at all (I might perhaps have been crucially misled by the promisee). A possible response to Dancy is that if a certain moral reason seems variable, then a further specification will reveal an invariable reason, e.g., that I have promised without being under duress or having been mislead by the promisee. Another particularist, John McDowell (1979), focuses on moral judgments rather than reasons and tends towards an even stronger form of holism, where the judgment of the practically wise person reflects a grasp of the situation as a whole rather than a balancing of reasons. Although certain features of the situation will be salient to such a person, his or her grasp of it need not be separable into distinct contributions by a set of reasons (not even a set of variable ones). See also > Atomism, Deontic Modalities, Emergence, Gestalt, Good Life, Perceptual Whole. Bibliographical remarks

Hooker, B.; Little, M. (eds.), 2000. Covers most aspects of particularism.

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McKeever, S.; Ridge, M., 2006. Thorough treatment of the role of rules in ethics. Rønnow-Rasmussen, T.; Zimmerman, M. J. (eds.), 2005. Part V contains several good papers on organic unities. References and further readings

Dancy, J., 2004, Ethics without Principles, Oxford: Oxford University Press. Hooker, B.; Little, M. (eds.), 2000, Moral Particularism, Oxford: Oxford University Press. Hurka, T., 1998, “Two Kinds of Organic Unity”, The Journal of Ethics 2: 299-320. Lemos, N. M., 1994, Intrinsic Value: Concept and Warrant, Cambridge: Cambridge University Press. Moore, G. E., 1903, Principia Ethica, Cambridge: Cambridge University Press. McDowell, J., 1979, “Virtue and Reason”, The Monist 62: 331-350. McKeever, S.; Ridge, M., 2006, Principled Ethics: Generalism as a Regulative Ideal, Oxford: Oxford University Press. Ross, W. D., 1930, The Right and the Good, Oxford: Clarendon Press. Rønnow-Rasmussen, T.; Zimmerman, M. J. (eds.), 2005, Recent Work on Intrinsic Value, Dordrecht: Springer. Johan Brännmark

Experience What are the spatial parts of human conscious experience? If we conceive the human mind on Cartesian premises as a non-spatial res cogitans that somehow interacts with the spatial, material world, we will obviously reject the question. However, if we set this Cartesian conception aside, we can explore two types of candidates for spatial parts of human experience: the parts of the human brain or, alternatively, the external objects that are experienced. Empirical evidence shows beyond reasonable doubt that human conscious experience is strongly correlated with physical events happening within the human brain; so much that it is often thought that human experience must be identical with some aspect of the human brain – its matter, its activity or its function, as described by various special sciences. A supporting fact for this assumption is that our experience is usually complexly structured, and so is the human brain. From this perspective it would appear natural to say that human conscious experience has spatial parts, namely, the spatial parts of whatever aspect of the human brain the experience is identical with. For example, if experience is said to be a complex system-wide electrochemical activity, then the spatial parts of human conscious experience are the parts of that activity: the individual firing of the individual nervecells, each composed of many individual gate-transmission events along the cell's membrane, etc.

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While some philosophers of mind accept this answer, others find it problematic, for the following reason. Human conscious experience includes feelings (‘qualia’, such as what it is like to be in pain or see something blue) as well as subjectivity (the fact that each ego has a unique point of view onto the world). It seems as if a scientific description can never really capture these mental facts properly. There are different variations of what exactly the worry is. Some authors point to an ‘explanatory gap’ (Levine 1983), because the scientific concepts cannot quite explain how feelings feel, or what it means to be a subject. Others argue that the scientific facts do not suffice to determine the mental facts, and certainly cannot be identical with the latter, given that we can conceive of a world in which all scientific facts are identical to our world, but there are no experiences at all. The possibility of such a ‘Zombie scenario’ would then prove that the scientific facts cannot be identical with the mental ones (Chalmers 1996). Subjectivity – and the fact that thanks to it we can access our own experiences in a special way – raises additional problems for claims that human experience can be identified with brain processes as described by science (cf. Kripke 2011). That human experience has spatial parts also follows from ‘panpsychism’, i.e., the claim that every natural thing has a mental aspect (cf. eg. Strawson 2009, Rugel 2013). If the atoms, molecules and nerve-cells of our brain themselves already have a mental aspect then this could perhaps

explain human experience. On the assumption of panpsychism human experience can be identified with natural entities or events, e.g., brain processes, but not all aspects of these natural entities or events are captured by scientific descriptions. The spatial parts of a human experience E would then be the mental aspects of those (differently located) natural entities or events Pi that are identical with E and would be located wherever the Pi are located. The issue of subjectivity, however, raises a mereological problem for panpsychism: the so-called the ‘combination problem’ (Coleman 2014, Goff 2006). It is hard to understand how the subjectivity – the point of view – of a human experience could arise from a combination of the subjectivities – the points of view – of smaller entities or events, e.g. atoms, even if these small things are the spatial parts of the human experience – the locus classicus for this observation is William James’ “Principles of Psychology” (James 1890: 226). The question of spatial mereology can also be addressed working from the phenomenology of experiences – an experience always ‘presents’ something. Instead of merely experiencing ‘green apple’, we experience ‘that green apple over there’. This observation has given rise to a variety of views which claim that the object of an experience somehow has to be considered a part of that experience (e.g. Tye 2009, Lycan 2001). Thus, if I veridically see an apple, the actual apple – and perhaps also the causal events and processes of information transfer connecting me to the

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apple – all together form the experience. This position, ‘phenomenal externalism’, holds that in two situations I can be in exactly the same brain state and yet have different experiences because I am in a different environment. On this construal of human experience the spatial part of my experience are the external items that I experience. Phenomenal externalism has difficulties accounting for the existence of illusions and hallucinations, which might either point to the fact that the objects that are part of the experience are not spatial after all (Harman 1990) or that veridical experiences have radically different parts from non-veridical ones – a view which is called ‘disjunctivism’ (Byrne and Logue 2008). Finally, Whitehead’s ontology, as laid out in his book “Process and Reality” (Whitehead 1929), unites panpsychism and phenomenal externalism. Here the basic entities of the world are understood in experiential terms, as panpsychism has it; but it is also claimed that there is a sense in which the objects of experience are proper parts of an experience, as phenomenal externalism has it. This approach can successfully explain the existence of illusions and hallucinations (Andrae 2014). The abovementioned combination problem for panpsychism does not arise for Whitehead’s account of human experience, since here it is assumed that a person’s conscious experience is merely one of the many different ‘small’ experiences that make up the world. This approach – reminiscent

of the ‘dominating monad’ in Leibniz’ thought – loses its absurdity once one adopts, as Whitehead does, a process-based metaphysics. For then one is not, like Leibniz, committed to holding that a person’s experience is that of the experience of one of the atoms of the person’s body. Rather, the claim is that a person’s experience is that of one of the centrally important processes that go on with and within a person’s body. See also > Conscious Experience, Descartes, Intentionality, Perceptual Whole, Whitehead’s Metaphysics. Bibliographical remarks

Brüntrup, G., 2012. A summary of the strengths and weaknesses of the different positions in the mind-body problem, with an in-depth look at panpsychism (in German language). Chalmers, D., 2010. A comprehensive collection of David Chalmers' papers about the mind-body relation and about the content of consciousness, arguing against physicalism. Dennett, D., 1991. Commentary on the mind-body problem that argues that what is really worth saying about the mind is what science has to say about it. Kim, J., 2010. Overview over the whole philosophical area, leaning towards physicalism and epiphenomenalism of the mental. Strawson, G., 2009. A speculative metaphysical essay that argues for a kind of panpsychism.

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References and further readings

Andrae, B., 2014, The Ontology of Intentionality, München: Philosophia Verlag. Brüntrup, G., 2012, Das Leib-Seele Problem, Stuttgart: Kohlhammer, 4. Edition. Byrne, A.; Logue, H., 2008, “Either/or”, in: Haddock, A.; Macpherson, F. (eds.), Disjunctivism: Perception, Action, Knowledge. Oxford: Oxford University Press, 57-94. Chalmers, D., 2010, The Character of Consciousness, Oxford: Oxford University Press.

Levine, J., 1983, “Materialism and Qualia: The Explanatory Gap”, Pacific Philosophical Quarterly 64: 354-61. Lycan, W., 2001, “The Case for Phenomenal Externalism”, in Tomberlin, J. (ed.), Philosophical Perspectives 15, Atascadero: Ridgeview Publishing Company, 17-35. Rugel, M., 2013, Materie – Kausalität - Erleben: Analytische Metaphysik des Panpsychismus, Münster: Mentis Verlag. Strawson, G., 2009, Selves, Oxford University Press.

Chalmers, D., 1996, The Conscious Mind, Oxford University Press.

Tye, M., 2009, Consciousness Revisited, MIT Press.

Coleman, S., 2014, “The Real Combination Problem”, Erkenntnis 79: 19-44.

Whitehead, A. N., 1929, Process and Reality: An Essay in Cosmology, abbreviated as “PR”, cited from the “Corrected Edition”, edited by David R. Griffin and Donald W. Sherburn, New York: The Free Press, 1979.

Dennett, D., 1991, Consciousness Explained, Little, Brown and Company. Goff, P., 2006, “Experiences don't sum”, Journal of Consciousness Studies 13 (10-11): 53-61. Harman, G., 1990, “The Intrinsic Quality of Experience”, Philosophical Perspectives 4: 31-52. James, W., 2007, “The Principles of Psychology, Vol. 1”, 1890. Quoted from the Cosimo Classics Edition, Cosimo Inc. Kim, J., 2010, Philosophy of Mind, Westview Press, 3. Edition, 2010. Kripke, S., 2011, “The First Person”, in: S. Kripke, Philosophical Troubles, Oxford: Oxford University Press, 292-321.

Benjamin Andrae

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F Facts Early in the 20th century Russell characterised facts – which he also called verifiers – as what make true statements true. Thus, late in the 20th century analytic metaphysics one hears of ‘truth-makers’ (a phrase apparently going back to Bolzano) or, more accurately, of ‘truth grounds’ and of truth theory as an ontological account of the existents in virtue of which true judgments are true. Truth theory was no longer merely a matter of the dealing with the logical problems surrounding the occurrence of a truth-predicate in a logistic schema. These phrases and their uses highlight the contemporary rehabilitation of metaphysics and ontology – of seeking what kinds of entities there are that compose our world and are required for our categorisation of it. Obviously there are things – objects of various sorts. It is also obvious, though not to all, that there are, in addition to things, properties of things and relations that things stand in. The simplest notion of a fact is of something having a property and of things being in relation. A fact is then readily taken as the reason or ground for statements or propositions being true ascriptions of predicates to things as being true.

Yet their role as grounds of truth does not exhaust the reasons for recognising facts, since facts are included among the kinds of entities that are apprehended in experience, along with things, properties and relations. As one apprehends two objects – say a circle and a square on a board – one apprehends that the objects are related in certain ways. That obvious situation must be accounted for in any adequate account of our experience and of what we experience. Thus nominalists who reject entities that are not particulars typically construct arguments that deny both facts and relations. The rejection of facts by nominalists, as well as by those who reject metaphysical issues in general, sometimes makes use of the idiom of Tarski’s Convention-T that is familiar from earlier concerns with a truth predicate. Following the Tarski pattern, truisms such as (T1) ‘Snow is white’ is true iff snow is white and (T2) ‘Abelard loves Heloise’ is true iff Abelard loves Heloise are taken to cite truth conditions, and not truth grounds or makers of truth. Supposedly, providing a sentence to the right of the biconditionals, as in (T1) and (T2) above, simply provides a truth-functional equivalent of the sentence referred to by the semantic ‘name’ on the left side. In this manner some think of Convention-T as involving a dis-quotation device for providing an expression of one semantic level with an equivlent ex-

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pression of another (‘lower’) level. Such matters involve neither truth grounds nor questions about the apprehension of facts or situations. But, the two are easily mixed due to a further use of the pattern of T-sentences. Classical arguments for the existence of universal properties and relations often begin with our apprehending an apparent sameness of attribute that two objects or two facts have, and seeking to account for that sameness or likeness. The revival of metaphysics in the second half of the 20th century is crucially connected to the rejection of the extreme nominalism and pragmatic idealism of Quine, Goodman and Sellars, and their legions of followers. A key factor in the disputes was the nominalist focus on taking the pattern of T-sentences as providing all that was necessary for handling the concept of ‘truth’ or providing a theory of truth. This was buttressed by logicians and semanticists taking a proposed definition of truth for a suitably specified set of sentences of a formal schema to be satisfactory if and only if it enabled one to derive a T-sentence for each sentence of the schema of the appropriate type. Hence, some philosophers followed suit and took there to be no need to consider ‘truth grounds’ or ‘makers of truth’ – Tsentences would suffice. To speak of truth in terms of facts was not to speak of anything significant at all. At best, to speak of a fact making a sentence true was simply to repeat that the sentence was true. Thinking along such lines, one understood the right side of the bi-

conditionals of T-sentences to be ‘about’ things – like snow, in one above case, and Abelard and Heloise, in the other. It was in virtue of those things that the predicate ‘white’ was ‘true of’ snow and ‘loves’ of Abelard and Heloise (or an ordered pair of the latter). Thus, in specifying such truth conditions for sentences, via a definition of a truth predicate involving familiar patterns in logic and semantics, only objects, predicates and sentences – not properties, relations or facts – were acknowledged. When philosophers awakened to question the dogmas of nominalism and the fashionable focus on classes or sets as the prototype abstract objects of the time, a natural target became the prevalent talk of truth conditions and the total ignoring of what Russell had called the ‘makers of truth’. A truth condition, one must recall, was simply given by citing a sentence, and not by taking the sentence to indicate a fact or situation or state of affairs. Russell, along with Moore and Wittgenstein, focused on the role of facts. This was due in part to his recognition of the importance of relations and, not as importantly, of monadic properties. Monadic properties of individual objects could be construed, and explicitly were by Moore, as particulars that were constituents of the objects they characterised. In short they were the familiar individual accidents of medieval philosophy. As medieval philosophers who took there to be such accidents often also acknowledged universal ideas (Platonic ideas, in effect) in God, so

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Moore also took there to be a universal monadic attribute that accounted for the ‘likeness’ of individual accidents of a kind. Russell’s focus was directed at relations. Partially because of his work on the logic of relations and partially in that the argument that he and Moore and developed for the existence of universals was an argument for relations as universals – the key relation of property likeness or exact similarity of accidents of a kind. While Moore had taken the universal color property to ground the likeness of tropes in Plato’s fashion, contemporary trope theories simply assert that the accidents are exactly similar in virtue of being what they are – i. e. exactly similar tropes. Nothing else is required. This pattern has been the hallmark of nominalism through the centuries – whether of the extreme nominalism that even denies properties as individual accidents or the apparently more moderate nominalism that accepts accidents as particulars. The one compresses facts into things, tropes and complexes of tropes, the other compresses them into the application of predicates to ordinary things. In his celebrated and formidable argument for relational universals, Russell does not speak of grounding true statements but of accounting for the immediate apprehension of color similarity. This went along with his noting that his taking relations to be apprehended involved the analysis of what is involved in the immediate apprehension of a fact, such as one tone preceding another, and his claim of direct acquaintance with temporal

precedence. There are numerous cases where facts, as well as things, are taken to be directly apprehended – just consider familiar arrangements of color patterns in modern paintings or of sound patterns or of both at once. At its simplest, just think of seeing a circle within a square as opposed to seeing a square within a circle. One experiences arrangements as well as the items arranged. This is not to argue that facts are required as truth grounds but simply to note what we take ourselves to experience. To claim that we are deluded about relations requires an argument – but none are generally forthcoming; what we are offered are merely claims that we can do without relations by packing relational facts into things via the absurdity of monadic relational properties – a move one finds in Scotus and earlier in others, and a move that both Abelard and Ockam found ludicrous long ago. Thus one takes Abelard to have the individual accident of loving Heloise and fathering Astrolabe and Heloise and Astrolabe of having the correlated accidents of being loved by Abelard and of being fathered by Abelard, respectively. All this is accomplished by the individual accidents alone, without there being any relations. Simons (2010) has currently revived the medieval peculiarities of such monadic relational properties. The recognition of elementary or ‘atomic facts’ led to questions about logically complex facts – negative, general and conjunctive facts, for example. Are there general facts, such as that all things that have a certain property have another specified

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property? Are general statements of law true in virtue of general facts? Or, are there causal relations among properties that form facts grounding the truth of such statements of law? Are there existential facts – conjunctive facts – negative facts? Attempts to formulate systematic answers to such questions, addressed over the span of the 20th century, came to be labeled ‘truth-maker theory’ towards the century’s end. One of the specific issues raised about facts helps to clarify a general issue about the appeal to facts. Consider the case of supposed causal necessity to account for the difference between laws and accidental generalities. The question we start with is how to account for that difference. Hume and Galileo set out one pattern; Plato another, which has enjoyed a revival in the 20th century – a connecting of Forms or universal properties forming higher order facts, as one now says. It is not an accident, I think, that an empiricist like D. M. Armstrong who advocates forces of causal necessity has sought to convince himself and others that we experience such causal forces or connections. Pressing on your own back is an example he uses – for one supposedly experiences the pressing as the cause of the feeling of pressure. Regarding such a claim about experience we cannot argue, as Hume noted, just as we cannot argue about experiencing relational situations, such as experiencing a sound and a flash of color occurring simultaneously or in sequence. Hume, in his way, did exactly what Armstrong does, in ap-

pealing to experience, though he found the opposite answer. In reflecting on our experience of relational situations, as well as in arguing for the need to recognise facts as truth grounds, it is worth noting that we do not postulate that there is a theoretical entity called a ‘fact’ – aping physicists and cosmologists who theorize about things from particles to a megaverse – a megauniverse of universes – to play a theoretical role. To note that we experience situations as well as things is simply to note that we do not simply experience things, attributes and relations, but objects standing in relations and instantiating properties. The latter are taken to be, or described or classified as, facts as entities of a particular kind recognised in a philosophical or metaphysical account. Just as we experience properties and relations, but do not experience them as universals or as tropes or as abstract Forms – that they are universals, we experience things having properties and being in relations, but not as being of an ontological kind – facts or states of affairs. That is a matter for dialectical argument – showing that alternative views are unsatisfactory and how taking things as being universals or facts or individuating particulars purportedly resolve certain traditional metaphysical issues that alternatives fail to resolve. The question of whether objects instantiate relations opens the dispute between moderate nominalists (tropists) and realists about universals and facts. One who claims to apprehend universals, as

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Russell did claim, might be prone to mix two claims: one about apprehending properties and relations; the other about apprehending that such a property or a relation is a universal.

tences or truths are ‘basic’ – that nothing further is required, as a defense of what Armstrong has labeled ‘ostrich nominalism’, seems to leave one stranded on the rock of language.

Compare the above claims about facts and universals with claims about tropes and their natures. The trope in its particularity differs numerically from all other particulars. But, then, the trope’s is declared to be such that it symmetrically relates itself, retaining its particularity, to another trope. Its particularity is joined with its exact similarity, so to speak. Here we have the first step in its Trinitarian multiplicity that would delight Scotus – that which is the trope, its tropiness, so to speak, and its being of a specific kind or sort, a color shade, say. To note this is to see that several facts are compressed into the entity that is a trope, that tropes are simply facts transformed into things.

The truth grounds of the two sentences, if we consider such things, must differ. Tropes meet that need, apparently, and supply the F-ness of oj and the G-ness of oj. This goes back to the early paper of K. Mulligan, P. Simons and B. Smith that brought the term ‘truth-maker’ into English speaking philosophy. It also fit with their phenomenologically inclined outlook that is tied to questions about parts, wholes and dependency relations and the role of ‘things’, not Russell’s facts, as makers of truth.

Consider an obvious line of argument against the extreme nominalism of Quine expressed in taking the objects o1…n… to be ‘what there is’ that the predicates ‘F’ and ‘G’ are ‘true of’. The Quinean claim, in its most recent variant, comes to stopping with the purported truths – ‘F’ is true of oj and ‘G’ is true of oj —in variants of the pattern of T-sentences. It is then obvious to point out that all that is acknowledged as being there, as what is referred to, is the object oj. The rest are linguistic items – two predicates and two sentences. Thus the ground of truth is one and the same in both cases – in terms of what there is. To add, as some do, that the two sen-

It is easy to see how the truth of statements like ‘oj = oj’ and ‘oj exists’ lead some to speak of objects sufficing as ‘truth-makers’ of existential and identity statements. Thus the ignoring of facts by focusing on things begins and appears ready-made for talk of things that make truths true. It continues with the idea that an object suffices for the predications of properties by introducing components that make up what the complex object is, and thus becomes a variant of a view that takes an object as a bundle of properties. Such a view is often developed in a mereological style framework along with discussions of parts, wholes and dependency. This fits the taking of component parts – tropes – as the basic elements of the ontology. On such views tropes are clearly seen as miniature facts – particulars carrying one property – that,

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as Scotus put it, is ‘contracted’ into a particular instance. Relations are either denied, much in the fashion of extreme nominalism – with the objects sufficing as truth grounds for relational predications – or they are also stuffed into the bundle of properties after being transformed into monadic relational properties in a familiar medieval fashion. Recall Abelard having the property of being a lover of Heloise. One sometimes hears that ‘truth supervenes on being’. Such embellished language, taken more simply, merely expresses that truths are such in virtue of the facts (or ‘things’) – alternatively, that the grounds of truth are the key ‘beings’ involved, or that the primary entities are the makers of truths and not the truths or bearers of truth, such as thoughts or propositions. See also > Logical Atomism, Nominalism, Order, Russell, Segelberg, Tropes. References and further readings

Armstrong, D. M., 1978. Universals and Scientific Realism, 2 vol. Cambridge University Press: Cambridge.

D. (ed.) (gen. eds. D. M. Gabbay, P. Thagard, J. Woods), Elsevier, Amsterdam, 449-496. Moore, G. E., 1953, Some Main Problems of Philosophy, London: Allen & Unwin. Mulligan, K.; Simons, P.; Smith, B., 1984, “Truth Makers”, Philosophy and Phenomenological Research 44 (1984): 287-321. Quine, W. V. O., 1948, “On what there is”, The Review of Metaphysics 2, 5: 21-36. Russell, B., 1912, The Problems of Philosophy, London: Allen & Unwin. Russell, B., 1918/19, “The Philosophy of Logical Atomism”, The Monist 28, 29: 177-281. Russell, B., 1919, “On Propositions: what they Are and how they Mean”, Proceedings of the Aristotelian Society Supplement 2: 285-320. Simons, P., 2010, “Relations and Truthmaking,” Aristotelian Society Supplementary Volume vol. 10, 1: 199-213. Wittgenstein, L., 1961, Tractatus Logico Philosophicus, (trans. D. F. Pears and B. F. McGuinness, London: Allen & Unwin. Herbert Hochberg

Hochberg, H., 1967, “Nominalism, Platonism and Being-true-of,” Nous 1, 3: 413-19. Hochberg, H., 1978, Thought Fact and Reference, Minneapolis: University of Minnesota Press. Hochberg, H., 2007, “Logicism and its Contemporary Legacy”, Jacquette,

Fiction Philosophy’s sensitivity to mereological considerations is often more evident in practice than in formal recognition. Philosophical theories of fic-

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tion are a case in point. Although some fictionality theorists expressly invoke the name of mereology − for example, Risto Hilpenen in conversation − this is by a wide margin the exception, not the rule. Two of fiction’s most dominant mereological problems are the full-story problem for fictional texts (Parsons, 1980, Woods, 2017) and the incompleteness problem for fictional objects. Following a suggestion of Currie (1990), we will say that the sentences of a fictional text sentences of a fictional text are fictive sentences. On the other hand, metafictive sentences have no textual occurrence, but are, even so, made true or true-like in virtue of relations that obtain between them and the story’s fictive sentences. In a rough and ready way, these relations can be thought of as implicational and presuppositional. Thus the fictive sentence ‘Holmes waved our strange visitor into a chair’ somehow implies the metafictive sentence ‘Holmes bade his visitor to sit’, and the fictive sentence ‘“Good”, said Holmes, “Excellent!”’ presupposes the metafictive sentence ‘Holmes expressed his satisfaction to someone’. Other metafictive sentences include ‘Holmes was a detective’, ‘Holmes lived in London’, ‘Holmes solved the case of the speckled band’, ‘Holmes patronised Watson and was smarter than Le Strade’. Metafictive sentences are part of the full story. Also important is a category that seems to have escaped Currie’s attention. It contains sentences true of the real world which a story inherits, which serve in conjunction with fictives and metafictives to derive fur-

ther sentences of the full story. For example, it is true in the real world that people who walk and sit have spines. Since the world of a story in the real world save for auctorial provisions otherwise, we have it as part of the full story that Sherlock has a spine, and an oesophagus too. These I call implicit sentences. Currie’s third category is made up of transfictive sentences, which express relations between ourselves and the objects and events of fiction. They are true of what’s true in the story but are not themselves part of it; e.g. ‘Agatha Christie admired Holmes more than any other detective’. The principle on which a story inherits the world, the world-inheritance principle, can be questioned, but not I think to advantage. If Holmes can’t be understood as operating in the London of the 1880s as London was then, if he cannot be vouchsafed a spine and an oesophagus and his other vital parts, his author’s stories would have no readers. Sir Arthur Conan Doyle could not have died a rich man. Of course, stories don’t absorb the world just as they come. The world they inherit is both an auctorially mandated restriction and extension of it. World-inheritance is the price paid by stories for their readability and their readers. This links in a telling way to fiction’s incompleteness problem. Let us say that the full story of a fictional text is the set-theoretic union of its fictive, implicit and metafictive ones. This characterisation resembles what Parsons (1980) calls a story’s maximum account. The task of specifying a fictional text’s full story is a

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mereological one. It requires an orderly specification of its parts and the considerations in virtue of which those parts come together as a linked and unified revelation. As we have it so far, a full story’s constituents are supplied by our three classes of sentences, and the story’s wholeness derives from the varying intersentential implicational, presuppositional and world-inherited relations over them. This takes us to the question of incompleteness. Although, by my lights, our fourfold distinction doesn’t quite capture the categories of taxonomic interest, we now have enough taxonomy to be turning our minds to the incompleteness problem. The incompleteness problem is the problem of determining the properties possessed by fictional entities beyond those expressly conferred by the author’s text. Everyone will agree that Holmes had a hand and a knee, if only in light of the fictive sentence, ‘Holmes reached out his hand for the manuscript and flattened it upon his knee’. But does he have two knees? Do they have kneecaps? Does he have a larynx? Does he have a mole on his left shoulder blade? Did he cry at the party celebrating his third birthday? Did he, for that matter, have a mother? The incompleteness problem can be seen as a special case of the full-story problem. We may define an object as complete if and only if with respect to every property predicable of things of its kind the object either has or lacks it. This in turn motivates a distinction between concrete and abstract objects. An object is concrete only if it is complete

in respect of its kind. If an object is incomplete with respect to its kind, it is an abstract object of that kind. Many theorists of fiction are of the view that fictional objects are incomplete, hence in this sense abstract (e.g. Parsons 1980; Jacquette 1996; Thomasson 1998). Others, emphasising a distinction between ontic and epistemic incompleteness, propose that although it is not known – or knowable – whether Holmes cried at his three-year birthday party, it is a fact that he did or that he didn’t (e.g., Woods 1974; 2017). This raises the question of what role, if any, is played by the properties that an object lacks? In raising it, we see that a correct answer to the full-story question doesn’t fully answer the completeness question. The reason why is that it is an individuating part of Sherlock that he doesn’t exist in reality and yet does exist in the stories. The first of these facts is not part of the stories, whereas the second is a fully established one there. With this comes the more general question of whether the absence of a property of a thing implies the presence of its complement? Does the fact that someone’s tallness precludes his being short suffice to install nonshortness as one of his identitymaking attributes? It is a condition on his tallness that he not concurrently be short. The question that remains unsettled is whether satisfied conditions are inherently propertyconferring. The incompleteness question for fiction occasions mereological considerations of its own, of which perhaps

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the most important is the relationship between the properties constituting an object and the metaphysical kind that the object instantiates. In cases such as Sherlock’s or Baron Charlus’ there is a complication. Sherlock and the baron are each two things at once. They are objects of the fictional kind, and they are objects of the human kind. On the face of it, it might be thought that this leaves perfectly respectable room for the conjecture that the difference between them and us is that, with them, epistemic incompleteness trumps ontic completeness, whereas, with us, it is the other way round. Either way, we see that the merelogical identity of a whole cannot be determined in isolation from kindedness. It is precisely here that the world-inheritance principle earns its keep. If we rejected it, Sherlock and Charlus would be ontic freaks, and readership would be nil. If we adopt it, we spare fiction the embarrassment of conferring freakiness on its instantiations solely by being objects of the fictional kind. (Of course, fiction would retain its capacity to make the hunchback of Notre Dame, who may well have been a freak in the common meaning of that term, without having to endure the unbearable costs of his ontic incompleteness). Even so, some philosophers resist this. There would be, they say, no fact of the matter about what Sherlock weighed after a story-unspecified lunch in a story-unspecified place in a story-unspecified place in London on an unspecified day in 1886. No knowledge of it (of course) but no fact about it one way or the other. If we bought this implication, we might

want to seize the moment and declare that, while not true of Sherlock the man that he fails to have determinate weight, it is definitively true of Sherlock the fictional man. To which I reply that the inherit-the-world principle uproots that hope. It stocks the Sherlock canon with whatever the world’s ontic completeness provisions were in the 1880s and a bit later, save only for Doyle’s auctorial defections. From which we have it that Sherlock was as ontically whole in the world of his stories as Doyle was in the world in which he wrote them. The world-inheritance principle carries with it the necessity to recognise a fifth member of fiction’s taxonomy, and with it in turn a mereologically important distinction for the fullstory question. If stories inherited the unedited world save for auctorially sanctioned deviations then, without further exception, stories would inherit everything else that’s true in the world in the 1880s and briefly afterwards. It inherits the fact of Caesar’s murder and the fact that Ottawa is the capital city of Canada. It inherits the fact that London is some thousands of miles east of Ottawa. But not the most concretely-disposed mereologist of fiction could countenance the idea that these facts form part of the full story of Sherlock’s doings. This motivates a distinction between world-facts that are part of the world of the story and story-world facts which, even so, aren’t part of the full story itself. There is nothing puzzling about this. Stories are narratives and, like all real-world narratives, most of what holds of the world it narrates

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doesn’t make it into the story. It is part of the world that narrates the situation in Aleppo in 2016 that Ottawa is the capital of Canada, but it simply can’t be part of the dreadful story of Aleppo’s destruction. In Woods (2017) sentences such as these are called rest-of-the-world sentences. They are true of the world that Doyle’s stories inherited, but not at all true in the stories themselves. One of the more interesting problems for the logic of fiction is specifying truth conditions for transfictive sentences, sentences such as ‘Agatha Christie greatly admired Holmes more than any other detective’ (which could be true), and ‘Jeremy Brett caught nuances of Homes’ personality better than any other actor to date’ (which isn’t at all true). Difficult as the transfictive problem certainly is, it is not a general problem for the full-story question or for the property-constitution of Holmes. Even if these sentences were true, they too wouldn’t be part of the Holmes stories. Neither would the properties they ascribe to Holmes – being admired by Agatha Christie and being revealingly portrayed by Jeremy Brett. For these are not object-constituting properties. But consider the transfictive sentence ‘Holmes was created by Doyle’. This is certainly so. It states an objectconstituting property but it’s not part of the story that this is so. Similarly, it is certainly true that Holmes could not have existed (in the story) in the absence of the property of being created by Doyle, but is the property of having been created by Doyle an object-constituting property of Holmes?

The world-answer – i.e., the answer to the question ‘what is part of the story-world that Doyle’s story inherits? – is in the affirmative. The storyanswer – i.e., the answer to the question ‘what is part of the full story that Doyle created?’ – is decidedly in the negative. See also > Abstract, Art, Artifact, Brentano, Medieval Mereology, Propositions, Universal. References and further readings

Currie, G., The Nature of Fiction, New York, Cambridge, 1990. Jacquette, D., 1966, Meinongean Logic: The Semantics of Existence and Nonexistence, Berlin: De Gruyter. Parsons, T., Nonexistent Objects, New Haven: Yale University Press, 1980. Reicher, M. E., “The Ontology of Fictional Characters”, in Eder, J.; Jannidis, F.; Schneider, R. (eds.), Characters in Fictional Worlds: understanding imaginary beings in literature, film and other media. Revisionen. Grundbegriffe der Literaturtheorie. Bd 3 (2010): 111-133. Thomasson A. L., Fiction and Metaphysics, New York: Cambridge University Press, 1998. Woods J., The Logic of Fiction, second edition with a Foreword by Nicholas Griffin, volume 23 of Studies in Logic, London: College Publications, 2009; first published with the subtitle Philosophical Soundings of

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Deviant Logic, The Hague and Paris: Mouton, 1974. Woods J., “Truth in Fiction: Rethinking its Logic”. Forthcoming in 2017. John Woods

Fractals Fractals are geometrical figures that exhibit increasingly fine structures at every successive level of magnification. Fractals are produced either through physical processes or by algorithm via recursively iterated functions. The most striking and beautiful examples are constructions produced by computer color graphics executing repeated applications of relatively simple operations at every level of scale that in principle ramify indefinitely, presenting thereby an appearance like that of a naturally occurring object. The natural-seeming aspect of fractals has encouraged the use of fractal geometry in studying the mathematics of many physical and organic phenomena, such as structures of dendrites, crystal growth, neural network articulation, and biological, geographical and cosmic developmental events related to chaos theory. The word ‘fractal’ was introduced in 1975 by the mathematician Benot Mandelbrot to designate structures satisfying the above description, adapting the term from the Latin word ‘fractus’ for ‘broken’, ‘fractured’, or ‘shattered’. Lewis Fry Richardson and Mandelbrot paved

the way for the more exact theoretical investigation of the properties of fractals in the 1960s by studying the mathematics of natural coastlines and replicating self-similarities. Although the term ‘fractals’ is of recent origin, many objects that are properly described as fractals have been known in nature and art from ancient times. Classic examples of fractals, some of which are named by their inventors or discoverers, include, among others, Cantor sets, the Cantor square, Cantor dust, the Koch curve, Koch snowflake, Sierpinski triangle, Sierpinski hexagon, Sierpinski carpet, Sierpinski sieve, Menger sponge, Carotid-Kundalini fractal, Levy fractal, Levy tapestry, Barnsley fern, dendrite fractal, Mira fractal, Cesaro fractal, San Marco fractal, box fractal, space-filling curve, dragon curve, Peano curve, Peano-Gosper curve, limit sets of Kleinian groups, Lyapunov fractal, Julia sets, Fatou set, Mandelbrot tree, and Mandelbrot sets. It is only recently, however, that computer graphics have made possible the dramatic realisation of fractal constructions. In one of the simplest and easily visualised examples, the frequently discussed Koch snowflake involves an operation that begins with an equilateral triangle, and then successively, for each process step, replaces each line segment in the evolving figure at its precise center with an equilateral open-bottomed angle consisting of two segments whose length from base to apex is one-third the length of the segment which they supplant. The resulting figure is thereby transformed in a consecutive series of

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steps from a triangle to a six-pointed star of David to an increasingly delicately articulated snowflake, each of whose six branches is indefinitely finely brachiated. The Koch snowflake illustrates an important feature common to fractal structures. The perimeter or boundary edge of the figure increases to infinity with recursive iterations of the transformation function in the unlimited production of its component line segments, while the figure’s internal area remains finite, scarcely expanding beyond that contained with the figure after the first operation producing the star of David shape. More complex fractals, sometimes brilliantly colored for striking effect, are generated by more elaborate functions. Fractals are accordingly classified into three main categories, depending on the algorithms used to produce them or to describe the mathematical features of their naturally occurring figures. These are isoiterative or exact self-similarity, like the Koch snowflake, in which the fractal pattern is identical at every magnification level; quasi-isoiterative or quasi-self-similarity, including fern-like structures, in which the fractal pattern is only approximately identical at different magnification levels; and, at the farthest end of the spectrum of true fractals that are the least exactly replicative, statistically iso-iterative or mere statistical self-similarity, typified by the Mandelbrot set, in which ramifications of the fractal pattern are only preserved to a statistically limited degree, and otherwise appear ‘chaotic’ despite displaying an internal reg-

ularity of iterative construction. Fractals of different types provide appropriate models for specific subjects of mereological analysis. The fact that the edge of a fractal figure becomes longer, increasing in principle or in the abstract to infinity, the shorter the length of its iterated parts or line segments of which it is composed or described as being composed, is known as the coastline paradox. See also > Chaos, Homeomerous and Automerous, Mereotopology. References and further readings

Edgar, G. A., ed., 1993, Classics on Fractals, Reading: Addison-Wesley. Mandelbrot, B. B., 1977, Fractals: Form, Chance and Dimension, San Francisco: W.H. Freeman. Rietman, E., 1989, Exploring the Geometry of Nature: Computer Modeling of Chaos, Fractals, Cellular Automata, and Neural Networks, New York: McGraw-Hill. Yamaguti, M.; Hata, M.; Kigami, J., 1997, Mathematics of Fractals, Providence: American Mathematical Society. Dale Jacquette

Frege, Gottlob Frege was not particularly interested in mereology. This is due to his conviction that numbers cannot be ap-

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plied to wholes or aggregates, but only to concepts and their extensions (classes). But, interestingly, Frege explained in a letter to P. Jourdain from 23.09.1902 that he was interested in the difference between wholes and classes at the time he worked on the Begriffschrift, although these reflections leaved no mark ‘in the printed version’. After Russell’s discovery of the contradiction in his system, Frege was compelled to analyse different kinds of collections and totalities. The main question was whether every collection should be considered to be an object or not. In a letter from 28.7.1902 to Russell, Frege distinguished three kinds of totalities. In the first kind the unity is only apparent. In ‘Socrates and Plato are philosophers’, for instance Socrates and Plato are put together in a single sentence only by means of linguistic convenience. They clearly do not constitute a single object. This case is uninteresting. In the second kind, the parts form jointly a real single system, as in ‘Bunsen and Kirchhoff founded the spectral analysis’ or ‘The Romans conquered Gaul’. Nations (united, say, by a single culture), armies and physical bodies are totalities of this kind. Contrary to the first case, the second case is not analysable by means of a simple conjunction, for it entails plural predication. In the third kind, finally, the totality is an object that is not composed of parts. Examples of this kind are sets or classes. The class of prime numbers – to take his example – is a single object, but not a totality composed of prime numbers. Frege’s main interest was to distinguish these

two last kinds, which, in contemporary terminology, correspond to (genuine) wholes (called ‘systems’ or – indistinctly – ‘aggregate’ by Frege) and sets (‘classes’), respectively. The first distinction concerns the relations between the constituents. The distinguishing mark of wholes is that their parts are united by essential relations: thus, an army is destroyed as soon as its unity is dissolved, even when all soldiers keep alive. In the case of classes, the members are not united by essential relations. The relations between them are irrelevant for the corresponding class. The class of all soldiers of an army remains intact even when its unity is dissolved. The second distinction is the determinate granularity of the constituents. On the one side, because of the transitivity of the part-whole relation, it is fully undetermined which entities we should consider as constituents in the case of the wholes. Corps, divisions, brigades and battalions are all parts of the army. On the other side, only prime numbers are the constituents of the class of prime numbers – the class of prime numbers of the form 4n+1 is not a constituent of that class, since it is not a prime number. Classes and wholes can coincide, but even in this case they must be distinguished: the class of the atoms of this chair is not the chair. Wholes are physical objects (when the constituents are physical entities), while classes are always logical objects. Although, as we said, Frege was not particularly interested in mereology, he made use of a mereological analy-

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sis to criticize Wittgenstein’s Tractatus. In a letter from 28.6.1919 to Wittgenstein, Frege accuses him to offend against the basic principle of the transitivity of the part-whole relation. According to the Tractatus, objects are parts of states of affairs, and existing states of affairs (facts) constitute the world – thus, how can Wittgenstein claim (in TLP 1.1) that the world is not composed of objects? Of course, Frege seems to be unfair to Wittgenstein, since the meaning of TLP 1.1 is just that the world is not composed of ‘free floating’ objects, but composed of objects necessarily organised in states of affairs. It becomes also clear that Frege did not recognise the peculiar and absolute simplicity of the Tractarian objects, when he asks in the same letter if the larva of the Vesuvius is part of a state of affairs of which Vesuvius is a constituent. More important are two tensions in Frege’s account of the mereological structure of thoughts. Here is the first one. Frege proposed that thoughts (Gedanken) are to be considered as senses of sentences. Such thoughts are complex senses composed of the senses of the parts of the sentence. Now, should we conceive of thoughts as composed of the senses of the parts of sentences, or should we conceive of the senses of these parts as resulting from a decomposition of the sense of the whole thought? The Principle of Compositionality suggests the first and the Principle of Context the latter. Frege Scholars disagree on this point. Frege himself, in many passages, seems to give priority to the whole. At the end of his

live, he declared “I therefore do not begin with concepts that I put together into thoughts or judgments. Rather, I obtain thought-components by analysing thoughts” (Aufzeichnung für Ludwig Darmstaedter, NS, p. 273). Even if this view seems not to fit to our normal intuitions of how language works, it is absolutely reasonable from the mereological perspective. It is a widely recognised mereological fact that sometimes the whole has priority over the parts, just as my body is prior to my hands. The second tension concerns the decomposition of thoughts into their parts. Take these two sentences (1) a is parallel to b. (2) The direction of a is (identical to) the direction of b. Do (1) and (2) express the same thought? On the one hand, both sentences seem to be strictly equivalent and to have exactly the same truth conditions. Hence they must express the same thought. But, on the other hand, the sense of (1) is a composition of the senses of the ‘a’, ‘b’ and ‘is parallel to’, while the sense of (2) is composed of the senses of ‘the direction of a’, ‘the direction of b’ and ‘is’ (identity). Thus, it seems that in each case we have different objects and different concepts. Indeed, as Bell noted (1996), there is a tension between two claims in Frege’s system, namely: thesis A: We can distinguish parts in the thought corresponding to the parts of a sentence, so that the sentence can serve as a model of the structure of the thought; and thesis B: two structurally differ

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ent sentences can express one and the same thought.

Wien, Hölder-Pichler-Tempsky, 95100.

Bell discussed this tension and concluded that we must reject an additional principle that is generally assumed, namely, that if one sentence involves a concept that another sentence does not involve, the two sentences cannot express the same thought. Or, expressed positively, we must accept that sentences can involve different concepts and, despite this, express the same thought. The thought expressed by (1)/(2) is thus neither dependent on the objects a and b, nor on the objects the direction of a and the direction of b. It is also neither dependent on the concept is parallel to nor on the concept is identical to. For our purposes, the interesting result is the following: thoughts are not rigidly dependent on specific objects or concepts as their parts, because different logical decompositions are possible.

Frege, G., 1976, Nachgelassene Schriften, Hermes, H. et al. (eds.), Hamburg: Meiner.

See also > Facts, Propositions, Russell. References and further readings

Bell, D., 1987, “Thought”, Notre Dame Journal of Formal Logic, vol. 28 (1): 36-50. Bell, D., 1996, “The Formation of Concepts and the Structure of Thoughts”, Philosophy and Phenomenological Research, 66: 583-596. Burkhardt, H., 1990, “Wittgensteins Monadologie”, Akten des 14. Internationalen Wittgensteins-Symposiums.

Frege, G., 1879, Begriffsschrift, eine der arithmetische nachgebildete Formelsprache des reinen Denkens, Halle: Von Louis Nebert. Frege, G., 1976, Wissenschaftlicher Briefwechsel, in Gabriel, G.; Hermes, H.; Kambartel, F.; Thiel; C.; Veraart, A. (eds.), Hamburg: Meiner. Kemmerling, A., 1900, “Gedanken und Ihre Teile”, Grazer Philosophische Studien 37: 1-30. McKay, T., 2006, Plural Predication, Oxford: Oxford University Press. Wittgenstein, L., 1960, Tractatus Logico-Philosophicus, Frankfurt am Main: Suhrkamp. Guido Imaguire

Fusion ‘Fusion’ is a philosophical term of art, with a variety of uses. First, it is often a synonym for ‘sum’. In this sense, a is a fusion of b, c and d iff b, c and d are parts of a, and every part of a shares a part with b, c or d. So a cat is a fusion of the cells which compose it, and the same cat is a fusion of the molecules which compose it. Relatedly, ‘fusion’ can refer to the occurrence of such composition: philosophers disagree about whether fusion is widespread, about whether it can be a vague matter, and so on.

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There is a second main use of ‘fusion’, according to which fusions are sums which inherit their temporal and modal identity conditions from their parts. In this sense, if a is now the fusion of b, c and d, then it is the fusion of b, c and d whenever it exists, and at every world in which it exists. Thus if a exists at a time (or world), then so do all of b, c and d. In addition, it is usually presupposed that the mere existence of b, c and d at a time (or world) suffices for the existence of a at that time (or world), no matter how scattered they are.

A third, more expansive sense of ‘fusion’ is due to Leśniewski. In this sense, a fusion of bs is a sum of at least some bs. For example, consider all the people currently alive. In Leśniewski’s sense, there are many fusions of these people, including the sum of the Welsh, the sum of the Europeans, and the sum of the lefthanders. As Peter Simons explains, “if there is any sum [of bs], there is exactly one, whereas if there is more than one b, there is more than one fusion of bs, of which the sum is the largest” (1987: 65).

These two senses of ‘fusion’ are not always clearly distinguished, but the difference between them is important. Anything which is a fusion in the second sense is also a fusion in the first sense: a sum with distinctive identity conditions is nonetheless a sum. But not everything which is a fusion in the first sense is also a fusion in the second sense: everyday complex objects are fusions in the first, but apparently not the second sense. It is not clear whether anything is a fusion in the second sense; if there are such things, presumably some of them coincide temporarily or contingently with ordinary objects. The phrase ‘mere fusion’ (or ‘mere sum’) may be used to mark this second sense of ‘fusion’. Unlike an ordinary object, a mere fusion satisfies no sortal, whence the idea that its identity conditions must be inherited from its parts. Moreover mere fusions obey the principles of classical extensional mereology (for discussion, see Simons 1987).

This third, Leśniewskian sense of ‘fusion’ is not now widely recognised, and it is controversial whether the second sense has any application. So the remainder of this article will focus upon fusions in the first sense of the term. Two main questions arise. First question: is fusion unrestricted? That is, does every plurality of objects have at least one fusion? Classical extensional mereology includes a principle of unrestricted fusion, sometimes called the ‘fusion axiom’, or even just ‘Fusion’ (Simons 1987 section 3.2.3, Casati and Varzi 1999 chapter 3). But this commits us to a vast array of unfamiliar scattered objects and ‘arbitrary sums’ (van Inwagen 1990, Markosian 1998 and forthcoming). Fans of unrestricted composition argue that restrictions upon which pluralities have fusions are inevitably either vague or arbitrary; in addition, they reconcile their position with common sense by conceding that we do not usually quantify over arbitrary sums (Lewis 1986

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pp. 211-213, and Sider 2001 section 4.9.1). Debate about unrestricted fusion usually focuses on pluralities of material objects, or pluralities of material objects which exist in the same world, or at the same time. But we may also wonder whether fusion extends across ontological categories, combining abstract and concrete objects, or universals and particulars, or events and continuants. Second question: is fusion unique? That is, does a given plurality of objects have at most one fusion, or could there be two different objects wholly composed of the same parts? Classical extensional mereology is committed to the uniqueness of fusion; indeed, the term ‘extensional’ alludes to this very feature (Simons 1987 section 3.2.4, Casati and Varzi 1999 p. 40 ff). Moreover, uniqueness complements the attractive idea that the properties of a whole are determined by those of its parts. However, it looks as if the very same plurality can have different fusions at different times, and in different possible worlds. Perhaps the molecules which currently compose you once composed Julius Caesar; perhaps they could have composed Jarvis Cocker right now. Fans of unique composition have two options. First, they may argue that you do not literally share parts with either Caesar or Cocker: Caesar is composed of earlier temporal parts of the molecules whose later parts compose you now, whilst a possible Cocker is composed of counterparts of your molecules (Lewis 1986, chapter 4). Second, they may instead

retreat to the claim that a given plurality of objects has at most one fusion at a given time and world. But there are apparent counterexamples even to this restricted uniqueness principle. The statue and its constituent lump of clay appear to be made of the same parts at the same time, and in the same world. Yet apparently they are distinct objects, with different modal, historical and perhaps aesthetic properties. Fans of unique composition must somehow explain away these differences (for discussion see Baker 2000, Bennett 2004, Fine 2003, Olson 1996 and Wasserman 2002). Both main questions about fusion are ontological questions, questions about what exists. Those who disagree about whether fusion is unrestricted disagree about how many distinct complex objects exist, as do those who disagree about whether fusion is unique. Such disagreements remind us that classical extensional mereology is not an innocuous formalism: it is a theory which has conditional but substantive consequences about what there is. See also > Coincidence, Collectives and Compounds, Common Sense Reasoning about Parts and Wholes, Gestalt, Material Constitution, Persistence, Sum. Bibliographical remarks

Baker, L. R., 2000. Argues against uniqueness.

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Simons, P., 1987. Sets out the principles of classical extensional mereology, then rejects some of them. Sider, T., 2001. Section 4.9.1 advocates unrestricted composition; section 5.3 advocates uniqueness. Van Inwagen, P., 1990. Argues for a very stringent restriction on composition. References and further readings

Baker, L. R., 2000, Persons and Bodies: A Constitution View, Cambridge: Cambridge University Press. Bennett, K., 2004, “Spatio-Temporal Coincidence and the Grounding Problem”, Philosophical Studies 118: 339-371. Casati, R.; Varzi, A., 1999, Parts and Places, Cambridge, MA: MIT Press. Fine, K., 2003, “The Non-Identity of a Material Thing and its Matter”, Mind 112: 195-234. Lewis, D., 1986, On the Plurality of Worlds, Oxford: Blackwell. Lewis, D., 1991, Parts of Classes, Oxford: Blackwell. Markosian, N., 1998, “Brutal Composition”, Philosophical Studies 92: 211-249. Markosian, N., forthcoming, “Restricted Composition”, in Hawthorne, J.; Sider, T.; Zimmerman, D. (eds.) Contemporary Debates in Metaphysics, Oxford: Blackwell. Olson, E., 1996, “Composition and Coincidence”, Pacific Philosophical Quarterly 77: 374-403.

Sider, T., 2001, Four-Dimensionalism, Oxford: Oxford University Press. Simons, P., 1987, Parts: A Study in Ontology, Oxford: Clarendon Press. Van Inwagen, P., 1990, Material Beings, Ithaca, NY: Cornell University Press. Wasserman, R., 2002, “The Standard Objection to the Standard Account”, Philosophical Studies 111: 197-216. Katherine Hawley

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G Gestalt When we perceive complex structures such as spatial figures or melodies, what is it that we perceive? And what is the ontological status of such perceptual complexes? Until the 19th century Hume’s atomistic psychology still dominated the scene, according to which the perception of a complex can only be accounted for as a summation of simple perceptions, each having its own simple object. The central problem for such an account is to explain the characteristic unity of complexes that we perceive. Against the background of Ernst Mach’s attempt to solve this problem within the atomistic framework, by appeal to further sensations, ‘muscular sensations’ as he called them (“Bemerkungen zur Lehre vom räumlichen Sehen”, Zeitschrift für Philosophie und philosophische Kritik, N.F. 46, 1865: 1-5), the Austrian philosopher Christian von Ehrenfels presented a new proposal. Ehrenfels, a student of Franz Brentano, argued that we have to recognise special ‘Gestalt qualities’ of complexes of elementary data given in experience (Über Gestaltqualitäten, Vierteljahrsschrift für wissenschaftliche Philosophie, 14, 1890: 242292). These Gestalt qualities are determined by the constituent elements

of the associated complexes. But the Gestalt is not a whole embracing the individual elements as parts. Rather it is an additional unitary object or attribute, a special sort of invariant structure, existing in consort with the sensational elements upon which it is founded. Edmund Husserl, in the Philosophy of Arithmetic (1891), independently put forward an account of our perception of structural wholes which is rather similar to that of Ehrenfels. Reflecting on configurations of objects in the visual field such as a heap of apples or a flock of birds, Husserl points to certain characteristic qualities, ‘figural’ or ‘quasi-qualitative moments’, of the unitary total perception of such sensory collections. He emphasises that the unity of such structural wholes is immediately given in perception, grasped in a single glance, not the product of any analysis. This immediacy is explained by the concept of ‘fusion’, which means the absence of phenomenal discontinuities between the relevant parts. The parts and the relations between them become fused together creating that distinctive unity which is the figural moment. Later, in the third of his Logical Investigations (1901), Husserl elaborated his earlier account. He argued that the typical unity of Gestalten can arise in two different ways. Either the objects in question do not need any additional objects to form a unitary whole, or they can be unified only by the presence of some additional object. Such unifying objects may again be of two kinds. They may be inde-

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pendent objects, capable of existing apart from the given whole, or they may be dependent objects, ‘moments of unity’, capable of existing only together with the objects they serve to unify. Moreover, Husserl here draws the important distinction between phenomenological and objective moments of unity. Finally, Husserl came to believe that relations of foundation or dependence can be applied quite generally to all objects and systems of objects whatsoever, and so presented the first systematically worked out general theory of wholes and parts, which he regarded as a formal ontology. Husserl was strongly influenced by his mentor Carl Stumpf, from whom he took over the central notion of fusion. Stumpf already distinguished between complex and Gestalt, the former being a whole of sensory contents, the latter being an articulated whole in which there are distinct parts, a unitary network of relations between contents. In contrast to ‘whole-properties’, which can be had by wholes in general, ‘Gestaltproperties’ can only be had by phenomenally articulated wholes. Moreover, he maintained that there is a system of a priori structural conditions of possibility among perceptual or sensory contents (Über den psychologischen Ursprung der Raumvorstellung, 1873, Leipzig: Hirzel). Thus Stumpf distanced himself definitely from atomistic presuppositions. The most famous members of the Berlin school of Gestalt theory, Max Wertheimer, Wolfgang Köhler and Kurt Koffka, all were students of

Stumpf. They rejected the assumption of earlier theories that a Gestalt is a special kind of quality of certain phenomenal complexes. Rather, according to them, a manifold of sensory data does not have a Gestalt but is itself a Gestalt, a special kind of whole whose parts can exist only as parts of a whole of the given sort. They held that it is Gestalten alone that are given in experience, sensations being merely abstractions from them. In their view, we simply do not have pure sensory experiences. And so we do not have to suppose that special mental activities, cognitive processing, is necessary to produce the various Gestalten, as Alexius Meinong and his followers of the Graz school, above all Vittorio Benussi and Gaetano Kanizsa, believed. What we perceive are Gestalten whose parts are not any discrete and independent entities that could in principle be experienced in isolation. Thus the Berlin theory rejected the two-storey ontology of their predecessors. With the development of mereology, however, new formal tools became available to address the problem of the integration of perceptual contents that Gestalt theories tried to solve. Building on Husserl, Stanislaw Leśniewski elaborated a formal mereology (Leśniewski, S., 1929), which became the point of departure for further mereological-topological studies by his pupil Alfred Tarski (Tarski, A., 1935). In this way, mereology gradually entered logic, mathematics and analytical philosophy as a whole. Above all, however, it was Nelson Goodman, who, together with

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Henry Leonard, developed and refined the system of classical extensional mereology: the calculus of individuals (“The Calculus of Classes and its Uses”, Journal of Symbolic Logic 5, 1940: 45-55). Later, in The Structure of Appearance (1951), Goodman revised the theory and made the pioneering proposal that mereology should replace set theory as a methodological foundation for the treatment of qualities, order, measure, time etc. The Structure of Appearance is a contribution to constructional philosophy in the tradition of Bertrand Russell’s Our Knowledge of the External World (1914), C.I. Lewis’ Mind and the World Order (1929), and Rudolf Carnap’s Der logische Aufbau der Welt (1928). As the title of the book indicates, Goodman is primarily concerned with the analysis of phenomena and with the study of phenomenalistic systems. He dissociates himself, however, from the foundationalist aims of traditional phenomenalism, and is even one of the sharpest critics of the empiricist dogma that all knowledge about the world can be build up from a sensory basis of unconceptualised elements that are simply given in experience. Rather, methodologically and ontologically Goodman espouses pluralism. Appearance has no unique structure. Goodman distinguishes between physicalistic or phenomenalistic systems. A system is physicalistic, if it takes perceptible physical objects as its basic units, and it is phenomenalistic, if it takes perceptible phenome-

nal individuals, ‘qualia’, as its basic units. Examples of qualia are single colours, sounds, moments, and visual locations. Phenomenalistic systems are then further divided into realistic and particularistic systems. Realistic systems regard qualia as nonconcrete or abstract entities, universals in the sense of being repeatable, while particularistic systems take concrete spatially and temporally bounded particulars as their starting-point. Criticising the particularism and platonism of Carnap’s Der logische Aufbau der Welt, Goodman opts for realism and nominalism. So the objects of Goodman's universe are qualia and sums of them. A sum of qualia is called a concretum if it is a smallest concrete part of the stream of experience. The basic concreting relation is a symmetric relation of togetherness among qualia, expressed by the two-place predicate ‘being with’. This relation obtains between any two qualia belonging to some one concretum. Finally, Goodman develops a theory of qualitative order, which attempts to construct, for each category of qualia such as color, sound, space and time, a map that will assign to each quale in the category a unique position and that will represent relative likeness of qualia by relative nearness of position. To this purpose, he introduces the predicate ‘matches’, which applies between qualia if and only if they are not discriminable on direct comparison. The problem for phenomenalism is, however, that transitivity does not hold for the matching-relation. Two qualia x and

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y may exactly match, and yet be distinct because there is a third quale z that matches one but not the other. Hence matching qualia are not always identical. This problem can be solved by postulating that two qualia to be identical if and only if they match all the same qualia. See also > Brentano, Carnap, Goodman, Husserl, Perceptual Whole, Structure of Appearance, Stumpf. References and further readings

Grossmann, R., 1983, The Categorial Structure of the World, Bloomington: Indiana University Press. Hempel, C. G., 1953, “Reflections on Nelson Goodman’s 'The Structure of Appearance'”, Philosophical Review 62: 108-116. Hochberg, J., 1978, Perception, 2nd ed., Englewood Cliffs, N.J.: Prentice Hall. Husserl, E., 1891, Philosophie der Arithmetik. Psychologische und logische Untersuchungen, Halle: Niemeyer.

Amin, I., 1973, Assoziationspsychologie und Gestaltpsychologie: Eine problemgeschichtliche Studie mit besonderer Berücksichtigung der Berliner Schule, Bern/Frankfurt: Lang.

Husserl, E., 1901, Logische Untersuchungen. Zweiter Teil: Untersuchungen zur Phänomenologie und Theorie der Erkenntnis, Halle: Niemeyer.

Bühler, K., 1913, Die Gestaltwahrnehmungen: experimentelle Untersuchungen zur psychologischen und ästhetischen Analyse der Raumund Zeitanschauung I, Stuttgart: Spemann.

Husserl, E., 1891, Philosophie der Arithmetik. Psychologische und logische Untersuchungen, Halle: Niemeyer.

Casati, R. & Varzi, A., 1999, Parts and Places: The Structures of Spatial Representation, Cambridge/MA: MIT Press. Eberle, R. A., 1970, Nominalistic Systems, Dordrecht: Reidel. Elgin, C. Z. (ed.), 1997, The Philosophy of Nelson Goodman, New York: Garland Science. Goodman, N., 1951, The Structure of Appearance, Cambridge/MA: Harvard University Press.

Jackendoff, R., 1983, Semantics and Cognition, Cambridge/MA: MIT Press. Kanizsa, G., 1979, Organization in Vision: Essays on Gestalt Perception, New York: Praeger. Köhler, W., 1969, The Task of Gestalt Psychology, Princeton: Princeton University Press. Leśniewski, S., 1929, “Grundzüge eines neuen Systems der Grundlagen der Mathematik”, Fundamenta Mathematicae XIV: 1-81. Lewis, D., 1991, Parts of Classes, Oxford: Blackwell.

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Martin, R. M., 1988, Metaphysical Foundations. Mereology and Metalogic, München & Wien: Philosophia Verlag. Moltmann, F., 1997, Parts and Wholes in Semantics, Oxford: Oxford University Press. Ridder, L. 2002, Mereologie. Ein Beitrag zur Ontologie und Erkenntnistheorie, Frankfurt: Klostermann. Simons, P., 1987, Parts: A Study in Ontology, Oxford: Oxford University Press. Smith, B. (ed.), 1988, Foundations of Gestalt Theory, München & Wien: Philosophia. Tarski, A., 1935, “Zur Grundlegung der Booleschen Algebra. I”, Fundamenta Mathematicae 24: 177-198 (Eng. trans. by J. H. Woodger: “On the Foundations of the Boolean Algebra”, in Tarski, A., Logics, Semantics, Metamathematics, Papers from 1923 to 1938, Oxford: Clarendon, 1956: 320-341). Whitehead, A. N., 1929, Process and Reality. An Essay in Cosmology, New York: Macmillan. Richard Schantz

God Philosophers have usually spelled out the claim that there is a God as the claim that there is a bodiless person who is omnipotent, omniscient, eternal, and perfectly good and free. This claim is called ‘Theism’. It is also claimed by theism that God is the

creator and sustainer of the universe. That God is a person means that he can act intentionally and for reasons. As Christian doctrine claims that God consists of three persons, sometimes instead it is said that God is a ‘personal being’. That God is eternal can be understood either as the claim that God is outside of time or that he is everlasting, i.e. exists at all times. Thus theism includes quite different views of God. However, both views, the timelessness view as well as the everlastingness view, maintain that God is imperishable, that he is the creator of the universe, that he can answer prayers, and that he can perform miracles such as raising a man from the dead. If omnipotence and omniscience were understood as meaning that God can do and knows ‘everything’, meaning everything that can be described, then certain paradoxa would result, such as ‘God can create a square circle’ or ‘God can create a stone that is too heavy for him to lift’. There is a consensus that ‘everything’ must be spelled out so that it expresses just the idea that God’s power and knowledge is maximal and not limited by any lack. God has the power to do everything that he can possibly do, and he knows everything that he can possibly know. But there is much debate about how exactly this is to be formulated and what exactly this includes, for example whether this includes infallible knowledge of all future free actions. Belief in the existence of God can be justified through evidence or through perception. (Swinburne 2004; Alston

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1994; against theism: Mackie 1982) Alvin Plantinga and the ‘Reformed Epistemology’ have emphasised that belief in God can also be justified without evidence. Plantinga (2000) has furthermore argued that if belief in Christian doctrine is true then it probably also has ‘warrant’ and is knowledge. Does God have parts? Certainly God does not have concrete parts, i.e. parts which could be separated from the other parts. In this he is like other non-embodied persons, like human souls or angels. But philosophers particularly in the Latin tradition (especially Augustine, Boethius, Anselm of Canterbury, and Thomas Aquinas), who held that God is outside of time, developed the view that there are no parts or distinctions of any kind in God. This is the doctrine of divine simplicity. Thus Anselm wrote: ‘Life and wisdom and the other [attributes], then, are not parts of You, but all are one and each one of them is wholly what You are and what all the others are.’ (Proslogion, § 18) Thomas Aquinas claimed that God is neither composed of matter and form, nor of subject and nature, nor of essence and existence, nor of subject and accident. (ST I, Q 3) Uncontroversial about this is that God has no concrete parts. Also human souls are supposed to be simple in this sense. More controversial is the claim that while human souls have different properties, God does not. The following assumptions are possible motives for the doctrine of divine simplicity: 1. There are property universals. They exist in God’s

mind. Therefore God himself does not have properties. 2. If God were not simple, then he would not be perfect. 3. Non-temporal entities are not composed of properties. See also > Boethius, Subject/Person, Thomas Acquinas, Whitehead’s Metaphysics. Bibliographical remarks

Brower, J., 2008. Defence of the doctrine of divine simplicity. Mackie, J.L., 1982. Defence of atheism. Swinburne, R., 1990. Investigation of the attributes of God. Swinburne, R., 2004. Defence of theism. References and further readings

Alston, W. P., 1989, Divine Nature and Human Language, Cornell UP. Alston, W.P., 1991, Perceiving God, Cornell UP. Bocheński, J. M., 2003, Gottes Dasein und Wesen: Logische Studien zur Summa Theologiae I, qq. 2-11, München: Philosophia. Brower, J., 2008, “Making Sense of Divine Simplicity”, Faith and Philosophy 25: 3-30. Gale, R. M., 1991, On the Nature and Existence of God, Cambridge UP. Hughes, Ch., 1989, On a Complex Theory of a Simple God, Cornell UP.

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Leftow, B., 1990, “Is God an Abstract Object?”, Noûs 24: 581-598. Leftow, B., 2006, “Divine Simplicity”, Faith and Philosophy 23: 365380. Mackie, J. L., 1982, The Miracle of Theism: Arguments for and against the existence of God, Oxford: Clarendon Press. Plantinga, A., 1980, Does God Have a Nature?, Marquette UP, 37-61. Plantinga, A., 2000, Warranted Christian Belief, Oxford: OUP. Pruss, A., 2008, “On Two Problems of Divine Simplicity”, in Kvanvig, J. (ed.), Oxford Studies in Philosophy of Religion 1, Oxford University Press, 150-167. Swinburne, R., 1990, The Coherence of Theism, Oxford: OUP. Swinburne, R., 1994, The Christian God, Oxford UP, ch. 7. Swinburne, R., 2004, The Existence of God (Second Edition), Oxford UP. Wierenga, E. R., 1989, The Nature of God: An Inquiry into Divine Attributes, Cornell UP. Wolterstorff, N., 1975, “God Everlasting”, in L. B. Orlebeke and C. J. Smedes, ed. God and the Good: Essays in Honor of Henry Stob, Eerdmans Publishing Company. Wolterstorff, N., 1991, “Divine Simplicity”, Philosophical Perspectives 5: 531-552. Daniel von Wachter

Good Life, The The question of wherein lies the good life, well-being, welfare, happiness, or eudaimonia has been at the centre of moral philosophy since antiquity. These notions might be defined to mean different things, but there is no consensus in the literature on anything but the fact that they are at least closely related. A notion like ‘happiness’ has a subjective ring and might sometimes refer to something like a feeling, but when philosophers study the matter, irrespective of whether they do so in terms of ‘happiness’ or some other notion, they tend to be concerned with how well a person’s life is going prudentially, i.e., for the sake of the person leading the life. With respect to the issue of parts and wholes there are two main concerns. The first arises for any account of the human good: what is the relation between momentary well-being and the having of a good life? The second arises only for pluralist accounts: what is the relation between having realised different kinds of goods in one’s life and leading a good life on the whole. There are many different views on what constitutes a good life, but at least in modern moral philosophy the most common approach is atomistic. The ambition is to list a number of basic goods the accumulation of which make our lives go better (or, conversely, the reduction of which will make our lives take a turn for the worse). The best example of this approach is hedonism, which holds that the goodness of a life is simply the net balance of pleasure over pain in that life. More complex

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atomistic theories might provide longer lists of basic goods, but they share the assumption that certain proper parts of any given life contribute certain amounts of value – the goodness of that life as a whole simply being a sum of those. On this picture, a human life as a whole is more or less like a container, preferably holding as many units of prudential value as possible. While such units might from a non-evaluative point of view still be composites (e.g., pleasures vary according to duration, intensity, and perhaps tone), they are from an evaluative point of view mereological atoms. Several objections can be raised against this simple picture. The most common one is that monistic accounts do not capture all relevant aspects; more relevantly, a number of writers have questioned the way the relation between parts and wholes is understood on the simple atomistic approach. An early example is F. H. Bradley (1927), who criticised hedonism for not being able to account for when we reach a point where we lead a good life: there is just more or less of pleasure and although some point could be selected at which we have reached the threshold for leading a good life, this selection would be arbitrary. There is never any real ground for saying that one has achieved enough to count oneself happy. This type of problem might also arise for pluralists about the good, prime examples of which are contemporary Aristotelians like Martha Nussbaum (1993), especially if they think that there is such a thing as leading a complete life. T. H. Irwin

(1999), in an effort to reconstruct Aristotle’s position, has argued that there is a difference between a good being realised as a general feature of a life and as a particular instance of that good. A human life is complete when all relevant goods are realised as general features of that life, but even a complete life can to some extent be better or worse depending on how many instances of the relevant good that are realised within it. The Bradleyan concern is however still relevant: how does one nonarbitrarily identify a point where one has enough of something for it to be a general feature of one’s life? The pluralists are, however, somewhat better off since they can make use of the idea of balancing different goods within the confines of a human life, whereas monists need to resolve this issue within a purely maximising account of achieving the good. Even a pluralist might think that an assessment of the overall quality of life is reached by adding all instances of momentary well-being. This would mean that for any given life a happiness curve could, at least in principle, be plotted along a time axis, and the total well-being would simply be the size of the area below that curve. In recent years, however, David Velleman (1991) and Johan Brännmark (2003), among others, have emphasised the narrative shapes of our lives, arguing that the order in which things occur and the overarching structures of our lives matter for how well our lives are going. For example, the overall tendency of the development of our lives matters for how well we take ourselves to fare in

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the early versus the later stages; if we fare better in the later stages, then that upward progression in itself adds value to one’s life. Some atomists have argued that such things can be accounted for in atomistic terms. For instance, Fred Feldman, a leading hedonist, has suggested that the importance of narrative features of our lives can be fully accounted for by the pleasure or discontent we tend to take in certain such features. The increased value introduced by the upward tendency might be explained simply by the fact that we tend to take extra pleasure in having that type of development. Of course, holistically inclined writers will say that Feldman is putting the cart before the horse, that the reason that we take pleasure in an upward development is that it is in itself something good. Among holistically minded authors there are differences in how far they distance themselves from simple atomism. Velleman maintains that while momentary goods are not everything, they cannot be altered by later events, while Brännmark argues that certain events, taken as momentary goods, can retroactively have their value changed by the way they are combined with later events. A more strongly holistic philosopher like Bradley understands the good life as a concrete whole in a way that not only stands in contrast to atomism but also casts doubts on the idea that happiness as a whole can at all be decomposed into bearers of prudential value on a level lower than that of the life as a whole.

See also > Atomism, Collectives and Compounds, Emergence, Ethics, Gestalt, Subject/Person, Structure. Bibliographical remarks

Chappell, T. D. J., 1998. Explores a narrative conception of the good life. Griffin, J., 1986. Excellent general book on happiness. Kagan, S., 1994. Seminal paper on the relation between the person and the life she is leading. Slote, M., 1983. Early contemporary proponent of the importance of structural features. References and further readings

Aristotle, 1999, Nicomachean Ethics, tr. T. H. Irwin, Indianapolis: Hackett. Bradley, F. H., 1927, Ethical Studies, 2nd Ed., Oxford: Clarendon Press. Brännmark, J., 2003, “Leading Lives: Happiness and Narrative Meaning”, Philosophical Papers 32: 321-43. Chappell, T. D. J., 1998, Understanding Human Goods, Edinburgh: Edinburgh University Press. Griffin, J., 1986, Well-Being: Its Meaning, Measurement, and Moral Importance, Oxford: Clarendon Press. Irwin, T. H., 1999, “Permanent Happiness: Aristotle and Solon”, in Sherman, N. (ed.), Aristotle’s Ethics: Critical Essays, Lanham, Md.: Rowman & Littlefield, 1-33.

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Kagan, S., 1994, “Me and my Life”, Proceedings of the Aristotelian Society 94: 309-324. Nussbaum, M. C., 1993, “NonRelative Virtues: An Aristotelian Approach”, in Nussbaum, M. C.; Sen, A. (eds.), The Quality of Life, Oxford: Clarendon Press, 242-269. Slote, M., 1983, Goods and Virtues, Oxford: Clarendon Press. Velleman, J. D., 1991, “Well-Being and Time”, Pacific Philosophical Quarterly 72: 48-77. Feldman, F., 2004, Pleasure and the Good Life, Oxford: Oxford University Press. Johan Brännmark

Goodman, Nelson Henry Nelson Goodman (1906– 1998) ranks among the most influential American philosophers of the latter half of the twentieth century. Goodman had wide-ranging philosophical interests: from formal logic and the philosophy of science to the philosophy of art. He made significant and highly original contributions to all of these diverse fields (see Cohnitz and Rossberg 2006). Goodman was born on August 7, 1906, in Somerville, Massachusetts (USA), to Sarah Elizabeth (Woodbury) Goodman and Henry L. Goodman. In the 1920s, he enrolled at Harvard University, where he studied under C.I. Lewis (who would later become his Ph.D. supervisor), Alfred North Whitehead, Harry

Scheffer, W.E. Hooking, and Ralph Barton Perry. Goodman completed his BA at Harvard in 1928. From 1928 until 1940, while working as director of the Walker-Goodman Art Gallery at Copley Square, Boston, he studied at the graduate level in Harvard. During this time he was a regular participant in W.V. Quine's seminars on the philosophy of the Vienna Circle and, in particular, the philosophy of Rudolf Carnap. Goodman also worked closely with Henry S. Leonard, who wrote his Ph.D. dissertation at the same time under Alfred North Whitehead's supervision. Leonard and Goodman collaborated on the development of mereology. Goodman applied their system in his monumental dissertation A Study of Qualities (1941, which he later rewrote substantially and published as The Structure of Appearance, Goodman 1951). In this work Goodman applies the mereological techniques in a ‘constructional system’ inspired by Carnap’s constitution program in Der Logische Aufbau der Welt. The technical aspects of this mereological system were probably mainly developed by Leonard (Rossberg 2009; see also Leonard’s Comments on the “Calculus of Individuals and its Uses”, published in this Handbook). The result of Goodman and Leonard’s project was presented in 1936 at a meeting of the Association of Symbolic Logic, and published 1940 as “The Calculus of Individuals and Its Uses”. This seminal paper introduced Anglophone philosophy to mereology. The Calculus of Individuals is equivalent to Stanisław Leśniewski’s system, but presented

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in the language of Whitehead and Russell’s Principia Mathematica. In their joint article, Leonard and Goodman also applied mereology to solve a technical problem in Carnap’s Aufbau. Together with W.V. Quine, Goodman used mereology to construct a finitist and nominalist syntax for the language of set theory in their famous article, “Steps Toward a Constructive Nominalism” (1947). Goodman’s philosophical defense of nominalism and mereology can be found in his “A World of Individuals” (1956) and the brief “On Relations that Generate” (1958). After military service, Goodman taught for a short while as ‘instructor in philosophy’ at Tufts College, and was then hired as associate professor (1946–51), and later as full professor (1951–64), at the University of Pennsylvania. He was Harry Austryn Wolfson Professor of Philosophy at Brandeis University (1964–67). He finally returned to Harvard in 1968, where he taught philosophy until 1977. At Harvard, he founded a center to study and improve education in the arts called Project-Zero. Besides being an art gallery director as a graduate student, and private art collector throughout his life, Goodman was also involved in the production of three multimedia-performance events, Hockey Seen: A Nightmare in Three Periods and Sudden Death (1972), Rabbit, Run (1973), and Variations: An Illustrated Lecture Concert (1985) (Scholz 2009). In addition to mereology, Goodman’s most famous contribution to philosophy is probably the ‘grue-paradox’

(1954), which points to the problem that in order to learn by induction, we need to make a distinction between projectible and non-projectible predicates. Other important contributions include his description of the methodology that would later be called ‘reflective equilibrium’, his investigation of counterfactuals (both in his 1954), his ‘irrealism’ (1978), a nominalistic account of logical syntax (with W.V. Quine, 1947), his contribution to the cognitive turn in aesthetics, and his general theory of symbols (both in his 1968). Aged 92, Goodman died on November 25, 1998, in Needham, Massachusetts, after a stroke. See also > Comments on “The Calculus of Individuals and Its Uses”, Carnap, Gestalt, Nominalism, Philosophy of Mathematics, Russell, Structure of Appearance, Whitehead. Bibliographical remarks

Scholz, O., 2009. A short introduction to the life and work of Nelson Goodman. Cohnitz, D.; Rossberg, 2006. A general introduction to the philosophy of Nelson Goodman Elgin, C. (ed.), 1997. Collections of the most relevant articles on different aspects of Goodman’s work. References and further readings

Cohnitz, D.; Rossberg, M., 2006, Nelson Goodman, London: Routledge.

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Elgin, C. (ed.), 1997, The Philosophy of Nelson Goodman, New York: Garland, 4 volumes. Ernst, G., Steinbrenner, J.; Scholz, O. (eds.), 2009, From Logic to Art: Themes from Nelson Goodman, Berlin: de Gruyter. Goodman, N., 1941, A Study of Qualities, PhD thesis, Harvard University. First published New York: Garland, 1990 (Harvard Dissertations in Philosophy Series). Goodman, N., 1951 [1977], The Structure of Appearance, third edition, Boston, MA: Reidel. Goodman, N., 1954 [1983], Fact, Fiction, and Forecast, fourth edition, Cambridge, MA: Harvard University Press. Goodman, N., 1956, “A World of Individuals”, in Bocheński, I. M.; Church, A.; Goodman, N. (eds.), The Problem of Universals: A Symposium, Notre Dame, IN: University of Notre Dame Press, 13–31; reprinted in N. Goodman, 1972, Problems and Projects, Indianapolis, IN: BobbsMerrill, 155-171. Goodman, N., 1958, “On Relations that Generate”, Philosophical Studies 9: 65-66; reprinted in Goodman, N., 1972, Problems and Projects, Indianapolis, IN: Bobbs-Merrill, 171172. Goodman, N., 1968 [1981]. Languages of Art: An Approach to a Theory of Symbols, second edition, Indianapolis, IN: Bobbs-Merrill. Goodman, N., 1978. Ways of Worldmaking. Indianapolis, IN: Hackett.

Goodman, N.; Quine, W.V., 1947, “Steps Toward a Constructive Nominalism”, Journal of Symbolic Logic 12: 105-22; reprinted in N. Goodman, 1972, Problems and Projects, Indianapolis, IN: Bobbs-Merrill, 173198. Leonard, H. S.; Goodman, N., 1940, “The Calculus of Individuals and its Uses”, Journal of Symbolic Logic 5: 45-55. Rossberg, M., 2009, “Leonard, Goodman, and the Development of the Calculus of Individuals”, in G. Ernst et al. (eds.), From Logic to Art: Themes from Nelson Goodman, Berlin: de Gruyter, 51-69. Scholz, O., 2009, “The Life and Opinions of Nelson Goodman: A Very Short Introduction”, in Ernst, G. et al. (eds.), From Logic to Art: Themes from Nelson Goodman, Berlin: de Gruyter, 1-32. Daniel Cohnitz Marcus Rossberg

Grammar The part-whole relation has enjoyed a prominent place in linguistics since at least the turn of the twentieth century; indeed, it has – under the label constituency – come to represent perhaps the fundamental relation in grammar. Almost all modern theories of grammar incorporate it in a prominent place, and analyse grammatical structures fundamentally in terms of this relation. Thus, Edward Sapir’s famous the farmer kills the duckling

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is analysed as a single whole consisting of parts, such as (according to some theories) the farmer and kills the ducking or (in other theories) the farmer, kills, and the duckling. Other more abstract analyses are adopted in other theories, which might identify parts that have no direct representation in terms of linguistic form. The part-whole relation allows us to identify parts of a whole as in turn wholes, so that it is possible to analyse sentences into parts-of-parts-ofparts, until one gets down to the fundamental unanalysable grammatical elements, the morphemes. Thus we can further analyse the farmer into words the and farmer, and the latter word into farm and -er. The partwhole relations permit us then to represent the entire structure of the sentence in terms of a tree, in which the part-whole relation is represented by branches – see ‘Syntax’. Analyses such as these were quite popular in the first half of the twentieth century, and underlie most modern grammatical theories. They suffer, however, from numerous difficulties. Among them are the following (see e.g. McGregor 1997:54-58; Hudson 1976; 1984:92ff; and below ‘Syntax’ for discussion of various problems): (a) there is evidence supporting both alternative divisions of Sapir’s sentence, raising the question of how to decide between them; and (b) the relation between the three parts the farmer, kills and the duckling and the wholes they belong to are not distinguished, and thus presumably identical. As to (a), there is no consensus amongst linguists as to

the evidential backing for one alternative over the others, and a wide range of analyses, along with justifications (both in terms of language facts, and theoretical prescriptions), can be found in the literature. As to (b), one resolution is to introduce labelling of the component parts (e.g. to call the farmer and NP) and/or the part-whole relations (e.g. to label the relation of the farmer in the whole sentence as ‘subject’). The former labelling concerns the category-type of the part; the latter, its function or role in the whole. Most modern theories accept both solutions in some form or other, although there is much disagreement concerning the labels. Can all of the grammatical patterning of human languages be described and accounted for with the part-whole relation, augmented by these additional categories? Perhaps the majority of linguists would think not. Due to space limitations, I outline here one model that aims to resolve some of the remaining inadequacies, semiotic grammar (McGregor 1997). The essence of the proposals in this work is that not all part-whole relations are amenable to relational labelling; some are fundamentally different. For instance, if we take Mary ate the meat raw, it is clear that raw belongs to the sentence, and in that sense is a part of it; it does not belong in an NP with the meat, however. But it seems implausible to claim that raw serves in any significant role in the entire sentence; rather, it evidently relates to the NP the meat (which does serve a relation in the sentence). What is important grammatically is the relation between the two parts, the meat

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and raw; this is what McGregor (1997) refers to as a part-part or dependency relation. Like part-whole relations, this relation links two grammatical categories (an NP and an adjective) and is a specific type of relation (attribution, as in the meat was raw (when Mary ate it)). McGregor (1997) argues further that not only do we need part-whole and part-part relations, but we also need to recognise whole-whole relations in grammar. If we consider the farmer probably killed the duckling, probably is evidently a part of the whole sentence, although again it does not serve a significant function within it. Rather, what we have are two whole sentences the farmer killed the duckling and the farmer probably killed the duckling. These two wholes are related via probably, which modalises the proposition expressed: it holds the whole sentence the farmer killed the duckling in its scope, indicating that it is possibly, though not certainly, true. These three types of relation, partwhole, part-part, and whole-whole, are all semiotically significant, according to McGregor 1997: they are linguistic signs. The relations themselves, that is, have both forms (signifiers) and meanings (signifieds). Indeed, they express meanings of three generic types, respectively: experiential (to do with the structuring of conceptionalisations of the world); logical (concerning the relations amongst things in the world of experience); and interpersonal (concerning the relations amongst human beings). Figure (1) provides diagram-

matic representation of the three fundamental types of grammatical relation according to the semiotic grammar model. X Part-whole relations: Constituency (signifier) Experiential (signified)

Y Part-part: Dependency (signfier) Logical (signfied)

Whole-whole: Conjugation (signifier) Interpersonal (signified)

Z

X

Y

X

Y

Figure (1)

This ternary classification of grammatical relations situates part-whole relations in a paradigm of other types of relation of a fundamentally different nature. Figure (2) below provides diagrammatic representation. Reductionist attempts to replace the three types by just one (e.g. Hudson 1984, who argues that dependency alone is adequate) are possible only if grammatical relations are viewed as purely formal objects – already shown to be problematic by Haas (1954). Whole1

Whole2

Part1

Part2

Figure (2)

Relations hold among ‘things’ or units of the grammar of human languages, and depend on how those

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units are construed, whether as parts or wholes. Parts that serve in semiotically significant relations in the wholes to which they ‘belong’ enter into part-whole relations, and may be understood as in inalienable relations to the wholes. Parts that do not serve in such semiotically significant relations in the wholes they ‘belong’ to can be likened to alienable possessions. As argued in McGregor (2003), what has gone wrong in modern linguistics is that the relations between units have been replaced by the units themselves (and thus dubbed constituents, heads, dependents, etc.) as the objects of investigation and analysis. Part-whole relations play an important role in grammar, but they tell only part of the story.

Halliday, M. A. K., 1985, An Introduction to Functional Grammar. London: Edward Arnold. Hudson, R., 1976, Arguments for a Non-transformational Grammar. Chicago: Chicago University Press. Hudson, R., 1984, Word Grammar. Oxford: Basil Blackwell. McGregor, W. B., 1997, Semiotic Grammar. Oxford: Clarendon Press. McGregor, W. B., 2003, “A Fundamental Misconception of Modern Linguistics”, Acta Linguistica Hafniensia 35: 39-64. Longacre, R. E., 1960, “String Constituent Analysis”, Language 36: 6388. Wells, R. S., 1947, “Immediate Constituents”, Language 23: 81-117. William B. McGregor

See also > Linguistic Structures, Proposition, Structure, Syntax. Granularity Bibliographical remarks

Haas, W., 1954. Function to be taken into account in constituency relations. Hudson, R., 1984. Arguments against the adequacy of constituency, and for its replacement by dependency. McGregor, W. B., 1997. Sign status of key grammatical relations. References and further readings

Haas, W., 1954, “On Defining Linguistic Units”, Transactions of the Philological Society: 54-84.

We may describe a given situation at different levels of precision. For example, the starting time of an event may be rounded to the nearest hour, minute, or second. A building’s location may be listed as a city or as a street address. A biological process may be described in terms of organ processes or in terms of the underlying cellular processes. A building may be classified as a public-use facility or, more specifically, as a hospital. Granularity is the level of precision which results from limiting the collection of basic entities (granules) in

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terms of which information of a given type is formulated. The granules used to fix a granularity level may be, for example, spatial regions (Donnelly, 2001), (Galton, 2002), (Schmidtke, 2003), (Worboys, 1998), temporal regions (Bettini et al., 2000), (Bittner, 2002), (Schmidtke, 2005), events (Pianesi andVarzi, 1996), or classes (Bittner, 2004). They may also be the components of a mathematical structure used to represent information such as a graph (Stell, 1999) or cell complex (Puppo and Dettori, 1995).

specifies granularity levels by fixing a range of temporal lengths or place sizes. Granules at a given level are of uniform size, may overlap, and cover the entire (temporal or spatial) domain. A similar approach is used in (Donnelly, 2001). In (Pianesi and Varzi, 1996), granules are special events called ‘divisors’ which may have diverse temporal lengths, but which are all maximal in the sense that they divide all non-overlapping events into those occurring before the divisor and those occurring after the divisor.

Besides taking different types of entities as granules, analyses of granularity differ in whether the granules associated with a given granularity level must i) be pairwise discrete, ii) cover the entire domain under consideration, or iii) satisfy some sort of uniformity constraint (requiring, e.g., that all granules are approximately the same size). One common approach is to associate each granularity level with a partition of the domain under consideration (Bittner, 2002), (Stell, 2003), (Worboys, 1998). The cells of the partition (the granules) are pairwise discrete and cover the entire domain, but size or other uniformity constraints are not generally imposed. Note, however, that the ‘granular partitions’ of (Smith and Brogaard, 2002) are not partitions in the standard sense – the cells of granular partitions are not pairwise discrete. In (Bettini et al., 2000), time granules at a given level are pairwise discrete, may be different sizes, and need not cover the entire temporal domain – there may be gaps within or between granules. (Schmidtke, 2005)

Researchers have proposed methods for formulating granularity-dependent descriptions of the entities in a given domain. Much of the work in the partition approach to granularity levels is based on the techniques of rough sets (Pawlak, 1991). Rough representations of spatial (Worboys, 1998), temporal (Bittner, 2002), or spatiotemporal (Stell, 2003) regions specify for a given region which cells of the partition are included in that region and which overlap that region. These representations may also take into account the relation of the region to boundaries between cells (Bittner, 2002) or distinctions between different modes of spatio-temporal overlap (Stell, 2003). Approximate relations between regions and approximate operations on regions are introduced in terms of their rough representations. Other approaches to granularity have been used for formulating different kinds of granularity-dependent descriptions. (Pianesi and Varzi, 1996) introduce a granurity-dependent

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predicate which distinguishes an event as punctual when it is part of a granule which has no other granule as a part. (Schmidtke, 2003) gives classifications of two-dimensional regions (as punctual, linear, planar, extended, or local) which depend both on grain-size and a designated context location. (Donnelly, 2001) introduces granularity-dependent distance and diameter measurement functions for spatial regions. An important goal of research on granularity is to develop systematic ways of linking descriptions formulated at different granularity levels. A first step in this direction is to introduce relationships between different granularity levels. The most important such relation links finer levels of granularity to coarser levels of granularity. Generally, granularity level G1 counts as finer than granularity level G2 if and only if each of G1’s granules are included in all of G2’s granules. But this relation is instead sometimes defined in terms of mappings from the granules at one level to those at another level (Stell, 1999), (Puppo and Dettori, 1995). (Schmidkte, 2005) defines a finerthan relation in terms of granule size. See (Bettini et al., 2000) for several additional relations between granularity levels. Methods for transferring information to finer or coarser granularity levels have be proposed in (Bettini et al., 2000), (Bittner, 2004), (Stell, 2003). Mereology has been used as a basis for several theories of granularity. The structure of the granular partitions of (Smith and Brogaard, 2002)

is specified in terms of two different parthood relations – one holding between the cells of the granular partition and one holding between the objects in the world to which the cells project. In (Bittner, 2004), mereological relations are used to distinguish levels of granularity within a granular partition, to define the finer-than relation between granularity levels, and to introduce the rough representations of regions. (Pianesi and Varzi, 1996) develop their theory of temporal granularity in a mereotopology that includes predicates for distinguishing divisors at different levels of granularity. Mereological relations are also used, but play a less central role in, (Donnelly, 2001), (Schmidtke, 2002), (Stell, 1999). Whether or not mereology is used as a basis for formulating a theory of granularity, a theory of granularity may provide tools for representing, at a given granularity level, approximate mereological relations between objects. For example, (Duntsch et al., 2001) introduces two different approximate parthood relations. Given the standard rough set representation of a region, we can use the two approximate parthood relations to distinguish between mereological claims about the original regions that can be inferred from their rough representations and mereological claims that are merely compatible with, but cannot be derived from, the rough representations. See also (Polkowski and Showron, 1996) and (Worboys, 1998). Taking a somewhat different approach, (Galton, 2003) supposes that spatial regions are represented at various granularity levels, not as

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rough sets, but as ordinary subsets of a discrete space. He shows that, even given reasonable conditions on the representations, mereological relations holding between regions may be altered in their granular representations. See also > Mereotopology, Order, Point, Structure, Transitivity. Bibliographical remarks

Demri S. and Orlowska E., 2002. Formal methods of data analysis and inference inspired by rough set theory. Euzenat J. and Montanari A., 2005. A detailed survey of work on temporal granularity. Hobbs J., 1985. An influential early work on granularity. References and further readings

Bettini, C.; Jajodia, S.; Wang, S., 2000, Time Granularities in Databases, Data Mining, and Temporal Reasoning, Berlin: Springer.

Donnelly M., 2001 “Introducing Granularity-Dependent Quantitative Distance and Diameter Measures in Common-Sense Reasoning Contexts”, in Welty, C.; Smith, B. (eds.) Formal Ontology in Information Systems. Collected Papers from the 2nd International Conference, New York: ACM Press, 321-332. Duntsch, I.; Orlowska, E.; Wang, H., 2001, “Algebras of Approximating Regions”, Fundamenta Informaticae 46: 71-82. Euzenat, J.; Montanari, A., 2005, “Time Granularity” in Fisher, M.; Gabbay, D.; Vila, L. (eds.), Handbook of Temporal Reasoning in Artificial Intelligence, Amsterdam: Elsevier, 59-118. Hobbs J., 1985, “Granularity” in Proceedings, Ninth International Joint Conference on Artificial Intelligence, 432-435. Galton A., 2003, “GranularitySensitive Spatial Attributes”, Spatial Cognition and Computation 3: 97-118. Pawlak Z., 1991, Rough Sets: Theoretical Aspects of Reasoning about Data, Dordrecht: Kluwer.

Bittner, T., 2002, “Approximate Qualitative Temporal Reasoning”, Annals of Mathematics and Artificial Intelligence 35: 39-80.

Pianesi, F.; Varzi, A., 1996 “Refining Temporal Reference in Event Structures”, Notre Dame Journal of Formal Logic 37: 71-83.

Bittner, T., 2004, “A Mereological Theory of Frames of Reference”, International Journal of Artificial Intelligence Tools 13: 171-198.

Polkowski L.; Skowron, A., 1996, “Rough Mereology: A New Paradigm for Approximate Reasoning”, International Journal of Approximate Reasoning 15: 333-365.

Demri, S.; Orlowska, E., 2002 Incomplete Information: Structure, Inference, Complexity, Berlin: Springer.

Puppo E.; Dettori, G., 1995, “Towards a Formal Model for Multiresolution Spatial Maps” in Egenhofer,

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M.; Herring, J. (eds.), Advances in Spatial Databases: Lecture Notes in Computer Science 951, Berlin: Springer, 152-169. Schmidtke H., 2003, “A Geometry of Places: Representing Extension and Extended Objects” in Kuhn, W.; Worboys, M.; Timpf, S. (eds.), Spatial Information Theory: Foundations of Geographic Information Science, Berlin: Springer, 235-252. Schmidtke H., 2005, “Granularity as a Parameter of Context” in Dey, A.; Kokinov, B.; Leake, D.; Turner, R. (eds.), Modeling and Using Context, 450-463. Smith, B.; Brogaard, B., 2002, “Quantum Mereotopology”, Annals of Mathematics and Artificial Intelligence 35: 153-175. Stell, J., 1999, “Granulation for Graphs” in Freksa, C.; Mark, D. (eds.), Spatial Information Theory. Cognitive and Computational Foundations of Geographic Information Science, Berlin: Springer, 417-432. Stell, J., 2003, “Qualitative Extents for Spatio-Temporal Granularity”, Spatial Cognition and Computation 3: 119-136. Worboys, M., 1998, “Imprecision in Finite Resolution Spatial Data”, Geoinformatica 2: 257-280. Maureen Donelly

Grossmann, Reinhardt S. In the phenomenological tradition the problem of wholes looms large.

Grossmann (1931-2010), a disciple of Gustav Bergmann and later professor at Indiana University Bloomington continues that tradition. He is also influenced by G.E.Moore and Russell. His final ontology is particularly close to that of the phenomenologist Meinong. Gestalt psychology, Grossmann explains, originates from phenomenological ontology. C.v. Ehrenfels, the founder of Gestalt psychology, was a student of Meinong. Grossmann’s ontology starts from Russell’s logical atomism but in the course of its development increasingly adopts and adapts Meinongian views. A case in point is Grossmann’s category of structures which is related to Meinongian complexes. However, they resemble also the structures of abstract algebra in consisting of a domain and relations which hold in the domain. Grossmann talks correspondingly of the proper parts and the characteristic relations of a structure. In contrast to abstract algebra his characteristic relations are not sets of ordered ntuples but relational universals. Since they have constituents or parts structures are complexes. Classes (or sets) are another category of complexes in Grossmann's ontology. Classes do not have characteristic relations and thus are not structured. A chair, e.g., is a spatial structure, according to Grossmann. Its proper parts, i.e., legs, back etc. stand in certain spatial relations which belongs to the chair. A chair is ‘more than’ the class of its spatial parts. Grossmann thinks that the discovery of structures and the clarification of the contrast between

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classes and structures is the main starting point and theme of Gestalt psychology. He shows that with respect to Ehrenfels. Thus Grossmann thinks that Gestalt theorists mean classes when they speak of ‘sums’ and that what they describe as wholes are structures. Grossmann insists, against Gestalt theorists, that wholes are analysable although they are not classes. He also rejects the Gestalt view that wholes determine the nature of their parts. But he agrees with Gestalt theorists that wholes have emergent properties, i.e., properties not attributable to their parts. Grossmann seeks to find further characteristics of structures and differences between structures and classes. He notes that there exists no parallel to class inclusion between structures. That is also an objection to those set theorists such as P. Halmos who views classes as wholes, that is, in Grossmann’s view, as structures. Another point of difference between classes and structures is the unit class of a class, i.e., the class having this class as its only member. There is nothing similar in the category of structures. A peculiarity of structures which Grossmann emphasises are the similarities between structures with totally different proper parts, for example, similarities between structures of numbers and structures of pieces of wood. These are structural similarities such as isomorphisms. Grossmann points to such structural similarities to question the Gestalt explanations of similarities in terms of a common Gestalt quality. He concedes that there are such qualities of wholes and similarities with respect

to them. For example, two squares, one consisting of four triangles, the other consisting of four smaller squares, share the property of being square. However, Grossmann does not agree with Ehrenfels’ explanation of the similarity between two instances of the same melody by a common Gestalt property. He claims that the two successions of tones are instances of the same melody because they contain the same relations between corresponding tones. Ehrenfels cannot accept this explanation because he presupposes that relations are not perceptible and that they are created by mind through acts of comparison. Grossmann advocates the perceptibility of relations. Otherwise, he could not hold that structures are perceived. He subscribes to the principle that all categories of an ontology must be empirically given in at least one case. Structures are crucial in Grossmann's ontology. He analyses all physical things as spatial and temporal structures. Moreover, universals are ordered, according to him, by relations into dimensions and dimensions are categorised as structures. There are furthermore structures in Grossmann’s ontology which he calls “descriptions” and which are in his view denoted by “description expressions”. Unlike Frege Grossmann assumes that description expressions are not ways of representation but refer to what is represented. That supports his epistemological realism. This epistemological position is also supported by Grossmann's third category of complexes, namely, the category of states of affairs. He holds that

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each cognitive state is related by the intentional relation to a certain state of affairs and that its truth depends on the existence of that state of affairs. Meinong’s and Grossmann’s breakthrough to epistemological realism hinges on taking objects of cognition to be complexes, rather than as simple objects, as traditionally conceived. See also > Collectives and Compound, Gestalt, Meinong, Segelberg, Structure. References and further readings

Grossmann, R., 1973, Ontological Reduction. Part Three, Bloomington: Indiana University Press Grossmann, R., 1974, Meinong, London: Routledge & Kegan Paul. Grossmann, R., 1977, “Structures versus Sets: The Philosophical Background of Gestalt Psychology”, Revista Hispanoamericana de Filosofia 9: 3-12. Grossmann, R., 1983, The Categorial Structure of the World, Bloomington: Indiana University Press, Chap. 5 Erwin Tegtmeier

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H Holes What is a hole? The answer to that question has spawned remarkably different answers, and yet those answers have something in common: they all appeal to mereology. The first systematic attempt to think about holes began with the Lewises (Lewis and Lewis 1970). Their account was a revisionary one, according to which much of what we say about holes turns out to false, because holes are not located where we think they are, nor do they have the dimensions we tend to attribute to them. On their view, holes are to be identified with what they call hole-linings: the relevant part of the surface of an object that we would normally say ‘lines’ the hole. To have a hole is nothing more than to have a surface with a particular topology, and since the surface of an object is, for Lewis and Lewis, a proper part of the object, holes are proper parts of objects. Varzi and Casati disagree with Lewis and Lewis. They want more of what we say about holes and their properties to turn out to be true. Thus they propose that holes are a particular kind of immaterial being, they are immaterial beings that are located at the surface of material objects, and which occupy roughly the regions we

would typically attribute to them in our talk of holes (2004; 1994). What is ontologically noteworthy about holes is that their existence is asymmetrically dependent on the existence of the material beings that “host” them: the material beings that have the holes. Very generally, Varzi and Casati endorse a view according to which for every material body, there exists an immaterial being that shares exactly the same location, and there is an immaterial being or beings that occupies the complement of all of the material beings. Varzi and Casati provide a very detailed mereological account of the relationship between holes and hole hosts (1994; appendix). The most important features of their account are that a hole is not a part of its host or hosts, and its host is not part of it, or indeed, part of any hole. Instead, a hole that is spatially contiguous with the external complement of the material beings is part of that complement, while a hole that is not contiguous in this way is part of neither the complement nor of the hole host (Varzi and Casati 1994 p 36). Immaterial beings turn out always to be parts of other immaterial beings, but their existence at the surface of some material beings makes true our everyday claim that that object has a hole even though the hole is not part of the object. Hoffman and Richards agree with Varzi and Casati about the rough location of holes. But they disagree about the mereological characterisation of the relationship between holes and hosts. They raise the idea that

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objects might have both positive and negative parts, with holes being negative parts of the objects that host them. Positive parts are those that are bounded by negative extrema of a principle curvature, and negative parts are those bounded by positive extrema of a principle curvature. That is, roughly, positive parts of an object are ones that are bounded by what would be, if smoothed, a concave contour, negative parts are ones that are bounded by what would be, if smoothed, a convex contour. Since Hoffman and Richards are interested in the ways in which our perceptual systems in fact divide objects into ‘natural’ parts, their account remains silent on any further details. For instance, it is unclear whether there is any metaphysical difference between negative and positive parts, or whether the difference is purely topological. Ultimately, perhaps the best way to think of this view is not that the hole is part of the hole-host, but rather, that the hole and the hole-host jointly compose a material object, and that material object has a hole in virtue of having some negative part. Finally, Miller has defended the view that holes are material beings that are located at the surfaces of material hole-hosts. On this view, every actual region is filled, and every filled region contains a material being, and for any two material beings, there exists a mereological sum of those beings. Then the question of whether the hole is part of the hole-host, or part of the complement of hole-hosts (or material beings) is viewed as a purely semantic matter. For any holehost and hole on its surface, there is a

mereological sum of the two objects, S, which has each of them as proper parts. There is also the sum S* of the complement of the hole-host, and the hole itself, of which the hole is a proper part. So the hole is neither part of the hole-host, nor part of the complement. Rather, it is part of both the sum S, and the sum S*. It is a semantic matter whether to then construe that, in ordinary language, something has a as the claim that the hole is part of the host or part of the complement. That is, it is a debate about whether when we refer to the hole host, we are referring to S, or when we refer to the complement, we are referring to S*. See also > Cosmology, Mereotopology, Privation, Shadows. Bibliographical remarks

Casati, R; Varzi, A. C., 2004. This is a nice overview of the debate that is very accessible. Casati, R.; Varzi, A. C., 1994. This offers an in depth account of the nature of holes including particularly with respect to providing mereological axioms as they apply to holes. Balashov, Y., 1999. A more difficult paper that considers whether there can be absences. Jackson, F., 1977. Defends an eliminativist view about holes. Lewis, D. K.; Lewis, S. R., 1970. The original introduction to holes. Miller, K., 2007. A new account of the nature of holes.

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References and further readings

Balashov, Y., 1999, “Zero-Value Physical Quantities”, Synthese 119(3): 253-286. Casati, R.; Varzi, A. C, 1994, Holes and Other Superficialities, Cambridge (MA): MIT Press. Casati, R.; Varzi, A. C., 2004, “Counting the Holes”, Australasian Journal of Philosophy 82(1): 23-27. Hoffman, D.; Richards, W., A., 1984, “Parts of Recognition”, Cognition 18: 65-69. Jackson, F., 1977, Perception: A Representative Theory, Cambridge: Cambridge University Press. Lewis, D., K.; Lewis, S., 1970, “Holes”, Australasian Journal of Philosophy 48: 206-212. Miller, K., 2007, “Immaterial Beings”, The Monist 90(3): 349-71. Kristie Miller

Homeomerous and Automerous The ancient Greek adjective ‘ὁµοιοµερής’ (homoeomerēs) – literally: ‘of like parts’ – and the associated noun ‘ὁµοιοµέρεια’ (homoeoméreia) – literally: ‘likepartedness’ – seem to have their first systematic use in Aristotle. Mourelatos (1998, p.336f) notes, however, that both “occur – probably tendentiously, under the influence of Aristotle’s usage – also in our ancient sources for a pre-Aristotelian philosopher, Anaxagoras of Clazomenae, with reference to the constituent “things” (chrema-

ta) involved in the latter’s scheme of universal mixture”. The Greek terms are transliterated into English as 'homoeomerous' (or simplified as 'homeomerous' or even 'homomerous') and 'homoeomery' (or 'homeomereity'), respectively. Aristotle does not offer an explicit definition of homeomereity. But the central elucidations of the term put focus on the synonymy of whole and parts: (H0) A whole is homeomerous just in case its parts “have the same name” as the whole (Parts of Animals, 655 b 23). Homeomereity thus states a semantic relationship between a whole and its parts that is inferentially relevant. Besides this logical sense, the term ‘homeomerous’ also has a complex systematic meaning within Aristotle’s theory of matter – “the homeomerous bodies are made up of the elements, and all the works of nature in turn of the homeomerous bodies as matter” (Meteorology, 389 b 26). Aristotle observes that the kinds of parts of a substance depend on the perspective of division; we can consider an animal as a whole composed of elements (fire, water, earth, air), or as a whole composed of “homeomerous parts”, e.g., “flesh, bone, blood”, or finally as a whole composed of “anhomeomerous parts, viz. face, hand, foot” (Parts of Animals 640 b 20). The focus of the present article will be entirely on the logical sense of homeomereity. Homeomereity versus homogeneity.

Version (H0) of the condition of homeomereity can be read in three dif-

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ferent ways. The first and the second reading assume that by ‘having the same name’ we mean ‘being of/belonging to the same (proximate) kind K’, where ‘kind’ is here used to cover both kinds of countable items (e.g., cat) and kinds of stuff (e.g., water). Then we can restate (H0) as: (H1) A whole of kind K is homeomerous just in case all its parts are of the same kind as the whole, i.e., of kind K. Since many of Aristotle’s examples for anhomeomerous wholes have parts that differ from each other – e.g., the face has a nose, mouth, eyes etc. – one could form the impression that what actually matters in (H0) is the following condition of homogeneity: (H2) A whole of kind K is homeomerous just in case all the parts of a whole of kind K are all of kind K*, which may differ from kind K. For example, a whole of the kind pyramid of wet sand is not homeomerous, since none of the parts are of this kind; however, asssuming that we divide no further than a grain of sand, the whole is homogenous, since all parts are of the kind sand. This impression may be reinforced by the fact that two main English translations render Aristotle’s term ‘ὁµοιοµερής’’ mostly as ‘homogeneous’ (J. Barnes) or as ‘uniform’ (A.L. Peck). But much indicates that Aristotle understood the term in the sense of (H1). He states that “Anaxagoras posits as elements the ‘homoeomeries’, viz. bone, flesh, marrow, and everything else which is such that

part and whole are synomymous” (Gen. and Corr., 314 a 20), refers to homeomerous parts of an organism as “parts [such as blood] uniform with themselves” (History of Animals, 487 a 2, 489 a 27), and explains that he discusses “the organs” horn, hoof, nails, claws in the same sections as the homeomerous parts of animals such as blood and marrow since “both the organs and their parts have the same name” (Parts of Animals 655 b 23; see also the comment by Simplicius, in Phys. 551,32-3: “Also every genus is predicated synonymously of all the species, but the whole only of the homoeomeries, and of those not in virtue of being a whole”). Moreover, among Aristotle’s illustrations –“the stuffs that are mined” such as gold, silver, stone, but also what animals and plants are “made of” such as flesh, bones, blood, serum, lard, milk, wood, bark, juice (cf. Metereology 388 a 12-21, Parts of Animals 647 b 10-20, On Plants, 818 a 21 – there is none that would comply only with (H2) but not with (H1). The properties defined by conditions (H1) and (H2), which shall hereafter be called ‘homeomereity’ and ‘homogeneity’, respectively, are not equivalent (see also Mourelatos 1998). Homeomereity implies homogeneity but the converse does not hold. Both homeomereity (H1) and homogeneity (H2) depend on the size of the parts selected for comparison, i.e., on the selected ‘grain size’ or ‘granularity’, in current terminology; accordingly, in both cases we can say that the condition holds to a larger or lesser degree – i.e., that something is

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more or less homeomerous or homogeneous – depending on which parts, larger-sized or smaller-sized, have been selected for comparison. For example, fruitsalad is less homeomerous and less homogeneous than rice. (Note that Aristotle seems to acknowledge degrees of homeomereity or homogeneity in his distinction between the elements and non-elemental matter. Cf. On the Heavens 302 b17: “Not every homoeomerous body can be an element; only…that which is not divisible into bodies of a different form”). But the degrees of homeomereity and homogeneity are often quite different. A wooden cube that is made up of nine wooden cubes is homeomerous precisely for one ‘grain size’, the size of the constituting wooden cubes, while it is homogeneous down to the size of cellulose molecules (discounting boundaries as parts). Moreover, whether something is homeomerous or homogeneous, and to what degree, depends on not only on the spatiotemporal size of the parts selected for comparison, but also on the kind terms used in the comparison. In naïve common sense reasoning – uninformed by ancient theories of elements or modern chemistry – the stuff water is maximally homeomerous and maximally homogeneous since there are no kind terms for stuffs that would allow us to identify parts smaller than the size of the smallest drop of water. In contrast, from a scientific viewpoint, the stuff water is homeomerous only down to the size of the smallest drop of water, but homogeneous down to the size of

an H2O molecule (see ‘Elements’ for adjustments of this latter claim). Homeomereity and homogeneity are typical properties of stuffs, but they may also be exhibited by collectives. For example, we may say that a (large) herd of is made up of (smaller) herds of cattle; as long as these are the only parts considered, the entire herd would be homeomerous (and homogeneous). If we were to count individual cattle also as parts, the herd would be merely homogeneous but not homeomerous. Primary substances in Aristotle’s sense (e.g., a cat or tree) are anhomeomerous and inhomogeneous. A quantity, e.g., 1 liter of water, is anhomeomerous, since no part of that quantity can be the same quantity; but quantities are always homogeneous. Homeomereity versus automereity. In

contemporary linguistic and ontological research homeomereity (in the sense of (H1) or (H2)) has played a role in the semantics of noun phrases (mass terms, terms for collectives), “action types” (see ‘activity’), quantities, and stuffs (see e.g. Bunt 1985, Cartwright 1970, Laycock 1979). In these contexts, authors also often resort not to Aristotle’s term ‘homeomerous’ but to N. Goodman’s notion of ‘dissectivity’ (1951: 53): a dissective predicate applies to all parts of anything to which is applies. It is important to note, however, that while dissectivity is co-extensional with homeomereity in the sense of (H1), the ‘semantic ascent’ of the rephrasal of (H1) as dissectivity is by no means ontologically innocent. Dissectivity is the feature of a predi-

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cate, and expresses how a property is distributed. (Alternatively, to stay with the parlance of nominalistic semantics, dissectivity specifies which filled spacetime regions satisfy a certain predicate expression). In either the Platonist or the nominalist interpretation of predication, the ‘traditional notion’ of individuals as concrete particulars (objects or filled spacetime regions) can be retained. As long as we formulate homeomereity as (H1) or as (H2), we will be prone to understand it as a condition holding of entities – a whole and its parts – understood as concrete bounded particulars with determinate spatial (and temporal) location. However, there is a third reading of condition (H0) that opens up a heuristic path to a new notion of individual; this reading is obscured if homeomereity is assimilated to dissectivity: (H3) A whole is homeomerous just in case what is named when we name the whole also is in all parts of the whole. In combination with Aristotle’s illustrations, (H3) can be read as a statement about the identity or ‘recurrence’ of an entity across spatial regions. To throw the difference to (H1) into stronger relief, condition (H3) has also been called “automereity” (Seibt 2004: 41). (AUT) An entity E is automerous just in case (all of) E exists or occurs in every part of the spatial region R in which E occurs. Automereity – as stated here, for its generalisation see below – is a prop-

erty that says something about the way in which an entity occurs in space – while a heteromerous whole (i.e., a whole all parts of which are of different kinds) like a human body occupies a spatial region, an automerous entity permeates a spatial region. This is the case for stuffs like gold or water but also – as noted in the discussion of “action types” – for activities such as melting or snowing (Mourelatos 1978, Roberts 1979, Seibt 1997). However, the categorisation of E in (AUT) creates a problem that challenges our theoretical habituations in ontology. E cannot be an individual in the traditional sense, viz., a concrete particular entity, since particulars are by definition not recurrent in space. On the other hand, while general entities (universals, types) are by definition entities that can recur in space and time, E cannot be a traditional universal, for two reasons. Universals are taken to be ‘predicable’ in Aristotle’s classification and in some fashion dependent entities that exist only if there is a particular instantiating them (or which they are predicated of). But terms for stuffs (and activities) occur in both predicative and in subject position, displaying an amphibious nature in terms of standard logical grammar. The “protean character” (Quine 1960: 95) of stuff terms has been noted variously in the contemporary discussion but most authors have been following Aristotle’s (Topics 103 a 14-23) strategy of fitting stuffs into the category-theoretic mold of universals, i.e., treating them as general, predicable entities, in order to uphold the

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traditional category dualism of concrete particular individuals and general predicable entities, abstract or concrete, called universals. The difficulties that arise when the ontology of stuffs is forced into the category dualism of traditional ontology (and the standard interpretation of quantificational logic) are investigated in detail in Laycock (2006). Seibt has argued in a series of articles from 1990 onwards that (AUT) is one ‘data point’ among many that could motivate ontologists to shed the “myth of substance” and abandon the idea that we must conceive of individuals as particulars, i.e., as entities that at any time necessarily have a unique spatial location and thus can be individuated (or at least differentiated) in terms of their spacetime locations. By contrast, an automerous entity is individuated in terms of its functionality (in a wide sense of the term) and thus can recur in space or time. Seibt differentiates homeomereity and automereity with respect to spatial and temporal dimension and introduces a basic graduation – minimal, normal, or maximal homeomerity and automerity, respectively – depending on whether none, some, or all parts (here in the sense of ‘proper part or equal’) of a whole E fulfill the required condition. For each of the types of entities countenanced in common sense reasoning – stuffs, things, events, activities, etc. – there is a characteristic “mereological signature of spatiotemporal distribution” that can be spelled out in terms of these ontological predicates, holding out the prospect for a monocategoreal ontology based on concrete non-

particular individuals (see e.g. Seibt 2015). (AUT) is an inferential principle that governs mass terms (and gerunds of activity verbs) of Indo-European languages. Since in these languages “count nouns” such as ‘dog’ or ‘chair’ dominate (i.e., nouns for entities that have discrete boundaries, do not overlap spatially, and thus can be identified by their location), speakers of Indo-European languages might thus be cognitively more disposed to tie individuality to particularity. However, other languages, especially classifier languages, seem to prime the cognitive dispositions of speakers more towards an understanding of individuality that is oriented toward an item’s materiality or functionality (Lucy 1992). Thus an ontology based on non-particular individuals may be more suitable as a general framework for the ontological interpretation of concepts (i.e., inferential roles of expressions) in a wide variety of languages (see e.g. Seibt 2015). Automereity and multilocation. Au-

tomereity as formulated in (AUT) is not the claim that something has itself as a part or that an entity E could wholly occur in one of its parts – these are paradoxical claims that are in conflict with the irreflexivity of the (proper) part relation. Temporal varieties of these paradoxical claims have been taken to follow from the endurance account of persistence (Barker & Dowe 2003); however, while the endurance account is a not committed to assertions about relationships between temporal parts and wholes (Ruse & Beebee 2003), pro-

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ponents of endurance owe an explanation of how enduring entities exist multiply in time or are ‘multilocated.’ Similarly, proponents of automereity should explain not only which category of entity is multilocated in space or time, but also how to conceive of such multi-locatedness or recurrence. A mereological explanation of multi-location can be given as follows, assuming clear distinctions between various senses of parts (spatiotemporal, material, functional, constructional, etc.). Assume that every entity E is defined in terms of its functional parts (taking functionality in the widest sense). Assume further that every filled spatiotemporal region SE, which is either a region occupied or permeated by E, is the whole of two functional parts: the extensional region R and the ‘filling’ E. If E is automerous, i.e., if E permeates SE, then each of the spatial or temporal parts of R of SE, namely, filled regions riE, has E as a functional part. But that E is a functional part of a spatiotemporal region (i.e., that E is what is ‘going on’ in that region, constituting the region’s individuality) does not imply that E is a spatiotemporal part of that region. The distinction between spatiotemporal or extent parts and functional parts blocks the paradoxes of multilocation set out by (Barker & Dowe 2003; 2005); these paradoxes capitalize on the narrow reading of ‘part’ as spatiotemporal parthood, and proceed from the assumption that locatedness is a matter of being a spatiotemporal part of a region. But an ontology that countenances automerous entities can define location in other ways, com-

bining functional and spatiotemporal parthood (Seibt 2008). We can say that an entity E is multilocated just in case (i) there are filled spatiotemporal regions S and S*; and (ii) S has as functional parts E (‘filling’) and R (extensional region), while (iii) S* has as functional parts E and R*, and (iv) R and R* are different since they have different spatiotemporal parts. Further kinds of ‘mereities.’ Homeo-

mereity, homogeneity, and automereity are theoretical predicates stating relationships of sameness – sameness in kind, or identity – among concrete entities and their parts. This definitional schema not only can be worked out, as mentioned above, by differentiating dimensions and introducing degrees, it also can be varied by requiring not sameness but relationships of similarity, difference, causal relatedness, or dependence etc. among (the kinds of) parts and entities involved. To give two tentative examples, one might define ‘heteromereity of degree d’ in terms of a similarity relation defined on kinds {K1…Kn} in order to classify mixtures or collections. Or again, one might characterise wholes with spatial or temporal patterns (e.g., repetitive molecules or a clockwork mechanism, respectively) in terms of their ‘cyclomereity’ in space or time: an entity E is cyclomerous in space (time) just in case (i) the spatial (temporal) parts P1…Pm of E are of kinds K1…Kn and form a linear spatial (temporal) sequence S of length m with a repetition index k, so that for any part Pi in this order, if Pi is of kind Ki then Pi+k is of kind Ki.

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See also > Activities, Aristotle’s Theory of Parts, Common-Sense Reasoning about Parts and Wholes, Continuants and Occurrents, Elements, Granularity, Persistence, Transitivity. References and further readings

Barker, S.; Dowe, P., 2003, “Paradoxes of Multi-Location”, Analysis 63: 106-14. Barker, S., Dowe, P., 2005, “Endurance is Paradoxical”, Analysis 65: 69-74. Beebee, H.; Rush, M., 2003, “Nonparadoxical Multi-location”, Analysis 63: 311–17. Bunt, H., 1985, Mass Terms and Model Theoretic Semantics, Cambridge: Cambridge Univ. Press. Cartwright, H., 1970, “Quantities”, The Philosophical Review 79: 25-42. Dowty, D., 1979, Word Meaning and Montague Grammar, Dordrecht: Reidel. Galton, A., 1984, The Logic of Aspect: An Axiomatic Approach, Oxford: Oxford University Press.

the Linguistic Relativity Hypothesis, Cambridge: Cambridge University Press. Mourelatos, A. 1978, “Events, Processes, and States”, Linguistics and Philosophy 2: 415-434. Reprinted with corrections in Tedeschi P. J.; Zaenen, A. (eds.), Tense and Aspect, New York, Academic Press, 1981: 191-212. Mourelatos, A., 1998, “Homoeomereity”, in: Audi, R. (ed.) The Cambridge Dictionary of Philosophy, Cambridge University Press, 336337. Quine, W.v.O., 1960, Word and Object, Cambridge, MA: MIT Press. Rapp, C., 1992, “Ähnlichkeit, Analogie und Homonymie bei Aristoteles”, Zeitschrift für philosophische Forschung 46: 526-544. Roberts, J. H., 1979, “Activities and Performances Considered as Objects and Events”, Philosophical Studies, 35: 171-185. Seibt, J., 1995, “Individuen als Prozesse: Zur prozess-ontologischen Revision des Substanzparadigmas”, Logos 2: 352-384.

Laycock, H., 1979, “Theories of Matter”, in Pelletier 1979: 89-120.

Seibt, J., 1997, “Existence in Time: From Substance to Process“, in: Faye, J.; Scheffler, U.; Urs, M. (eds.), Perspectives on Time. Boston Studies in Philosophy of Science, Dordrecht: Kluwer, 143-182.

Laycock, H., 2006, Words Without Objects – Semantics, Ontology, and Logic, Oxford University Press.

Seibt, J., 2004, “Free Process Theory: Towards a Typology of Processes”, Axiomathes 14: 23-57.

Lucy, J., 1992, Grammatical Categories and Cognition: a Case Study of

Seibt, J., 2005, General Processes – A Study in Ontological Category

Kearns, S., 2011, “Can a Thing be Part of itself?”, American Philosophical Quarterly 48: 87-93.

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Construction, Habilitationsschrift at the University of Konstanz. Chapter 2 and 3 republished as Activities, Berlin: DeGruyter (forthcoming 2018). Seibt, J., 2015a, “Ontological Scope and Linguistic Diversity: Are There Universal Categories?”, The Monist 98: 318-343. Johanna Seibt

Husserl, Edmund Edmund Husserl (1859-1938) studied philosophy under Franz Brentano and was professor of philosophy in Göttingen and Freiburg. As founder of the school of phenomenology he established new standards in philosophy and counts as one of the most influential philosophers of the 20th century. His major works are Logische Untersuchungen (19001901), Ideen I (1913), Formale und transzendentale Logik (1928), and Meditations cartesiennes (1931). A. Reinach, R. Ingarden, and H. Conrad-Martius are Husserl’s students, but he also greatly influenced several independent thinkers like M. Heidegger, M. Scheler or N. Hartmann. When he announced the program of ‘transcendental idealism’, many of his followers turned away, much to his disappointment. Husserl refers to Twardowski’s theory of parts and wholes, trying to give it a more formal exposition by introducing definitions and theorems and improving on many formulations. In the IIIrd Logical Investigation, enti-

tled “On the theory of wholes and parts”, Husserl presented his provisional reflections on the topic but never elaborated these more fully afterwards; in several places he stressed however that the outlined theory has great importance for phenomenology. Analysis of the notions of independence and dependence. In accordance

with Twardowski’s distinction of physical and metaphysical parts Husserl introduces his notions of ‘pieces’ (Stücke) and ‘moments’ (Momente). Pieces are parts in the common sense of the term and moments are abstract components of a whole. ‘Each part that is independent relatively to a whole W we call a Piece (Portion), each part that is dependent relatively to W we call a Moment (an abstract part) of this same whole W.’ And further: “Abstract parts can in their turn accordingly have pieces and pieces – abstract parts”. “It makes no difference here whether the whole itself, considered absolutely, or in a relation to a higher whole, is independent or not” (B 266-7 – the pagination of 2nd German edition, quotations from J. N. Findlay’s English translation). To avoid the undesirable and counterintuitive effect of an endless multiplication of parts which affected Twardowski’s theory, Husserl does not regard every relation between parts of a whole as another part of this whole. In particular, all logical or formal relations between parts, like difference or dependence, are no longer regarded as parts of a qualitatively characterised whole. He proudly announces “advances which our notion [of a whole] promises as re-

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moving […] a difficulty in the theory of wholes. This concerns the endlessly complicated part-relations […], [a view based on the opinion that] wherever two contents form a real unity, there must be a peculiar part […], that binds them together” [B 280]. Similarly: “Our conception avoids these endless regresses of parts which are always splitting into further series. Nothing really exists – in a sense of being a possible object of sense perception – beyond the aggregate of a whole’s pieces, together with the sensuous forms of unity which rest on these pieces conjointly [so called moments of unity]” [B 281] The distinction between piece and moment ties in with the definitions of dependence and its negation – independence. A content x is dependent [unselbständig] just in case it is not possible that x exists separately. “The content is by its nature bound to other contents, it cannot be, if other contents are not there together with it […] they form a unity with it.” [B 236]. The species/genus relationship of a content supplementing a given one must be determinate: nothing can be dependent on something else without determining this complement. This claim provokes a question about contents (objects) whose dependencies may ‘alternate’, as in the case when somebody’s basic needs (say, power) can be fulfilled by something else (e.g. sexual fulfillment). Moreover, nothing is dependent on an individual. If color is dependent on extension (since a colored object has to be extended), any particular shade depends on there being some exten-

sion, but not any particular extended item. Husserl writes: “Dependent objects are objects belonging to such pure Species as are governed by a law of essence to the effect that they only exist (if at all) as parts of more inclusive wholes of a certain appropriate Species” [B 240]. Necessary connections like that between color and extension are, in Husserl’s opinion, of a synthetic a priori nature—he claims that the essence of a color does not contain relational connotations. He distinguishes such synthetic connections from purely analytic ones as, e.g., the necessity of being a trilateral for a triangle, the necessity of having parts in case of being a whole, etc. A particular and very important kind of a dependent part is a so-called moment of unity (Einheitsmoment): “we mean by it a content founded on a plurality of contents, and on all of them together and not on some of them simply” [B 281]. These moments help to resolve the chief problem of a part-whole theory that remained unsolved by Twardowski, namely, the unification into one whole of several independent elements. Husserl proposes that moments of unity that are founded collectively on independent pieces connect these into whole. In a case of a table its top and legs are objects able of independent existence but unified by a shape which they make in the proper configuration. This shape is their common moment of unity – it cannot exist without all of them. Six ‘theorems’ of parts and wholes.

In order to reconstruct Husserl’s outline

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of a theory of parts and wholes I shall use the standard language of the Ist order predicate logic and additional abbreviatory symbols defined as follows: x Boolean Algebra, Metamathematics of Mereology, Mereotopology, Philosophy of Mathematics, Topology.

Forrest, P., 1996, “From Ontology to Topology in the Theory of Regions”, The Monist 79(1): 34-50.

Bibliographical remarks

Johnstone, P.T., 1984. Succinct introduction into pointless topology for mathematicians. Mac Lane, S., Moerdijk, I. 1992. Basic text on pointless spaces (locales) and category-theoretical generalisations of points. Ridder, L., 2002. A good source for questions concerning the role of points in mereology and related disciplines such as mereotopology. Russell, B., 1914. Classical philosophical text advocating the elimination of points in favor of events, lacking mathematical precision however. Tarski, A., 1956. Classical source of pointless geometry. References and further readings

Bennett, B.; Düntsch, I., 2007, “Axioms, Algebras and Topology”, in Aiello, M.; Pratt-Hartmann, I. E.; van Bentham, J. F. A. K. (eds.), Handbook of Spatial Logic, Springer, 99159.

Clarke, B. L., 1985, “Individuals and Points”, Notre Dame Journal of Formal Logic 26: 61-75.

De Laguna, T., 1922, “Point, Line, and Surface, as Sets of Solids”, Journal of Philosophy 19: 449-461. Gerla, G., 1995, “Pointless Geometries”, in Buekenhout, F. (ed.), Handbook of Incidence Geometry, Amsterdam: Elsevier. Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., 2003, Continuous Domains and Lattices, Cambridge: Cambridge University Press. Heath, T. L. (ed.), 1956, Euclid. The Thirteen Books of the Elements, Second Edition, New York: Dover Publications. Hilbert, D., 1899 (2001), Foundations of Geometry, Chicago and LaSalle: Open Court. Hilbert, D., 1935, Gesammelte mathematische Abbhandlungen III, darin (von Otto Blumenthal), Lebensgeschichte, 403: Springer. Johnstone, P. T., 1984, “The Point of Pointless Topology”, Bulletin (New Series) of the American Mathematical Society 8: 4-54. Mac Lane, S.; Moerdijk, I. 1992, Sheaves in Geometry and Logic, A

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First Introduction to Topos Theory, New York: Springer. Mormann, T. 1998, “Continuous Lattices and Whiteheadian Theory of Space”, Logic and Logical Philosophy 6: 45-54. Ridder, L., 2002, Mereologie. Ein Beitrag zur Ontologie und Erkenntnistheorie, Frankfurt/Main: Vittorio Klostermann. Roeper, P., 1997, “Region-Based Topology”, Journal of Philosophical Logic 26: 251-409. Russell, B., 1914 (1995), Our Knowledge of the External World as Field for Research in Scientific Philosophy, London: Routledge. Stone, M. H., 1936, “The Theory of Representations for Boolean Algebras”, Transactions American Mathematical Society 40: 37-111. Tarski, A., 1956, “Foundations of the Geometry of Solids”, in Tarski, A., Logic, Semantics, and Metamathematics, Oxford: Clarendon Press. Whitehead, A. N., 1929, Process and Reality: An Essay in Cosmology, New York: Macmillan. Vickers, S., 1989, Topology via Logic, Cambridge: Cambridge University Press. Thomas Mormann

Possession and Partitives Investigations of part-whole relations from a linguistic perspective can focus on one of three different aspects:

- on X as being a part of Y – see (1a), - on Y as having X as a part – see (1b), or - on the relation itself between X and Y – see (1c). (1) (a) the man’s blue eyes (b) the man with the blue eyes (c) The man has blue eyes. Although languages have a wide range of constructions for encoding these meanings, hardly any of those is dedicated exclusively to the encoding of part-whole relations. Thus, the examples in (2a–c) are structurally parallel to those in (1a–c), but refer to relations of legal ownership rather than to part-whole relations. (2) (a) the man’s big house (b) the man with the big house (c) The man has a big house. In what follows we will concentrate on linguistic encoding of X as part of Y, as used in literal speech (for metonomy and synecdoche see partwhole relation in non-literal language use). Example (1a) illustrates a very common and also the most explicit and direct way of doing this – both the part and the whole are mentioned and directly syntactically related to each other within one and the same noun phrase, where the part is referred to by the head noun (eyes) and the whole by an attribute (the man’s). The conceptual relatedness between the part and the whole is thus iconically reflected in the syntactic relatedness of the corresponding expressions. Noun phrases like (1) are called ‘possessive noun phrases’, since they encode a broad range of relations covered by the typ-

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ical possessive verb like ‘have’ (cf. 1c and 2c), among others, the relation of legal ownership, as in (2a). An interesting and complicated question concerns the relationship between part-whole relations and possessive relations. Although a substantial subset of all possessive noun phrases in a language like English does involve reference to body-parts of animate beings and, more generally, to concrete parts of concrete entities, prototypical part-whole relations (spatial containment) are hardly evoked in such cases as the man’s big house, Mozart’s sonatas or Monday’s performance. It has been suggested that the common semantic (or pragmatic) denominator in the majority of possessive noun phrases is the function of the attributes, i.e. possessors, as anchors (Fraurud 1990), or reference point entities (Langacker 1995) for identification of the head’s referents. In other words, knowing which man we are talking about, we can easily identify his eyes and his house, and the same goes for Mozart’s sonatas and for Monday’s performance. Languages can split up the domain of possessive relations by using different types of constructions for their different subtypes. Co-occurrence of several different possessive constructions in the same language is in fact frequent, but languages differ considerably as to how many such constructions they have, what motivates the choice among those and what relations each of them may cover. For instance, the Semitic language Maltese uses two structurally different constructions as correspondences

to the three structurally similar English examples Peter’s hand (2a) and Peter’s son (2b) vs. Peter’s chair (2c): only the latter involves the preposition ta’ which explicitly marks the relation between the chair and Peter. Maltese (Semitic) (2) (a) id Pietru hand Peter ‘Peter’s hand’ (b) bin Pietru son Peter ‘Peter’s son’ (c) is-siġġu ta’ Pietru DEF-chair of Peter 'Peter's chair' The Maltese distinction has numerous cross-linguistic parallels and is traditionally called the opposition between inalienable (2a, b) vs. alienable possession (2c). There have been many attempts to characterise this opposition in semantic terms, as reflecting some basic difference between typically inalienable concepts and others, e.g. in the way their referents are conceived of. Thus, the connection between a person and his/her body-parts and relatives is inherent, while a connection between a person and his/her chair has to be established in a special way (e.g., by the act of buying). The closer conceptual connection in the former case is also iconically reflected in the absence of the explicit marker ta’ for marking the relation between the head and the possessor. Since partwhole relations seem to play a crucial role for inalienable possession characterised in such a semantic way, there have been further attempts to

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extend part-whole relations to other cases of inalienable possession. A person’s relatives can, for instance, be viewed as belonging to, or being parts of his/her personal sphere (Chappell & McGregor 1996). This solution makes the notion of the whole somewhat elusive. Even more importantly, semantic characterisations of the distinction between inalienable and alienable possession turn out to be too general from a crosslinguistic perspective. The whole distinction seems to originate in historical processes which are only indirectly related to semantic issues. In addition, languages often have different constructions for human possessors expressed by pronouns, proper names and kin terms, or just definite noun phrases, on the one hand, and other types of possible possessors (cf. the boy’s foot vs. the foot of the mountain in English). This distinction is rooted not so much in the semantics of the relations covered by each of the construction types, but rather in the relative easiness with which different entities can provide clues for the identification of other entities. Furthermore, many languages use possessive constructions for expressing non-anchoring relations, in which the nominal dependent classifies, describes or qualifies the class of entities denoted by the head rather than identifies it, cf. a house of stone (material), or a man of honour (quality). In addition to possessive noun phrases, languages have an array of constructions where the part-whole relations are encoded in less explicit

and / or less direct ways. This is particularly common for relations between body parts and the whole person. Body parts are special in that their state and any changes in it have direct relevance for the whole organism. In most contexts, body part terms have a low discourse status: what really matters is not so much the body part as such, but rather the affected person or animal. The task is here two-fold: we talk about X as being a (body) part of Y, but our main concern is Y. Since possessive noun phrases normally solve only the first part of this task, there are special syntactic constructions for these purposes. Example (3a) illustrates one case within the broad and crosslinguistically well-spread class of phenomena called, among other things, external possession (Haspelmath & König 1998, Payne & Barshi 1999), where both the whole and the body part are coded as dependent on the verb, rather than belonging to one and the same noun phrase. Example (3b) gives an idea of another phenomenon, called ‘body-part incorporation’, in which the verb is made more specific by incorporating the expression for a particular body part, while the expression for the whole is directly dependent on the complex verb (Chappell & McGregor 1996). (3) (a) I hit him on the head. (b) I head-hit him. In both these examples the partwhole relation is encoded in a less explicit and direct way than in normal possessive noun phrases. Finally, languages normally have ways of encoding relations between

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parts separated or separable from the whole. The so-called partitive constructions involve expressions for various kinds of parts (fractions) for encoding quantification, e.g., How much does [a slice of this cake] cost?, I need a [quarter of an hour], or Give me a glass of [this good juice]. As the last example shows, the notion of ‘part (fraction)’ is in this case fairly broad. Partitive constructions are semantically and often structurally related to pseudopartitive constructions, e.g., a slice of bread, or a glass of juice. While partitive constructions involve a presupposed set of items or a presupposed entity (the cake, an hour, this good juice) from which a subset or a subpart is selected (a slice, a quarter, a glass), pseudo-partitive constructions merely designate quantification over the kind of entity (‘bread’, ‘juice’) (Koptjevskaja-Tamm 2006; 2009).

Koptjevskaja-Tamm, M., 2009. An overview over the structural types and semantics of partitive and pseudo-partitive constructions in the European languages

See also > Body Parts, Grammar, Non-Literal Language Use and PartWhole Relations, Syntax.

Fraurud, K., 1990, “Definiteness and the Processing of Noun Phrases in Natural Discourse”, Journal of Semantics 7: 395-433.

Bibliographical remarks

Chappell, H. & McGregor, W. eds., 1996. A collection of insightful papers on the cross-linguistic variation in how languages encode relations between body parts and the whole body and other part-whole relations, Koptjevskaja-Tamm, M., 2002. An overview over the structural types and semantics of possessive constructions in the European languages.

McGregor, W., 2009. A collection of nine original articles on the expression of possession at various levels of grammar, morphological, phrasal, and syntactic, and from a typologically diverse range of languages. Taylor, J. R., 1996. A detailed semantic analysis of English possessive constructions within a coherent theoretical framework (Cognitive Grammar). References and further readings

Chappell, H.; McGregor, W. (eds.), 1996, The Grammar of Inalienability: A Typological Perspective on Body Part Terms and the Part-Whole Relation, Berlin: Mouton de Gruyter.

Haspelmath, M.; König, E., 1998, “Les constructions à possesseur externe dans les langues de l’Europe”, in J. Feuillet, ed. Actance et valence dans les langues de l’Europe. Berlin: Mouton de Gruyter, 525-606. Heine, B., 1997, Possession. Cognitive Sources, Forces, and Grammaticalization, Cambridge: Cambridge University Press. Hoeksema, J. (ed.), 1996, Studies on the Syntax and Semantics of Partitive

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and Related Constructions, Berlin/ New York: Mouton de Gruyter. Koptjevskaja-Tamm, M., 2002, “Adnominal Possession in the European Languages: Form and Function”, Sprachtypologie und Universalienforschung 55: 141-172. Koptjevskaja-Tamm, M., 2006, “Partitives”, in K. Brown, ed., Encyclopedia of Languages and Linguistics, Second Edition, Oxford: Elsevier, 218-221. Koptjevskaja-Tamm, M., 2009, “A Lot of Grammar with a Good Portion of Lexicon”: Towards a Typology of Partitive and Pseudo-partitive Nominal Constructions. In Helmbrecht, J., N. Yoko, S. Yong-Min, S. Skopeteas & E.Verhoeven (eds.), Form and Function in Language Research. Berlin: Mouton de Gruyter, 329-346. Langacker, R. W., 1995, “Possession and Possessive Constructions”, in Taylor J. R.; MacLaury, R. E. (eds.), Language and the Cognitive Construal of the World. Berlin/New York: Mouton de Gruyter, 51-79. McGregor, W., 2009. The Expression of Possession, Berlin: De Gruyter Payne, D. L.; Barshi, I. (eds.), 1999, External Possession. Amsterdam/ Philadelphia: John Benjamins. Seiler, H., 1983, Possession as an Operational Dimension of Language, Tübingen: Gunter Narr Verlag. Taylor, J. R., 1996, Possessives in English: An Exploration in Cognitive Grammar, Oxford: Clarendon Press. Maria Koptjevskaja-Tamm

Powers Dispositions or powers are ascribed to objects, substances and persons. An object can be fragile, a substance soluble and a person moody. For much of the twentieth century, empiricist philosophers struggled to understand dispositions in terms of conditional relations between events rather than accepting dispositions as a class of properties in their own right (Ryle 1949). In recent years, however, realists about dispositions have offered an alternative to the empiricist event ontology in which powers can be basic and irreducible and can account for many other troublesome metaphysical phenomena such as causes, laws, modality and properties (Bird 2007, Molnar 2003 and Mumford 1998; 2004). Powers can be instantiated in everyday macroscopic objects, substances and persons but also in their component parts. It seems, therefore, that powers can stand in part-whole relations though it is not clear that these are the traditional part-whole relations of standard mereology. Relatively macroscopic powers can be instantiated in virtue of relatively microscopic powers. The larger-scale powers are thought of as derivative or non-fundamental. Their component powers might themselves be derivative as well, having even lowerlevel powers as their own constituents. It is an open philosophical question whether there are any absolutely fundamental, non-derivative powers, but the powers of the sub-atomic particles, the smallest known components of anything, are often thought

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to be fundamental. The spin, charge and mass of arguably structureless particulars would be fundamental powers. Component powers can combine to produce higher-level powers that are not to be found at the lower levels. Hence, an airplane has the power of flight. This power derives from various component powers, such as the power of the engine to provide propulsion, the power of the fuel to drive the engine. The engine itself has numerous components, each adding their own powers to the whole. Flight would not be possible without these components, as shown when one or more of them fails. But the power of flight is a power of the airplane as a whole. The engine cannot itself fly unless it is suitably related to the other components, each manifesting their powers. The shape of the wings provided uplift but they could not fly on their own without the forward thrust the engines contribute. Hence, there will be many constituent powers working together to produce a power of flight that is not to be found among the constituent powers themselves. So far it looks as though powers can stand in some sort of mereological relation because it seems they can stand to each other as parts and wholes. That is so. However, mereology standardly would commit to: A. For any collection of parts, there is only one whole that they can compose. B. If a collection of parts exists, so does the whole composed of them.

A and B do not seem to hold true if the wholes we are talking of are composed powers. For the powers to come together to form a higher-level power, some further, non-mereological relation will have to be involved. For example, in an aircraft hangar, I may have all the component parts of an airplane, each of them in good working order. The parts taken as a simple mereological whole, however, do not have the power of flight simply in virtue of being a whole. As a mere collection of parts, they may have their individual powers intact but for them to come together and gain as a whole a higher-level property of flight, they must have the correct arrangement and connectedness. This will usually be a matter of them standing in the correct spatial relations. The fuel pump, for example, must be connected to the fuel tank at one end and the engine at the other. It would be correct to assume that the fuel pump is fully empowered to pump fuel whether or not it is so connected. Whether it actually does so, of course, depends on there being this correct spatial arrangement. This shows what is crucial about powers: that they can exist unmanifested and in the absence of their appropriate stimuli. From this, we can conclude that the correct combination of powers to form higher-level powers is not a matter of pure mereology but requires some other suitable enabling relation. This is sufficient to show that B does not hold if the parts and wholes in question are powers. The component powers are not alone sufficient to ensure the higher-level

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powers that can be composed from them. But it is also arguable that A fails to hold: there is not just one single higher-level power that can be made from a collection of component powers. There is at least the logical possibility of taking the same component powers but arranging them in a new way such that they produce a power as a whole that is different from a power that they can have if differently arranged. A clever mechanic, for example, might be able to understand the powers of all the constituent parts of a machine and then rearrange them. A vacuum cleaner might be able to have its motor and other parts rearranged so that it expels instead of sucks air, thus functioning as a makeshift hand-drier. This hand-drier contains all the same component parts, each empowered exactly as they were before the modification. But those parts now produce a power as a whole that exactly those same parts did not have while they stood in a different arrangement. It is not the case for powers, therefore, that there is but a single whole for any collection of constituent powers. The nonmereological component in the composition of powers remains vital, therefore. Given that the composition of powers is not a simple matter of addition, this means that such composition should be understood as a nonlinear matter (Mumford and Anjum 2011). The powers of a whole are not merely the sum of the powers of the parts as the following simple case illustrates. Sodium has the powers to float

on water and to ignite spontaneously. Chlorine is poisonous. Were composition of powers to be only a matter of addition one might think sodium chloride to be a highly dangerous compound. As common salt, however, its powers are relatively innocuous and some salt is essential for human health. See also > Causation, Collectives and Compounds, Dispositions, Emergence, Structure. References and further readings

Bird, A., 2007, Nature’s Metaphysics, Oxford: Oxford University Press. Molnar, G., 2003, Powers, A Study in Metaphysics, Oxford: Oxford University Press. Mumford, S., 1998, Dispositions, Oxford: Oxford University Press. Mumford, S., 2004, Laws in Nature, Abingdon: Routledge. Mumford, S.; Anjum, R. L., 2011, Getting Causes from Powers, Oxford: Oxford University Press. Ryle, G., 1949, The Concept of Mind, London: Hutchinson. Stephen Mumford

Praedicabilia At the end of the 12th Century, when the term praedicabile began to be used as what is or is not suitable to be predicated of many, the logicians of that time faced the task to establish,

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in conformity with Porphyry (c. 233 305), five items that expressed the different ways through which the predicates could be predicates. These are: property, accident, genus, species and differentia. In Isagoge, property is seen by Porphyry as that which belongs to only one species and from that species it is predicated convertibly, although it does not express its essence. Thus, the ability to laugh, for instance, is a property of man, as the ability to neigh is the property of a horse. In this work, accident is variously described as that which can belong or not to something or as that which comes and goes without the destruction of its subject. According to Porphyry, an accident can be separable, as in the case of a man and a horse, or inseparable as with black in relation to a crow or an Ethiopian. On the other hand, genus is that which is predicated in reply to the question ‘What is it?’ (Quid est) of many specifically distinct items. Consequently, the genus of man as well as of an ox or horse, for example, is animal. Additionally, species is that which is predicated in reply to the question ‘What is it?’ of many numerically distinct items such as in the case of man, that whilst species, must be predicated of Socrates and Plato. Ultimately, differentia is described as that which is predicated in reply to the question ‘What is it like?’ (Quale est) of many specifically distinct items. There are at least two ways through which the differentiae may be distinguished. Some of them, given that they divide the genus, are called divisive differentiae, whereas others, given that they con-

stitute a species, are called constitutive differentiae. However, there are situations when the same differentia simultaneously divides a genus and constitutes a species. E.g. rational is at the same time dividing the genus animal and constituting the species man. In spite of the huge controversy regarding the ontological status of the predicables – which to some would designate only linguistic predicates, while to others they would also apply to extralinguistic universals – a major part of medieval logicians recognised at least three basic criteria through which each predicable could undeniably be characterised. Based on the principle: ‘For some x, if x is S, then x is P and if x is P, then x is S’, the criterion of conversion or reciprocal implication between subject and predicate allows us to establish if a specific predicable is predicated convertibly or not. Another no-less relevant criterion operates with the distinction between predicating something in respect to what the subject is and predicating something in respect to what characteristics the subject has, so as to specify which predicables answer the question ‘What is it?’, and which answer the question ‘What is it like?’ Finally, there is also the criterion that enables to clarify whether a predicable to express a certain characteristic that is or fails to be contained in the essence of the respective subject, is either essentially or merely accidentally predicated of it. Historically, there have been at least two influential accounts of the division of predicables. During the first half of the 14th century, Walter Burley (c. 1275 - 1344) and others

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defended the view that all predicables would be predicated either in quid, i.e. in reply to the question ‘What is it?’ or in quale, i.e. in reply to the question ‘What is it like?’ Among the predicables that would be predicated in quid, species would differentiate themselves from the genera in that they would constitute the essence of the respective individuals. Those, however, that would be predicated in quale would be predicated essentially, to result in a differentia, or they would be predicated accidentally to be identified either as properties, in case they were convertibly predicated, or as accidents, in the opposite case. Other logicians such as John Buridan (c. 1292 - 1363) questioned the legitimacy of such division and proposed to adopt another instead. According to them, the predicables would be predicated denominatively or essentially. Whereas the first would include what is convertibly predicated, i.e. the accident, the latter would be divided into predicables to be predicated in quale – as the differentia – and into predicables to be predicated in quid, which would include both the genus and the species.

was very influential until the first half of the 12th century.

See also > Medieval Mereology, Nominalism, Universal.

Buridan, J., 1995, Summulae de praedicabilibus, ed. by L.M. de Rijk, Nijmegen: Ingenium.

Bibliographical remarks

Porphyry, 1887. Work that served as basis for the medieval discussions regarding predicables. Boethius, 1906. The treatment of the predicables in Boethius’ Second Commentary on Porphyry’s Isagoge

Buridan, J., 1995. A commentary on the Tractatus of Peter of Spain, who defends a nominalist approach to the predicables. Burley, W., 2004-5. A realistic, early-14th century account on the predicables. Peter of Spain, 1972. It’s section on the predicables was widely used and commented until the 16th century. References and further readings

Baumgartner, H. M.; Kolmer, P., 1972, “Prädikabilien, Prädikabilienlehre” in Ritter, J. (ed.), Historisches Wörterbuch der Philosophie. Basel: Schwabe, Band 7, 1179-1186. Boethius, 1906, In Porphyrii Isagogen commentarium editio duplex, Corpus Scriptorum Ecclesiasticorum Latinorum XLVIII. Vienna: Tempsky. Brunschwig, J., 1986, “Sur le système des ‘prédicables’ dans les Topiques d’Aristote” in Energeia. Études aristotéliciennes offertes à Mgr Antonio Jannone. Paris: Vrin, 145-157.

Burley, W., 2004-5, Expositio super Universalia Porphyrii (1337), in M. Vittorini, Predicabili e categorie nell'ultimo commento di Walter Burley all'Isagoge di Porfirio. University of Salerno, Ph.D. diss., 429-495.

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Dahlstrom, D., 1980, “Signification and Logic: Scotus on Universals from a Logical Point of View”, Vivarium 18: 81-111. De Libera, A., 1996, La querelle des universaux. Paris: Éditions du Seuil. De Libera, A., 1998, “Introduction”, in A. de Libera & A.-P. Segonds (trad.), Porphyre Isagoge, Paris: J. Vrin, VII-CXLII.

Privation Two basic conditions for wholes to be deprived of parts. The basic mereo-

logical intuition underlying the notion of privation concerns missing parts of wholes. There are, however, some restrictions concerning which parts are eligible for privation.

Henry, D. P., 1982, “Predicables and Categories”, in Kretzmann, N.; Kenny, A.; Pinborg, J. (eds.), The Cambridge History of Later Medieval Philosophy, Cambridge: Cambridge University Press, 128-142.

Aristotle (Metaphysics, 1022 b 22-3; 1046 a 34-5; Categories, 12 a 31-4) and the commentators (Simplicius In Aristotelis Physicorum libros commentaria, p. 821; Simplicius In Aristotelis De caelo, pp. 101-2; Alexander of Aphrodisias In Aristotelis Metaphysica commentaria, p. 253) restricted privation only to missing parts for which it would be natural (pephykota) to be possessed by the whole that actually lacks them. Aristotle regarded privations, i.e. missing parts in the above sense, as missing capacities. In Categories, 11 b 17-9, he opposed privation to (available) capacity (hexis – the very word stems from echein: to have). In this sense, the occurrence of a privation in a whole disables a capacity of the kind that it would be natural for the whole to possess (ibid, 12 b 41-13 a 13).

Peter of Spain, 1972, Tractatus called afterwards Summulae logicales, Assen: Van Gorcum.

Thus, the first Aristotelian condition for a whole to be said to be deprived of some parts/capacities is that

Porphyry, 1887, Isagoge, Berlin: G. Reimer.

1. it would be natural for the whole to possess these parts/capacities.

Stump, E., 1988, “Categories and Predicables” in Stump, E. (trad.), Boethius’ In Ciceronis Topica, Ithaca: Cornell University Press, 244-255.

In condition 1, the reference to parts/capacities which are natural for the whole to possess, may not be understood as to the parts/capacities which are essential to the whole. The whole can in essence ‘survive’ the loss of the parts/ capacities men-

de Rijk, L. M., 2002, “The Categories and the ‘Paradigms of Assignment’ (Predicables)” in L.M. de Rijk, Aristotle: Semantics and Ontology Vol. I, Leiden: Brill, 476-498. Evangeliou, C., 1985, “Aristotle’s Doctrines of Predicables and Porphyry’s Isagoge”, Journal of the History of Philosophy 23: 15-34. Gambra, J., 1988, “La lógica aristotélica de los predicables”, Anuário Filosófico 21: 89-118.

Guilherme Wyllie

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tioned in condition 1. I.e. the second condition of privation is that 2. the parts/capacities of which a whole is deprived, may not be essential to the whole. Some examples given by Aristotle Metaphysics, 1022 b 25-7 to fulfil conditions 1 and 2 are: Blind humans, and this includes babies who are blind since birth, are said to be deprived of sight since the capacity to see is natural for them – still not essential to be human. Also moles are said to be deprived of sight. Although this is not the same sense of being deprived as the sense in which blind humans are said to be deprived of sight, it does not refute condition 2. On the one hand, it is natural for moles to be blind qua moles, on the other, it would be natural for moles to be able to see qua animals endowed with eyes. This shows that it is not essential to the mole to be deprived of sight. As one sees, also in the mole example the lost capacity is not essential. An example of a whole which cannot be said to be deprived of its parts because one of the already mentioned conditions does not hold is the heap, in which condition 1 is not fulfilled: a heap of stones cannot be said to be deprived of a stone which happened to fall out of the heap, since it is not natural for the stone to be part rather of this than of any other or of no heap. Already the right use of the privativa (i.e. adjectives like: ‘blind’, ‘naked’, ‘dark’, and also those which in English usually have the prefix ‘un-‘ or

the suffix ‘-less’) shows that natural language prefers talk of privation of parts that are not essential, nevertheless natural for the relevant whole to possess. For example, one does not speak of a wall being blind, since blindness belongs to the indeterminable (aoriston) number of parts/capacities which would be unnatural for a wall to have in the first place (Alexander of Aphrodisias In Aristotelis Metaphysica commentaria, p. 327). Therefore blindness does not form a privation for walls. But, of course, one speaks of humans being blind or sick, blindness and sickness being obviously privations for humans. Every human capacity corresponds to a human privation and the number of names for capacities which are natural for a human to have can be easily determined. Therefore, the number of names for human privations can be easily determined (Alexander of Aphrodisias In Aristotelis Metaphysica commentaria, p. 493). Privation as a cause of transmuting and constructing wholes. Along with

matter and form, in Aristotelian physics (Aristotle Metaphysics, 1070 a 6-9; b 18-9; William of Ockham, Expositio in libros Physicorum Aristotelis, lib. V, cap. I, § 3, vol. 5, pp. 325-6), privation was thought to be one of the principles and, at the same time, one of the causes (principium transmutationis) of alteration of substances. The individual substance: Socrates survived the alteration from curly-and-long-haired to bald. That is, we would not be inclined to say that Socrates was negated existence by losing hair and a new individual

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without hair emerged. This is because a capacity’s (having hair) giving way to the corresponding privation (baldness) does not cause the negation of existence of a substance but the alteration thereof. Generally, any alteration of an individual is caused by the individual’s having possessed something the individual is now deprived of, or by the individual’s having been deprived of something the individual possesses now (cf. Aristotle Physics, 192 a 25-7; 225 b 3-5; Simplicius In Aristotelis Physicorum libros commentaria, 820; Alexander of Aphrodisias In Aristotelis Metaphysica commentaria, 623; Leibniz 1966: 504). For example, healthy organisms become sick by being deprived of a part or a capacity without ending up in being another organism, and sick organisms become healthy by retaining the part or the capacity. Augustine of Hippo (Confessions, lib. VII, cap. 12; Contra epistulam Manichaei quam vocant fundamenti, cap. 35-36; 38 = Patrologia Latina 42, coll. 200-202; 203-204) applies this pattern of thought to ethics. Like ignorance is only lack of education due to the defection (corruptio) of the intellect and disease is lack of health due to the defection of the body, evil is lack of being – a privation – due to the defection of nature. Thomas Aquinas (Quaestiones disputatae de malo, passim) adopted this teaching of Augustine adding, ibid, q. 1, a. 4, ad 10, that not every privation is an evil (i.e. an evil is necessarily a privation but a privation is not necessarily an evil). This explains, according to Thomas, why human inability to do good

works is an evil and entails a penalty while human inability to fly is not an evil and entails no penalty: to human beings, namely, succumbing in the struggle against sin is a privation, non-volatility, however, is not. In the more special case in which matter is deprived of a form, privation was considered to be a cause of the construction of wholes. The fact that bronze is deprived of the statueform causes the making of the bronze statue. If bronze were not deprived of the statue-form, the sculptor would not be able to give this form to it. But a wall’s having no eyes will never cause the wall’s having eyes, since, as mentioned above, walls are not deprived of the eyes which they do not have (Alexander of Aphrodisias In Aristotelis Metaphysica commentaria, pp. 493-4). In the case of the bronze’s being deprived of the statue-form a – rather minor – problem emerges. Bronze is always deprived of the one or the other form: When it is a box, then it is not a statue, a cauldron and so on. When it is a statue, it is not a cauldron, a box and so on. Moreover, when it is a statue of Caesar, it is not a statue of Aristotle, of Bismark, of Claudia Cardinale and so on. The language has not so many names for all these privations. Alexander, ibid, complains that there is no Greek word for a stone’s or the bronze’s being deprived of the form of a statue. The large number of the words needed to express all these privations, should not mislead us to think that the case with bronze being deprived of the statue-form (which is a

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case of privation, although there is no word for it) is parallel to the case of the wall being eyeless (not a case of privation). Alexander maintained, ibid, that, when the matter’s lacking a form can cause someone to construct from this mattera whole endowed with the missing form, then the matter was deprived of the form, even if natural language does not provide a word for this privation. Generally, avoiding to make privation dependent on the lexicalisation of privative predicates is a good idea, since sometimes natural language misleadingly has non-privative expressions to offer which refer to privations. An example for this would be a hand with six fingers. The English expression ‘six-fingered hand’, let alone the Greek expression ‘hexadaktylos’ (the Greek adjective assigns the property in question to the whole human being!) sounds as if the person in question had more than it would be natural for her to have, although in reality her hand is deprived of a certain form (See also Collectives and Compounds). Privation and nominalist mereology.

Conditions 1 and 2 seem to suggest that a mereology which rejects universals and accepts only aggregates, i.e. a nominalist mereology, should eliminate privation. To a mereological nominalist, the construction of wholes like universals is equally arbitrary as the construction of heaps. Since heaps of stones and foxes are arbitrary wholes on this view, it makes no sense to distinguish between the fox’s being deprived of his (missing) tail on the one hand, and

the heap’s not including a stone it formerly did include. Indeed, nominalists usually thought that privation is not embedded in nature (William of Ockham Summa Physicorum, pars I, cap. 10; Gassendi Exercitationes paradoxicae, lib. I, ex. 6, cap. 1). One century after Ockham (and two centuries before Gassendi) antiAristotelianism went as far as to deny the Aristotelian distinction between privation and negation (Valla Retractatio, pp. 114-5). This is nominalist in spirit but, as the following argument shows, it is doubtful whether it can be coherently maintained. Aristotle (Categories 13 a 37-b 21; Metaphysics, 1004 a 14-6) and Simplicius (In Aristotelis Physicorum libros commentaria, p. 820) had thought that there is a clear difference between privation and negation simpliciter. A privation is about a missing part for which it is natural to be possessed by its whole, but a negation is about anything not being there. The missing tail of a tailless fox is a privation. But a human’s missing tail is not a privation since the human had no tail in the first place. I.e. whereas tailless foxes are deprived of their tails, the predicate of tail-having is negated (simpliciter) to humans – at least this is the Aristotelian view. But Aristotle himself mentioned some cases in which privation is not distinguished from negation simpliciter. For example, he spoke of plants as being ‘deprived’ of sight, although there is no sense in which it would be natural for a plant to be endowed with sight (Metaphysics, 1022

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b 23-4). But Aristotle probably thought that this is simply a deviant use of the Greek verb stereisthai (= to be deprived of something). So, it spoils nothing to say that Aristotle withheld the distinction between privation and negation simpliciter in every philosophically relevant context. To break with the Aristotelian distinction between privation and negation simpliciter, one should be a radical nominalist who holds that individuals (in a way very analogous to Spinoza’s substance) do not undergo alterations. What for others would be a minimal inessential alteration of an individual, the radical nominalist would have to regard as the emerging of a new individual. But then, Socrates vanished and a new individual emerged after Socrates lost his hair; the moon vanished and a new satellite of the earth emerged after the youngest lunar crater was formed. A whole’s being deprived of a part would come to the same, then, as the whole’s existence being negated. But as the examples above show, this position is very hard to maintain. Further conditions of privation. Con-

dition 1 will not suffice to give an exhaustive account of privation. For example, one can say that the Venus of Milo in the Louvre is deprived of her arms. But suppose that the arms of the statue are actually resting right now undiscovered under the ground, in the vicinity of the place where the statue was discovered a long time ago – in Milo that is, far away from the Louvre. We would not say, of course, that the arms are deprived of

Venus’s rump. Although the arms do fulfill condition 1: it is natural for them to be connected to the rump since they would not have been constructed at all if they were not about to be parts of the original statue, they are not deprived of the rump, since their losing their rump does not fulfill condition 7 (cf. below): the rump, not the arms, formed the substance of the Venus of Milo after the arms were broken – in fact, their losing the rump does not fulfill condition 2 either (cf. above), but condition 2 presupposes faith in essences and a nominalist would dismiss this anyway. Consequently, it is the rump of the Venus of Milo which is mutilated and deprived of the arms, but the arms are neither mutilated (at least not because of their being divided from the rump) nor deprived of the rump. Aristotle discusses in Metaphysics 1012 a 11-28 at length the conditions, which this special case of privation: mutilation (or amputation – kolobosis in Greek), fulfills. Mutilations and amputations presuppose that: 3. there is a continuous whole (Aristotle remarks that a musical scale is not mutilated if one does not play the last note, insinuating that there is a mutilation if one does not play the last note of a bar in a musical piece). 4. the parts of this whole are more than two (i.e. it is not a mutilation if you take away one of two halves). From 4 follows 4´: 4´. the whole consists of proper parts. 5. the parts are structured in a certain way to form the whole and not arbi-

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trarily (i.e. heaps of stones, water and fire cannot be mutilated – the two last examples are Aristotle’s). 6. an extremity of the whole is lost (Aristotle’s examples: a hole in a jug is not a mutilation, and loss of the spleen is not an amputation). 7. the substance of the whole remains after the mutilation or amputation (Aristotle’s example in Metaphysics 1024 a 15-6, for a lack of a part that does not fulfill condition 7, is number: a number cannot be mutilated, since loss of any of its parts would result in another number. The ‘mutilated’ number would lose its substance. Today we would express the same idea by pointing out that a given number is the set of all those sets which have the cardinality which corresponds to the number – being eo ipso equivalent – and that if a certain amount of elements, the same for all, is subtracted from each of the latter sets – this would be Aristotle’s “mutilation” of a number – the result is another number). 8. the lost part cannot be regenerated (Aristotle’s example: cutting hair is not an amputation). 1 through 8 form conditions of mutilation/amputation. But privation simpliciter fulfills only conditions 1, 2, 4´, 5 and 7. Aristotle Categories, 13 a 32-6, pointed out some privations: blindness, baldness and teethlessness which are no mutilations but do fulfill condition 8. However, there are many counter-examples of privations simpliciter, which do not fulfill conditions 6 and 8. One which was

pointed out by Simplicius In Aristotelis Physicorum libros commentaria, 821, is nakedness. Nakedness is a privation, no mutilation, and fulfills neither condition 6 nor condition 8: clothes are no extremities and a naked person can get dressed again. Another ready example of a privation which is not a mutilation and fails to fulfill conditions 6 and 8 would be pennilessness. A couple of interesting examples and features concerning some of Aristotle’s conditions follow. Concerning condition 5: Milk and the universal: fox are wholes which are not structured in a certain way, consequently they cannot be deprived of parts. Like water (Aristotle’s example for condition 5), milk is not deprived of spilt milk parts, and the universal: fox is not deprived of some fox shot dead. In such cases, the structure of the whole is not affected. However, small amounts of milk and single foxes are thought of as structured. To affect these structures (for example by depriving milk of some cream, and every fox of his tail) is to enable privations (of milk cream, of fox tails). By condition 5, the wholes will continue to be milk and the universal: fox respectively. Concerning condition 7, a whole which can be deprived of something, is a substance, but mereological essentialism, i.e. the view that every single of its parts is essential to it, cannot hold for it. According to Moses ben Maimon, a Jewish author of the 12th-century, usually Privations as nonexistent objects.

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referred to as Maimonides, a privation is an accident of a substance in much the same way as its corresponding capacity (Führer der Unschlüssigen, i.e. Moreh nebukim, lib. I, cap. 73, prop. 7). There are good reasons to embrace this postulate. Like any capacity, a privation endows its whole with real accidents. For example, it is true to say of a fox’s missing tail that it is bunchy, and it is untrue to say of it that it is string-like, although the tail does not exist. But then, it is true to say that the fox possesses the real accident of being deprived of his bunchy tail (and he does not possess the accident of being deprived of a string-like tail). On the same issue (although without direct reference to Maimonides), the 15th-century Thomist Thomas Cajetan Super librum De ente et essentia Sancti Thomae, pp. 299-300, conceded that privations give the impression of being real accidents of a substance, independently from the perceiver. But this is, Cajetan thought, the false impression. A blind man and a ship without a pilot are deprived of nothing if no one perceives their missing parts/capacities as privations. One century later and in the same line of argument, Francisco Suárez (Disputationes metaphysicae, disp. LIV, sectio III, 4-8) maintained that privations are perceived as entia rationis: as entities, that is, which do not exist absolutely but are expected to exist from the perceiver’s point of view (respective). The examples Suárez gives are the sight in the case of blindness and the deceased father in the case of or-

phanhood. This last example shows that privations can also be missing relations, i.e. failing binary predications rather than only failing unary predications. An orphan is deprived of a not-anymore-existing relation to her deceased father. For Suárez, moreover, privations presuppose that something exists. In any case a subiectum: the blind, the orphan family – the whole, that is, which is deprived of the missing part. But, what is more interesting about Suárez’s views on the topic is that privations presuppose the existence, in a sense, of the missing part itself as the ens rationis which the subiectum is thought to possess. Blind people are thought of as possessing blindness. Orphans of father are thought of as having something: no father (cf. the English expression: ‘She grew up with no father’). See also > Aristotle’ Theory of Parts, Aristotle’s Theory of Wholes, Collectives and Compounds, Mereological Essentialism, Nominalism, Persistence, Substance, Thomas Aquinas. Bibliographical remarks

Burkhardt, H.; Degen, W., 1990. See pp. 7-8 for the notions of true wholes and privation in Leibniz. Davis, D. P., 1991. An overview of some historical discussions of privation, however without reference to the whole/part relation and its meaning for privation.

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Fritsche, J., 1989. Representative collection of citations from the history of the issue. Goddu, A., 1984. For Ockham’s account of privation in physics. Nauta, L., 2003. A discussion of Valla’s views on privation on pp. 63742. Wolfson, H. A., 1946. Contains an account of the views of some Jewish Kalam-authors concerning privations as nonexistent objects. References and further readings

Alexander of Aphrodisias, 1891, In Aristotelis Metaphysica commentaria, ed. by Hayduck, M., Commentaria in Aristotelem Graeca, vol. 1, Berlin: Reimer. Aristotle, Physics. Aristotle, Metaphysics. Aristotle, Categories. Augustine of Hippo, 1886, Contra epistulam Manichaei quam vocant fundamenti, Patrologia Latina 42, coll. 173-206. Augustine of Hippo, 1992, Confessions, Vol. 1, Oxford: Oxford University Press. Burkhardt, H.; Degen, W., 1990, “Mereology in Leibniz’s Logic and Philosophy”, Topoi 9: 3-13. Davis, D. P., 1991, “Privation”, in Burkhardt, H.; Smith, B. (eds.), Handbook of Metaphysics and Ontology, vol. 2 (L-Z), Munich: Philosophia, 724-5.

Fritsche, J., 1989, “Privation”, in: Ritter, J.; Gründer, K. (eds.), Historisches Wörterbuch der Philosophie, vol. 7 (P-Q), Darmstadt: Wissenschaftliche Buchgesellschaft, 137883. Gassendi, P., 1649, Exercitationes paradoxicae adversus Aristoteleos, lib. I, ex. 6, cap. 1, Amsterdam. Goddu, A., 1984, The Physics of William of Ockham, Leiden/Köln: Brill. Leibniz, 1966, Akademie-Ausgabe, vol. VI/2. Moses ben Maimon, 19953, Führer der Unschlüssigen, Hamburg: Meiner. Nauta, L., 2003, “William of Ockham and Lorenzo Valla: False Friends. Semantics and Ontological Reduction”, Renaissance Quarterly 56, 613-51. Simplicius, 1894, In Aristotelis De caelo commentaria, ed. by J.L. Heiberg, Commentaria in Aristotelem Graeca, vol. 7, Berlin: Reimer. Simplicius, 1882, In Aristotelis Physicorum libros commentaria, ed. by H. Diels, Commentaria in Aristotelem Graeca vols 9 and 10, Berlin: Reimer. Suárez, F., 1960–1966, Disputationes metaphysicae, ed. by S. Rábade, S. Caballero, A. Puigcerver, Madrid: Gredos (Biblioteca hispanica de filosofía, 24). Thomas Aquinas, 1982, Quaestiones disputatae de malo, Opera omnia iussu Leonis XIII P.M. edita, vol. 23, Rome and Paris: J. Vrin. Thomas Cajetan, 1590, Super librum De ente et essentia Sancti Thomae,

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Opuscula omnia, Bergomi: Typis Comini Ventura. Valla, L., 1982, Retractatio totius dialectice cum fundamentis universe philosophie, (= Repastinatio dialectice et philosophie, vol. 1), ed. by G. Zippel, Padua: Antenore. William of Ockham, 1985, Expositio in libros Physicorum Aristotelis, ed. F. E. Kelley et al., Opera philosophica, vols 4 & 5, St. Bonaventure/N.Y.: St. Bonaventure University. William of Ockham, 1984, Summa Physicorum, ed. by S. Brown, Opera philosophica, vol. VI, St. Bonaventure/N.Y.: St. Bonaventure University. Wolfson, H.A., 1946, “The Kalam Problem of Nonexistence and Saadia’s Second Theory of Creation”, The Jewish Quarterly Review 36/4: 371-91. Stamatios Gerogiorgakis

Propositions There are many different definitions and characterisations of propositions. They are complex senses or intensions expressed by a sentence, assertive complexes of concepts and primary bearers of truth-values, referents of that-clauses, or sets of possible worlds. In most cases, they are characterised as objective non-mental and non-linguistic complexes, although they are grasped by a thinker and expressed by sentences. If propositions are wholes, they are clearly not aggregates, but structured wholes composed of parts, and they are often

thought to be structured in a manner analogous to the way in which sentences are structured wholes composed of words. A mereological theory of propositions must have at least two parts: the mereology of simple, or atomic, propositions, and the mereology of complex, or molecular, propositions. In this entry both parts are treated as quite distinct. In the mereology of simple propositions I will assume sub-propositional parts (as the a and the F in the proposition a is F) as atoms and analyse how these constituents are unified in a proposition, whereas in the mereology of complex propositions I shall assume simple propositions as mereological atoms and analyse exclusively mereological relations between them (and eventually connectives). Of course, one could easily connect both, e.g. assuming that, by transitivity, if a is part of Fa and Fa part of Fa∧Gb, than a is part of Fa∧Gb. Let us begin with the analysis of (extensional) atomic propositions, i.e. propositions that do not contain propositions as proper parts. What are the parts of an atomic proposition? The answer to this question depends on the notion of proposition one defends. There are basically two main positions: the defenders of the Fregean propositions and the advocates of the Russellian propositions. A Fregean proposition is a structured complex composed of intensional entities like senses, i.e. entities that have some cognitive value. A Russellian proposition, on the other hand, is a complex composed of universals and concrete objects: “in spite of all its snowfields Mont Blanc itself is a

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component part of the proposition that Mont Blanc is more than 4.000 metres high” (Russell’s correspondence to Frege from 12.12.1904). For classical extensional mereology (Fregean and Russellian) propositions are problematic because they seem to be more than a mere aggregate of their terms. Plato loves Socrates and Socrates loves Plato are constituted by the same parts, but are clearly different propositions. Thus, a proposition is a whole with some kind of intensional unity such as logical form or something similar. Russell was for a long time worried about the question of how to explain the unity of a proposition, but he never achieved a satisfactory solution. His most famous attempt to explain this unity was the so called ‘multiple-relations theory’. Strictly speaking, this is more a theory about judgments than a theory about objective propositions because it presupposes a thinking subject. In its standard form, the theory claims that when a thinking subject S judges that aRb, this fact has the form B(S, a, R, b) (where B stands for the belief relation). But this fact has exactly the same parts as when S believes that bRa (or even when a believes that SRb, provided that a is a possible possessor of beliefs). The decomposition of the whole directly to the atomic parts is not sensitive to the propositional structure. Thus, for distinguishing S believes that aRb from S believes that bRa, we must establish that aRb is a (non-atomic) part of the first proposition, but not of the second one, i.e. we must abandon extensional mereology and also ac-

cept non-atomic parts. This nonatomistic mereological analysis is necessary not only for Russell’s theory, but a general requirement for every mereology of simple propositions. Since the relation being a non-proper part is reflexive, even without the addition of a believing subject S and the relation of believing B, nonatomistic mereology could be able to distinguish between aRb and bRa because aRb is a (non-proper) part of the former, but not of the latter. Another issue in the mereology of propositions concerns the analysis of its parts relative to the sentences used to express them. This complex question makes clear that the mereology of sentences and the mereology of propositions are quite different. The analysis of the parts of sentences does not yield immediate insight into the mereology of propositions. The divergences go in both directions: sentences have parts without correspondents in the expressed propositions, and vice-versa. Frege pointed out that a proposition (Gedanke) qua logical entity lacks any subjective elements (Färbungen) contained in sentences. Thus, a sentence like ‘John is unfortunately not at home’ just expresses the proposition that John is not at home, i.e. the objective proposition does not have any part that corresponds to the word ‘unfortunately’. On the other hand, it is possible to introduce into the proposition parts that do not correspond to any part of the sentence. Thus, uttering the sentence ‘it’s raining’ we express (in standard situations) the proposition that it is raining here and now. These elements (in the example

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here and now) are in general introduced into the proposition by the context of utterance. The positions concerning the parts of propositions relative to the uttered sentences vary from minimalism, according to which the proposition only contains parts with explicit correspondents in the sentence, from radical contextualism, according to which the proposition contains many (‘context-sensitive’) elements without explicit correspondents in the sentence. Bell (1996) also noted that there is a tension between two claims in Frege’s system, namely: (i) the thesis that we can distinguish parts in the thought corresponding to the parts of a sentence, so that the sentence can serve as a model of the structure of the thought; and (ii) the thesis that two structurally different sentences can express one and the same thought. For take the sentences ‘a is parallel to b’ and ‘the direction of a is (identical to) the direction of b’. Do they express the same proposition? On the one hand, both are strictly equivalent and have exactly the same truth conditions Hence, they must express the same proposition. On the other hand, the sense of the first sentence is a composition of the senses of the ‘a’, ‘b’ and ‘is parallel to’, while the sense of the second sentence is composed of the senses of ‘the direction of a’, ‘the direction of b’ and ‘is’ (identity). According to Bell, for solving this tension we must give up the principle that is generally assumed, namely, that if one sentence involves a concept that another sentence does not involve, the two sentences cannot express the same

thought, i.e. we must accept that sentences can involve different concepts and, despite this, express the same thought. Now, what are the parts of a complex proposition? Some cases seem to be clear: p is a part of p∧q, and p∧q is a part of (p∧q)→r. Unfortunately, things are not so simple. Central for any mereology is the specification of a suitable notion of parthood as it applies to propositions. One possible way of determining the parts of propositions is to do it in terms of logical entailment. The obvious problem of conceiving parthood simply as logical entailment is the anti-intuitive conclusion that a single simple proposition like p has infinitely many propositions as parts (¬¬p, p∨q, q→p, and so on). Further, if p∨q were part of p, and q (intuitively, not logically) part of p∨q, every proposition would be part of every proposition. Finally, when we assume logical connectives as genuine parts of propositions, we should conclude that a single atomic proposition has all connectives as genuine parts. Thus, although mereological analysis of propositions must take logical analysis into account, we cannot identify propositional parthood with logical entailment. Actually, there are two interconnected problems here: one concerning the propositions as such (i.e. which propositions are parts of other propositions?), the other concerning the connectives (i.e. are logical connectives parts of complex propositions?). These will be discussed in sequence.

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For avoiding the mentioned inflation of infinitely many parts, one could appeal to the notion of explicit occurrence. Thus, p and q are proper parts of p∧q because they explicitly occur in it, but p∨q is, despite being a logical consequence, not a proper part of p because p∨q does not explicitly occur in p. This is a superficial solution because ‘explicit occurrence’ is something well defined for sentences qua complex signs (‘p’ explicitly occur in ‘p∧q’), but not for propositions qua abstract entities expressed by sentences; and, as we already saw, mereological analysis of sentences is not a good guide for mereological analysis of the corresponding proposition. Further, even when we accept a notion of explicit occurrence derived by simply inspection of sentences (or signs), should we conclude that q is a genuine part of p∧(q∨¬q)? Since p∧(q∨¬q) is logically equivalent to p, the part (q∨¬q) seems to be a parasitic, and hence, logically inert part. Of course, we can accept that a parasitic part is also a part, and conclude that some complex propositions are what we might call ‘organic wholes’ – that is, wholes where some parts are essential and others not – and others are integral wholes, where each part is essential. An example of the former might be p∧(q∨¬q), since (q∨¬q) is an inessential part of p∧(q∨¬q). An example of the latter might be p∧q. Interestingly, if all this is correct, we have here a curious case in which an inessential part of a whole has, in its turn, an essential part, since (q∨¬q) is an inessential part of p∧(q∨¬q), but q is essential to

both. In any case, no proposition is a mere aggregate. One consequence of accepting logically inert parts would also be that the mereological identity of propositions diverges from logical equivalence: from a logical point of view p∧(q∨¬q) is equivalent to p, but from a mereological point of view they are different propositions. But, in fact, such a divergence is welcome, since accepting logical equivalence as a criterion for the mereological identity of propositions would thrown us back to mereological inflation: every proposition would be part of a single proposition. Another strategy consists in accepting that the kind of parthood needed in a mereology of propositions can be defined by means of logical entailment plus complexity-reduction. It seems natural to suppose that, when a proposition is a proper part of another proposition, the latter must be more complex than the first. Thus: p is a (proper) part of p∧q because p∧q logically implies p and is more complex than p, p∨q is not a part of p because, although p implies p∨q, this later is more complex than the first. The problems for this suggestion are clear. First, there is no obvious criterion for measuring the complexity of propositions (again, propositions are not sentences, so we cannot read propositional complexity directly from signs). Any decision concerning complexity of propositions depends on a clear account of mereology of propositions, and thus this suggestion is circular. Second, (p∧q)∧(q∧p)∧ (¬¬r→(q→p)) is more complex and implies p∨s, but it seems absurd to

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conclude that the latter is a part of the former. The third alternative is radically atomistic. According to it, all and only atomic propositions that are logically derivable from a complex proposition are really parts of it. Thus, p and q are the only parts of (p∧q)∧(¬r∨r); neither r nor (p∧q) are parts of it, the first because it is not logically entailed, the second because it is nonatomic. Indeed, this seems the most suitable strategy for solving the inflation problem. But a problem still remains: in ¬p∧q should we take p or ¬p as a genuine part? Strictly speaking, ¬p is not atomic, because it contains a non-eliminable connective (contrary to ‘¬¬’). Thus, ¬p is not a part because of the atomicity requirement; p alone is a part. On the other hand, p cannot be a part according to the derivability requirement, since p is not logically entailed by ¬p∧q. A possible, although somewhat ad hoc, solution would consist in claiming that all and only atomic propositions – in positive or negative form – that are logically derivable from a complex proposition are really parts of it. Whether the negation qua connective is a part of propositions is a particular case of the general question whether connectives are genuine parts of complex propositions. Concerning this question, three alternatives are possible. The first one consists in simply denying that connectives are genuine parts of propositions. According to this, p∧(p→q) and p∧(¬p∨q) are logically and mereologically equivalent; both have two

parts: p and q. The main problem of this solution is that mereological analysis will not be sufficiently finegrained. For example, p∨q and p∧q will be mereologically indistinguishable. The second solution consists in adding some strong intensional element, like Carnapian intensional isomorphy, for individuating propositions. Thus, although p→q and ¬p∨q are logically equivalent, they are mereologically different: the first proposition contains implication, but not negation and disjunction as parts, the second the reverse. The problem for this analysis is that it is in one respect too fine-grained, and in another respect not fine-grained enough. It is too fine-grained because it distinguishes logically equivalent propositions like p→q and ¬p∨q; it is too coarse-grained because it identifies logically different propositions like p→q and q→p (both have exactly the same parts). Finally, the third solution appeals to a peculiar notion of logical form (present in Wittgenstein’s Tractatus): p→q, ¬p∨q, ¬(p∧¬q), and all the other equivalent propositions contain three constituents, namely, p, q and a single common logical form. The two most obvious problems of this solution are, first, that this additional part, the logical form, is a mysterious constituent and, second, that it is unclear how one can, using this account, distinguish p→q from q→p. Both have as parts exactly the same simple two propositions and the same logical form but are obviously different propositions. Thus, a general difficulty of every mereological analysis of propositions is that, whatever strate-

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gy is adopted, it is not possible to give an analysis with the correct granularity: either the analysis is too fine or, in most cases, too coarse. Of course, one could also view this not as a weakness, but as a virtue. Mereological analysis does not have the same goal as logical analysis; otherwise a mereology (of propositions) would be superfluous. We should neither aim to reduce logical analysis to mereological analysis, nor to do the opposite. See also > Metaphysical Atomism, Logical Atomism, Carnap, Fact, Frege, Russell, Structure, Syntax. References and further readings

Bell, D., 1987, “Thought”, Notre Dame Journal of Formal Logic, vol. 28 (1): 36-50. Bell, D., 1996, “The Formation of Concepts and the Structure of Thoughts”, Philosophy and Phenomenological Research, 66: 583-596. Carnap, R., 1956, Meaning and Necessity, 2nd ed. with supplementary essays, Chicago: University of Chicago Press. Frege, G., 1984, “Thoughts” in Collected Papers on Mathematics, Logic, and Philosophy, McGuinness, B. (ed.), Oxford: Basil Blackwell, 35172. Gaskin, W., 2008, The Unity of Propositions, Oxford: Oxford University Press. Imaguire, G., 2001, Russells Frühphilosophie: Propositionen, Realismus

und die Sprachontologische Wende, Hildesheim: Olms. Kemmerling, A., 1900, “Gedanken und ihre Teile”, Grazer Philosophische Studien 37: 1-30. King, J., 1995, “Structured Propositions and Complex Predicates”, Nous 29(4): 516-535. King, J., 2007, The Nature and Structure of Content, Oxford: Oxford University Press. Moore, G., 1899, ‘The Nature of Judgment’, Mind. Reprinted in Regan, T., 1986, G.E. Moore – The Early Essays, Philadelphia: Temple University. Recanati, F., 2007, Perspectival Thought: A Plea for (Moderate) Relativism, Oxford: Oxford University Press. Russell, B., 1903, The Principles of Mathematics, London: Norton. Russell, B., 1913, Theory Knowledge, London: Routledge.

of

Wittgenstein, L., 1960, Tractatus Logico-Philosophicus, Frankfurt am Main: Suhrkamp. Guido Imaguire

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Q Quantum Mechanics Introduction.* Quantum mechanics is

a huge a territory including nonrelativistic quantum mechanics, relativistic quantum field theory, condensed matter theory, and quantum gravity. Each of these domains has multiple unique implications for mereology. Within each of these domains there are several different formalisms, each with its own potential implications for mereology. In addition, different interpretations of quantum mechanics provide different mereological ontologies: Quantum entanglement and superposition raise further questions concerning physical composition. If the quantum state completely describes a system but never collapses then it seems likely that the only system with definite physical properties is the entire universe – a dramatic failure of the properties of the whole to be determined by those of its parts! Moreover the radical indistinguishability of quantum ‘particles’ often associated with the (anti)- symmetrisation of the quantum state of a set of (fermions) bosons of the same species threatens to undermine their claim to exist as individual parts of the fusion of

*

The author would like to thank Peter Lewis and Fred Kronz for very helpful comments on this entry.

that set into a whole (Healey and Uffink, 2013: 20).

Therefore this entry will largely (but not exclusively) be restricted to nonrelativistic quantum mechanics and will focus on superposition and entanglement sans larger interpretative schemas. Quantum Superposition and Entanglement. It is well known that in clas-

sical physics compound systems are generally characterised as consisting of separable, distinct parts that interact by means of forces encoded in the Hamiltonian function of the overall system. And that if the full Hamiltonian is known, maximal knowledge of the values of the physical quantities pertaining to each one of these parts yields an exhaustive knowledge of the whole compound system in principle. Classical systems appear to obey the following compositionality principle: Compositionality Principle: The states of any spatio-temporally separated subsystems S1, S2, ..., SN of a compound system S are individually well defined and the states of the compound system are wholly and completely determined by them and their physical interactions including their spatio-temporal relations (cf. Howard, 1885: 989, 1992; Healey, 1991).

Many people have argued that quantum entanglement (and other features of quantum mechanics that cannot be dealt with here) entails, in one way or another, a failure of the compositionality principle. Therefore we will be concerned with the following questions: 1) Does quantum entanglement imply that wholes such as entangled states have properties that are not determined by the properties (wheth-

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er intrinsic or relational) of the particles that enter into such states? 2) What is the best ontological characterisation of the relationship between parts and wholes in entangled states? 3) In what sense if any does quantum mechanics tell against part/whole reductionism? And in what sense if any does quantum phenomena suggest some kind of mereological emergence? More specifically, does quantum entanglement tell against any of the following principles: Separability principle: any two systems A and B, regardless of the history of their interactions, separated by a non-null spatio-temporal interval have their own independent real states such that the joint state is completely determined by the independent states. Locality principle: any two spacelike separated systems A and B are such that the separate real state of A let us say, cannot be influenced by events in the neighborhood of B. Realisation principle: the properties or causal powers of wholes are synchronically determined at a time t by the properties of its proper or contemporaneously existing parts (Silberstein 2012). This is an acausal determination relation that is coextensional with mereological supervenience (ibid). Dynamical closure of the physical domain principle: every physical event is dynamically determined (whether uniquely or stochastically) by a) the laws governing it such as the Schrödinger equation plus b) the

values of the relevant properties of the antecedent state of the system. Western Metaphysicians have often opted for atomistic or Lego-like conceptions of reality. Let us look at two such philosophical statements of classic modern atomism: Humean supervenience is named in honor of the greater [sic] denier of necessary connections. It is the doctrine that all there is to the world is a vast mosaic of local matters of fact, just one little thing and then another…. We have geometry: a system of external relations of spatiotemporal distance between points. Maybe points of spacetime itself, maybe pointsized bits of matter or aether fields, maybe both. And at those points we have local qualities: perfectly natural intrinsic properties which need nothing bigger than a point at which to be instantiated. For short: we have an arrangement of qualities. And that is all. All else supervenes on that (Lewis 1986, p x).

Humean supervenience says that the complete physical state of the world is determined by (e.g., supervenes on) the intrinsic physical state of each spacetime point or each pointlike entity such as classical particles and the spatiotemporal relations between those points. Thus all fundamental properties are local properties (in the supervenience or mereological sense) and spatiotemporal relations are the only fundamental external physical relations. Humean supervenience is thus an affirmation of the separability principle given above. Whether or not it’s local in the (dynamical) locality principle sense is trickier because it’s unclear whether it rules out instantaneous action at a

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distance (e.g. classical Newtonian gravity). Mereological supervenience: systems with an identical total microstructural property have all other properties in common. Equivalently, all properties of a physical system supervene on, or are determined by, its total microstructural property (Kim 1993: 7).

The idea behind mereological supervenience is that the intrinsic properties of the basic parts and/or their relations determine (e.g., realize) the properties of wholes. Kim’s principle also adds a modal claim involving microphysically identical systems. Most do not doubt that quantum entanglement tells against Humean supervenience and mereological supervience in some essential way, as many of us have argued strenuously (Hawthorne and Silberstein 1995; Silberstein 1999 and 2012). As Butterfield puts it, At the end of Sect. 3.1, I noted the uniformity of the rules, in classical and quantum physics, for defining a composite system’s state-space and its quantities; viz.for state-spaces, Cartesian products and tensor products respectively. The uniformity of the rules served my purpose there: namely eulogising the power of reduction. But on the other hand, several philosophers have argued that the quantum rules harbour a very different moral: namely, that the existence of entangled states in the tensor product of two Hilbert spaces, makes for important, indeed pervasive, cases of emergence combined with a failure of supervenience (and so of reduction). What am I to make of this? I can simply agree with these authors (2011: 955).

On most interpretations of quantum mechanics, entanglement does violate Humean and mereological supervenience as well as the realisation principle in at least the following respect: entangled states are neither formally (cf. the non-factorizability of such states) nor empirically (cf. experimentally confirmed distinct quantum probabilities for outcomes in EPR-correlations) a function of the properties of the particles that “make them up” (Silberstein 2002). Likewise, most agree that quantum entanglement surely must tell against the separability principle and/or locality principle depending on one’s interpretation of quantum mechanics. As Healey notes, This [separability] principle could fail in one of two ways: the subsystems may simply not be assigned any states of their own, or else the states they are assigned may fail to determine the state of the system they compose. Interestingly, state assignments in quantum mechanics have been taken to violate state separability in both ways (2008).

Einstein himself was most concerned about possible violations of one or more of the preceding principles in quantum mechanics, as the following letter to Max Born makes clear: If one asks what, irrespective of quantum mechanics, is characteristic of the world of ideas of physics, one is first of all struck by the following: the concepts of physics relate to a real outside world, that is, ideas are established relating to things such as bodies, fields, etc., which claim 'real existence' that is independent of the perceiving subject- ideas which, on the other hand, have been brought into as secure a relationship as possible with the sense-data. It is further charac-

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teristic of these physical objects that they are thought of as arranged in a spacetime continuum. An essential aspect of this arrangement of things in physics is that they lay claim, at a certain time, to an existence independent of one another, provided these objects ‘are situated in different parts of space’. Unless one makes this kind of assumption about the independence of the existence (the ‘being-thus’) of objects which are far apart from one another in space - which stems in the first place from everyday thinking - physical thinking in the familiar sense would not be possible. It is also hard to see any way of formulating and testing the laws of physics unless one makes a clear distinction of this kind. This principle has been carried to extremes in the field theory by localising the elementary objects on which it is based and which exist independently of each other, as well as the elementary laws which have been postulated for it, in the infinitely small (four-dimensional) elements of space. The following idea characterises the relative independence of objects far apart in space (A and B): external influence on A has no direct influence on B; this is known as the ‘principle of contiguity’, which is used consistently in the field theory. If this axiom were to be completely abolished, the idea of the existence of (quasi-) enclosed systems, and thereby the postulation of laws which can be checked empirically in the accepted sense, would become impossible (Born 1971: 170-1).

Let us take a moment to appreciate in more detail why quantum mechanics is believed to have these antiatomistic implications. Beginning with J. S. Mill, many nineteenthcentury British philosophers held intuitions that contrasted ‘resultant’ from ‘emergent’ mereological properties in the following sense: result-

ant properties share an additive parts/whole relationship, i.e. the property of the aggregate is simply the sum of properties of its constituents, whereas emergent properties are non-additive. What was meant by ‘resultant’ was directly inspired by vector and scalar addition in Newtonian mechanics. (Kronz & Tiehen 2002: 331). For example, four hydrogen atoms combine to form a helium atom in the sun. The mass of the helium atom is non-additive or emergent, with respect to the masses of the hydrogen atoms, according to Newtonian mechanics. However, according to special relativity, additivity is restored, insofar as mass and energy are interchangeable (the mass deficit of the helium atom is converted to energy in the fusion of the hydrogen atoms). Most consider the traditional distinction along the lines of additivity and non-additivity at best oversimplifying, if not outright misleading. For instance, Krontz and Tiehen (2002) state: [I]t appears that a central claim of the British empiricists, that additivity is the mark of resultant (i.e., non-emergent) properties, is wrong…the mark of a nonemergent property of composite systems in quantum mechanics crucially involves a multiplicative operation …[However] [t]he situation is different for evolution. A non-separable evolution is a product rather than a superposition [i.e. addition]... This may provide a way to partially vindicate the British emergentists. (333)

To illustrate this point in greater detail, Kronz & Tiehen examine quantum mechanical entanglement in the

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case of a triplet state or three spin-1/2 systems. ‘Entangled’ states (a term first coined by Schroedinger in the 1920s) are quantum mechanical phenomena arising out of combinations of addition and multiplication of state vectors in the formalism of quantum mechanics. Contrast this with classical mechanics, wherein states are represented by points and not vectors; talk of adding or multiplying points is nonsensical. Quantum mechanics has cases in which a state description or an observable quantity cannot be ‘factored’ into the product of simpler constituents thus violating the compositionality principle. Such states are referred to as ‘entangled’, as the simpler terms represent the properties of the system’s constituents. Thus if no such factoring can take place, then no information can be extracted concerning the properties of the system’s basic constituents. In such cases one can only have information concerning the properties of whole system, not of its basic parts. In contrast to classical mechanics, quantum mechanics has a nonBoolean logical structure and quantum properties have a noncommutative algebraic structure. Such entangled states exhibit stronger degrees of correlation than the rules of classical probability theory allow, as delimited by the so-called Bell Inequalities. Experimentally confirmed EPR-correlations tell us that such correlations apparently do not drop off with distance, obtaining even at space-like separation: this is the so-called phenomena of quantum non-locality.

More formally, in the case of quantum mechanics the three state spaces of particles 1, 2, 3 (described respectively by the two-dimensional spinor spaces (H1, H2 H3) combine to form a tensor product (versus a direct sum ⊕ in the case of classical mechanics) eight-dimensional composite space: H1 ⊗ H2 ⊗ H3. Similarly, the system Hamiltonian ĤS combines via the rules of tensor product and superposition. Now, in principle, ĤS can evolve in time to become fully entangled, that is to say, the (8-dimensional) matrix representing ĤS cannot be factored into the (8-dimensional) representations of the Hamiltonian matrices representing particles 1, 2, 3 respectively (represented accordingly by Ĥ1 = ĥ1 ⊗ I2+3, Ĥ2 = I1 ⊗ ĥ2 ⊗ I3, Ĥ3 = I1+2 ⊗ ĥ3). In other words, in such a case, no such factorisation exists, which would allow one to state that ĤS = Ĥ1 ⊗ Ĥ2 ⊗ Ĥ3. Instead, let us denote the fully entangled triplet (pure state) case with the Hamiltonian: Ĥ1+2+3. Other possibilities include ĤS evolving into a superposition of partially entangled mixed doublet states, with respect to, say, systems 1 & 2: ĤS = ĥ1+2 ⊗ I3 + I1+2 ⊗ ĥ3 ≡ Ĥ1+2 + Ĥ3. Finally, the system Hamiltonian can evolve into a superposition of (fully non-entangled) Hamiltonians: ĤS = Ĥ1 + Ĥ2 + Ĥ3. Again, the preceding facts about entangled particles obtain even at space-like separation! It seems that even though each particle in our entangled system presumably occupies a region disjoint from the other, it is not the case that each has its own intrinsic spin state. Nor is it the case

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that the spin-state of the composite system is determined by the states of the particles taken individually, together with the spatiotemporal relations between them. No ‘pure’ state for a single particle yields the same predictions as the ‘Singlet State’. Were one to ascribe a pure state to each of the electrons, their joint state would be a product state rather than an entangled state. The joint state of the pair cannot be analysed into pure states for each of the components. Accounts of quantum superposition and entanglement. There is very little

disagreement about anything said so far but there are several differences of opinion beyond this. Before we get to that however it is important to note that quantum entanglement does not entail either non-separability or nonlocality. For example, retrocausal interpretations of quantum mechanics such as Price (2011) can avoid non-locality with the construction of a local hidden variable theory by utilising forwards and backwards causal links (within their respective light-cones). Such retrocausal dynamics can explain any correlation between spacelike separated events without adverting to non-local connections (Silberstein, Stuckey and Cifone, 2008). Such approaches rely on quantum contextuality to avoid other no-go theorems. Keeping it as general as possible let us say that an observable is contextual if and only if the measured value depends in some way on how the measurement is performed. If a property or observable is contextual that typically implies that it is

not an intrinsic property. However in retrocausal accounts, contextual properties can be intrinsic properties, simply because the state of a system in such cases can dynamically depend on the details of measurements performed on it in the future. Whether or not retrocausal accounts can banish non-separability is less clear. Possibly one could avoid all nonseparability with contextuality alone. For example, two entangled particles could be specified by their respective intrinsic properties despite the formalism of quantum mechanics, but because of the dependence of those properties on the measurements that will be performed on them in the future, the most useful way to write down their state given that you don't know what measurements will be performed on them is as an entangled state of the pair. Finally, it also seems that retrocausal accounts need not violate either Humean supervenience or mereological supervenience. The state of a quantum system right now supervenes on the properties of its proper parts right now. The fact that these properties might depend dynamically on future measurement interactions does not obviously violate either form of supervenience. Let us now return to some key points of general disagreement about what quantum entanglement implies about the relationship between parts and wholes in such systems. Answers to this question range from the relatively conservative to the more radical. Generally we can ask: does entanglement entail merely the non-separability of states or the more

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radical non-separability of systems; the latter implies that space-like separation is not sufficient for the distinctness or individuality of systems (Howard 1992). The answer depends on exactly the nature of the failure of the separability of states. Does entanglement imply the more radical claim that there are no separate (distinct) states for say space-like separated EPR-correlations or does it imply only that the joint state for such systems is not completely determined by their separate states as if the case on some modal interpretations for example. If the former, then the parts (subsystems) of the compound system considered in isolation will be assigned no definite states whatsoever, leaving scepticism as to the very existence of such contemporaneous parts especially if one has a properties bundle-type view about objects/entities. If the latter then in an entangled compound system S, each subsystem S1 [S2] can be regarded as having a state, but only specifiable with reference to the partner subsystem S2 [S1], via the total information contained in S. Therefore, each subsystem (contemporaneous part) derives its existence only from its role within the whole and cannot be characterised apart from the whole. Only the former more radical failure of state separability obviously entails systems non-separability. Perhaps one could also say that the parts have definite states relative to some but not all of their properties. So for example, the particles in a singlet state have definite positions but not definite spins. Different interpretations of quantum mechanics will have differ-

ing implications on these and related matters, as we will see. More specifically, by way of concrete examples, one can find the following positions (accounts of entanglement) progressing from the more to less radical in the literature (these are meant to be illustrative not exhaustive): A) Entanglement as Ontological Structural Realism (OSR): Ladyman et. al (2007) and Silberstein et. al (2008) argue for a conception of entanglement as a case of OSR. OSR rejects the idea that reality is ultimately composed of things, i.e., selfsubsisting entities, individuals or trans-temporal objects with intrinsic properties and ‘primitive thisness’, haecceity, etc. According to OSR the world has an objective modal structure that is ontologically fundamental, in the sense of not supervening on the intrinsic properties of a set of individuals. In Einstein’s terminology, given OSR, particles do not have their own ‘being thus’. The objective modal structure of the world and the abstract structural relations so characterised are fundamental features of reality relative to entities such as particles, atoms, etc. This is not antirealism about objects or relata, but a denial of their fundamentality. Rather, relations are primary while the things are derivative, thus rejecting ‘building block’ atomism or Legophilosophy. Relata inherit their individuality and identity from the structure of relations. According to OSR, entities/objects and their properties are secondary to relational structure. As Kuhlmann puts it, “so proponents

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of ontic structural realism say we might as well dispense with things and assume that the world is made of structures, or nets of relations” (2013: 46). While the standard conception of structure is either set theoretic or logical, OSR holds that graph theory provides a better formal model for the nature of reality because relations (links) are fundamental to nodes therein. Certainly, it is difficult to think about structure without ‘hypostatising’ individuals or relata as the bearers of structure, but it does not follow that relata are truly ontologically fundamental. The point is not that there are no relata, but that relata are not fundamental. Entanglement as OSR is especially radical in the sense that it denies socalled ‘compound systems’ (including and especially entangled systems) are in any fundamental way composed of parts. Unlike the ‘fusion’ relation characterised below, given OSR, it is not as if there ever were any pre-existing autonomous parts with independent existence and intrinsic properties whose behaviour somehow gets subsumed by properties of the whole system (higher-level properties) at some later time t. Kuhlmann again, “instead of considering particles primary and entanglement secondary, perhaps we should think about it the other way round” (ibid). However OSR-like interpretations of quantum mechanics are becoming more prevalent (Kuhlmann 2013). For example Carlo Rovelli’s relational interpretation of quantum me-

chanics (1996) holds that a system’s states or the values of its physical quantities as standardly conceived only exist relative to a cut between a system and an observer or measuring instrument. As well, on Rovelli’s relational account, the appearance of determinate observations from pure quantum superpositions happens only relative to the interaction of the system and observer. Rovelli is rejecting absolutely determinate relata. Rovelli’s relational interpretation of quantum mechanics is inspired by Einstein’s theory of special relativity in two respects. First, he makes the following analogy with special relativity: relational quantum mechanics relativises states and physical quantities to observers the way special relativity relativises simultaneity to observers. Second, Einstein does not merely provide an interpretation of the Lorentz formalism, but he derives the formalism on the basis of some simple physical principles, namely the relativity principle and the light postulate. Another closely related example is Mermin’s Ithaca interpretation (1998) which tries to “understand quantum mechanics in terms of statistical correlations without there being any determinate correlata that the statistical correlations characterise.”. According to Mermin, physics, e.g., quantum mechanics, is about correlations and only correlations; “it’s correlations all the way down”. It is not about correlations between determinate physical records nor is it about correlations between determinate physical properties. Rather, physics is about correlations without correla-

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ta. On Mermin’s view, correlations have physical reality and that which they correlate does not. Mermin claims that the physical reality of a system consists of the (internal) correlations among its subsystems and its (external) correlations with other systems, viewed together with itself as subsystems of a larger system. Mermin also claims inspiration from special relativity. Both these interpretations reject the notion of absolute states and properties. It must be said however that both relational interpretations of quantum mechanics are themselves open to multiple interpretations, e.g., is Mermin making an ontic claim or a methodological one. For a detailed OSR-like model of fundamental physics see Silberstein et al (2008: 2013) and Stuckey et al (2014), wherein we propose an interpretation of quantum mechanics and quantum field theory whereby the fundamental building blocks of Nature are not fundamental particles or excitations of a field interacting via fundamental forces. By contrast, in our relational ontology, the fundamental ontological constituents of Nature are not properties localised *in* space and time, but are elements *of* space, time, and sources (spacetimesources). Why spacetimesource matter? Because the metric is not independent of the matter-energy content of spacetime, rather the SCC (a global ‘self-consistency criterion’) leads to the self-consistency of a graphical spacetime metric and its relationally defined sources. Thereby, properties are fundamentally relational, not in-

trinsic, and the worldtubes of transtemporal objects (TTOs) ultimately emerge contextually per these spacetimesource elements. These ‘spacetimesource’ elements are modeled by gradients in the action of the transition amplitude (i.e., the fundamental computational element in the path integral approach to quantum field theory), and constitute what we mean by ‘relations’. These relations must give rise to interacting TTOs with their worldtubes of (seemingly) intrinsic properties distributed in space and identified through time in Lorentz invariant fashion. To do this, we underwrite the transition amplitude via a global ‘self-consistency criterion’ (SCC) instead of a law for time-evolved entities. This SCC entails gauge invariance for quantum field theory and underwrites a divergence-free stress-energy tensor germane to the construct of TTOs in classical physics. The divergencefree stress-energy tensor reflects conserved quantities that, by Noether’s theorem, entail symmetries of the action. Thus, the time-evolved dynamics based on symmetries of the action of classical physics is ultimately underwritten relationally by an adynamical SCC of a theory fundamental to quantum field theory (Silberstein, Stuckey and McDevitt, 2017, chps. 4 and 5). Essentially, Nature is fundamentally a spatiotemporal ‘micrograph’ from which one may construct statistically a much coarser spatiotemporal ‘macrograph’. Quantum physics then describes how a particular spatiotemporal region of the macrograph can be decomposed into various micrographs. On this

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view quantum entanglement is best characterised as a kind of global spacetimematter contextuality, it’s not a dynamical phenomenon. Instead of explaining quantum entanglement with forwards and backwards causal/dynamical links in spacetime as retrocausal accounts do (forcing them to look for new dynamical laws to account for this), we employ acausal and adynamical global constraints (the SCC) that reside in a theory fundamental to the quantum. Thus on our view quantum entanglement violates neither locality nor separability. B) Entanglement as ‘Fusion’: Humphreys (1997) characterises a type of property emergence he calls ‘fusion’ and claims that quantum entanglement is a paradigmatic example. In order to explain the dynamics of fusion, Humphreys makes use of the notion of levels: (L) There is a hierarchy of levels of properties L0, L1, …, Ln, … of which at least one distinct level is associated with the subject matter of each special science, and Lj cannot be reduced to Li, for any i < j. Note that events are property instantiations at a time for Humphreys and that so understood are the relata of causation. He formally represents events as follows: Pmi(xri) denotes an i-level entity (i.e., xr) instantiating an i-level property (i.e., Pm), for i > 0. Properties and entities are indexed to the first level at which they are instanced. Now let “*” denote the fusion operator. If Pmi(xri)(t1) and Pni(xsi)(t1) are i-level events (i.e., the event of x r's exemplifying Pm at t1,

etc.), then the fusion of these two events, [Pmi(xri)(t1)*Pni(xsi)(t1)], produces an i+1-level event, [Pmi*Pni][(xri)+(xsi)](t2), which can also be denoted as Pli+1[(xri)+(xsi)](t2). The fusion operation is a diachronic, dynamical one. For our purposes the essential feature of a fused event [Pmi*Pni][(xri) + (xsi)](t2) is that it represents a nonseparable whole in that its causal effects cannot be truly characterised in terms of the separate causal effects of its basic parts. More specifically, within the fusion the original property instances Pmi(xri)(t1) and Pni(xsi)(t1) no longer exist as separate contemporaneous entities and they no longer have all their i-level causal capacities available for use at the i+1-level. Hence Humphreys’ account of entanglement as fusion represents a radical kind of non-separability but not as radical as OSR for the reasons given above. C) Entanglement as Nonseparability of the Hamiltonian: Kronz and Tiehen (2002) argue that entanglement as OSR or fusion are unnecessarily radical and opt instead for the less radical conception discussed above wherein entangled systems have contemporaneous parts but the parts cannot be characterised or specified independently of their role in the whole. More formally, the time evolution of the density operator that is associated with a part (subsystem) of a composite entangled system cannot be characterised in a way that is independent of the time evolution of the whole (345-46). All three of the preceding accounts, to one degree or another, tell against

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Humean supervenience, mereological supervenience, or the realisation principle. None of the preceding accounts tells against the dynamical closure principle. From the preceding discussion it is obvious that there are several different ways to conceive of and explain quantum entanglement and quantum phenomena more generally. Obviously, such interpretive choices will have a major impact on questions of mereology. Another question is whether entanglement is best explained and characterised synchronically in terms of the failure of the compositionality principle – the nonseparability of quantum states at a time t (as superficially suggested by Humphreys’ fusion relation which upon closer examination is a diachronic one) or, diachronically and dynamically in terms of the linear evolution of the Schrödinger equation (Hüttemann 2005). The question here is whether it is the Schrödinger dynamics that best explains quantum entanglement or rather something inherently synchronic about quantum mereology. One may ultimately find this distinction suspect however; for example, it is widely regarded that the essential properties of entangled states (their non-separability) are a function of the linearity of the dynamical Schrödinger equation. Or take the case of entanglement characterised as the nonseparability of the Hamiltonian. The Hamiltonian or the evolution operator is non-separable if it can be written only as a superposition of tensor products of the Hamiltonians or evolution operators for the parts (subsystems). Unlike a non-separable Hamil-

tonian, a separable one gives time evolutions of the parts of the compound system that are independent of each other. If Kronz and Tiehen gave us some reason to believe that the non-separability of quantum states at a time t was a function of the nonseparability of the Hamiltonian then the distinction between the synchronic and diachronic characterisations would be little more than pragmatic. While it is almost certainly true on most accounts of quantum entanglement that the compositionality principle will fail and therefore mereological reduction will fail for such systems, this does not settle the question of reduction entirely. Hüttemann (2005) notes that while quantum entanglement clearly tells against synchronic microexplanation if only because of non-factorizability (the clearly determinate state of the compound cannot be explained by determinate states of the parts in the case of superposition or entanglement because no attribution of pure states to (some of) the parts is possible in such cases), it does not obviously tell against dynamical microexplanation. The former explains the state of a compound system at t in terms of the states of the constituents at t. The latter explains the state of a compound system at t in terms of earlier state of the compound system plus the dynamics of the system. The latter is generally based on the following: laws for the dynamics of the parts considered in isolation, laws of composition and when necessary laws of interaction. The key point here is that states of the parts play no necessary role in the explanation of

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the dynamics of compound systems. As Hüttemann puts it, “Diachronic microexplanation does not require the states of the constituents to be specifiable” (119). Hüttemann goes on to show that quantum mechanics is no worse off than classical mechanics with respect to successful diachronic microexplanation: A vector in Hilbert space represents the state of a quantum mechanical system at a time t. The Schrödinger equation describes its dynamics, i.e., its time evolution. All this requires that one determine the quantum mechanical Hamiltonian. In the case of an isolated one-particle system, the classical Hamiltonian has to be replaced by the quantum mechanical Hamiltonian H p P2/2m, where P is the momentum operator of the particle. The behavior of the system of two non-interacting particles is determined by the same procedure as in the classical case. A quantum mechanical law of composition is invoked that requires that we take the tensor product of the two Hilbert spaces so as to gain a new Hilbert space in which the twoparticle system can be represented. The Hamiltonian for the combined system is the sum of those for the isolated subsystems. Microexplanation in quantum mechanics is therefore very much the same as in classical mechanics. The dynamics of compound quantum mechanical systems can be explained in terms of the dynamics of the components considered in isolation (plus laws of composition and interaction). The mathematical tools we use to describe the system and subsystems changes, but

that is it. Quantum entanglement, i.e., the failure of synchronic microexplanation, does not undermine diachronic microexplanations. In quantum mechanics the same sort of ‘completely general’ microreductive strategies are available and employed as in classical mechanics; Hamiltonians are built according to the same procedure as in classical mechanics. Hüttemann notes that: Analysing the dynamics of a compound quantum mechanical system in terms of the parts (plus laws of composition and interaction) does not commit us to the claim that while the parts constitute the compound, they are still identifiable as parts. What we are committed to is this: First, there is some sense in which we legitimately talk about the parts of a compound system as systems of their own. For instance, the parts are systems of their own in the following sense: they were identifiable before they constituted the compound. Second, we are committed to give some kind of interpretation of the terms in the Hamiltonian. For instance, the kinetic energy terms refer to how the constituents would have developed if they were isolated. Such a counterfactual claim does not commit us to any claims about what the parts actually do while they are constituting the compound. The upshot is that the quantum mechanical explanation of the dynamics of compound quantum systems is just as reductionist as its classical counterpart (123).

Obviously Hüttemann’s claim about reduction a la diachronic microexplanation is consistent with the failure of the realisation principle, the separability principle, etc. Quantum entanglement in a relativistic setting. Which of the preceding

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accounts of quantum entanglement the scientific and philosophical community accepts will depend ultimately upon considerations that transcends non-relativistic quantum mechanics, such as considerations from relativistic quantum field theory and quantum gravity. Take quantum field theory. As Maudlin notes in his excellent 1998 (56-57) article mereological matters are far worse in relativistic settings. A whole is presumably made up of parts that exist together at the same time. Indeed, that is what is meant by a whole: each and every point in the region of spacetime designated as ‘the whole’ is simultaneous with every other point. Think now of our space-like separated EPR-systems. Suppose that by the ‘whole’, occupied by region R, we mean: the composite made up of the spacetime regions R1, occupied by particle 1 (corresponding to some point on particle 1’s worldline) and R2, (corresponding to some point on particle 2’s worldline). Keep in mind that particle 1 and 2 are in an entangled state. Now assume that our entangled system is embedded in Minkowski spacetime, we then lose the relation of absolute simultaneity between any two (non-identical) spacetime points. So when we try to relate the points along the worldlines of distinct particles, there is no way to specify uniquely which two are absolutely simultaneous with each other. That is, we can choose to relate different stages of each of the particle’s worldlines to each other, thus constructing many wholes, none being preferred as far as special relativity is concerned. It then follows

that the ‘whole’ in question (being a composite of R1+R2) can be divided along many, equally physically plausible, space-like hypersurfaces: some divisions have R1 preceding R2 in time, others R2 preceding R1 in time and so on. The situation gets thorny when we start asking questions about the state of particle 1 or 2 after a measurement on one of the particles has taken place. Suppose that, in the life of particle 1, it is part of the product state S (the ‘singlet’ state), but after a measurement we discover that 1’s spin is zup. The state is now described as |zup, 1>|z-down, 2> (for perfectly anticorrelated particles). So, before measurement, particle 1 was nonseparable with particle 2. However, after the measurement, we have a completely factorizable state. (A caveat about the Everett interpretation. The state remains non-factorizable after measurement in Everett, but the state in each branch is (FAPP) factorizable.) Now, chose a point P on the worldline of particle 2 and we get a strange result: on one hypersurface, particle 2 (at P) is part of a nonseparable whole described by a nonfactorizable product state S. However, given the relativity of simultaneity, it is equally true that: point P on particle 2’s worldline is also part of a completely factorizable state which tells us that 2’s state is decidedly |zdown, 2>. Now, if we have created the particles in an entangled state and they have moved out to some distance, and we know that the physical states of each particle must be regarded as part of a larger whole, then

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which whole is it? As Maudlin dramatises it: … Relativity transforms [quantum] holism from a mere surprise to a real theoretical problem, It now seems not just that a particle may not have an intrinsic state, not just that one must supplement the physical description of the parts with a further specification of the whole, but that the parts may be attributed seemingly conflicting states depending on the whole that is considered (1998: 60).

It is well known that in quantum field theory the location of a particle depends on one’s frame of reference. Indeed, that the particle has a location at all is frame dependent. Weirder still, the number of particles is relative to reference frame (Stuckey, Silberstein and McDevitt, 2016). Philosophers of physics are fond of cooking up interpretations of nonrelativistic quantum mechanics, but at the end of the day any interpretation of physical reality must be consistent with and driven by such wider considerations. See also > Emergence, Natural Science, Quantum Mereology, Tropes, Whitehead, Whitehead’s Metaphysics References and further readings

Born, M., 1971, The Born-Einstein Letters, Macmillan Press. Butterfield, J., 2011, “Emergence, Reduction and Supervenience: A Varied Landscape”, Foundations of Physics, doi: 10.1007/s10701-0119549-0.

Hawthorne, J.; M. Silberstein, 1995, “For whom the Bell Arguments Toll”, Synthese 102: 99-138. Healey, R., 1991, “Holism and Nonseparability”, Journal of Philosophy, 88: 393-421. Healey, R., 2007, Gauging what's Real, Oxford: Oxford University Press. Healey, R., 2008, “Holism and Nonseparability in Physics”, Stanford Encyclopedia of Philosophy. Healey, R.; Uffink, J., 2013, “Part and Whole in Physics: An Introduction”, Studies in History and Philosophy of Modern Physics 44: 20-21. Howard, D., 1985, “Einstein on Locality and Separability”, Studies in History and Philosophy of Science 16: 171-201. Howard, D., 1989, “Holism, Separability and the Metaphysical Implications of the Bell Experiments”, in Cushing and McMullin (eds.), 22453. Howard, D., 1992, “Locality, Separability and the Physical Implications of the Bell Experiments”, in van der Merwe, A.; Selleri, F.; Tarozzi, G. (eds.), Bell's Theorem and the Foundations of Modern Physics. Singapore: World Scientific. Humphreys, P., 1997, “How Properties Emerge”, Philosophy of Science, 64:1-17. Hütterman, A., 2005, “Explanation, Emergence and Quantum Entanglement”, Philosophy of Science 72: 114-127.

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Kim, J., 1993, Supervenience and Mind, Cambridge University Press. Kronz, F.; Tiehen, J., 2002, “Emergence and Quantum Mechanics”, Philosophy of Science 69(2): 324347. Kuhlmann, M., 2013, “What Is Real?”, Scientific American, August, 40-47. Ladyman, J.; Ross, D., 2007, Everything Must Go: Metaphysics Naturalized, London: Oxford University Press. Lewis, D. K., 1986, On the Plurality of Worlds, Blackwell. Maudlin, T., 1994, Quantum NonLocality & Relativity, Oxford: Blackwell. Maudlin, T., 1998, “Part and Whole in Quantum Mechanics” in Castellani, E. (ed.) Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton University Press, 46-60. Maudlin, T., 2007, The Metaphysics within Physics, Oxford: Oxford University Press. Mermin, D., 1998, “What is Quantum Mechanics Trying to Tell Us?” American Journal of Physics 66: 753-767. Price, H.; Wharton, K.; Miller, D., 2011, “Action Duality: A Constructive Principle for Quantum Foundations”, Symmetry 3 (3): 524-540. Rovelli, C., 1996, “Relational Quantum Mechanics”, Int. J. of Theor. Phys. 35: 16-37.

Silberstein, M., 1999, “The Search for Ontological Emergence”, Philosophical Quarterly 49: 182-200. Silberstein, M., 2002, “Reduction, Emergence, and Explanation”, in Machamer, P.; Silberstein, M. (eds.) The Blackwell guide to the philosophy of science, Malden, MA: Blackwell, 203-226. Silberstein, M.; Stuckey, M.; Cifone, M., 2008, “Why Quantum Mechanics Favors Adynamical and Acausal Interpretations such as Relational Blockworld over Backwardly Causal and Time-Symmetric Rivals” in a focus issue of Studies in the History and Philosophy of Modern Physics on Time-Symmetric Approaches to Quantum Mechanics, Huw Price and Guido Bacciagalupi, editors. Volume 39, Issue 4, 732-747. Silberstein, M., 2012, “Emergence and Reduction in Context: Philosophy of Science and/or Analytic Metaphysics”, Metascience, DOI 10.1007/s11016-012-9671-4. Silberstein, M.; Stuckey, M.; Cifone, M., 2013, “Being and Becoming: A Dialogue between Relational Blockworld and the Implicate Order” in Foundations of Physics. Special issue in honor of Basil Hiley. Volume 43 Number 4, 2013. With Mark Stuckey and Timothy McDevitt. Stuckey, M.; Silberstein, M.; McDevitt, T., 2016, “An Adynamical, Graphical Approach to Quantum Gravity and Unification”, in Licata, I, Space, Time and Quantum: Beyond Peaceful Coexistence, London: Imperial College Press, 499-544.

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The Algebric Way: Space, Time & Quantum. Institute of Physics. Silberstein, M.; Stuckey; McDevitt, 2017, Beyond the Dynamical Universe: Unifying Block Universe Physics and Time as Experienced, Oxford: Oxford University Press. Michael Silberstein

Quantum Mereology A ‘quantum mereology’, that is, a mereology that could be used in the ontology of quantum physics, should address at least the following four questions. The first question is methodological. In the standard formalism that uses Hilbert spaces, any objects of discourse proper (particles, say) have disappeared. The formalism speaks of the states of physical systems and not of quantum objects. Thus it has been suggested by some that ontological questions are irrelevant for physics, since physics describes why the world is as it is, and not what exists in the world. For instance, Steven Weinberg said that “the aim of physics at its most fundamental level is not just to describe the world, but to explain why it is the way it is”(Weinberg 1993, p.175). Thus, if mereology is the study of part-whole relations holding among objects, quantum mereology needs to accommodate the fact that quantum objects properly speaking do not appear as parts of the formalism. On the other hand, one might insist with

Sunny Auyang that “physical theories are about things” (Auyang 1995, p.152) and postulate that there is a “reality” behind the formalism, composed of quantum objects of some kind, yet a “reality” that may remain veiled, to use B. d’Espagnat’s suggestion (2006, pp.236ff). The second question pertains to the identity conditions of quantumphysical objects. Suppose that we have a quantum system whose state is described by the symmetric vector |ψ12〉 = 1/√2 (|ψ1A〉⊗(|ψ2B〉 + (|ψ2A〉⊗(|ψ1B〉).

We can say that it is a whole composed of two parts, namely, those parts described by the product vectors |ψ’〉 = 1/√2 (|ψ1A〉⊗(|ψ2B〉) and |ψ’’〉 = 1/√2 (|ψ2A〉⊗(|ψ1B〉). The first vector says that there is a quantum object 1 in a state A and another quantum object 2 in a state B, while the second one says that 1 is at B and 2 is at A. But, according to quantum physics, we cannot distinguish these objects: the state |ψ12〉 is totally symmetric, it is a superposition (linear combination) of the two parts. This is a typical quantum situation, with has no correlate in classical physics, so that according to quantum mechanics we may say that the “real” cannot any more be considered as localised [we could say, “distinct”] elements embedded in space-time (d’Espagnat op.cit., p.454). These “parts” of the whole described by the product vector are absolutely indiscernible, and the state (the object) does not change if we permutate its “parts”. This is expressed in the theory by the so-called Indistinguishability Principle, which roughly speaking

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says that the expectation values (which give the relevant probabilities) of the measurement of any observable does not change if we take the system before or after a permutation of similar quantum objects. The situation is problematic even if we consider quantum fermions in an anti-symmetric state |ψ12〉 = 1/√2 (|ψ1A〉⊗(|ψ2B〉 – (|ψ2A〉⊗(|ψ1B〉).

Here, a permutation (of quantum 1 by quantum 2 and vice-versa) changes the sign of the state vector, but its square, which gives us the relevant probabilities, is the same before and after the permutation. In other words, as before, permutations are not regarded as observable; again, we can’t say (even in principle) which quanta is which. A quantum mereology thus will need to allow for the fact that the whole does not change significantly (in a physical sense) when its parts are substituted by “other” indiscernible parts (that is, by quantum objects of the same sort); to translate the situation into the domain of classical physics, a quantum mereology will need to treat me as the same whole whether or not I have my right hand were amputated and substituted with another right hand. Thus here we can no longer treat a whole as a sum of its parts, for we have no longer access to the parts of a whole (a composite system) beyond indiscernible permutations. In standard (extensional) mereologies, objects having the same parts are the same object; so, by classical logic, if some part is replaced, we no longer have the same whole. This

identity principle, the Proper Parts Principle, corresponds to the conception of identity used in standard mathematics (say that built in the Zermelo-Fraenkel set theory with the axiom of choice, ZFC); here the Axiom of Extensionality postulates that sets are equal (identical) if and only if they have the same elements. But in the quantum case it is questionable whether this conception of identity is still applicable. In fact, it is not completely clear what we mean when we say that two electrons (protons, neutrons, etc.) are the same. Since states can be the same (for instance in an entangled system), the corresponding quantum objects would be also the same if we assume classical logic, for the vector states (it is assumed), concentrate all information we have about the ‘realities’ described (to avoid the parlance of hidden variables). As acknowledged a long time ago (1808) by John Dalton, “[w]hether the ultimate particles of a body, such as water, are all alike, that is, of the same figure, weight, etc. is a question of some importance. From what is known, we have no reason to apprehend a diversity in these particulars: if it does exist in water, it must equally exist in the elements constituting water, namely, hydrogen and oxygen. Now it is scarcely possible to conceive how the aggregates of dissimilar particles should be so uniformly the same. If some of the particles of water were heavier than others, if a parcel of the liquid on any occasion were constituted principally of these heavier particles, it must be supposed to affect the specific gravity of the mass, a circumstance not known. Similar observations may be made on other substances. Therefore we may conclude that the ultimate particles of all homogeneous bod-

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ies are perfectly alike in weight, figure, etc. In other words, every particle of water is like every other particle of water, every particle of hydrogen is like every other particle of hydrogen, etc” (Dalton 1808, 10, pp.142-3).

Dalton’s claim was realised by quantum physics. In fact, suppose the ionisation of a Helium atom He in its fundamental (neutral) state. Informally speaking, ionisation rules out one of the electrons, originating a positive ion He+. Then we can make the ion absorb an electron, turning it neutral again, call it He’. Is He the same as He’? To answer that, we need to suppose that the electrons ruled out and the electron absorbed are the same electron, but this simply makes no sense at all. Even if it would be possible to apprehend the very same electron, the resulting effect would not be different if we had apprehended another electron. For all physical purposes, the “two” atoms He and He’ are empirically the same. However, if we describe this scenario more precisely, using classical logic (or mathematics using a set theory like ZFC), and represent the two Helium atoms, as two sets, we would need to say that the two electrons and, consequently, the two atoms would be necessarily identical, for it is supposed that there are no differences between them. Classical logic is Leibnizian, and does not admit indiscernible but non-identical objects. Moreover, it is important to note that spatial location cannot come to the rescue here and individuate quantum entities, since some physicists conjecture that the notion of space does not make sense at all due to entan-

glement (Gisin 2009). If we work with the topology of the differentiable manifold and say that the represented objects lie in different points of the manifold, so that they belong to distinct open sets and consequently are different, we make again use of classical mathematics (and logic), and thus again with the identity of indiscernibles. A quantum mereology thus needs to come to terms with the problem of individuation (a detailed discussion about identity and individuality in quantum physics can be seen in French & Krause 2006). The third fundamental question for a quantum mereology concerns the formation of wholes. Suppose the chemical substance C2H6O. What is it? Well, it depends. A chemical compound is generally not only a collection of atoms. It is neither a set of forming substances nor a simple mereological sum of its parts. Really, in the quantum case, the nature of this process of forming wholes from parts by summation is something to be further explored. In the given example, depending on the structural arrangement of the elements, we can have either ethylic alcohol or methylic ether, depending on how the component atoms are arranged. So, the third question of quantum mereology concerns the structure of the wholes and of its parts. This is of course an enormous problem from the formal point of view. Apparently, there is no way other than to postulate that the structure is something to be introduced case by case or, as logicians would say, from model to model.

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The fourth question to be addressed by a quantum mereology is the problem of quantum holism. Broadly speaking, holism says that a whole cannot be understood by looking at its parts. It is more. In quantum physics, quantum holism, which some address under the label of nonseparability, says that the state of a whole cannot be taken as constituted by the states of its parts (Healey 2009). Nonseparability is a typical consequence of entanglement. If we cannot individuate the parts of a whole, and perhaps there is no sense in speaking of them, because a whole simply has no parts in the usual sense, how can we form wholes? One might answer as follows. Take certain fundamental elements, say two quantum objects described by their states (e.g., |ψ’〉 and |ψ”〉 of our first example) and then form the joined system by stating the sum vector (in our example: |ψ12〉 = |ψ’〉 + |ψ”〉. Then we have a “whole” with no more “parts” at all, but “composed” of something we know quite well when considered in isolation. We know the wholes, we know their parts but we don’t know the whole and its parts. It seems that when parts meet, they lose their individuality as entities of some kind to become parts of something bigger. (For some suggestions for how to addesss this situation see Krause 2011). See also > Elements, Emergence, Natural Science, Quantum Physics, Tropes

References and further readings

Auyang, S., 1995, How is Quantum Field Theory Possible? Princeton: Princeton Un. Press. Dalton, J. 1808, A New System of Chemical Philosophy, Printed by S. Russell. d’Espagnat, B., 2006, On Physics and Philosophy, Princeton: Princeton Un. Press. French, S.; Krause, D. 2006, Identity in Physics: A Historical, Philosophical, and Formal Analysis. Oxford, Oxford Un. Press. Gisin, N. 2009, “Quantum Nonlocality: How Does Nature Perform the Trick?” http://arxiv.org/abs/0912.1475v1. Healey, R., “Holism and Nonseparability in Physics”, The Stanford Encyclopedia of Philosophy (Spring 2009 Edition), Zalta, E. (ed.), http://plato.stanford.edu/archives/spr 2009/entries/physics-holism/ Krause, D. 2011, “A Calculus of Non-Individuals (Ideas for a quantum mereology)”, in Dutra, L. H. A.; Meyer Luz, A. (eds.), Linguagem, Ontologia e Ação. Col. Rumos da Epistemologia Vol 10, pp. 82-106. Florianópolis, NEL/UFSC. Weinberg, S. 1993, Dreams of a Final Theory: The Search for the Fundamental Laws of Nature, London, Vintage. Décio Krause

RADULPHUS BRITO

R Radulphus Brito In Quaestiones 9-12 of his commentary on De differentiis topicis of Boethius Radulphus Brito (Raoul de Breton, died 1320) discusses the partwhole relation. In Quaestio 9 he distinguishes between two kinds of integral whole. One kind is the homogeneous whole, the essence or nature of which is inherited by every part. Examples of homogeneous wholes are water and flesh. The other kind of integral whole is the heterogeneous whole. An example of heterogeneous whole, is a house – the parts of the house, for example a wall or roof, are not themselves houses. Brito uses Boethius’ terms: ‘to hold constructively’ and ‘to hold destructively’, to signify confirmation or negation of the existence of the antecedent, respectively. In combination with the inferences from whole to part and from part to whole we get four possible inferences. (1) The whole is (exists) ergo: the part is (exists). (2) The whole is not ergo: the part is not. (3) The part is ergo: the whole is. (4) The part is not ergo: the whole is not.

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Brito then seeks to establish the valid kinds of inference for the two kinds of whole. For a heterogeneous whole the inference from the whole to the part is only constructively (1) valid, because the whole is nothing else than an aggregation of its parts. Destructively (2) it is not valid, because the parts can exist independently of the whole. The inference from the part to the whole on the other hand is only destructively (4) valid, because the destruction of any part destroys the whole. Thus, we have the following valid inferences for heterogeneous integral wholes. (HET 1) The heterogeneous whole is (exists), ergo: any part is (exists). (HET 4) Any part is not, ergo: the whole is not. Thus for example, (HET 1) implies that if the house exists, then the wall exists, and (HET 4) implies that if the wall does not exist, then the house does not exist. If we look at the homogeneous whole, we have to distinguish between a quantitative and a qualitative aspect. Quantitatively there is no difference between a heterogeneous and a homogeneous whole. As examples Brito uses the sentences: Instance of HOM-Quant 1: The whole of water is, ergo: any part of water is Instance of HOM-Quant 4: The part of water is not, ergo: the whole of water is not. But if we take in account the quality: i.e. the essence or nature of a homogeneous integral whole, the inference

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from whole to part is only destructively (2) valid and not constructively (3). So the following is not valid: ‘Water is, ergo this part of water is’. The inference from part to whole, on the contrary, is only constructively (3) valid. Thus we have the following examples: Instance of the HOM-Qual 2: The whole of water is not, ergo: any part of water is not. Instance of the HOM-Qual 3: The part of water is, ergo: water is. In order to distinguish between homogeneous and heterogeneous wholes, then, we have to look at the inferences between part and whole constructively and destructively from a qualitative viewpoint. Inferences valid for heterogeneous integral wholes are in general not valid for homogeneous integral wholes and vice versa. It is worth noting Brito’s first counterexample, namely ‘having the value of 100 marks’, because in this case it is not merely the existence of whole and part that is the subject of the topic or inference, but the predication of a property. It is intuitively clear that exact quantitative expressions like ‘to have the value of 100 marks’ or ‘to have the weight of 100 pounds’ are not communicable from the whole to its parts. See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Boethius, Homeomerous and Automerous

References and further readings

Green-Pedersen, N. J. (ed.), 1978, “Radulphi Britonis Quaestiones super libro Topicorum Boethii”, in: Cahiers de L´Institut du Moyen-Age Grec et Latin, Copenhague 1-92. Hans Burkhardt

Raimundus Lullus and Lullism Life and Education. Raimundus Lul-

lus (1232-1315) was a philosopher and theologian. He was born in Palma de Majorca, where he also died. He belonged to a noble family, and taught philosophy in Paris and Montpellier as well as in Naples. He was also the first to publish his scientific ideas, poems and mystical reflections in Catalane language, because he preferred his language to Latin. In contradistinction to other savants of his time he was fluent in Arabic and studied not only Christian theology but also Islamic and Judaic culture and theology. Lull was the founder of combinatorics and consequently this art was called the ‘Art of Lull’. In 1276 James II of Majorca had founded in Miramar a monastery where Franciscans could study Arabic and Lull’s art to prepare for their mission in Islamic lands. Philosophy and Theology. Lull’s basic position in philosophy was neoplatonic rationalism and actionism, but he was primarily a theologian and therefore had above all theological aims. Thus, the character of his philosophy was purely apologetic, and his main aim was to convert Pagans,

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Moslems and Jews to the Catholic faith. According to Lull, God, insofar he can be known to men, consists of a series of divine attributes, or ‘dignities’, which in his later works Lull identified as God’s goodness, greatness, eternity, power, wisdom, will, virtue, truth, and glory. These attributes are the absolute principles of Lull’s Art, since they are the instruments of God’s creative activity, the causes and archetypes of all created perfection. The essence of the art does not consist in demonstration, but in the metaphysical reduction of all created things to the dignities, which are the principles of knowing as well as of being, and in the comparison of particular things between themselves in the light of the dignities, by means of such relative predicates as difference, agreement, contrariety, beginning, middle, end, majority, equality, and minority. The absolute and relative predicates together form the self-evident principles common to all the sciences. These principles are combined in circular figures, where letters are substituted by their names. Traditional metaphysics is characterised by three relations: Substanceaccident, cause-effect, especially the efficient cause and the final cause, and part-whole. In the history of philosophy, and especially medieval philosophy, the substance-accident relation and the four kinds of causes are dominant, and the part to whole relation is usually reduced to them. Thanks to his combinatoric art Lull is the one of the rare philosophers who tries to reduce the other metaphysical relations to the part-whole relation.

Combinatorics and the Art of Lull. To reach this aim he developed combinatorial methods which enabled him to construct sentences and syllogisms for arguments. The art based on combinatorics he called ars magna (‘great art’), and this ars magna was a part of his logica nova (‘new logic’). In the tradition of Avicenna the ars magna is conceived by Lull as a Scientia Generalis, the true fundamental science upon which both metaphysics and logic depend. From this it follows that Lull’s logica nova is not a ‘logic’ in the traditional and true sense, but rather a combination of logic, ontology and methodology strongly influenced by theological ideas and concepts, especially the Trinity.

Lulls’ logica nova took its starting point from several traditional texts, including Aristotles’ Categories and De Interpretatione, and Peter of Spain’s Summule Logicales. However, Lull’s logic contains new aspects, so for example he emphasises the category of relations to such an extent that relations not only figure as the most important accidents, they are almost essential entities. Indeed the basic unity of this Art – inspired by the concept of the Trinity – is a relation between three items. For Lull a relation always has to do with action and it consists of an agent, a patient and the action itself. The subjects or terms of the relation are understood as parts. For example the relation ‘A warms B’ is a warming activity or process (calefacere), and it has as terms and parts a warm object (calefactivo), A, and an object to be warmed (calefactibili) B. All

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Lull’s treatises on the different sciences are applications of his general Art. The basic idea of Lull’s Art is that every whole or compound object can be produced by combining the smallest and simplest parts and it can be analysed into such smallest parts by an opposite procedure. Lull already applied this Art to sentences, parts of sentences and syllogisms. Thanks to this procedure he was the first logician in the history of logic who tried to base logic on combinatorics and who therefore also used a combinatorial approach to syllogistic. But the letters used by Lull in his systems were not variables as it was the case of later combinatoric thinkers. On the contrary they stand for material concepts stemming from the Christian, Islamic and Cabbalistic tradition. In Lull’s time the Art had nothing in common with mathematics. Lull, for example, never calculated the number of possible combinations in different contexts. The mathematisation of combinatorics was above all the work of Christoph Clavius S.J. (1537-1612) and Leibniz (16461716) who is perhaps the most famous and successful follower of Lull. In his Dissertatio de arte combinatoria (1666) Leibniz used his combinatorial method to construct twentyfour valid syllogisms, six in each of the four figures. The acceptance of the fourth figure by Leibniz, which we do not find in the logica nova, is a necessary consequence of the application of combinatorics on syllogistics, i.e. of combinatorial completeness. Like Lull himself, Leibniz re-

garded the art of combinatorics as more basic than traditional logic. One of the typical problems to be solved by combinatorics, even in a mechanistic way, was the inventio termini medii, that is, the discovery of the middle term. Without this middle term no syllogism can be constructed. This inventio was a typical topic in medieval logic and a part of the ars inventiva, and many logicians were eagerly looking for a method of discovery that would be a ‘real’ method, one that is independent from intuition. Lull and Aristotle. Lull discusses the part-whole relation in standard Aristotelian terms, derived from, for example, Aristotle’s Physics and Metaphysics.

In Physics 210a 14-24 we learn that there are many ways in which something is ‘in’ another. According of the first mode of being in, something is said to be in something, as an integral part is in its whole, e.g. a finger is in a hand, a wall is in a house. According to the second mode, an integral whole is in its parts, e.g. a house is in its wall, roof and foundations. According to the third mode, a species is in a genus, e.g. man is in animal. According to the fourth mode, a genus is in a species, e.g. animal is in the man. According to the fifth mode, a form is in a matter. According to the sixth mode, something is in a prime cause, e.g. a reign in a regent. According to the seventh mode, something is in an end or goal, e.g. virtue is in happiness. According to the eighth mode, something is in a

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container, and generally what is placed is in a place. Let us look at some examples of part to whole items in the logica nova, inspired by the analysis in Aristotle’s Physics. According to the third mode, a species is in a genus. We find the third mode in Lull: The genus itself is portioned in its species like the whole is portioned in its parts. We find also a variant of the fourth mode. According to the fourth mode, a genus is in a species. Lull considers not the intensional parts of the species, but rather those of the genus. The genus contains general parts which are the conceptual parts or ingredients of the genus, for example body which itself is a genus. The invisible and unimaginable body consists of general form and matter of the body, which both are invisible and unimaginable. According to the fifth mode, a form is in matter. This fifth mode was subdivided by Boethius into two modes, because there are both (1) substantial forms, (e.g. the soul is the substantial form of man), and (2) accidental forms, (e.g. the whiteness of a man). Now the former is properly said to be ‘in’ something else, in the sense that the form is in the matter, and the soul is in the body; the latter is said to be “in” something, in the sense that the accident is in the subject, as the whiteness is in a wall, and colour is in the body. Following Aristotle, Lull’s analysis of the part to whole relation concerns both the relation de subjecto dici and the relation in subjecto esse. Thus, we find in Lull’s logica nova a formulation of the fifth mode of Aristo-

tle which, following Peter of Spain, is divided into two sub-modes that conform in Boethius’s subdivision. Man is man together with his essential parts, i.e. the soul is in the body and vice versa the body is in the soul. The substance is in its accidents and vice versa the accidents are in their substance. An interesting aspect of this formulation is that Aristotle in the Physics takes individual accidents as parts of the substance seen as a whole. In Book Z of his Metaphysics, Aristotle offers a linguistic example of the relation part to whole. A word is with its syllables as parts and a sentence is with its words as parts, i.e. syllables are present in words and words are present in sentences. Aristotle seems to conceive words and sentences as true wholes, and these examples show that wholes are ontologically prior to their parts. In contradistinction, for Lull the parts and the wholes, that is, both words and syllables, and sentences and words are on the same level. There is no semantic gap between linguistic wholes and their parts. This is a consequence of his combinatoric approach. Lullism. In addition to Clavius and

Leibniz, in the following centuries Lullism had a profound effect upon philosophers, scientists and theologians of quite different origins, capacities and interests. For example, Lull’s work influenced the mystic and first German commentator Agrippa von Nettesheim (1486-1535), Giordano Bruno (1548-1600), the encyclopaedist Johann Heinrich Alsted (1588-

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1638), the Jesuit philosopher and theologian Athanasius Kircher (16021680), and Johannes Hospinianus (Gast) (1515-1575), who successfully applied the art of combinatorics on syllogistics. Some of these men were attracted to Lulls’ mystical approach to numbers, others to his application of the art of combinatorics on terms, sentences and syllogisms, and still others to the mathematical application of his art of combinatorics. See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Common Sense Reasoning About Parts and Wholes, Medieval Mereology, Propositions, Substance References and further readings

Burkhardt, H., 1980, Logik und Semiotik in der Philosophie von Leibniz, Munich: Philosophia. Hillgarth, J. N., 1967, “Ramon Lull”, in: The Encyclopedia of Philosophy. (Edited by Paul Edwards) Vol. 5 and 6, London – New York, The Macmillan Company & the Free Press, 107108. Hillgarth, J. N., 1971, Ramon Lull and Lullism in Fourtenth Century France, Oxford: Clarendon Press Johnston, M. D., 1987, The Spiritual Logic of Ramon Lull, Oxford: Clarendon Press Knobloch, E., 1973, “Die mathematischen Schriften von Leibniz zur Kombinatorik”, Studia Leibnitiana Supplementa Band XI, Wiesbaden, Franz Steiner Verlag.

Lohr, C. D. 1991, “Raymond Lull”, in Burkhardt H.; Smith, B. (eds.) Handbook of Metaphysics and Ontology. Vol. 2 Munich: Philosophia, 760-61. McMahon W. E., 1996, “The Semantics of Ramon Llull”, in Angelelli, I.; Cerezo, M. (eds.), Proceedings of the III Symposium on the History of Logic. Studies on the History of Logic. Berlin – New York: Walter de Gruyter, 155-171. Platzeck, E. W., 1962, Raimund Lull. Sein Leben, Seine Werk. Die Grundlagen seines Denkens, (Prinzipienlehre). Vol. 1. Düsseldorf: L. Schwann. Prantl, C., 1867, Geschichte der Logik im Abendlande. Dritter Band, Leipzig, S. Hirzel, 145-177. Raimundus Lullus, 1985, Die neue Logik. Logica nova. Textkritisch herausgegeben von C.Lohr. Übersetzt von W. Büchel und V. Hösle, Hamburg, Felix Meiner. Risse, W., 1964, “Die Lullische Tradition. Kombinatorik. Lingua universalis. Mathematisierung” in: Die Logik der Neuzeit 1. Band 1500– 1640, Stuttgart-Bad, Cannstatt: Friedrich FrommannVerlag, 532-560. Thiel, C., 1984, “Raimundus Lullus”, in: Enzyklopädie Philosophie und Wissenschaftstheorie Band 2 (ed. J. Mittelstraß) Mannheim, Wien, Zürich: Bibliographisches Institut Wissenschafts-verlag, 725-26. Hans Burkhardt

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Reduplication Reduplication concerns the qualification of statements so as to talk about certain aspects of them. In Aristotelian philosophy reduplications were typically marked by ‘qua’, ‘in virtue of’ etc.; hereafter ‘qua’ will be used for the purpose of a generic illustration. Such qua phrases generally restrict the scope of the statement to a part of what is stated originally. Sometimes Aristotle uses the term ‘reduplication’ (ἐπαναδίπλωσις) in discussing them, as his examples of qua propositions generally had a repetition, or ‘reduplication’, of one of the terms. E.g., “the good is known, in that (ὅτι) it is good” (An. Pr. 49 a 11 – An. Pr. 49 a 35). By medieval times all qua propositions were called ‘reduplicative’. Like other conjunctions such as ‘if’ and ‘because of’, a qua connective links up sentences, clauses, and phrases. Qua connectives occur at important points in the work of many philosophers: in Avicenna’s threefold distinction of quiddity, held by many to be the main medieval solution to the problem of universals (AlMadkhal, 15,1-15); in the supposition of subject terms in sentences like 'man is the worthiest of creatures' according to William of Sherwood (Introductiones ad logicam, 77, 1828); in the analysis of the Incarnation by Aquinas (Sentences III.XI.l; Summa theologiae III. 16.8-10), and Scotus (Sentences III.XI.2). The qua connective also occurs in Leibniz's formulation of the identity principles and in his reduction of relations (Mugnai 1978; 1982); in one of Rus-

sell's solutions to Russell’s Paradox (Principles of Mathematics I.X. 104); and in Martin Heideggers’s discussion of ‘als’ in Sein und Zeit. The reasons for the ubiquity of reduplication are fairly obvious: whenever senses of concepts are to be distinguished, whenever different aspects and modes of a thing are to be singled out and abstracted, whenever an assertion is to be qualified in a certain respect, the appearance of qua (or functionally cognate linguistic expressions) is nearly inevitable. Reduplicative propositions abound in Aristotle’s philosophy. He uses qua phrases to distinguish the essential (‘in virtue of itself’; καθ’ αὑτό) and the accidental (“in virtue of an accident”; κατὰ συµβεβηκός) (Posterior Analytics I.4). He defines motion as “the fulfilment of the movable qua movable” (Physics 201 a 28-9). Above all, his Metaphysics (IV.1) concerns being qua being. While presenting no systematic theory, Aristotle himself had logical doctrines about such qua phrases. When they appear in syllogisms, they should be attached to the predicate (Prior Analytics I.38). Aristotle discusses when qua phrases could and could not be dropped validly from propositions in his fallacy of secundum quid ad simpliciter (later called ‘converse accident’). His example of that fallacy became the standard: the Ethiopian is white in respect of his teeth; therefore the Ethiopian is white (Sophistical Refutations 167 a 7-9). The Latin medievals above all developed Aristotle’s doctrines into a fullblown theory of reduplication. Gen-

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erally they distinguished two logical types of propositions having qua phrases: the reduplicative proper and the specificative. The reduplicative contained those qua propositions deemed respectable in Aristotelian logic and science; the specificative contained those liable to occasion fallacy. An exhaustive analysis of reduplicative propositions was given by such philosophers as William Ockham and Walter Burleigh (De puritate artis logicae tractatus longior). Thus, to take the standard medieval example, ‘man qua rational as risible’ is to be analysed as: ‘man is rational, and man is risible, and every rational thing is risible, and if something is rational, it is risible’. On the causal analysis, a fifth exponent, ‘being rational is the cause of being risible’ is added (William Ockham, Summa Logicae II.16). As was recognised later on explicitly by those such as John Wyclif, however, some of the conjuncts of these expositions are redundant; thus the basic reduplicative analysis of ‘S qua M is P’ may be reduced to: ‘S is M, and being M entails being P’ (Tractatus de logica, I.5). Generally specificative propositions were not analysed further, except that explanations were offered in such a way as to make their meaning plainer. Here the logical analysis centered more on the semantics, how the qua phrase changes the reference of the unqualified subject into something related to it. This discussion was generally put in terms of parts and wholes; e.g., by Albert the Great (De

Sophisticos Elenchos I.III.6). Thus, as teeth are an integral or material part of a whole human body, ‘in respect of his teeth’ when attached to 'the Ethiopian' changes the reference from the whole, the human body, to the integral part, the teeth. Ockham offered a more syntactic analysis in his treatment of the fallacy of secundum quid et simpliciter (in Sophisticos Elenchos). Although they did not speak of ‘the reduplicative’ and ‘the specificative’ nor present a general theory, Islamic philosophers also had extensive treatments of qua propositions, especially in sophisms. In particular Avicenna offered a general treatment of predication in terms of qua phrases: a simple categorical proposition of the form ‘S is P’ has the logical structure, ‘S is existent qua P’. Avicenna works out squares of opposition in conformity with this analysis. Along these lines, perhaps by coincidence, some Vedanta systems, like the Nyāya, make the distinction of respects the foundation of their ontology. For any thing, say, a dog, ‘is’ in one respect (‘is a dog’); ‘is not’ in another respect (‘is not a cat’), ‘is and is not’ in yet another (‘is a dog and is not a cat’) etc. From the texts that we have it is not clear how detailed this theory of aspects is. There was much discussion of reduplicative propositions in the postmedieval period, though not as much originality (Lax, Tractatus exponibilium). In the modern period, with the decline of interest in formal logic, reduplicative propositions fell into

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obscurity. Still, in the twentieth century, with the renewal of interest in logic and linguistics, interest in reduplication revived (Fine, 1982; Simons, 1987; Wiggins, 2001). See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Medieval Mereology, Non-literal Language Use, Possessives and Partitives, Propositions, Syntax. References and further readings

Albertus Magnus, 1632, Opera ad logicam pertinentia, Opera omnia, vol. I, Venice. Aristotle, 1986, The Complete Works, Barnes, J. (ed.), Princeton, N.J.: Princeton University Press. Avicenna, 1952ff., Al-Shifā‛, Badawī et al., eds., Cairo: Government Press: Part 1 vol. 1: Al-Madkhal (= Logica, in Opera Avicennae 1501, Venice; repr. Frankfurt, 1961). Bäck, A., 2000, Aristotle’s Theory of Predication, Leiden: Brill.

Lax, Gaspar, 1512, Exponibilia. Lear, J., l982, “Aristotle’s Philosophy of Mathematics”, The Philosophical Review 91. Mugnai, M., l978, “Intensionale Kontexte und ‘Termini Reduplicativi’ in der Grammatica Rationis von Leibniz,” Studia Leibnitiana 8. Mugnai, M., l982, “La expositio reduplicativarum chez Walter Burleigh et Paulus Venetus,” in English Logic in Italy in the l4th and l5th Centuries, ed. A. Maierù, Naples: Bibliopolis. Ockham, W., 1979, Expositio super Libros Elenchorum, ed. F. del Punta, St. Bonaventure, N.Y.: St. Bonaventure University Press. Ockham, W., 1974, Summa logicae, St. Bonaventure, N.Y.: St. Bonaventure University Press. Peter of Spain (non-Papa), 1992, Syncategoremata, ed. L. M. de Rijk & trans. J. Spruyt, Leiden: Brill. Simons, P., 1987, Parts, Oxford: Oxford University Press.

Bäck, A., 1996, On Reduplication, Leiden: Brill.

Wiggins, D., 2001, Sameness and Substance Renewed, Cambridge: Cambridge University Press.

Banarsidass, M., 1998, ed. & trans., Classical Indian Metaphysics, 2 Vols., Chicago: Open Court.

Wycliffe, John, l896, Tractatus de Logica, ed. M. H. Dziewicki, London.

Burleigh, Walter, l955, De puritate artis logicae tractatus longior, St. Bonaventure, N.Y.: Franciscan Institute. Fine, K., 1982, “Acts, events, and things”, in Leinfellner, W. et al. (eds.), Language and Ontology, Dordrecht: D. Reidel.

Allan Bäck

Reinach, Adolf Adolf Reinach (1883-1917) was one of the main representatives of early phenomenology. He studied in Mu-

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nich before moving to Göttingen, where he worked with Edmund Husserl. Even though Reinach did not elaborate a mereological theory in its own right, he made an innovative use of some mereological concepts originally developed by Husserl in the third of his Logical Investigations (“On the theory of wholes and parts”). States of affairs. Reinach’s ontology

of states of affairs allowed him to clarify some aspects of Husserlian mereology. According to Husserl, all real (real) predicates denote a part of the object that is referred to by the subject of these predicates. For instance, in the sentence: (1) the rose is red the predicate ‘red’ refers to a part of the rose. But not all predicates are real. In the sentence: (2) the rose is one the predicate ‘one’ does not point to any part of the rose, as one is not a real predicate, but rather a ‘categorial’ or ‘formal’ predicate. (Formal predicates are empty concepts which ‘group around’ the equally empty notion of ‘object as such’, Logical Investigations II, p. 19.) All parts (Teile) of a whole are either pieces (Stücke) or moments (Momente). Pieces are independent or concrete parts, which means that these parts can exist even if they were not contained in some whole. As an example, consider the following sentence: (3) the rose has petals.

Here the petals are pieces of the rose as they can exist without the rose. By contrast, the predicate ‘red’ in (1) points at a moment, which is a nonindependent, or abstract, part of the rose. The particular red nuance of the rose would not exist without the rose, and the same applies to all moments: they can exist only in relation to a whole. Moments, Husserl contends further, might be related to a whole by necessity. For instance, in the following sentence (4) this shade of red is extended the red’s extension is not a part that inheres in the shade of red by matter of contingency: rather, extension inheres in the shade of red by necessity. Husserl argues that this necessity is grounded in the essence of red (cf. Mulligan 2004), for this shade of red is extended in virtue of the fact that it is a case of the essence of red. In other words, not only this particular nuance of red but all nuances of red are extended, and it is a law of essence (Wesensgesetz) that the species of red (or of colour as such, for that matter) requires extension. If, however, “objective necessity is as such tantamount to a being that rests on an objective law” (Logical Investigations II, p. 12), then which element properly bears the modal property of necessity? More precisely, is the bearer of necessity the object, denoted by the subject of sentence (4), or the nonindependent part of the object, referred to by the predicate? Reinach’s theory of states of affairs offers a clear and original answer to this question (cf. Mulligan 2004: 400).

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According to Reinach, modal bearers are never objects; only states of affairs (Sachverhalte) can be qualified as necessary, probable, or possible (Reinach 1989b: 115). Therefore, necessity can neither be found in the moments taken as such nor in the objects of which these moments are parts, but rather in states of affairs (e.g., in the state of affairs that makes sentence (4) true). Within Reinach’s (and Husserl’s) formal ontology the shade of red and its extension are objects (Gegenstände). However, the union of copula and predicate, ‘is extended’, is not an object, but rather an incomplete fragment requiring supplementation (ergänzungsbedürftig) of a state of affairs (the being-extended of this red shade). This incomplete entity does not coincide with the red’s extension, i.e., it does not coincide with the moment of extension belonging to this red shade. Nor does the being-extended of this nuance of red correspond to the referent of the complex singular term ‘this extended red shade’. Singular terms of this latter kind denote complex formation (Gebilde) called by Reinach 'factual materials’ (sachliche Tatbestände) (cf. Smith 1987). Factual materials underlie (zugrundeliegen) states of affairs, but differ from them ontologically. For instance, only the being-extended of this red shade has a negative counterpart, i.e. its not-being-extended (Reinach 1989b: 121), whereas no negative objects exist and therefore no negative objects can function as predicates or as subjects. Accordingly, expressions like ‘not-extended’, ‘not-red’ or ‘non-smoker’ should be

considered as linguistic abbreviations for syntactically more complex expressions referring to states of affairs (cf. Reinach 1989b: 138). Like negation, modal properties can pertain only to states of affairs. In particular, if objects grounded in the essence of the objects in subject position, then the corresponding states of affairs are necessary (Reinach 1989a: 70). Thus, all so called “essential connections (Wesenszusammenhänge)” can be described in terms of necessary states of affairs (Reinach 1989a: 70): this leads to the idea that, since to be an object is to be an object with a certain essence, all objects “found” (fundieren) some necessary states of affairs. In conclusion, Reinach’s clarification of the ontological notion of state of affairs can be seen as a refinement of Husserl’s mereology, since it specifies the notion of necessity as a modal property instantiated only by states of affairs. States of affairs do not exist at the same ontological level as objects (i.e. wholes and their parts), and they instantiate specific properties that objects do not – and cannot – instantiate (inter alia modal properties). See also > Facts, Husserl, Segelberg, Tropes. References and further readings

Mulligan, K., 2004, “Essence and Modality: The Quintessence of Husserl’s Theory”, in Textor, M. (ed.), Semantik und Ontologie, Heusenstamm: Ontos, 387-418.

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Reinach, A., 1989a [11911], “Kants Auffassung des Humeschen Problems”, in Reinach, A., Sämtliche Werke. Textkritische Ausgabe in 2 Bänden, Schuhmann, K., Smith, B. (eds.), München/Hamden/Wien: Philosophia, 67-94. Reinach, A., 1989b [11911], “Zur Theorie des negativen Urteils”, in Reinach, A., Sämtliche Werke. Textkritische Ausgabe in 2 Bänden, K. Schuhmann, B. Smith, eds., München/Hamden/Wien: Philosophia, 95140. Smith, B., 1987, “On the Cognition of States of Affairs”, in: K. Mulligan, ed., Speech Act and Sachverhalt. Reinach and the Foundations of Realist Phenomenology, Dordrecht/ Boston/Lancaster: M. Nijhoff, 189227. Alessandro Salice

Rhetoric A definition of the part-whole relation is not in the scope of rhetoric. From a rhetorical viewpoint all that matters is transition from whole to part, whole to whole, part to part, and part to whole. From classical rhetoric – roughly speaking the rhetoric of Aristotle, the Rhetorica ad Herennium, Cicero and Quintilian – up to the baroque and enlightenment era, transitions between part and whole were dealt with in the theory of tropes. As far as questions of adequate style and expression are concerned, it was part of the elocutio, in the classical period one of the educated speaker’s (vir

bonus) duties (officia orationis) supposed to ensure sufficient clarity and perspicuity. As a result, the partwhole relationship was seen as a relatively special topic in the field of rhetoric, appearing under the head of synecdoche, i.e. a set of tropes for replacement of a part by a whole or the reverse. Synecdoche can occur in six different types: 1) pars pro toto; 2) (the reverse) totum pro parte; 3) genus pro specie (transition from special to general concepts, e.g. ‘mortals’ for ‘man’); 4) (the reverse) species pro genere; 5) plurale tantum (transition from plural to singular: ‘the German’ instead of ‘the Germans’); 6) (the reverse) pluralis pro singulari (‘Germans are never scared’ for a special German). The first case, the pars-totum synecdoche is much more common in the literature than the other types, the genus-species and the numerus synecdoche. The Rhetorica ad Herennium displays the full theory. In the sentence (1) ‘Were not those nuptial flutes reminding you of his marriage?’ (Ad Herennium 1989, IV, XXXIII, 44) the entire marriage ceremony is suggested by the flutes as a special sign, which is a case of pars pro toto. Another example reflects the opposite direction: (2) ‘You display your riches to me and vaunt your ample treasures’. Here a part, or special feature of wealth is to be understood as starting from the whole. These examples show that the no-

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tions of part and whole are used in classical rhetoric in an intuitive way. There is no such thing as a rhetorical concept of the part-whole relation. The above distinction between the pars-pro-toto and the totum-pro-parte synecdoche does not work at a semantic level of language analysis. There is no obvious reason why sentence (2) shouldn’t be seen as a case of pars pro toto. Also genus-species and numerus synecdoche rather reflect special contexts in which a partwhole relation is established and the predicates ‘part’ and ‘whole’ are omitted, than logically different relations. Quintilian, however, seems to distinguish the numerus synecdoche from the others by extensionality: “It is where numbers are concerned that synecdoche can be most freely employed in prose”, as in (3) “The Roman won the day”, where one Roman stands for the total number of all Romans (Quintilian 1986, VIII, 6, 19). Some modern authors hold however that the idea of a quantitative shift is specific for all types of synecdoche (Koch & Winter-Froemel, 2009: 357). The rhetorical approach has its advantages if the pragmatics of part and whole are to be scrutinised. Rhetorical tropes can help to discover hidden mereology, especially in ordinary and poetic language, in which concepts of part and whole are often used implicitly for the sake of brevity or elegance (Cicero 1992, III, XLII, 168). The legitimate use of tropes in classical rhetoric is restricted to intelligible forms of deviation from the proper meaning of words. The idea that words (verba) have a proper meaning

(res) was not to be sustained, however, since a clear-cut distinction between synecdoche and catachresis (the inexact use of a like and kindred word in place of the precise and proper one, cf. Ad Herennium 1989, IV, XXXIII, 44) was never at hand. Apart of these special cases, partwhole relations play a more general role in rhetoric, insofar as ornament is mixed up with forms of argument in many instances. Cicero distinguishes between topics (common places) inherent in the nature of the subject and others which are brought in from without. Inherent arguments often depart from a definition of the major term from which a judgment about the minor term is derived. Deduction is thus (applying genus synecdoche) defined as subordination of parts under a whole (Cicero 1993, II, 8). The idea that ornamentation was to be restricted to calculated deviation from proper use, mirrored by the virtues of correctness and clarity (Rowe 1997: 122-123) attributed to it, dominated rhetoric throughout the Middle Ages. The concept of synecdoche was transmitted to the Latin West by the Fourth Book of the Rhetorica ad Herennium, the Grammar of Donatus, and Isidor of Seville’s Etymologica (Murphy 1974: 183189; Schindel 1975: 226-228). It was not until modern times that rhetoric overcame the “proper meaning superstition” of the res-verbum theory, as I.A. Richards has shown contrasting Enlightenment rhetoric (Home 2005) with Coleridge’s idea of organic semantics (Richards 1965: 11; 17; 103-112; Coleridge 1816). Neither what an author or speaker

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means to say nor what he really says is something fixed, known to the recipients before they hear his words. On the contrary, it is the result of an “interplay of interpretative possibilities of the whole utterance” (Richards 1965: 55). What Richards and, after him, Burke (1969) develop is an account of meaning depending on the idea that metaphor is omnipresent in language. Words cannot be meaningful in isolation, but only in an organic totality of interinanimating words. Language users only grasp in part the inexhaustible meaning of an utterance. Only the “full use” some poets can make of language approximates the totality of meaning possibilities that are dormant in language (Richards 1965: 85). Here, a rhetoric notion of organic semantic totality is at work, and one that is underlying the understanding of parts of meaning entailed in it, but at the same time does not allow for composition from the parts. It is a totality of expressed meaning that is underlying partial understandings of utterances in ordinary and poetic language (the case of full access to meaning is restricted to special instances of scientific language). Alongside with this, a “dramatist” view of action interprets actions as “things contained” in the whole dramatic scene from which agent, agency and purpose can be understood (Burke 1955). A New Rhetoric is considered to be an analysis of processes of identification of meaning by language users, by which recipients and orators identify with one another (Burke 1969: 19-27, 5559). Transitions from part to whole, in classical times a special branch of

the theory of tropes, has developed to the cornerstone of a rhetorical account of meaning. Today, the idea to use part-whole tropes to determine ways of categorisation plays an important role in Prototype Theories of categorisation (Rosch 1978). G. Lakoff (1987: 283 f.) has suggested that complex models of mental spaces, which are image-schematic concepts without spatio-temporal reference, structure our cognition of abstract domains. Categories in general are to be understood as container-schemas. Part-whole and up-down schemas are used to conceptualise hierarchical structure. Other image-schematic concepts are the link schema, the centre-periphery, front-back and linear-order schemas. Spatialisation requires itself metaphorical mapping from a source to a target domain. Metaphorical mapping is also understood in terms of image schemas. On the one hand, image schemas are concepts understood directly, on the other hand they are transferred metaphorically to structure other complex concepts. Cognitive models have complex mereological structures in two ways. They are extensional “building-block structures” in which the meaning of the whole “is a function of the part” (Lakoff 1987: 284), or otherwise, with reference to gestalt theory, nonextensional gestalt structures with at least some elements that are dependent on the gestalt – i.e. elements not all of which exist independently or whose meaning is not predictable from its parts. This shift from an a priori account of classical categories, disembodied ones in terms of objec-

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tivism, towards a new theory of a categorisation physiologically and psychologically accessible (embodied, cf. Lakoff & Johnson 1999) is not restricted to propositional attitudes but also includes imagination (Johnson 1987). From a rhetorical point of view, this amounts to an application of the synecdoche to cognitive science. See also > Common Sense Reasoning about Parts and Wholes, Gestalt, Non-literal Language Use and PartWhole Relations. References and further readings

Ad Herennium, 1989, On the Theory of Public Speaking, translated by Harry Caplan, Cambridge (Mass.), Harvard University Press. Burke, K., 1955, A Grammar of Motives [1945], New York: George Braziller. Burke, K., 1969, A Rhetoric of Motives [1950], New York (reprint Berkeley and London 2007, University of California Press). Cicero, M. T., 1992, De Oratore, Book III, De Fato, Paradoxa Stoicorum, De Partitione Oratoria, with an English translation by H. Rackham, Cambridge (Mass.): Harvard University Press [1942]. Cicero, M. T., 1993, De Inventione, De Optimo Genere Oratorum, Topica, with an english translation by H. M. Hubbel, Cambridge (Mass.), Harvard University Press [1949].

Coleridge, S. T., 1816, The Stateman’s Manual, London. Home, H.; Kames, L., 2005, Elements of Criticism [1761], ed. with an Introduction by P. Jones, 6th edition, Indianapolis, Liberty Fund. Johnson, M., 1987, The Body in the Mind, The Bodily Basis of Meaning, Imagination, and Reason, Chicago: Chicago University Press. Koch, P.; Winter-Froemel, E., 2009, “Synekdoche”, in G. Ueding (ed.), Historisches Wörterbuch der Rhetorik, vol. 9, Tübingen: Niemeyer, 356366. Lakoff, G., 1987, Women, Fire, and Dangerous Things. What Categories Reveal about the Mind, Chicago: Chicago University Press. Lakoff, G.; Johnson, M., 1999, Philosophy in the Flesh, The Embodied Mind and its Challenge to Western Philosophy, New York: Basic Books. Quintilian, M.F., 1986, Institutio Oratoria, with a translation by H. E. Butler, Cambridge (Mass.), Harvard University Press. Murphy, J. J., (1974), Rhetoric in the Middle Ages. A History of Rhetorical Theory from Saint Augustine to the Renaissance. Berkeley, Los Angeles, London, University of California Press. Rowe, O. G., 1997, Ch. 5, “Style”, in Porter, S. E. (ed.), Handbook of Classical Rhetoric in the Hellenistic Period 330 B.C.-A.D. 400, Leiden, New York, Brill.

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Richards, I. A. 1965, The Philosophy of Rhetoric [Byrn Mawr, 1936], New York: Oxford University Press. Rosch, E., 1978, “Principles of Categorization”, in, Rosch, E. and Lloyd, eds., Cognition and Categorization. Hillsdale, N.J. Lawrence Erlbaum Associates, 27-48. Schindel, U., 1975, Die lateinischen Figurenlehren des 5. bis 7. Jahrhunderts und Donats Vergilkommentar (mit zwei Editionen), Göttingen. Ueding, G. (ed.), 2005, Rhetorik, Begriff - Geschichte - Internationalität, Tübingen: Niemeyer. Temilo van Zantwijk

Russell, Bertrand Russell’s first contact with mereology occurred probably in the time around 1899 during his investigations on the philosophy of Leibniz. In the early idealistic phase (1895-1899) he did not reflect much about the parthood relation, because it “has been wrapped in obscurity – though not without certain more or less valid logical reasons – by writers who may be roughly called Hegelian” (1903: §133). The main ‘logical reason’ for this is the idealistic doctrine Russell begins to reject at this time, namely, that analysis (decomposition of a whole in its parts) implies falsification, i.e. only a theory of the reality as a unified whole can be true (see Russell 1918: 178). But even recognising some importance of mereological analysis after 1900, he never accepted it to be so fundamental as set

theory or predicate logic (1903: § 139). One first peculiarity of Russell’s conception of mereology (of course, he did not use the term ‘mereology’, he just did talk about the whole-part relation) is that he uses the word ‘whole’ to refer only to things with proper parts: atoms are not wholes. Whatever is not a class is for Russell a unit; some units are simple other complex. A complex unit is a whole (1903: §133). He discussed in Principles the possibility of reducing the notions of whole and part (of concepts) to notions of pure logic. Since for every x ‘x is a man’ implies ‘x is mortal’, for mortal is a (‘intensional’, we should say) part of man we could be tempted to conclude that A is a (proper) part of B when B is implies A is, but A is does not imply B is. Russell rejected this reduction because there are counter-examples: take ‘A is greater and better than B’ implies ‘B is less than A’. The converse implication does not hold, but the latter proposition is not part of the former. This controversial example depends, of course, on his thesis concerning irreducibility of relations, in particular, that the converse relations greater and lesser are not reducible to each other. His second example is ‘A is red’ implies ‘A is coloured’, but not the reverse. Nevertheless, we should not say that red is more complex that coloured, for both concepts are equally simple. This example is suspect too, as coloured could be considered an intensional part of red. In any case, for Russell, there are many kinds of whole-part relation,

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and at least one of them – the intensional one – is indefinable. Russell distinguished three kinds of the whole-part relation. The first one corresponds roughly to the set theoretical ∈: the whole, in this case, is called ‘aggregate’, and its parts ‘terms’. Contrary to the other kinds, an aggregate is definite as soon as its terms are known. Terms have no direct connection inter se, but only the indirect connection of being parts of the same whole. Despite appearance, aggregates are not collections without any logical need or relevance. Even if such a collection is fully determined when the terms are given, it is different from its members, and its autonomy is needed for some purposes of mathematics, such as theory of fractions and theory of extensive quantity. The second part-whole relation corresponds roughly to setinclusion ⊂, and is definable in terms of implication: if for every x, x is a v implies x is a w, and the converse does not hold, then v is a proper part of w. Parts in this sense are genuinely called ‘parts’ in Russell’s view (we would call them ‘subsets’, of course). Only this sense admits logical addition and multiplication (in the Boolean sense). The third part-whole relation, finally, is basically the special case of the relation between the parts of a proposition and the whole proposition: A, B and difference are parts of the proposition A differs from B. These parts are called ‘constituents’, and the constituents of a proposition are directly connected to each other. Propositions are not simple extensional aggregates, for A is greater than B and B is greater than A are

identical qua aggregates, but not qua propositional wholes. This kind of wholes, i.e. propositions, are the main object of logical analysis. Because of these various senses, the whole-part relation is not transitive: if A is part of B in one sense, while B is part of C in another sense, we cannot infer that A is part of C. (1903: §135). From this, we could extract a solution – a restriction – for the problem of transitivity of part-whole relation: transitivity is only given when the part-whole relation involved is not ambiguous. Russell defended the actual existence of infinite wholes. Space, time and the series of natural numbers are examples of infinite wholes of the first kind, i.e. they are aggregates (1903: §140). But he left unresolved the question if there could be propositions with infinite complexity. His early theory of denoting concepts (before 1905) served, under other purposes, to enable propositions of finite complexity – with expressions like any, all, and every – to deal with infinite classes of terms. Thus, a finite unity or proposition can be about an infinite aggregate. Russell’s mature (after 1908) solution for the famous paradox of logic and semantics he investigated is based on a mereological intuition. He suggests the vicious circle principle, which forbids a certain kind of ‘mereological circularity’. According to this principle, one should not define or tacitly assume wholes, whose existence would entail the existence of parts of the same whole that are only definable in terms of it. Gödel (1989:

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135) noted that Russell’s principle (‘no totality can contain members definable only in terms of this totality, or members involving or presupposing this totality’) has three different interpretations corresponding to the phrases ‘definable in terms of’, ‘involving’ and ‘presupposing’. The second and third are more plausible than the first. Russell’s basic intuition seems to be that the relation of ontological dependence between a whole and its parts cannot be symmetric. Thus, the whole and its parts cannot simultaneously ‘presuppose’ the existence of each other. But some logicians have argued that there is no reason for rejecting that some wholes and some parts (or members) are only definable in terms of each other, unless we suppose that these entities are ‘created’ by their definitions, like in some versions of constructivism. See also > Logical Atomism, Metaphysical Atomism, Collectives and Compounds, Propositions. References and further readings

Gödel, K., 1989, “Russell’s Mathematical Logic” in Schilpp, P.A. (ed.), The Philosophy of Bertrand Russell, Illinois: Open Court. Hylton, P., 1990, Russell, Idealism and the Emergence of Analytic Philosophy, Oxford: Clarendon Press. Imaguire, G., 2001, Russells Frühphilosophie: Propositionen, Realismus und die Sprachontologische Wende, Hildesheim: Olms.

Russell, B., 1900, A Critical Exposition of the Philosophy of Leibniz, London: Routledge, 1992. Russell, B., 1903 [1992], The Principles of Mathematics, London: Routledge. Russell, B., 1908, “Mathematical Logic as Based on The Theory of Types”, in Logic and Knowledge. London: Routledge, 1994. Russell, B., 1910/1997, Principia Mathematica. Volume I., Cambridge: Cambridge University Press. Russell, B., 1918, “The Philosophy of Logical Atomism”, in: Logic and Knowledge. London: Routledge, 1994. Guido Imaguire

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S Scherzer, Johann Adam Johann Adam Scherzer (1628-1683) taught philosophy in Leipzig. Of historical importance to mereology is the chapter of his Axiomata resoluta on the subject (Scherzer 1686). This work influenced his pupil Leibniz who treated Scherzer along with Joachim Jungius as sources of his mereological analyses in a number of texts. Scherzer’s mereology chapter is an introduction to the part-whole relation for undergraduates. The author treats the part-whole relation as one of the three basic ontological relations – together with substanceaccident and cause-effect. The text bears resemblance to the obligationes, introductions to modal theory written for students by the postmedieval scholastic logicians and philosophers. Scherzer’s approach to mereological structures is typical for his time and also for similar medieval and postmedieval texts. Mereological discussions of this kind are not to be found for centuries to follow. Mereological structures were, at any rate, not explicitly discussed in later textbooks. Scherzer discusses different kinds of wholes, ontological structures and the

difference between the ontological and the epistemic approach. He formulates eleven rules: I. The simple is prior by nature to the composite whose part it is. The simple is not always temporally prior to the whole because sometimes the simple and the composite emerge simultaneously. E.g. when God created prime matter he added form to it at the same moment. (Prime matter is traditionally seen as ens rationis or more exactly as ens rationis cum fundamento in re.) The simple is prior by nature necessarily. Scherzer points out that in the following conditional the consequent does not follow from the antecedent: “There is a part, therefore there is a whole” whereas the following conditional is always true: “There is a whole, therefore there is a part”. II. As far as being as such is concerned, the part is prior to the whole (by nature and order of generation) Every part insofar as it is a (proper) part with respect to the composite, is something simple, but not every simple entity is a (proper) part. Scherzer uses a reduplicative sentence in order to characterise the relation between the part and the composite: every part is simple with respect to its composite, but being simple alone is not a sufficient condition for being also a (proper) part. Again, Scherzer makes clear that this rule does not state that the part is temporally prior. Like all related things, the whole and the parts are simultaneous because nothing is (or begins to be) a composite unless

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there are parts. The category of relation already elucidates this fact. The addition ‘by nature’ says that in contradistinction to linguistic expressions a part cannot be ontologically or metaphysically prior to the whole. With respect to nature’s intention, the whole is prior to the parts. Nature primarily intends to generate a man, not a head. But since nature first produces and forms limbs, by order of generation, the part can be said to be prior. A special question is: Teeth as integral parts of an animal are posterior to their whole. But which is the whole whose parts are teeth? This appears to be the dentated mouth. The teeth are posterior to the dentation of the mouth because they are temporally later. III. The whole (quantitative or integral) is greater than its parts (not taken simultaneously) To support this rule, Scherzer produces examples from arithmetic and geometry: the number eight is greater than the numbers four or three, its parts. A six-feet line is longer than a two-feet line. This rule is restricted to quantitative and integral wholes. In an essential whole this rule fails, because a man as a whole is not greater than his soul. Taken simultaneously the parts are equal to the whole, because the whole is nothing else than the parts together. In this case the whole is conceived as a mereological sum of its parts.

Scherzer does not discuss the mereology of universals, such as man or animal because it is not relevant for beginners and perhaps also too difficult. IV. The (essential) whole and its parts are in the same category. E.g. the whole man and his parts, soul and his body, belong in the category of substance. In this case, the essential whole is conceived in contradistinction to the accidental whole for which the rule fails. E.g. the parts of white things, musical things and other accidental unities, belong to two categories: the ultimate subject, whether milk or robe, belongs to the category of substance; whiteness, however, belongs to the category of quality. V. Whatever is a part of a part is also a part of a whole. This rule, one that expresses the transitivity of the part-whole relation, has no counterexamples. E.g. the finger is part of a hand and the hand is part of the human body. Therefore the finger is also part of the human body. VI. The part is not predicated of the whole (which is per se one, integral, and essential physical thing). It is neither correct to say: ‘Man is a soul’, ‘Man is a body’, ‘Man is a foot’ (‘…a hand’, ‘…a head’, ‘…a nose’ etc.) nor is it correct to say: ‘The soul is a man’, ‘The foot is a man’, ‘The hand is a man’. In an accidental whole only concrete things can be predicated of each other. E.g. ‘This body is white’, ‘This white thing is a body’. One can nei-

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ther predicate abstracts of concretes (e.g. ‘Whiteness is a white thing’) nor abstracts of abstracts (e.g. ‘Whiteness is sweetness’) although one may very well say that the white thing is sweet. This is an important insight. Leibniz later says that a triangle is trilateral, but trilaterality is not triangularity. In this context, Leibniz uses the term reduplication to distinguish between three semantic levels: word signs, concepts and entities or objects. It is important to restrict this rule to essential physical things as the wholes in question because an essential logical thing can be predicated of its parts after all – e.g. ‘Man is an animal’. In this case animal is a universal whole, man is a subjective part. Here, Scherzer appears to conceive of natural kinds as intensional wholes. VII. Whatever is predicated of a part of a universal is also predicated of the whole universal and vice versa. E.g. ‘Man as part of animal is a substance’. Since this rule is not valid of other entities, its restriction to universals is important. Notice that the following implications are false: ‘A certain part of a man is a soul; therefore the whole of man is a soul’; ‘The soul is intelligent, therefore the universal man is intelligent’. The rule holds only when it is taken to pertain to entities taken absolutely. Notice that the following is not valid: ‘A part is a part, therefore the whole is a part’. Also the following is not valid: ‘Man is a species of animal distinct from brute, therefore also

animal is a species of animal distinct of brute’. These predications are formally and reduplicatively made of a part insofar as it is a part. The inverse implication (from the universal whole to its parts) holds without restrictions. It is interesting to note that Scherzer uses in this context the word ‘reduplication’. VIII. Having defined all the parts whatever they may be and whenever they are combined, the whole that is constituted by those parts is also defined. For example, a natural whole is defined once matter and form are combined. Whiteness having been united to milk, a white thing is defined. ‘Whatever they may be’ means that the parts may be either essential or accidental. If the former; the corresponding wholes are essential, if the latter they are accidental. The clause ‘that is constituted by those parts’ is documented by examples. ‘Having been combined’ is added because a union is required. Parts might not yet create a composite even when they are close to each other. IX. If any part is subtracted, the whole whose part this was either ceases to exist or fails to remain the same. Scherzer’s examples for this are essential and integral wholes. If the soul is taken from a man, the man ceases to exist. If a hand or head is

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taken from a human body, the integral body does not remain the same. X. Given a whole, all its parts are defined.

all status of an entity, it would be necessary that the body and the soul of the cow perish simultaneously.

This is simply the inverse of the last rule and is self-explanatory. E.g. if there is a triad, it will have three units.

See also > Medieval Mereology, Reduplication, Persistence, Transitivity.

XI. If the whole (no matter what kind) is removed (with respect to relative being), it is not necessary that all the parts are removed.

References and further readings

‘No matter what kind’ means that the rule holds for all wholes, whether they are homogeneous or heterogeneous, essential or accidental wholes. E.g. for homogeneous wholes: when a given stone is divided into three stones, the whole ceases to exist but all the parts remain. E.g. for heterogeneous wholes: when a man’s hand is amputated, his body does not remain integral or the integral whole it was before the amputation but the parts survive. E.g. for essential wholes: if a man dies, his body and soul still survive. E.g. for accidental wholes: if the whiteness is deleted, the tablet can still remain. ‘With respect to relative being’ means according of the whole’s being of the kind with respect to which its parts are said to be removed. If the whole, conceived as being as such or being qua being, ceases to exist, it would be necessary that every part be taken away simultaneously. An example is annihilation: if God annihilated a cow, so that it lost

Scherzer, J.A. (1654a), Vademecum sive Manuale philosophicum, Pars III, Axiomata resoluta, Leipzig: Weidmann, 1686. Scherzer, J.A. (1654b), Axiomata resoluta, translated by John Kronen and Jeremiah Reedy, introduction and explanatory notes by John Kronen, Munich: Philosophia, 2016. Hans Burkhardt Stamatios Gerogiorgakis

Segelberg, Ivar Ivar Segelberg was a unique figure in Swedish philosophy at the middle of the 20th century. Born in Sweden in 1914, he published his doctoral dissertation, Zeno’s Paradoxes, in 1945 and became a docent in theoretical philosophy at Uppsala University in 1947 with the publication of his second book, Properties. The latter, continuing themes from the earlier book, developed, in great detail, a systematic theory of what are now commonly called ‘tropes’. From his student days in philosophy Segelberg was influenced by the Swedish philosopher Adolph Phalén. Phalén, as Segelberg was to also do, combined the analytic

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metaphysics and ontology of C. D. Broad, whom Segelberg knew from Broad’s visits to Swedish universities, Russell, and Moore with the phenomenology of Husserl and theory of objects of Alexius Meinong. Appointed to the chair of theoretical philosophy at Gothenburg University in 1951, he set out a unique blending of basic themes from the diverse traditions in his last book Consciousness and the Idea of the Self of 1953. Until his retirement in 1979 he was the center of a lively and productive group of philosophers whose focus on the philosophical issues that preoccupied the Cambridge philosophers, on the one hand, and the phenomenological movement, on the other, was unique in Sweden. For in Sweden logic, ethics and modal logic were the dominant areas of interest in the 1950s, 60s and 70s. Segelberg died in Gothenburg in 1987. Segelberg’s focusing on fundamental problems posed by ontology and intentionality, along with his Moorean manner of considering concrete cases and taking linguistic usage to provide clues as well as evidence for philosophical distinctions, contributed to his distinctive place in Swedish philosophy. These characteristics would continue to stamp the productive group that formed around him and produced numerous works on major figures of the analytic and phenomenological traditions. (Segelberg’s blending of the two traditions is reflected in his noting in 1953 that he now used “…the word ‘ontology’ in the same sense as I used ‘phenomenology’ in Zeno’s Paradoxes…in the sense of … Meinong’s Gegenstand-

stheorie.” (Segelberg, 1999: 239) Zeno’s Paradoxes contains one of the most detailed examinations of the paradoxes in recent philosophical literature. An early modern advocate of a tropist account of qualities (and of ordinary relations, as opposed to ‘connections’ or logical relations – part of, exact similarity), Segelberg also developed a system of mereological combination. He distinguished, as basic and inexplicable ‘urphenomena’, complex unities from collections as two kinds of complexes. Complex unities do not involve their constituent ‘parts’ being combined by a relation or nexus but exist given their parts. Consequently, a complex unity consisting of the parts x, y differs from a corresponding collection with the same constituents, x + y, without there being a difference in ‘content’. (Segelberg, 1999: 265) No two complex unities of the same ‘kind’ can so differ. (Related is Segelberg’s proposal that universals differ from particulars in that it is possible for diverse particulars to be exactly similar but that was not possible for diverse universals. Complex unities are ‘structured’. This led him to become one of the first philosophers, if not the first, since Russell (in his 1913 unpublished manuscript, Theory of Knowledge), to address the philosophical problem posed by the need to account for ‘order’ in structured complexes. In the case of complex unities, he recognised a ‘structure’ that contained ‘places’ filled by constituent parts. This is reminiscent of

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Russell’s ‘logical forms’ of facts. Segelberg, however, did not employ his notion of a structure in his analysis of facts. Segelberg’s views about complexes also fit with his rejection of Wittgenstein’s Tractarian doctrine that all complexes are facts. Facts for Segelberg involve relations, complex ‘objects’ do not. Recognising facts, as ‘higher order’ entities in the phenomenological tradition, as well as ‘structured’ complex objects, he took nonsymmetric relations to have a ‘plus’ and a ‘minus’ side. This is like Russell’s sometimes taking such relations to have a ‘direction’. Again, like Russell, he took symmetric relational facts not to involve relational order, for the same fact would provide the ground of truth for ‘aRb’ and ‘bRa’ in such a case. Like his friend Gustav Bergmann, but years before, Segelberg distinguishes collections from classes. Yet, unlike Bergmann who takes classes to have elements yet not be complexes, he not only takes elements to be parts of classes but requires that classes have a ‘form’. The form, like a structure in a fact, provides for the unification of the elements into a further entity. Thus he easily obtains the null class as being simply the ‘class form’. Nevertheless, he then takes classes to be ‘pure fictions’. Segelberg follows Meinong and takes the connection between a conscious state – a thought of the Eiffel tower, for example – and its intentional object to be necessary or internal. For

him, a mental state is usually a complex composed of tropes that ‘intend’ the trope-qualities of the objects; the complex unity composed of such intentional tropes and an ‘instance’ (moment, trope) of ‘consciousness’ stands in the basic intentional relation to its intended object (provided the latter exists). Alternatively, Segelberg suggests that in some cases the ‘intending’ quality can be simple, though its intentional object is complex. He dismisses, rather than resolves, a problem often associated with Meinong regarding non-existent objects and states of affairs – objects beyond being and non-being. Segelberg’s books are rich in metaphysical analysis and suggestions. In addition to themes indicated above, his books contain proposals for basic propositions for various kinds of part-whole relations, insightful discussions of phenomenological and physical space and time, detailed analyses, and proposed solutions to, the puzzles posed by the concepts of the ‘self’ and of ‘substance’ and illuminating analyses of Husserl’s ‘laws of essence’. He also, at various points, addresses issues posed by materialism, phenomenalism and perception, and the ontological grounds of logic and of causal laws and engages in pointed critical examinations of writings of C.D. Broad, E. Husserl, G. E. Moore and G. F. Stout. See also > Logical Atomism, Bergmann, Compounds and Collectives, Gestalt, Intentionality, Meinong, Reinach, Russell, Structure, Tropes.

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References and further readings

Hochberg, H., 1999, Complexes and Consciousness, Stockholm. Segelberg, I., 1945, Zenons Paradoxer, Stockholm. Segelberg, I., 1947, Begrippet Egenskap, Uppsala. Segelberg, I., 1953, Studier over Medvetandet och Jagidén, Stockholm Segelberg, I., 1999: Three Essays in Phenomenology and Ontology, (trans. Hochberg, H.; Hochberg, S. R.) Stockholm. (English translations of 1945: 1947 and 1953.) Svennerlind, C., 2004, “Ivar Segelberg on Relations”, in Ursus Philosophicus: Essays Dedicated to Björn Haglund, 153-171. Herbert Hochberg

Shadows A shadow is a two-dimensional region of space from which light is blocked and hence within which light is absent due to the presence of a three-dimensional object coming between the region and the light source. A shadow is therefore a kind of absence: light is not merely absent from the region, but the shadow is itself an absence of light that spatially marks out the region. Two things, apart from the region, are required for a shadow to exist – a light source and a caster that blocks or occludes the light. This makes a shadow a kind of ontologically dependent entity (Lowe 1994: 1998) as

opposed to an individual substance such as the caster itself. The shadow’s existence depends on that of the caster and the light source. As absences, shadows might seem strange, but they have good company and it seems difficult to eradicate absences from ontology. Holes are best thought of as absences (Casati and Varzi 1994). Illness is the privation of health (Aristotle, Metaphysics, Book 9). Blackness is arguably an absence of chromatic colour (Sorensen 2008, ch.12). Cold is the absence of heat, since heat is a form of motion. Negative facts seem to play a role in semantics as well as metaphysics that is difficult to eliminate (Russell 1985 [1918]). Since shadows are absences of light, how can we see them? For sight seems to require light, and moreover, given the causal theory of perception (Grice 1961), how can a mere absence be an object of vision if it does not cause anything? If seeing always requires the transmission of light from the thing seen, it is hard to explain how we see other absences such as holes or an object such as a silhouette, which is the dark surface of a backlit object. Sometimes we can see an object simply by virtue of its effects (an astronomical black hole). With silhouettes and shadows, it is possible that we see the object due to its contrast with other things we see that do transmit light (Sorensen 2008). Arguably, we see holes not so much contrastively but by virtue of their not being pure absences, rather absences combined with illuminated surfaces that ipso facto transmit light.

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The causal theory of vision, if it were to stand in the face of such entities as shadows, would need significant refinement. It would be ill-advised to suggest we merely see that there is a shadow by inference from other things we see directly, since our sight of shadows seems every bit as direct as our sight of their casters (when we can see them). Perhaps the causal theorist is required to allow causation by absences, or else to amplify the causal element so that what does the causing that produces sight of X need not be X itself. Shadows seem to exhibit many of the features of ordinary material objects. They appear to move, to rotate, and to change shape. Yet we cannot analyse these features in exactly the way we do for material objects (For distinct approaches see Casati and Varzi 1994, Sayan 1996). When shadows interact spatially they do not cause changes in each other in the way cars do when they collide. There is no causal dependence between a shadow’s being in one place and its being in another when it moves: its being located in each place are effects of a common cause, namely the caster and the light source. Perhaps shadows are a kind of ‘pseudo-process’ (Salmon 1984), in which each part of the apparent process is causally independent of the other. Yet what could the parts of such a pseudo-process be? It is tempting to follow the temporal parts theorist and hold that shadows persist only in virtue of having temporal parts. Even a stationary shadow is really a compound of smaller and smaller shadows being produced by the light source and the

caster in rapid succession, by analogy with the composition of a film by frames. It is, however, hard to see where the non-arbitrary terminus of such analysis could be. The movie is made of frames, the latter different in kind from the former. But the temporal parts of shadows are themselves shadows, so are we to say that every shadow is ultimately constituted by instantaneous shadows? Not only is it difficult to see how instantaneous objects could ever constitute a persisting object, but as Sorensen points out (2008: 86-7), instantaneous shadows would be invisible, so how could a series of invisible entities constitute something visible? We might say that shadows are infinitely divisible only in potentiality, in which case shadows would be homeomerous composites (having parts of the same nature as the whole), each part being a non-instantaneous shadow. There would be no simples or instantaneous entities underlying shadow composition. Still, shadow motion, rotation, deformation, and related features would have to be analysed only analogically to the way such features are genuinely manifested in persisting objects that lacked temporal parts, as the threedimensionalist insists in the case of material objects. In keeping with the idea that shadows, whilst real, are ontologically dependent entities requiring special analysis that can only be analogous to our analysis of material objects, we note that shadow parts do not function in the same way as the parts of their casters. Parts of material objects causally depend on and sub-

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serve each other and their wholes, but shadow parts do not. The existence and identity of shadow parts depend on the parts of their casters. As purely geometrical objects, shadow parts can be individuated spatially, by carving up regions of the shadow. But to identify a shadow part as, say, a shadow arm, we need to know about the human being (for example) whose arm casts the shadow. In other words, the complex interplay of shadow parts as effects of a common cause cannot be understood independently of their casters. In particular, shadow geometry requires an analysis of the geometrical correlations between shadow and caster (Knill et al. 1997). Shadows need to be distinguished from similar but distinct entities, such as: para-reflections, in which a dark object reflects a small amount of light that looks like but is not a shadow; para-refractions, in which a nonopaque object such as a flame or a plume of smoke allows some light through onto a nearby surface, again giving the appearance of a shadow; and filtows, i.e. bodies of filtered light that, though grey or near-black, are not shadows (Sorensen 2008, chs. 6, 7, and 9). See also > Continuants, Ontological Dependence, Holes, Perception, Persistence, Privation, Quantum Mereology. Bibliographical remarks

Casati, R.; Varzi, A., 1994. Useful material on shadows but also a very

good book on absences and negative entities generally. Knill, D.C.; Mamassian, P.; Kersten, D., 1997. A detailed analysis of shadow optics. Sorensen, R., 2008. The first fulllength philosophical study of shadows; essential reading. References and further readings

Aristotle, Metaphysics, in Ross, W.D. (ed.), 1928, vol. VIII of The Works of Aristotle, Oxford: Clarendon Press. Casati, R.; Varzi, A., 1994, Holes and Other Superficialities, Cambridge, MA: Bradford Books/MIT Press. Grice, H. P., 1961, “The Causal Theory of Perception”, Proceedings of the Aristotelian Society Supplementary Volume 35: 121-52. Knill, D. C.; Mamassian, P.; Kersten, D., 1997, “Geometry of Shadows”, Journal of the Optical Society of America 14: 3216-3232. Lewis, D.; Lewis, S. R., 1997, “Holes”, Australasian Journal of Philosophy 48: 206-12. Lowe, E. J., 1994, “Ontological Dependency”, Philosophical Papers 23, 31-48; substantially reprinted in ch.6 of Lowe 1998. Lowe, E. J., 1998, The Possibility of Metaphysics, Oxford: Clarendon Press. Martin, C. B., 1996, “How it Is: Entities, Absences, and Voids”, Australasian Journal of Philosophy 74: 5765.

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Russell, B., 1985 [1918], The Philosophy of Logical Atomism, Chicago: Open Court. Salmon, W., 1984, Scientific Explanation and the Causal Structure of the World, Princeton, NJ: Princeton University Press. Sayan, E., 1996, “A Mereological Look at Motion”, Philosophical Studies 84: 75-89. Sorensen, R., 2008, Seeing Dark Things: The Philosophy of Shadows, Oxford: Oxford University Press. Taylor, R., 1952, “Negative Things”, The Journal of Philosophy 19: 43349. David S. Oderberg

Society, Individualism and Holism (Collectivism) in the Study of, The individualism-collectivism discussion in general concerns the question if social wholes are fully reduced to the individuals, who the wholes consist of. The individualist gives an affirmative answer to this question (Hayek 1942: 286-9; Popper 1957: 127; Popper 1945: 98; Watkins 1957: 105-6), the collectivist gives a negative one (Mandelbaum 1955: 307-9; Ruben 1985, chapter 3; Weldes 1989: 362). The original methodologically individualistic position implied no special engagement of an ontology (cf. the retrospectives by Agassi 1975: 145 and Arrow 1994: 1-4), but soon the issue was given an ontological twist. Thereafter individualists often tended to see social wholes as

mereological sums of their parts, whereas holists focussed on the existence of social wholes in their own right, i.e. as not (just) mereological sums (Agassi 1960; Quinton 1975: 20-7; Ruben 1982: 295; Tuomela 1990: 139). Some arguments pro and contra were the following: • As parthood, unlike membership, is a transitive relation (i.e. every part of an entity that is a part of a mereological whole, is at the same time part of the same mereological whole), organisations cannot be studied as mereological wholes. Otherwise every part of every country that belongs to the International Monetary Fund would have to be a part of the IMF. This is clearly not the case: Bavaria, Greater London and California are not ‘parts’ of the IMF, although Germany, the UK and the US belong to it (Ruben 1983: 236). • Human individuals are mereotopological wholes, but the clubs, organisations and associations that they constitute are not, in the sense that their (scattered) members can fluctuate immensely whereas the wholes remain the same (Ruben 1983: 224). Quinton 1975: 21 argued that this disqualifies social objects as ordinary material objects but is not an obstacle to their being aggregates. • According to French 1984: 7-10; 24-5; 28-30 mereological individualism would have to see every individual member of a social whole as accountable for a certain action just in case the social whole is accountable for it; still for most social wholes this does not hold. It has been objected that the methodological individualist

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does not need to support such a strong mereology and may allow that social entities are reducible to individuals (entity individualism), although properties like accountability are not (property collectivism). Popper 1957: 126-7 had conceded that some ‘Gestalt’-properties (‘aspects’) of the whole may be irreducible to parts, but this, the argument goes, is not a denial of individualism. See also > Collectives and Compounds, Gestalt, Material Constitution, Persistence, Subject. Bibliographical remarks

French, P. A., 1984. A discussion of the main issues of the methodological individualism from a mereological point of view on pp. 1-30. Quinton, A., 1975. Vindication of the idea that social objects are wholes with human parts - with reference to methodological individualism. Ruben, D.-H., 1983. Condemnation of the mereologic conception of social wholes and a plea for ontological holism. References and further readings

Agassi, J., 1960, “Methodological Individualism”, The British Journal of Sociology 11: 244-70. Albert, H., 1990, “Methodologischer Individualismus und historische Analyse”, in Acham, K.; Schulze, W. (eds.), Teil und Ganzes, München:

Deutscher Taschenbuch Verlag, 21939. French, P. A., 1984, Collective and Corporate Responsibility, New York: Columbia University Press. Hayek, F. A. v., 1942, “Scientism and the Study of Society”, Economica 9: 267-91. Lukes, S., 1968, “Methodological Individualism Reconsidered”, The British Journal of Sociology 19: 11929. Mandelbaum, M., 1955, “Societal Facts”, The British Journal of Sociology 6: 305-17. Nagel, E., 1952, “Wholes, Sums and Organic Unities”, Philosophical Studies 3: 17-32. Popper, K. R., 1957, The Poverty of Historicism, London: Routledge. Popper, K. R., 1945, The Open Society and Its Enemies, Vol. 1, London: Routledge. Quinton, A., 1975, “Social Objects”, Proceedings of the Aristotelian Society 76: 1-27. Ruben, D.-H., 1982, “The Existence of Social Entities”, The Philosophical Quarterly 32: 295-310. Ruben, D.-H., 1983, “Social Wholes and Parts”, Mind 92: 219-38. Ruben, D.-H., 1985, The Metaphysics of the Social World, London: Routledge and Kegan Paul Tuomela, R., 1990, “Methodological Individualism and Explanation”, Philosophy of Science 57: 133-40.

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Watkins, J. W. N., 1957, “Historical Explanation in the Social Sciences”, The British Journal for the Philosophy of Science 8: 104-117. Weldes, J., 1989, “Marxism and Methodological Individualism: A Critique”, Theory and Society 18: 353-86. Stamatios Gerogiorgakis

Stoics When asked whether we humans have parts, Chrysippus (d. ca. 206 BCE), the third head of the Stoic school, is reported to have distinguished an ‘inexact’ sense of the question, according to which our parts are head, trunk and limb, and another (presumably exact) sense in which the question is about ultimate parts. In this latter sense, Chrysippus replied, “we must not ... concede any such things, but must say neither of what parts we consist, nor, likewise, of many, either infinite or finite” (Long and Sedley 50C). The Stoics believed that something was present in all of its parts and subparts, through and through, in every part of our body, namely the soul: “None of the soul lacks a share in the body which possesses the soul” (Long and Sedley 45H, 48C). In a similar way, Mind penetrates every part of the cosmos: “The cosmos is administered by mind and providence (as Chrysippus says in book 5 of his On Providence and Posidonius in book 13 of his On Gods) since mind penetrates every part of it just as soul

does us” (Inwood and Gerson Text 25 p.52). They thought of God as “an intelligent and fiery pneuma which does not have a shape but changes into whatever it wishes and assimilates itself to all things” (Inwood and Gerson Text 31.1 p.79). These conceptions of Soul, Mind and pneuma require a non-atomistic mereology. The Stoics distinguished different kinds of mixture. One of these, which they called ‘blending’, characterises the type of mixture which Soul and Mind form with what they pervade. Blending satisfies the following condition. “Certain substances and their qualities are mutually extended through and through ... so that no part among them fails to participate in everything contained in such a mixture ...” (Long and Sedley 48C). Blending is different from juxtaposition (which involves physical contact between the constituents) and from fusion (in which the constituents interact in such a way as to lose their original properties): “ ... ‘blending’ resembles ‘fusion’ in that the constituents are related to one another ‘through and though’ and not merely at their surfaces. But it differs from ‘fusion’ and resembles ‘juxtaposition’ in that its constituents retain all their original properties in the mixture and can be separated out again” (Long and Sedley vol.1 p.293). If blending is not to be a kind of juxtaposition, we cannot conceive of it merely as the interspersion of one body in gaps within another body. And if blending is not to be a kind of fusion, a blend has to be a kind of mereological sum. The mereological sum of x and y is the individual such

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that everything having a common part with it has a common part with either x or y. Thus everything having a common part with a blend of x and y has a common part with either x or y. Since the parts of a whole need not be contiguous, this would not entail any assumptions about the spatial location of the ingredients in the blend. What kind of mereological sum is fusion? Among the ancients, some like Alexander of Aphrodisias argued that Aristotelian hylomorphism had greater explanatory power than the Stoic theory of pneuma (Inwood and Gerson text 68, p.95); and some moderns regard the Stoic theory as untenable (e.g. Long 1974 p.160). Daniel Nolan interprets the Stoics as saying that there is a sense in which we, and the cosmos, are composed of ‘gunk’, i.e. of parts each of which has proper parts (Nolan p.162). If x is the Soul (or Mind), ∀y(PPyx→∃zPPzy). Nolan argues that Chrysippus held that all parts of material bodies have proper parts. He shows how this interpretation fits the Stoic concept of blending whereby two substances can fully interpenetrate each other in such a way as not to be merely juxtaposed and to be separable. He also shows how his interpretation is consistent with the Stoic view that no time is fully present but is invariably partly past and partly future. Nolan argues that on a minimal interpretation of Stoic blending, it just means that “there could be a mixture which had a division into a privileged set of parts M, such that each member of M had parts of each of the

original mixed substances in it, and the members of M were further such that each member of M had proper parts that were also among the members of M” (Nolan n.11). This is a purely mereological interpretation, expressed solely in terms of the partwhole relation and other notions definable through it. Nolan also proposes a stronger interpretation, which along with mereological notions includes notions that concern the spatial position of parts and wholes. According to this interpretation, “every part of the blend which fills a region occupied by the mixture contains parts of all the original substances mixed. Furthermore, ... for any magnitude equal or less to the total magnitude of the blend, there is a part of the blend which is of that size and has as parts some of the parts of each of the original substances which are blended” (Nolan 170). In favour of this stronger interpretation, it can be pointed out that some of the ancient texts presuppose the notion of a continuous region occupied by a whole, and thus are more specific than the modern mereological notion of a whole (whose parts may be dispersed in space). For example, “If wholes completely extend through wholes and the smallest through the largest right up to the limits of extension, whatever place is occupied by the one will be occupied by both together” (Von Arnim ii 477).

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See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Ancient Greek Atomism, Conscious Experience, Subject, Whitehead’s Metaphysics. References and further readings

Gould, J. B., 1970, The Philosophy of Chrysippus, Leiden: Brill. Inwood, B.; Gerson, L. P. (eds.), 2008, The Stoics Reader: selected writings and testimonia, Indianapolis: Hackett. Long, A. A., 1974, Hellenistic Philosophy: Stoics, Epicureans, Sceptics, London: Duckworth. Long, A. A.; Sedley, D. N., 1987, The Hellenistic Philosophers 2 vols, Cambridge University Press. Nolan D., 2006, “Stoic Gunk”, Phronesis 51: 162-183. Sambursky, S., 1959, Physics of the Stoics, London: Routledge and Kegan Paul. von Arnim, H. (ed.), 1903-1924, Stoicorum Veterum Fragmenta 4 vols, Leipzig. Paul Thom

Structure The notion of structure is of central importance to mereology (for the following see also Koslicki 2008, ch. IX). Historical contributions had this clearly in view; for example, as is brought out in Harte 2002, Plato in numerous dialogues grapples with

the question of how a whole which has many parts can nevertheless be a single unified object and ultimately endorses a structure-based response to this question (see Plato). Wholes, according to Plato’s mature views (as developed primarily in the Sophist, Parmenides, Philebus and Timaeus), have a dichotomous nature, consisting of both material as well as structural components; it is the job of structure to unify and organize the plurality of material parts that are present in a unified whole. A similar conception is taken up and worked out further by Aristotle who famously believed that ordinary material objects, such as houses, are compounds of matter (viz., the bricks, wood, etc.) and form (viz., the arrangement exhibited by the material components for the purpose of providing shelter). In contrast, due to the development in the early 20th century of a theory often referred to as ‘standard mereology’, based on the work of Stanislaw Leśniewski and Alfred North Whitehead (see also Tarski 1937: 1956; Leonard and Goodman 1940), the notion of structure has been largely absent from more recent mereological frameworks. (A notable exception, however, is the Third Logical Investigation of Husserl 1900-1.) Because the founders of standard mereology were primarily interested in providing a nominalistically acceptable alternative to set theory, according to standard mereology wholes (also known as ‘mereological sums’, ‘fusions’ or ‘aggregates’) are conceived of as completely unstructured entities. On analogy with the axiom of extensionality in set theory, the

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existence and identity of a mereological sum is determined exclusively on the basis of the existence and identity of its parts; the arrangement or configuration of these parts is immaterial to the existence and identity of the sum they compose. In fact, because standard mereology does not recognise a distinction analogous to that between subset and membership, mereological sums are, if anything, even more unstructured than sets, since all the entities recognised by standard mereology are of the same ontological type, viz., so- called ‘individuals’. Finally, as a result of its endorsement of the now controversial principle of Unrestricted Composition (according to which any plurality of objects itself composes a further object, viz., their mereological sum), standard mereology is committed to a plenitude of potentially gerrymandered objects, such as David Lewis’ notorious ‘trout-turkey’, an object composed of, say, the (still attached) upper half of a trout and the (still attached) lower half of a turkey (see Lewis 1986). Because standard mereology has been and perhaps still is the most well-worked out and widely accepted conception of parthood and composition in recent history, it was thought that, insofar as ordinary material objects are wholes (i.e., composite objects made up of parts), they must therefore be conceptualised as mereological sums in the standard sense. This seemingly universal consensus among contemporary metaphysicians, however, is now beginning to be called into question by the arrival of some dissenting voices, who have

turned their attention to the development of alternative non-standard mereological frameworks, and in particular to the re-introduction of the notion of structure into the analysis of parthood and composition, especially as it aims to capture the mereological characteristics of ordinary material objects (see for example Fine 1982; 1994; 1999; Harte 2002; Johnston 2002; Koslicki 2008; Simons 1987). To these theorists, it seems quite clear that the material objects we encounter in ordinary life and scientific practice cannot have the conditions of identity and individuation that are attributed to mereological sums by standard mereology: for, unlike mereological sums, not only are these objects quite obviously capable of surviving changes with respect to their parts, while mereological sums (like sets) have their parts essentially; but, in contrast to the completely unstructured nature of mereological sums, the existence and identity of these objects is also evidently tied to the arrangement or configuration of their parts. For example, as is pointed out in Fine 1999, a ham sandwich does not in fact come into existence until a slice of ham is placed between two slices of bread; and the ham sandwich does not remain in existence unless the parts in question continue to exhibit this arrangement. Given the apparent clash between the conditions of identity and individuation of material objects, as we ordinarily conceive of them, and those of mereological sums in the standard sense, there seems to be plenty of room, then, for the devel-

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opment of alternative structure-based mereologies. One obstacle that has stood in the way of the pursuit of such alternative systems is that the notion of structure, given its traditional affiliation with Platonic forms or Aristotelian essences, in the minds of many contemporary metaphysicians and mereologists inherits much of the philosophical baggage that is associated with its historical precursors. Aristotle already criticised Platonic forms for being so far removed from the sensible particulars whose characteristics they were supposed to explain that they became, in his view, causally inert and explanatorily useless. Plato’s invocation of the participation relation, which was meant to connect Platonic forms to sensible particulars, did not improve the situation, in Aristotle’s mind since he found this relation to be utterly unexplained and mysterious. In reaction to the Platonic model, Aristotle made an effort to connect his own explanatory and causal principles much more intimately to the matter/form compounds whose behavior and characteristics they were supposed to make comprehensible. However, in the course of doing so, Aristotle’s own conception of form or essence became associated with philosophically loaded notions such as his actuality/potentiality distinction and the accompanying Homonymy Principle (according to which an ‘axe’ that cannot cut, for example, is an ‘axe’ in name alone), which in turn made Aristotelian forms or essences acceptable only to philosophers who share his general teleological outlook.

When we look more closely at the various disciplines in which the notion of structure obviously plays a central and significant role, however, we realize that Aristotle’s notion of structure as form need not be conceived of as the causally and explanatorily inert metaphysical invention ridiculed by Descartes and others. Rather, in such disciplines as mathematics, logic, chemistry, linguistics, and music, for example, we find that the notion of structure is alive and well, whatever exactly its metaphysical status turns out to be. Although the notion of structure, as it is applied in each case, is tailored to the particular concerns of each such discipline, we can nevertheless recognise certain general characteristics that go along with any such domainspecific conception of structure. (The general characteristics I am about to single out will be illustrated shortly by means of examples from particular disciplines.) First, structures in general are entities that make available ‘slots’, positions or nodes for other objects to occupy. In order to be admissible occupants of these positions, the objects in question must satisfy two different sorts of constraints: (i) constraints concerning the type of object which may occupy the position in question; and (ii) constraints concerning the configuration or arrangement which must be exhibited by the occupants of the positions made available by the structure. Secondly, a particularly noteworthy characteristic of structures or structural features across different do-

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mains is that the numerical identity of the particular objects occupying the positions made available within a structure inevitably tends to be immaterial to the question of whether the structure or structural feature in question is implemented. As long as the occupants in question satisfy the two constraints just mentioned, they are considered indistinguishable and hence interchangeable from the point of view of the structure. Thus, the notion of a structure or structural feature should be thought of as going along with a distinction between what is considered to be variable and what is considered to be invariable within a given domain or context; variability, in this connection, amounts to the interchangeability of objects in the domain relative to certain admissible transformations which leave the structural features at issue unchanged. Finally, in each case, the discipline in question is interested in particular in capturing, usually by means of a system of laws, axioms, and the like, the characteristics and behavior of those features that are taken as invariable, i.e., the structural features within the domain in question. The particular nature of those elements that occupy the positions made available by a given structure, i.e., elements which are considered to be variable within the domain at issue, on the other hand, tends not to lie within the purview of the significant generalisations formulated by the theory in question. For these elements in any case are taken as interchangeable as far as the structure is concerned, provided that the type and configuration

constraints imposed by the structure remain satisfied. I now turn to the illustration of these general principles governing the notion of structure by means of examples taken from particular disciplines. structure. Structures within mathematics are defined as ordered n-tuples consisting of a set of objects (the universe or domain of discourse) along with “a list of mathematical operations and relations and their required properties, commonly given as axioms, and often so formulated as to be properties shared by a number of possibly quite different specific mathematical objects” (Mac Lane 1996: 174). Widely studied examples of mathematical structures include for example groups, metric spaces, topological spaces, rings, fields, orders and lattices. Mathematical structures can be compared and contrasted by means of various relations, such as embedding, homomorphism, isomorphism, and the like. As any two isomorphic structures satisfy the same axioms and are thus indistinguishable from the point of view of the theory in question, structures are often said to be describable only ‘up to isomorphism’. Mathematical

Logical structure. A logically valid

argument is one that is not only necessarily truth-preserving, but is so in virtue of its logical form or structure. To illustrate, while the first requirement is satisfied in the argument, ‘Roses are red; therefore, roses are colored’, the second is not. The notion of logical form makes sense only relative to a particular choice of logical vocabulary; for example, because

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of the meaning assigned to the logical constant, ‘and’, any instance of the axiom schema ⟔p and q; therefore q is valid within classical sentential logic. The role of p and q in this argument schema is merely to mark places that may be occupied by any non-logical expression of the right grammatical category (viz., in this case, a sentence). Thus, as far as the validity of the argument schema in question is concerned, the interpretation of the non-logical vocabulary may vary, while that of the logical vocabulary stays fixed. The inference-rules of a logical system aim in particular to describe the role played by the logical vocabulary in generating valid argument patterns. Chemical structure. The chemical

structure of a compound is determined on the basis of (i) the types of constituents of which it consists, viz., its formula; and (ii) the spatial (i.e., geometrical or topological) configuration exhibited by these constituents. In the 18th and 19th century, it was discovered, in connection with the phenomenon of ‘isomers’ or ‘chiral’ (‘handed’) molecules, that chemical substances which are composed of the same constituents, i.e., have the same chemical formula, can nevertheless exhibit dramatically different behavior under certain circumstances, if these constituents are arranged differently. (Cases in point are for example silver cyanate and silver fulminate as well as racemic and tartaric acid.) This discovery led to a three-dimensional conception of molecular shape, which is still to this day widely employed across many of the natural sciences to explain the

processes undergone by organic and inorganic compounds. Linguistic structure. Linguistic struc-

ture bears a remarkable similarity to chemical structure. For example, the syntactic structure of a linguistic compound is similarly determined on the basis of (i) the types of constituents of which it consists (e.g., nounphrases, verb phrases, modifiers, and the like) as well as (ii) the hierarchical arrangements exhibited by these constituents; the latter is typically represented by means of a spatial (i.e., geometrical or topological) vocabulary, consisting of such notions familiar for example from the tree-diagrams used within the Chomskyan tradition as ‘being to the left of’, ‘being higher up than’, ‘being connected via a continuous downward path to’ and so on. These two aspects of syntactic structure help explain why linguistic compounds that on the surface look very similar (e.g., ‘John is reluctant to leave’ versus ‘John is likely to leave’) may nevertheless exhibit very different behavior under certain transformations (e.g., ‘*It is reluctant that John leaves” versus “It is likely that John leaves’). The numerical identity of the lexical items filling the various positions within a syntactic structure is again irrelevant from the point of view of the structure, as long as the syntactically relevant features mentioned in (i) and (ii) remain unchanged. Thus, insofar as two lexical items belong to the same syntactic category and fit into the same hierarchical arrangements, they are indistinguishable from the point of view of the syntax and are hence inter-

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changeable without affecting the grammaticality of the resulting construction. Musical structure. Musical structure,

unlike the other examples considered thus far, of course concerns a perceived or phenomenal order, a kind of ordering or organisation which comes about when sound waves interact with creatures like us who are equipped with the sort of cognitive apparatus required to hear sound as music. The experience of hearing sound as music sets up in such hearers certain expectations as to how the tones they hear are going to be organised with respect to the principles of pitch, rhythm, melody and harmony. Relative to certain musical traditions, e.g., the Western tradition of ‘tonal music’, it is even possible to speak (though somewhat metaphorically no doubt) of a system of ‘laws’, e.g., the laws of tonality, which constrain how smaller musical units (e.g., tones) may be organised into larger musical wholes (e.g., chords, patterns, motifs, melodies, and the like) relative to the principles of composition that govern a particular musical tradition. The sorts of arrangements into which individual tones enter are again characterised by means of a quasi-threedimensional vocabulary invoking space and motion, e.g., ‘high’, ‘low’, ‘fast’, ‘slow’, etc. The study of structure, as this concept is relevant in particular to the development of non-standard systems of mereology, confronts five important metaphysical questions which at this point remain relatively underexplored, especially in the context of

the contemporary literature on parthood and composition. (1) Ontological Category. To what ontological category do structures belong? Are they objects, properties, relations, or something else entirely? (2) Grounding Problem. How is the modal or essential profile of a structured whole connected to the structure that is present within it? That is, what sorts of contributions does the presence of a structure within an object make to the nature of that structured whole? (3) Mereological Constraints. What sorts of mereological constraints do structures impose on the wholes they organize? To what extent and in what way do they dictate the mereological make-up of a structured whole? (4) Individual vs. Species Forms. What sorts of structural features are shared by the members of a single kind or species? To what extent should structures be thought of as incorporating haecceitistic features that are peculiar to individual members of a kind? (5) Structural Change. To what extent can structured wholes change with respect to their structural features? Through what sorts of structural changes can they persist? The resolution of these questions would contribute much to the advancement of alternative structure-based systems of mereology vis-a-vis standard mereology. See also > Aristotle’ Theory of Parts, Aristotle’s Theory of Wholes, Chemistry, Gestalt, Goodman, Husserl, Linguistic Structures, Order, Segelberg, Structure of Appearance, Syntax, Whitehead’s metaphysics.

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Bibliographical remarks

Fine, K., 1999. A neo- Aristotelian theory of the nature of material objects and their parts. Harte, V., 2002. A historical study of different conceptions of parthood and composition considered in the works of Plato. Johnston, M., 2002. A neoAristotelian approach to mereology. Leonard, H.; Goodman, N., 1940. The classic text introducing standard mereology to the English- speaking world. Rescher, N., and Oppenheim, P., 1955. A Gestalt-theoretic exploration of the concepts of part, whole and structure. Tarski, A., 1966. An analysis of what it means to be a logical notion in terms of invariance under a sufficiently wide conception of transformations. References and further readings

Fine, K., 1982, “Acts, Events and Things”, Language and Ontology, Proceedings of the 6th International Wittgenstein Symposium, Wien: Hölder-Pichler-Tempsky, 97-105. Fine, K., 1994, “Compounds and Aggregates”, Nous 28 (2): 137-158. Fine, K., 1999, “Things and Their Parts”, Midwest Studies in Philosophy 23: 61-74. Harte, V., 2002, Plato on Parts and Wholes: The Metaphysics of Structure, Oxford: Clarendon Press.

Husserl, E., 1900-1, Logische Untersuchungen, 1st edn., Halle, Germany: M. Niemeyer Verlag. Johnston, M., 2002, “Parts and Principles: False Axioms in Mereology”, Philosophical Topics 30 (1): 129166. Koslicki, K., 2008, The Structure of Objects, Oxford: Oxford University Press. Le Poidevin, R., 2000, “Space and the Chiral Molecule”, in Bhushan, N.; Rosenfeld, S. (eds.), Of Minds and Molecules: New Philosophical Perspectives on Chemistry, New York: Oxford University Press. Leonard, H.; Goodman, N., 1940, “The Calculus of Individuals and its Uses”, Journal of Symbolic Logic 5: 45-55. Lésniewski, S., 1916, “Podstawy ogólnej teoryi mnogości I” [Foundations of a General Theory of Manifolds], Prace Polskiego Koła Naukowe w Moskwie, Sekcya matematyczno-przyrodnicza, 2, Moscow. Lésniewski, S., 1927-30, “O Podstawach Matematyki” [On the Foundations of Mathematics], Przeglad Filozoficzny, 30 (1927), 164-206; 31 (1928), 261-291; 32 (1929), 60-101; 33 (1930), 75-105, 142-170. Lewis, D., 1986, On the Plurality of Worlds, Oxford: Blackwell. Mac Lane, S., 1996, “Structure in Mathematics”, Philosophia Mathematica 4 (3): 174-183. Nozick, R., 2001, Invariances: The Structure of the Objective World,

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Cambridge, MA: Harvard University Press.

mer University, Copenhagen: Ejnar Munksgaard.

Rescher, N.; Oppenheim, P., 1955, “Logical Analysis of Gestalt Concepts”, The British Journal for the Philosophy of Science 6 (22): 89-106.

Woolley, R. G., 1978, “Must a Molecule Have a Shape?”, Journal of the American Chemical Society 100, 1073-1078.

Scruton, R., 1997, The Aesthetics of Music, Oxford: Clarendon Press. Sider, T., 2001, Four-Dimensionalism: An Ontology of Persistence and Time, Oxford: Clarendon Press. Simons, P., 1987, Parts: A Study in Ontology, Oxford: Oxford University Press. Tarski, A., 1936, “On the Concept of Logical Consequence”, in J. H. Woodger, transl. and ed. with an introduction by John Corcoran, Logic, Semantics, Metamathematics, 2nd Edition, Indianapolis: Hackett, 1983: 409-420 (article was first published in Polish and German in 1936). Tarski, A., 1937, “Appendix E”, in Woodger, J. H. (ed.), The Axiomatic Method in Biology, Cambridge: Cambridge University Press, 161172. Tarski, A., 1956, “Foundations of the Geometry of Solids”, in: A. Tarski, Logic, Semantics and Metamathematics, transl. by J.H. Woodger, Oxford: Clarendon Press, 24-29. Tarski, A., 1966, “What Are Logical Notions?”, History and Philosophy of Logic 7 (1986): 143- 154. Tranöy, K. E., 1959, Wholes and Structures: An Attempt at a Philosophical Analysis, Interdisciplinary Studies from the Scandinavian Sum-

Kathrin Koslicki

Structure of Appearance, Goodman’s The Structure of Appearance (1951, short SA) is perhaps Nelson Goodman's main work, although it is less widely known than, for example, Languages of Art (1986). It is, in fact, a heavily revised version of Goodman's Ph.D. thesis, A Study of Qualities (Goodman 1941, short SQ). SA presents a ‘constructional’ system that, just like the constitution system in Rudolf Carnap's Der logische Aufbau der Welt (1928), shows how from a basis of primitive objects and a basic relation between those objects all other objects can be obtained by definitions alone. In SA (and already in SQ) Goodman applies a mereological system, the Calculus of Individuals, which he developed jointly with Henry Leonard (first published in Goodman and Leonard 1940). The use of mereology allows him to avoid certain technical problems that Carnap’s system encounters. In the Aufbau, Carnap investigates the example of a world built up from primitive temporal parts of the totality of experiences of a subject (the socalled ‘elementary experiences’ or just ‘erlebs’) and thus faces the prob-

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lem of abstraction: how can qualities, properties and their objects in the world be abstracted from our phenomenal experiences. Erlebs, which are time slices of the totality of our experiences, can be part-similar with each other in a variety of ways. Perhaps two slices are similar with respect to what is in our visual field at the time in question, or they are similar with respect to what we hear or smell. However, since the time-slices are primitives in the system, we cannot yet even talk about these respects or ways in which the slices should be similar in order to be considered experiences of the same feature (for example, the same color). Carnap’s ingenious idea is to group exactly those erlebs together that are mutually part-similar, thereby grouping exactly those that (pre-theoretically speaking) share a property. Carnap tries to show that by using this method of ‘quasi-analysis’ all the structure can be retained from the basis if the erlebs are indeed ordered by a simple relation of partsimilarity. Although the individual temporal slices of our totality of experience are not structured (and thus have no parts), we can, via quasianalysis, get to their quasi-parts, the ‘qualities’ they share with other timeslices with which they are partsimilar. Goodman’s SA offers a critical discussion of the Aufbau. Goodman notes in particular two difficulties for Carnap’s system. He shows that Carnap’s quasi-analysis leads to (intuitively) wrong results when one property always co-occurs with another

(in which case quasi-analysis fails to distinguish between the two), and when the part-similarity relation happens to interconnect a group of erlebs even though no one property is shared by all erlebs in the group (but quasi-analysis would assign a property to the group). The first problem is called the ‘difficulty of constant companionship’, the second is the ‘difficulty of imperfect community’ (note that from correspondence between Carnap and Goodman it appears that Carnap was aware of both of these problems (cf. Seibt 1997)). In contrast to Carnap, Goodman begins from a realist basis, considering the example of a system built on phenomenal qualities, so-called qualia (phenomenal colors, phenomenal sounds, etc.) and thus faced the problem of concretion: how can concrete experiences be built up from abstract particulars? In the visual realm, a concretum is a color-spot moment, which may be construed as the sum of a color, a visual-field place and a time, all of which stand in a peculiar relation of togetherness. Goodman adopts this relation as a primitive and then shows how one can, in terms of this relation, define the concept of the concrete individual as well as the various relations of qualification in which qualia and certain sums of qualia stand to the fully or partially concrete individuals that exhibit them. After this is done, Goodman turns to his second major constructional objective, namely, to order the qualia into different categories. The prob-

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lem is to construct for each of the categories (color, time, place, and so forth) a map that assigns a unique position to each quale in the category, and that represents the relative likeness of qualia by nearness in position. The solution to the problem requires in each case the specification of a set of terms by means of which the order at hand can be described, and then the selection of primitives suitable to define them. Goodman thereby shows how predicates referring to size and shape of phenomenal concreta may be introduced, and he suggests some approaches to the definition of the different categories of qualia by reference to their structural characteristics. Since Goodman does not deal with time slices of experiences, which come already with a temporal order, but has phenomenal time among the basic qualia, he has to reconstruct temporal order as well as any other ordering of qualia. This is done in the final chapter of SA, “On Time and Eternity”, which deals with the ordinal aspects of phenomenal time, the linguistic devices used to express temporal relations and some connections between phenomenal and physical time. Goodman shows in SA how using a mereological system can help to avoid the difficulty of imperfect community for a system built on a realist base (such as SA), as well as for a system built on a particularistic base (such as in the Aufbau). The constant companionship difficulty, on the other hand, does not arise in SA because no two concreta can have all their qualities in common.

See also > Comments on “The Calculus of Individuals and its Uses”, Carnap, Gestalt, Goodman, Mereotopology, Nominalism, Perceptual Whole, Stumpf, Temporal Parts. Bibliographical remarks

Goodman, N, 1951. The third edition contains a highly recommended introduction by Geoffrey Hellman. Elgin, C. (ed.), 1997. A collection of papers concerning Goodman’s nominalism and constructivism, first developed in SA. Cohnitz, D.; Rossberg, 2006. Introduction to the philosophy of Nelson Goodman, explaining SA in the context of his other work References and further readings

Carnap, R., 1928 [1961], Der logische Aufbau der Welt; Scheinprobleme in der Philosophie, Hamburg: Meiner. Cohnitz, D., 2009, “The Unity of Goodman’s Thought”, in Ernst, G. et al. (eds.), From Logic to Art: Themes from Nelson Goodman, Berlin: de Gruyter, 33–50. Cohnitz, D.; Rossberg, M., 2006, Nelson Goodman, London: Routledge. Elgin, C. (ed.), 1997, The Philosophy of Nelson Goodman, Volume 1: Nominalism, Constructivism, and Relativism in the Work of Nelson Goodman.New York: Garland.

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Goodman, N., 1941 [1990], A Study of Qualities, PhD Thesis, Harvard University, first published New York: Garland. Goodman, N., 1951 [1977], The Structure of Appearance, third edition, Boston, MA: Reidel. Goodman, N., 1968 [1981]. Languages of Art: An Approach to a Theory of Symbols, second edition, Indianapolis, IN: Bobbs-Merrill. Rossberg, M., 2009, “Leonard, Goodman, and the Development of the Calculus of Individuals”, in G. Ernst et al., eds. From Logic to Art: Themes from Nelson Goodman, Berlin: de Gruyter, 51–69. Seibt, J. (1997), “The Umbau: From Constitution Theory to Constructional Ontology”, History of Philosophy Quarterly 14 (3): 305-348. Daniel Cohnitz Marcus Rossberg

Stumpf, Carl Carl Stumpf (1848-1936) was a German philosopher and psychologist. He studied in Würzburg with Franz Brentano and in Göttingen with Hermann Lotze. His contributions to phenomenology, Gestalt psychology, aesthetics, musicology and acoustics (among other disciplines) make him a significant protagonist of the cultural and intellectual milieu at the turn of the nineteenth and twentieth century. Stumpf never articulated a mereological theory in a strict sense, but in one

of his first publications, Über den psychologischen Ursprung der Raumvorstellung (1873), he draws some conceptual distinctions about the notion of part that should prove to be extremely influential for the future of the discipline. One of the most important developments of Stumpf’s ideas is Edmund Husserl’s mereology, as documented in the Third Logical Investigations. The psychological origin of the presentation of space. The book of 1873

does not investigate the mathematical notion of space, but it exclusively focuses on perceived space, that is, visual space (Gesichtsraum) or tactile space (Tastraum). Explaining the psychological origin of the presentation of space (in the sense just specified) hence requires explaining whether the mental presentation of space has a genuine content, and under which circumstances this presentation can occur. First, the term ‘content’ is used to indicate what is presented (cf. Stumpf 1873: 14, 25) and it does not bear metaphysical import: “[…] here we consider only the space that we present and how we present it, regardless of whether (gleichviel ob) something alike or similar corresponds to it or not, or even whether something real corresponds to it or not” (Stumpf 1873: 13). Put another way, Stumpf’s investigation is concerned only with space as this is presented to the mind, but does not explore the question as to whether anything real in the mind-independent world corresponds to the presentation of space. Second, ‘presentation’ is used to denote either perception (‘concrete [wirkliche] presentation’)

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or fantasy (‘fantastic presentation’ [Phantasievorstellung]). Finally, should the investigation indicate that space is a genuine content, the relationship with other contents has to be clarified. More precisely, since perception and imagination seem to present space always in combination with other qualities (in particular, visual qualities like colours), the relation between these contents needs to be ascertained. (Even imagining a black surface does not deliver the pure content of space as such: the surface only comes into sight together with the quality of blackness.) Stumpf scrutinises four possible views about the nature of space (cf. Stumpf 1873: 7): (a) either space is not a genuine content of presentation, but it merely is an aggretation of different other contents (e.g. colour sensations); or (b) space is a genuine content apprehended by some particular sense (hence, it is a sensual quality [Sinnesqualität] just like, for instance, colour is the sensual quality apprehended by the sense of sight) and it is connected with other qualities in a purely extrinsic way (in the same sense in which, e.g., the presentation of a colour can – but does not have to be – accompanied by a coexistent presentation of a sound); or (c) space is a genuine content, but it is not a sensual quality for it does not stem from senses, as it is rather caused by some mental disposition; or, finally, (d) space is a genuine content, it is not a sensual quality, but it is connected with sensual qualities in an intrinsic way (and, thus, it is not the product of some mental activity). After discussing these different ideas,

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Stumpf rejects the first three and embraces the last one. At this juncture, Stumpf turns to the problem of how contents of a presentation can be related to each other in order to determine the specific connection between the content of space and that of colours. The theory of psychological parts.

Stumpf describes different ways in which contents can be connected to each other. Firstly, opposite contents (Entgegegensetztes) can be aggregated and hence be presented together. As an example, consider the thought expressed by the sentence: (1) wooden iron is impossible. Understanding this sentence presupposes bringing together the ‘opposite’ contents of wooden and of iron, or so Stumpf claims. Another possibility is the concomitance of contents: for instance, enjoying an opera performance presupposes the coexistent presentations of sounds and colours. Here, the presentation has different contents, which are presented by distinct senses, but these contents are not opposite to each other as in the first case. In these two initial cases, the presented contents do not show any positive affinity (positive Verwandtschaft) among each other. By contrast, such affinity characterises the contents of the third case: if we listen to a chord, we present different sounds together with one single sense. Here the sounds belong to the same species, and yet they are different contents. To put this differently, we have the possibility of presenting every single sound as such separately. Now, such

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separability of contents is impossible within the framework of the fourth and last case discussed by Stumpf: in one and the same presentation (of a tone, e.g.) delivered by one single sense the tone is given together with its intensity, quality, duration, etc. Here, the contents can be distinguished only by means of abstract consideration as it is impossible to (perceptually or imaginatively) present contents like intensity, quality or duration as such and in isolation from the content of the tone. These considerations motivates Stumpf in distinguishing between independent contents (selbstständige Inhalte) and partial contents (Theilinhalte). Independent contents, as discussed in the first three examples, are such that can be presented in isolation from each other. By contrast, partial contents like those illustrated in the fourth example can be presented only in connection with other contents. To come back to the initial question about the relation between the content of space and that of colour, it can now be said that these contents belong to the class of partial contents as they can be presented only together. Content and space, that is, exist only as parts of a whole. (Other instances of this kind of contents are movement and velocity, colour and intensity, etc.) Partial contents and predication. In distinguishing independent from partial contents Stumpf claims that the separability and independence of a content is not due to a mental operation (such as association) or to a convention; contents are independent or

partial “according to their nature (ihrer Natur nach)”, and, “[i]n fact, what is logically possible or necessary, has to be the same always and for everyone” (Stumpf 1873: 111). This is remarkable, since Stumpf seems here to be committed to mindindependent (or ‘objective’) properties of psychological contents. This impression is reinforced by Stumpf’s consideration of the relation between a substance and its accidents. Stumpf argues that the substance and its accidents should not be considered as separate and autonomous entities, since they build a unity (Stumpf 1873: 114). Likewise, he adds further, space and colour should not be considered as separate and independent contents, but rather as parts of a single whole. The peculiarity of the link between partial contents is indicated by the fact that one content is predicable of the other (as accidents are predicable of a substance), for instance in the judgments: (2) this colour is extended (or it has an extension) and (3) this extension is coloured. Predicability does not apply to independent contents or to substances: to use Stumpf’s own example, even if Schiller, Goethe, Beethoven and music sheets are presented together, no one would ever consider predicating the one content of the other. See also > Brentano, Gestalt, Husserl, Intentionality, Perceptual

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Whole, Reinach, Structure of Appearance. References and further readings

Mulligan, K.; Smith, B., 1982, “Pieces of a Theory”, in Smith, B. (ed.), Parts and Moments. Studies in Logic and Formal Ontology, Munich and Vienna: Philosophia Verlag, 15-108. Kaiser El-Safti, M., 1994, “Carl Stumpfs Lehre vom Ganzen und den Teilen”, Axiomathes 1: 87-122. Stumpf, C., 1873, Über den psychologischen Ursprung der Raumvorstellung, Stuttgart: S. Hirzel Verlag. Alessandro Salice

Subject, Person, Self The subject (Latin subiectum) is that which encounters the object (Latin obiectum), or equivalently it is what can be referred to the object. An object is anything to which some reference could be made. It is everything that is thrown ‘in front of’, or ‘opposite of’, the subject. Thus, every subject in this sense can be an object, because we can refer to it. However, the subject moves, grows, thinks, feels, sees, calculates, loves, etc. Therefore the subject in its ontological form, as phenomenologists used to say, is the fulfiller of acts. There is some dispute over whether the subject is part of the world or not. The majority view is that the subject is a part of the world. Wittgenstein,

on the other hand, seems to express the opposite view. In his Tractatus he says that “the subject does not belong to the world; rather, it is a limit of the world” (Tractatus 5.632); and he continues: “What brings the self into philosophy is the fact that ‘the world is my world’. The philosophical self is not the human being, not the human body, or the human soul, with which psychology deals – but rather the metaphysical subject, the limit of the world – not a part of it” (ibid 5.641). No matter what its status with respect to the world may be, it is clear enough that the subject exists and constitutes itself in time through its acts and in its acts. The central part of the subject is a ‘point-like’ source of all his or her acts (in the case of multiple personality disorder, for example, the point-source may be more than one if the personalities are not discrete but overlap). In this sense, the subject is contractible to the ‘point-like’ source. But the subject is not a mereological ‘point-like’ atom. Besides the central source, the subject may have many other parts: consciousness, body, personality, character, temperament, etc. According to the Brentano tradition, consciousness is, speaking somewhat figuratively, the ‘stream’ of acts or the ‘bundle’ of experiences. Consciousness is intentional; that is to say, it refers to something, it is directed towards something, it turns to something. An important mereotopological problem is the question whether consciousness (and subjectness) has some spatiality. Philoso-

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phers are divided on this issue into at least two groups. The first group, epitomised by for example Descartes and Ingarden, believe that consciousness has no parts (it is simple) and in no way can be extended. The second group, here represented by J. Szewczyk and K. Lewin, thinks that in some sense consciousness (Szewczyk) and personhood (Lewin) has extension. Consciousness stretches out towards the world, hitting the objects to which it intends, it approaches them, meets them in a way. It is a “universally infinite (infinitely spreadable) field of fields, finite as well as infinite, establishing the certain closed in itself and internally integral unity of existence, not being the material part of nature, neither connected to it causally nor structurally, centered in its subjective source. This field spouts or radiates from ‘Self’ in all directions (therefore undoubtedly three-dimensionally) and constitutes itself in an unceasingly and multi-dimensionally undulating medium. It makes the specific environment or plasma of intentionality, having its own temporality, and therefore, necessarily, its own spatiality or phenomenological quasi corporeality” (Szewczyk 1987: 36). According to Lewin and his topological approach: “(…) psychology, dealing with manifolds of coexisting facts, would be finally forced to use not only the concept of time but that of space too. Knowing something of the general theory of point sets, I felt vaguely that the young mathematical discipline ‘topology’ might be of some help in making psychology a real science” (Lewin 1936: vii). The

person “is represented as a differentiated region of the life space; however in the first approximation he can be represented as an undifferentiated region or a point” (Lewin 1936: 216). Among the parts of the person Lewin distinguished (a) the motorperceptual region; (b) the innerpersonal region and its parts: (c) peripheral regions, (d) central regions. For detailed description of the mereology and topology of the person, see his Principles of Topological Psychology. The notion of a subject is systematically related to the notion of a person. In our common reasoning about persons – excepting the legal sense of person that also applies to corporations – every person must be a subject but not every subject must be a person. Some examples of subjects that are not persons may be computers, some animals, and fictional characters. Being a person is a dependent part (or, what Husserl would call, a ‘moment’) of being a subject. The subject could be a person, if it performs (or engages in) a special kind of acts, namely, personal acts. There are at least two classes of such acts: acts satisfying moral values (according to Stróżewski) and second order volitional acts (according to H. Frankfurt). We will briefly elaborate on both of these kinds of acts in what follows. According to W. Stróżewski, it is not the case that something either is a person or is not a person. We are constantly becoming persons. The current intensity of being a person

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depends on the quantity and the quality of the positive moral values which have been satisfied by this person. There are two extreme kinds of being an individual: total dispersion in being (i.e. being nowhere dense, being scattered) and being intensely (i.e. being dense, integrated). Between these two there is a continuum of ways of being a person. As Stróżewski writes: “The human being is being in a valuable way. Realisation of values consists in saturating, as if ‘illuminating’ his own being by them. (…) We are prone to say that the human living in this way really exists. But it means only that he exists in the way intensified by value. The value valuates the existence” (Stróżewski 1981: 84-85). Frankfurt suggests that person should be able to form second (or higher) order volitions. Persons “should be capable of wanting to be different, in their preferences and purposes, from what they are” (Frankfurt 1971: 7). First-order volitions are merely desires to do something or not. If somebody wants to have a certain desire of the first order to be his will or not to have a certain desire of the first order to be his will, then he has second order volitions. For example, our volitions of wanting not to want to eat the sweets are second order volitions. In Frankfurt’s view, having second order volitions is a necessary condition of being a person. Just as not every subject is a person, upon closer consideration it might also be questionable whether every subject is a self. The issue arises especially if one adopts a Humean ac-

count of personal identity and conceives of a person as a sequence of momentary total states of consciousness that are suitably related. Derek Parfit explores in a much discussed fission argument (Parfit 1975; 1984) whether a subject’s survival, understood as the relationship between your current self and your future self, can be defined in terms of the ‘Rrelation’, i.e., a relationship of ‘psychological continuity.’ In Parfit’s setting, psychological continuity is the transitive closure of ‘strong psychological connectedness’, where psychological connectedness is a nontransitive relation that holds between two total states of consciousness just in case they share at least one item of subjective content (here taken in the wide sense to include contents of beliefs, intentions, hopes, feelings etc.). Two total states of consciousness are more strongly connected, if they share a larger number of subjective contents. If two sequences of states of total consciousness are psychologically continuous, these belong to the ‘same self’ – a future state thus is the ‘future self’ of a present state; if they are only weakly connected, the future state is merely a ‘descendent self’ (Parfit 1984: 226f). Parfit shows personal identity and sameness of self can come apart. That is, the Rrelation is not one-to-one – it allows for sequences of person-stages to branch and to fuse, and in the course of these fissions and fusions to overlap for some time. Parfit argues that what matters from an ethical perspective is not personal identity but survival – relatedness to a subject with whom we are psychologically con-

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tinuous with. In Parfit’s view our ethical deliberations turn on whether we, as subjects and present selves, make the right choice with respect to future selves of subjects we may or may not be identical with as persons. Using the constructional ideas of the Humean approach we can also define a person’s time-variant self-conception in terms of overlapping collections of subjective contents at and across times. For example, using settheoretic and mereological concepts, one might formulate conditions under which collections of subjective contents (experiences, perceived capacities, and valuative commitments) form identificatory narratives, of groups or individuals. An individual’s personal ‘identity’ (selfconception) can then be defined as a selecting function on overlapping identificatory narratives, i.e., as the outcome of choices, at and across times, of subjective contents representing personal, ethnic, and cultural identificatory narratives (identities) (Seibt and Nørskov 2012). Whether it might be possible to use mereotopological concepts to define types and degrees of integration of the identificatory narratives constituting a selfconception, and thereby different forms of self-conceptions and personal integration (ego-centered versus group-centered etc.), is yet to be explored. See also > Conscious Experience, Ingarden, Intentionality, Medicine, Mereotopology, Persistence, Society, Stoics, Substance, Substrate, Topology.

References and further readings

Frankfurt, H., 1971, “Freedom of the Will and the Concept of a Person”, The Journal of Philosophy 68: 5-20. Lewin, K., 1936, Principles of Topological Psychology, translated by Fritz and Grace Heider, New York: McGraw-Hill. Parfit, D., 1975, “Personal Identity”, in Perry, J. (ed.), Personal Identity (Berkeley: University of California Press), 199-226. Parfit, D., 1984, Reasons and Persons, Oxford Clarendon Press. Parfit, D. 1995, “The Unimportance of Identity”, in Harris, H. (ed.), Identity, Oxford University Press, 13-45. Pervin, L. A., John O. P., 2001, Personality: Theory and Research, 8th ed. John Wiley&Sons, Inc. New York. Seibt, J., 2000, “Fission, Fusion, and Survival: Parfit’s Branch Line Case Revisited”, Metaphysica 2: 106-134. Seibt, J.; Nørskov, M., 2012, “Embodying’ the internet: Towards the Moral Self Via Communication Robots?” Philosophy and Technology 25: 285-307. Stróżewski, W., 1981, Istnienie i wartość, (The Existence and Value), Znak, Kraków. Szewczyk, J., 1987, O fenomenologii Edmunda Husserla, (Edmund Husserl’s Phenomenology), Warszawa. Bartłomiej Skowron Johanna Seibt

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Substance Substances are traditionally considered to be independent and fundamental entities; they are said to be the ‘bearers’ of properties and relations, as well as participants in events, or even the locus where changes happen. Some elements of this conception can be traced back to the Presocratics and Plato, who considered as primary entities either material elements (mostly earth, water, air, and/or fire) or transcendent forms (the ‘Platonic ideas’), respectively. However, the first comprehensive theoretical development of this conception is to be found in Aristotle, who integrated the extant component lines of thought into a coherent category theory that was to become the most influential approach in the history of ontology so far. As philosophical terminology was yet to be established, the philosophers contributing to the ancient debate about substance used different Greek terms for more or less related ideas. These terms also acquired other meanings, and when they were translated first into Latin and then into modern languages, translators often used different terms for different contextual accentuations (as can be seen from the English terms ‘substance’, ‘essence’, ‘nature’, ‘subject’ and ‘substrate’), and sometimes words changed their meanings considerably in this process. For these reasons, the history of philosophical thinking about substances and the arguments pertaining to it cannot be narrated as the history of something denoted by a single technical term. The terms we

find both in Aristotle are ousia (a derivation from einai, ‘to be’, already used in Plato’s Eutyphro, 11 a 7) and hypokeimenon (verbally translated as ‘substrate’), while the English word “substance” is a verbal translation of hypostasis, which got into philosophical use with the Stoics and was popular with Plotinus and the church fathers. In Aristotle, substance (ousia) is the first and primary ontological category (cf. esp. Categories 5, Metaphysics V 8 and VII-VIII). Substance terms like ‘man’ or ‘horse’ answer to questions of the form ‘What is that?’ – as opposed to questions like ‘How is it?’ or ‘How large is this?’ which correspond to the categories of quality and quantity, respectively (Kahn 1978). For this reason, Aristotle also uses the technical term ‘the what-itis-to-be-this’ (to ti ên einai) as a name for his first category, which is often translated as ‘essence’. Hence, substance terms transfer information about a thing’s essence as opposed to its ‘accidents’ (features it has at some time but may fail to have at another time), like its size, its properties of appearance, or its location. For Aristotle, individual substances (so-called ‘primary substances’) are the primary kind of beings; they are ontologically prior with regard to their accidents, because, on Aristotle’s account, no quality, quantity, relation, or process can exist without the substances that are their bearers or participants, respectively. All other particulars, like particular qualities or particular quantities (the particular white colour of this horse or the read-

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ing skill of Socrates) depend for their existence on one or more substances in which they inhere. Next to concrete substances like a human or a horse we can distinguish the abstract feature that makes them a substance of their kind, like Socrates’ humanity or Bucephalos’ horseness, which is sometimes also called ‘substance’, but more often its ‘essence’ or ‘nature’, and which can thus be conceived of as an abstract particular comprising the essential properties of that entity. Finally, Aristotle calls the species or genus of a concrete substance a ‘second substance’ (deutera ousia), and kinds of substances are not particulars but universals. According to Aristotle, substance universals such as mankind or horseness do not exist without any particular substances (e.g., Socrates or this horse) that instantiate them (Categories 5). (In contrast, Plato thought that universals have an existence independent from their instances; cf. Symposion 211b.) Substances are also logically or conceptually prior to accidents, because a definition of a substance does not need to mention accidents, but a definition of an accident has always to contain at least an implicit reference to a substance as the bearer of the accident – a quality is always a quality of a substance, and a relation is always a relation between substances (Metaphysics IV 2). Substances are basic objects of reference; they are, as Aristotle puts it, a “this such-and-such” (tode ti). For Aristotle, a substance (ousia) fulfils the characteristics connected with

being a substrate (hypokeimenon): Substances are the subjects of predications; they are “not asserted of a subject but of which everything else is asserted” (Metaphysics VII 3, 1029 a 8): Various properties can be ascribed to properties but Socrates is never ascribed to something as a property. These logical features of a substance dovetail with their ontological role as bearers of properties (because they do not themselves need a bearer in order to exist), and as that which underlies change, i.e., is the substrate of change. A well-known doctrine of Aristotle’s Metaphysics is hylomorphism, i.e. the thesis that concrete substances are compounds out of some matter (hylê) and a form (morphê, eidos), where the latter can be identified with the thing’s essence or nature – and, in the case of living beings, with their soul. It is, however, debated whether hylomorphism can be formulated as a mereological claim. Some philosophers think that matter and form can indeed be understood in terms of mereological parts of a thing (e.g., Koslicki 2008). It should be noted, though, that there is a peculiar ontological interdependence between matter and form, which is atypical for standard mereology. For which particular chunk of matter belongs to a thing is determined by the thing’s form, and the form of this thing, as a particular, could not exist without that matter (Gill 1989, Scaltsas 1994). In standard mereology, a sum of bricks exists independently from how these bricks are spatially related to each other. Thus it would not be possible to account for the difference

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between a heap of bricks and a house. Referring to the contrast between a heap and a house, Aristotle points out that, on pain of a regress, the unity of a concrete thing cannot be accounted for by adding a further part at the same level as the material constituents (Metaphysics VII 17). Aristotle’s solution to this problem is to postulate the form of things as their unifying principle. Some authors suggested identifying Aristotelian form with the sum of all structural relations between the material constituents, and hence identifying concrete substances with the sum of their material parts plus the structural relations connecting these (Koslicki 2008). Still, this suggestion cannot account for Aristotle’s observations that the part often exists potentially only and that the whole is often prior to its parts—e.g., a body part like a finger is only a finger in the strict sense when attached to the body and fulfilling its proper function (Metaphysics VII 16; Marmodoro 2013). While this argument applies only to functional parts, mere bits and pieces of matter like atoms or molecules may pre-exist before their integration into the organism. Then, however, the mereologist has to deal with the ‘growing argument’ that goes back at least to the Presocratic philosopher Epicharm: assuming that the adult organism contains more and other atoms than the infantile organism, the organism cannot be timelessly identical with any of the sums of atoms of which it is composed (Barnes 1988: 271; for a scholarly exposition of Aristotle’s solution to this “paradox of unity” see Gill 1989).

Entities from other categories are considered to be accidental because a thing can survive the change of, e.g., its qualities and its quantity – a thing can change its colour or acquire new capacities, and it can grow or shrink. In the Categories, Aristotle points out that it is one of the most distinctive marks of substances that they – successively – admit of contrary properties without ceasing to exist (Categories 5, 4 a 10). However, a thing cannot survive all changes – when a vase breaks, it ceases to be a vase, i.e. it ceases to exist. Similarly, when a horse dies, it ceases to be a horse and it ceases to exist altogether. In such cases, a thing’s essence or form gets lost, and only its matter remains. When a thing comes into existence – e.g., when a house is constructed or a horse is born – the form of the house (or horse respectively) is imposed upon a new chunk of matter. Matter, thus, is the substrate for substantial changes. Reviewing the different ontological roles that a substance is to play in Aristotle’s conception, one should note that what is meant by the term ‘substance’ changes somewhat from one theoretical context to another. For example, if we say that a ‘substance’ is that which comes into being, or ceases to be, we are referring to a particular, concrete, individual including all its accidental properties that is named according to its essence, i.e. with a term that could be used as an answer to what-is questions. However, if we say that a substance is bearer of properties, and if we take this characterisation literally,

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we must refer to such a concrete entity minus its accidental properties. Starting from the Aristotelian conception of substance, many medieval philosophers, e.g. Thomas Aquinas used to distinguish between three kinds of ontologically independent things (Jansen 2008). First, there are the concrete material things known to us by the senses, analysed by Aristotle as composites of matter and form. These substances differ in their capabilities – while some have only physical properties, others can grow, reproduce, perceive and move by themselves, i.e. they are living beings. These capabilities are conferred on living beings by their souls, i.e. by their particular forms. Very special among these soul-forms is the human soul, which not only confers on humans the ability to think, but is also thought to be immortal. It is this peculiarity that makes human souls the lowest inhabitants of the second group of ontologically independent things, the immaterial intelligences, which also comprise the entities called ‘angels’ in religious language. While the existence of these nonembodied thinking substances has been taken for granted up until modern times, their ontology was hotly debated. Aquinas does conceive of them as composites, but not as composites of matter and form, but of essence and existence, i.e., of whatthey-are and that-they-are. Again, it is doubtful whether these can be seen as parts in the sense of mereology. A consequence of this analysis is that there cannot be more than one angel with a specific form, i.e., not more of one angel of a certain kind. Hence

every angel is of its own species. Thirdly, there is God who is not only ontologically independent from all other entities, but is also cause and principle of all other entities. According to Aquinas, God is an absolute non-composite; He is neither composed out of matter and form, nor out of essence and existence: God’s essence is identical with his existence. While God’s ontological independence seems to have been uncontroversial for Aquinas, he was less certain about the question of whether God is a substance – in earlier writing he tends to affirm the claim, later he denies that God is a substance. His reason is that if substance is the first category, then it is a highest genus, and then a specific difference is needed in order to distinguish God from other substances. As God, however, is conceived as an indivisible unity, there is no possibility of Him being composed of a genus and a specific difference. Hence, God cannot share any genus with other entities. Later characterisations of substance build on these foundational considerations from the Aristotelian tradition. Descartes, for example, takes ontological independence as the defining mark of a substance, when he defines a substance as “a thing which exists in such a way as to depend on no other thing for its existence” (Principia I 51). He stays within traditional lines when he says that only God is absolutely independent, and that among created substances there are two kinds, bodies and minds, which each need only God in order to exist, and each have exactly one essential

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attribute, i.e., extension for bodies and thinking for minds. Spinoza, however, gives his definition of substance an epistemic twist, recurring to the definitional priority of substance, characterising substance as “what is in itself and is conceived through itself, that is, that whose concept does not require the concept of another thing, from which it must be formed” (Ethica I, def. 3). Kant then turns his focus on substance as the ultimate substrate of change while seeing the distinction between substance and attribute merely as a cognitive category that is projected by the mind as a form onto the unstructured sense perceptions. According to Kant, substance is conceived by us as that which persists in all changes of appearances (Critique of Pure Reason, B 224). Substance, hence, is conceived to exist at all times, such that in the realm of appearance substance can never be thought of as coming to be or ceasing to exist. In effect, Kant presents an epistemologised account of Aristotelian prime matter, not an account of independent concrete entities. The broad variety of conceptions of and criteria for substances continued to be discussed after Kant, and is still debated in contemporary ontology. See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Whole, Category, Continuants and Occurrents, Ingarden, Subject,, Structure, Substratum, Thomas Acquinas.

References and further readings

Aristotle, The Complete Works. The Revised Oxford Translation, ed. by J. Barnes, Princeton NJ: Princeton University Press 1984. Aquinas Th., On being and essence, transl. with an introd. and notes by Armand Maurer, 2., rev. ed., Toronto: Pontifical Institute of Mediaeval Studies 1983. Barnes, J., 1988, “Bits and Pieces”, in: Barnes, J.; Mignucci, M. (eds.), Matter and Metaphysics, Naples: Bibliopolis, 224-294; repr. in: J. Barnes, Method and Metaphysics. Essays in Ancient Philosophy I, Oxford: Clarendon Press 2011: 429-483. Descartes R., “Principles of Philosophy”, in: Cottingham, J.; Stoothoff, R.; Murdoch, D. (eds.), The Philosophical Writings of Descartes, vol. 1, Cambridge: Cambridge University Press 1985: 177-291. Gill, M.-L., 1989, Aristotle on Substance. The Paradox of Unity, Princeton: Princeton University Press. Hoffman J.; Rosenkrantz, G. S., 1994, Substance among Other Categories, Cambridge: Cambridge University Press. Jansen, L., “Die Struktur der Substanz bei Thomas von Aquin”, in: Gutschmidt, H.; Lang-Balestra, A.; Segalerba, G. (eds.), 2008, Substantia – Sic et Non. Eine Geschichte des Substanzbegriffs von der Antike bis zur Gegenwart in Einzelbeiträgen, Frankfurt: Ontos, 181-209. Halfwassen, J.; Wald, B.; Arndt, H. W.; Trappe, T.; Schantz, R., 1998, “Substanz; Substanz/Akzidens”, in: J.

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Ritter et al., eds., Historisches Wörterbuch der Philosophie, vol. 10, Basel: Schwabe, coll. 495-553.

thrown under something’) it has been used to translate the Greek noun hypokeimenon (‘the underlying’)

Kahn Ch., 1978, “Questions and Categories”, in Hiz, H. (ed.), Questions, Dordrecht/Boston: Reidel.

Historically, the Greek term is used to express two related ideas. In a linguistic context, it is used to refer to the grammatical subject of a sentence as opposed to the predicate of that sentence. In an ontological context, however, it is used to refer to the underlying bearer of a property or attribute. The ontological use is related to the linguistic use, as a bearerattribute relation can be expressed by sentences with a matching subject and predicate, which can, in turn, be made true by that bearer-attribute relation. To use a classical example, that Socrates is the bearer of a certain whiteness accounts for the truth of the sentence ‘Socrates is white’, bringing together the subject-term ‘Socrates’ and the predicate ‘is white.’ (However, not every grammatical subject denotes a bearer of attributes, as shown by sentences like ‘Nessie is green” and ‘Red is a colour.’ Empty names like ‘Nessie’ do not refer to anything, a fortiori not to bearers of attributes. Neither is colour an attribute born by red; rather, red is a determinate property of the determinable property colour.) While the term ‘subject’ was prevalent for both the grammatical and the ontological aspect long into modern times, nowadays it is normally used for the linguistic aspect (among other things), while the term ‘substrate’ can be reserved for the ontological aspect expressing the relation between concrete things and their abstract properties.

Kant I., Critique of Pure Reason, transl. by Guyer, P.; Wood, A. W., Cambridge: Cambridge University Press, 1998. Koslicki, K., 2008, The Structure of Objects, Oxford: Oxford University Press. Loux, M. J., 1978, Substance and Attribute. A Study in Ontology, Dordrecht/Boston: Reidel. Marmodoro, A., 2013, “Aristotle’s Hylomorphism without Reconditioning”, Philosophical Inquiry 36: 1-22. McCall, R. E., 1956, The Reality of Substance, Washington DC: The Catholic University of America Press. Spinoza, B., A Spinoza Reader: The Ethics and Other Works, transl. E. Curley, Princeton NJ: Princeton University Press 1994. Scaltas, Th., 1994, Substances and Universals in Aristotle’s Metaphysics, Ithaca. Ludger Jansen Substrate In Latin, the noun substratum is derived from the participle of the verb substernere (‘underlying’) and denotes that which is laid (stratum) under (sub) something. Along with the word subiectum (‘that which is

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Much of the tradition has been shaped by iterated receptions of Aristotle’s writings and further development of Aristotelian ideas, especially from the Categories, the Physics, and the Metaphysics. Following these texts, the term “substrate” and its cognates from other languages are used for (1a) for the substrates of predication, i.e. for grammatical subjects; (1b) for substrates of properties, i.e. for the bearers of dependent entities; (2) for the substrates of change, i.e. that which persists during change: (1) In his Categories, Aristotle introduces two logically independent criteria to set up a cross-classification of entities into four classes: particular substances, particular accidents, substance universals, and accident universals. These two criteria correspond to the linguistic and ontological use of the term hypokeimenon, namely predicability and inherence: (1a) Something is, first, predicable, if its name can be ‘said of something as a subject (hypokeimenon)’. E.g., colour is predicable, as ‘colour’ can be predicated of ‘white (‘White is a colour’), and man is predicable, because ‘man’ can be predicated of Socrates (‘Socrates is a man’). It is, however, not so clear, whether Aristotle wants to draw a linguistic or an ontic distinction; in any case he seems to employ a linguistic criterion. (1b) Second, something inheres in something else, if it is ‘in something as a substrate (hypokeimenon), though not as a part, and cannot exist separately from what it is in’, like the reading ability is in the soul or a cer-

tain bit of white is in a certain body (Categories 2). The latter phrase indicates that inherence is thought of as a certain kind of ontological dependence. Later in the text it becomes clear that paradigmatic substrates are in the category of substance, because individual substances (‘first substances’ in Aristotle’s terminology) can neither be predicated of something else nor do they inhere in something else (Categories 5). (2) The criteria from the (probably early) Categories are accompanied by the general analysis of movement (kinesis) and change (metabolê) in the Physics. A change needs three distinct elements: Something that comes into existence, the nonpresence of this something before the change starts, and the underlying substrate of this something and its privation, which undergoes the change and persists through it (Physics I 6-7 and 9). When it is a quality, quantity or location that changes, the underlying substrate is the concrete entity that is bearer of the qualities or quantities in question or that inhabits the locations in question. In case of ‘substantial change’ (i.e., generation or corruption of a substance) it is the matter that can take on or lose a certain substantial form. In general, the substrate of change must be able to be a bearer of contrary properties or forms. As this is a defining element of matter (hylê), some have assumed that the ultimate material constitutent is the ultimate substrate. The notion of a substrate is controversial in a number of regards. First, many authors question the idea of a

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correspondence between grammatical subjects and ontological substrates, which some criticize as the fallacious derivation of ontology from the surface properties of Indo-European languages. However, while grammatical structure might have had some heuristic role in the history of metaphysics, arguments for the existence of substrates can well be independent from linguistic facts. We can easily construct formal languages without singular terms (Russell 1905; Quine 1948), but it is hard (though maybe not impossible) to imagine properties without bearers or processes without participants. Second, the exact nature of the substrate of change is often debated. With respect to this issue, Aristotle mostly focuses on the proximate substrate of change – when Socrates’ hair becomes white, Socrates’ hair is the proximate substrate of change; when a house is built, a certain amount of bricks and logs is the proximate substrate. However, other philosophers, like Locke and Kant, focus on the ultimate substrate of properties and the ultimate substrate of change. This would correspond to what Aristotle calls prime matter – the ultimate material constituent of all material beings that has only potential properties or potencies (Metaphysics IX 7, 1049a 24-27; cf. On generation and corruption II 5). It should be clear that prime matter is a metaphysical abstraction. In reality, there is never such a chunk of matter without actual properties around. It is tempting to see concrete particular things as mereological wholes consisting of a substratum and properties as parts. A concrete thing then

would be a mereological complex out of a chunk of prime matter and a number of properties. Instead of talking about chunks of prime matter, some philosophers have postulated ‘bare particulars’, i.e. particulars without any properties, as the substratum of individual things (Suarez, Metaphysical Disputations V; Bergmann, Realism). On this picture, concrete things are sums of a bare substrate and a number of properties. Both ideas, however, might be conceptually incoherent, for if we, e.g., abstract from a thing’s colour and retain only the potency to be coloured, this potency still is a property of the thing (and so would be a potency for a potency, and so on). In addition, the idea of such an ultimate substrate is threatened by a dilemma. Either the ultimate substrate has properties or it does not have them. If it has properties, it is not the ultimate substrate. However, if it has no properties, not only is it not knowable, it is also causally irrelevant. Some philosophers have responded to this dilemma by distinguishing between ‘naked’ and ‘nude’ particulars. Naked particulars have no properties at all, while nude particulars do have properties, but do not have natures. On the latter view, nude particulars necessarily have properties, but they do not have necessary properties (Baker 1967, Garcia 2014). Other responses to the dilemma operate with proximate substrates instead of ultimate ones, or with a distinction between the predication of essential and accidental properties. We can then say that a substrate is ultimate if we abstract from it all accidental

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properties, while still being able to formulate true predicative sentences about the substrate, which still has its essential properties. Similarly, some distinguish between ‘thin’ and ‘thick’ particulars, where the ‘thin’ particular is a thing ‘taken apart from its properties’, which is linked to its properties by instantiation, while the ‘thick’ particular comprises all its properties and thus ‘enfolds both thin particulars and properties’ (Armstrong 1989: 94-96). A concrete particular thing can thus be seen as a mereological sum of a bare/naked/thin particular and its various properties, or of a mereological sum of its substantial core plus its various accidents, or (non-mereologically) as a naked particular that already contains all its properties plus a set of persistence conditions that makes some of these properties necessary properties. Such problems, in combination with a desire for ontological parsimony, have led some to abandon the idea of substrates and conceive of concrete things not as complexes out of a substrate and properties, but as a ‘bundle’ of properties only. The most promising bundle account in recent ontology is known as ‘trope ontology’, where tropes are particular properties (like the colour of Socrates’ hair or the size of my body now). What we normally think of as concrete entities are then complexes of co-located tropes. However, trope theory faces the challenge of having to explain why there is no swapping of tropes from one thing to another. The traditional answer would be that particular properties are ontologically dependent on the substrate that is

their bearer and can thus not exist without them. Tropists have the option to refer to an ontological dependence of tropes on other tropes that form something like a ‘nucleus’ of a trope bundle (Simons 1994). But in this way tropists actually redefine substrates rather than eliminate them, and so the substratum will continue to occupy ontologists. See also > Aristotle’s Theory of Parts, Category, Material Constitution, Mereological Essentialism, Ontological Dependence, Persistence, Subject, Substance. References and further readings

Aristotle, The Complete Works. The Revised Oxford Translation, ed. by J. Barnes, Princeton NJ: Princeton University Press, 1984. Armstrong, D. M., 1989, Universals. An Opinioated Introduction, Boulder CO: Westview. Baker, R., 1967, “Particulars: Bare, Naked, and Nude”, Noûs 1: 211-212. Bergmann G., 1967, Realism, Madison: University of Wisconsin Press (esp. sect. 1-2). Garcia, R. K., 2014, “Bare Particulars and Constituent Ontology”, Acta Analytica 29 (2): 149-159. Kaufmann M., 1998, “Substrat”, in Ritter, J. et al. (eds.), Historisches Wörterbuch der Philosophie, vol. 10, Basel: Schwabe, coll. 557-560. Loux, M., 2012, “Substances, Coincidentals, and Aristotle’s Constituent

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Ontology”, in Shields, C. (ed.), The Oxford Handbook of Aristotle, Oxford: Oxford University Press, 372399. Quine W. V. O., 1948, “On what there Is”, in Review of Metaphysics 2, 21-38; repr. in: ibd., From a Logical Point of View, Cambridge MA: Harvard University Press 1953: 1-19. Russell B., 1905, “On Denoting”, Mind NS 14, 479-493; repr. in: ibd., Logic and Knowledge, ed. Robert Marsh, London: Allen & Unwin 1956: 39-56. Simons, P., 1994, “Particulars in Particular Clothing: Three Trope Theories of Substance”, Philosophy and Phenomenological Research 54: 553575. Stolzenberg J., 1998, “Subjekt”, in Ritter, J. et al. (eds.), Historisches Wörterbuch der Philosophie, vol. 10, Basel: Schwabe, coll. 373-399. Suarez, F., Metaphysical Disputations V: Individual Unity and Its Principle, transl. Garcia, J. J. E., Milwaukee WN: Marquette University Press 1982. Ludger Jansen

Sum Starting with overlaps as primitive, we can define part as follows: x is part of y if and only if everything that overlaps x also overlaps y. Something that overlaps nothing, according to this definition, is part of everything. To avoid this and similar strange consequences, the domain of

mereology is properly restricted to items – call them ‘overlappers’ – that overlap something. (Every overlapper overlaps itself.) Although this made-up term appears only in this paragraph, readers may assume that colorless words such as ‘thing’, ‘everything’, and so forth, refer to overlappers. Coincidence is a central notion: x and y coincide when they overlap exactly the same things. In terms of ‘part’, x and y coincide when they are parts of each other. Some definitions of ‘sum’ are in terms of ‘part’, ‘distinct’, ‘overlap’, and combinations of these. The following definition is purely in terms of ‘overlap’: x is a sum of the y’s if and only if anything that overlaps any y overlaps x and anything that overlaps x overlaps some y. Some authors put definitions in terms of set and member. ‘For a nonempty set A, x is a sum of A if and only if every member of A … &c.’ Definitions in terms of set and member have advantages and disadvantages. One motive for using the notion of a sum is to replace the notion of set and avoid puzzles and paradoxes of sets. (For example, there is no sum without a part; but if there are sets, there is a set without a member.) Some theorists emphasize that mereology is a calculus of individuals, items of the lowest logical type, and can thus avoid admitting abstract objects such as properties and sets. Van Cleve, 2008 (who refers to van Inwagen, 1990) shows how using plu-

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ral noun phrases as above avoids reference to sets.

inclined to reject both please contact the author.)

Definitions of the sum-part relation by means of the set-member relation help make an important point more explicit. Just as members of a set need have no shared properties or particular non-set-theoretic relations between them, parts of a sum can be heterogeneous in quality and spatiotemporally scattered.

This Handbook entry refrains from attempting to support one of these positions.

Tarski’s formulation of mereology has two postulates. The first says that ‘part’ is transitive: if x is part of y, and y is part of z, then x is part of z. (This provides another contrast between parts of sums and members of sets. Set membership is not transitive.) The above definition of ‘part’ in terms of ‘overlap’ entails transitivity, which makes a transitivity postulate superfluous. On the other hand, the definition quietly depends on another postulate, namely that ‘overlap’ is symmetric: if x overlaps y, y overlaps x. Tarksi’s second postulate conjoins separate claims.

See also > Coincidence, Fusion, Metamathematics of Mereology, Non-Well-Founded Mereology, Philosophy of Mathematics. Bibligraphical remarks

Lewis, D., 1991. The summation postulates in this entry emulate Lewis’s formulations on p. 74. Simons, P., 1987. Part I is a standard reference for extensional mereology. Parts II and III treat further topics and question some assumptions of extensionality. Has an extensive bibliography. References and further readings

Unrestricted Summation. Any things have at least one sum.

Cartwright, R., 1975, “Scattered Objects”, as reprinted in his Philosophical Essays, 1987, Cambridge, MIT Press, 171-86.

Unique Summation. Any things have at most one sum.

Lewis, D., 1991, Parts of Classes, Oxford: Blackwell.

Without treating the status of postulates generally, we assume that one accepts postulates only when one is willing to accept all their consequences. Philosophers disagree sharply about accepting these two summation postulates. Some accept both; some accept only the first; some accept only the second. (Any

Sanford, D. H., 2003, “Fusion Confusion”, Analysis 63: 1-4. Sanford, D. H., 2011, “Can a Sum Change its Parts?”, Analysis 71: 2359. Simons, P., 1987, Parts: A Study in Ontology, Oxford: Clarendon Press.

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Tarski, A., 1929, “Foundations of the Geometry of Solids” as reprinted in his Logic, Semantics, Metamathematics, Oxford: Oxford University Press, 1959: 24-29. Van Cleve, J., 2008, “The Moon and Sixpence: A Defence of Mereological Universalism”, in Sider, T. et. al. (eds.) Contemporary Debates in Metaphysics, Oxford: Blackwell: 321-40. van Inwagen, P., 1990, Material Beings, Ithaca: Cornell U. P. van Inwagen, P., 2006, “Can Mereological Sums Change their Parts?”, The Journal of Philosophy 103: 61430. Wiggins, D., 1980, Sameness and Substance, Cambridge, Harvard U. P. David H. Sanford

Syllogism Before Aristotle ‘syllogism’ had the general sense of ‘reckoning up’ or ‘inference’ (Herodotus, Histories 2.148; Plato, Republic 516B). But Aristotle narrowed the usage. The precise nature of what counts as a syllogism for Aristotle is obscure. It does not include all the arguments that would be reckoned valid today in a system of predicate logic, e.g. those having no or only one premise (µονολήµµατα). On the other hand, Aristotle, outside of his syllogistic proper, does not restrict ‘syllogism’ to arguments having only two premises in the canonical forms (A, E, I, O) standardised in later textbooks (Prior Analytics 24 b 19-20; Topics

100 a 25-7; Sophistical Refutations 164 b 27-165 a 2; Rhetoric 1356 b 16-8; Smith 1989, xvi). He does make that restriction once he discusses his categorical syllogistic with its three figures (Prior Analytics, 25 b 32-5; 41 b 36-8; 42 a 30-3; 44 b 6-8). Alexander, 8, 19-24; 20, 24-6, is right to insist that Aristotle develops his syllogistic to construct the logical theory for his demonstrative science. So perhaps he focuses on syllogisms having those forms of use to him. Aristotle is best known for his categorical syllogistic. He also develops in much detail a modal syllogistic. He makes some remarks about other categorical forms, later said by Galen to belong to a fourth figure of syllogisms (Łukasiewicz 1957: 38-42), hypothetical syllogisms, and the reduction of more complex arguments to syllogistic form. Later the Stoics developed an elaborate hypothetical syllogistic for conditional, disjunctive and conjunctive propositions (Sextus Empiricus, Adversus Mathematicos, VIII; Mates 1961; Boethius, De hypotheticis syllogismis). The medievals constructed elaborate syllogistics for many other types of propositions: tensed ones; ones with opaque operators like ‘knows’; exponibles, like ones containing ‘except’ or ‘qua’. They also constructed syllogisms for obverse forms like ‘Every non-S is a non-P’, which Aristotle recognised but did not use in his syllogistic (e.g. Albert, Opera ad logicam pertinentia; Ockham, Summa logicae; Lax, Tractatus syllogismorum). In modern times, with the rise of the more powerful notation of predicate logic, syllogistic has been

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relegated mostly to textbooks discussing only the categorical forms. Aristotle’s syllogistic has mereological features. The canonical premises are negative or affirmative, particular or universal (‘Every S is (not) P’; ‘Some S is (not) P’). Aristotle (Prior Analytics, 24 a 17-20) calls the universal καθόλου and the particular κατὰ µέρος or ἐν µέρει; that is, ‘of the whole’ or ‘in part’. Usually he expresses the relation between subject and predicate by ὑπάρχει, ‘belongs to’, as in ‘Animal belongs to every swan’ (Mignucci 1961: 480-481; Kahn 1982: 323). His main principles for the syllogism, the dicta de omni et de nullo, are expressed in terms of parts and wholes: “Whenever three terms are so related to one another that the last is in the middle as in a whole, and the middle is either in, or not in, the first as a whole, the extremes must be related by a perfect syllogism” (Prior Analytics, 25 b 32-5). In modern mereology, the dictum de omni is an instance of M3, the axiom of the transitivity of parts: If Pxy and Pyz, then Pxz. Those who developed modern mereology had strong interests in Aristotle’s syllogistic. Like Łukasiewicz, Twardowski and Leśniewski had an interest in the validity of Aristotle’s syllogisms (Leśniewski 1928; Dambska 1978: 123; Poli / Libardi 1999: 187-188). Some peculiar doctrines of Aristotle’s syllogistic may also appeal to parts and wholes. Take Aristotle’s claim that sometimes a necessary

conclusion follows from a necessary major and a categorical minor premise, as with INAANA (If it is necessary that every B is A, and every C is B, then it is necessary that every C is A) (Prior Analytics, 30 a 17-23). This was rejected as invalid already by Aristotle’s student Theophrastus. However, Aristotle seems to find the validity of the inference obvious. He says only that C is one of the B’s. At 30 a 40 to prove INAINI valid, Aristotle says, apparently equivalently, “C is under B”, which often signifies the dictum de omni in both Analytics. It is worth noting that the former phrasing is a typical formulation for ἔκθεσις (exposition) and also occurs regularly in the Posterior Analytics. Along these lines, Rescher (1974: 3; 14-5), proposes understanding the validity of this syllogism to come from the minor premise justifying the introduction of a new, complex term (‘BC’), the C species of B. Then it is necessary for the C species of B that it be B, and this plus the original major premise validly infers a necessary conclusion. Rescher’s explanation follows the interpretation of exposition given by Łukasiewicz. He developed a view of particular propositions based on this conception of expostion. (Łukasiewicz 1957: 59-67; Patzig 1968: 118, 166-180). On this view, ‘Some A is B’, say, is to be read as making an existential claim of a general, complex term: ‘(There) is an BA’. Then (I) ‘Some A is B’ iff ‘(∃M)(Every M is A and every M is B)’ (Łukasiewicz 1957: 61; Patzig 1968: 161). Likewise, (O) ‘Some A

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is not B’ iff ‘(∃M)(No M is A and no M is B)’ (Patzig 1968: 171). However, Smith (1982: 118) and Mignucci (1991: 14-17) object to this interpretation of exposition because it does not work for Darapti (Aristotle, Prior Analytics 28 a 22-6), where the exposition is on universal premises. (This might not be a problem since the universal affirmative A implies the particular affirmation I.) Yet some who reject this approach admit that it works better sometimes. (Mignucci 1991: 17-8) Łukasiewicz’s understanding of exposition is not the customary one. As Aristotle (Prior Analytics 49 b 33-50 a 3) uses it, ‘exposition’ involves “putting forward” an instance of a universal claim, and even states that it does not constitute proof nor give additional premises (cf. Alexander, 32, 32-33,15; 99, 19-100, 26). Alexander suggests that ἔκθεσις consists not in substituting a part of S for S: for what would be the difference? Rather, according to Aristotle (Prior Analytics 33,1-5), ἔκθεσις consists in putting forward an instance of S given by present sense perception. He says that ἔκθεσις then demands a special assumption, sc. of there being that sense perception. Failing that assumption, a proof by ἔκθεσις fails. He says that ἔκθεσις is proper to the third figure as it concerns the particular and not the universal, and only in the third figure do valid syllogisms by the middle term’s being tied to some individual (some R) arise. Still Łukasiewicz’s view has historical precedents. Robert Kilwardby (d. 1279), says about INAANA:

... the conclusion is part of the major, and mostly in regard to the predicate, which they share. With regard to the subject, it is part of the minor. And so it follows the minor in features affecting the subject (such as universality and particularity), and the major in features affecting the predicate (such as affirmative and negative, assertoric and modal). (Quoted by Thom 2007: 154)

So perhaps this mereological approach to exposition and the modal syllogistic should be reconsidered. See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Jurisprudence, Leśniewski, Mereological Triangle, Structure, Universal. References and further readings

Albertus Magnus, 1890, Opera Ad Logicam Pertinentia, in Opera Omnia, ed. A. Borgnet, Vol. I, Paris: Vivès (Venice 1632 and Lyons 1651 were consulted). Alexander of Aphrodisias, 1883, Alexandri Aphrodisiensis in Aristotelis analyticorum priorum librum I commentarium, ed. M. Wallies, in Commentaria in Aristotelem graeca Vol. 2.1, Berlin: Georg Reimer. Alexander of Aphrodisias, 1991, On Aristotle’s Prior Analytics I: 1-7, trans. & comm. by Barnes, J.; Bobzien, S.; Flannery, K.; Ierodiakonou, K., London: Duckworth. Alexander of Aphrodisias, 1999, On Aristotle’s Prior Analytics I.8-13, trans. & comm. by Mueller, I.;

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Gould, J., Ithaca: Cornell University Press.

matematyki”, Przeglad filozoficzny 34: 142-170).

Avicenna, Al-Qīyās (Syllogism), ed. Zayid, S.; Madkour, I., Cairo: alMatba“a I-Amiriyya, 1964. (Volume I, Part 4 of al-Shifa’)

Łukasiewicz, J., 1957, Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, second edition, Oxford: Clarendon Press.

Boethius, M. S., De Hypotheticis Syllogismis, in Opera Omnia, ed. J.-P. Migne, Patrologia Latina, Vol. 64, Paris, 1891.

Malink, M., 2013, Aristotle’s Modal Syllogistic, Cambridge Mass.: Harvard University Press.

Dambska, F., 1978, “Brentano et la pensée philosophique en Pologne: Casimir Twardowski et son école”, Grazer philosophischen Studien 5: 117-129. Herodotus, Historiae. Henry, D. P., 1958, Medieval Logic and Metaphysics, London: Hutchinson. Kahn, C., 1976, “Why Existence Does not Emerge as a Distinct Concept in Greek Philosophy”, Archiv für Geschichte der Philosophie, Vol. 58.4; repr. in Morewedge, P. (ed.), 1982, Philosophies of Existence, New York: Fordham University Press. Lax, G., 1509, Tractatus syllogismorum, Paris. Leśniewski, S., 1983, “On the foundations of mathematics”, Topoi 2: 7-52 (English translation of 1928, “O podstawach matematyki”, Przeglad filozoficzny 34: 142-170). Leśniewski, S., 1992, “On the Foundations of Mathematics”, in Surma, S. J.; Srzednicki, J. T.; Barnett, D. I.; Rickey, V. F. (eds.), Collected Works, Kluwer: Dordrecht (English translation of 1928, “O podstawach

Mates, B., 1961, Stoic Logic, Berkeley: University of California Press. Mendell, H., 1998, “Aristotelian Demonstration”, Oxford Studies in Ancient Philosophy 16: 161-225. Mignucci, M., 1961, Gli analitici primi, traduzione, introduzione e commento di Mario Mignucci, Naples: Luigi Loffredo. Mignucci, M., 1991, “Expository Proofs in Aristotle’s Syllogistic”, Oxford Studies in Ancient Philosophy 9: 9-28. Ockham, William, 1974, Summa Logicae, in Opera Philosophica et Theologica, Vol. I, St. Bonaventure, NY. Patzig, G., 1969, Aristotle’s Theory of the Syllogism, Dordrecht: Reidel (J. Barnes’s English translation of Die Aristotelische Syllogistik, 1959). Plato, Republic. Poli, R.; Libardi, M., 1999, “Logic, Theory of Science, and Metaphysics

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according to Stanisław Leśniewski”, Grazer Philosophische Studien 57: 183-219. Rescher, N., 1974, “A New Approach to Aristotle’s Apodeictic Syllogisms”, in Studies in Modality, Oxford: Oxford University Press. Simons, P., 1994, “Discovering Lesniewski”, History and Philosophy of Logic 15: 227-235. Smith, R., 1982, “What Is Aristotelian Ecthesis?”, History and Philosophy of Logic 3: 113-127. Smith, R., 1983, “Completeness of an Ecthetic Syllogistic”, Notre Dame Journal of Formal Logic 24: 224232. Smith, R. [trans. & comm.], 1989, Aristotle’s Prior Analytics, Indianapolis: Hackett. Smith, R., 1983, “An Ecthetic Syllogistic”, Notre Dame Journal of Formal Logic 24: 224-232. Thom, P., 1981, The Syllogism, Munich: Philosophia Verlag. Thom, P., 1991, “The Two Barbaras”, History and Philosophy of Logic 12: 135-150. Thom, P., 1976, “Ecthesis”, Logique et Analyse 74-6: 299-310. Thom, P., 1993, “Apodeictic Ecthesis”, Notre Dame Journal of Formal Logic 34: 183-208. Thom, P., 1996, The Logic of Essentialism, Dordrecht: Kluwer. Allan Bäck

Syntax The conceptual tool of part-whole relations resolves a paradox: it allows for something to be viewed as both ‘one’ and ‘many’ at the same time. On the one hand, it enables us to analyse one thing as more than one by decomposing it into parts. On the other hand, it allows us to view many things as one if we posit a single whole that they are parts of. Both analytic steps facilitate generalisations: wholes that appear distinct may turn out to be similar by sharing some of their parts, and assemblages of different things may form wholes that are in some respects alike. Both directions of mereological analysis have found abundant application in the description of sentence structure. First, it is useful to break sentences into words: the two sentences Bill arrived and Bill left are distinct as wholes but similar in that they both include the word Bill as their subject. Second, sequences of words can act jointly as single units, such as when an entire clause functions as the subject of the sentence [That he arrived] surprised me. All syntacticians posit sentences as wholes and words as their parts and most will posit clauses and phrases in between as well (cf. Jackendoff 1991). Mereological structure in syntax is called constituent structure or phrase structure. (1) shows a traditional phrase structure tree of a sentence complete with category labels.

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noun phrase, rules of which word sequences can be subjects and objects and which can be replaced by pronouns or left out would have to be stated by listing all the parts over and over again, as in (3). The concept noun phrase simplifies these rules, as in (4). (3) (a) A subject must be a noun, or an article and a noun, or an article and an adjective and a noun, etc. (1) Symptoms of syntactic wholes phrases or clauses - include joint recurrence in and across sentences, joint non-occurrence (replaceability or omissibility), and adjacency of the parts. These properties of mereological structure are illustrated in (2) for the noun phrase. (2)(a) Recurrence in and across sentences (i) [The guests]NP read [the book]NP. (ii)[The guests]NP are tired. (b) Joint replaceability by a pro-form [The guests]NP arrived and they left immediately. (c) Joint omissibility [The guests]NP arrived and __ left immediately. (d) Adjacency (i) [The guests]NP arrived. (ii) *The arrived guests. The utility of positing such wholes is shown by what happens if we do not posit them. Without the concept of

(b) A pronoun may replace a noun, or an article and a noun, or an article and an adjective and a noun, etc. (4) (a) A subject must be a noun phrase. (b) A pronoun may replace a noun phrase. (c) A noun phrase is a noun or an article and a noun or an article and an adjective and a noun, etc. While the mereological analysis of sentences enhances the generality of descriptions, it introduces problems of its own. On the one hand, not all syntactic part-whole relations are simple; on the one hand, criteria for wholeness may be in conflict. Complexity. Simple partonomic systems would involve only two sister parts for each whole, no parts within parts, parts that are equal, each part uniquely assigned to a single whole, and wholes that are compositional (i.e., their properties are the sum of the properties of the parts and their relations) (cf. Moravcsik 2006; 2009). There are, however, syntactic constructions that violate one or the other of these simplicity criteria. For example, in a noun phrase like the

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ripe cherries, the adjective ripe and the noun cherries are not on equal footing in that the noun phrase is grammatical without the adjective (the cherries) but not without the noun (*the ripe). Another complex syntactic mereological pattern is a part simultaneously belonging to two wholes. Thus, in Jill expects him to succeed, the pronoun him belongs both to the main clause Jill expects him and to the subordinate clause him to succeed. Such instances of what is called multiple motherhood have been dealt with differently in various syntactic theories. In Word Grammar, him is represented as simultaneously dependent on both verbs: expects and succeed (Hudson 1984: 112). Other approaches in turn solve the problem by an additional application of partonomy: rather than assuming a single structure, they assign two constituent structures to each sentence. Thus, in some versions of transformational grammar, him is the subject of the subordinate clause in underlying structure and is subsequently “raised” into the main clause by a transformation (Postal 1974). Conflicting criteria. A common way

in which wholeness criteria are in conflict involves adjacency: words of what otherwise appears to be a phrase or a clause may not be contiguous. For example, (5) and (6) show that the English noun and relative clause may or may not be adjacent. (5) (a) The man that came to see me is German.

(b) The man is German that came to see me. Yet, for purposes of replaceability, noun and relative clause form a phrase whether they are adjacent or not. (6) (a) The man that came to see me is German. He was kind. (b) The man is German that came to see me. He was kind. Discontinuous constituency has been a central problem in syntactic theorising (Huck & Ojeda 1987). One solution is once again by splitting a single structure into two layers. For sentences like (5b), surface structure would have the relative clause separated from the noun but in the underlying structure, the clause is adjacent to the noun, just as in (5a). Underlying structure thus regularises the linearly deviant construction (see for example Jacobs & Rosenbaum 1968). Other analyses (e.g. Sadock 1987) assume that the non-adjacent noun and relative clause do not form a phrase in syntax but they do so in semantics. In sum: mereological analysis solves some problems in syntax and poses others. Many of the differences among syntactic theories boil down to varying attempts to represent the exact nature of constituent structure and to resolve its complexities and contradictions. See also > Grammar, Linguistic Structures, Paradoxes, Possessives and Partitives, Structure.

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Huck, G. & Ojeda, A., 1987. Papers on discontinuous constituents.

Jackendoff, R., 1991, “Parts and Boundaries”, Cognition 41, 1-3: 945.

Jacobson, P., 1996. A synopsis of theories of syntactic constituent structure.

Jacobs, R. A.; Rosenbaum, P. S., 1968, English Transformational Grammar, Waltham, MA: Blaisdell.

Jackendoff, R., 1991. A partonomic analysis of semantic and syntactic structures.

Langacker, R., 1997, “Constituency, Dependency, and Conceptual Grouping”, Cognitive Linguistics 8:1, 1-32.

Langacker, R., 1997. A less constrained concept of constituent structure.

Leffel, K.; Bouchard, D. (eds.), 1991, Views on Phrase Structure, Dordrecht: Kluwer.

Moravcsik, E., 2009. A general overview of part-whole relations in syntax.

Moravcsik, E., 2006, An Introduction to Syntax, Fundamentals of Syntactic Analysis, London: Continuum.

Tversky, B., 1990. An overall analysis of partonomic structures in language

Moravcsik, E., 2009. “Partonomic Structures in Syntax”, in Vyvyan, E.; Purcell, S. (eds.), New Directions in cognitive Linguistics, 269-285. Amsterdam/Philadelphia: Benjamins

References and further readings

Postal, P., 1974, On Raising. One Rule of English and its Theoretical Implications, Cambridge, MA: MIT Press.

Bibliographical remarks

Bergs, A., 2003, “Holismus und Individualismus in der Linguistik: ein Überblick”, in Bergs, A.; Curdts; S. I. (eds.), Holismus und Individualismus in den Wissenschaften, Frankfurt a. M.: Lang, 143-161. Huck, G.; Ojeda, A. (eds.), 1987, Discontinuous Constituency. Syntax and Semantics 20, Orlando, FL: Academic Press. Hudson, R., 1984, Word Grammar, Oxford: Blackwell. Jacobson, P., 1996, “Constituent Structure”, in Brown, K.; Miller, J. (eds.), Concise Encyclopedia of Syntactic Theories, Oxford: Elsevier, 5467.

Sadock, J. M, 1987, “Discontinuity in Autolexical and Autosemantic Syntax”, in Huck & Ojeda, ed. 283-301. Tversky, B., 1990, “Where Partonomies and Taxonomies Meet”, in Tsohatzidis, S. L. (ed.), Meanings and Prototypes. Studies in Linguistic Categorization, 334-344, London: Routledge. Edith Moravczik

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T Tarski, Alfred Alfred Tarski (original surname Tajtelbaum) (1902-1983) was born in Warsaw, Poland, and died in Berkeley, California, having immigrated to the United States in 1939. Despite prodigious mathematical abilities, Tarski began studying biology at the University of Warsaw in 1918. Soon after, he came under the influence of the brilliant Polish logician Stanislaw Leśniewski, and applied his energies to work in logic, attending lectures by Leśniewski, Jan Łukasiewicz, Stefan Mazurkiewicz, Waclaw Sierpinski, and Tadeusz Kotarbinski, and eventually completing a doctoral dissertation with Leśniewski. Having earned his advanced degree, Tarski taught logic at the Polish Pedagogical Institute, and mathematics at the University of Warsaw, where he was hired to work as Łukasiewicz’s assistant, as well as accepting teaching assignments at a local secondary school. During this early time in his career, Tarski wrote several textbooks and papers in logic and mathematics, but failed in several efforts to obtain university professorships in Poland. Undeterred by these setbacks, Tarski began an association with the so-called Vienna Circle of logical positivists, lecturing there in 1930 and meeting Kurt Gödel; in 1935 he returned to Vienna under a

fellowship and presented some of his ideas to the first meeting of the International Congress for the Unity of Science in Paris. A subsequent meeting of the Congress at Harvard University in 1939 enabled Tarski to avoid Nazi persecution after the Third Reich’s occupation of Poland, traveling on the last boat permitted to leave the country. In the United States, Tarski taught at Harvard, City College of New York, and was a Guggenheim Fellow at the Institute for Advanced Study at Princeton University, where Gödel had also fled after the Vienna Circle diaspora. Tarski joined the mathematics department at Berkeley in 1942, where he spent the rest of his life, and became a US citizen in 1945. He lectured at a number of institutes and universities worldwide, was honored with prestigious academic awards, and served as president of the Association for Symbolic Logic and International Union for the History and Philosophy of Science. Throughout his career, Tarski made important contributions to logic, formal and philosophical semantics, mathematics and the foundations of mathematics. Of greatest relevance to the field of mereology, is the landmark essay “Sur la décomposition des ensembles de points en parties respectivement congruentes” from 1924, written in French and coauthored with Stefan Banach, where Tarski proved the paradox known as ‘doubling the ball’, since referred to also as the Banach-Tarski or Hausdorff-Banach-Tarski Paradox. The proof appeals to the axiom of choice in standard set theory and the

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abstract process of geometrical translation. The axiom of choice states that a set can be defined from or composed by randomly selected members of any number of sets containing at least one member. Rotation, in turn, is the movement of every point in a geometrical figure at a constant distance in a specified direction. Banach and Tarski’s proof demonstrates that by assuming the axiom of choice it is possible to take a solid three-dimensional ball, divide it into finitely many (precisely six, non-measurable) pieces, and then, redistributing the parts exclusively by means of translation and rotation, reassemble the pieces into two balls of the same radius as the original, in effect producing two solid balls of the same size out of one. Banach and Tarski presented the proof as a reductio ad absurdum of the axiom of choice, but it has often been interpreted instead as showing merely that the axiom has surprising implications as a ‘topological curiosity’. In his 1929 paper, Les fondaments de la geometrie des corps, Tarski proved that much of Euclidian solid geometry could be recovered as a first order theory whose objects are limited to spheres, a single primitive binary part-whole containment relation, and two axioms implying that containment partially orders the spheres. By extending the language to include solid geometrical figures other than spheres, Tarski provides a more elegant formalisation of mereology than Lesniewski’s. After 1933, Tarski embarked on the research program that is more widely

recognised in philosophical logic and semantics. Here he undertook to provide formally rigorous definitions of such concepts as truth and logical consequence. Confronting the liar paradox, which arises in a classical bivalent truth-value semantics for a sentence that implies its own falsehood, Tarski draws a number of important distinctions for avoiding inconsistency in formal languages. He distinguishes the sentences of lowlevel object languages about a class of objects from the sentences of higher-level metalanguages, arranged in a hierarchy in which it is possible to predicate in a metalanguage L* the truth or falsehood of sentences in the object language or in metalanguages lower than L* within the hierarchy. Tarski then proposes satisfaction schemata wherebya formula is satisfied by an assignment of objects to variables in the formula just in case taking each free variable in the formula as a name of an object assigned to the formula’s variables makes the formula into a true sentence. Tarski exploits a recursive compositionality relation according to which the satisfaction conditions of arbitrarily complex formulas are determined by the satisfaction conditions of its component subformulas. In this way, he covers all possibilities from the standpoint of a limited surveyable specification of satisfaction requirements. The key to Tarski’s efforts to explicate the concept of satisfaction for formal languages is its reliance exclusively on set theory and a language’s syntax, while satisfaction and the assignment of objects to variables belong to semantic theory. Ul

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timately, Tarski does not offer a theory of truth as such, but of the truth conditions for the sentences in a particular formal language; he provides a related family of truth conditions based on quantifier elimination principles. Tarski’s criterion of logical consequence is equally syntactical. He regards an inference as a sequence of sentences in which truth is preserved when we hold the logical constants in the sentences fixed and uniformly substitute any arbitrary choice of other nonlogical terms for the sentence’s nonlogical terms. Tarski was aware of the problems in trying to distinguish logical from nonlogical terms on which his analysis of the concept of logical consequence depends, and offered suggestions for syntactical factors whereby the two categories of terms can be properly isolated. Tarski made further valuable discoveries in mathematics. His studies led him in particular to cardinal and ordinal algebra. He developed a formal decision method for elementary geometry and algebra, and proved, again by quantifier elimination, that first-order real number theory closed under addition and multiplication and Abelian groups are formally decidable. In contrast, Tarski demonstrated that lattice theory, abstract projective geometry, and closure algebras, and non-Abelian groups are formally undecidable. See also > Boolean Algebra, Leśniewski, Mereotopology, Para-

doxes, Philosophy of Mathematics, Points. References and further readings

Banach, S.; Tarski, A., 1924, “Sur la décomposition des ensembles de points en parties respectivement congruentes”, Fundamenta Mathematicae 6: 244-277. Etchemendy, J., 1990, The Concept of Logical Consequence, Cambridge: Harvard University Press. Feferman, A. B.; Feferman, S., 2004, Alfred Tarski: Life and Logic, Cambridge: Cambridge University Press. Tarski, A., 1929, “Les fondaments de la geometrie des corps”, Ksiega Pamiatkowa Pierwszego Polskiego Zjazdu Matematycznego: 29-33. Tarski, A., 1984 [1956], Logic, Semantics, Metamathematics: Papers 1923-38, ed. and trans., Woodger, J.; Corcoran, J., Indianapolis: Hackett. Wagon, S., 1986, The Banach–Tarski Paradox, Cambridge: Cambridge University Press. Dale Jacquette

Temporal Parts Temporal parts are analogous to spatial parts: just as the conference has one spatial part which occupies the seminar room, and another which occupies the lecture hall, it has one temporal part which ‘occupies’ Friday and another which ‘occupies’ Saturday. These temporal parts of the

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conference have half-hour coffeebreaks as temporal parts of their own; these coffee-breaks are also temporal parts of the whole conference. It is relatively uncontentious that events, e.g. conferences, have temporal parts; perhaps only a presentist, who denies the existence of past and future, would reject this claim. It is much more contentious whether material objects, e.g. cats, cars and continents, have temporal parts too. There are two main theories of the persistence of material objects – perdurantism and endurantism. Of these, perdurantism is often identified with the claim that persisting objects have temporal parts, or indeed that each has an instantaneous temporal part (a ‘time slice’) at every moment at which it exists. Conversely, endurantism is identified with the claim that persisting objects are ‘wholly present’ – not just partially present – whenever they exist. Theodore Sider provides the following definition: x is an instantaneous temporal part of y at instant t =df (1) x exists at, but only at, t; (2) x is part of y at t; and (3) x overlaps at t everything that is part of y at t. He writes: This captures the idea that my current temporal part should be a part of me now that exists only now but is as big as me now. It should overlap my arms, legs – everything that is a part of me now. [Perdurantism] may then be formulated as the claim that, necessarily, each spatiotemporal object has a temporal part at every moment at which it exists. (2001: 59)

Sider argues that it is difficult to provide a positive characterisation of the central endurantist claim that persisting objects are ‘wholly present’ whenever they exist. Roughly, this is because endurantists are restricted to a notion of parthood-at-t rather than atemporal parthood. Suppose that a exists at t. Then it is trivial that anything which is a part-at-t of a is present at t (in that sense, it is trivial that a is wholly present at t). Yet provided a gains and loses parts, it is false that anything which is at any time part of a is present at t (in that sense, it is false that a is wholly present at t). Debate continues about how best to formulate the enduranceperdurance distinction, and about whether the notion of ‘temporal part’ should play a central rôle (McKinnon 2002, Crisp and Smith 2005). Supposing that we have grasped the central notion, why believe that persisting material objects have temporal parts? The main reason is that this enables us to solve puzzles about change, about coincidence, and about vagueness. Accordingly, those who reject temporal parts should show that they have solutions of their own to these puzzles, and/or show that the temporal-parts ‘solutions’ are ineffective. The first puzzle is the problem of change or ‘problem of temporary intrinsics’ as David Lewis called it (1986: 202-204). Ordinary persisting things change: the cat is first asleep then awake. Yet Leibniz’s Law tells us that one and the same object cannot have incompatible properties – properties like being awake and be-

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ing asleep. If the cat has temporal parts, the puzzle is resolved: an earlier temporal part of the cat is asleep, and a later temporal part of the cat is awake, and it is in virtue of this that we may truly say that the cat itself is first asleep and then awake. Those who reject temporal parts propose alternative explanations of how the passage of time makes it possible for one and the same object to have apparently incompatible properties (Merricks 1994 surveys some options). In addition, some claim that the temporal parts picture simply eliminates change, on the grounds that no single thing has both being awake and being asleep nonderivatively. The second puzzle is that of temporary coincidence. The sculptor moulds the lump of clay into a statue. Now the statue and the lump are distinct yet coinciding material objects. The puzzle is to explain how they can differ in their properties (e.g. their historical and sortal properties) whilst apparently sharing all their parts. If the objects have temporal parts, they do not in fact share all their parts, and the puzzle is resolved. Those who reject temporal parts may deny the existence of statue, lump or both (van Inwagen 1990), or else provide some other explanation of how objects may differ despite sharing all their parts (Baker 2002). They may also argue that the temporal parts solution does not generalise to similar puzzles in which apparently distinct objects coincide permanently.

The third puzzle concerns vagueness in persistence. Many objects, including ourselves, appear to have fuzzy beginnings or endings; some have trajectories which become obscurely entangled with those of other objects. If objects have temporal parts and if the various overlapping pluralities of such parts have sums, then there are very many nearly-coincident persistents. We can then blame vagueness either on our inability to discern which of the many overlapping persistents we refer to (an epistemic theory of vagueness) or else on our failure determinately to refer to any particular persistent (a semantic theory of vagueness). Those who reject temporal parts may propose an ontic theory of vagueness, according to which some objects are vague in and of themselves, or insist that we are simply ignorant of the exact trajectories of the few persisting objects there are, or posit a proliferation of coinciding objects (van Inwagen 1990 opts for ontic vagueness). To do this puzzle-solving work, temporal parts must be abundant (enough to underpin every intrinsic change), and readily combinable (enough to underpin a semantic or epistemic account of vagueness in persistence). Advocates of temporal parts typically accept that every persisting object has an instantaneous temporal part at every moment at which it exists, and that such temporal parts obey the principles of classical extensional mereology, in particular the principles of unrestricted fusion and unique fusion. Sometimes they also presuppose that instantaneous temporal parts are in some sense more basic

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than the persistents they compose. But these principles and presuppositions are not built into the very notion of a temporal part.

http://plato.stanford.edu/archives/win 2004/entries/ temporal-parts/

See also > Coincidence, Continuants and Occurrents, Homeomerous and Automerous, Medieval Temporal Parts, Mereological Essentialism, Persistence, Substance, Substrate.

Lewis, D., 1986, On the Plurality of Worlds, Oxford: Blackwell.

Bibliographical remarks

Lewis, D., 1986. Sets the terms of debate in three pages (202-204).

Merricks, T., 1994, “Endurance and Indiscernibility”, Journal of Philosophy 91: 165-84.

Sider, T., 2001. Provides definitions, examines arguments, and advocates temporal parts.

Sider, T., Dimensionalism, University Press.

Merricks, T., 1994. Against temporal parts.

Van Inwagen, P., 1990, Material Beings, Ithaca, NY: Cornell University Press.

Hawley, K., 2004. An extended overview, which includes a lengthy bibliography.

Heller, M., 1984, “Temporal Parts of Four-Dimensional Objects”, Philosophical Studies 46, 3: 323-334.

McKinnon, N., 2002, “The Endurance/Perdurance Distinction”, Australasian Journal of Philosophy 80: 288-306.

2001, Oxford:

FourOxford

Katherine Hawley

References and further readings

Theoretical Mereology

Baker, L. R., 2000, Persons and Bodies: A Constitution View, Cambridge: Cambridge University Press.

Mereology is widely defined as the theory of parthood, but as I shall explain it is better described as the theory of material parthood, where I use the adjective ‘material’ to mean concerned with the stuff of which things are made. Mereology was introduced by Leśniewski (1916) following his equally rigorous theories of protothetic and ontology (Simons 1987: 60–65) and, in a somewhat different and more accessible form as the Calculus of Individuals by Leonard and Goodman (1940), based on Leonard’s 1930 thesis ‘Singular Terms’.

Crisp, T. M.; Smith, D. P., 2005, “‘Wholly Present’ Defined”, Philosophy and Phenomenological Research LXXI. 2: 318-344. Hawley, K., 2001, How Things Persist, Oxford: Oxford University Press. Hawley, K., 2004, “Temporal Parts”, in Zalta, E. N. (ed.) The Stanford Encyclopedia of Philosophy (Winter 2004 Edition),

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These systems are different ways of developing what I call classical mereology, which is mathematically equivalent to a complete Boolean algebra with the minimum element deleted – deleted because there is no null thing. There are many expositions of classical mereology but not as much has been written on alternatives. (See, however, Simons1987: 81-92)). So in this article I shall briefly expound the classical theory, and then consider what sort of case can be made for or against classical mereology. A summary of classical mereology.

The way classical mereology is formulated will depend on the choice of the underlying language. Following Lewis (1991) I shall take an informal approach – but one capable of full rigour – which allows for plural reference. We can replace plural reference by reference to set, or by predicates provided we ensure enough of them. We have a choice of the basic mereological relation but I take it to be parthood, written x ≤ y. I follow the standard convention that anything is part of itself, and so define proper parthood, x < y as x ≤ y but x ≠ y. We define overlapping by saying that x and y overlap if they have a common part, and we define disjointness by saying that x and y are disjoint if they have no common part. Although not an axiom, it is assumed that there is no null thing, because if there were it would be part of everything and so overlapping would be trivial. An important definition in mereology is the fusion of some things, defined thus: w is a fusion of the Zs if, for all

x, x overlaps w iff x overlaps some Z. In classical mereology the fusion of the Zs is thought of as the technical explication of the intuitive idea of the sum of the Zs, that is, all the Zs put together. At least that is my terminology, but often the word ‘sum’ is used in a technical sense to mean what I mean by fusion. Furthermore Simons talks of ‘a fusion’ of the Fs to mean the fusion of all or some of the Fs (1987: 65). So beware! Although there are several different ways of formulating classical mereology the most usual is to require the transitivity of proper parthood and the principle that any Zs have a unique fusion. Another equivalent characterisation is that classical mereology is the theory of complete Boolean algebras with the minimal element (ø) deleted. We may, like Lewis (1991), allow quite unrestricted fusions of any things whatever, in which case there will be a fusion of everything there is. Or, more modestly, we might restrict classical mereology to some domain, such as all material objects, or all regions of Spacetime, in which case there will be a fusion of all things n that domain, the sum of all matter, or Spacetime itself. Here are a few of the interesting results in classical mereology: If w is the fusion of the Zs then every Z is part of w. If x < y then there is some z disjoint from x such that y = x + z. Unless y ≤ x there is some z ≤ y, such that z is disjoint from x.

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If every proper part of x is a proper part of y and vice versa then either x and y have no proper parts or x = y. Things without any proper parts are mereological simples; things that are not the sum of proper parts are atoms. Things that are the sum of proper parts but not the sum of disjoint proper parts may be called hemiatoms. In classical mereology there are no hemi-atoms and the atoms coincide with the simples. In classical mereology anything is either the sum of atoms, or has no atoms, or is uniquely the sum of two disjoint parts, one of which has no atomic parts and the other of which is the sum of atoms. Following Lewis something with no atomic parts is called atomless gunk, or gunk for short. Material parthood. Consider the following modification of the familiar clay/statue example. A stage magician throws seven coloured handkerchiefs into the air, claps her hands together and they turn into a multicoloured elephant made of seven pieces of coloured plasticine. Let us suppose that relying on some cunning chemistry the handkerchiefs really do turn into lumps of plasticine as they are banged together, making the elephant. Now there are, many philosophers would say, two objects: the sum of the seven pieces of plasticine and the plasticine elephant. They differ in that the elephant is necessarily elephant-shaped while the sum of the seven lumps of plasticine is not even necessarily connected – the trick could have failed. Yet the seven pieces of plasticine are surely parts of

the elephant. So the sum must be a part of the elephant too, and yet it has no complementary part, that is, nothing disjoint that is added on to it to make the whole elephant. Or at least if there is a complement it is not a further substance in the sense of something capable of existing by itself. This is similar to Brentano’s claim that the man was part of the sitting man with no complementary part (Brentano 1933, cited in Simons 1987: 26n). Now I shall query whether proper parts always have complements, but mereologists are not impressed by examples such as the elephant and the sum of the lumps of plasticine. Because we should be neutral on such questions as whether there are de re modalities, such as being necessarily elephantshaped, we should conclude that these considerations are irrelevant to mereology. For that reason I describe mereology as the theory of material parthood. I take it to be concerned with ‘matter’ in the sense of stuff. So in this context, ectoplasm counts as matter as do Space, Time and Spacetime. I say that two things are materially identical if they contain the very same matter, and when doing mereology we may speak as if materially identical things are strictly identical, and so speak of the elephant being the sum of the seven pieces of coloured plasticine. If we desire greater rigour then we should say that the elephant is a sum of the seven pieces and that the lump of multi-coloured plasticine is another sum and that they are materially identical. In formal mereology it is assumed that sums are unique, ex-

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hibiting the common discrepancy between formality and genuine rigour. But we should interpret uniqueness to mean uniqueness up to material identity. Mereology understood this way may be called extensional mereology, but extensionality is understood by Simons as requiring the Proper Parts Principle, namely that two things can have the very same proper parts only in the vacuous case in which they have no proper parts whatever. (Simons 1987:28) In defence of transitivity. Some phi-

losophers have queried the transitivity of the parthood relation. For instance it might seem plausible to think of the members of a set as parts but in that case a set such as {{b}, c} provides a counterexample because although b might be thought of as a part of {b} and {c} as a part of {{b}, c}, it seems less intuitive that b is a part of {{b}, c}. Another example, due to Rescher (1955) is that a nucleus is part of a cell, a cell part of an organ, but , he suggests, a nucleus is not part of the organ. There are several ways of defending transitivity. The first is to note that counter-examples are more persuasive if we use the phrase ‘a part’, as in ‘a nucleus is not a part of an organ’. But the phrase ‘a part’ often means ‘a salient part’ and salient parthood is not transitive. The second, similar defence, is to note that when we talk of ‘a part’ or of ‘the parts’ of something we usually have in mind just one decomposition into disjoint parts. Therefore such examples do not cast doubt on the transitivity of

parthood itself. The third reply is that even if proper parthood fails to be transitive it is acyclic and so we may consider the ancestral of parthood in place of parthood, which will be transitive. Here I note that putative examples of parthood cycles are highly controversial, One, noted by Sanford (1993: 222) and due to Borges is a story in which the Earth is seen in the Aleph and the Aleph in the Earth. The natural interpretation of this is not that of a cycle in which a < e < a, but rather a case in which x < y < z and y and z are qualitatively identical, both being alephs. So rather than being a counter-example to the transitivity of parthood this is an exotic counterexample to the Identity of Indiscernibles. My preferred defence is none of the above but rather to insist that mereology is a theory of parts and wholes for matter, meaning stuff of some kind. There are no putative counterexamples in which transitivity fails for portions of stuff considered as such. A qualified defence of summation. I have already indicated that sums might not really be unique but rather unique up to material identity. And if we are thinking of portions of stuff there is some intuitive support for restricting summation to portions of the same kind of stuff, and so not, for instance, considering the sum of a material object and the region it occupies. I defend, however, the principle that any things made of the same stuff have a unique sum up to material identity. For the question of just how one portion of stuff is separated

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from another so as to be an individual is one that concerns the extra structure something has, in addition to its being a portion of stuff, and so is not relevant to mereology. Consider again the example of the plasticine elephant. Its stuff happens to be arranged in elephantine form but it could have been in any other arrangement of the same quantities of the different colours of plasticine. Since it would be the same portion of stuff if arranged otherwise, the arrangement of the parts is not a necessary condition for there being a sum. Against this it might be suggested that the Zs fail to have a sum if every Z is either an X or a Y and no X either touches or overlaps any Y. This connectedness constraint seems to have been assumed by Whitehead when considering regions, and it has some initial plausibility (See Simons 1987:85) There are, however many geographic examples of scattered objects, such as the State of Michigan. To these I would like to add the consideration that a person might have a scattered body, as in Dennett’s example in which the brain is stored safely away while the rest of the body goes on a hazardous expedition (Dennett 1979). If there are no scattered objects either there are no persons or they are not even materially identical to their bodies. Neither conclusion is welcome so I conclude there are scattered objects. Yet again we could consider a two particle quantum system in which the spin of one particle is the opposite of that of the other, so that whatever happens to the one happens, in reverse, to the other. It would be curmudgeonly to

deny that these two particles form a single thing and so we should reject the connectedness requirement. Assuming, then, that things made of the same kind of stuff have a sum the uniqueness up to material identity follow from the concept of a sum as all that stuff put together. Sums and fusions. I have provided the

standard technical definition of a fusion, and a defence of the principle that some things (made of the same stuff) have a unique sum (up to material identity). I also accept transitivity of proper parthood. The quick path to classical mereology is then to explicate sum as fusion. But this is too swift. If classical mereology is correct and any things have a unique fusion then indeed the sum is the fusion, but until we have established classical mereology we have no reason to reject another explication of summation as the least upper bound or join. Here an upper bound of the Zs is some w that has every Z as a part. And the least upper bound of the Zs is an upper bound that is part of every upper bound of the Zs. It follows from the transitivity of proper parthood that if the Zs have an upper bound they have a unique least upper bound. Bearing in mind that there might be infinitely many Zs it is not self evident that the sum is the fusion rather than the least upper bound. Likewise it is not self-evident why, prior to the explication of summation as fusion, we would assume fusions are unique up to material equivalence. An example that illustrates the difference between least upper bounds and

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fusions is the hypothesis that the only parts of Time are intervals. (Cf Simons 1987: 32) Consider two intervals I and J, where I is from 0 to 1 second and J from 4 to 5 seconds. Their least upper bound is the interval from 0 to 5 seconds, but this is not a fusion because it overlaps the interval K from 2 to 3 seconds, yet K does not overlap either I or J. In this case there is no fusion of I and J and, I submit, no sum of them either. For the sum of I and J should not contain ‘stuff’ that is neither in I nor in J, hence it should not contain K. Generalising, we should require that a sum be a fusion. That does not, however, vindicate classical mereology because although the sum must be a fusion and although the sum is unique it does not follow that the fusion is unique. This motivates a theory weaker than classical mereology one in which proper parthood is transitive, any things (made of the same stuff) have a least upper bound, and in which the least upper bound is always a fusion. Equivalent to this last requirement is the principle that if x is disjoint from every Z then x is also disjoint from the least upper bound of the Zs. In that case we may explicate the sum of the Zs as their least upper bound. Because any Zs have a least upper bound any Ws with a common part have a greatest common part or meet, namely the least upper bound of all the common parts of the Ws. To turn the resulting system into classical mereology it is enough to suppose the principle that if unless y is part of x there is some z overlap-

ping y but not x. For in that case if there are two distinct fusions of the Zs one of then is not part of the other and we obtain a contradiction. But is that principle plausible? We can find countably many ball-shaped regions (balls for short) in Space whose total volume is only 1 cc, and which are dense in the sense that any ball however small overlaps at least one of them. Call these countably many balls regions the Ds. If we further suppose that every region contains some ball it follows that the whole of Space is a fusion of the Ds. Yet it is intuitive that the sum of the Ds has volume at most 1 cc. Hence the sum is not a unique fusion. Moreover if we take y to be the whole of space and x to be the sum of the Ds we can understand why even though x is a proper part of y there is no z overlapping y but not x. For such a region z would contain a ball and so could not avoid overlapping one of the Ds and hence their sum. (Forrest 2004) I conclude that the quick route to classical mereology fails. I now exhibit three other routes. Expanding the mereology. One way to

obtain classical mereology is to expand a non-classical mereology by adjoining enough extra parts. If we assume that proper parthood is transitive then parthood is a partial ordering and any partial ordering of some things may be embedded in a complete Boolean algebra generated by those things. One of the adjoined things is going to be the null thing ø, which we treat as a fiction. Ignoring that we obtain the classical extension of a non-classical mereology. We

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might treat the additional items as convenient fictions, the way the null thing is treated. Or we might take them to supervene on the things in the initial non-classical mereology. For instance in the above example of the countably many balls, the Ds, the complement of the sum of the Ds would be either a fictional or a supervenient entity and the intuition that every region be the sum of balls is treated as an intuition about the real or the subvenient regions respectively. To take another example if initially we thought that regions corresponded to topologically open sets of coordinate triples then their surfaces would either be fictions or supervenient. Contracting the mereology. Another way to obtain classical mereology is to restrict attention to the mereologically well-behaved things. Given a mereological system in which the least upper bound is always a fusion we may consider the fusions of a given thing u (or if you prefer the fusions of u with itself). Clearly u is one of the fusions of u, but it is easy to see that the any fusion of the fusions is a fusion and so the least upper bound of all the fusions is itself a fusion. It is the greatest fusion of u, call it u*. Then u ≤ u* and u** = u*, and we may consider the regular things, namely those u such that u = u*. Now in general there are many non-regular fusions of regular things but there is always just one regular fusion, which is the least of their regular upper bounds. So if we restrict attention to the regular things we obtain a classical mereology. If for instance we supposed that the regions

of Space were represented by all and only the non-empty topologically open sets of coordinate triples then the greatest fusion operator is represented by interior-of-closure and so the regular regions are represented by the non-empty regular open sets and are therefore precisely those regions that Tarski hypothesised (1956). If we now consider the dense balls D, each one is regular but their least upper bound is not. Their regular fusion is the whole of Space The a posteriori justification. Apart

from the strategies of either expanding or contracting the mereology there remains one way in which classical mereology might well be justified. The best scientific theory of Spacetime and its occupants might in fact imply classical mereology. For instance Spacetime might turn out to be discrete in the sense that every ball of finite radius is the sum of finitely many mereological simples. That implies classical mereology for the actual worlds. A further metaphysical argument might conclude that if Space is discrete it is necessarily discrete, in which case mereology is necessarily classical. Conclusion. Because the explication

of summation as fusion is not selfevident any resort to classical mereology must be justified, either by adjoining extra things, or by concentrating only on regular things or a posteriori. I diagnose the widespread unargued belief that classical mereology is correct as a combination of (1) respect for authorities on technical matters such as chemistry and (2) the belief

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that the mereology is a technical matter, like chemistry; supported by a belief in the uniqueness of fusion derived from (3) the intuitive plausibility of the uniqueness of summation (up to material identity) with (4) the case for saying that the sum must be a fusion. See also > Boolean Algebra, Comments of “The Calculus of Individuals and Its Uses”, Fusion, Material Constitution, Sum, Transitivity. References and further readings

Brentano, F., The Theory of Categories, (trans. Roderick Chisholm and N Guterman), The Hague: Nijhoff, 1981. Burkhardt, H.; Dufour, C. A., 1991, “Part/Whole I: History”, in Burkhardt, H.; Smith, B. (eds.), Handbook of Metaphysics and Ontology, Munich: Philosophia, 663-673. Casati, R.; Varzi, A. C., 1999, Parts and Places: The Structures of Spatial Representation, Cambridge (MA): MIT Press. Clay, R. E., 1981, Leśniewski's Mereology, Cumana: Universidad de Oriente. Dennett, D., 1979, “Where am I”, in Brainstorms: Philosophical Essays in Mind and Psychology, Hassocks: Harvester Press, 310-323. Eberle, R. A., 1970, Nominalistic Systems, Dordrecht: Reidel.

Forrest, P., 2004, “Grit or Gunk: Implications of the Banach-Tarski Paradox”, The Monist 87: 351-370. Hudson, H., 2004, “Simples”, The Monist 87: 3030-351. Leonard H. S.; Goodman, N., 1940, “The Calculus of Individuals and its Uses”, Journal of Symbolic Logic 5: 45-55. Leśniewski, S., 1916, “Podstawy ogólnej teoryi mnogości I”, (Foundations of a General Theory of Manifolds), Prace Polskiego Kola Naukowe w Moskwie. Lewis, D. K., 1991, Parts of Classes, Oxford: Blackwell. Luschei, E. C., 1965, The Logical Systems of Leśniewski, Amsterdam: North-Holland. Rescher, N, 1955, “Axioms for the Part Relation”, Philosophical Studies, 6: 8-11. Sanford, D., 1993, “The Problem of the Many, Many Composition Questions, and Naive Mereology”, Noûs 27: 219-228. Simons, P. M., 1987, Parts. A Study in Ontology, Oxford: Clarendon. Simons, P. M., 1991, “Part/Whole II: Mereology Since 1900”, in Burkhardt, H.; Smith, B. (eds.), Handbook of Metaphysics and Ontology, Munich: Philosophia: 209-210. Smith, B., 1982, “Annotated Bibliography of Writings on Part-Whole Relations since Brentano”, in B. Smith (ed.), Parts and Moments. Studies in Logic and Formal Ontology, Munich: Philosophia: 481-552.

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Smith, B., 1985, “Addenda to: Annotated Bibliography of Writings on Part-Whole Relations since Brentano”, in Sällström, P. (ed.), An Inventory of Present Thinking about Parts and Wholes, vol. 3, Stockholm: Forskningsrådsnämnden: 74-86. Tarski, A., 1956, “Foundation of the Geometry of Solids” in Logic, Semantics, Metamathematics, Tr. J. H. Woodger, Oxford: Clarendon Press: 24-9. Varzi, A. “Mereology”, Stanford Encyclopedia of Philosophy. Peter Forrest

Thomas Aquinas Mereological considerations play an important role in the metaphysics of Thomas Aquinas (1225-1274). Aquinas distinguished various sorts of wholes and described their properties. Homogeneous and heterogeneous wholes. According to Aquinas some

wholes consist of homogeneous parts and others consist of heterogeneous parts. A whole consists of homogeneous parts if each of its parts is qualitatively identical to the whole. A heterogeneous whole consists of parts that are different in species from the whole and from each other. Aquinas considered the nerves, bone marrow, bones (Commentary on Aristotle’s Metaphysics (i.e. In XII libros Metaphysicorum Aristotelis) I, lectio 4, n. 22), wood and the elements (ibid, V, lectio 4, n. 6), to be examples of homogeneous wholes. A

house would be an example of a heterogeneous whole. Universal and integral wholes. Aquinas distinguished, moreover, universal (or distributive) wholes from integral wholes. Universal wholes can be predicated of their parts. For example, every part of ‘animal’ (e.g. God, Socrates, horse) is itself an animal (cf. Commentary on Aristotle’s Metaphysics V, lectio 21, n. 16). An integral whole cannot be predicated of its parts when they are taken singularly. No single part of a house is a house (cf. Scriptum super libros Sententiarum I, dist. 3, q. 4, art. 2, resp. ad arg. 1). Aquinas held that the loss of certain parts of an integral whole does not entail the destruction of the whole (cf. Summa theologiae III, q. XVII, art. 2). This is a denial of mereological essentialism at least insofar as integral wholes are concerned. Aquinas also distinguished those integral wholes which remain the same regardless of the order of their parts, like a heap of stones, from those which are praeter elementa, i.e. something else than simply an aggregation of their parts or elements. An example of a whole that is something else than its elements would be flesh. The aggregate of the elements that make up flesh is not flesh. What makes flesh something other than its elements, is not an extra part that is added to them; rather, it is the substantial form of the animal, which transforms the elements into a new item (cf. Commentary on Aristotle’s Metaphysics VII, lectio 17, nn. 2733; compare Aristotle’s Metaphysics 1041b).

THOMAS AQUINAS 563 Potential wholes. Additionally, Aqui-

nas believes that there are potential wholes, which are intermediate wholes between the universal and the integral ones (cf. Scriptum super libros Sententiarum I, dist. 3, q. 4, art. 2, resp. ad arg. 1). Two senses of potential wholes are to be distinguished. The first pertains to wholes whose parts are potential in the sense that, in act, they are physically and functionally united (imagine, for example, parts of a compact stone). The second pertains to wholes consisting of powers (potentiae). Potential wholes in the first sense are continuous (Commentary on Aristotle’s Metaphysics V, lectio 21, n. 18) and contain imagined, arbitrary parts. Moreover, they are homogeneous and resemble universal wholes insofar as they are predicated of their parts (cf. Summa theologiae, I, q. 77, a. 1, ad 1). In the very same place, Aquinas’s example is found of what would be a potential whole in the second sense. The soul is a potential whole consiststing of exactly three capacities: memory, intelligence and the will, that are functionally distinct from each other. Each of these parts does correspond to the soul’s essence but it does not fulfill, taken alone, all the functions (virtus) of the soul. The powers of the soul resemble integral parts insofar as they have different functions from each other. Consequently, the soul is not predicated of its parts as strictly (proprie) as a universal whole would be predicated of its own parts. Transitivity. Transitivity of parthood

(i.e. the principle according to which every part of a whole is eo ipso a part

of every other whole of which the first whole is a part) holds according to Aquinas only as pertains to potential parthood. In his Commentary on Aristotle’s Metaphysics V, lectio 15, n. 1 Aquinas uses the following example: flesh is an integral part of a human body and the primary elements are integral parts of flesh. But the primary elements are not integral parts of a human body. That is, integral parthood is not transitive. However the primary elements are potential parts of a human body. Clearly, Aquinas thought that a human body can have any potential parts which contain, among others, flesh and that a primary element can be a potential part of such parts as well as of the body itself. That is, potential parthood is transitive. Irreflexivity. Generalising from what

he found in Aristotle’s examples of wholes and their parts (cf. Metaphysics 1023 a 14-7), Aquinas held that the whole is always greater than any part of it (Summa contra Gentiles I, c. 10, n. 2 and c. 18, n. 6; Commentary on Aristotle’s Metaphysics IV, lectio 6, n. 10). This implies that the whole-part-relation is irreflexive. In other words, Aquinas thought that wholes can have only proper parts. For example, I am not just my soul (Commentary on St. Paul’s First Letter to the Corinthians, c. 15, 17-9). Overlapping. Aquinas occasionally

spoke in favour of a non-overlapping mereology (cf. Scriptum super libros Sententiarum I, d. 8, q. 2, a. 1, arg. 4; Summa Theologiae III, q. 90 a. 3 arg. 2). In his commentary on Boethius’s On Trinity, q. 4, a. 3, Aquinas stated

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that nature provides no examples of bodies that exist in the same location in act and he pointed out several conceptual difficulties in accepting this idea, but he conceded that two potential bodies could overlap. This view is in agreement with Aquinas’s discussion of natural forms (e.g. alternative possible bodies, which would displace parts of, say, air to occupy space) in De virtutibus, q. 1, a. 8, co. There, Aquinas mentions Anaxagoras as holding natural forms to be present in matter in act and to become manifest (reducuntur ab occulto in manifestum) due to external agency. Moreover, he mentions Avicenna as claiming natural forms not to be in matter in any sense and to be generated by external agency. After rejecting the aforementioned views of Anaxagoras and Avicenna, Aquinas assents to Aristotle’s position, according to which many natural forms are at the same time preexistent in matter potentially but not in act. On Aristotle’s and Aquinas’s account then, two alternative statues which potentially preexist on the same pedestal (let people be undecided about which one to place atop the pedestal and let there be not enough room for both), have some (potential) parts in common. Aquinas even allowed that two actual bodies could overlap in the case of some miracles. This would allow, to use one of Aquinas’s examples in his commentary on Boethius’s On Trinity, q. 4, a. 3, for the passing of the resurrected Jesus through walls.

See also > Aristotle’s Theory of Parts, Aristotle’s Theory of Wholes, Coincidence, Homeomerous and Automerous, Medieval Mereology, Medieval Discussions on Temporal Parts, Mereological Essentialism, Privation, Totum potentiale, Transitivity. Bibliographical remarks

Brown, C., 2005. Discusses Aquinas’ understanding of the different senses of “whole” and “part”, and applies a Thomistic mereology to traditional problems pertaining to material constitution. Henry, D. P., 1991. For an account of Aquinas’ mereology considered from a Leśniewskian perspective, see chapter 3, pp. 218-328. The chapter includes an extensive selection of key Latin texts and their English translations. Klima, G., 2002. Contains an analysis of Aquinas’ mereology of the human nature. Thomas Aquinas, In XII libros Metaphysicorum Aristotelis. Book V, lectiones 20-1 contain a thorough analysis and systematisation of Aristotle’s views. English translation: Aquinas, Commentary on Aristotle’s Metaphysics, transl. by J. P. Rowan, Notre Dame, IN: Dumb Ox Books, 1995. See References for more English translations of works less easily accessible. Thomas Aquinas, Scriptum super libros Sententiarum I, dist. 3, qq. 4-5 contain a short exposition of Thom

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as’ main distinctions concerning the whole-part-relation. References and further readings

Brown, C., 2005, Aquinas and the Ship of Theseus. Solving Puzzles about Material Objects, London/New York: Continuum.

gy, transl. by A. Maurer, Toronto: Pontifical Institute of Mediaeval Studies, 1986. Thomas Aquinas, In XII libros Metaphysicorum Aristotelis. English translation: Aquinas, Commentary on Aristotle’s Metaphysics, transl. by J. P. Rowan, 1995, Notre Dame, in: Dumb Ox Books.

Cross, R., 2002, The Metaphysics of the Incarnation, Oxford: Oxford University Press. Henry, D. P., 1991, Medieval Mereology, Amsterdam/Philadelphia: B. R. Grüner. Klima, G., 2000, “Aquinas on One and Many”, Documenti e Studi sulla Tradizione Filosofica Medievale 11: 195-215. Klima, G., 2002, “Man=Body+Soul: Aquinas’s Arithmetic of Human Nature”, in Davies, B. (ed.), Thomas Aquinas: Contemporary Philosophical Perspectives, Oxford: Oxford University Press, 257-73. Thomas Aquinas, Opera omnia, editio Leonina, Rome: Polyglot Press, 1882 ff. Thomas Aquinas, Commentary on Aristotle’s Metaphysics, Notre Dame, IN: Dumb Ox Books, 1995. Thomas Aquinas, Commentary on St. Paul’s First Letter to the Corinthians, c. 15, 17-9, in Aquinas, Selected Philosophical Writings, ed. by T. McDermott, Oxford: Oxford University Press, 1993. Thomas Aquinas, Commentary on Boethius’ On Trinity, qq. I-IV, in Aquinas, Faith, Reason and Theolo-

Stamatios Gerogiorgakis

Topology Topology is a branch of mathematics that investigates formal definitions of notions such as space, continuity, convergence, connectedness, neighbourhood, boundary, closure etc. In philosophical terms, it may be said to provide logical analyses of spatial concepts, where ‘spatial’ is understood in a broad sense. Historically, topology may be considered as a generalisation of metrical geometry. In its modern sense it originated around 1900 and is grounded in the works of Georg Cantor, Felix Hausdorff, Maurice Fréchet, Henri Poincaré and others. Although it is useful to compare the conceptual content of topological notions with everyday geometric intuitions and experiences, topology studies spaces and spatial mappings that are quite different from the Euclidean space and the mappings we are acquainted with. Formally, a topological space is a set X together with a set OX of subsets of X that contains the empty set Ø and X, and is closed under finite intersections

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and arbitrary unions. The elements of OX are called ‘open sets’. A topological space is denoted by (X, OX) or simply by X, if the topology OX is understood. Open sets are defined relative to the system OX, i.e., U ⊆ X is open (relative to OX) iff U ∈ OX. There are no intrinsic features that characterise a subset of X as open per se, except, of course that Ø and X are always open. For the familiar Euclidean topology of the real line R typical open sets are the open intervals (a, b) := {x; a < x < b}. If X is not a singleton {x}, it may carry many different topological structures OX. For every X two extreme topologies may be defined: the discrete topology for which every subset of X is open, and the indiscrete topology, whose only open sets are Ø and X. An open set U containing p ∈ X is called an open neighborhood of p. The settheoretical complement (with respect to X) of an open set is called a closed set. A set that is open and closed is called clopen. The smallest closed set containing a set A is called the closure of A and denoted by cl(A); dually, the largest open set contained in a subset B is called the open kernel of B and denoted by int(B). The operators cl and int are interdefinable. A set C is called regular open iff C = int(cl(C)). The class of regular open sets of X is denoted by O*(X). The topological boundary bd(D) of D is defined as the set cl(D) - int(D). A map f between topological spaces X and Y is continuous iff for every open subset A of Y the set f-1(A) = {x; f(x) ∈A} is open in X. The concatenation of continuous maps is again continuous. Hence topological

spaces and continuous maps form a mathematical category (cf. Mac Lane 1986, Lawvere and Schanuel 1996). General topology may be defined as the study of topological spaces and continuous maps. Whether a settheoretical map f is continuous or not, depends on the topologies defined on X and Y: every map f is continuous if Y is endowed with the indiscrete topology, or X is endowed with the discrete topology. In general, it is difficult to prove or to disprove the existence of (non-trivial) continuous maps f. A famous case in question was Brouwer’s fixed point theorem according to which there is no continuous map from the closed disc D onto its boundary bd(D) mapping the boundary onto itself. Even for ‘elementary’ spaces such as spheres or projective spaces the totality of continuous maps is not fully understood today. Important progress in this area has been made by using the machinery of algebraic topology. The notion of a topological space is extremely general and not much can be said about all topological spaces. An important task of topology has been to single out interesting classes of topological spaces. Elementary examples are Euclidean spaces, and, a bit more general, metrical spaces (X, d), i.e., sets X endowed with a distance function d onto R, the real numbers: X x X ---
R, where f satisfies some familiar axioms and in particular the triangle inequality. A base of a topology OX is a subset of OX such that all elements of OX are generated as unions of elements of the base. The canonical metrical topology of a metrical space (X, d) has

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the base of open sets {y; d(x, y) < d, x ∈ X and d ∈ R+, R+ positive real numbers}. There are many different classes of topological spaces (cf. Steen and Seebach Jr. 1978); to mention some of the most familiar ones: (i) A topological space X is Hausdorff iff for any two distinct points p, q ∈ X there are disjoint open neighbourhoods U of p and V of q; (ii) X is compact iff when ∪Ui = X and Ui open, already finitely many of the Ui suffice to cover X; (iii) X is second countable iff it has a countable open base; (iv) X is connected if it is not the disjoint union of two non-empty open (or closed) subsets. It is a theorem that a second countable compact Hausdorff space (X, OX) is metrizable, i.e., there is a metric d on X whose canonical topological structure is OX. Euclidean spaces are Hausdorff and connected but not compact, spheres and projective spaces are connected, Hausdorff and compact. Mereology can be related to topology as follows: Generalising Lewis’s mereological interpretation of the subset relation (cf. Lewis 1991), we define the parts of a topological space X to be the elements of OX, i.e., for A, B ∈ OX, A is said to be a part of B iff A ⊆ B. The resulting mereological structure (OX, ⊆) is a complete Heyting algebra. Another interesting mereological structure arising for a topological space X is the complete Boolean algebra (O*X, ⊆). As is known by Stone’s representation theorem any complete Boolean algebra arises in this way, i.e., as the Boolean algebra O*X for some topo-

logical space X (cf. Davey and Priestley 1990). Thus, classical Boolean mereology can be considered as a part of topology. Since the 1970s it has become clear that topology need not be treated as a special branch of set theory, considering topological spaces as structured sets of points. Imposing certain weak conditions on the structure of OX a topological space (X, OX) is uniquely defined by OX, i.e., up to isomorphism, the point set X can be reconstructed from the mereological structure (OX, ⊆) (cf. Johnstone 1982, Mac Lane and Moerdijk 1992, Vickers 1989). Thus, if one is prepared to accept Heyting algebras as mereological systems, a large part of topology can be considered as mereology in disguise. Topological considerations have found applications in virtually all branches of mathematics. In this way, new disciplines such as algebraic topology, differential topology, low dimensional topology, combinatorial topology and others arose. More recently, topology has been applied also in physics and theoretical computer science (cf. Gierz et al. 2003, Vickers 1989). See also > Axiomatic Method, Carnap, Metamathematics, Mereotopology, Point, Russell, Tarski, Whitehead. Bibliographical remarks

Smith, B., 1996. Ten papers on various aspects of topology especially

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designed for the needs of philosophers. No mathematical prerequisites required.

Johnstone, P. F. 1982, Stone Spaces, Cambridge: Cambridge University Press.

Engelking, R., 1989. Classical textbook on set-theoretical topology.

Kunen, K.; Vaughan, J. E. (eds.), 1984, Handbook of Set-Theoretical Topology, Amsterdam: North-Holland.

Vickers, S., 1989. Non-Classical Topology for Theoretical Computer Science. Steen, L. A.; Seebach, J. A., Jr., 1978. Useful Collection of Examples and Counterexamples for Checking One’s Working Understanding of Topological Concepts. References and further readings

Lewis, D., 1991, Parts of Classes, Oxford: Blackwell. MacLane, S., 1986, Mathematics, Form and Function, Heidelberg and New York: Springer. Mac Lane, S.; Moerdijk, I., 1992, Sheaves in Geometry and Logic. A First Introduction to Topos Theory, New York: Springer.

Davey, B.; Priestley, H., 1990, Introduction to Lattices and Order, Cambridge: Cambridge University Press.

Nagata, J., 1985, Modern General Topology, 2nd edition, Amsterdam: North-Holland

Dugundji, J., 1966, Topology, Boston: Allyn and Bacon.

Roeper, P., 1996, “Region-Based Topology”, The Journal of Philosophical Logic 26: 251-309.

Engelking, R., 1989, General Topology, 2nd edition, Berlin: Heldermann. Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., 2003, Continuous Lattices and Domains, Cambridge: Cambridge University Press. Grosholz, E., 1980, “Two Episodes in the Unification of Logic and Topology”, The British Journal for the Philosophy of Science 36: 147-157. Hart, K. P.; Nagata, J.; Vaughan, J. E. (eds.), 2004, Encyclopedia of General Topology, Amsterdam: Elsevier. James, I., (ed.), 1999, History of Topology, Amsterdam: North-Holland.

Smith, B., 1996, “Topology for Philosophers”, The Monist 79 (1). Steen, L.A.; Seebach, J.A., Jr., 1978, Counterexamples in Topology, New York: Springer. Vickers, S., 1989, Topology via Logic, Cambridge: Cambridge University Press. Thomas Mormann

Totum Potentiale Adhering to the Aristotelian sense of ‘whole’ as something whose components form a unit, Boethius focuses in his De Divisione on the notion of potential whole, to later differentiate it

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not only from the integral whole but also from the universal whole. Based on the Boethian approach regarding this topic, most medieval thinkers, from the 12th century onwards, start to call potential whole any particular item divisible into powers or compounded by them. Although in a certain way they are to be taken as parts, these powers do not add up to create a whole and they cannot be separated from it. Aware of this fact, some thinkers such as Thomas Aquinas suggest the use of a certain notion of potential part which identifies ‘x is a part of the potential whole y’ with ‘the essence of y is totally present in x, but x involves only a fraction of the potential whole of y’. It is further assumed that this understanding of potential part suffice to ground the relationship. As potential parts seem to behave at times as parts of an integral whole and at other times as parts of a universal whole, Aquinas places the potential whole between the universal and the integral whole. Unlike universal wholes, potential wholes are essentially present in each of their parts, although not in conformity with their full power. Again, unlike integral wholes, potential wholes are essentially present in their parts. Subsequently, the human soul – a repeatedly cited example of potential whole – is usually conceived as a particular that cannot be exhaustively characterised by any of its powers but would be incomplete if it lacked any of them. See also > Aristotle's Theory of Parts, Aristotle's Theory of Wholes,

Boethius, Medieval Mereology, Power, Subject, Thomas Acquinas, Totum potentiale. Bibliographical remarks

Arlig, A., 2005. Detailed study of relevant doctrines that contributed to the development of mereology in the early Middle Ages. Boethius, 1998. Considerably influential in the Middle Ages, this work contains a significant part of the mereological ideas of Boethius. Henry, D. P., 1991. A pioneering work about medieval mereology. References and further readings

Albertus Magnus, 1913, Commentarii in librum Boethii De divisione, Bonn: P. Hanstein. Antonius Andreas, 1480, Scriptum in artem veterem Aristotelis et In divisione Boethii, Venice: Octavianus Scotus. Arlig, A., 2005, A Study in Early Medieval Mereology: Boethius, Abelard, and Pseudo-Joscelin, Columbus: The Ohio State University, Ph.D. dissertation. Arlig, A., 2011, “Is There a Medieval Mereology?”, in M. Cameron, J. Marenbon, ed. Methods and Methodologies: Aristotelian Logic East and West, 500–1500. Leiden: Brill, 16189. Arlig, A., 2011b, “Mereology”, in H. Lagerlund, ed. Springer Encyclopedia of Medieval Philosophy. Dordrecht: Springer, 763–71.

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Boethius, 1998, De divisione, Leiden: E. J. Brill. Burkhardt, H.; Dufour, C. A., 1991, “Part/Whole I: History”, in Burkhardt, H.; Smith, B. (eds.), Handbook of Metaphysics and Ontology. Munich: Philosophia Verlag, Volume 2, L-Z, 663-73. Henry, D. P., 1972, Medieval Logic and Metaphysics, London: Hutchinson & Co. Henry, D. P., 1984, That Most Subtle Question (quaestio subtilissima): The Metaphysical Bearing of Medieval and Contemporary Linguistic Disciplines, Manchester: Manchester University Press. Henry, D. P., 1991, Medieval Mereology. Amsterdam: B. R. Grüner. Lofy, C. A., 1959, “The Meaning of ‘Potential Whole’ in St. Thomas Aquinas”, The Modern Schoolman, 37: 39-48. Peter Abelard, 1954, “De divisionibus incipit”, in M. Dal Pra, ed. Scritti filosofici. Milan: Fratelli Bocco. Thomas Aquinas, 1888-1906, Summa theologiae, in Thomas Aquinas, Opera omnia. Vatican: Vatican Polyglot, vols 4-12. Guilherme Wyllie

Transitivity Transitivity is a formal property of relations. A relation R is transitive just in case for arbitrary instantiations of variables x, y, z the following inference schema holds: if x is R-

related to y and y is R-related to z, then it can be inferred that x is Rrelated to z. There are two ways in which a relation can fail to be transitive – it may be non-transitive or intransitive. For example, the relation ‘x is shorter than y’ is a transitive relation, while the relation ‘x is a friend of y’ is a non-transitive relation – for some but not all instantiations of x, y, z the inference is valid, in contrast, the relation ‘x is the biological mother of y’ is an intransitive relation, since there are no instantiations of x, y, z for which the inference is valid. Whether our reasoning about part-whole relationships is transitive, non-transitive, or intransitive is a significant issue since “failure of transitivity as a general partwhole principle would appear to have important philosophical ramifications” (Varzi 2006: 141). For example, philosophical analyses of material constitution and emergence, of spatio-temporal coincidence and transtemporal identity directly depend on this question. Yet in the contemporary philosophical discussion of formalisations of part-whole relationships the axiom of transitivity has received much less critical attention than other and less fundamental issues such as the extensionality of parthood or the existence of arbitrary sums. This may be the indication that transitivity of parthood is indeed a ‘law of thought’ of common sense reasoning; alternatively, it might reflect the contingent restrictions of a research tradition. Generally speaking, the issue whether our reasoning about part-whole relationships in common sense or

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science exhibits transitivity would need to be explored along various trajectories, including the following three questions. (T1) In common sense and scientific reasoning we express parthood relationships with many different linguistic means, with partitives, possessives and relational predicates or verbs such as e.g., ‘is part of,’ ‘is a part of,’ ‘has,’ ‘involves,’ ‘consists of,’ ‘is component of’, ‘is piece of,’ ‘is ingredient of,’ ‘has as phase,’ ‘belongs with’, etc. – should these parthood relationships be treated as different relations or as modifications of one basic relation, and which ones of these senses of parthood are transitive? (T2) Is there a basic sense of ‘is part of’ that is transitive? (T3) To what extent is transitivity domain-dependent, i.e., is there a certain class of entities for which some or all part-whole relations are transitive, or fail to be so? The contributions referenced here have made important headway towards addressing aspects of these questions but a comprehensive study of the transitivity of parthood relationships that investigates (T1) through (T3) is still a desideratum. In the context of the metamathematical discussion during the early decades of the 20th century, when the first formalisations of reasoning about parts and wholes were developed by S. Leśniewski, A. N. Whitehead, H. Leonard, N. Goodman, and A. Tarski, (T1) and (T3) were set aside. (T2), on the other hand, was

decided by fiat – working with intuitive models of spatial inclusion, the early mereologists simply stipulated that the relation they formalised as the ‘(proper) part relation’ was to be transitive. Since the primary goal was to construct an alternative to settheory, there was no need to engage in a discursive justification of this decision vis-à-vis the complexities of common sense reasoning, or, for that matter, philosophical discussions of part-whole relationships where transitivity failures had long be acknowledged in structured or “integral” wholes (cf. e.g., Brentano’s classification of types of part, Twardowski’s notion of “first order and secondorder parts, or Husserl’s notion of ‘mediate part’ in Logische Untersuchungen § 24, or in). The transitivity axiom of early formal mereology was taken up in all later logical studies of ‘classical extensional mereology’ (cf. the survey in Simons 1987). Transitivity of parthood is even accepted by proponents of so-called “intensional” mereologies who reject the Proper Parts Principle (i.e., the extensional principle of identity that says that items are identical just in case they have exactly the same parts). In view of counterexamples such as ‘(i) At time t, the lump of clay C has precisely the same parts as the statue S; (ii) but C and S have difference modal properties; (iii) thus C and S cannot be not identical’, proponents of intensional mereologies question the identity principle but not premise (i) and the transitivity axiom that supports this premise (as illustrated e.g. in Simons 1987, ch. 6, esp. 249f).

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The conviction that irreflexivity, antisymmetry, and transitivity are “partly constitutive of the meaning of ‘part’, which means that anyone who seriously disagreed with them has failed to understand the word” (Simons 1987: 11), became widely accepted in analytical ontology where mereological terminology (part, sum, fusion) has been used as conceptual tool since the mid-20th century. But it can be called into question whether any of these allegedly conceptually constitutive formal properties of the part-relation, each of which have substantive ontological implications, should be decided dogmatically. While dissident voices concerning irreflexivity and antisymmetry (e.g., Kearns 2011 and Cotnoir 2010), have been raised more recently, scepticism about transitivity has been more longstanding. In particular, from the 1950s onwards questions (T1) and (T3) began to gain momentum and it was increasingly questioned, mainly from researchers in linguistic semantics, whether the axiomatised relation of classical extensional mereology was sufficiently close to the common concept of parthood (cf. Rescher 1955). Transitivity failures were noted in implicit expressions of parthood such as ‘has’—the house has a door, and the door has a handle, but this does not entail that the house has a handle (Lyons 1977)—but also for some cases where parthood is explicitly expressed in term of ‘is part of’: (1) The mouth is part of the face. The face is part of the body. (*) The

mouth is part of the body (Miller/Johnson-Laird 1976: 241). (2) A platoon is part of a company. A company is part of a battalion. (*) A platoon is part of a battalion (Rescher 1955: 10). (3) The nucleus is part of the cell. The cell is part of the heart. (*) The nucleus is part of the heart (ibid). The inferences in these and similar examples were either cautiously glossed as “a bit strained” (Miller/Johnson-Laird 1976: 241) or treated as clearly “invalid” (Cruse 1979: 30; Rescher 1955: 10). A common feature of examples (1) through (3) is that the noun phrases all have generic reference – they do not refer to a concrete determinate particular entity but (e.g.) to a type of entity. But putative examples for transitivity failures can also be formulated with nouns that refer to concrete particulars (to particular individuals (4), particular collections ((5), and particular actions (6), respectively): (4) Cathode C is part of battery B, which is part of Max’ pacemaker. Max’ pacemaker is part of Max’ heart. (*) Cathode C is part of Max’ body. (5) In 2010 family Andersen was part of the Aarhus Golf Club. In 2010 the Aarhus Golf Club was part of the Danish Association of Sports Clubs. (*) In 2010 family Andersen was part of the Danish Association of Sports Clubs. (6) The spark’s misfiring on Christmas Eve was part of the engine’s

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running during 2015. The engine’s running during 2015 was part of the operating of the factory’s production line, which is part of the country’s industrial production during 2015. (*) The spark’s misfiring was part of the country’s industrial production during 2015. The discussion about the early stock of putative counterexamples (see Winston et al 1987, Iris et al 1988) was refueled in the 1990 when ontologists began to engage in conceptual modeling in information systems engineering and encountered problems for the application of classical mereology in databases (cf. e.g Artale 1996). Currently there is little controversy that common sense and scientific discourse is replete with usages of ‘is (a) part of’ that do not seem to lend themselves to the kind of reasoning captured by the transitivity axiom – here the relationship expressed may be inherited along some pairwise connections but not for chains of arbitrary length. There is also general agreement that transitivity failures are predominantly due to ‘functional aspects’ in part-whole organisations, i.e., the fact that a part plays a relevant role for the structure, for the constitution, or for the function of the whole. However, disagreement persists about how to diagnose the many instances where ‘is (a) part of’ appears in ordinary and scientific discourse but does not fit the mold of a strictly transitive relation, and how to react to this phenomenon. Here three lines of approach shall be distinguished.

The transitivist position. The first line

of approach, which shall be called here the ‘transitivist’ position, rejects examples along the lines of (1) through (6) as counterexamples to the transitivity of parthood. According to the transitivist position – typically championed by logicians – there is only one part-whole relation, the socalled “general parthood relation” (Casati and Varzi 1999). General parthood, so goes the claim, is adequately axiomatised as a transitive relation. Transitivists accept that in certain contexts ‘is part of’ is not transitive but claim that in these contexts the expression ‘is (a) part of’ does not denote the general parthood relation, but a different, restricted parthood-relation. “In general, if x is a φ-part of y and y is a φ-part of z, it may well be true that x is not a φ-part of z: the predicate modifier ‘φ’ may not distribute over parthood. But that shows the non-transitivity of ‘φ-part’ (e.g., of direct part, or functional part), not of ‘part’. And within a sufficiently general framework this can easily be expressed with the help of explicit predicate modifiers” (Casati and Varzi 1999: 34). In sum, transitivists claim that whenever transitivity fails in part-whole reasoning, these instances are irrelevant for our understanding of parthood, since implicitly they involve “a departure from the broader notion of parthood that mereology is meant to capture” (ibd). It is not quite clear, however, precisely how one is to understand the relationship between, on the one hand, the so-called ‘general’ or ‘broad’ parthood relation that transitivists take to be transitive and, on the other hand,

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relations of ‘modified-parthood’ that are denoted when ‘is part of’ is used with “implicit narrowness” (Varzi 2006: 141). As clarified by Johansson (2004), parthood and modifiedpart-hood cannot stand in a traditional genus-species relationship, since this would imply that the more specific modified-parthood relations inherit the properties that are definitional for the genus, general parthood, and thus should be transitive. But transitivists disagree on whether modified-parthood, e.g., functional parthood, is to be analysed in the form of a conjunction of two binary relational predicates, as ‘x is part of y and x is functional for y’ (Varzi 2006), or should better be understood as involving an implicit ternary predicate, as Johansson (2004; 2006) proposes: “x is functional part of y’ =def ‘x is part of y and there is a z such that x makes something happen to z that is relevant for x’s function in relation to y’” (2006: 159). Johansson’s analysis has the clear advantage that one can state conditions for local transitivity regions of otherwise nontransitive modified-parthood relations in terms of the ‘hidden’ third argument of the relation (2004: 174). In a seminal paper Winston et al. (1987) introduced a different variety of the transitivist position. The authors present (i) a taxonomy of six parthood-relations expressed by ‘is part of’: being a component of an integral object, a portion of a stuff, a stuff of an object, a member of a collection, a feature of an activity, a place in an area; (ii) they take all of these parthood relations to be transitive; and (iii) ascribe transitivity fail-

ures to an equivocation on ‘is part of’. That is, instances of the transitivity axiom can yield invalid conclusions, Winston et al claim, just in case occurrences of ‘is part of’ in the premises denote different parthood relations. Each of these three elements of Winston et al’s position has been criticised in the subsequent discussion – Johansson (2004) offers counterexamples to (iii), and the research industry of knowledge representation created by the internet from the 1990s onwards is replete with alternatives to (i) as well as counterexamples to the generalised transitivity thesis (ii). And yet, since Winston et al. rejected the logicians’ idea that the parthood relation formalised by classical mereology is the only one worthy of systematic investigation, the paper became the point of departure for a new line of approach based on a broader understanding of parthood. The pluralist position. This second

line of approach, here called ‘the pluralist position’, holds that there is a plurality of parthood relations; while some of these relations are nontransitive or only locally transitive and some even intransitive (e.g., ‘direct part’), they each represent our notion of parthood as it guides of common sense and scientific reasoning. The pluralist position has its beginnings in linguistics and psycholinguistics (Cruse 1979, Winston et al 19987, Iris et al 1988, Moltmann 1997). It received special momentum, however, in the late 1990s and early 2000s with the development of object-oriented languages and conceptual modeling in information sys-

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tems engineering, when it became clear that the systematic investigation of domain-dependent part-hood relations could have great practical significance in information retrieval (inference engines, search engines). Researchers working in the intersection of information systems engineering, cognitive science, and AI began to investigate part-whole reasoning as a “conceptual subsystem” that is accessed by other cognitive “modules” such as the “language module” and the “vision module” (Gerstl and Pribbenow 1996: 871). As such, however, the pluralist position is independent of specific assumptions in semantics or the theory of cognition. Typically proponents view themselves to be in the business of ontology, understood Carnap-Quine style as the philosophical discipline that reconstructs the inferential commitments of common sense and scientific reasoning, by means of formal domain theories. The central task for the pluralist position is to determine, in some systematic fashion, which parthood relations are transitive for which domains, and how parthood relations interact with other domain relations. Cruse’s early observation (1979) that an entity y’s parthood relations do not transfer to an entity z if y and z are associated with different “functional domains”, set an important impulse for both of the current main strategies for the pluralist approach. The first strategy aims to formulate general conditions for local transitivities of a certain non-transitive parthood relation, such as functional parthood (Johansson 2004, Vieu 2006, Vieu and Aurnague

2007, Guizzardi 2009). L. Vieu’s account (2006) has the particular asset that it can be used to analyse partwhole reasoning about ‘types’ (examples 1 through 3 above) as well as about concrete particular individuals (examples 4 through 6); it is also sensitive to changes in our classificatory concepts (e.g., due to scientific progress). The second strategy is to develop a taxonomy of parthood relations based on ontological categorisations of the relata of these relations. This strategy was initiated by Winston’s et al. taxonomy (1987) and, via its reworking by P. Gerstl and S. Pribbenow (1995; 1996, Pribbenow 2002), influential in knowledge representation and database ‘ontologies’. Here part-whole relations are taken to be “induced by the compositional structure of a whole” (Gerst and Pribbenow 1995: 887) and are analysed as relations that hold between entities of different ontological categories or “roles” of parts and wholes: as relations that hold between an element and a collection, between a component and a complex, between a quantity and a mass, between a portion and an object, and between a segment and an object. Part-whole reasoning in a certain domain thus can be modeled by the use of data representations that specify the roles of domain items. The non-transitivist position. There is

a third way to react to phenomena of transitivity failures – even though very few adhere to it, it shall be mentioned here as the ‘non-transitivist position.’ While transitivists assume that there is one, “general” or “broad”, parthood relation that is

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transitive, and pluralists take there to be several parthood relations some of which are transitive, some nontransitive, and some intransitive, the non-transitivist position holds that ‘is part of’ stands for one general parthood relation and that this relation is non-transitive. The non-transitivist position is by no means committed to abandoning formal mereology—one just needs to give up on the standard axiomatisation. Constructional ideas for a system with a non-transitive part-relation (“Leveled Mereology”) have first been presented by Seibt (2000; 2009; 2015). (In Leveled Mereology entities are represented by partitions (tree structures of direct parthood) and the non-transitive direct parthood relation is recursively extended (restricted transitive closures) across specific partition levels; since identity (defined via the proper parts principle) is also relativised to levels of depth in a partition, an entity can recur in a partition but in such a way that parthood remains acyclic.) Along quite different lines, and in full generality and formal detail, A. Pietruszczak (2014) has shown how to axiomatise domains where parthood is merely locally transitive. (Pietruszczak postulates that the domain can be divided into maximal sets closed under transitivity (MCT’s) so that these “intersect only at their extreme boundaries“ (Pietruszczak 2014: 369), i.e., so that the maximum of one MCT is the minimum of the other; in addition, he assumes the strong supplementation principle as well as a postulate of the disjointness of the minima of MCT’s. He defines sums on such domains,

but does not assume the existence of arbitrary sums.) The development of formal systems for non-transitive parthood – or more generally, of meta-mathematical research on “proto-transitive” relations defined on domains that are structured by non-transitive relations but contain transitive subregions (Mani 2012; 2016 – undermines the strongest argument of the transitivist position, namely, that only a transitive parthood can be axiomatised in domain-independent fashion. There are two additional sources of support for the non-transitive view. First, on the assumption of non-transitivity is it possible to connect part-whole reasoning to other basic conceptual relations such as similarity and prototypically and to ground taxonomies of part-whole relationships within a broader theory of concepts. This is argued, with rich empirical support, by Fiorini, Gärdenfors, and Abel (2014), who recently have shown, using Gärdenfors’ theory of conceptual spaces, how prototypical wholes can be defined and similarities between structural wholes can be measured. Second, there are also ontological arguments in support of the idea that ‘is part of’ is a very vague relational predicate expressing a nontransitive relation. First, transitivists claim that ‘is part of’ as used in ordinary discourse is transitive except in those cases when it is not transitive and implicitly modified; as long as no independent reasons are given when ‘is part of’ is used with “implicit narrowness” (Varzi 2006: 141) this has the air of an ad-hoc manouvre we simply got used for

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historical reasons. In contrast, nontransitivists can take linguistic surfaces at their face value – in common sense and scientific reasoning the expression ‘is part of’ (modulo cognates) is used to relate entities of widely divergent ontological category and allows for transitive inference chains of different length depending on subdomain. Second, while transitivists claim that they capture parthood in its “basic broad” sense (Simons 1987: 108), upon closer consideration transitivity only holds when the relata are geometric regions representing spatial or temporal extent. It is often claimed that classical mereology captures our notions of spatial or material parthood, but so far it has been overlooked in the discussion that sentences about spatial parthood or material parthood are ambiguous (Seibt 2015). Compare (7), (8), and (9):

‘x is a spatial part of y’ and ‘x is a material part of y’ express how x functions in y that disturb strict transitivity. Just as we are be “duped by grammar” (A. J. Ayer) in finding substance-attribute ontologies most ‘natural,’ we may be duped by geometry in believing that parthood in its most basic sense is transitive.

(7) Screw S is a spatial part of house H. / Screw S is a material part of house H.

Calosi, C.; Graziani, P. (eds), 2014, Mereology and the Sciences: Parts and Wholes in the Contemporary Scientific Context, Heidelberg: Springer.

(8) The geometric region representing the spatial extent of screw S is part of the geometric region representing the spatial extent of H. (9) A screw is a spatial part of a house. / A screw is a material part of a house. The claims in (7) are true only if we read them as in (8), as a claim stating a relationship between geometric regions representing the spatial extent occupied by the entities in question, and this relationship is surely transitive. However, as brought out by the claims in (9), already the notions of

See also > Activity, Body, Boethius, Bolzano, Common Sense Reasoning about Parts and Wholes, Homeomerous and Automerous, Husserl, NonWellfounded Mereology, Twardowski. References and further readings

Artale, A.; Franconi, E; Guarino, N; Pazzi, L., 1996, “Part-Whole Relations in Object-centered Systems: An Overview”, Data & Knowledge Engineering 20: 347-383.

Casati, R.; Varzi, A., 1999, Parts and Places – The Structures of Spatial Representation, Cambridge, MA: MIT Press. Cotnoir, A., 2010, “Anti-Symmetry and Non-Extensional Mereology”, The Philosophical Quarterly 60: 396405. Cruse, D., 1979, “On the Transitivity of the Part-Whole Relation”, Journal of Linguistics 15: 29-38. Fiorini, S.; Gärdenfors, P.; Abel, M., 2014, ”Representing Part-Whole Re-

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lations in Cognitive Spaces”, Cognitive Processing 15: 127-142.

– A Challenge to Philosophy, Frankfurt: Peter Lang, 35-45.

Gerstl, P.; Pribbenow, S., 1995, “Midwinters, Endgames, and Body Parts – A Classification of PartWhole Relations”, International Journal for Human-Computer Studies 43: 865-889

Johansson, I., 2006, “Formal Mereology and Ordinary Language— Reply to Varzi”, Applied Ontology 1: 157-161.

Gerstl, P.; Pribbenow, S., 1996, “A Conceptual Theory of Part-Whole Relations and its Applications”, Data & Knowledge Engineering 20: 305322. Guizzardi, G., 2009, “The Problem of Transitivity of Part-Whole Relations in Conceptual Modeling Revisited” in Van Eck, P.; Gordijn, J.; Wieringa, R. (eds.), International Conference on Advanced Information Systems Engineering, Heidelberg: Springer, 94-109. Iris, M. A.; Litowitz, B. E.; Evens, M., 1988, “Problems of the PartWhole Relation”, in Evens, M. (ed.), Relational Models of the Lexicon: Representing Knowledge in Semantic Networks, Cambridge: Cambridge University Press, 261-288. Johansson, I., 2004, “On the Transitivity of the Parthood Relations”, in Hochberg, H.; Mulligan, K. (eds.), Relations and Predicates, Frankfurt: Ontos Verlag, 161-181. Johansson, I.; Smith, B., 2005, “Functional Anatomy: A Taxonomic Proposal”, Acta Biotheoretica 53: 153-166. Johansson, I., 2006, “The Constituent Function Analysis of Functions”, in H. J. Koskinen, et al. (eds.), Science

Kearns, S., 2011, “Can a Thing Be Part of itself?”, American Philosophical Quarterly 48: 87-93. Lyons, J., 1977, Semantics I., Cambridge: Cambridge Univ Press. Mani A., (2012) “Axiomatic Granular Approach to Knowledge Correspondences”, in Li T. et al. (eds) Rough Sets and Knowledge Technology. RSKT 2012. Lecture Notes in Computer Science, vol 7414, Berlin: Springer, 482-487. Moltmann, F., 1997. Parts and Wholes in Semantics, Oxford: Oxford University Press. Motschnig-Pitrik, R. 1993. “The Semantics of Parts versus Aggregates in Data/Knowledge Modelling”, in Rolland, C.; Bodart, F.; Cauvet, C. (eds.), CAiSE ’93: Proceedings of Advanced Information Systems Engineering, Berlin: Springer, 352–373. Miller, G. A.; Johnson-Laird, P. N., 1976, Perception and Language, London: Cambridge University Press. Pietruszczak, A., 2014, “A General Concept of Being a Part of a Whole”, Notre Dame Journal for Formal Logic 55: 359-381. Pribbenow, S. 2002, “Meronymic Relationships: From Classical Mereology to Complex Part-Whole Relations”, in Green, R.; Bean, C. A.; Sung Hyon Myaeng (eds.), The Se

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mantics of Relationships, Berlin: Springer, 35-50. Rescher, N., 1955, “Axioms for the Part Relation”, Philosophical Studies 6: 8-11. Seibt, J., 2000, “The Dynamic Constitution of Things”, Poznan Studies in the Philosophy of the Sciences and the Humanities 76: 241-278. Seibt, J., 2009, “Forms of Emergent Interaction in General Process Theory”, Synthese 166: 479-512. Seibt, J., 2015, “Non-transitive Parthood, Leveled Mereology, and the Representation of Emergent Parts of Processes”, Grazer Philosophische Studien 91: 165-190. Varzi, A., 2006, “A Note on the Transitivity of Parthood”, Applied Ontology 1: 141-146. Vieu, L., 2006, “On the Transitivity of Functional Parthood”, Applied Ontology 1: 147–155. Vieu, L.; Aurnague, M., 2007, “Partof Relations, Functionality and Dependence”, in: Aurnague, M.; Hickmann, M.; Vieu, L. (eds.), Categorization of Spatial Entities in Language and Cognition. Amsterdam: John Benjamins, 307-336. Winston, M. E.; Chaffin, R.; Herrmann, D., 1987, “A Taxonomy of Part-Whole Relations”, Cognitive Science 11: 417-444. Johanna Seibt

Tropes Traditionally known as ‘individual accidents’ and called by various names – ‘quality moments’ (Segelberg, 1999), ‘perfect particulars’ (Bergmann, 1967 – quality instances, taken as particulars – particular to the specific objects they qualify – and hence not shared by diverse ‘ordinary’ objects, have increasingly come to be called ‘tropes’ (Williams, 1953). As entities, tropes appeal to philosophers for a variety of reasons. Supposedly they (1) allow for a uniform ontology recognising only one basic kind of entity, particulars that are spatio-temporal entities, as opposed to a ‘realist’ ontology that acknowledges universals and often facts, as well as particulars; (2) allow one to take an ordinary object like a red cross, to use Meinong’s example, as a complex entity composed of tropes which are construed as parts of the complex object. Taking tropes (a color trope, a shape trope) as parts of the red cross, one purportedly does not face the need for, or the problems posed by, an ‘instantiation’ relation. This, in turn, (3) allows one to avoid the notorious ‘Bradley’s regress (regresses) and mysterious ‘substrata’ in the analysis of ordinary particulars, since the latter can be taken to be ‘bundles’ of tropes, and not analysed in terms of an underlying substratum that instantiates universal properties (though some trope theorists follow the latter path). Tropes also (4) allow one to reject relations between objects by reintroducing foundational tropes (as internal to or constituents of an object), thus supposedly avoiding further problems posed by the

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exemplification of relations in relational facts. If Mary loves Pierre, a particular trope, an instance of loving Pierre, is taken to inhere in Mary, without any need for the relation loves being instantiated by Mary and Pierre. Early in the 20th century, Moore and Russell (Moore, 1901; Russell, 1911) set out purported refutations of quality-instance or ‘moderate nominalism’. (While moderate nominalists recognise qualities as particulars, not universals, extreme or immoderate nominalists recognise predicates – general words, but not qualities.) The argument basically claims that moderate nominalists are forced to recognise at least one relation, connecting instances of the same ‘kind’, as a universal – a relation of exact similarity or likeness that functions as an equivalence relation to generate an equivalence class or bundle of quality instances. This was recognised by Meinong quite early, but in recent years trope theorists have tried to counter the argument by insisting that such a relation of exact similarity, in so far as it is recognised at all, is an internal relation. Being an internal relation, they argue, implies that a statement that exact similarity holds between two tropes is, if true, ‘made true’ by the existence of the two tropes – and not by the tropes standing in or exemplifying a relation (whether that relation is construed in terms of tropes or not). It is the nature of the tropes that accounts for the truth of such a statement and not the purported ‘fact’, necessary or not, that the tropes instantiate a relation.

The quality instances, by themselves or their ‘natures’, supposedly suffice to ground the truth of statements of qualitative similarity, since it is their being the kind of instances that they are that provide sufficient truth grounds. Yet this purportedly does not mean that instances have natures distinct from the instances themselves – they are said to be their natures. This is reminiscent of views regarding God being his essence and existence – one might say his substantial form. But whereas such an identity was unique to God in the middle ages, for modern tropists every trope is so distinguished. It is as if each trope has its own specific form that somehow allows for distinguishing their kind, form or category. Tropes are also taken to necessarily be of the kind they are and are simples that are diverse but exactly similar. This will not do, though it is not as easily dismissed as the absurdly ad hoc version of the claim that simply, and senselessly, pronounces that some relations are ‘non-entity’ relations and that exact similarity is such a relation. It will not do since it simply brushes aside the problem of universals, and the appeal to ‘tropes being what they are’ is problematic in that it must, oddly, be reiterated for different purposes in the case of each trope. Two tropes being what they are grounds not only their being of a kind, and hence their exact similarity as tropes of a specific color shade, say, but also their being diverse, their being tropes, and not ordinary particular objects – such as Meinong’s red cross – and their being of a logical kind, monadic rather than relational

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tropes, for example. So the apparent triviality that ‘tropes are what they are’ resolves classic problems of diversity, qualitative ‘identity’ and logical form – for tropes just are the entities that resolve such problems by simply being what they are. Some will find it ironic that this argument is exactly the same argument that some realists about universals use to support the claim that a trope ‘is what it is’ in virtue of being of a kind, and hence that the tropist recognises ‘kinds’ as well as particular tropes. There is another reason for the trope theorist to avoid the relation of exact similarity or exact likeness – ES. Consider a trope, say, ϐ, and another kind of trope α, one that is not exactly similar to ϐ. Then we have ‘¬ ES(ϐ, α)’. But, ES cannot here be taken as a particular relational trope – a relational instance. For two tropes that are exactly similar will only stand in one particular instance of exact similarity to each other, assuming such a relation, and not in an indefinite number of other instances. Therefore the trope theorist has no way of stating the negation he needs without introducing a class or sum of exact similarity relations. For one must hold that there is no instance of exact similarity, no exact similarity trope, that holds between α and ϐ. But to do that one needs a prototype exact similarity relation to specify the elements of such a sum or class. Exact similarity, however, is precisely the one case where one cannot have such a prototype. For a universal exact similarity relation is needed to be able to refer to the prototype, in order to specify the exact similarity class or

sum. One must specify the exact similarity class as the class of all instances that stand in an exact similarity relation to the exact similarity prototype. There is no way to spell that out without appealing to a universal relation of exact similarity. Russell was right to hold that taking a quality or relation as a universal accounts for the sameness in diversity of two pairs of tropes being cases of exact similarity. For, so long as the connection of a universal to its instances is coherent, no further question arises about the ground of true attributions of one and the same predicate. That is why tropists appeal to the nature or natures of the two tropes as well as take exact similarity, if they recognise it at all, to be an internal, essential relation—and hence not a ‘real’ relation. A universal, being a common factor in certain facts containing α and ϐ, allows us to specify entities that provide truth grounds in virtue of which ‘α is red’ and ‘ϐ is red’ are true, and hence closes off the question about the truth grounds for the judgments. For given the existence of the grounds of truth, the statements are true. Attributes and relations are held to be universals if they are taken to possibly be constituents in more than one fact and exemplified in such facts. A particular, by contrast, can exemplify a property but cannot not be exemplified by anything. Universals and particulars are thus categorially different types of constituents of facts, and particularity and universality can be basically understood in terms of such a type of asymmetry of exemplifica-

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tion. While particulars, as well as universals, can occur in diverse atomic facts, it is the multiple occurrence (or possible multiple occurrence) of entities that occur as what is exemplified in facts that are taken as universals. Russell’s well known 1911 argument for the existence of universals makes implicit, but essential, use of the difference between what can be exemplified and what cannot. A trope sum is taken to be a kind of particular, since it is a sum of localisable particulars. But though such collections are said to be localisable in space and time, this does not mean that they are localisable in the sense in which a quality instance or an ordinary object may be said to be localisable. This becomes obvious if we ask what it would mean to say that one collection of quality instances is to the left of another or is prior to another. To respond that some element of the one collection is to the left of or prior to an element of the other collection is to reveal the affinity of such collections to the realist’s universals, for one can say the same sort of thing about a universal. By speaking of collections as spatial and temporal in such ways, one clearly abandons the normal spatio-temporal characterisation of ordinary localisable objects, which is what one does if one speaks of a universal quality being localised in space and time. In both cases objects, to which standard spatial and temporal laws do not apply, are taken to be spatial and temporal.

Prototype instances of qualities are required for a trope theorist to construct definite descriptions of collections or classes of quality instances, as was argued in the case of the true negation that we considered, ‘¬ES(c, b)’. Collections of quality instances, like the mereological sums or fusions spoken of in systems of mereology and the quality complexes of Stout, are taken to be individuals or particulars by trope theorists. But such entities differ significantly from other particulars, the quality instances themselves and ordinary objects. Such collections are not localised in time and space in an ordinary sense, and, hence, are not terms of temporal and spatial relations in a straightforward way, irrespective of taking them to be given locations by means of the locations of their constituents. Nor are they objects that are experienced, since one can hardly claim to be presented in experience with the collection or sum of instances of a color. To avoid universals, some trope nominalists accept various kinds of particulars; ordinary objects (temporal ‘slices’ of such), quality instances, and classes or mereological sums of such, as well as a variety of accompanying universal connections, albeit formal or abstract, and universal relations as constituents of facts. Other trope theories recognise special kinds of sums, that are neither classes nor mereological sums, of quality instances and are formed from a special kind of operator. This is one way of construing Stout’s variant of the quality instance view (Stout, 1923). Stout’s view raises a question as to

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whether sub-sums of a sum of quality instances are also further such special individuals. If they are not, the declaration seems arbitrary. If they are then, ironically, there are more subsums than there are individual tropes of a kind. In any case, whether or not trope theorists recognise classes they tend to explicitly recognise at least one entity corresponding to a universal quality – some kind of whole or sum of quality instances. ES is an equivalence relation, and its being such is crucial to the construction of quality sums or classes. Let ‘R’ represent a relation treated as a trope similarity class or sum containing relation instances or tropes, r1, r2, …, rn, of R. A problem arises when one tries to explicate the claim that ES is transitive, which it must be. For R to be transitive is for the similarity class to be such that if it contains two trope instances that relate objects α and ϐ and ϐ and γ, respectively, then it contains a third instance that relates α to γ. If ES is not taken as a class or sum, what it means to say that exact similarity is a transitive relation will be different from what it means to say that a relation like R is a transitive relation. On the other hand, if ES is construed in terms of a similarity class or sum an even more basic problem arises. The view is seen to be viciously circular, as we already noted in considering the case of a true negation earlier. For to specify the class of exact similarity tropes, like any similarity class, the trope theorist must do so in terms of the relation of exact similarity or by simply arbitrarily assuming he has a given class of exact similarity

tropes that does not need to be specified. But, in the special case of exact similarity itself, that relation is supposedly construed as the exact similarity class (or sum or complex) that is being specified. Thus what is to be specified is presupposed in its own specification. This forces the advocate of tropes to recognise that exact similarity cannot be construed in terms of tropes – it must be recognised as a universal relation or, as many do, rejected as a relation (sometimes along with all relations). The point is that one cannot give a definite description of the class or sum of exact similarity tropes without referring to (or using an embedded description of) that class or sum in the purported description. Even if we grant the existence of the totality of similarity tropes, we cannot specify the class. Interestingly the class would have to be an infinite class, since once you have two such tropes, there would be the third holding between them, and so on. But even if the class should be finite the point would be that giving it by enumeration does not show that it is an exact similarity class. An alternative way of handling relational predication leads some to recognise relational properties. Some trope theorists see relational properties as viable constituents of ordinary objects. Thus, for such philosophers, when ‘αRϐ’ is true, α supposedly contains or instantiates the relational property Rϐ, rather than standing in the relation R to ϐ. Relations are reduced to relational properties. Introducing Rϐ as a property is, as F. P.

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Ramsey once argued, absurd (Ramsey, 1931). All one does is take the fact that we can form a predicate abstract, such as ‘xRϐ’, from a relational sentence and hold that such an abstract stands for a property. Nevertheless the attraction of such ‘properties’ for tropists is understandable. A dyadic relation cannot be considered to be contained in only one of a pair of terms it holds of – as monadic quality instances may be taken to be contained in objects. Likewise, it cannot be contained in both, since then it would not be an instance of a relation but a common characteristic of two objects. Hence an instance nominalist is driven to deny that there are relations at all and to explain relations away in terms of monadic relational properties or ‘fundaments’ internal to objects. This medieval Aristotelian pattern is found in Brentano and those he influenced (Meinong, Mulligan, 1998) and is currently advocated by Campbell (1990: 97-110) and those he influences (Maurin, 2004). To get the ‘fundaments’ to account for a particular case, some take them in matching pairs. Thus if α loves ϐ, α is proclaimed to contain an instance of loving ϐ, while ϐ has an instance of being loved by α. What such relational properties are is never spelled out. Doing so would threaten to have a be a constituent of the property loved by α, which is, in turn, a constituent of ϐ. This is exactly how F. H. Bradley, by turning relations into relational properties came to the conclusion that everything was part of everything else – and hence developed one path to his Absolute –

the idea that every ‘thing’ was somehow a constituent of every other thing. Then, in a manner reminiscent of Leibniz, when one really comprehends ‘a’ thing one comprehends the totality of things (Reality). Another variation on the theme is found in the discussion of quantities. Introducing trope quantities, one takes statements about an apparent relation like larger-than to be true in virtue of the intrinsic nature of such trope quantities, which are constituent attributes of objects. Thus, to put the view in terms of the natural numbers, it is 12 and 5 that comprise the truth grounds for ‘12 > 5’, and not (assuming numbers are objects) that 12 stands to 5 in the relation >. This is an empty verbal solution which, if one thinks about it, packs the arithmetic of the natural numbers into each such number. Moreover, taking a collection of elements to form a series by ‘their nature’ does not furnish conditions for a serial order, given the denial of relations, including relations that are required to specify a serial ordering. You don’t even have the requisite relational instances to speak of a sum of such instances being transitive, etc. – recall the earlier discussion of ES and R being transitive. Yet, recognising relational instances raises questions – about their ‘localisation’ in time, for example – as to how mathematical relations could be treated in terms of such instances, taken as particulars. There is a further problem with instantiation that is endemic to trope theories. It is generally an article of tropist faith that tropes are necessari

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ly attached to the terms that they ‘in fact’ qualify or relate. On a bundle analysis of ordinary objects this means that various trope qualities are necessarily ‘compresent’ if they exist. To take an instantiation or exemplification relation in terms of trope instances would then yield the consequence that predications were necessary – a particular trope of exemplification would necessarily connect the terms it does, if it exists at all. A trope theorist who allows for relation instances might then introduce a categorially different exemplification connection or ‘tie’ to connect relation instances to objects to form relational facts – and seek to deal with the questions such a connection raises. Alternatively one might hold that a relation instance itself provides the connection in a relational fact – as Russell held about universal relations. In either case one raises a question about temporally ‘localised’ relation instances. To suggest that relation instances are temporally localised raises the obvious question about temporal relations. It is prima facie absurd to take them to be localised in time. Yet, to suggest that they are not so localised makes them ‘particulars’ in name only. See also > Gestalt, Goodman, Husserl, Meinong, Nominalism, Proposition, Substrate, Universal.

Campbell, K., 1990, Abstract Particulars, Oxford. Maurin, A. S., 2004, If Tropes, Amsterdam. Meinong, A., 1983, On Assumptions, 2nd ed., (trans. J. Heanue), Berkeley. Moore, G. E., 1900-01, “Identity,” Proceedings of The Aristotelian Society, 1. Mulligan, K.; 1998, “Relations – Through Thick and Thin”, Erkenntnis 48: 2-3. Russell, B., 1911, “On the Relations of Universals and Particulars”, in Logic and Knowledge: Essays 19011950, (ed. R. Marsh), 1956, London. Ramsey, F. P., 1931, “Universals”, in The Foundations of Mathematics, (ed. R. Braithwaite), London. Segelberg, I., 1999, Three Essays in Phenomenology and Ontology, (trans. H. Hochberg; S. Ringström Hochberg), Stockholm. Stout, G. F., 1923, “Are the Characteristics of Things Universal or Particular?”, Proceedings of the Aristotelian Society, suppl. vol. III. William, D. C., 1953, “The Elements of Being,” Review of Metaphysics 7 3-18: 171-192 Herbert Hochberg

Twardowski, Kazimierz References and further readings

Bergmann, G., 1967, Realism: A Critique of Brentano and Meinong, Madison.

The Polish philosopher Kazimierz Twardowski (1866-1938) was the first who proposed a fully developed theory of parts and wholes. He stud-

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ied philosophy under Franz Brentano in Vienna and founded the LvovWarsaw Philosophical School in Poland in 1895. His main philosophical work is the influential study Zur Lehre vom Inhalt und Gegenstand der Vorstellungen [On the Doctrine of Content and Object of Representations] (1894). This study contains in chapters 9-11 Twardowski’s partwhole theory, which, however, has various weaknesses and deficiencies. It was formulated with the aim of analysing the relation between an object and the content of a representation and distinguishes many different kinds of parts, but not all distinctions are supplied with intuitive examples. Even though it leaves many questions unanswered, it can serve as a rich source of inspiration rather than as a fully furbished theoretical tool. Twardowski subscribes to the broadest possible use of the term ‘a part of a whole’. Not only a detachable fragment of a whole, but also properties like shape or weight and even the arbitrary, even purely logical, partwhole relation itself are regarded by Twardowski as parts of the whole in question. Hence the first nonintuitive feature of the theory: if all relations between parts of a given whole are its parts as well, then all wholes – disregarding absolutely monolithic objects – have infinitely many parts, because arbitrary parts of a whole are always in some relation, e.g. they differ one from another. The first distinction between parts of a whole holds between non-relative and relative ones. Non-relative parts are called material and all relations betwen arbitrary

parts of a whole are called its formal parts as they constitute the form of a given whole. Another dichotomy is made between physical parts, which are concrete, and metaphysical parts, which are relations and properties of a whole, i.e., its general abstract aspects. This dichotomy does not play any role in subsequent investigations and further classifications. Material wholes differ with four following respects: (i) the order of a part, (ii) its simplicity or complexity, (iii) its possible manifold or univocal functioning in different wholes and (iv) the independence of a part with regard to its whole or dependence of its part with regard to some other parts of its whole. In the latter case dependence can be one-sided or mutual. A first order material part of a whole is its immediate part. A second order part of a given whole is an immediate part of an immediate part of the whole. For example an arm is a first order part of a body and fingers are its second order parts. An example of a part that can function differently in different wholes is – in Twardowski’s view – redness. It plays different functions as e.g color of a material thing and as a constituent of a light spectrum. It is not clear whether there is any kind of object that functions in different wholes in one unique manner only – it seems that one can always find different applications even for such a trivial object like a nail. Physical parts

Metaphysical parts

Material parts

Formal parts

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The distinctions of formal parts are more complicated. Firstly, Twardowski distinguishes relations among proper parts of a whole and relations between a proper part and its whole. The former are called formal parts of the secondary order and the latter formal parts of the primary order. Further, formal parts of the primary order differ with regard to their constitutive role for the whole (formal parts of the primary order in the proper sense) or their irrelevance for the existence of a whole (formal parts of the primary order in the improper sense). For example, ‘belonging to the whole’ is a formal part of this whole of the primary order in the proper sense while ‘being made of the same stuff as the whole itself’ is in many cases a formal part of the primary order in the improper sense. All relations between first order material parts of the whole or between such parts and the whole itself are called formal parts of the first order. When we investigate analogical relations inside a material part of the first order, we deal with formal parts of the whole of the second order and so on. Next, relations between nonrelational parts of a whole are called formal parts of the first degree. A relation between formal parts of the first degree is a formal part of the second degree and so on. Finally, if a relation is decomposable into other relations we get formal parts of higher ranks. For example, consider the decomposition of a relation of imilarity between two physical parts – if both parts have complex structure, their similarity can be decomposed into a set of similarities betwen re-

spective parts of these parts and so on.

The fatal flaw of this interesting theory is that it does not contain a definition of a whole. Twardowski claims that formal parts of the first order in a proper sense make one whole of many parts, as all parts are referred to a common whole. But these relations hold between parts and their whole, so they already as-

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sume the whole which they ought to constitute. See also > Brentano, Husserl, Leśniewski and Polish Mereology, Mally, Reinach, Segelberg, Stumpf. Bibliographical remarks

Rosiak, M., 1998. Detailed analysis of Twardowski’s PW theory, its shortcomings and the improvements proposed by Husserl. References and further readings

Betti, A., 2010, “Kazimierz Twardowski”, in Zalta E. N. (ed.), The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), http://plato.stan ford.edu/archives/sum2011/entries/tw ardowski/. Cavallin, J., 1997, Husserl, Twardowski and Psychologism. Phaenomenologica 142, Dordrecht: Kluwer. Poli, R., 1996, “Kazimierz Twardowski 1866-1938”, in Albertazzi L.; Libardi M.; Poli R. (eds.), The School of Franz Brentano, Nijhof International Philosophy Series, Dordrecht: Kluwer, 207-232 Rosiak, M., 1998, “Twardowski and Husserl on Wholes and Parts”, in Kijania-Placek K. and Wolenski J., eds, The Lvov-Warsaw School and Contemporary Philosophy, Dordrecht: Kluwer, 85-100. Twardowski, K., 1894, Zur Lehre vom Inhalt und Gegenstand der Vorstellungen - Eine psychologische Untersuchung, Wien (reprinted by

Philosophia Verlag, München-Wien, 1982). Eng. transl. On the Content and Object of Presentations (transl. R. Grossmann), The Hague: M. Nijhoff, 1977. Marek Rosiak

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U Universal Universals belong to those entities that have neither spatial nor temporal parts, and that therefore have neither a (literally) spatial nor a (literally) temporal localisation. Universals are either non-predicative or predicative. The non-predicative universals are also called types or type-objects (for example, the letter A). The predicative universals, in turn, are divided into the properties and the relations. Types are closely related to properties: there is a property p(T) corresponding one-to-one to each type T, such that x exemplifies/instantiates T if, and only if, x exemplifies/instantiates p(T). Predicative universals should be distinguished from concepts, just as states of affairs should be distinguished from propositions. But just as there is a certain analogy between states of affairs and propositions, so there is a certain analogy between predicative universals and concepts; in particular, there is an analogy between properties and monadic concepts, and an analogy between relations and polyadic concepts. The analogy is of such a kind that names for properties can also be used as names for monadic concepts, and names for relations also as names for

polyadic concepts. Thus, ‘love’ can both function as a name for a certain dyadic relation, and as a name for a certain dyadic concept. If the context does not already make it clear what is being referred to, then the name can easily be disambiguated: ‘the relation of love’, ‘the concept of love’. The situation is entirely the same in the case of states of affairs and propositions: ‘that the moon revolves around the earth’ can function both as a name for a state of affairs, and as a name for a proposition; putting ‘the state of affairs’ or ‘the proposition’ to the left of the ‘that’-phrase will make it clear, if need be, what is being referred to. Moreover, predicative universals and states of affairs belong together in a way that is analogous to the way in which concepts and propositions belong together. Concepts are prominent constituents in the composition of propositions. Analogously, a predicative universal U, together with the right number N of ordered entities X1, …, XN, each – in its place – of the right kind, constitutes a state of affairs: the state of affairs which is the composition of U with X1, …, XN, in short: [U, X1, …, XN]. The just-mentioned rightness in composition is dictated by the so-called type of U, by its composition-profile, so to speak; if that composition-profile is not respected, the composition-result will not be a state of affairs. For example, [Younger, Mack, Jack] – the composition of the dyadic relation Younger with, first, the individual Mack, and, second, the individual Jack – is a state of affairs because it respects the type of Younger: it is the

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state of affairs that Mack is younger than Jack. But neither [Younger, Mack] nor [Younger, Mack, Younger] are states of affairs, because they do not respect the type of Younger. A predicative universal U is exemplified by the entities X1, …, XN (in the given order) if, and only if, [U, X1, …, XN] is an obtaining state of affairs (a fact). Sometimes instantiation is distinguished from exemplification, such that an apple, for example, is taken to exemplify, but not instantiate, the property of being red, whereas a red-trope, found on that apple, is indeed taken to instantiate that property. It seems best to treat instantiation as a species of exemplification: instantiation is exemplification by individuals which are not substances (that is, by entities that are so-called individual accidents). Though universals have neither spatial nor temporal parts, this does not prevent them from having, and being, parts in some sense. In a sense, a universal is a part of all the state of affairs it helps to compose. But since universals and states of affairs differ in ontological category, it seems more appropriate to say that a universal is a constituent of all the states of affairs it helps to compose, than that it is a part of them. Part-relations between homocategorial entities differ vastly from part-relations between heterocategorial entities, and there seems to be a slight bias – at least – in favor of regulating ontological discourse in such a way as to reserve the word “part” for designating only part-relations between homocategorial entities, while the word “constitu-

ent” is to be used as the more general mereological term (such that every part is a constituent, but not vice versa). In any case, there is not only a heterocategorial part-relation between universals and states of affairs, but also a homocategorial part-relation between universals of the same type. Consider the simplest case: generally defined properties of individuals, that is, monadic (predicative) universals that compose a state of affairs with each individual, but not with any non-individual. Let P and P´ be two such properties; then P is an intensional part of P´ if, and only if, for all individuals X, (the state of affairs) [P, X] is an intensional part of (the state of affairs) [P´, X]. Accordingly, the property of being extended is an intensional part of the property of being colored. Or consider a slightly more complex case: generally defined dyadic relations between individuals, that is, dyadic (predicative) universals that compose a state of affairs with each ordered pair of individuals, but not with any ordered pair that has a non-individual as a component. Let R and R´ be two such relations; then R is an intensional part of R´ if, and only if, for all individuals X and Y, (the state of affairs) [R, X, Y] is an intensional part of (the state of affairs) [R´, X, Y]. Accordingly, the relation Beginningto-exist-earlier-than is an intensional part of the relation Being-aprogenitor-of. These examples are instances of a general principle, stating the general reducibility of intensional parthood

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for universals to intensional parthood for states of affairs: Universal U is an intensional part of universal U´ if, and only if, U and U´ compose states of affairs with the very same sequences of entities and, for every sequence Q of entities with which U composes a state of affairs, it is true that the state of affairs U composes with Q is an intensional part of the state of affairs that U´ composes with Q. Clearly, according to this, the principles of the intensional mereology of universals will be consequences of the principles of the intensional mereology of states of affairs. See also > Abstract, Facts, Nominalism, Propositions, Reinach, Segelberg, Structure, Tropes. References and further readings

Armstrong, D. M., 1978, Universals and Scientific Realism, 2 vols., Cambridge: Cambridge University Press. Armstrong, D. M., 1979, A World of States of Affairs, Cambridge: Cambridge University Press. Meixner, U., 1997, Axiomatic Formal Ontology, Dordrecht: Kluwer. Meixner, U., 2006, The Theory of Ontic Modalities, Heusenstamm: Ontos Verlag. Uwe Meixner

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W Whitehead, Alfred North Whitehead was one of the first logicians to engage in mereology. He did so in the course of his lifelong project to provide an adequate metaphysical account of the physical universe. A principal aspect of this was his endeavour to provide a properly, physically grounded geometry. This was understood by him as a theory of space in which the occupants of this space were not merely passively present at locations in a threedimensional Euclidean continuum, but rather formed an interrelated system, whose governing principles were such that the axioms of geometry would be derivable from them. Standard modern geometry has treated geometric figures such as lines, triangles, cubes and so on as sets of points, and mathematically this is nothing to object to, but Whitehead was always against taking points as the basis of geometry as they are necessarily imperceptible. Even in his early memoir “On Mathematical Concepts of the Material World” (1906) he prefers systems where the basic elements are lines rather than points. Later, prompted by ideas of Grassmann and using the logical tools he and Russell developed in Principia Mathematica, he reconcep-

tualised points and other geometric elements of zero thickness, such as lines and surfaces, as logical constructions out of three-dimensional items, and for this he needed the relation of (proper) part to whole. He developed his mereology to the extent required for his geometrical purpose. The unfinished fourth volume of PM, assigned to Whitehead alone, was to be on geometry, and this would surely have contained an axiomatised mereology. As it was, the project was shelved and the ideas and techniques plundered piecemeal for Whitehead’s logically less systematic writings in natural philosophy and process philosophy. The ordained destruction of Whitehead’s Nachlass after his death deprives us of a fully developed formal mereology from his pen, but we can gain a fair idea of the general outlines from the published work. Whitehead published three tranches of work on mereology, never for its own sake, but always in the service of his larger project. The first is in his 1916 essay “La théorie relationniste de l’espace” (TRE), based on a talk given in Paris in 1914 and containing material Whitehead was intending to incorporate into PM IV. The timing of the essay suggests that Whitehead may have been the first logician to formalise some kind of mereology, though he was preceded into print by Leśniewski. Noting in TRE that the word ‘part’ is used in several senses, Whitehead elects not to become embroiled in terminological disputes and instead defines a basic notion which he calls

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σ-inclusion: “We will now define a means by which the geometric concept ‘point’ can be defined for a σworld. The starting point for this definition is the definition of a derived relation Eσ, called ‘σ-inclusion’, and which in its formal properties is analogous to the relation of whole to part” (1916: 432). The definition is schematic, and proceeds as follows: Given a class of relations σ, the relation Eσ or σ-inclusion is defined as a σ-includes b iff (Def.) for some R in σ and some x, xRb, and for every R in σ and every x, if xRb then xRa. The letter ‘E’ in the definition is obviously meant to remind us of the term ‘extends over’: Whitehead’s use of ‘extend’ and ‘extension’ is clearly modelled on Grassmann’s term Ausdehnung. Whitehead habitually prefers a notation and terminology where the term for the whole precedes that for the part. As this is contrary to standard practice, we will reverse the order and say instead that b is a σ-part of a. Here ‘part’ is to be understood in the improper, reflexive sense. The motivation for the definition appears to be something like this. Imagine b is an object perceived by one or more observers, or acted upon by one or more agents. Then whenever they perceive or act upon b, they cannot help but similarly perceive or act upon the (intuitively) more encompassing a. The modes of perception or interaction being comprised by the relations R in σ, b is then a part of a from the vantage point of σ.

Whitehead notes that σ-inclusion (and therefore also its converse, σpart) is elementarily reflexive and transitive. This presumably constitutes the analogy to part–whole that he mentions. However, the generality of his definition means that antisymmetry does not follow, so σ-inclusion need not define anything stronger than a pre-order. As a theory of part and whole it lacks any supplementation principle: it is easy to generate models of σ-inclusion where one object is a proper σ-part of another without additional parts, and indeed to compose a linear chain of proper σ-parts. So the theory falls well short of capturing the essential properties of part–whole. Whitehead does mention the conditions which must obtain if there are to be no minimal σ-parts. He also explains, for the first time in his writings (Russell had sketched the ideas in Our Knowledge of the External World in 1914), the method of extensive abstraction in connection with geometry. Whitehead’s second excursion into mereology occurs in his writings on natural philosophy. The fullest and most satisfactory exposition is in his An Enquiry concerning the Principles of Natural Knowledge (PNK) of 1919: The Concept of Nature contains only a briefer variant of this. Once again, Whitehead’s reason for employing the concepts of mereology is to enable him to define central notions regarding space and time: “Every element of space or of time (as conceived in science) is an abstract entity formed out of this relation of extension … by means of a determi-

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nate logical procedure (the method of extensive abstraction)” (PNK 75). For Whitehead the terms of the relation K of extending over (the converse to the (here, proper) part relation) are events. What he calls ‘objects’ are enduring things recognised among events. The theory of extension, which is the essential preliminary to Whitehead’s crucial method of extensive abstraction, is thus a mereology of events. In PNK Whitehead does not trouble himself to give axioms for this mereology, but contents himself with stating “Some properties of K essential for the method of extensive abstraction” (PNK 102). Among the stated principles the following are sufficient to axiomatise the mereology (Simons 1991). Irreflexivity: Nothing is a part of itself. Transitivity: A part of a part is a part of the whole Density: Between any two events, one of which is part of the other, there is a third event which is part of the greater and has the lesser as part. Intersection: Events intersect which have a common part (Definition) Supplementation: If whatever intersects a intersects b then a is part of b or is identical to b. Upper Bound: Any two events are both parts of some event. Separate: Events are separate which have no common part (Definition)

Separated: A set of events is separated if its members are pairwise separate (Definition) Dissection: A set of events A is a dissection of an event a iff A is separated and a is the sum of the members of A, that is, whatever intersects a intersects some element of A and vice versa (Definition). Part-Dissection: Every part is a member of a dissection of the whole. We have used Whitehead’s terminology here: more standard terms for some of these are: intersects = overlaps separate = disjoint separated = discrete dissection = partition Among the consequences of the stated principles are that there is no maximal or universal event, and more importantly, there are no events without proper parts. Whitehead’s mereology is thus (as he designed it) anti-atomistic. This is not immediately obvious but follows from the principles because Whitehead’s definition of intersection (overlap) uses the notion of a common proper part, whereas the more usual definition involves proper-or-improper parts, and with this more usual definition of overlap the principle that every event has another as part has to be asserted independently. Whitehead then moves into topological territory. The most important notion is that of the junction of two events. Two events are joined when there is a third event that overlaps

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them both and there is a partition of the whole of which each member is a part of one or the other or of both. (PNK 102) Two events are joined if they either overlap or have a common boundary. Whitehead asserts that there are joined events which are nevertheless disjoint, that is, they have a common boundary but no common part: such events are externally joined or are adjoint . (PNK 103). There are also overlapping events which have a partly common outer boundary: these are called injoined (ibid). From this it follows that Whitehead’s mereology, unlike that of Leśniewski, does not accept the existence of disjoint sums. In Leśniewski’s mereology any two individuals are joined: take their sum as the third object and partition it into the common part (if any) and the remainders. Whitehead assumes (without explicitly stating so) that all events are continuous or connected, that is, do not consist of two or more non-joined parts. It is only thus that it could appear plausible to him that purely mereological concepts could be used to define topological concepts such as being connected. The whole point of the mereology in Whitehead is to serve as a tool in defining the abstract element of geometry such as points, lines, surfaces and boundaries. For this he employs his method of extensive abstraction. A set of events is an abstractive class when any two members are such that one is a (proper) part of the other, and there is no event which is a part of every event of the set. The idea is

that an abstractive class is like an infinite nest of (four-dimensional) Russian dolls without a smallest doll inside all the others. One class A covers another B when every member of A has some member of B as a part. Classes which cover one another are called K-equal, and K-equality is an equivalence relation among abstractive classes. As an example consider a square, inside which is inscribed a circle, inside which is inscribed another square, inside which is inscribed another circle, and so on without end. The squares and the circles are K-equal abstractive classes: intuitively, they both converge to the same point. But Whitehead turns this upside-down and defines the point as the set of all K-equal abstractive classes, of which these are but two. By varying the conditions constraining the relevant abstractive classes, Whitehead closes in on logical constructions of the most important elements of point-geometry and the idea of an instant of time. While the idea is only sketched and not worked out in detail, it is clearly an interesting and fruitful way to a geometry without infinitely small or thin elements. Whitehead’s third mereology occurs in his major metaphysical treatise Process and Reality (PR). Here the development is complicated over and above that of the earlier writing by two developments. The first is the separation of spacetime from its occupants. In the earlier philosophy, the mereological and topological relations among the primary beings, events, serve to ground physical geometry. However, by the time of PR, Whitehead had concluded – for rea-

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sons which need not concern us here – that while spacetime remains continuous and atomless, the basic events are atomic, having no proper parts. To mark this, Whitehead dubs such atomic events ‘actual occasions’. All such occasions are thus disjoint (separate) from one another. However, each occasion brings a quantum of spacetime into existence with it, which it fully occupies or “enjoys”. These quanta are themselves atomless, and they and their composites, which Whitehead calls regions, are the subject of Whitehead’s new mereology. The second development is that Whitehead’s composite of mereology and topology had come under criticism from Theodore de Laguna, and Whitehead adapts his development as a result, weaving mereological and topological concepts together into a single mereotopology. He replaces the notion of part as basic by that of (extensive) connection. Regions are connected if and only if they share a point. But since it is Whitehead’s aim, as before, to define the notion of a point via others, this is an intuitive guide only. As in PNK, the development in PR is informal, and indeed sloppily and elementarily inconsistent, though the inconsistencies are easily eliminated by minor modifications to the principles Whitehead states. A new mereological notion of containment is defined: Region a is contained in region b iff (Def.) whatever is connected to a is connected to b. Containment is transitive, asymmetric (according to Whitehead: it

should not be), anti-atomistic, and every region contains other regions which are not connected to one another. Whitehead goes on to define intersection of regions: an intersection is a maximal part where regions overlap, and two regions can have more than one intersection. Regions are connected externally when they are connected but do not overlap. A region is tangentially contained in another if it is contained in it and both are externally connected to a third region; non-tangentially contained in another if it is contained in it and they are not both externally connected to a third region. This completes the preparation for the method of extensive abstraction, which now employs the notion of non-tangential containment rather than part to overcome some technical difficulties in the earlier treatment. As before, Whitehead implicitly assumes all regions are connected (in the topological sense, not his). Without benefit of an axiomatically worked out treatment by Whitehead it is not quite clear how to mend his system. Subsequent developments of mereotopology by Bowman Clarke (1981) and others have remedied the defectsproduced systems free of such unclarities. See also > Leśniewski, Mereotopology, Philosophy of Mathematics, Point, Tarski, Topology, Whitehead’s Metaphysics

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References and further readings

Clarke, B. L., 1981, “A Calculus of Individuals Based on Connection”, Notre Dame Journal of Formal Logic 22: 204-218. Russell, B., 1914, Our Knowledge of the External World As a Field for Scientific Method in Philosophy, La Salle: Open Court. 2nd ed. London: Allen & Unwin, 1926. Simons, P. M., 1991, “Whitehead und die Mereologie”, in: Hampe, M.; Maaßen, H. (eds.), Die Gifford Lectures und ihre Deutung. Materialien zu Whiteheads Prozess and Realität, Band 2. Frankfurt/Main: Suhrkamp, 1991: 369-388. Revised translation: Whitehead and Mereology, in: Durand G.; Weber, M. (eds.), Les Principes de la connaissance naturelle d’Alfred North Whitehead/Alfred North Whitehead’s Principles of Natural Knowledge. Frankfurt/Main: Ontos, 2007: 215-233. Whitehead, A. N., 1906, “On Mathematical Concepts of the Material World”, Philosophical Transactions of the Royal Society Series A, 205, 465-525; reprinted in Northrop, F. C. and Gross, M. W., (eds.) Alfred North Whitehead. An Anthology. Cambridge: Cambridge University Press, 1953: 11-82. Whitehead, A. N., 1916, “La théorie relationniste de l’espace”, Revue de Metaphysique et de Morale 23: 423454 (TRE). Whitehead, A. N., 1919, An Enquiry Concerning the Principles of Natural Knowledge, Cambridge: Cambridge University Press. 2nd ed. 1925 (PNK).

Whitehead, A. N., 1978, Process and Reality: An Essay in Cosmology. Corrected edition, Griffin, D. R.; Sherburne, D. W. (eds.), New York: The Free Press, 1978. 1st eds. 1929: New York: Macmillan; Cambridge: Cambridge University Press. (= PR) Peter Simons

Whitehead’s Metaphysics The work of Alfred North Whitehead (1861-1947) can be divided into an early period, characterised by writings focused on logic and mathematics, and a late period, where philosophy and in particular metaphysics take centerstage. The pinnacle achievement in the early period are the three volumes of the Principia Mathematica which he co-authored with Bertrand Russell (Whitehead and Russell, 1910-1913), whereas the late period culminates in Process and Reality (Whitehead 1929), a book which has been hailed as “arguably the most impressive single metaphysical text of the twentieth century” (Simons 1998: 378). This article will be focused on Whitehead’s metaphysical ideas put forth in the late period, which are still fully relevant and interesting in their original form today. Two preliminaries are in order, on language and on method, respectively, which have proven to be the main obstacles in the appreciation of Whitehead’s philosophy.

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First, the terminology of Whitehead’s metaphysics is entirely idiosynchratic. Because Whitehead thought that “an old established metaphysical system gains a false air of adequate precision from the fact that its words and phrases have passed into current literature” (PR 13), he operated with neologisms whose meanings are axiomatically defined by how they are used within his system. What he achieved with this policy, however, was the unintended effect of giving his writings an air of being rather cryptic. In what follows, I introduce only a few items of Whitehead’s terminology and try to sketch basic ideas in common terms. For a proper encounter with Whitehead’s philosophy, however, it is necessary fully to understand Whitehead’s technical terms; a good starting point for this effort is (Cobb 2008). Second, Whitehead’s speculative metaphysics is justified by a fallibilist position. Any metaphysical system, he claims, aims to give adequate and coherent expression to the many very different facts that we human beings experience, but is most likely to do so imperfectly. This fallibilist position is presumably the reason why Whitehead is often credited with having “revived speculative metaphysics without ignoring the criticisms made by Kant” (Hampe 1998, in the summary, translation by the author). Both his fallibilism and holism, i.e., the ambition to include all aspects of human experience, connect Whitehead with the American Pragmatists – especially with William James, whom Whitehead actually praises as one of the ‘four great

thinkers’ of western literature, together with Plato, Aristotle and Leibniz (MT 2). Turning to the part-whole relationships in Whitehead’s metaphysics, the basic idea is this: the ultimate entities of the world are ”actual entities,” i.e., processes that are modeled on experiences, and contain their past as a proper part. To explain this basic idea, we need to answer the following three questions. (1) How can an ultimate entity even have parts in the first place?; (2) What is the past?; and (3), If the past is a proper part, what else is added? Regarding the first question, it must be said that ultimate metaphysical wholes composed of parts that themselves are not again such wholes is nothing new. Already Aristotle took living beings to by ultimate wholes, but held that their parts (parts of their bodies) are not themselves metaphysical wholes. Similarly, Leibniz postulates ‘monads’ as metaphysical ultimates with internal structure, but without being decomposable into their parts. Whitehead shares this idea and even goes so far as to say that “all other meanings of ‘composition’ are referent to this rootmeaning” (PR 147) of how the ultimate entities of the world are composed of parts that existentially depend on the existence of the whole. But what are these ultimate constituents of the world, ‘actual entities’, and what are their parts? Actual entities are ‘experiences’, and as such they are modeled on our own, conscious experiences. Whitehead expresses this by saying that “apart

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from the experiences of subjects there is nothing, nothing, nothing, bare nothingness” (PR 167). However, an experience or actual entity is a complex process called ‘concrescence’ that is the ‘growing together’ of a collection of ‘prehensions’ – the taking up of a ‘datum’. A prehension takes up as its ‘datum’ whatever happened in its past, freely deciding which aspects of the past to reinforce and carry on further, and which to de-emphasize or even block out, and how to integrate these elements into one more or less coherent or harmonious experience that is guided by overarching goals such as truth and attunement with external context. The processes of prehension that make up the process of the concrescence of a unified experiencing, i.e. actual entity, eventually come to an end, not so much in the temporal sense as in the sense of reaching their completion, which one might understand in analogy to the mathematical concept of the base of a vector space – once it is there, nothing more can be meaningfully added to it. Once an actual entity or experience is complete, it is no longer an experience, it loses its subjective moment of decision and becomes fixed – or ‘dead’ – and can thus serve as an objective datum for new ‘processes of experience’. Whitehead summarises all this in his famous dictum “the many become one, and are increased by one” (PR 21). All concrescences occur ‘in step’, i.e., the universe proceeds ‘stepwise’ – what exists at the shortest measurable interval are a genera-

tion of basic processes which provide data for the next generation of basic processes; while all such processing goes on in parallel, i.e., basic processes cannot influence each other directly, the next generation of basic processes takes up the total information of the past. Let us turn to the second and the third question. The past consists of processes of experience that are completed, and what is added to an experience, besides the past as objective datum, is a moment of free, subjective and creative decision of how to experience the past. This can be understood by saying that experiences happen “at the edges of the world” (Weber 2007, p.26), in a present moment where not everything is yet completely determined and objective. Thus, according to Whitehead the world has a very interesting partwhole structure: It consists of many different essentially relational entities, which include each other in a one-directional manner that gives rise to the objective appearance of an arrow of time. Whitehead’s metaphysics is thoroughly panpsychistic, in that each and every existing entity either is or was a subjective experience. Prima facie this appears absurd, but since most of these experiences are completely non-conscious, they can be thought of as causation – see e.g. (Basile 2009). This renders the panpsychism perhaps somewhat less counter-intuitive. Another worry, put forth by Timothy Sprigge (1983: 230), concerns the status of the past – to say that the past consists of ‘expe-

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riencings’ that are no longer going on as experiencings – seems a contradiction in terms. There is no obvious rebuttal to this charge – all that can be done here is to accept the continued existence of a completed experience as a basic postulate. In the end, one has to keep in mind that the justification for such any metaphysics is to be “sought in its general success, and not in the peculiar certainty, or initial clarity, of its first principles” (PR 8). See also > Conscious experience, Emergence, Experience, Locke, Stoics, Whitehead. Bibliographical remarks

Andrae, B., 2014. Whiteheadian methodology and metaphysics in the application to the problem of intentionality in the philosophy of mind. Brüntrup, G., 2010. Discusses how the concept of an enduring person fits into the very impermanent-seeming Whiteheadian Ontology. Emmet, D., 1932. The first and most direct commentary on Whitehead’s philosophy, written by one of his students. Ferre, F., 1998. A reminder that Whitehead’s philosophy also has a crucially epistemological dimension. Jones, J., 1998. A study of why – in Whitehead’s Ontology – experiences happen in the first place. Whitehead, A. N., 1967. Sometimes called ‘Process and Reality light’,

this book is a welcome rephrasing of many of Whitehead's main ideas. References and further readings

Andrae, B., 2014, The Ontology of Intentionality, München: Philosophia Verlag. Basile, P., 2009, Leibniz, Whitehead and the Metaphysics of Causation, Palgrave Macmillan. Brüntrup, G., 2010, “3.5Dimensionalism and Survival: a Process-ontological Approach”, in: Gasser, G. (ed.), Personal Identity and Resurrection: How Do we Survive our Death? Aldershot: Ashgate. Cobb, J., 2008, Whitehead Word Book, Claremont: P&F Press. Emmet, D., 1932, Whitehead’s Philosophy of Organism, London: Macmillan. Ferre, F., 1998, Knowing and Value, Albany: State University of New York Press. Hampe, M., 1998, Alfred North Whitehead, München: C.H.Beck. Jones, J., 1998, Intensity. An Essay in Whiteheadian Ontology, Nashville and London: Vanderbilt University Press. Simons, P., 1998, “Metaphysical Systematics: A Lesson from Whitehead”, Erkenntnis 48: 377-393. Sprigge, T., 1983, The Vindication of Absolute Idealism, Edinburgh: Edinburgh University Press.

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Weber, M., 2006, Whitehead’s Pancreativism: The Basics, Heusenstamm: Ontos Verlag. Whitehead, A. N.; Russell, B., 19101913, Principia Mathematica, 3 vols, Cambridge: Cambridge University Press. Whitehead, A. N., 1929, Process and Reality: An Essay in Cosmology, abbreviated as “PR”, cited from the Corrected Edition, edited by Griffin, D. R.; Sherburn, D. W., New York: The Free Press, 1979. Whitehead, A. N., 1933, Adventure of Ideas, available as ‘New York: The Free Press, 1967’. Whitehead, A. N., 1938, Modes of Thought, abbreviated as “MT”, available as ‘New York: The Free Press, 1968’. Benjamin Andrae

World, Actual Mereology tells us about the relationship between wholes and parts. Since it is natural to think of the actual world as a whole, or totality, with parts, it is likewise natural to think that we can explicate some aspects of the actual world in mereological terms. Since the actual world is concrete, it is tempting to suppose that it is a mereological sum of concrete particulars. If a concrete particular is one that has spatio-temporal location, and if a world includes all that there is, then we can define the actual world as a maximal mereological sum of spatio-temporally connected

particulars. As it turns out, this is David Lewis’ (1986 p 69) view about possible worlds in general, of which the actual world is but one. Call this the Lewisian conception of the actual world. Though Lewis has his own views about mereology (Lewis 1991), as stated the Lewisian conception remains silent about many issues in mereology, and thus considered independently of any additional mereological commitments it leaves open a number of possibilities regarding the nature of our world. The Lewisian conception does not foreclose the issue of whether or not our world has mereological atoms: simple particulars that lack proper parts. Since the Lewisian approach is top-down, it is consistent with it that our world is one in which for every proper part of our world, that part has some further proper part. If this were true, our world would be gunky (Zimmerman 1996; Hudson 2007). Equally, it is consistent with the view that our world ultimately bottoms out in mereological atoms (Markosian 1998; 2004; Simons 2004). In turn, either of these possibilities is consistent with our world being one in which the world itself is the fundamental entity, and its parts are nonfundamental dependent entities, a view sometimes known as priority monism (Schaffer 2007) or with the view that the parts of the world are fundamental, and the world is dependent on the parts, a view known as priority pluralism (Sider 2007). The Lewisian conception is also consistent with a number of different

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views about the relationship between being a part and being an object. Read straightforwardly, the view seems to be committed to the idea that the parts of the world are themselves particulars. That is, it is committed to what Smith calls mereological actualism, the view that the proper parts of a whole are themselves particulars (Smith 1994 p 78). This is in contrast to mereological potentialism, the view that a proper part is potentially a particular, in that it would be a particular if it were to be separated from the whole of which it is a part (Smith 1994 p 78). Lewis, at least, is committed to the idea that every way that a part of a world could possibly be, is a way that some part of some world is, and that we can patch together parts of different possible worlds to yield some other possible world (Lewis 1986 pp. 8788). This is the view that the parts of a world can be pulled apart and or recombined in different ways. If we allow that this is possible, then the potentialist could read ‘particular’ as ‘potential particular’, so that for her, a world is a maximal mereological sum of potential particulars that are spatio-temporally connected. Further, the Lewisian conception remains neutral as to whether, for any two or more proper parts of our world, there is some further part of our word that has each of those as proper parts, this is the view known as unrestricted mereological composition or mereological universalism, defended by, among others, Lewis (1986 p 213) Sider (2001 p 121) and Heller (1990 p 49). Heller extends something very like Lewis’ concep-

tion by arguing that our world is a maximal mereological sum of fourdimensional volumes of space-time, where any two or more volumes of space-time have a mereological sum (Heller 1990 ch 1). This means that for any sub-volume of our world, there exists a particular (or a potential particular) that occupies that and only that region. It is clear, however, that those who defend restricted mereological composition, such as van Inwagen (1987; 1981) and Markosian (2005) could still adopt the Lewisian conception of the actual world. The Lewisian conception also remains neutral as to whether mereology is extensional. So it is consistent with the idea that two or more distinct particulars might share all the same proper parts, as for instance, three-dimensionalists about persistence contend (Johnston 1992; Lowe 1995). Equally, it is consistent with the idea that if P and P* share the same proper parts, then P and P* are identical (Noonan 1993). Finally, this characterisation is neutral with respect to whether or not objects are distinct from regions of space-time. If regions of space-time are themselves particulars, that is, if substantivalism is true (a view defended by Hoefer 1996 and Maudlin 1988) then it is an open question whether ordinary particulars like tables and chairs are identical to certain regions of space-time, or whether they are distinct particulars that occupy those regions. In general, depending on which additional mereological axioms one accepts, the Lewisian view will offer a

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different conception of the actual world. One of the few things that the view rules out is that the actual world could have been empty: that there could have been nothing rather than something (Lewis 1986 p 73). Since mereology does not allow that there are empty sums, this falls directly out of the Lewisian characterisation of a world. Though there are those who share this view (Lowe 1996 p 118; Armstrong 1989 p 93) others find it objectionable (Baldwin 1996). See also > Metaphysical Atomism, Emergence, Fusion, Mereological Essentialism, Non-Wellfounded Mereology, Sum, Temporal parts, Transitivity. Bibliographical remarks

Armstrong, D. M., 1997. This is a good introduction to a related idea, that we might think of worlds as being composed of states of affairs, but where something more than just mereology is required. Cameron, R. P., 2008. A good discussion of whether there are good arguments for the idea that the actual world must bottom out in a fundamental level.

Lewis, D., 1986. This is the locus classicus of Lewis’ view about worlds, possible and actual. References and further readings

Armstrong, D. M., 1997, A World of States of Affairs, Cambridge University Press. Baldwin, T., 1996, “There Might Be Nothing”, Analysis 56.4: 231-238. Cameron, R. P. (2008). “Turtles all the Way down: Regress, Priority and Fundamentality”, Philosophical Quarterly 58 (230): 1-14. Efird, D.; Stoneham, T. (2005). “Genuine Modal Realism and the Empty World”, European Journal of Analytic Philosophy 1(1): 21-37. Heller, M., 1990, The Ontology of Physical Objects: Four Dimensional Hunks of Matter, Cambridge: Cambridge University Press. Hoefer, C., 1996, “The Metaphysics of Space-Time Substantivalism”, The Journal of Philosophy 93(1): 5-27. Hudson, H., 2007, “Simples and Gunk”, Philosophy Compass 2 (2): 291-302. Johnston, M., 1992, “Constitution Is not Identity”, Mind 101: 89-105.

Efird, D. and Stoneham, T., 2005. Efird and Stoneham argue that the modal realist can accommodate the idea of an empty world.

Lewis, D, 1991, Parts of Classes. Oxford: Blackwell.

Hudson, H., 2007. Nice accessible overview of the topic of simples and gunk.

Lowe, E. J., 1995, “Coinciding Objects: In Defence of the Standard Account”, Analysis 55: 171-178.

Lewis, D, 1986, On the Plurality of Worlds. New York Blackwell Press.

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Markosian, N., 1998, “Simples”, The Australasian Journal of Philosophy. 76(2): 213-229.

Van Inwagen, P., 1987, “When Are Objects Parts?”, Philosophical Perspectives 1. Metaphysics: 21-47.

Markosian, N., 2004, “Simples, Stuff and Simple People”, The Monist 87: 405-428.

Varzi, A. C., 2003, “Perdurantism, Universalism and Quantifiers”, Australasian Journal of Philosophy 82 (2): 208-214.

Markosian, N., 2005, “Restricted Composition” in Hawthorne, Sider, T.; Zimmerman, D. (eds.), Contemporary Debates in Metaphysics. Basic Blackwell. Maudlin, T, 1988, “The Essence of Space-Time”, in Fine, A.; Forbes, M. (eds.), Philosophy of Science Association 1988 Volume 2. Noonan, H., 1993, “Constitution is Identity”, Mind 102: 133-146. Schaffer, J., 2007, “From nihilism to monism”, Australasian Journal of Philosophy 85(2): 175-191. Sider, T., 2001, Four-dimensionalism: An Ontology of Persistence and Time, Oxford University Press. Sider, T., 2007, “Against Monism”, Analysis 67(293): 1-7. Simons, P., 2004, “Extended Simples: A Third Way between Atoms and Gunk”, The Monist 87: 371-84. Smith, B., 1995, “More Things in Heaven and Earth”, Grazer Philosophische Studien, 50: 187-201. Smith, B., 1994, Austrian Philosophy, Chicago: Open court publishing. Van Inwagen, P., 1981, “The Doctrine of Arbitrary Undetached Parts”, Pacific Philosophical Quarterly 62: 123-137.

Zimmerman, D. W., 1996, “Could Extended Objects Be Made out of Simple Parts? An Argument for ‘Atomless Gunk’”, Philosophy and Phenomenological Research 56: 129. Kristie Miller

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Index

A priori 232, 494 synthetic 263 Abilities 41, 190, 191, 549 Absence 26, 165, 220, 222, 231, 282, 384, 439, 443, 505 Absences 505 Abstraction 50, 173, 266, 432, 520, 536, 594, 595, 596, 597 extensive 129, 173 Accident 49, 59, 121, 123, 124, 131, 215, 216, 236, 321, 430, 445, 446, 453, 483, 485, 487, 499, 530, 535 Accidental form 342, 485 Accidents individual 112 Action 28, 29, 30, 32, 69, 180, 189, 191, 208, 235, 326, 494, 572 types of 32, 34, 257, 258 Activity 31, 32, 33, 34, 35, 36, 37, 38, 42, 62, 96, 153, 157, 161, 162, 164, 165, 166, 209, 257, 259, 275, 302, 328, 401, 414, 483, 523, 574 Acts 29, 107, 121, 158, 183, 185, 250, 295, 325, 326, 330, 339, 525, 526 mental 28, 75, 186 Actuality 35, 40, 49, 51, 57, 58, 82, 108, 197, 514 Actuality vs. potentiality 514 Additivity 464 Adherence 112 Adjacency 545, 546 Adverbial clauses 294 Adverbs 294 Aesthetics 62, 65, 241, 522 digital 65 Agency 37, 180, 494, 564

Agent 29 Aggregate 22, 57, 58, 134, 163, 225, 263, 324, 331, 342, 352, 353, 380, 429, 456, 458, 464, 481, 497, 562 infinite 497 Albert the Great 488 Alexander of Aphrodisias 56, 447, 448, 449, 454, 511, 542 Algebra Boolean 114, 116, 118, 156, 291, 358, 361, 413, 436, 555, 559, 567 Heyting 567 non-degenerate Boolean 366 semi-Boolean 118 Al-Ghazali 82 Amputation 451, 452, 502 Analysis 169, 231, 360, 415, 419, 423, 456, 494 conceptual 122, 162 logical 73, 129, 460 mereological 64, 108, 110, 224, 226, 456, 457, 458, 459, 460, 496, 544, 545, 546 semantical 493 syntactic 488 Anatomy 42, 369 functional 323 topographic 323 Anaxagoras 255, 256, 564 Animal 41, 42, 43, 51, 52, 53, 54, 55, 59, 61, 83, 104, 120, 255, 323, 325, 329, 440, 445, 484, 500, 501, 562 parts of 50, 54, 57, 256 Anselm 45, 46, 47, 48, 236, 341 Antisymmetry 97, 384, 385, 572, 594 Aquinas, Thomas 192, 236, 331, 336, 338, 339, 341, 343, 344, 345, 449,

608 INDEX

453, 454, 487, 532, 533, 562, 563, 564, 565, 569, 570 Arguments a parte ad totum 278 a partibus 278 a toto ad partem 278 Aristotelianism 21, 39, 55, 56, 57, 58, 60, 70, 78, 81, 83, 84, 87, 94, 119, 120, 168, 182, 187, 198, 200, 218, 240, 255, 271, 278, 282, 297, 314, 331, 340, 342, 377, 392, 412, 434, 447, 448, 450, 451, 484, 487, 488, 507, 509, 511, 514, 518, 531, 532, 533, 535, 543, 544, 568, 569, 584, 585 Aristotle 32, 33, 35, 41, 42, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 71, 72, 78, 81, 82, 83, 98, 103, 119, 120, 131, 133, 143, 144, 168, 170, 192, 197, 198, 199, 238, 255, 256, 257, 258, 287, 313, 321, 325, 328, 329, 331, 333, 334, 335, 336, 338, 356, 370, 400, 402, 405, 409, 415, 429, 432, 433, 447, 448, 449, 450, 451, 452, 484, 485, 487, 492, 505, 512, 514, 529, 530, 531, 532, 535, 536, 540, 541, 542, 562, 563, 564 Art 51, 62, 63, 64, 65, 223, 227, 241, 339, 482, 483, 484, 486, 562, 563 Artifact 35, 68, 69, 342, 343 complex 69 dynamic 68 static 68 Artificial intelligence 152, 153, 575 Aspect 294 Aspectology 34 Assymmetry 384 Atomic number 199 Atomism 70, 71, 73, 80, 81, 84, 85, 88, 108, 146, 177, 200, 208, 218, 238,

239, 336, 385, 400, 460, 498, 504, 508, 512 existence 87 mereochronological 336 priority 87 Atomless 117, 118, 556, 597 Atoms 36, 70, 71, 72, 73, 75, 76, 81, 82, 83, 84, 86, 87, 88, 108, 117, 144, 150, 151, 173, 174, 175, 186, 198, 199, 210, 211, 225, 238, 300, 316, 317, 372, 375, 376, 394, 395, 400, 402, 413, 415, 417, 421, 422, 423, 425, 455, 464, 467, 478, 496, 531, 556, 602 formal 84 geometrical 82 material 82 mathematical 82 theory of 198 Attractor 139, 141 Attributes 76, 94, 162, 170, 183, 184, 185, 188, 216, 220, 236, 439, 483, 534, 584 Augustine 46, 236, 449, 454, 495 Automereity 34, 38, 224, 258, 259 Avicenna 483, 487, 488, 489, 543, 564 Axioms 61, 89, 90, 91, 114, 115, 179, 254, 288, 289, 290, 362, 363, 364, 373, 374, 375, 384, 385, 394, 413, 414, 415, 416, 417, 422, 515, 550, 566, 593, 595, 603 Baumgarten, A.G. 62, 63, 66 Bearer 95, 168, 191, 321, 330, 490, 530, 531, 534, 535, 537 Bergmann, G. 75, 77, 80, 93, 94, 95, 171, 249, 504, 536, 537, 579, 585 Bertalanffi, L.von 322, 327, 330 Biology 43, 45, 96, 97, 99, 138, 164, 200, 292, 315, 327, 329, 549 Blending 510

INDEX 609

Body 23, 29, 30, 39, 42, 44, 53, 54, 55, 58, 59, 61, 64, 70, 71, 72, 83, 84, 87, 97, 99, 100, 101, 102, 103, 104, 110, 111, 120, 131, 155, 156, 159, 162, 166, 183, 186, 187, 188, 189, 197, 198, 199, 211, 226, 253, 257, 258, 289, 299, 300, 303, 305, 315, 316, 317, 318, 319, 321, 323, 325, 326, 383, 430, 439, 440, 441, 449, 477, 485, 488, 500, 502, 510, 525, 531, 535, 537, 558, 563, 572, 586 contiguous 58 continuous 58 cross-cultural conception of the 99 dynamic representation of 99 functional parts 317 segmental parts of 100 segmentation of 99 semantic-conceptual representation 99 structural representation of 99 systemic parts of 100 Boethius 21, 23, 41, 46, 48, 103, 104, 105, 106, 236, 278, 282, 285, 286, 337, 338, 339, 341, 343, 344, 446, 447, 481, 482, 485, 540, 543, 563, 565, 568, 569, 570, 577 Boolean lattice 365 Bosanquet, B 77 Boundaries 50, 58, 123, 170, 246, 257, 259, 315, 316, 319, 321, 322, 323, 325, 326, 355, 356, 358, 361, 391, 397, 547, 576, 596 Bradley, F.H. 77, 238, 239, 403, 408, 584 Bradley’s regress 76, 579 Brain 26, 60, 154, 155, 160, 161, 162, 163, 164, 165, 166, 192, 209, 210, 211, 315, 316, 318, 319, 321, 322, 323, 324, 325, 327, 328, 329, 330, 341, 401, 408, 558

Brentano School 275 Brentano, F. 30, 57, 61, 119, 120, 121, 122, 123, 124, 125, 132, 133, 170, 171, 231, 234, 262, 268, 271, 275, 276, 287, 325, 345, 347, 349, 405, 406, 522, 524, 525, 543, 556, 561, 562, 571, 584, 585, 586, 588 Broad, C.D. 31, 38, 202, 203, 205, 503, 504 Bundle 51, 93, 94, 217, 218, 467, 525, 537, 580, 585 Bundles of tropes 380, 579 Burgess 413, 419, 420, 421, 423, 424 Burleigh, W. 488, 489 Cajetan 453, 454 Cancer 315, 319, 320 Capacities 161, 303, 447, 448, 453, 470, 485, 528, 531, 563 Cardinals 420 Carnap, R. 32, 75, 80, 128, 129, 130, 233, 234, 240, 241, 460, 519, 520, 521, 567, 575 Catachresis 493 Categorematic 40, 338 Categorial frameworks 57 Categories 22, 32, 35, 36, 37, 41, 43, 57, 59, 75, 93, 94, 95, 102, 120, 122, 131, 132, 133, 143, 150, 152, 153, 154, 156, 162, 168, 170, 179, 219, 220, 224, 229, 233, 243, 244, 249, 250, 258, 259, 260, 278, 293, 294, 295, 296, 297, 316, 323, 340, 402, 437, 483, 494, 500, 516, 517, 520, 521, 529, 531, 532, 533, 535, 544, 551, 566, 575, 577, 580, 590 exclusivity of 132 exhaustivity of 132 functional modifier 296 Categorisation

610 INDEX

prototype theories of 494 Category, mathematical 423 Causal closure 163, 203 Causality 26, 29, 30, 52, 82, 88, 96, 97, 98, 133, 134, 135, 136, 161, 163, 164, 169, 188, 189, 190, 194, 201, 203, 204, 210, 216, 260, 272, 316, 317, 325, 376, 428, 429, 462, 466, 470, 488, 504, 505, 506, 514 downward 203 negative 134 Cause common 506, 507 efficient 60, 84, 483 final 55, 60, 483 formal 60, 84 material 60, 83, 84 Cells 43, 96, 191, 209, 210, 227, 246, 247, 316, 319, 321, 375 Chalmers, D. 89, 161, 163, 166, 210, 211, 212 Chamber, R. 43 Change 23, 24, 28, 33, 36, 43, 57, 64, 68, 70, 82, 83, 84, 102, 139, 141, 148, 149, 156, 163, 169, 170, 188, 193, 277, 299, 319, 322, 325, 328, 341, 351, 376, 409, 410, 476, 477, 506, 517, 530, 531, 533, 535, 536, 539, 552, 553 intrinsic 553 mereological 341 Chaos deterministic 204 theory 138 theory 139 theory 223 Chatton, W. 82, 85 Chisholm, R. 40, 123, 124, 125, 179, 180, 181, 332, 333, 337, 349, 350, 351, 391, 392, 393, 406, 431, 561

Choice, axiom of 291, 436, 477, 549, 550 Cicero 103, 104, 105, 278, 282, 492, 493, 495 Circularity 145, 398, 411, 419, 497 Class form 504 Classes 80, 178, 225, 230, 233, 234, 249, 292, 337, 395, 424, 435, 539, 561, 568, 596, 604 Classical Extensional Mereology 27, 28, 37, 96, 228, 229, 230, 233, 381, 402, 430, 456, 553, 571, 572 Classical First Order Mereology 115 Classification 33, 34, 118, 119, 131, 153, 154, 156, 157, 159, 244, 258, 277, 282, 315, 329, 339, 370, 535, 571 basic-level 156 of part-whole relations 156 Closure conditions 414 Cognition 131, 166, 251, 295, 494, 575 spatial 99 Cognitive models 494 Cognitive science 99, 430, 495, 575 dynamical 166 Coincidence 23, 53, 148, 149, 312, 313, 346, 488, 552, 553, 570 material 147 Collection 22, 36, 65, 93, 94, 96, 101, 102, 105, 106, 107, 108, 109, 110, 111, 112, 123, 145, 150, 151, 153, 193, 205, 211, 225, 245, 301, 305, 342, 352, 380, 381, 384, 402, 429, 441, 443, 444, 454, 478, 497, 503, 521, 574, 575, 582, 584, 600 cross-categorical 108 mind-independent 108 Combination problem 161, 210, 211 Combinatorial method 484 Combinatorics 482, 483, 484, 486 Common-sense reasoning 152, 155, 158

INDEX 611

Compatibilism 411 Complement absolute 114 relative 116 Completeness 32, 90, 94, 136, 154, 220, 221, 417, 484 Complexes 77, 93, 94, 150, 151, 169, 215, 231, 232, 249, 250, 251, 308, 310, 316, 317, 345, 346, 347, 455, 503, 504, 537, 582 perceptual 231 Complexions 122, 307, 308, 309, 310, 311 Complexity 69, 102, 158, 188, 278, 305, 318, 458, 497, 586 infinite 497 Composite 21, 40, 76, 107, 147, 148, 298, 307, 312, 338, 342, 343, 353, 463, 464, 465, 466, 470, 473, 477, 499, 501, 513, 532, 597 Composition 21, 22, 48, 59, 60, 61, 62, 76, 87, 109, 143, 147, 148, 149, 151, 199, 226, 227, 228, 229, 230, 297, 298, 299, 301, 302, 303, 304, 305, 328, 380, 381, 402, 414, 432, 433, 434, 444, 457, 461, 471, 472, 494, 506, 513, 517, 518, 589, 599, 603, 605 problem of 59 relations 432 unique 229 unrestricted 385, 513 Composition, principles of 148 compositionality principle 461, 465, 471 Compositionality principle 226 Compounds 52, 143 matter/form 514 Computer science 180, 567 Concepts 26, 27, 32, 35, 36, 37, 40, 48, 58, 64, 77, 107, 108, 111, 113, 122, 129, 131, 132, 133, 142, 171, 172,

173, 174, 175, 180, 194, 210, 225, 226, 227, 259, 271, 288, 289, 291, 297, 323, 329, 355, 356, 358, 359, 397, 398, 401, 419, 430, 434, 439, 455, 457, 463, 483, 484, 487, 490, 492, 493, 494, 496, 497, 501, 504, 518, 528, 550, 565, 575, 589, 594, 596, 597 Conceptualization 152, 155, 156 Concretion, problem of 520 Concursus Dei 183, 190 Connectionism 201 Connectives 455, 457, 459, 487 Consciousness 121, 160, 161, 162, 163, 164, 165, 166, 167, 168, 185, 211, 273, 305, 318, 322, 324, 325, 326, 327, 330, 401, 402, 406, 504, 525, 526, 527 content of 165 Consecutive 58 Consequentialism 206 Constant companionship difficulty 520 Constituency 242, 245, 426, 546 Constituents 51, 53, 55, 60, 71, 73, 75, 76, 93, 123, 134, 214, 220, 225, 245, 249, 288, 299, 300, 346, 347, 376, 391, 442, 455, 459, 464, 465, 469, 471, 472, 497, 503, 510, 516, 531, 547, 579, 581, 582, 583, 589, 599 determined 346 undetermined 346 Constitution material 23, 311, 312, 313, 341, 564, 570 Constitution as identity 147, 313 Constitution theory 128, 129 Constructional system 240 Contemplation, aesthetic 65

612 INDEX

Contents 50, 123, 165, 184, 185, 232, 263, 266, 382, 402, 428, 523, 524, 527, 528, 595 independent 524 partial 524 Context, principle of 226 Contextualism 457 Contextuality global spacetimematter 470 Contiguity 58, 101, 359, 464 Contiguity, principle of 464 Contiguous 58, 100, 253, 359, 511, 546 Continuant 36, 168, 169, 170, 322 Continuity, space-time 418 Continuos psychologically 527, 528 Continuous 21, 22, 51, 52, 58, 59, 83, 99, 103, 104, 119, 123, 142, 183, 193, 273, 298, 300, 355, 357, 360, 373, 374, 400, 405, 417, 451, 511, 516, 563, 566, 596, 597 Continuous maps 566 Continuum 57, 58, 59, 81, 82, 90, 123, 174, 175, 177, 186, 346, 415, 464, 527, 593 Convergence 565 Copyright theory 430 Cosmic epochs 177 Cosmos 86, 87, 88, 171, 175, 177, 510, 511 Count aspect 36 Coupling 138 Creation 22, 183 continuous 82 Dalton 198, 477, 478, 479 Dancy, J 208, 209 Danto, A. 64, 65, 66 Darwin, C. 43, 44 Database ontology 575

Davidson, D. 28, 29, 30, 133, 136 Decidability 89, 118, 551 Decomposition 110, 147, 164, 198, 226, 297, 456, 496, 557, 587 logical 227 multiple 413 uniqueness 413 Definite descriptions 47, 74, 79, 582 Definiteness 294, 297, 420 Democritus 70, 73, 85, 86, 87, 400 Deontologism 206, 207, 208 Dependence 184, 185, 188, 189, 217, 244, 245, 345, 391, 547 essential 389, 390, 391, 392 existential 389, 390 Descartes, R. 41, 42, 43, 44, 119, 182, 183, 184, 185, 186, 187, 188, 189, 190, 211, 514, 526, 532, 533 Desideratives 345 Designation plural 108 singular 108 Destruction 35, 61, 222, 319, 341, 445, 481, 562, 593 Detachability 121 Determination 99, 122, 136, 141, 280, 308, 309, 310, 311, 405, 462 relation 462 synchronic 201, 202 Developments 34, 35, 36, 37, 46, 181, 522, 596, 597 Dialectical School 71 Diatoms 421, 422 Difference 114 Differentiae 21, 104, 120, 445 Differential equation 193 Dignitatives 345 Dimension ontological 359 Diodorus Cronus 71, 73

INDEX 613

Diseases 315, 319, 320, 322, 323, 324, 325, 326, 327 Disjointness 114, 555, 576 Disjunctivism 211 Dispositions 28, 190, 191, 192, 202, 259, 442 Dissectivity 34, 257, 258 Divisibility infinitely 71, 81, 174, 506 Division 51, 59, 132, 225, 243, 266, 277, 339, 428, 429, 473 Dualism 163, 182, 259 Duration 33, 34, 168, 169, 238, 298, 300, 301, 332, 336, 524 Duties 180, 208, 492 Dynamical microexplanation 471 Dynamical systems theory 164, 193, 194, 195, 201 Dynamics chaotic 139 deterministic 138 non-chaotic 138 Effects 29, 30, 97, 156, 162, 184, 191, 315, 407, 470, 505, 506, 507 resultant vs. emergent 202 Einstein, A. 172, 463, 467, 468, 474 Elements 70, 87, 118, 143, 144, 145, 151, 175, 188, 197, 198, 199, 200, 255, 256, 257, 288, 298, 322, 425, 477 chemical 198, 199 Emergence 37, 64, 71, 73, 161, 167, 194, 200, 201, 202, 203, 204, 205, 463, 470, 570 diachronic 204 mereological 161, 462 modest kind of 202 radical 202 strong 161 strong synchronic 203

synchronic 201 Emergentism, British 202 Empedocles 197 Endurantism 386, 410 Energeiai 32, 35 Entailment logical 26, 457, 458 Entanglement 161, 461, 462, 463, 464, 466, 467, 468, 470, 471, 472, 473, 474, 478, 479 Entia rationis 453 Epicureans 72, 512 Episodes, mental 28, 29 Epistemology 57, 106, 128, 298 EPR-correlations 463, 465, 467 Equivalence logical 458 Erlebs 519, 520 Essence 53, 60, 182, 183, 236, 243, 263, 264, 266, 267, 300, 302, 303, 305, 389, 420, 445, 446, 447, 481, 483, 490, 491, 504, 514, 529, 530, 531, 532, 533, 563, 569, 580 law of 490 sameness in 22 Essentialism mereochronological 332, 334, 335, 337 mereological 40, 105, 124, 304, 332, 333, 334, 337, 349, 350, 351, 430, 452, 562 Eternalism 172 Eudaimonia 237 Events 28, 29, 35, 36, 38, 82, 97, 133, 134, 135, 136, 157, 158, 168, 174, 190, 209, 210, 219, 223, 229, 239, 241, 246, 259, 261, 291, 295, 320, 330, 371, 437, 442, 462, 466, 470, 489, 529, 552, 595, 596, 597

614 INDEX

Evolution 43, 138, 139, 193, 194, 204, 324, 329, 464, 470, 471, 472 novelties in 204 Exemplification 29, 76, 94, 275, 580, 582, 585, 590 Existence 41, 51, 54, 74, 77, 78, 79, 86, 87, 88, 105, 112, 115, 121, 122, 123, 124, 134, 152, 182, 183, 184, 185, 201, 211, 228, 235, 253, 263, 295, 298, 300, 321, 330, 331, 333, 334, 335, 350, 352, 362, 364, 391, 410, 413, 429, 433, 448, 464, 481, 482, 498, 505,507, 530, 532, 535, 536, 552, 553, 566, 580, 581 objective 184 Experience conscious 26, 160, 161, 162, 163, 164, 165, 166, 209, 210, 211 elementary 128, 129, 519 spatial parts of human 209 Extended-phenomenological-cognitive systems 166 Extension 50, 87, 115, 141, 175, 176, 183, 184, 186, 188, 219, 263, 280, 298, 300, 301, 397, 490, 491, 511, 524, 526, 533, 559, 594 theory of 173, 595 Extensionality 114, 387, 477 Strong Principle of 147 Strong Principle of 147 Time-independent Principle of 148 Weak Principle of 147 Extensionality axiom counterexample 313 Extensive connection 354 Extensive quantity 497 Externalism phenomenal 211 Facts

atomic 76, 77, 79, 80, 215, 582 general 78, 80, 215, 216 mental 210 negative 78, 79, 80, 216, 505 non-actual 79 scientific 210 Factual material 491 Fallacy 49, 93, 487, 488 mereological 162 Feelings 210, 527 Fictional objects 219, 220 texts 219 Fictions 121, 123, 309, 333, 504, 560 Field theory 461, 464, 469, 473, 474 Field, H. 418, 419 Fields scalar 418 vector 418 Figurative use of language 382 Filters 436 maximal 436 Finitism 72, 82, 241, 412 First-person 160 Formal ontology 232, 273, 354, 355, 359, 491 Formalism 412, 414 Foundation, relation of 264, 392 Foundational closure 356 Four-dimensionalism 349, 410, 411 Fractal 139, 223, 224 Fractions 441, 497 Free ontological lunches 76 Freud, S. 325 Function 62, 93, 94, 117, 176, 224, 278, 293, 295, 317, 318, 328, 362, 418, 419, 422, 436, 463, 471, 491, 528, 566, 574 Hamiltonian 461 recursive iterated 223

INDEX 615

representational 62 Fusion 76, 114, 115, 116, 118, 132, 143, 198, 227, 228, 229, 231, 232, 357, 362, 413, 421, 422, 461, 464, 468, 470, 471, 510, 511, 555, 558, 559, 560, 561, 572 axiom 228 mere 228 unique 229, 553, 555, 558, 559 unrestricted 228, 229, 553, 555 Galen 42, 44, 540 General relativity 175 Genetics 43 Genitives 294 Genus 21, 50, 51, 52, 53, 59, 103, 104, 105, 120, 122, 256, 263, 278, 280, 281, 285, 338, 339, 353, 445, 446, 484, 485, 492, 493, 530, 532, 574 division of the 104 Genus synecdoche 493 Geometry 72, 173, 291, 355, 360, 367, 371, 412, 413, 416, 417, 418, 423, 426, 435, 437, 462, 500, 507, 550, 551, 565, 577, 593, 594, 596 Euclidean 72, 415, 417 Euclidian solid 550 fractal 223 point free 415 point-based hyperbolic plane 417 pointless 435 relational 435 synthetic 417 two-dimensional spherical 417 Gerard of Odo 84 Gestalt 62, 66, 100, 208, 229, 231, 232, 234, 235, 239, 241, 249, 250, 251, 276, 320, 321, 347, 348, 349, 377, 405, 408, 495, 504, 509, 517, 518, 519, 521, 522, 524

Gestalt principles of grouping 100 Gestalt qualities 231 Glosses 31, 277 God 22, 25, 46, 47, 48, 82, 83, 112, 182, 183, 184, 185, 188, 214, 235, 236, 237, 341, 350, 483, 499, 502, 510, 532, 562, 580 Gödel, K. 419, 497, 498, 549 Goethe, J. von 43, 524 Golden Ratio 63 Goodman, N. 34, 94, 129, 214, 232, 233, 240, 241, 257, 291, 318, 362, 393, 394, 412, 414, 415, 422, 512, 519, 520, 521, 554, 571 Grammar 25, 74, 242, 244, 245, 258, 383, 441, 442, 546, 577 structural-functional theories of 293 Granularity levels 150, 245, 247, 315, 460 Granules 245, 246, 247 Graph theory 164, 468 Grounds of truth 75, 213, 218, 581 Groups 42, 107, 155, 156, 198, 199, 223, 426, 432, 515, 526, 528, 551 Guise 320, 321 Guise theory 320 Gunk 88, 173, 174, 175, 383, 511, 556, 602 Gunky 417 Haecceitas 134, 467, 517 Harvey, W. 42, 44 Head noun 294, 295 Heap 57, 58, 231, 273, 319, 352, 353, 397, 428, 429, 448, 450, 531, 562 Hedonism 237, 238 Henry of Harclay 82 Henry, D.P. 21, 23, 24, 25, 45, 46, 47, 56, 61, 82, 85, 105, 106, 189, 233, 237, 240, 332, 333, 334, 337, 338,

616 INDEX

340, 341, 344, 447, 519, 543, 564, 565, 569, 570 Hierarchy 59, 63, 100, 280, 303, 347, 381, 398, 426, 427, 470, 550 Hilbert space 463, 472 Holism 194, 208, 239, 378, 474, 479, 509, 599 about reasons 208 quantum 479 Holon 57, 58, 402 Homeomereity 33 Homeomorphisms 355 Homoeomeries 143, 256 Homogeneity 255, 256, 257, 260 Homomorphism 117, 419, 515 Homonymy 54 Homonymy Principle 514 Homotopy theory 356 Husserl, E. 63, 95, 120, 122, 124, 131, 133, 231, 232, 234, 262, 263, 264, 265, 266, 267, 268, 269, 271, 275, 276, 287, 348, 351, 356, 357, 359, 360, 380, 391, 392, 393, 395, 399, 405, 428, 431, 490, 491, 503, 504, 512, 517, 518, 522, 524, 526, 528, 571, 577, 588 Hylemorphism 82, 186 Hylomorphism 511, 530 Hypokeimenon 529, 530, 534, 535 Hypostasis 529 Ideal 116 Identity modal 228, 349 numerical 22, 343, 515, 516 of quantum objects 476 personal 23, 298, 527 principle of 405, 571 Image schemas 494 Imagination 184, 347, 495, 523

Immaterial being 253 Imperfect community dificulty 520, 521 Inactuality 334 Inclusion 97, 98, 129, 132, 250, 279, 280, 282, 283, 285, 353, 394, 497, 571, 594 Incommensurability 72 Incompleteness problem 219, 220 Independence 65, 77, 98, 134, 262, 264, 271, 322, 429, 464, 524, 532, 586 Indiscernibility 476, 478 Indiscernibility, principle of 410, 476 Individual 25, 35, 36, 44, 69, 94, 95, 97, 105, 138, 233, 258, 259, 291, 326, 327, 329, 339, 366, 369, 374, 375, 380, 393, 394, 395, 413, 414, 421, 422, 433, 446, 451, 467, 468, 508, 509, 513, 520, 528, 538, 572, 575, 582, 583, 590, 596 biological 35 non-particular 35, 259 Individualism vs. collectivism 508 Individuation 95, 134, 136, 370, 401, 478, 513 Individuator 95 Indivisibility 51, 59, 70, 71, 81, 82, 83, 84, 86, 162, 185, 400, 532 Induction, mathematical 420 Inesse 123 Inessentialism mereological 350 Inference-rules 89, 90, 516 Inferential intuitions 152 Inferiora 307, 310, 345, 346, 347 Infinitesimal 172, 173, 174, 415 Infinity 106, 107, 112, 175, 224 Infinity, Axiom of 422 Information systems 573, 575, 578 Information theories integrated 162

INDEX 617

Ingarden, R. 63, 65, 67, 131, 133, 262, 268, 271, 272, 273, 274, 526, 528 Ingredients 22, 193, 198, 199, 361, 415, 485, 511 Inherence 59, 184, 410, 411, 535 Instantiation 33, 76, 170, 379, 380, 537, 579, 584, 585, 590 Instants 81, 83 Intellectual property 430 Intelligence 42, 563 Intensional unity 456 Intentionality 28, 31, 79, 122, 251, 274, 275, 405, 504, 525 Intentions 29, 186 Interactionism in biology 97 Intersection mereological 114 Interspersion 510 Interval 33, 34, 36, 138, 139, 142, 145, 157, 384, 417, 462, 559, 600 Intervals, open 566 Intransitivity 570 Involvement rigid essential 389 Irrealia 123 Irreducibility 201, 203, 208, 285, 340, 442, 509 Islamic philosophers 82, 488 Isomorphism 164, 165, 417, 419, 420, 515, 567 Ithaca interpretation 468 Jungius, J. 278, 286, 499 Junk 385 Jurisprudence 277, 278, 282, 284, 285 Juxtaposition 76, 510 Kant, I. 42, 43, 44, 65, 67, 120, 130, 131, 132, 169, 208, 533, 534, 536, 599

Kenny, A. 32, 33, 38, 84, 183, 190, 447 Kim, J. 31, 86, 88, 134, 136, 137, 202, 205, 206, 211, 212, 463, 475 Kinēseis 32, 35 Kripke, S. 179, 210, 212, 391, 392 Lamarck, B. 43 Language ideal 74 metalanguage 550 object 550 perspicuous 74, 75 Lattice 116, 117, 118, 365, 366, 551 factor 117 locally distributive 116 Lavoisier 143, 198, 200 Laws 95, 136, 142, 144, 149, 161, 188, 204, 216, 277, 278, 279, 284, 285, 442, 462, 464, 470, 471, 472, 504, 515, 517, 582 heteropathic 202 homopathic 202 Layers 69, 293, 294, 295, 296, 546 Leibniz, G. 57, 87, 88, 106, 119, 132, 189, 211, 278, 330, 354, 410, 411, 415, 449, 453, 454, 484, 485, 486, 487, 489, 496, 498, 499, 501, 552, 584, 599, 601 Leonard, H. 94, 233, 240, 241, 242, 291, 362, 366, 381, 386, 512, 518, 519, 522, 554, 561, 571 Leśniewski, S. 48, 172, 174, 228, 232, 234, 240, 287, 288, 289, 290, 291, 292, 293, 361, 362, 380, 381, 398, 399, 402, 512, 541, 542, 543, 544, 549, 551, 554, 561, 571, 588, 593, 596, 597 Levels 37, 38, 41, 99, 100, 144, 175, 224, 245, 246, 247, 266, 272, 296,

618 INDEX

297, 315, 317, 318, 426, 441, 443, 470, 501, 576 Lewis, C.I. 240 Lewis, D. 169, 174, 199, 223, 229, 253, 331, 332, 409, 420, 422, 462, 513, 555, 556, 567, 602, 603 Life 42, 43, 237, 304, 329 artificial 201 human 238 Linking relation 394 Living beings 329, 350, 530, 532, 599 Locality principle 462 Location 25, 60, 83, 187, 245, 247, 253, 258, 259, 260, 261, 295, 300, 312, 342, 403, 411, 412, 474, 478, 511, 529, 535, 564, 602 Locology 358, 359 Logic 73, 240 ancient 103 deontic 307 modal 179, 180, 413, 420, 422, 503 non-classical 398 of fiction 222 plural 413 predicate 26, 74, 77, 90, 91, 264, 496, 540 second-order monadic 418 standard deontic 179 Logical analysis 497 Logical consequence 458, 550, 551 Logical form 79, 456, 459, 515, 581 Logical type 538 Logicism 74 Loops 383 Łukasiewicz, J. 287, 540, 541, 542, 543, 549 Lustre 403, 404, 408 Mackie, J.L. 134, 135, 136, 137, 236, 237 Magnitudes 71, 72, 112, 418

Maimonides 82, 453 Mass 36 Mass production 68, 69 Mass terms 52, 257, 259 Matter 31, 50, 52, 53, 54, 57, 59, 60, 61, 78, 86, 104, 108, 110, 141, 142, 143, 149, 151, 161, 172, 173, 174, 175, 186, 187, 188, 189, 209, 255, 257, 271, 272, 299, 339, 341, 343, 350, 351, 391, 450, 461, 462, 469, 484, 485, 530, 531, 532, 535, 536, 555, 556, 557, 564 Matter and form 49, 53, 55, 59, 60, 104, 105, 236, 271, 353, 377, 448, 485, 501, 512, 514, 532 Matter, prime 78, 82, 83, 84, 499, 533, 536 Meanings 201, 244, 293, 438, 529, 599 Mechanics, synthetic 413 Mechanisms 42, 161, 164, 165, 401 Medieval logical curriculum 103 Megethology 413, 422, 423 Meinong, A. 120, 122, 124, 179, 180, 232, 249, 251, 275, 276, 307, 308, 310, 311, 345, 346, 347, 348, 349, 406, 503, 504, 579, 580, 584, 585 Melissus 400 Mendeleev 144, 198, 200 Mental phenomena 121, 122, 324, 325 Mental spaces 494 Mental states 164, 274, 275 Mereological continents 132 Mereological maxima 132 Mereological minima 132 Mereologies abstract 26 applied 68, 241, 292, 371, 372, 373, 375, 378 Mereology basic 114

INDEX 619

Boolean 567 minimal 114 Minimal Extensional 363 non-overlapping 563 structure-based 514, 517 Mereotopology 171, 172, 174, 247, 354, 355, 356, 357, 358, 359, 360, 437, 597 metaphor 163, 383, 494 Metaphor 382 Microbiology 43 Microexplanation diachronic 472 synchronic 472 Micro-reduction 194 Microstructure 202, 203 Mill, J.S. 29, 133, 134, 135, 137, 202, 206, 464 Mimesis 62, 63 Mind 29, 93, 161, 162, 183, 187, 189, 194, 209, 302, 325, 510 parts of the 186 Mind-immanent 275 Minima naturalia 83 Minkowski 410, 473 Mixture 142, 200 separable 198 theory of 197 Modalities 179, 180, 311, 556 deontic 179 formal systems of 179 Modes of being 131, 185, 275 Modes of division 103 Modifiers 294, 295, 296, 297, 376, 516, 573 Moments 83, 131, 231, 232, 233, 262, 263, 266, 267, 332, 335, 356, 357, 380, 392, 428, 490, 491 of moments 267 Monism 77, 86, 88, 161, 172, 189, 201, 400, 602, 605

existence 88 neutral 161, 166 physical 201 priority 88 Mono-sortality 150 Moods 322, 325 Moore, G.E. 73, 74, 75, 77, 78, 79, 80, 81, 207, 208, 209, 214, 215, 218, 249, 460, 503, 504, 580, 585 Morphemes 243, 425 Morphosyntax 296 Motion 42, 72, 84, 85, 188, 195, 299, 508 Mourelatos 33, 36, 38, 255, 256, 258, 261, 400 Multi-location 259, 380, 385, 410 in time 410 Multiplicity 121, 123, 217 indefinide 123 infinite 123 Multi-sortality 150 Multiverse 176 Natural kinds 58, 353, 374, 501 Naturalism 37, 171 Nead noun 438 Necessitation generic 389 rigid 389 rigid essential 389 Neighbourhood 420, 565 Neuroscience 163, 164, 165, 166, 325 Newton, I. 44, 85, 86, 89, 187, 299, 306 Nexus 93, 94, 503 Nicholas of Autrecourt 84, 85 Nihilism mereological 86, 87 Nominalism 214, 215, 217, 218, 233, 241, 291, 297, 379, 380, 413, 450, 521 bit 380

620 INDEX

class 380 in mathematics 414 moderate 215, 580 Non-additivity 464 Non-existence 134, 333, 449, 451 Non-locality 465, 466 Non-separability 466, 467, 470, 471 Non-transitivity 37, 376, 431, 527, 570, 574, 575, 576 Non-wellfoundedness 383, 384, 385 Noun phrases 257, 293, 294, 295, 296, 297, 438, 439, 440, 441, 539, 572 Now 332, 335, 336, 456, 457, 466, 552 Numerical Rigidity vs. Flexibility 150 Object aesthetical 63 complex 228, 229, 319, 504 dependent 232 determinant 308, 310 higher-order 307, 310 of experience 210 quantum 476, 477, 479 single unified 512 spatial 354 Objecta 307, 308, 309, 310, 311 Occasionalism 82, 189 Occurrences 28, 31, 32, 37, 157, 411, 574 Occurrents 36 Ockham, W. 25, 84, 331, 335, 336, 338, 341, 343, 344, 448, 450, 454, 455, 488, 489, 540, 543 Olivi, P.J. 331, 336, 337 One per se 58 Ontological dependence 183, 390, 391, 392, 498, 535, 537 Ontological parsimony 133, 537 Operator 133, 179, 355, 356, 357, 358, 364, 470, 471, 472, 560, 582

modal 179 Optimality Theory 426, 427, 428 Order definite 51 relational 504 Ordered pairs 393, 414, 421 Ordinals 420 Organic unities, principle of 207 Organisations 508 Organism 43, 256, 298, 304, 315, 318, 319, 321, 322, 323, 327, 328, 329, 330, 375, 440, 449, 531 hierarchical structure 315 Organs 41, 55 Ousia 168, 529, 530 Overlap 22, 40, 50, 114, 115, 129, 132, 246, 259, 323, 340, 343, 355, 359, 362, 369, 384, 421, 525, 527, 538, 539, 552, 555, 559, 564, 595, 596, 597 Panprotopsychism 161 Panpsychism 161, 210, 211, 600 Paradox Hausdorff-Banach-Tarski 549 of multi-location 260 Russell’s 74, 287, 288, 290, 398, 399, 550 set-theoretic 398 Zeno’s 502, 503 Paradoxe sorites 397 Parameters 137, 138, 194, 402 Parfit, D. 527, 528 Parmenides 26, 28, 52, 70, 71, 72, 170, 399, 400, 432, 433, 434, 435, 512 Part accidental 40 antecedent to the wholes 60 autarkic 322 biological 96, 97

INDEX 621

canonically necessary 155 configurational 155 dependent 263, 264, 265, 267, 526, 575 detachable 120 distinctional 122 encapsulated 155 ex-changeable 155 formal 53, 55, 60, 586, 587 functional 52, 155, 260, 319, 323, 375, 376, 573, 574 homeomerous 155 homogeneous 51, 61, 143, 154, 155, 282, 283, 300, 301, 305, 562 improper temporal 332 independent 264, 265, 405, 428, 490 intensional 485 interdependent 429 interior 355 intermediate 266 labeled 101 logical 26, 27, 120, 121, 122, 123 mandatory 155 material 52, 53, 54, 60, 61, 311, 312, 313, 342, 512, 531, 587 mediate 266, 267, 268 metaphysical 121, 132, 262, 586 mutually pervading 122 non-spatial 195 non-uniform 61 of a definition 53 of animals 54 of organs 316, 321 of sentences 456 of the person 526 of the species 53 physical 120 posterior to the wholes 60 potential 40, 50, 51, 55, 273, 563, 569

proper 315 psychological 523 removable 155 segmental 155 separable 155, 441 separated 441 shadow 506, 507 shareable 155 simultaneous to the wholes 60 standardized 69 structural relations between 433 substantial 339, 343 temporal 25, 26, 33, 37, 147, 148, 169, 170, 229, 259, 260, 320, 322, 331, 332, 333, 335, 336, 337, 346, 349, 371, 373, 391, 410, 411, 506, 519, 551, 552, 553, 554, 589, 590 ultimate 174, 266, 272, 510 uniform 61 Part relation approximate 247 atemporal 552 determinate 150 general 573, 576 granular 150 material 554, 556, 577 non-extensional 434 Parthood-at-t 552 Participation 379, 434, 514 Particle 71, 84, 86, 145, 147, 148, 149, 150, 161, 176, 216, 303, 320, 321, 350, 395, 442, 461, 462, 463, 465, 466, 467, 468, 469, 472, 473, 474, 476, 477, 478, 558 Particular 35, 95, 503 abstract 379, 520, 530 bare 95, 536 ordinary 579, 603 perfect 579 Particularism 208

622 INDEX

Partition 38, 102, 155, 246, 247, 576, 595, 596 Partonomic systems 545 Partonomy 102, 546 Part-part relation 244 Part-similarity 520 Peano-Dedekind axioms 419 Perdurance 169, 170, 410, 552 Persistence 23, 24, 35, 147, 148, 149, 151, 169, 259, 341, 349, 407, 409, 410, 411, 537, 552, 553, 603, 605 Person 23, 45, 64, 100, 110, 151, 152, 154, 155, 156, 162, 175, 207, 208, 211, 235, 237, 239, 284, 294, 303, 322, 326, 327, 346, 347, 397, 439, 440, 442, 450, 452, 526, 527, 528, 558, 601 Personality 222, 525 Personhood 526 Phalén, A. 502 Phase space 193, 194, 195 Phases 69, 123, 141, 142, 144, 322 Phenomenology 163, 166, 210, 262, 287, 406, 489, 503, 522 Philosophy of art 240 Philosophy of science 240 Phonology 287, 296, 383, 424, 425, 427 Physical domain principle 462 Physicalism 163, 204, 211 Piece/part distinction 430 Pieces 125, 262, 267, 367, 431, 490, 525, 533 absolute 267 moments of 267 of moments 267 of pieces 267 relative 267 Piece-whole relation 429, 430 Places 95, 102, 230, 234, 249, 360, 371, 378, 561, 577

Plants 23, 41, 43, 104, 256, 304, 450 Plato 26, 49, 52, 55, 62, 63, 71, 215, 216, 225, 325, 340, 432, 433, 434, 435, 445, 456, 512, 514, 518, 529, 530, 540, 543, 599 Platonism 183, 218, 423 Pleasure 207, 208, 237, 238, 239 Plenum principle 172 Plotinus 87, 89, 529 Pluralism 233, 237, 238, 423, 574, 575, 602 Pluralities 22 Pneuma 510, 511 Poetic language 493, 494 Points 57, 81, 82, 83, 94, 170, 172, 173, 174, 175, 193, 354, 358, 359, 360, 412, 415, 416, 417, 418, 435, 436, 462, 465, 473, 567, 593, 596 Points of space 86 Politics 42 Porphyry 285, 445, 446, 447 Portions 50, 51, 55, 70, 153, 266, 315, 316, 370, 557 Possession 101, 297, 438, 441, 442 alienable 439, 440 external 440 inalienable 439, 440 Possessives 295, 571 Possessors 439, 440 Possibility 43, 76, 78, 79, 88, 122, 133, 145, 176, 177, 189, 210, 232, 296, 302, 315, 350, 385, 423, 444, 496, 523, 532 Possible worlds 229, 455, 602, 603 Potency 51, 57, 58, 120, 191, 536 Potentiality 35, 41, 49, 57, 82, 186, 506 Power set 413, 436 Power Sets, Axiom 422 Powers active 191

INDEX 623

component 442, 443, 444 composition of 444 fundamental 443 macroscopic 442 microscopic 442 passive 191 Predicables 389, 535 division of 445 Predicate 26, 33, 34, 47, 49, 74, 75, 77, 78, 79, 90, 91, 117, 120, 131, 142, 213, 214, 233, 247, 257, 258, 264, 274, 275, 355, 373, 375, 376, 379, 398, 402, 445, 450, 487, 490, 491, 496, 501, 534, 540, 541, 542, 550, 573, 574, 576, 581, 584 dispositional 191 mass 142, 143 Predication 33, 34, 94, 191, 225, 258, 374, 376, 379, 482, 488, 524, 535, 536, 583 verbal aspects of 33 Present, specious 401 Presentation content of 523 Presentism 170, 552 Preservation 183 Presocratics 529 Prime matter 499 Priority causal 135 conceptual 530 logical 530 metaphysical 500 of the parts 87 Processes 35, 36, 37, 38, 69, 97, 144, 158, 164, 168, 193, 199, 210, 211, 223, 245, 317, 320, 321, 322, 371, 373, 402, 411, 431, 440, 494, 516, 536, 599, 600 chaotic 204

deterministic 204 indeterministic 204 Propensities 190 Properties 197 Archimedean 417 categorical 191 contrary 531, 535 elemental 144 emergent 152 higher-level 443 intrinsic 312, 410, 416, 462, 463, 466, 467, 468, 469 modal 148, 313, 491, 571 non-corporeal 324 non-structural 204 phase 141, 142, 143 relational 76, 215, 218, 428, 583, 584 resultant 152 system 201 systemic 201 Propositions 26, 27, 28, 49, 75, 78, 84, 107, 108, 118, 179, 180, 213, 218, 284, 339, 374, 455, 456, 457, 458, 459, 460, 487, 488, 497, 504, 540, 541, 589 complexity of 458 Fregean 455 Russellian 455 unity of 456 Prosodic units 426 Provability 89 Prudential value 238, 239 Pseudo-partitive constructions 441 Pseudo-process 506 Psychiatry 323, 324 Psychic boundaries 325 Psychognosy 121 Psychology 102, 119, 121, 140, 152, 153, 166, 249, 250, 347, 377, 406, 522, 525, 526

624 INDEX

atomistic 231 descriptive 121 ecological 166 Qua operator 487 Qualia 160, 162, 210, 233, 234, 310, 311, 520, 521 eliminativism about 160 ordering of 521 Qualities intrinsic 65 secondary 184, 299, 303 sensory 37 Quality moments 579 Quantification plural 417, 420, 421, 422, 423 Quantifiers class 420 plural 420 Quantities intensive 132 Quantum 50 Quantum mechanics 461, 465 non-relativistic 461 relational interpretation of 468 Quasi-analysis 129, 520 Quasi-parts 129, 520 Quine, W.V. 32, 89, 214, 217, 218, 240, 241, 242, 258, 261, 379, 380, 381, 412, 414, 415, 422, 424, 536, 538, 575 Rationalism 284, 482 neoplatonic 482 Realism 73, 75, 186, 233, 250, 251, 275, 379, 467, 468, 604 Ontological Structural 467 Reality 47, 53, 75, 87, 162, 164, 184, 185, 189, 220, 277, 278, 298, 300, 302, 336, 389, 400, 401, 403, 405,

409, 422, 450, 462, 467, 468, 469, 474, 476, 496, 536 formal 184, 185 grades of 184 Realization principle 462 Recurrence 170, 258, 260, 545 Recursivity 224, 426, 550 Reduction 144 token-token 191 type-type 191 Reductionism 82, 144, 145, 163, 202, 205, 377, 389, 390, 472 material substance 343 part-whole 462 Reduplication 487, 488, 499 Region bounded 417 connected 417 Reichenbach, H. 135, 137 Reismus 123 Relata 308, 311, 467 Relation 76 bearer-attribute 534 connection 354 intentional 274 internal 94, 580 irreducibility of 496 missing 453 qua objectum 309 quad objective 309 theory of 111, 414, 422 Relative clauses 294, 295 Replacement, Axiom of 422 Representation knowledge 152, 430, 574, 575 Theorem 418, 567 Rescher, N. 35, 38, 354, 375, 378, 518, 519, 541, 544, 557, 561, 572, 579 Roles 68

INDEX 625

Ross, D.W. 180, 182, 208, 209, 337, 352, 475, 507 Russell, B. 73, 74, 75, 76, 77, 78, 79, 80, 81, 87, 89, 111, 128, 130, 174, 175, 178, 213, 214, 215, 217, 218, 225, 227, 233, 241, 249, 287, 288, 289, 290, 347, 348, 349, 395, 398, 399, 416, 436, 437, 438, 456, 460, 479, 487, 496, 497, 498, 503, 504, 505, 508, 536, 538, 567, 580, 581, 582, 585, 593, 594, 598, 602 Ryle, G. 29, 31, 32, 162, 191, 193, 442, 444 Sameness numerical 312, 313 Satisfaction schemata 550 Scholastics 81, 82, 119 Searle, J. 163, 164, 165, 166, 167, 274, 276 Segment 223, 331, 335, 370, 394, 425, 575 Self-organization 201 Self-similarities 223 Semantic representation 296 Semantic totality 494 Semiotic grammar 243 Sensation 41, 121, 184, 191, 297, 305, 408 Separability 86, 117, 118, 121, 122, 123, 131, 153, 155, 182, 198, 208, 325, 352, 353, 445, 461, 464, 470, 471, 473, 511 Separability principle 462 Separation characteristics 117 sequence 117 Sequences 394 Series 111 Set

open 566 regular open 357, 566 rough 246, 248 theory 94, 107, 174, 233, 241, 248, 288, 289, 362, 380, 381, 398, 413, 414, 417, 418, 419, 420, 422, 423, 430, 477, 478, 496, 512, 549, 550, 567 Zermelo-Fraenkel Theory 420 Set theory vs. mereology 414 Shadows 359, 505, 506, 507 Similarity exact 215, 217, 503, 580, 581, 583 Simons, P. 34, 37, 70, 91, 97, 99, 101, 103, 112, 113, 114, 115, 118, 119, 120, 124, 125, 133, 153, 160, 172, 175, 178, 215, 217, 218, 228, 229, 230, 235, 269, 276, 292, 293, 314, 361, 362, 363, 366, 367, 369, 371, 376, 377, 378, 380, 381, 385, 386, 389, 391, 392, 393, 395, 411, 412, 430, 431, 489, 513, 519, 537, 538, 539, 544, 554, 555, 556, 557, 558, 559, 561, 571, 572, 577, 595, 598, 601, 602, 605 Simples 75, 86, 87, 175, 176, 177, 298, 301, 506, 556, 560, 580, 604 extended 86, 175, 176 Simplicity 149, 185, 197, 236, 237 divine 48, 236 Simplicius 49, 52, 53, 55, 56, 256, 447, 449, 450, 452, 454 Simulations 137, 138, 139, 141, 204 complete 204 Singleton 107, 289, 390, 413, 422, 566 Solidarity 58, 59 Solution 143 Somatic substrate 325 Sophism 39, 40

626 INDEX

Soul 21, 23, 42, 53, 61, 83, 84, 100, 103, 104, 189, 322, 323, 324, 325, 330, 339, 432, 485, 500, 501, 502, 510, 525, 530, 532, 535, 563, 569 Sounds 424 Space Euclidean 435 Hausdorff 567 infinite-dimensional state 193 non-mereological theories of 435 part of 171 region-based theories of 354 Relational Theory of 95 topological 357, 358, 436, 515, 566, 567 Spacetimesource 469 Special Relativity, Theory 464, 468, 469, 473 Species 21, 40, 43, 50, 51, 52, 53, 59, 103, 104, 120, 122, 138, 256, 263, 266, 278, 280, 281, 282, 285, 303, 312, 323, 329, 333, 339, 340, 345, 353, 392, 445, 446, 461, 484, 485, 490, 492, 493, 501, 517, 523, 530, 532, 541, 562, 574, 590 Spinoza, B. 189, 451, 533, 534 Squares of opposition 488 Stability 68, 194, 329 Stages 58, 239 States brain 162 conscious 162, 163, 166, 504 States of affairs 26, 28, 134, 216, 226, 250, 274, 275, 346, 347, 490, 491, 504, 589, 590, 591, 604 necessary 491 negative 134 Stoics 131, 198, 510, 511, 512, 528, 529, 540

Stone, M.H. 129, 130, 436, 438, 567, 568 Story 165, 219, 220, 221, 222, 245, 272, 557 Strict partial order 362, 363 String Theory 175, 176, 178 Strings 89, 175, 176, 415, 425 Structure 52, 93 changes of 517 chemical 516 closed 327 complex 231, 317 constituent 544, 546, 547 electronic 144, 145, 199, 200 invariant 231 layered 295 linguistic 516 logical 80, 128, 515 mathematical 193, 246, 357, 360, 515 mereological 99, 100, 102, 122, 169, 226, 307, 315, 327, 361, 362, 363, 364, 365, 366, 545, 567 modal 467 molecular 144, 145 musical 517 non-Boolean logical 465 non-commutative algebraic 465 phrase 544, 547 sentence 544 similarity 129 syntactic 516 Stuffs 31, 33, 36, 37, 256, 257, 258, 259 Stumpf, C. 30, 120, 122, 124, 232, 234, 521, 522, 523, 524, 525, 588 Suárez 453, 454 Subjectivity 160, 210 Substance 21, 22, 23, 42, 47, 49, 50, 52, 53, 54, 55, 57, 59, 60, 61, 82, 83, 87, 88, 94, 121, 122, 123, 124, 131, 136,

INDEX 627

141, 142, 143, 144, 153, 168, 182, 183, 184, 185, 186, 187, 188, 189, 201, 255, 259, 298, 300, 302, 303, 316, 320, 339, 343, 409, 442, 448, 449,451, 452, 453, 478, 483, 485, 499, 500, 501, 504, 505, 524, 529, 530, 531, 532, 533, 535, 556, 577 alteration of 448 chemical 141 corporeal 183, 186 elemental 198 extended 186 finite 184 infinite 184 intelligible 53 mode of 184 primary 529 second 530 simplicity of 198 thinking 183 Substantial form 82, 83, 84, 339, 342, 343, 485, 535, 562, 580 Sum 39, 76 unique 557, 558 Summation unique 539 unrestricted 539 Superius 307, 347 Superposition 461, 464, 465, 471, 476 Supervenience 148, 149, 162, 163, 312, 377, 462, 463, 466, 471, 474, 560 Humean 462 mereological 202, 462 Supplementation 114, 132, 300, 356, 491, 594 strong 114, 115, 576 weak 114, 373, 377 Syllables 426, 427, 485 Syllogisms 483, 484, 486, 487, 540, 541, 542

Syllogistic categorical 540 modal 540, 542 Symmetry 63, 469 Syncategorematic 40, 336, 338 Synecdoche 382, 492, 493 Synholon 59 Syntax, formal 414, 415 Systems artificial 327 auto-regulative 315, 328 biological 315 component 195 deterministic 330 dynamical 164, 193, 194, 195, 201 mechanical 138, 144, 472 medical 315 organ 316 taxonomic 329 Teleology 42, 272, 433, 514 Temporal order 521 Temporal scales 162, 166, 195 Temporary intrinsics, problem of 552 Tense 33, 294 Theism 235, 236 Theory of truth 214, 551 coherence 77 Three-dimensionalism 349, 603 Time subjective 401, 408 Time segments 331 Time-A 330 Time-B 330 Tode ti 530 Togetherness 233, 394, 520 Topology 99, 253, 291, 354, 355, 356, 357, 358, 359, 369, 384, 412, 415, 435, 478, 526, 565, 566, 567, 568, 597 discrete 566

628 INDEX

indiscrete 566 pointless 355, 360, 435, 437 Topos theory 423 Totality 51, 52, 53, 225, 282, 301, 302, 305, 328, 352, 353, 402 Totum 40, 338, 492, 493 totus 338 Totus 338 Transcendent object 275 Transformations 355, 357, 515, 516, 518 Transitivity 97, 101, 108, 110, 114, 156, 157, 158, 159, 225, 226, 233, 265, 289, 313, 370, 373, 374, 375, 378, 379, 429, 455, 497, 500, 539, 541, 555, 557, 558, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579 Transitivity failures 572 Trinity 24, 45, 103, 105, 392, 483, 563, 565 Tropes 112, 135, 215, 217, 311, 380, 503, 504, 537, 579, 580, 581, 582, 583, 584, 585, 590 foundational 579 Truthmakers 35, 191 Truth-value 398, 550 bearer 455 Tuples 249, 414, 515 Twardowski, K. 120, 262, 263, 266, 268, 269, 271, 287, 541, 543, 571, 585, 586, 587, 588 Unification 263, 285, 504 Uniqueness 557 Unity accidental 342, 343, 500 complex 503 composite 62 degrees of 169, 342 finite 497

moment of 263, 266 organic 51, 207 phenomenal 162 Universals 21, 26, 35, 53, 54, 95, 102, 103, 131, 186, 215, 216, 217, 229, 233, 236, 249, 250, 258, 259, 329, 331, 332, 333, 374, 375, 379, 380, 445, 450, 455, 487, 500, 501, 503, 530, 535, 579, 580, 581, 582, 589, 590, 591 collection account of 340 concrete 379 Universals, problem of 54 Universals, problems of 582 Unpredictability 137, 139, 140, 201, 204 Unstability structural 194 Vacuum 83, 444 Vagueness 397, 552, 553 ontic 553 Validity 90 Value-atoms 207 van Inwagen, P. 45, 86, 89, 97, 98, 145, 149, 150, 228, 312, 314, 350, 351, 352, 381, 382, 387, 391, 393, 538, 540, 553, 603 Vedanta 488 Vendler, Z. 32, 33, 39, 133, 137, 158, 160 Vesalius, A. 42 Vicious Circle Principle 497 Volitions 29, 527 Water 22, 31, 36, 38, 51, 52, 68, 83, 141, 142, 143, 144, 145, 146, 150, 151, 180, 188, 190, 191, 197, 255, 256, 257, 258, 317, 374, 375, 376, 389, 444, 452, 477, 478, 481, 482, 529 Well-being 237, 238

INDEX 629

Well-foundedness converse 385 Whole 207 aesthetic 64 arbitrary 301, 380, 450 artificial 51 categorematic 39 continuous 21, 50, 51, 273, 451 continuous integral 339 discrete 22, 51 discrete integral 339 division of 104, 105 essential 58, 352, 353, 500 essentially structured 433 finite quantitative 40 general 283 good 207 heterogeneous 317, 481, 562 heterogeneous integral 342 homogeneous 481 homogenous 266 in a respect 339 in place 339 in quantity 339 in time 339 infinite 40, 41, 497 infinite quantitative 40 integral 21, 23, 331, 333, 334, 341, 342, 353, 356, 377, 458, 481, 482, 500, 501, 562, 569 intensional whole 501 material 64, 272 maximal 323 mereotopological 508 natural 51, 501 organic 458

perceptual 402 phenomenally articulated 232 potential 338, 339, 341, 563, 568, 569 presentational 402 psychic 315, 330 qualitative 40, 41, 318 qualitative accidental 40 qualitative essential 40 quantitative 40, 51, 283, 318 social 508, 509 somatic 315, 329 structural 231 structured 22, 53, 455, 517 substantial 40 successive 333 syncategorematic 39 temporal 331, 332, 333, 334, 336 undifferentiated 356, 357 universal 21, 284, 285, 338, 339, 340, 501, 563, 569 Whole-whole relations 244 Will 185 William Crathorn 82 Wittgenstein, L. 30, 31, 74, 75, 76, 77, 78, 79, 80, 81, 162, 214, 218, 226, 227, 331, 459, 460, 504, 518, 525 World-inheritance principle 219, 221 Wyclif, J. 83, 84, 85, 488 Xenocrates 71 Yablo, S. 135, 136, 398 Zeno 71, 174, 397, 399, 433, 502, 503 Zombie scenario 210