Logic in Religious Discourse 9783110319576, 9783110319187

Table of contents :
PREFACE: RELIGIOUS LOGIC AS APART OF PHILOSOPHICAL LOGIC Andrew Schumann
LOGIC IN INDIAN THOUGHT Subhash Kak
ON TWO QUESTIONS OF THE NEW LOGIC OF INDIA Kamaleswar Bhattacharya
THE USE OF FOUR-CORNERED NEGATION AND THE DENIAL OF THE LAW OF EXCLUDED MIDDLE INNĀGĀRJUNA’S LOGIC Dilipkumar Mohanta
A PLEA FOR EPISTEMIC TRUTH: JAINA LOGIC FROM A MANY-VALUED PERSPECTIVE Fabien Schang
REMARKS ON ANCIENT CHINESE LOGIC Jerzy Pogonowski
TALMUDIC HERMENEUTICS Avi Sion
OCKHAM AND ORATIO MENTALIS Francesco Bottin
ANALOGY IN THOMISM Petr Dvořák
TOWARDS A LOGIC OF NEGATIVE THEOLOGY Paweł Rojek
REASONING ABOUT THE TRINITY: A MODERN FORMALIZATION OF A MEDIEVAL SYSTEM OF TRINITARIAN LOGIC Sara L. Uckelman
LATE MEDIEVAL TRINITARIAN SYLLOGISTICS: FROM THE THEOLOGICAL DEBATES TO A LOGICAL TEXTBOOK Paloma Pérez-Ilzarbe
INEFFABILITY PERFORMANCE: CRITIQUE AND CALL Timothy Knepper

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Andrew Schumann (Ed.) Logic in Religious Discourse

Andrew Schumann (Ed.)

Logic in Religious Discourse

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CONTENTS Andrew Schumann Preface: Religious Logic as a Part of Philosophical Logic……………….10 Subhash Kak Logic in Indian Thought……………………………………………….....20 Kamaleswar Bhattacharya On Two Questions of the New Logic of India………………………… ...34 Dilipkumar Mohanta The Use of Four-Cornered Negation and the Denial of the Law of Excluded Middle in Nāgārjuna’s Logic…………………………………..44 Fabien Schang A Plea for Epistemic Truth: Jaina Logic from a Many-Valued Perspective……………………………………………………………......54 Jerzy Pogonowski Remarks on Ancient Chinese Logic……………………………………...84 Avi Sion Talmudic Hermeneutics…………………………………………............104 Francesco Bottin Ockham and Oratio Mentalis…………………………………………...132 Petr Dvořák Analogy in Thomism…………………………………………………....164 Paweł Rojek Towards a Logic of Negative Theology………………………………...192

Sara L. Uckelman Reasoning about the Trinity: a Modern Formalization of a Medieval System of Trinitarian Logic……………………………...216 Paloma Pérez-Ilzarbe Late Medieval Trinitarian Syllogistics: From the Theological Debates to a Logical Textbook............................................240 Timothy Knepper Ineffability Performance: Critique and Call…………………………….262

PREFACE: RELIGIOUS LOGIC AS A PART OF PHILOSOPHICAL LOGIC Andrew Schumann Department of Philosophy and Science Methodology, Belarusian State University Minsk, Belarus [email protected] Between religion and logic there is a strange and deep connection. On the one hand, religious practice and faith in God draw us away from the limits of the real, perishable and changeable world and allow us to feel eternity and give us the feeling of membership in something limitlessly-free, great, unbounded, and unending. It is connected to the circumstance that religious practice sets up special social technologies, which could be said to be eternal. These technologies organize our social life to repeat the same social cycles (e.g., religious feasts, ways of life, rituals, and social mores) and to keep them year in, year out and from previous generations to present ones. On the other hand, the same feeling of eternity comes into being in us through studies of logic and mathematics. The following estimation given by the famous logician, Bertrand Russell (1872 – 1970), eloquently speaks about this fact: I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe – because, like Spinoza's God, it will not love us in return.

A similar description belongs to Georg Cantor (1845 – 1918), the founder of set theory: I am far from assigning my discoveries to personal merits because I am just a tool of a higher force which will work after me, also, in the same manner as it was passed down, thousand years ago in Euclid and Archimedes.

Cantor’s main intuition is a feeling of never-ending freedom which has been presented by logic and mathematics: The essence of mathematics lies in its freedom.

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Logic is a unique science, which for the whole history of its official existence (it is more than two and a half thousand years old) has not varied in its foundations in any way. Even mathematics has essentially changed, in the beginning it was based on geometry (the Ancient Greek mathematics), then on algebra (the classical mathematics), and now it is grounded in mathematical logic, in particular in computational or algebraic logic. Any scientific knowledge can be regarded as perishable and changeable with the exception of logic. Only this science defines the limits of my thinking, the limits outside of which there is nothing, just silence: The limits of my language mean the limits of my world (L. Wittgenstein, Tractatus LogicoPhilosophicus).

Logic is an eternal science. Therefore many religious thinkers tended to believe the statement that the eternal social technologies, set by religious practice, are built on the basis of logic. For example, it is affirmed that logic is one of the ‘five big sciences’ (among which there are as follows: (i) the science of reasoning and logic, (ii) the science of craft and architect, (iii) the science of healing and medicine, (iv) the science of language and grammar, (v) and the science of Buddhist culture and Buddhist philosophy) which Buddha knows thoroughly and perfectly. Further, more logic is one of the six darśanas, which are the classical schools of Indian religious philosophy. In Thomism (the official ideology of Roman Catholics), logic is the basis of doctrinal reasoning. In Judaism, logic, set up by Rabbi Ishmael’s 13 inference rules, is the key for understanding the Torah, a key that was orally passed down from God to Moses. However, between religious faith and logic there is also an essential distinction. Religious faith is blind, it assumes a thoughtless acceptance, it is based on emotions and existences, and logic demands a strict substantiation and it is based on pure reason. It was shown in the most impressive way, probably, in Christianity. One of the first thinkers who wrote about an illogical nature, and the absurdity of religious faith, was Tertullian (155/165 – 220/240): Natus est Dei Filius, non pudet, quia pudendum est; et mortuus est Dei Filius, prorsus credibile est, quia ineptum est; et sepultus resurrexit, certum est, quia impossibile. The Son of God was born: there is no shame, because it is shameful. And the Son of God died: it is wholly credible, because it is unsound. And, buried, He rose again: it is certain, because impossible (Tertullian, De Carne Christi V, 4).

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The following passage, regarded as a quintessence of similar reasoning, was afterwards assigned to Tertullian: ‘Credo quia absurdum’ which means ‘I believe because it is absurd.’ The phrase of the great Russian writer, Fyodor Dostoyevsky (1821 – 1881), became the best literary paraphrase of that passage: If someone proved to me that Christ is outside the truth and that in reality the truth were outside of Christ, then I should prefer to remain with Christ rather than with the truth.

In the 20th century Roman Catholic Church there was a special intellectual movement seeing the light of an emotional side of religious ministering. It is called Worshipping Divine Mercy. This movement had a lot of prophets, one of them was the Polish nun and canonized saint, Sister Faustyna Kowalska, known as the “Apostle of Mercy,” who lived from 1905 to 1938. This intellectual movement for the cleanliness of emotional experience in faith was headed by Pope John Paul II (1920 – 2005). Due to his diligence, Sister Faustina was sainted in 2000. The same year, Pope John Paul II officially and universally instituted the Feast of the Divine Mercy for the Catholic Church. The essence of this movement is a devotion focused on the mercy of God and His power, particularly as a form of thanksgiving and entrusting of oneself to God’s mercy. It is obvious that religious faith cannot be based in logic. Faith is really nothing more than a personal belief in things that cannot be proved or disproved. The aim of faith consists of a mechanism for the human mind to rationalize the unknowable by means of acceptance of statements which cannot be proved. For this reason, many people from the world of logic look at religion as a form of mental defect that undermines the principles of science. The criticism of religious faith as unreasonable knowledge was generated for the first time in the 18th century Enlightenment (the first phase of scientific revolutions) when an anti-traditional, anti-religious way of thought, a worldview that concentrated largely on critical and rational thinking was formed in societies. This movement grew in anticlericalism, i.e. in the negation of the significance of the church as a social institution. The anticlericalism, however, did not exclude recognizing the existence of God as a divine principle which gave the world physical laws according to which the world exists ever now. One of the most prominent followers of anticlericalism and deism was Voltaire (1694 – 1778). He estimated

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theology as a cultural phenomenon, instead of a phenomenon of knowledge: We shall not extend our views into the depths of theology. God preserves us from such presumption. Humble faith alone is enough for us. We never assume any other part than that of mere historians (Voltaire, Church, [in:] Voltaire, Philosophical Dictionary, Part 2, 1764).

Since the Enlightenment, attitudes towards logic have considerably changed. Before that, logic, gnoseology (the theory of cognitions) and ontology had been inseparable. Recently logic has generally been accepted to be formal, in that it aims to analyze and represent the form of any valid argument type. Thereby the subject of logic has been considerably reduced. An ontological and epistemic dimension of reasoning has been disregarded and defied. It takes place thanks to the fact that in logic just the form of reasoning has been accepted and considered and the content of reasoning has been fully ignored. Formal logic is now considered to be a unique science about reasoning. Through the process of time, only the German philosophers, such as Kant (1724 – 1804) and Hegel (1770 – 1831), tried to show the significance of informal philosophical logic: transcendental logic, speculative logic, and dialectical logic. In such systems, logic was connected with ontology and cognitive science. For example, Kant distinguished formal logic as an a priori science in that forms of analytical arguments/judgments are considered and transcendental logic as an a priori science in that the content of synthetic arguments/judgments and its typology are proposed. Formal logic defines laws of our thinking and transcendental logic defines ways of cognizing the world, ways of filling up naked circuits, skeletons, and forms of reasoning, which are presented by formal logic. However, such a splitting of the two logics eventually lost all sense; formal logic became a unique authentic logical science. It is explained by the fact that since the Enlightenment, scientific arguments have been understood differently. They became value-neutral without any axiological dimension, therefore there has been a requirement to build up logic without any ontological dimension as a simple game of sign-combinations. After the Enlightenment, societies were laicized, i.e. they were formed with a certain vector of emancipation of social actions and behaviors from many religious and ethical standards. Since the end of the 19th century, a strategic action has been presented as the main kind of social behavior. Differences of strategic actions from other forms of behaviors exist in that in their frames only the individual’s aim and goal projections

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corresponding to concrete physical referents are taken into consideration. While these projections, which were carried out by actors of Ancient and Medieval behavior standards, assumed the presence of non-physical referents (like ‘God,’ ‘good,’ ‘spirit,’ etc.), in a strategic action nonphysical referents are already more likely to be ignored. Accordingly, in logical semantics the main emphasis is placed on defining models, i.e. on an effective setting of classes of referents/denotations. Such a logical task could not be put forward, neither by Greek, nor Medieval philosophers in virtue of the fact that they were not actors of our strategic behavior, whereas in modern formal logic this task is considered predominant. In formal logic, possible models of logical relations and also ways of setting up these models are studied. At the same time, in these models a corresponding relation between the structure of objects for interpreting (referents) and syntactic laws of language are defined. As a result, the ideal objects expressing structural properties of a collection of referents are implicitly defined only as correlatives of appropriate classes of referents. Taking this into account, just relations between referents of the first-order logic and finite arithmetics, i.e. their models, are effectively (by means of recursions) defined. These objects, according to Hilbert’s (1862 – 1943) program [9], [10], should appear, in turn, as basic referents for all other logical systems. Thus, a hierarchy, where all logics are various extensions of the first-order logic, should be examined as appropriate extensions of first-order logic. At first sight, these extensions should present the whole modern logic theory. Hence, formal logic assumes to hold the following strong presupposition: the form of an argument should be displayed by representing its sentences in the formal grammar of a logical language to make its content usable in formal inference. However, there exist arguments which can seem cogent and demonstrative in natural language and cannot be translated into the logical language of symbolisms owing to the limited possibilities of the latter language. For additional legitimation of formal logic, Jan Łukasiewicz (1878 – 1956), the great Polish logician, initiated a special program of looking at the history of logic through the glasses of formal logic. The history of logic has been presented by him as a linearly developing process, in which the old logical systems are regarded as predecessors of modern formal logic. According to Łukasiewicz it is not fair to fault or condemn traditional logic; such a blameworthy attitude was quite popular among the originators of modern logic who, including Frege and Russell, maintained that modern logic completely broke with the past.

15 Łukasiewicz thought that not everything was wrong in the past. In fact, argued Łukasiewicz, it was Descartes who basically caused the degeneration of logic and pushed it to psychologism. Even Leibniz could not stop this process, although he should be considered as a predecessor of modern mathematical logic. Łukasiewicz's program meant that every revolution is doing the history of logic. Łukasiewicz himself discovered that the Stoics constructed propositional logic. He also rehabilitated the logical inventions of the medieval Schoolmen. Another of Łukasiewicz’s results was a reinterpretation of Aristotelian syllogistics in terms of modern logic. He also pointed out that many-valued logic is rather more non-Stoic than non-Aristotelian, because the Stoics very strongly defended the principle of bivalence, but Aristotle doubted this principle as far as it concerned statements about the future [14].

Łukasiewicz’s program was supported by another famous Polish logician, Fr. Józef Maria Bocheński OP (1902 – 1995), one of the first experts in formal logic who paid attention to the possibilities of the analysis of Medieval thinkers’ religious reasoning from the standpoint of modern formal logic [1] – [4]. Since then a huge number of scientific works on formalizing religious reasoning have been written. Thereupon it is possible to speak about the appearance of a new branch in modern logic in which the logic of religious discourse is studied. This branch has most accurately to be considered as a part of philosophical logic [5] – [8]. By analogy with the name ‘philosophical logic,’ it is possible to call the new branch religious logic. Some tasks of the latter are as follows: 1. The construction of consistent logical systems formalizing religious reasoning that at first sight seems inconsistent. This research is carried out within the limits of modal logic, paraconsistent logic and many-valued logic. For example, in Christian theology there are two inconsistent claims about God’s unity and trinity simultaneously. However, the following conclusion is a paralogism for both believers and non-believers: The Father is God. God is the Son. ——————————————– Therefore, the Father is the Son. It is possible to assert that God’s trinity is a subject of blind faith, therefore ordinary rules of inference simply do not apply. But it is possible to construct logical systems in which the individual’s (God’s) unity and trinity do not imply paralogisms like the above-mentioned one. Another example is the substantiation of apophatic theological reasoning from the point of view of modern formal logic, namely the proof within the limits of a logical system

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that the following two statements are valid simultaneously: (i) God has negations of all positive properties and (ii) God has negations of all negations of positive properties. In this connection, the logical theories of Indian philosophy proving the validity of inconsistency of some statements are very interesting, too. For example, Nāgārjuna in Vigrhavyāāvartanīī v. 57 offers a special type of negation called prasajya pratiùedha, the ‘pure’ and ‘simple’ negation. In such a negation when you negate a thesis, ‘p,’ as false, you need not accept the counter-thesis, ‘not-p,’ as true. These ideas of Nāgārjuna undoubtedly allow us to estimate this thinker as the forerunner of paraconsistent logic. A further example: in Jaina logic, the possibility of a relative, non-one-sided view of truth, is proved. 2. An illocutionary analysis of religious discourse rooted in investigating logico-pragmatical aspects of religious reasoning. This research is carried out in frames of illocutionary logics [13]. For instance, it is possible to speak about special illocutionary characteristics of mystical speeches, an example is Tertulian’s statement quoted above. In these speeches we find an attempt to ‘state’ the aporia of transcendence by disrupting the rules and conventions of ordinary language. 3. A formalization of the Ancient and Medieval logical theories used in the theology of an appropriate religion. The greatest interest is evoked by theories which cannot be formalized within the limits of conventional formal logics. They could be studied only within the limits of unconventional logics, such as non-monotonic logics, non-well-founded logics [11], etc. Probably, studying the thirteen inference rules (midot) of Rabbi Ishmael (the main logical theory in Judaism) and the Thomist theory of analogy (the logical theory for the substantiation of divine attributes in the Roman Catholic theology) should be evaluated as belonging to unconventional (non-standard) logical theories. They admit softer methods of output than those which are now available in modern formal logic. Religious logic has a number of semantic presuppositions which are characteristic of all Ancient and Medieval logical theories in the same measure. These semantic assumptions do not hold in modern formal logic, but without their explication it is impossible to treat and formalize these theories adequately, in particular with reference to religious discourse. We may give these assumptions: 1. In these doctrines there is no distinction between logic and ontology, and respectively between theory and metatheory, therefore logic is examined as a resource of conceptual description of the whole real universum. In this connection it is possible to call this logic real. In this logic one

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distinguishes two dimensions (but not levels): the mental and written/oral ones. In both dimensions, any human intellectual manifestation is reflected. For example, in his logical studies, Ockham carried out a complete process of grammaticalization of mental language carefully outlining distinct specific rules for the written, spoken and mental language. In these languages, by Ockham’s plan, all aspects of describing universum should be mirrored. In Indian logic there is the same distinction between the mental language (‘conclusions for oneself’) and written/oral language (‘conclusions for others’). This distinction is given not within the frameworks of different logics, but within one and the same system, the same as in Ockham’s approach. 2. These real logics do not assume that they are set within the frameworks of a finite or countable list of axioms (accordingly, they are not axiomatized). The point is that real logics describe syntactic and semantic conditions of all true statements in relation to the dynamically varying world. Such logics contain an uncountable number of true statements. There is a very interesting fact that it is possible to anticipate ideas of massive-parallel computations in these logics. Therefore it is obvious that in such logics there cannot be an assumption of the substitution rule and Leibniz’s identity principle. All these circumstances explain why real logics (to which, for example, the thirteen inference rules of Rabbi Ishmael belong) cannot be formalized from the standpoint of conventional formal logics. All existing formalizations of Aristotle’s syllogistics do not reflect the logical theory which was conceived by Aristotle. For instance, the fragments of his lost manuscripts, which were then collected in books of the Metaphysics, are in fact fragments of Aristotle’s lost logical works, in which a logical ontology is described. But this ontology on which Aristotle’s logic is based, is not presented in any formalizations of Aristotle’s logic. 3. In real logic it is possible to detect the anticipation of ideas of unconventional computation. The world is organized in such a way that it is exhaustively described by logic. The world lives by logic. Each natural process may be regarded as a substratum of logical computing. On the other hand, each logical concept (an interval of epistemic space, a logical image of reality) may be presented as a way of substance distribution. Therefore logic cannot be evaluated as a product of human intelligence, it is the result of peering into the world, the result of revelation. Logic is a semantic frame of space, Purushi’s body (in Hinduism) or Adam Kadmon’s body (in Judaism). Thereupon it is possible to speak about an

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assumption of the special subject of logical research, a semantic continuum, i.e. a class of all true statements. New mathematical research, which may have resulted from the abovementioned assumptions, could kindle an appreciable interest in modern formal logic, e.g.: • in the development of a proof theory without axioms [12], • in the development of dynamic (behavior) logic, whose statements hold on behavioral models. The present book entitled “Logic in Religious Discourse” is the first attempt to consider religious logic as a holistic phenomenon without respect to religious community membership. In the book, a wide spectrum of logical theories used in Hinduism, Confucianism, Jainism, Buddhism, Judaism, and Christianity is presented and studied. The book is lost in thought as the first step towards the development of general theories of religious (real) logic. References [1] Bocheński J. M. A history of formal logic, English translation by Ivo Thomas. Notre Dame, University of Notre Dame Press, 1961. [2] Bocheński J. M. Ancient formal logic. Amsterdam, North-Holland, 1951. [3] Bocheński J. M. Formale Logik. München, Alber, 1956. [4] Bocheński J. M. О “relatywizmie” logistycznym, [in:] Mysl katolika wobec logiki wspóiczesnej. Poznan, 1937, 87 – 111. [5] Gabbay D., Guenthner, F. (eds.). Handbook of philosophical logic. Vols. I—IV. D. Reidel, Dordrecht, 1983 — 1989. [6] Gabbay D., Guenthner, F. (eds.). Handbook of philosophical logic. Vols. 1—18. 2nd Edition. Kluwer Academic Publishers, Dordrecht, 2001 — 2004. [7] Gabbay D., Woods J. (eds.). Handbook of the History and Philosophy of Logic. Vol. 1. Greek, Indian and Arabic Logic. North Holland, Elsevier, 2004. [8] Gabbay D., Woods J. (eds.). Handbook of the History and Philosophy of Logic. Vol. 2. Mediaeval and Renaissance Logic. North Holland, Elsevier, 2004.

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[9] Hilbert D. and Bernays P. Grundlagen der Mathematik, Vol. 1. Zweite Auflage. Springer-Verlag, 1968. [10] Hilbert D. Mathematical Problems, Bulletin of the American Mathematical Society, 8, 1902, 437–479. [11] Schumann A. Non-well-foundedness in Judaic Logic, Studies in Logic, Grammar and Rhetoric, 13 (26), 2008, 41 – 60. [12] Schumann A. Towards Theory of Massive-Parallel Proofs. Cellular Automata Approach, Bulletin of the Section of Logic, 2009. [13] Searle J. R. and Vanderveken D. Foundations of Illocutionary Logic. Cambridge: Cambridge University Press, 1984. [14] Woleński J. Mathematical logic in Poland 1900 —1939: People, circles, institutions, ideas, Modern Logic, 5, 1995, 363 – 405.

LOGIC IN INDIAN THOUGHT Subhash Kak Department of Computer Science Oklahoma State University Stillwater, USA [email protected] This paper presents an overview of the Indian tradition of logic. The paper starts with Vedic ideas and goes on to summarize the relevant contributions of the formal schools of philosophy which included one devoted principally to logic. The Indian tradition of logic reached its peak in the Navya Ny¯ aya school of medieval India.

1. Introduction This article is a general survey of the tradition of logic (¯ anv¯ıks.ik¯ı, ny¯ aya, or tarka in Sanskrit) in India. This tradition is very old and can be seen in its beginnings in the R.gveda, the earliest text available from India and dated to about 2000 BCE. The Vedic system looked at reality at two levels. At the ordinary level of apprehension it was rational and, therefore, it needed logic to describe it; but at a higher level it had a transcendental basis. The transcendental nature was expressed in statements that were paradoxical such as the individual self was equivalent to the cosmic self (¯ atman equals brahman) or fullness is present everywhere, it arises from itself, and when subtracted from itself it remains full ¯sa¯v¯ (I´ asya Upanis.ad). Some have taken this latter statement to imply an intuition of the idea of infinity. The R.gvedic hymn 10.129, speaking of creation, mentions a time that was neither existent nor non-existent, suggesting the beginnings of representation in terms of various logical divisions that were later represented formally in Indian logic as the four circles of catus.kot.i: “A,” “not A,” “A and not A,” and “not A and not not A.” Amongst the early sources of textual evidence for Indian logic are the vari-

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ous schools of philosophy including Ny¯ aya and Vai´ses.ika, dealing respectively with linguistic and physical objects. The epic Mah¯ abh¯ arata mentions different schools of logic. The grammar of P¯ an.ini (5th century BCE) uses logical categories and the rich grammatical tradition continued to influence logic and other philosophical thought. Early modern reviews of the subject are by Vidyabhusana [1],[2]; for general reviews, see the edited volumes by Potter [3],[4]; for a broad historical context, see [5],[6],[7]. The tradition of Indian logic, which developed in the background of the Vedic theory of knowledge, was divided by the historian Vidyabhusana [2] into three periods: ancient (up to 400 CE), medieval (400 CE – 1200 CE), and modern (1200 CE – 1850 CE). He saw the Ny¯ aya S¯ utra of Aks.ap¯ ada Gautama (or Gotama) (c 550 BCE) as the foremost, if not the earliest, representative of the ancient period; Pram¯ an.a-samuccaya of Dign¯ aga as representative of the medieval period; and Tattva-cint¯ aman.i of Ga˙nge´sa Up¯ adhy¯ aya as representative of the modern period. The medieval period produced many important glosses on the ancient period and much original thought. For example, Bhartr.hari (5th century CE) presented a resolution to the problem of self-referral and truth (Liar’s paradox) [8]. In the modern period philosophers took up new issues such as empty terms, double negation, classification, and essences. 2. Dar´sanas and the Ny¯ aya Sutra ¯ Logic is one of the six dar´sanas, which are the classical schools of Indian philosophy. These six schools are the different complementary perspectives on reality, which may be visualized as the views from the six walls of a cube within which the subject is enclosed. The base is the system is the broad system of the tradition (P¯ urva M¯ım¯ am a), and the ceiling represents the large questions . s¯ of meaning related to the objective world and the subject (Uttara M¯ım¯ am a or . s¯ Ved¯ anta); one side is analysis of linguistic particles (Ny¯ aya), with the opposite side being the analysis of material particles (Vai´ses.ika); another side is enumerative categories in evolution at the cosmic and individual levels (S¯ am . khya), with the opposite side representing the synthesis of the material and cognitive systems in the experiencing individual (Yoga). Logic is described in Kaut.ilya’s Artha´sa¯stra (c. 350 BCE) as an independent field of inquiry ¯ anv¯ıks.ik¯ı [9]. The epic Mah¯ abh¯ arata, which is most likely prior

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to 500 BCE because it is not aware of Buddhism in its long descriptions of religion [10], declares (Mah¯ abh¯ arata 12.173.45) that ¯ anv¯ıks.ik¯ı is equivalent to the discipline of tarka. Clearly, there were several equivalent terms in use in India for logic in 500 BCE. The canonical text on the Ny¯ aya is the Ny¯ aya S¯ utra of Aks.ap¯ ada Gautama [11]. The most important early commentary on this text is the Ny¯ aya Bh¯ as.ya of V¯ atsy¯ ayana which is estimated to belong to 5th century CE. The physician Caraka, in his Sam a, speaks of the importance of the use of . hit¯ logic in medicine just as it was also essential to other sciences. The Ny¯ aya S¯ utra speaks of three kinds of debate: • kath¯ a (literally, speech), where a thesis and a counter-thesis are argued by the protagonists based on evidence and argument; • jalpa, which may entail equivocation and false reasoning; • vitan.d.a, which is characterized by the absence of a counter-thesis. The Ny¯ aya also calls itself pram¯ an.a ´s¯ astra, or the science of correct knowledge. Knowing is based on four conditions: (i) The subject or the pramatr.; (ii) The object or the prameya to which the process of cognition is directed; (iii) The cognition or the pramiti; and (iv) the nature of knowledge, or the pram¯ an.a. The four pram¯ an.as through which correct knowledge is acquired are: pratyaks.a or direct perception, anum¯ ana or inference, upam¯ ana or analogy, and ´sabda or verbal testimony. The function of definition in the Ny¯ aya is to state essential nature (svar¯ upa) that distinguishes the object from others. Three fallacies of definition are described: ativy¯apti, or the definition being too broad as in defining a cow as a horned animal; avy¯apti, or too narrow; and asambhava, or impossible. Gautama mentions that four factors are involved in direct perception: the senses (indriyas), their objects (artha), the contact of the senses and the objects (sannikars.a), and the cognition produced by this contact (j˜ n¯ ana). The five sense organs, eye, ear, nose, tongue, and skin have the five elements light, ether, earth, water, and air as their field, with corresponding qualities of color, sound, smell, taste and touch. Manas or mind mediates between the self and the senses. When the manas is in contact with one sense-organ, it cannot be so with another. It is therefore said to

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be atomic in dimension. It is due to the nature of the mind that our experiences are essentially linear, although quick succession of impressions may give the appearance of simultaneity. Objects have qualities which do not have existence of their own. The color and class associated with an object are secondary to the substance. According to Gautama, direct perception is inexpressible. Things are not perceived as bearing a name. The conception of an object on hearing a name is not direct perception but verbal cognition. Not all perceptions are valid. Normal perception is subject to the existence of (i) the object of perception, (ii) the external medium such as light in the case of seeing, (iii) the sense-organ, (iv) the mind, without which the sense-organs cannot come in conjunction with their objects, and (v) the self. If any of these should function improperly, the perception would be erroneous. The causes of illusion may be dos.a (defect in the sense-organ), samprayoga (presentation of only part of an object), or sam ara (habit based on irrelevant recollection). . sk¯ Anum¯ ana (inference) is knowledge from the perceived about the unperceived. The relation between the two may be of three kind: the element to be inferred may be the cause or the effect of the element perceived, or the two may be the joint effects of something else. The Ny¯ aya syllogism is expressed in five parts: 1. pratij˜ n¯ a, or the proposition: the house is on fire; 2. hetu, or the reason: the smoke; 3. dr..s.t¯anta the example: fire is accompanied by smoke, as in the kitchen; 4. upanaya, the application: as in kitchen so for the house; 5. nigamana, the conclusion: therefore, the house is on fire. This may be represented symbolically as [12]: 1. A 2. Because B 3. B goes with A always; witness C 4. It is a case of B

24

5. Therefore, A The Ny¯ aya syllogism recognizes that the inference derives from the knowledge of the universal relation (vy¯ apti) and its application to the specific case (paks.adharmat¯ a). There can be no inference unless there is expectation (¯ ak¯ an˙ ksh¯a) about the hypothesis which is expressed in terms of the proposition. The minor premise (paks.adharmat¯ a) is a consequence of perception, whereas the major premise (vy¯ apti) results from induction. But the universal proposition cannot be arrived at by reasoning alone. Frequency of the observation increases the probability of the universal, but does not make it certain. Ga˙nge´sa, a later logician, suggested that the apprehension of the universal requires alaukika pratyaks.a (or nonsensory apprehension). It was also argued that the major premise (vy¯ apti) should be formulated negatively to ensure that the process of inference does not involve petitio principii. Let A be what has a; whatever does not differ from non-A, does not have a. The five-part syllogism would then run as: 1. Not A 2. Because not B 3. A goes with B always; witness C 4. It is not so (not a case of B) 5. Therefore, it is not a case of A The Ny¯ aya system lays stress on antecedence in its view of causality. But both cause and effect are viewed as passing events. Cause has no meaning apart from change; when analyzed, it leads to a chain that continues without end. Causality is useful within the limits of experience, but it cannot be regarded as of absolute validity. Causality is only a form of experience. The advancement of knowledge is from upam¯ ana, or comparison, with something else already well-known. The leads us back to induction through alaukika pratyaks.a as the basis of the understanding. ´ Sabda, or verbal testimony, is a chief source of knowledge. The meaning of words is by convention. The word might mean an individual, a form, or a type, or all three. A sentence, as a collection of words, is cognized from the trace

25

(sam . sk¯ara) left at the end of the sentence. Knowledge is divided into cognitions which are not reproductions of former states of consciousness (anubhava) and those which are recollections (smr.ti). The Ny¯ aya speaks of errors and fallacies arising by interfering with the process of correct reasoning. The Ny¯ aya attacks the Buddhist idea that no knowledge is certain by pointing out that this statement itself contradicts the claim by its certainty. Whether cognitions apply to reality must be checked by determining if they lead to successful action. Pram¯a, or valid knowledge, leads to successful action unlike erroneous knowledge (vipary¯ aya). 3. Object and subject The Ny¯ aya propositions assume a dichotomy between object and subject. The objective world is open to logical analysis since it maps to linguistic categories; the subjective world can suffer from invalid perception for a variety of reasons. This is consistent with the Vedic view that the although the inner world maps the outer, the mind can be clouded by habits or wrong deductions owing to incorrect assumptions. The S¯ am . khya [13], attributed to the legendary rishi Kapila, is the background to be considered when speaking of Indian logic. Its concern is the enumeration of categories as they arise in the space of the mind with the objective of obtaining discriminative knowledge of the manifest (vyakta), the unmanifest (avyakta) and the knower (purus.a). In S¯ am . khya, evolution occurs due to changing balance and proportion both in the objective and the subjective worlds. The three gun.as or fundamental modalities are sattva, tamas and rajas, and they operate both at the large scale as well as in quick transformation. The normative “thing” behind this ceaseless change is the witness, or self, who is viewed in the singular for the entire universe. At the objective level, tamas is inertia, rajas is action or transformation and sattva is the relative balance or equilibrium between tamas and rajas. The interplay between the three sets up oscillations in the objective and the mental levels. In Yoga, the objective is to achieve the cessation of the fluctuations of the mind. Consciousness or pure awareness is by definition not an object and therefore it does not have attributes. It must for the same reason be beyond the categories

26

of the living or dead. It must be beyond inertia, or change or fluctuations. It is extraordinary that in this analysis the qualities that are associated with objects become describable by an internal order. The gun.as do not admit of any further breakdown. This defines a position that is different from that of Aristotelian physics [14]. The three gun.as are present in all objects and we can isolate one only in terms of the momentary strength of one in relation to the other in a process. Their fluctuations mark the universal “internal clock” of worldly processes. In the S¯ am . khya, the effect is the cause in a new form, and this is why the system is also called parin.¯ amav¯ ada, or theory of transformation. Between the cause and effect is a relation of identity-and-difference, that is identity of stuff but difference of form (bhed¯ abheda). The method at the basis of the S¯ am . khya and the Ny¯ aya S¯ utra may be seen in the Yoga S¯ utra as well. In the Yoga S¯ utra 3.13 three aspects of change are identified: transformation of a thing (dharmi) into a property (dharma), transformation of a property into a mark (laks.an.a), and the transformation of a mark into a condition (avasth¯ a). 4. The form of the Ny¯ aya syllogism The five parts of the Ny¯ aya syllogism spring from the idea of bandhu that is fundamental to Vedic thought. The bandhu is the equivalence between two different systems, which ordinarily are the microworld, the macroworld, and the individual’s cognitive system [5]. The Ny¯ aya syllogism first sets up the propositional system with its two components (two parts) and then identifies another well known system to which the first is supposed to have a bandhu-like relationship (third and fourth parts). The conclusion (fifth part) can be made only after the preliminaries have been formally defined. The appeal to the bandhu in the syllogism is to acknowledge the agency of the subject who can be, without such knowledge, open to invalid perception. One can see how in systems that do not accept transcendental reality (such as Aristotle’s or Buddhist), a simplification from the five-part to the three-part syllogism would be most natural. The Ny¯ aya considers the following five elements essential to correct reason [12]:

27

1. The reason (evidence) must be present in the case under consideration; 2. It must be present in another case similar to the one under consideration; 3. It must not be present in cases dissimilar to the case under consideration; 4. It must be such that the proposition it tries to establish is not contradicted by another already established truth; 5. It must be such that there should not be another evidence or reason establishing the opposite thesis, to counterbalance the thesis it tries to establish. 5. Vai´ses.ika and other views This school of “individual characteristic” is supposed to have been founded uka [15],16]. Vai´ses.ika S¯ utras describe a system of by Kan.a¯da, the son of Ul¯ physics and metaphysics. Its physics is an atomic theory of nature, where the atoms are distinct from the soul, of which they are the instruments. Each element has individual characteristics (vi´ses.as), which distinguish it from the other non-atomic substances (dravyas): time, space, soul, and mind. The atoms are considered to be eternal. There are six fundamental categories (pad¯ artha) associated with reality: substance (dravya), quality (gun.a), motion (karman), universal (s¯ am¯ anya), particularity (vi´ses.a), and inherence (samav¯ aya). The first three of these have a real objective existence and the last three are products of intellectual discrimination. Each of these categories is further subdivided as follows. There are nine classes of substances, some of which are nonatomic, some atomic, and others all-pervasive. The nonatomic ground is provided by the three substances ether (¯ ak¯ as´a), space (di´s), and time (k¯ ala), which are unitary and inde¯ water (¯ structible; a further four, earth (pr.thiva), apas), fire (tejas), and air (v¯ ayu) are atomic composed of indivisible, and indestructible atoms (an.u, param¯ an.u); self (¯ atman), which is the eighth, is omnipresent and eternal; and, lastly, the ninth, is the mind (manas), which is also eternal but of atomic dimensions, that is, infinitely small. There are seventeen qualities (gun.a), listed in no particular order as color or form (r¯ upa), taste (rasa), smell (gandha), and touch (spar´sa); number (sam a), . khy¯ size or dimension (parim¯ an.a), separateness (pr.thaktva), conjunction (sam . yoga),

28

and disjunction (vibh¯ aga); remoteness (paratva) and nearness (aparatva); judgment (buddhi), pleasure (sukha), pain (duh.kha), desire (icch¯ a), aversion (dves.a), and effort (prayatna). These qualities are either physical or psychological. Remoteness and nearness are interpreted in two different ways: temporally or spatially. This list is not taken to be comprehensive because later sound is also described as a quality. But there is a fundamental difference between sound and light. Sound is carried by the non-atomic a¯k¯ as´a, whereas light, implied by r¯ upa, is carried by tejas atoms. But even sound is sometimes seen as a specific characteristic of atoms. There are five different types of motion (karman) that are associated with material particles or the organs of the mind: ejection, falling (attraction), contraction, expansion, and composite motion. Universals (s¯ am¯ anya) are recurrent generic properties in substances, qualities, and motions. Particularities reside exclusively in the eternal, non-composite substances, that is, in the individual atoms, souls, and minds, and in the unitary substances ether, space, and time. Inherence (samav¯ aya) is the relationship between entities that occur at the same time. This provides the binding that we see in the various categories so that we are able to synthesize our experience. The Vai´ses.ika atomic structure characterizes four of the five S¯ am a. khyan mah¯ bh¯ utas; the fifth, ether, is non-atomic and all-pervasive. Some of the Vai´ses.ika gun.as correspond to the S¯ am atras. In S¯ am atras come . khyan tanm¯ . khya the tanm¯ first, in Vai´ses.ika atoms are primary. In the medieval period, Dign¯ aga (c. 500 CE) argued that inference is a function of three terms: the property to be inferred (s¯ adhya), the inferential mark (s¯ adhana), and the locus (paks.a). Kum¯ arila Bhat.t.a (c. 700 CE) argued that language can generate cognition of non-existent entities in what are empty terms (such as “horned rabbit” or the “son of a barren woman”). Udayana (10th century) refuted the Buddhist view of the momentariness of all entities. 6. Navya Ny¯ aya In the thirteenth century, the Navya Ny¯ aya (New Logic) system was founded by Ga˙nge´sa Up¯ adhy¯ aya of Mithil¯ a [17],[18],[19]. Its development was influenced by the work of earlier philosophers V¯ acaspati Mi´sra (10th century) and Udayana. It developed a highly technical language to formulate and solve problems in logic and epistemology.

29

Ga˙nge´sa’s book Tattvacint¯ aman.i (“Thought-Jewel of Reality”) dealt with important questions in logic, set theory, and epistemology, which improved the Ny¯ aya scheme. It systematized the Nyaya categories of perception (pratyaks.a), inference (anum¯ ana), analogy (upam¯ ana), and testimony (´sabda). A property with an empty domain was taken to be fictitious or unreal and nonnegatable. Negation was considered a valid operation only on real properties. This could be considered to generate a three-valued table. If P, N, and U represent “positive”, “negative”, and “unnegatable”, then we have the truth table [12]: w not-w P N N P U U Knowledge was taken to be analyzed into three kinds of epistemological entities in their interrelations: “qualifier” (prak¯ ara); “qualificand”, or that which must be qualified (vi´ses.ya); and “relatedness” (sam . sarga). For each of these was the corresponding abstract entity: qualifierness, qualificandness, and relatedness. The knowledge expressed by the judgment “This is a red flower” was then analyzed in the following sense: “The knowledge that has a qualificandness in what is denoted by this is conditioned by a qualifierness in red and conditioned by another qualifierness in flowerness.” Various relations were introduced, such as direct and indirect temporal relations, pary¯ apti relation (in which a property resides in sets rather than in individual members of those sets), svar¯ upa relation (which holds, for example, between an absence and its locus), and relation between the cognition of a knowledge and its object. The concept of “limiterness” (avacchedakat¯ a) was put to many different uses. If a field has fire in one region and not in another, the Navya Ny¯ aya proposition would be expressed as: “The field, as limited by the region A, possesses fire, but as limited by the region ¬A possesses the absence of fire.” In the manner, limitations of time, property, and relation were also described. The notion of negation was developed beyond specifying it with references to its limiting counterpositive (pratiyogin), limiting relation, and limiting locus.

30

Questions such as the following were asked: Is one to recognize, as a significant negation, the absence of a thing A so that the limiter of the counterpositive A is not A-ness but B-ness? Ga˙nge´sa believed that the answer to these three questions was in the negative. He, however, believed that the absence of an absence itself could lead to a new property. 7. Concluding remarks The important role that logic has enjoyed in the Indian cultural area is due to belief in the overarching Vedic system that knowledge is of two kinds: objective and subjective. All objective knowledge is governed by logic, whereas the knowledge related to the experiencing subject is extra-logical or transcendent. Since ordinary science must be objective, logic is essential in all philosophy and scientific disciplines. The philosophical systems of the Ny¯ aya and the Vai´ses.ika, together with the philosophical school related to grammar, kept up a continuing debate over centuries that led to the consideration of several subtle problems. The culmination of the Indian logical tradition was in the Navya Ny¯ aya school. References [1] K.H. Potter (ed.), Indian Metaphysics and Epistemology: The Tradition of Nyaya-Vaisesika Up to Gangesa. Delhi: Motilal Banarsidass and Princeton: Princeton University Press, 1977. [2] K.H. Potter and S. Bhattacharyya (eds.), Indian Philosophical Analysis: Nyaya-Vaisesika from Gangesa to Raghunatha Siromani. Delhi: Motilal Banarsidass and Princeton: Princeton University Press, 1993. [3] S.C. Vidyabhusana, A History of Indian Logic. Calcutta: University of Calcutta, 1921. [4] S.C. Vidyabhusana, History of the Medieval School of Indian Logic. Calcutta: University of Calcutta, 1909. [5] S. Kak, Aristotle and Gautama on Logic and Physics. arXiv:physics/ 0505172.

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[6] S. Kak, The Architecture of Knowledge. Delhi: CSC and Motilal Banarsidass, 2004. [7] T. McEvilley, The Shape of Ancient Thought: Comparative Studies in Greek and Indian Philosophies. New York: Allworth Press, 2002. [8] J.E.M. Houben, Bhartrhari’s solution to the Liar and some other paradoxes. Journal of Indian Philosophy, 23, 1995, 381–401. [9] R.P. Kangle, The Kaut.il¯ıya Artha´s¯ astra. Delhi: Motilal Banarsidass, 1986. [10] K.M. Ganguly, The Mahabharata. New Delhi: Munshiram Manoharlal, 1991. [11] S.C. Vidyabhusana, The Ny¯ aya S¯ utras of Gotama, revised and edited by Nandalal Sinha. Delhi: Motilal Banarsidass, 1990. [12] B.K. Matilal, Logic, Language and Reality. Delhi: Motilal Banarsidass, 1985. [13] S. Dasgupta, A History of Indian Philosophy. Cambridge: Cambridge University Press, 1932. [14] S. Kak, Greek and Indian cosmology: review of early history, [in:] The Golden Chain. G.C. Pande (ed.). New Delhi: CSC, 2005; ArXiv:physics/ 0303001. [15] B.K. Matilal, Nyaya-Vaisesika. Otto Harrassowitz, Wiesbaden, 1977. [16] S. Kak, Physical concepts in Samkhya and Vaisesika, [in:] Life, Thought and Culture in India (from c 600 BC to c AD 300), edited by G.C. Pande, New Delhi: ICPR/Centre for Studies in Civilizations, 2001; arXiv: physics/0310001. [17] B.K. Matilal, Perception: An Essay on Classical Indian Theories of Knowledge. Oxford: Clarendon Press, 1986. [18] K.K. Chakrabarti, Some Comparisons Between Frege’s Logic and NavyaNyaya Logic. Philosophy and Phenomenological Research, 36, 1976, 554–563.

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[19] B.K. Matilal, The Word and the World. India’s Contribution to the Study of Language. Oxford: Oxford University Press, 1990.

ON TWO QUESTIONS OF THE NEW LOGIC OF INDIA Kamaleswar Bhattacharya Indology Department University of Bonn Bonn, Germany [email protected] Formal logic was born in two – and only two – cultural spheres: the western and the Indian. There was a ‘formalism,’ too, in Indian logic. The last phase of this logic, the ‘new logic,’ created a new language which, as with the ‘new logic’ of the West, became the language of other disciplines as well. But, while the new logic of the West followed the mathematical model, that of India followed the linguistic model: in that they followed two distinct traditions. Unlike the western, the Indian new logic did not construct an ‘artificial language,’ consisting in a system of symbols, but formulated its definitions and solved various logical problems with different combinations of concepts in natural language. Very little work has been done so far on the subject in western and Indian languages. The present paper deals with two questions which have been the subjects of discussion in recent years.

So far as our present knowledge goes, formal logic was born in two – and only two – cultural spheres: the western and the Indian [15, p. 13]. I call Indian logic ‘formal’ in the sense in which I. M. Bocheński did: Daß es sich um eine formale Logik handelt, sieht man daraus, daß die indischen Denker Formeln aufstellen, welche sich auf die Grundfrage der Logik beziehen, nämlich auf die Frage, ob etwas aus etwas anderem folgt oder nicht – und zwar so, daß diese Formeln als allgemeingültig gedacht werden [15, p. 516].

It has been shown that there was a ‘formalism,’ too, in Indian logic, in the sense in which Łukasiewicz understood it. ‘Formalism requires that the

35

same thought should always be expressed by means of exactly the same series of words ordered in exactly the same manner’ [20, p. 16], also see [9]. Thus, ‘India is the only country which developed independently a formal logic somewhat comparable with that due to the Greeks’ [14, p. 117]. The last phase of this logic, named Navya-nyāya, ‘New logic,’ began around the 13th century. Its creative period continued up to the 18th century and, to some extent, until recently. With the ‘new logic’ of the West, it has this in common, that it created a new language which became the language of other disciplines as well, such as philosophy, law, grammar, and poetics. But, while the new logic of the West followed the mathematical model, that of India followed the linguistic model: in that they followed two different traditions.1 Unlike the western, the Indian new logic did not construct an ‘artificial language,’ consisting in a system of symbols; instead, fully exploiting the extraordinary power of abstraction of the Sanskrit language and the acute linguistic theories developed by the ancient Indian linguists, it created a language of its own, which is not ‘artificial’ in the strict sense but which is free from many of the ambiguities of ordinary language – a language which even the best knowers of Sanskrit must learn [11]. Thus, Navya-nyāya formulates its definitions and solves various logical problems with different combinations of concepts in natural language [12]. To be able to read an original Navya-nyāya text, one must be familiar with these concepts. But ‘the small store of western tools for the interpretation of Navya-nyāya,’ as Daniel H. H. Ingalls, one of the very few western Sanskritists ever interested in the subject, used to put it, is not adequate for that purpose. Nor are the Indian tools in the form of textbooks such as the Siddhāntamuktāvalī, the couple of dictionaries, one in Sanskrit2 and another in Bengali,3 the exposition in Sanskrit by Mahāmahopādhyāya

1

See K. Bhattacharya’s paper [8] with the references. The well-known Nyāyakośa by Mahāmahopādhyāya Bhīmācārya Jhalakīkar (revised and re-edited by Mahāmahopādhyāya Vāsudev Shāstrī Abhyankar, Poona: Bhandarkar Oriental Research Institute, 4th edition, 1978). 3 Bhāratīya Darśana Kosa by Srimohan Bhattacharya and Dinesh Chandra Bhattacharya, Vol. I, Calcutta: Sanskrit College 1978. 2

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Maheśacandra Nyāyaratna,4 the scholar who in the 19th century taught scholars such as E. B. Cowell, and its two supplements by Mahāmahopādhyāya Kālīpada Tarkācārya, the teacher of Ingalls in 1941.5 To be familiar with these concepts, one has, therefore, often to go back to the texts themselves, with the help of the traditional scholars who, over generations, have transmitted an immense store of knowledge. Now the number of these scholars has considerably diminished in recent years, and a great number of Navya-nyāya texts have remained unpublished so far. Even when they are published, no critical edition is available. This is the case, among others, with what is considered to be the basic text of the school, the Tattvacintāmani of Gaṅgeśa (14th century). However, old commentators point out variant readings. Particularly important for following the evolution of the concepts are the commentaries on the Tattvacintāmaṇi that preceded Raghunātha’s Dīdhiti (15th century), which eclipsed them all. Not only are many of them lost, but even those which have survived, sometimes in unique manuscripts, have not been fully or properly edited.6 Texts widely read were often manipulated, with the consequence that some of them have come down to us in different versions, making it impossible to determine with certainty what were the texts written by the authors. Old manuscripts contemporaneous with the authors were sometimes noticed in the past; but they are no longer traceable.7 Traditional scholarship hardly attaches any importance to these textual problems; instead, scholars prefer 4

Brief Notes on the Modern Nyāya System of Philosophy and its Technical Terms, Calcutta 1891 (read at the Ninth International Congress of Orientalists, 1891). 5 Nyāyadarśanabinduh. Vārānasī: Vārānaseyasamskrtaviśvavidyālayah samv. 2021 (AD 1964). Navyanyāyabhāsāpradīpah, containing Nyāyaratna’s text with Tarkācārya’s Bengal translation and commentary and Tarkācārya’s Pariśista with Bengali translation. Calcutta: Sanskrit College 1973. Also useful is Venīmādhava Śukla’s Pariskāradarpana with the notes by his son Rājanārāyana Śukla (Benares 1934). 6 See, e.g., [7], on the Vienna edition of Yajñapati Upādhyāya’s Tattvacintāmaniprabhā (Anumānakhanda), the earliest commentary known so far on the Tattvacintāmani. Also bad is the Baroda edition (1999) of Narahari Upādhyāya’s Anumānadūsanoddhāra (Gaekwad’s Oriental Series 179). 7 Thus, the manuscript of the Jagadīśī mentioned by D. C. Bhattacharya in [3, p. 171].

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to speculate on readings handed down from generation to generation, even when they are basically wrong.8 In these circumstances, I wish to elucidate two questions of Navya-nyāya, which have been the subjects of discussion in recent years. I. Ubhayābhāva ‘absence of both’. Navya-nyāya often operates with this concept (see, e.g., [12]). There is another concept, anyatarābhāva ‘absence of either.’ Both the expressions are ‘rather peculiar,’ as Saileswar Sen in his Amsterdam dissertation, published in 1924, observed. Nevertheless, he clearly distinguished between these concepts. There is ‘absence of both,’ he said, when one of the two entities is absent, and there is ‘absence of either’ when neither is present [26, p. 26]. The former interpretation is in accord with the principle that Even when one of the entities is present, there is the apprehension: ‘The two do not exist together.’9

Ingalls, however, did not agree. He tried to demonstrate that there is also ‘absence of both’ when both the entities are absent, while recognizing that there is ‘absence of either’ only when both the entities are absent [17, pp. 64—65]. V. N. Jha opposed this interpretation, to uphold that of Sen [19]. Now, the truth is that Ingalls was wrong in his demonstration, but right in holding the opinion he held. There are, indeed, Navya-nyāya texts where ubhayābhāva ‘absence of both’ is found used to cover both the cases where there is absence of only one of the entities and those where there is absence of both the entities [10, p. 237 n. 62], [2, p. 199]. Jha himself cites a ‘puzzling’ case where ubhayābhāva ‘absence of both’ is used for what is anyatarābhāva ‘absence of either’ [19, p. 243]. Furthermore, Nīlakaṇṭha, the famous commentator on Annaṃbhaṭṭa’s textbook Tarkasaṃgraha(dīpikā), says in his comment upon the passage referred to

8

One such case was studied by me thirty years ago. See [5], [6, 292 n, p. 34; pp. 321—322]. For the traditional interpretation, see also [18, pp. 104—105] – one of the most eminent scholars of the 19th and 20th centuries. 9 ekasattve ’pi dvayam nāstīti pratīteh, Siddhāntamuktāvalī, [27, p. 221]. Cf. Tarkasamgraha(dīpikā), see [28, p. 187].

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above:10 ‘In a case like “there are not pot and cloth,” the triad potness, clothness and bothness is the delimiter of the counterpositiveness; for there is such a notion with respect to what has only a pot or only a cloth or any [other] two things.’11 As the Bhāskarodayā commentary by his own son makes it clear, in the last part of this sentence Nīlakaṇṭha indicates that there is ‘absence of both’ when there is absence of either of the two things or of both the things.12 Here is also an example from an 18th-century text on poetics, using the Navya-nyāya language. Simile (upamā) is defined there as: yatsādṛśyapratiyogitāyām upameyatāvacchedakāvacchinnatvasvāśrayamātravṛttisvānavacchedakadharmasāmānādhikaraṇyobhayābhāvaḥ sopamā Simile is that [relation of] similarity,13 in the adjunctness of which there is absence of both the fact of being delimited by the delimiter of the property of being the object of comparison and the fact of sharing a locus with a property which resides only in its [= of the adjunctness] locus and which does not delimit itself [= the adjunctness].14

To explain: In a simile such as candra iva mukham ‘face like the moon,’ the adjunct (pratiyogin)15 of the relation of similarity (sādṛśya) is the moon. In this adjunct, i.e. the moon, resides the adjunctness (pratiyogitā) which is delimited by moonness (candratva), the uncommon property of the moon which is apprehended as its determinant (viśeṣaṇa).16 Now, if the figure were ananvaya (self-comparison), e.g. candraś candra iva ‘the moon is like the moon,’ the adjunctness of the relation of similarity would be delimited by the delimiter of the property of being the object of 10

Note 9. ghatapatau na sta ityādau ghatatvam patatvam ubhayatvam caitat tritayam pratiyogitāvacchedakam, kevalaghatavati kevalapatavati yatkimcidubhayavati ca tathā pratyayāt. Tarkasamgraha [28, p. 187]. 12 anenobhayābhāvo ’nyatarābhāvaprayuktah kvacid ubhayābhāvaprayukta iti sūcitam [28, p. 188]. 13 This is one of the interpretations possible, according to the commentary. 14 Alamkārakaustubha [1, p. 5]. 15 See [17, p. 44]. 16 [4, p. 130]. See also here below. 11

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comparison, i.e. the same moonness (candratva). In order to avoid ‘overpervasion’ (ativyāpti) of the definition to this figure is therefore said that there is in the adjunctness absence of the fact of being delimited by the delimiter of the property of being the object of comparison (upameyatāvacchedakāvacchinnatva). On the other hand, if the figure were vyatireka (difference), the adjunctness residing in the adjunct, i.e. the moon, would share a locus (samanādhikaraṇa)17 with, e.g., the property stainedness (kalaṅkitva) = stain (kalaṅka),18 which resides only in the moon, the locus of the adjunctness, but does not delimit the adjunctness (svāśrayamātravṛttisvānavacchedakadharma), -- a property which would show the inferiority of the standard of comparison, the moon, to the object of comparison, the face. To avoid overpervasion to this figure, therefore, it is said that there is in the adjunctness absence of the fact of sharing a locus with such a property. Here it is obvious that the term ubhayābhāva has been used to mean both the absences together. However, our author was not unaware of the distinction between ubhayābhāva and anyatarābhāva: the delimiter of the counterpositiveness (pratiyogitāvacchedaka) of the ‘absence of both’ is ‘bothness’ (ubhayatva), whereas the delimiter of the counterpositiveness of the ‘absence of either’ is ‘eitherness’ (anyataratva); the former property resides in both together (vyāsajyavṛtti), while the latter resides in two separately [19, p. 244]. Our author, therefore, makes clear in his commentary that by ubhayābhāva here is meant an absence having ‘both’ as counterpositive (pratiyogin) but the counterpositiveness of which is delimited by ‘eitherness’ (ubhayapratiyogiko 19 ’nyataratatvāvacchinnapratiyogitāko ’bhāvaḥ). II. The other question I wanted to deal with is that of the concept or concepts which underlie the so-called substitution rule traditionally transmitted without being recorded – so far as I am aware – in any ancient 17

On samānādhikarana (‘sharing a locus’) and sāmānādhikaranya (‘community of locus’) – to be distinguished from the same terms in Grammar, where they mean ‘coreferential,’ ‘co-referentiality’ – see [4, p. 99 n. 2]. 18 See below. 19 On pratiyogin, pratiyogitā in the context of absence or negation, see [17, p. 55].

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text: tadvattvaṃ tad eva, i.e. the property of being that which possesses something is the very thing possessed. On this rule are based such abbreviations as dhūmavattva ‘property of being that which possesses smoke’ = dhūma ‘smoke,’ which Navya-nyāya frequently uses. Actually, it is not confined to Navya-nyāya. Thus in poetry are to be found amplifications (not abbreviations) such as vegitā ‘property of being that which has velocity’ = vega ‘velocity’ [25, p. 36]. Only the neo-logicians are alone to express it in this form. There is another rule, more trivial, which is noted by Ingalls: tattvavat tad eva ‘that which has thatness is that itself.’ Ingalls cites as example pṛthivītvavat ‘possessor of earthness’ = pṛthivī ‘earth’ [17, p. 36]. But, by a strange confusion, he also brings under this rule saṃyogitva ‘property of being that which has contact’ = saṃyoga ‘contact,’ [17, p. 114], cf. [13, p. 175] which should come under the rule being considered here. Despite Matilal,20 the question involved here is not ontological, but semantical and logical. All the elements of explanation were given to Toshihiro Wada by his Indian teacher or teachers, and he cites them in a little note, [29, p. 43 n. 32] but he was not able to understand them – as is shown also by his developed study on the question.21 Nor was Claus Oetke, who used them. He therefore declared: ‘It appears almost certain that the tenet of the identity ‘X = X-vatta’ is the outcome of some serious confusion’ [24, p. 59 n. 2], see also [31, p. 50 n. 12]. The confusion, rather, lies in the fact that neither Wada nor Oetke sees that the statement (cited by Wada) viśiṣṭād bhāvapratyayo viśeṣaṇam abhidhatte ‘an abstract suffix introduced after a word denoting a determined entity expresses the determinant’ does not mean exactly the same thing as Kātyāyana’s vārttika 5 on Pāṇini’s Grammar V, 1, 119 (also cited by Wada). The latter says that the abstract suffixes –tva- and –tā- are used to express the quality (guṇa = viśeṣaṇa ‘determinant’) thanks to the presence of which a word is applied to a thing (yasya guṇasya bhāvād 20

[22, pp. 29—30] = [16, pp. 212—213]. See also [30, p. 361]. Wada’s reference ([30, p. 357] to Matilal [21, p. 101], where the author speaks of Kātyāyana’s vārttika, is a result of the confusion that is going to be spoken of below. 21 [30]. This seems to have been the case also with the late Justice A. K. Mukherjea: [23, p. 49 n. 12].

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dravye śabdaniveśas tadabhidhāne tvatalau), in other words, to express the ‘cause of the application of a word’ (śabdapravṛttinimitta). Now, starting from this vārttika, the statement in question makes clear how, e.g., dhūmavattva ‘property of being that which possesses smoke’ = dhūma ‘smoke:’ dhūmavattva is the ‘determinant’ (viśeṣaṇa) to dhūmavat ‘that which possesses smoke,’ as dhūmatva ‘smokeness’ is the determinant to dhūma ‘smoke;’ but dhūma ‘smoke’ is already the determinant to dhūmavat ‘that which possesses smoke;’ dhūmavattva is, therefore, identified with dhūma. This is what is meant by the statement ‘an abstract suffix introduced after a word denoting a determined entity expresses the determinant.’ All that I succinctly stated as early as 1977, following the explanation of my teacher Paṇḍitarāja Badarīnātha Śukla [4, p. 105 n. 9]; but it was not noticed by anybody. It is true that it was in French. References [1] Alaṃkārakaustubha of Viśveśvara. Edited by M. M. Śivadatta and Kâshînâth Pâṇdurang Parab. Bombay: Nirṇaya-Sâgara Press 1898 (Kāvyamālā 66) [2] Bhaṭṭācārya Vāmācaraṇa. Vivṛti on the Siddhāntalakṣaṇa. Vārāṇasī, 1933. [3] Bhattacharya, Dinesh Chandra. Baṅge Navyanyāycarcā (in Bengali). Calcutta, 1952. [4] Bhattcharya, Kamaleswar. Le Siddhāntalakṣaṇaprakaraṇa du Tattvacintāmaṇi de Gaṅgeśa, avec la Dīdhiti de Raghunātha Śiromaṇi et la Ṭīkā de Jagadīśa Tarkālaṃkāra. Journal Asiatique CCLXV, 1977, 97 – 139. [5] Bhattcharya, Kamaleswar. On a Passage of the SiddhāntalakṣaṇaJāgadīśī. Ludwik Sternbach Felicitation Volume, Lucknow: Akhila Bhāratīya Sanskrit Parishad, 1979, 479 – 483. [6] Bhattcharya, Kamaleswar. Le Siddhāntalakṣaṇaprakaraṇa … (suite). Journal Asiatique CCLXVIII, 1980, 275 – 322. [7] Bhattcharya, Kamaleswar. Yajñapatyupādhyāyaviracitāyāṃ Īśvaravādavyākhyānam (in Sanskrit). Tattvacintāmaṇiprabhāyām

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Lokaprajñā: Prof. N. S. Rāmānuja Tātācārya Felicitation Volume, Puri, 1988, 275 – 294. [8] Bhattcharya, Kamaleswar. Sur la base grammaticale de la pensée indienne. Langue, style et structure dans le monde indien. Centenaire Louis Renou, Paris, 1996, 171 – 185. [9] Bhattcharya, Kamaleswar. A Note on Formalism in Indian Logic. Journal of Indian Philosophy 29 (Ingalls Festschrift), 2001, 17 – 23. [10] Bhattcharya, Kamaleswar. Le Siddhāntalakṣaṇaprakaraṇa … (suite). Journal Asiatique 293, 2005, 213 – 244. [11] Bhattcharya, Kamaleswar. On the Language of Navya-Nyāya: An Experiment with Precision through a Natural Language. Journal of Indian Philosophy 34, 2006, 5 – 13. [12] Bhattcharya, Kamaleswar. On the ‘Generosity’ of a Natural Language. Journal of Indian Philosophy 35, 2007, 413 – 416. [13] Bhattacharyya, Sibajiban. Some Aspects of the Navya-Nyāya Theory of Inference, [in:] [16], 162 – 182. [14] Bocheński, Innocentius Maria (later Józef Maria). Review of Ingalls, Materials…, Journal of Symbolic Logic 17, 1952, 117 – 119. [15] Bocheński, Józef Maria. Formale Logik. 5th edn. Freiburg / München: Verlag Karl Albert, 1996. [16] Ganeri, Jonardon (ed.), Indian Logic: A Reader. Richmond, Surrey: Curzon Press, 2001. [17] Ingalls, Daniel Henry Holmes. Materials for the Study of NavyaNyāya Logic. Cambridge, Mass, 1951 (Harvard Oriental Series 40). [18] Jha, Dharmadatta (Baccā). Siddhāntalakṣaṇatattvālokaḥ. Vārāṇasī, 1925. [19] Jha, Vashishtha Narayan. On Ubhayābhāva, Anyatarābhāva and Viśiṣṭābhāva. Annals of the Bhandarkar Oriental Research Institute LXIII, 1982. 239 – 244. [20] Łukasiewicz, Jan. Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, 2nd edn. Oxford: Clarendon Press, 1957. [21] Matilal, Bimal Krishna. Epistemology, Logic, and Grammar in Indian Philosophical Analysis. The Hague-Paris: Mouton, 1971. [22] Matilal, Bimal Krishna. The Character of Logic in India. Edited by Jonardon Ganeri and Heeraman Tiwari. Albany: State University of New York Press, 1998 (SUNY Series in Indian Thought: Text and Studies).

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[23] Mukherjea, A. K. The Definition of Pervasion (Vyāpti) in NavyaNyāya. Journal of Indian Philosophy 4, 1976, 1 – 50. [24] Oetke, Claus. Indian Logic and Indian Syllogism. Indo-Iranian Journal 46, 2003, 53 – 69. [25] Renou, Louis. Sur la structure du Kāvya. Journal Asiatique CCXLVII, 1959, 1 – 114. [26] Sen, Saileswar. A Study on Mathurānātha’s Tattvacintāmaṇirahasya. Wageningen, 1924. [27] Siddhāntamuktāvalī: Nyāyasiddhāntamuktāvalī of Viśvanātha Pañcānana Bhaṭṭācārya with Dinakarī (Prakāśa) Commentary by Mahādeva Bhaṭṭa and Dinakarabhaṭṭa and Rāmarudrī (Taraṅgiṇī) Commentary by Rāmarudra Bhaṭṭācārya & Pt. Rājeśvara Śāstrī, edited by Harirāma Śukla Śāstrī. Vārāṇasī 1972: Chowkhamba. [28] Tarkasaṃgraha of Annaṃbhaṭṭa. In: Bhāskarodayā, ed. Mukunda Jha. Bombay: Nirṇaya-Sâgar Press. Fourth edition, revised by Wâsudev Laxmaṇ Śâstrî Paṇśîkar, 1926. [29] Wada, Toshihiro. Invariable Concomitance in Navya-Nyāya. Delhi: Sri Satguru Publications, 1990. [30] Wada, Toshihiro. A Rule of Substitution in Navya-Nyāya: x-vat-tva and x. Nyāya-Vaśiṣṭha: Felicitation Volume of Prof. V. N. Jha, Kolkata: Sanskrit Pustak Bhandar, 2006, 356 – 369. [31] Wada, Toshihiro. The Analytical Method of Navya-Nyāya. Groningen: Egbert Forsten 2007 (Gonda Indological Studies XIV).

THE USE OF FOUR-CORNERED NEGATION AND THE DENIAL OF THE LAW OF EXCLUDED MIDDLE IN NĀGĀRJUNA’S LOGIC Dilipkumar Mohanta Philosophy Department University of Calcutta Kolkata, India [email protected], [email protected] In this paper I propose to argue that Nāgārjuna has a different meta-level presupposition for using ‘negation’ in the ‘non-relational sense,’ because only this can enable him to reject the validity of the so-called law of excluded middle without ‘inconsistency-phobia.’ For him, the law of excluded middle does not convey any necessary truth and therefore, its denial is consistently possible. As a corollary of this denial, he would obviously deny the validity of the law of double negation. It is only an indication not to allow any categorical view about the world. I shall try to develop this thesis in and through our interpretation of the use of ‘negation’ by Nāgārjuna in connection with the four-fold patterns of ‘theory-making’ in Indian philosophy.

Indian logicians have used two different types of negation. They have their different meta-level presuppositions. There are logicians who believe that the world can be exhaustively divided into two categories, namely, positive and negative and the one is the negation of the other. This type of negation is called parjudāsa pratiùedha in Sanskrit. It is ‘relational negation.’ It is also called ‘nominally bound term negation.’ Philosophers who believe in two valued logic use this type of negation. The users of this type of

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negation believe that everything in epistemology is either true or false, in metaphysics either real or unreal. If ‘something is true’ is meant by ‘p’, then ‘it is not the case that something is true’ would be meant by ‘~p’. It is their meta-level presupposition. A large section of philosophers and the common people at large use this type of negation in order to understand the world. They accept ‘p’ and ‘~p’ as mutually exclusive and understand the statement ‘p exists’ as ‘p exists unconditionally,’ in absolute sense; they understand ‘~p’ as ‘p does not exist’ also in absolute sense. To them what is real is unconditionally real and what is called unreal is unconditionally / absolutely unreal. Any conjunction of ‘p’ and ‘~p’ is an explicit contradiction. In order to avoid the contradiction they accept the law of excluded middle as necessarily valid. For them ‘p ∨ ~p’ cannot be negated and they claim that ‘p ∨ ~p’ can explain the world in an exhaustive way. As a corollary, they also accept ‘double negation’ as a valid principle of reasoning. Gautama and other realist philosophers, like the Mīmāmsakas and the Buddhist realists believe in this type of negation. The Nyāya championed this relational negation. The Nyāya philosophers claim that there exists a necessary relation of universality between ‘know-ability’ and ‘name-ability.’ Nāgārjuna in Vigrhavyāvartanī v. 57 reduced this claim to absurdity using another type of negation called prasajya pratiùedha, the ‘pure’ and ‘simple’ negation. In contrast to parjudāsa pratiùedha, which is relational negation, it is non-relational [8, p. 298]. In such a negation when you negate a thesis, e.g. ‘p’ as false, you need not accept the counter-thesis, e.g. ‘not-p’ as true. It is a negation without any kind of assertion or commitment. Gorisse [2] calls it ‘denegation.’ This type of negation has been used by Sañjaya (the 6th century BCE) in Amarāvikùepavāda and Nāgārjuna in prasaïgāpādāna. It is used in the four-cornered negation. It is contextual negation, the negation of the opposite in a certain context. The users of this type of negation have a different kind of meta-level presupposition. And for them, there is no rationale to say that the truthvalue must be limited to two only; it may be as many as the context demands. We cannot be sure about the nature of a thing and therefore we cannot claim that we know more things than we cannot know. This is mostly the logical stand of the skeptics who use dialectics in philosophizing in ancient India. For them, what is real is beyond the reach of these four typical patterns of human know-ability. We are

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epistemologically not in a position to claim anything in absolute terms. The world of our know-ability is a fluid one with all its fuzzy and definitely indefinable character and therefore, any kind of theory-making with absolute claim about the world is an exercise in dogmatism. Nāgārjuna, who does not subscribe to two valued logic, used negation as a tool of philosophizing. In Nāgārjuna’s philosophy the use of negation is integrally connected with his refutation of four different possible patterns (koti-s) by which usually philosophical theories are being built up. These are bhāva, abhāva, bhāvābhāva and na-bhāva-naivābhāva – may be expressed symbolically as the denial of ‘p’, ‘~p’, p & ~p, p ∨ ~p respectively. The aforesaid four patterns are usually considered as the possible ways of understanding what is real. Without entering into the controversy regarding this, let us see how it looks like if we represent the four different philosophical claims (which may be called T1, T2, T3 & T4) in the following table. A:T1:

A:T3: p

p &~p

p ∨ ~p

~p

B:T4:

B:T2:

It is interesting to see that Nāgārjuna negates all these four possible ways of explaining the world. But no philosopher, who believes in two valued logic, who believes in relational negation, can negate T4 (i.e., p ∨ ~p). This would be clear if we compare the following two, e.g., A in the right side and B in the left side. A represents T3 and the negation of T3 leads to the acceptance of T4. Again, B represents T4 and the negation of B leads to the acceptance A (i.e. T3). But no consistent philosopher can accept T3 as a thesis. Therefore, it is not possible to negate both T3 and T4. But

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Nāgārjuna negates all these four. We must have to assume that he uses ‘negation’ in a different sense as his meta-level pre-supposition is different. For him, truth values cannot necessarily be confined to two. His use of negation is known as ‘four-cornered negation’ among the interpreters of Indian philosophy. He would rather believe in many-valued logic – how many, maybe, we cannot claim in absolute terms. In order to get a comprehensive account of Nāgārjuna’s use of ‘negation’ we are to explain all these four cases of negation in the paragraphs to follow. T1: Since everything is conditionally originated, nothing can be said to have self-nature. Everything is ‘essence-less.’ When we say that ‘p’ exists, we do not mean that ‘p exists’ unconditionally, because it cannot be explained without the notion of ‘~p’. What is essentially, unconditionally existent cannot have the origination and that which has no origination cannot have destruction. In other words, a thing with self-nature can neither have a beginning nor have an end. In this world everything is devoid of self-nature (svabhāva). So there is no difficulty in negating the thesis, ‘p exists’ because it presupposes ‘~p’. Nothing exists unconditionally. Now let us consider ‘~p’ in T2. It can also be negated without any difficulty. It says, p is unreal in absolute sense and this is also an extreme view. But our experiences of the furniture of the world make it clear that nothing in this world is categorically unreal. Barren woman’s child, rabbit’s horn, sky-flower etc. are some of the classical examples of what is unconditionally unreal. If we admit that what conditionally exists is as good as what is absolutely unreal, then there would be a context of admitting nihilism (uccedavāda, sarvābhāvavāda). But both eternalism and nihilism are extreme theories. The world of conditional, relative existence cannot be called fictitious. What is evident here is that neither T1 nor T2 is acceptable as an independent koti, as an adequate ground for explaining something as unconditionally existent or unconditionally non-existent. Since both T1 and T2 are meaningful depending on each other, both of these alternatives are devoid of any intrinsic nature (niþsvabhāva). So the negation of ‘~p’ like the negation of ‘p’ does not create any difficulty. It rather emphasizes that ‘p’ represents eternalism and ‘~p’ represents nihilism and the nature

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of the world cannot be explained by any of these extreme views. Both ‘p’ and ‘~p’ are defective interpretations and therefore unacceptable. Now consider the third alternative, T3, i.e. ‘p & ~p’. In relational logic (which is otherwise known as propositional logic) T3 can be true if and only if the following three conditions be fulfilled. a) If ‘p’ and ‘~p’ – both are unconditionally true; both are having self-nature. b) If ‘p’ and ‘~p’ – both are not contradictory to each other. c) If between ‘p’ and ‘~p’ there exists no common property even in some instances. None of the afore-said conditions, according to Nāgārjuna is satisfiable in case of T3 (i.e. ‘p ∨ ~p’). He now tries to show absurdity in opponents’ own thesis in the light of their own norms. In Mūlamadhyamakakārikā, 12/9 Nāgārjuna explicates that the conjunction of both ‘self’ and ‘not-self’ can be the cause of suffering if in disconjoined positions they can be the cause of suffering. What seems to be evident is that T3 being a mechanical combination of the afore-said two extreme views which are defective, T3 contains the defects of those two and therefore, according to Nāgārjuna, is to be refuted. Even in accordance with those who divide the world into two exclusive compartments of ‘is’ and ‘is-not,’ T3 represents an explicit form of contradiction and therefore, it is to be rejected. To admit the categorical, unconditional existence of something as well as the non-existence of that thing simultaneously is to admit self-stultification. But there seems to be a very specific sense in which T3 can be rejected, according to Nāgārjuna. Candrakīrti in his commentary titled Prasannapadā on Mūlamadhyamakakārikā, 18/8 hinted towards a new interpretation of T3 and if we keep in mind Candrakīrti’s interpretation, we can apply negation without any difficulty to both T3 and T4. In other words, the rejection of the law of excluded middle does not necessarily lead to accept T3, the law of contradiction and the denial of T3, the law of contradiction does not necessarily lead to accept the law of excluded middle as valid. The text is as follows: Tatra bālajanāpekùayā sarvametattathyam. Āryajñānāpekùayā tu sarvametanmçùā (there ‘it is true’ under the condition of understanding of the lay persons whereas ‘ it is not true’under the condition of understanding of the noble persons) [6, vol 2, p. 72].

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In T3, according to Candrakīrti’s interpretation, ‘p’ is ‘true for’ a special class of individual and ‘~p’ is ‘true for’ a different class of individual. Since the conditions are different there is no difficulty even if we arrive at T3 by denying T4 and since T3 does not represent any categorically absolute sense of existence of either ‘p’ or ‘~p’, it can safely be said that the conjunction of ‘p & ~p’ is also devoid of its own nature, that is, independent existence. T4: Now it is to be seen how it is possible to deny ‘p ∨ ~p’ and the thesis is represented as T4. The T4 is the law of excluded middle. Nāgārjuna was not a believer in any exhaustive demarcation in exclusive terms of the world into two halves rejects T4 and this rejection as a matter fact, according to him, does not lead to any contradiction. Take for example, the application of the word ‘red.’ When we say that something is red, we also admit that there exists something ‘non-red’ which includes all colored objects other than the class of red. But if we mean only ‘red or non-red’ objects then the class of colorless objects would be left outside the scope of the colored objects, i.e. ‘red or non-red.’ Analogically, the text argues that all the furniture of the world cannot judiciously be classified into two exclusive classes of being ‘eternal or non-eternal.’ In Dṛùtiparīkùā section of Mūlamadhyamakakārikā, 27/18 Nāgārjuna shows the non-acceptability of the law of excluded middle when he says: Aśāśvata§ śāśvatañca prasiddhamubhaya§ yadi/ Siddhe na śāśvataṁ kāma§ naivāśāśvatamityapi// (If both the eternal and the non-eternal are [claimed to be] established, then neither eternal nor non-eternal is assertible) [8, vol. 2, p. 271].

We cannot say that eternal, non-eternal etc. are independent koti-s, alternatives. The meaning of one is always relative to the grasping of the meaning of the other. ‘A child of a barren woman’ cannot be classified either as eternal or non-eternal. Our experiential data caution us that there is no sufficient ground to classify the world in a clear and distinct way in absolute terms. In other words, all objects of the world do have relative, conditional existence. It is also called functional existence (vyāvahārika sattvā). Always there is the principle of interdependence and conditionality operating in the actual world and on the basis of this only we can explain

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the fact of arising, enduring, and ceasing. So in actual world there is no fruitful application of the law of excluded middle. We cannot deny that there are ‘border line’ cases like ‘colorless objects’ or ‘fictitious class of objects’ for the explanation of which the law of excluded middle is not only inadequate but also unnecessary. It is not necessary, because the denial of this does not lead to any contradiction as we have seen in Candrakirti’s interpretation. In relational logic though T4 (i.e. p ∨ ~p) is valid, but in non-relational logic, which denies that all possibilities within the universe of discourse as exhaustive, it is not valid. In other words, the law of excluded middle is not acceptable for the logic to which Nāgārjuna would subscribe. The 18th verse of Dṛùtiparīkùā section of the 27th chapter of Mūlamadhyamakakārikā expresses Nāgārjuna’s rejection of the validity of the law of excluded middle. In case of the application of ‘negation’ in T3, the problem is ‘how to make sense of what is denied.’ But in case of T4, our concern is ‘how the denial can make sense’ [1, p. 13]. This interpretation can be defended from the following sentences of Candrakīrti’s Prasannapadā on Mūlamadhyamakakārikā, 18/8: Yadyepyeva§ tathāpi vyavahārasatyānurodhena laukikatathyā daya bhypagamavattasyāpi samāropato lakùaõamucyatāmiti (though it appears as categorical, but in actual function it is conditional and this is said to be the defining characteristics of it) [6, vol. 2, p. 72].

We cannot deny that in actual world there are indistinct fuzzy areas about which no absolute truth-claim can be made and T4 for this reason does not convey any sense of ‘necessary truth.’ There is no guarantee that all objects of the world belong to, e.g. either ‘green or not-green’ classes, that is to say, the class of colorless objects does not belong to either of the classes of ‘green or not-green.’ All objects of the world have relative, context-bound, and interdependent existence. For a comprehensive explanation of the world the law of excluded middle is thus neither necessary nor sufficient. From this it is evident that we cannot say that the negation used by Nāgārjuna represents any ‘single pattern of alternatives, rather it includes a variety of patterns which differ from each other.’ Since the negation T4 (i.e., p ∨ ~p) does not invite any self-contradiction, we are to understand that the law of double negation would not be valid to Nāgārjuna. He would, on the contrary, think that when our ordinary

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language of day-to-day use is examined through philosophical scrutiny, then any absolute and extreme claim about reality would be turn out to be ‘non-sensical’ Candrakīrti’s interpretation brings novelty here – in case of T3 it means ‘true for’ which represents conditional, relative sense here. And in case of T4 it is to be understood as ‘border-line fuzzy cases.1 In case of T4, the concepts used are not defined in absolutely sharp and clear manner. Since there are fuzzy cases having ‘borderline’ intermingling of definitions, we cannot accept T4 as an adequate explanation of what is real. To emphasis this point Candrakīrti in his Commentary on Mūlamadhyamakakārikā 18:8 argues that ‘one who has to uncover some more lumps of mud to reach the extreme end of the roots of a tree cannot also clearly and distinctly characterize it.’2 This reminds us Frege’s explanation of the non-applicability of the law of excluded middle to ‘illdefined, vague or fuzzy areas,’ where we are incapable to say with certitude ‘whether the concept or its negation is applicable to it’ [3, p. 159]. Again, when nothing in this world is sharply defined, there is no sense in saying that the negation of T4, as a matter of fact, compels us to accept T3. T3 is nothing but a conjunction of two contradictory terms and there is no application of this possibility in the world. In other words, it is meaningless. It is non-sense also to say that it is the combination of two defective possibilities. The negation of both T3 and T4 is possible. The value-assignments in these two cases are different i.e. in T3, it is to be understood as ‘true for’ and in T4, it means ‘border line cases.’ That the two need different concepts to explain how they can be rejected, shows that the two rejections, even when regarded as negations, cannot be contradictory. It is only by ignoring how Nāgārjuna has actually argued for the rejection of these two positions in different places, that puzzlement over his theory can arise [1, p. 13].

Nāgārjuna’s use of negation is ‘pure negation.’ Any thesis whatever, if it is put forward before him, he would reject it without making any commitment to the counter-thesis. Here lies the clue of his art of nonasserting any thesis without contradiction. It is possible for him, because of 1

Nānārtho’steti nānārtha-bhinnārtha na nānārtho’ nānārthomithyarthaþ, see Prasannapadā on Mūlamadhyamakakārikā 18/9, see [8, vol. 2, p. 73]. 2 keùā§ cittvaticirābhyastatattvadarśanānā§ ki§ cinmātrānutsātavaraõatarumūlānā§ naivatathya§ naiva tathya§ taditi deśitam [8, vol. 2, p.72].

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his use of ‘negation’ with a different sort of ‘meta-level’ presupposition. Nāgārjuna negates different possibilities separately and in different senses. For him the world is not exhaustively divided into two absolute categories, namely ‘is’ and ‘is-not’. His dialectics (prasaïga) aims at showing inner contradiction in opponents’ thesis using their own logical norms and this enables him to negate any thesis without advancing any thesis of his own. Here lies the difference between Nāgārjuna’s dialectics (prasaïga) and the Nyāya method of tarka, indirect reasoning. The Nyāya describes the 5th type of tarka as ‘tadanya-vādhitārtha prasaïga’. In the Nyāya use of tarka there are only two alternative possibilities and the denial of one indirectly establishes the other. This also pre-supposes an exhaustive and sharp conceptual definition of the actual state of affairs. But since in actual state of affairs concepts cannot be sharply defined and divided into two alternatives in absolute sense, in the sense of only two possibilities, Nāgārjuna’s methodology cannot subscribe to it. He has shown that philosophers can use dialectics rejecting the basis of universality of the Nyāya tarka (indirect / hypothetical reasoning) into a ‘reduction to absurdity.’ His use of prasaïga is the rejection of all possible views about reality or description about the world. It seems to be a case of ‘deconditioning’ instead of ‘deconstruction.’ Negation of all theses is not one more thesis. It is like a ‘verbally bound predicate-negation.’ The distinction between prasajya pratiùedha, and parjudāsa pratiùedha roughly corresponds to Johnson’s understanding of difference between ‘S is not-P’ and ‘S is non-P’. Nāgārjuna’s negation of four alternatives is ‘commitmentless denial’ [5, p. 66]. His use of dialectics is a case of ‘unrestricted principle of reduction ad absurdum.’ As Nāgārjuna uses negation in each four possible positions (koti-s) separately, no question of affirming or denying of another alternative position can arise, as it arises in case of the ‘restricted use of ad absurdum principle.’ It is rather a methodological device of the relativistic reasoning by showing absurd consequences of the philosophical theses of the opponents. Nāgārjuna’s logic has its own meta-logical presupposition which is different from the meta-level presupposition of the Nyāya logic. References

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[1] Bhattacharya, Sibajiban. Some Unique Features of Buddhist Logic, [in:] S.R. Bhatt (ed.). Glimpses of Buddhist Thought and Culture. New Delhi, 1984. [2] Gorisse, Marie-Helene. The Art of Non-asserting: Dialogue with Nāgārjuna, [in:] R. Ramanujam and S. Sarukkai (eds.). Logic and Its Applications. ICLA, Springer Berlin Heidelberg, January, 2009. [3] Greach, P.T. and Black, Max (eds.). Frege’s Philosophical Works, Oxford University Press, 1970. [4] Johnson, W.E. Logic. Dover Pub., Inc. N.Y., 1964 (the 1st pub. 1921, Cambridge University Press). [5] Matilal, B.K. Perception. Oxford University Press, 1986. [6] Nāgārjuna. Mūlamadhyamakakārikā with Akutobhayā, Prasannapadā, and other Commentaries, 2 vols, ed. R.N. Pandey, Motilal Baranasidass, Delhi, 1988 & 1989. [7] Nāgārjuna. Vigrhavyāvartanī (Bengali translation and annotation by Dilipkumar Mohanta in Madhyamakadarśaner rūpa-rekhā O NāgārjunakŸta SavŸttivigrahavyāvartanī), Mahabodhi Book Agency, Kolkata, 2006. [8] Śabdakalpadruma, Vol. 3, Motilal Banarasidass, Delhi, 1963.

A PLEA FOR EPISTEMIC TRUTH: JAINA LOGIC FROM A MANY-VALUED PERSPECTIVE Fabien Schang LHSP Henri Poincaré, University Nancy 2 Nancy, France Technological University of Dresden Dresden, Germany [email protected] We present the Jaina theory of sevenfold predication as a 7-valued logic, in which every logical value consists in a 3-tuple of opinions. A questionanswer semantics is used in order to give an intuitive characterization of these logical values in terms of opinion polls. Two different interpretations are plausible for the last sort of opinion, depending upon whether “nonassertability” refers to incompleteness or inconsistency. It is shown that the incomplete version of JLG is equivalent to Kleene’s logic K3, whereas the inconsistent version of JLM is equivalent to Priest’s Logic of Paradox LP. Finally, it is argued that the Indian logics depart from Western logics by conflating truth and justified belief; the different preconditions for belief-assignment accounts for the difference between pluralist and skeptic logics.

1. Presentation: Jaina Logic Indian logics haven’t been so much scrutinized until now, in the widespread Western community of philosophers and logicians. The same could be said for other Eastern doctrines like Chinese, or even Arabic logic. Two reasons may be advocated for this state of affairs: not only are Indian logics more like a theory of judgment than a theory of consequence; but also, this family of Eastern logics has been often taken to be an

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irrational treasure of human thought that no Western reasoner should be able to understand. That Hegel has been largely influenced by Eastern works about contradiction and dialectics may have appeared as a sufficient reason not to take them seriously. But despite this negative outlook, a recent sample of modern philosophers and logicians attempted to show that, to the contrary, the peculiar case of Jaina logic does make sense once the appropriate modern tools are used to describe its rules (see the references). It will be argued in this same vein that an Indian logic can be translated within a first-order or modal language. The Eastern legacy of Jaina logic could be then translated by means of Aristotelian logic (for the first-order language) and Stoic logic (for the propositional language), so that the old-fashioned myth of an unavoidable gap between Eastern and Western philosophies should be definitely kept aside. Jaina logic is only one doctrine within a large family of philosophies that flourished from the 6th century BCE to the 17th century CE; let us mention among these doctrines the Catuskoti of Nagarjuna (2nd century BCE), the Nyāya doctrine of Gotama (2nd century BC), or the later Buddhist logic (from the 5th to the 14th century CE). The roots of Jaina logic can be traced back to the 6th century CE and have been developed until the 17th century. Like the ancient Greek schools, most of these Indian doctrines include ontologies, theories of judgment and associated grammars, and epistemologies. Jainism is a contemporary religion whose theoretical content served to make a rational sense of compassion and tolerance between opposite views. It includes both a theory of knowledge and a corresponding theory of judgment, which can be called a “logic” in the same sense as the Aristotelian syllogistic. The Jaina theory of knowledge is called nayavāda: it is a theory of standpoints that aims at identifying the various sorts of evidence by means of which a sentence can be assessed. Given the complex organization of the world in terms of substances, qualities or modifications, one and the same sentence can be assessed in a large number of different standpoints or naya. The doctrine of syādvāda is the logical part of Jainism that purports to identify the different kinds of resulting standpoints; it will be the central point of the present paper, since the Jaina doctrine of relative truth amounts to claim that no single sentence is absolutely true or false. This logical counterpart of a doctrine of relative

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truth requires some preliminary words, because it is not obvious prima facie why relativism should rise to a different form of logic. What is the peculiarity of that Indian logic, by contrast to the more familiar ancient systems of Aristotelian logic and Stoic logic? The main difference is described by a threefold distinction [2, p. 268], concerning the preconditions for a truth-assignment within a theory of justification: Doctrinalism is the view according to which it is always possible, in principle, to discover which of two inconsistent sentences is true, and which is false. Aristotelian logic endorses this dogmatic approach of truth, in the sense that the two logical principles of non-contradiction and excluded middle forbid a sentence and its negation to be both true and both false. The conjunction of these principles entails a consistency principle, according to which every sentence of a given language is considered as either true (if its negation is taken to be false) or false (if its negation is taken to be true). The classical bivalent logic of the Frege-Russell tradition subscribes to doctrinalism, as well as the recent AGM mainstream theory of belief revision from the 1970’s. In a nutshell, every consistent and complete logical system implicitly obeys this one-sided approach of truth. Skepticism argues that the existence both of a reason to assert and a reason to reject a sentence itself constitutes a reason to deny that we can justifiably either assert or deny the sentence. Pyrrhonism can be seen as the main school that supports this stringent condition for truth-assignment. It is worthwhile to note that the Greek skepticism is historically related to Pyrrho’s trip to India with his master Anaxarchus, where he would have been deeply impressed by the passionless behavior of the Gymnosophists. Nagarjuna’s Catuskoti (or the Tetralemma) is commonly assimilated with the skeptic approach of truth,

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and it is because of their strict refusal to anything true that the Buddhists dubbed them “eel-wrigglers.” No affirmation should be made so long as no conclusive evidence could be given correspondingly, whereas a current practice of scientific research should advise to make a choice between truth and falsity and subscribe to the “most plausible” evidence for or against a given sentence. Doctrinalism is preferred to skepticism with this respect, in the sense that the scientific discourse doesn’t pretend to produce standing or necessary truths. Pluralism, finally, purports to find some way conditionally to assent to each of the sentences, by recognizing that the justification of a sentence is internal to a standpoint. Pluralism can be viewed as the opposite side of skepticism, insofar as the latter rejects any contradictable statement from its theory whereas the former includes some of them at once. It seems more difficult to find a counterpart of pluralism in the ancient Greek schools; the unclear trend of eclecticism may be the case in point, although its main task is more extracting the best parts of different theories than retaining their whole content without any care for consistency. The logical trouble that may arise with pluralism concerns consistency: how to hold a meaningful theory in which any theoretical conflict between an affirmation and its opposite shouldn’t lead to an exclusive choice (doctrinalism) or a provisory suspension of judgment (skepticism)? A logical explanation of this objection to pluralism is the case of Explosion: any theory which contains both a given sentence p and its negation ~p is accused to entail the truth of any other sentence q, and that should be one the main reasons why Jaina logic had been taken to be an irrational source of the allegedly “Eastern thought.” Now a large family of logical systems is supposed to avoid such a disastrous consequence of contradiction: the so-called “paraconsistent” systems, where a theory can be inconsistent without entailing the truth of any sentence; we will see how a separation of contradiction and explosion can be devised within Jaina logic in order to support its epistemological doctrine of “non-one-sidedness” (anekāntavāda).

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The relative, non-one-sided view of pluralist truth can be finally depicted by a widespread tale of Eastern philosophy (from India to Japan). Six monks are asked what an elephant looks like by feeling different parts of its body. The blind man who feels a leg says the elephant is like a pillar; the one who feels the tail says the elephant is like a rope; the one who feels the trunk says the elephant is like a tree branch; the one who feels the ear says the elephant is like a hand fan; the one who feels the belly says the elephant is like a wall; and the one who feels the tusk says the elephant is like a solid pipe. Neither of them agrees with any other one; but a wise man explains to them that all of them is right. The reason every one of them is telling it differently is because each one of them touched a different part of the elephant. Actually the elephant has all the features they mentioned, and this accounts for the Jaina pluralist view of truth: this resolves the conflict, and is used to illustrate the principle in living in harmony with people who have different belief systems that truth can be stated in different ways. These different ways give rise to the syādvāda and its sevenfold predications. 2. Formalization of Jaina Logic The sevenfold theory of predication, or saptabhan۟gī, was presented by Vādiveda Sūri (1086 – 1169) by means of a basic sentence: “it exists,” which means that some object occurs with a specific property. According to Vādiveda Sūri, there are seven possible statements about this sentence: (1) Syād asty eva: arguably, it (some object) exists (2) Syān nāsty eva: arguably, it does not exist (3) Syād asty eva syān nāsty eva: arguably, it exists; arguably, it does not exist (4) Syād avaktavyam eva: arguably, it is non-assertable (5) Syād asty asti eva syād avaktavyam eva: arguably, it exists; arguably, it is non-assertable (6) Syān nāsty eva syād avaktavyam eva: arguably, it does not exist; arguably, it is non-assertable (7) Syād asti eva syān nāsty eva syād avaktavyam eva: arguably, it exists; arguably, it does not exist; arguably, it is non-assertable

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A short reflection is sufficient to see that each of these predications (or statements) is a combination of three basic statements, namely: assertion (truth-claim), denial (falsity-claim), and an unclear non-assertability. The meaning of the fourth predication is usually given by making a distinction between successive and simultaneous statements: (1) is a plain assertion and (2) is a plain denial, whereas (3) is a successive assertion and denial and (4) is described as a simultaneous assertion and denial; (5) is a combination of an assertion and a simultaneous assertion and denial; (6) is a combination of a denial and a simultaneous assertion and denial; finally, (7) is a combination of the three basic statements of assertion, denial, and non-assertability. The difference between (3) and (4) can be made explicit by using logical symbols of truth-values. Let v(p) = t mean that the sentence p is true or, in accordance with the Jaina epistemic view of truth, that p is held to be true. Then the three basic statements assign either t, f, or u to p and state that p is held to be either true, false, or non-assertable. The seven predications can be rephrased accordingly in terms of valueassignments: (1) v(p) = t; (2) v(p) = f, (3) v(p) = tf,

(4) v(p) = u, (5) v(p) = tu, (6) v(p) = fu, (7) v(p) = tfu.

The third predication assigns truth and falsity to p in a successive way, and this makes a crucial difference with (1) and (2) that respectively assign only truth and only falsity to p. That is: (1) means that v(p) = t and v(p) ≠ f, while (2) means that v(p) = f and v(p) ≠ t. More generally, each of the seven predications (except for the last one) excludes one or two truth-value assignments among t, f, and u, i.e. those not occurring in the above definitions. But such a semantic difference doesn’t settle a list of six main problems for Jaina logic, namely: What does syād mean? What does avaktavyam mean? Why are there seven values? How to define the logical connectives? How to single out the designated values? And, finally: can Jaina logic be a modal logic?

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2.1. What does syād mean? It is worthwhile to note that the expression syād is prefixed to any predication and may occur repeatedly, as in the combined cases (3), (5), (6) and (7). This helps to account for the seven predications and to disentangle the very idea of a logic with seven “truth-values.” For syād is read as “arguably,” “maybe,” or “in some respect;” accordingly, a formal interpretation of “syād” should amount to a definition of truth relative to a theory, a possible world, or a belief set. In accordance to Gokhale [5], the basic predication (1) can be rendered by an existentially quantified statement such as “there is a standpoint that makes p true,” or “there is an agent x that holds p to be true:” ‘∃xBx(p),’ where ‘B’ is a belief operator that proceeds like a truth-claim. Another symbolization of (1) can also be ‘◊’ in place of ‘∃xBx’, where the prefixed diamond expresses the modality of possibility. This results in the following reformulation of the seven predications, where the fourth case still remains unclear: Jain predication (1) v(p) = t (2) v(p) = f (3) v(p) = tf (4) v(p) = u (5) v(p) = tu (6) v(p) = fu (7) v(p) = tfu

First-order translation Modal translation ◊p ∃xBx(p) ∃xBx(~p) ◊~p ◊p ∧ ◊~p ∃xBx(p) ∧ ∃xBx(~p) ? ? ∃xBx(p) ∧ ? ◊p ∧ ? ◊~p ∧ ? ∃xBx(~p) ∧ ? ◊p ∧ ◊~p ∧ ? ∃xBx(p) ∧ ∃xBx(~p) ∧ ?

Each sentence p is thus interpreted in the metalanguage, e.g. as “it is believed by x that p” with (1). Again, the belief-operator B is not a doxastic counterpart of the modal box in our present translation; it just serves to express a propositional attitude that proceeds as a value-claim, so that a statement is said to be “true” or “false” whenever x believes or disbelieves p. That there are three basic statements in the Jaina logic is clearly reminiscent of a 3-valued system with t, f and u as its truth-values. A combination of these three value-claims should mean that the sevenfold predication can be reconstructed by means of some former 3-valued logics;

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but whether one of these is appropriate still depends upon the meaning of avaktavyam in the fourth Jain predication. 2.2. What does avaktavyam mean? The meaning of avaktavyam vacillates between an incomplete and inconsistent reading of a statement: “neither believed nor disbelieved,” or “both believed and disbelieved.” A sentence is said to be “non-assertable” if it also appears to be “indescribable.” A natural understanding of this fourth statement should be that the corresponding sentence cannot be asserted and, then, cannot be held true by any agent or any theory. But it cannot be denied either, as the meaning of (4) differs from (2). Thus a sentence is non-assertable whether positively or negatively, i.e. if the content of this sentence cannot be ever assessed by a given theory (non-assertability as indeterminacy), or even if it doesn’t make sense in a given language (non-assertability as meaninglessness). Such is the understanding of Ganeri [2], where a sentence is said to be “non-assertable” whenever no plausible belief can be entertained about it in some respect. The sentence p: “love is the greatest value” cannot be assessed from the standpoint of a physical theory, for instance, so that p is non-assertable in this perspective. Let us symbolize by a non-belief this first, incomplete reading of the fourth Jaina predication: (4)G v(p) = u

∃x(~Bx(p) ∧ ~Bx(~p))

The other, rival interpretation is supported by the way in which avaktavyam is equally described, that is: as a simultaneous assertion and denial. The latter clearly differs from Ganeri’s suspension of judgment, insofar as whoever both asserts and denies a given sentence does believe its being true and false. Is the fourth predication a case of incomplete or inconsistent reading? According to Ganeri [2], two sorts of inconsistent readings have been endorsed in the literature. On the one hand, Bharucha & Kamat [1] claimed that avaktavyam referred to a single inconsistent belief in the form

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(4)BK v(p) = u

∃xBx(p ∧ ~p)

whereas Matilal took it to be a conjunction of two inconsistent beliefs like: (4)M v(p) = u

∃x(Bx(p) ∧ Bx(~p))

Ganeri saw a slight difference between the two sorts of inconsistent interpretation, because (4)M wouldn’t entail (4)BK if one rejects the adjunction rule: p, q |= (p ∧ q).1 This is correct only if p and q are metavariables that stand for ∃xBx(p) and ∃xBx(q), respectively. But this is not the case in our own interpretation, where the inconsistent interpretation of (4) is assumed simultaneously by a single agent and departs from the successive statements of (3). The difference between (4)BK and (4)M is thus immaterial, both meaning that p and ~p are in the scope of one and the same belief or standpoint. It is also immaterial to Ganeri, but for a different reason that concerns a collapsing argument. While recalling that a proper formalization of Jaina logic should do justice to the difference between each of the seven logical values (or combined truth-values), Ganeri adds that (4)BK entails a collapse of two statements and is thus a sufficient reason to reject any inconsistent interpretation of (4). Thus he asks: What is the fifth value, tu? If Bharucha and Kamat are right then it means that there is some standpoint from which ‘p’ can be asserted, and some from which ‘p ∧ ~p’ can be asserted. But this is logically equivalent to u itself. The Bharucha and Kamat formulation fails to show how to get a seven-valued logic [2, p. 271].

Now this argument seems to rely upon a conflation of two distinct standpoints: to state that p is asserted from one standpoint and both asserted and denied from another standpoint doesn’t entail that p is merely asserted and denied, unless the crucial syād is suddenly removed from the 1

Thus Ganeri adds that “Matilal’s interpretation is a little weaker than Bharucha and Kamat, for he does not explicitly state that the conjunction ‘p ∧ ~p’ is asserted, only that both conjuncts are. Admittedly, the difference between Matilal and Bharucha and Kamat is very slight, and indeed only exists if we can somehow make out the claim that both a proposition and its negation are assertable without it being the case that their conjunction is” [2, p. 272].

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meaning of a statement. But it could not be so, and Ganeri unduly commits the following simplification: p ∧ (p ∧ ~p) ≡ (p ∧ ~p). To say that tu = u is to say that the two main conjuncts p and (p ∧ ~p) are stated by one and the same agent; but this needn’t be the case whenever the right interpretation of (5)BK is ∃xBx(p) ∧ ∃x(Bx(p ∧ ~p)) and not ∃xBx(p ∧ (p ∧ ~p)). For this reason, Ganeri’s collapsing argument fails and the inconsistent interpretation of (4) remains open. An intuitive account of (4)BK should be one in which the belief-claim ‘B’ is weakened and doesn’t have the exclusive sense of an assertion: p would be merely conjectured by the agent without excluding its possibility of being eventually false. No matter how an inconsistent reading of (4) could be supported, actually: the various combinations of value-assignments require a clear-cut distinction between successive and simultaneous statements, without which the seven logical values couldn’t be distinguished from each other. 2.3. Why seven? The number of values in the Jaina system might appear as somehow surprising or unnatural, however: most of the well-known many-valued logics count either three or four distinguished ‘truth-values,’ but we’ve already noted at the same time that the Jaina valuation includes three basic predications among its various combinations. Such a cardinality can be easily grounded in terms of a question-answer game: let Q = 〈q1; q2; q3〉 be an ordered set of questions about a sentence, and let A = 〈a1; a2; a3〉 be the corresponding set of yes-no answers. The three single questions proceed as statement-forming operators upon sentences, that is: q1(p) =

‘is p asserted?’2

‘∃xBx(p)?’

q2(p) =

‘is p denied?’

‘∃xBx(~p) ?’

‘v(p) = f ?’

q3(p) =

‘is p non-assertable?’

either ‘∃x(~Bx(p) ∧ ~Bx(~p)) ?’ or ‘∃x(~Bx(p) ∧ ~Bx(~p)) ?’

‘v(p) = u ?’

2

‘v(p) = t ?’

Or “is p affirmed?”, if ‘B’ takes the weaker sense of a conjecture for the inconsistent interpretation of (4)BK.

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Two exclusive answers are possible for these questions: either an affirmative or a negative one, assuming that one and the same question cannot be answered in both ways at once. a(p) = 1

‘yes, p is …’

‘∃x(…))’

‘v(p) = …’

a(p) = 0

‘no, p is not …’

‘~∃x(…))’

‘v(p) ≠ …’

These single questions stand for the three basic predications, and the exhaustive combination of ordered answers corresponds to the powerset of Q minus one element: P(Q) – 1 = 23 – 1 = 7 logical values. (1) = 〈1;0;0〉 (5) = 〈1;0;1〉

(2) = 〈0;1;0〉 (6) = 〈0;1;1〉

(3) = 〈1;1;0〉 (7) = 〈1;1;1〉

(4) = 〈0;0;1〉 (8) = 〈0;0;0〉

The eighth combination is the empty subset, and such a logical value is cancelled from the sevenfold predication: the set of questions Q is exhaustive, in the sense that whoever answers negatively to any two questions is led to answer positively to the third one. The clause that every answer is exclusive (either yes or no) explains how Jaina logic may be consistent while allowing some inconsistent statements: its metalanguage of questions-answers is consistent, while its object language gives rise to a weaker form of paraconsistency. As Ganeri puts it: The degree to which the Jaina system is paraconsistent is, on this interpretation, restricted to the sense in which a proposition can be tf, i.e. both true and false because assertable from one standpoint but deniable from another. It does not follow that there are standpoints from which contradictions can be asserted [2, p. 272].

Although we previously contended that a given standpoint might be inconsistent (by assuming that the corresponding sentence is not asserted but merely conjectured), this means that the relation of contradiction can occur at three distinct levels: between sentences (both p and ~p), statements (both true and false), or answers (both yes and no). Ganeri claims that inconsistent statements cannot be made from one single standpoint, but a weaker reading of the yes-answer (qua affirmation) seems to do justice to (4)BK or, equivalently, (4)M. At the same time, a commonly

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prohibited contradiction is that obtaining between answers: no answer to a statement-forming question should be both positive and negative (the total number of the logical values would exceed seven, otherwise), and this precondition makes the Jaina system still minimally consistent. Another classical feature of the Jaina system is its peculiar bivalence, with respect to the behavior of the answers a. Ganeri also remarked this but, again, at another level of the Jaina language: Using many-valued logics in this way, it should be noted, does not involve any radical departure from classical logic. The Jainas stress their commitment to bivalence, when they try to show, as Vādiveda Sūri did above, that the seven values in their system are all products of combining two values. This reflects, I think, a commitment to bivalence concerning the truth-values of propositions themselves. The underlying logic within each standpoint is classical, and it is further assumed that each standpoint or participant is internally consistent [2, p. 274].

Although we acknowledge that bivalence still holds in the Jain system, we disagree with Ganeri when he claims that each standpoint should be classical and internally consistent: not only does the incomplete reading of avaktavyam depart from a classical, i.e. complete and consistent interpretation; but also, each answer A includes three elements and resorts to a 3-valued basic system that is either incomplete or inconsistent. Priest [6] quoted other sources from the ancient Jainas that seems to support an inconsistent reading of avaktavyam, and the situation about (4) remains undecided as it stands. The collapsing argument Ganeri called for to undermine the inconsistent reading is incorrect, at any rate, and we’ll accept a twofold interpretation of the third element a3 in the sense of either (4)G or (4)M. Nevertheless, the Jaina bivalence appears in another respect: there are two available answers to a corresponding statement-forming question, and only one can be given at once. In other words, such a bivalence concerns the number of answers relatively to questions about truth-values, and not the number of truth-values themselves. 2.4. How to define the logical connectives?

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Just as our reconstruction of the seven values seems somehow intuitive by preserving bivalence within a question-answer semantics, it is an artificial project to reconstruct the logical connectives of the Jaina logic. Such an enterprise is both artificial and anachronistic, as pointed out by Priest: What are the semantic values of such compounds? Such a question is not one that Jaina logicians thought to ask themselves, as far as I know. So we are on our own here. There are probably several possible answers [6, p. 268].

Priest suggested two distinct semantics: one Type 1 (3-valued) semantics, and one Type 2 (7-valued) semantics. Our precondition to respect the cardinality of the sevenfold predication forces us to follow the second line, albeit in a sensibly different way from Priest’s. The Jain constants can be rendered by a many-valued logic JL with the structure , where: • L is a language with sentential variables {p, q, r, …}; • Q is a question-operator such that Q(p) gives a sense to the sentence p; • A is a valuation function from L to V7 such that A(p) gives a reference to p; • C = {~, ∧, ∨, →} is the set of logical connectives of type {1, 2, 2, 2}; • V7 is a set of seven logical values; • D is a subset of designated values in V7. Two logics JLG and JLM can be set up for the Jaina system, depending upon the interpretation of u. Some general features can be displayed irrespective of this ambiguity, however. For one thing, our existential reformulation of the logical values entails that JL follows the clauses of the existential quantifier ∃ in first-order logic. On the other hand, the logical connectives have been specified by quasivalue-functional definitions: the value of a compound sentence partly depends upon the value of its components, according to the formulation of

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the Jaina sentences in terms of existential statements.3 Quasi-valuefunctionality occurs in the introduction and elimination rules that define the aforementioned set C. An introduction rule is a rule that helps to infer a compound sentence from its components, while an elimination rule is the converse rule that infers the components by means of the compound formula. Given that each Jaina statement (α)JL is expressed in the metalanguage by an existential sentence, not every logical value of a compound sentence can be fully determined by the logical values of its components. This clearly appears in the following correspondences: Elimination rule for conjunction (p ∧ q)JL ---------------(p)JL (q)JL

∃xBx(p ∧ q) ------------------------∃xBx(p) ∃xBx(q)

v(p ∧ q) = t ----------------v(p) = v(q) = t

a1(p ∧ q) = 1 -------------------a1(p) = a1(q) = 1

Introduction rule for disjunction (p)JL (q)JL ---------------(p ∨ q)JL

∃xBx(p) ∃xBx(q) ------------------------∃xBx(p ∨ q)

v(p) = t v(q) = t ----------------------v(p ∨ q) = t

a1(p) = 1 a1(q) = 1 -------------------------a1(p ∨ q) = 1

These introduction and elimination rules don’t equally apply to conjunction and disjunction, respectively: whereas the value of a1(p ∨ q) is determined by the values of a1(p) and a1(q), the value of a1(p ∧ q) cannot be determined by the values of a1(p) and a1(q). Indeed, ∃xBx(p) and ∃xBx(q) may be verified while ∃xBx(p ∧ q) being falsified in a given model if no agent x believes both p and q in this model. On the basis of these partial valuations, due to the existential translation of the Jaina sentences, the inference rules of first-order logic can be applied to define the four logical connectives of JL. (A detailed proof of the following characterizations is given in the Appendix.) 3

About the origins of quasi-truth-functional logics, see Rescher [7]. We prefer here the expression “quasi-value-functional” in order to avoid any terminological ambiguity, given that our logical values are not the usual Fregean truth-values of truth and falsehood but correspond to yes-no answers.

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Let us consider first the Jaina connectives “à la Ganeri:” CG = {~G, ∧G, ∨G, →G}, where the third basic element u is read incompletely; then the Jaina connectives “à la Matilal:” CM = {~M, ∧M, ∨M, →M}, where u is interpreted inconsistently. Two main results about JL are established in the Appendix, namely: • CG and CM are identical, i.e. the matrices for JLM and JLG uniformly characterize the logical connectives; • JLG is equivalent to Kleene’s 3-valued logic K3, and JLM is equivalent to Priest’s logic of paradox LP.4 The first result leads to matrices that are quasi-identical to the Type 2 Semantics in Priest [6]: each partial value of JL corresponds there to a single value that is the greatest one (e.g. the partial value (7) – (6) corresponds there to (7)). Moreover, it makes irrelevant the construction of an enriched 15-valued logic including the two competing interpretations of u, where A(p) = 〈a1(p); a2(p); a3(p); a4(p)〉, a3: ‘∃x(~Bx(p) ∧ ~Bx(~p) ?’, and a4: ‘∃x(Bx(p) ∧ Bx(~p) ?’.5 Given that a3 and a4 yield the same valuations, the 15-valued logic could be reduced to a 7-valued logic. But a difference remains between JLG and JLM, which concerns the subset of the designated values in V7. 2.5. How to single out the designated values? The designated values D of a many-valued system help to characterize its subset of theorems, in the sense that these preserve the consequence relation between logically true formulas. This means that, for any formulas A, B: A |= B if and only if B∈D whenever A∈D. Now the question is what a designated value should be in JLG and JLM. Assuming that the Jaina system relies upon an epistemic view of truth, this means that truth is 4

About K3, see Kleene, S., Metamathematics, North Holland Amsterdam, 1952; about LP, see Priest, G., The Logic of Paradox, Journal of Philosophical Logic 8, 1979, 219 – 241. The invalidity of Explosion means that (p ∧ ~p) |= q is invalid, that is: there is some counter-model where the premise (p ∧ ~p) is designated and the consequence q is not designated. It must be noted that Explosion is not to be confused with its conditional form (p ∧ ~p) → q, which is valid in LP (and JLM). 5 Indeed, such a logic would have P(A) – 1 = 4² – 1 = 15 logical values. A similar case has been actually entertained by Sylvan [8].

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synonymous with assertion or truth-claim; if so, then any formula is designated whenever asserted. Hence the subsets of designated values: D = {(1), (3), (5), (7)} in JLG, and D = {(1), (3), (5), (6), (7)} in JLM. The number of designated values in JL sounds rather surprising, because it includes most of the available logical values and seems to imply to the subset of theorems is more extended than usual. But it shouldn’t be so surprising, and this for two reasons: on the one hand, Jaina logic is the formal expression of a philosophy that supports tolerance between competing judgments and a relative doctrine of truth; on the other hand, the subset of the corresponding theorems shouldn’t be so much extended with respect to some further well-known logical systems, given our second result of the preceding section to the effect that JLG and JLM are equivalent to K3 and LP. This means that the 7-valued logic JL is an extension of these two 3-valued logics, although neither of these can be plainly equivalent to JL. The equivalence between JLG and K3 means that the former shares the same properties as the latter, among which the fact that JLG includes no logical truths: no Jaina formula is designated under any valuation of its components. So is the case even for the law of identity: p → p, where a counter-model for it is A(p) = (4). Nevertheless, a minimal list of axioms for JL might be found in another level of discourse: that of modalities, assuming that “syād” proceeds like a modal operator before any sentential content. 2.6. Can JL be a modal logic? Ganeri [2] rightly noted a resemblance between Jaina logic and Jaśkowski [3]: truth is synonymous with relative truth or truth from one particular standpoint in the latter, so that a formula is taken to be false if and only if no agent believes p to be true. The introduction of quantifiers in the definition of truth naturally led to a further comparison with modal logic: it clearly appears that (α)JL = ◊α. And correspondingly, the Jaina logical value (1) could be assimilated with necessity: if A(p) = (1), then someone believes p and no one disbelieves it at all (whether by believing its negation ~p or by disbelieving both p and ~p). Thus everyone believes p to

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be true and, assuming the logical correspondences ∃ :=: ◊ and ∀ :=: …, A(p) = (1) could be rendered by …p. Similarly, the logical value (4) can be viewed as a many-valued counterpart of the modality of contingency in JLG: that an agent disbelieves both p and ~p entails that not everyone believes ~p and not everyone believes p, respectively, and this is the exact definition of contingency: Vp :=: ~…p ∧ ~…~p. But such an analogy between our quantified reading of the Jaina sentences and modal logic is not so much promising. On the one hand, the analogy with Jaśkowski’s discussive logic D2 must stop here because of the difference in translation: in the case of negation ~, it is to be noticed that (~p)D2 = (~p)JL = ◊~p (for a modal translation of JL). But the case of double negation shows that this identical result is as much misleading as 2 + 2 and 2 × 2 in arithmetic: that both amount to the same result doesn’t mean at all that + and × mean the same relation. Thus (~~p)D2 = ◊~◊~p = ◊…p, whereas (~~p)JL = ◊~~p = ◊p. It is so because, unlike JL, the modality of possibility ◊ (or, equivalently, the existential quantifier) is part and parcel of the definition of negation in D2. In other words: for any n number of iterated negations (~n(p))D2 = ((◊~)n)p, whereas (~n(p))JL = ◊(~n)p. On the other hand, the behavior of negation in JL entails that no iterated modalities do make sense in the Jaina system. Indeed, the iteration of modalities in D2 is due to the iteration of negation while any Jaina sentence is prefixed by the constant operator of possibility ◊ (expressing the relative syād). It follows from this that no modal translation of JL can give expressions like …◊p, ……p or ◊…p; ◊p means that a1(p) = 1 and …p means that A(p) = (1), i.e. 〈1; 0; 0〉. But that’s all, concerning the analogy between JL and modal logic; this gives rise to a weak range of modal systems, given that S4 and S5 don’t make sense at all in our reading of JL and, also, T would mean that p is true when believed by every agent.6 6

T: …p → p, whose existential translation is ∀x(Bx(p)) → p. The latter formula somehow corresponds to the consensus theory of truth formalized by Jerzy Łoś’ logic of assertion. See Łoś, J., Logiki wielowartościowe a formalizacja funkcji intensjonalnych (Many-valued logics and the formalization of intensional functions), Kwartalnik Filozoficzny 17, 1948, 59 – 78. That a sentence is true (and not only taken to be so) when believed by everyone could be a correct characterization of the Jaina’s view of objective or absolute truth, if an idealist epistemology is attached to it.

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More ultimately, the problem of comparing the Jaina system with a modal logic creates a syntactic confusion: we must keep in mind that every nonmodal sentence p of the object language is translated in our metalanguage by a possible counterpart ◊p; that doesn’t mean that modal and non-modal formulas both occur in a common object language, as our modal translations occur in the metalanguage and don’t give rise to mixed formulas like p → ◊p or …p → p. Now assuming the difference between both levels of language, a formula like p → ◊p could be charitably read as ‘if p, then p is taken to be true’ or ‘if p, then someone takes p to be true.’ With this interpretation at hand, where any mixed formula talks about the relation between modal sentences of the metalanguage and non-modal sentences of the object language, the closest modal counterpart of JL might be like S0.5, that is the non-normal system that includes T and excludes the rule of necessitation from K (since not every classical theorem is believed by every agent in JL). By the way, T would now naturally mean that any sentence that is believed by every agent does hold in the Jaina system. This results in the following syntactic “meta-system” for JL and its corresponding set of axioms: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

(α ∧ β) → (β ∧ α) (α ∧ β) → α (α ∧ β) → β α → (α ∧ α) α ∧ (β ∧ γ) → β ∧ (α ∧ γ) (α → ◊α) ◊(α ∧ β) → ◊α (α ∧ β) → (◊α ∧ ◊β) …(α ∧ β) → (…α ∧ …β) …α → α

Despite this plausible adaptation from Jaina to modal logic, it sounds not only ambiguous but too weak to be very efficient or expressive: the failure of iterated modalities might give a sufficient reason to keep the Jaina system apart from modal logic and stick it to a mere case of manyvaluedness.

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3. Conclusion: on truths and beliefs Our exploration of Jaina logic with modern tools leads to four main theses. These concern the opposition between ontological and epistemological truth; the opposition between propositional and sentential logics; the difference between inconsistency and paraconsistency; and the emergence of two rival “epistemic logics” within the realm of Indian logic. While, in an ontological view of truth, the so-called truthmaker is a fact (as in the Aristotelian tradition of the correspondence view of truth), it cannot be so in an epistemological view of truth as in Jaina logic. What makes a sentence true is a given theory or standpoint that is supported by a number of agents, and only one agent is sufficient to do so with our existential translation of truth-claims. It remains to see whether truth and truth-claim are on a par in the Jaina relative doctrine of truth: a pluralist approach is to the effect that reality is the syncretistic result of a manifold description of the same object by means of opposite judgments; but to what extent two contradictory standpoints can be completed to each other is not clear: either reality is defined as a complete description of an object from one and the same perspective, and the syādvāda is useless for this purpose; or the latter gives an ultimate access to reality, in which case no absolute or onesided judgment could be ever given. Such a tolerant view of epistemic truth overcomes the threat of triviality, as described in Plato’s Theaetetus: that one agent feels a given wine to be sweet, while another feels it to be bitter when sick, doesn’t mean that truth is gone if everything is somehow right. The Socratic method of reductio ad absurdum does presuppose an ontological or one-sided view of truth, so that the difference between truth and falsity doesn’t vanish in a pluralist doctrine of truth like the syādvāda. A corollary of this epistemic view of truth is that JL is a logic of sentences, rather than propositions. Following the definition of “proposition” as a standing sentence whose truth-value is invariable, it clearly appears that no such proposition is evaluated in the Jaina system; to the contrary, any sentence is a mere symbolic device that stands for a context-relative statement, doesn’t refer to any ideal entity like the Fregean Gedanke and, consequently, cannot be evaluated irrespective of the agents’ opinion. Actually, JL is a non-Fregean logic: the truth-bearers are not propositions but sentences, and these don’t refer to abstract truth-values but concrete

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ordered answers that occur as the new logical values of a many-valued logic. The best approximation of a Fregean proposition would be a sentence whose truth-value is the same for every agent, i.e. a sentence p such that A(p) = (1) or (2). But an examination of the Jaina texts shows that no meaningful sentence can be so: (1) and (2) refer to one-sided sentences whose very statement is contrary to the Jaina philosophy, and there is always at least one evidence for opposite views. Recalling the earlier distinction of Ganeri [2] between skepticism and pluralism, we can say that Vādiveda Sūri reasons like Pyrrho while gathering opposite conclusions: there is no one-sided judgment, even in non-empirical domains like arithmetic; but this is not a sufficient reason to defeat and reject any truth-assignment, because doing so would be accepting onesidedness as a precondition of truth. However, a main difficulty with pluralism is how a distinction could be made between opinion and mistake: if any opinion holds as it stands, does there still make sense to talk about wrong judgments? A way to assess a given judgment might lead to a reliabilist view of justified beliefs; but such a topic would go beyond our logical paper. Another moral of our examination is that a clear-cut difference ought to be made within paraconsistency between inconsistency and contradiction. Contrary to what is often claimed about paraconsistent systems (including the Jaina logic), inconsistency doesn’t challenge the Aristotelian principle of contradiction to the effect that contradictory sentences must be so with the same respect. This is clearly not the case in JL (and D2), where an inconsistent value like (3) points at two different standpoints. A genuinely contradictory sentence ‘p ∧ ~p’ should be stated with one and the same respect, and our question-answer semantics wouldn’t render this situation by a case in which p is both asserted and denied: a1(p) = a2(p) = 1. Rather, a proper contradiction would be one in which one and the same question about a sentence results in opposite answers: a(p) = 1 and 0. This is the precise sense in which contradiction has been removed from JL (see section 2.3). A final remark concerns the occurrence of two competing “epistemic logics” in the realm of Indian logics, that is: the relativist Jaina logic versus the skeptic logic of Catuskoti, also well-known with the case of Four-Fold Negation. Both can be viewed as two internally epistemic logics, given that

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their sentences and logical connectives are implicitly charged with propositional attitudes or truth-claims. According to Nagarjuna, some sentences can be (a) neither asserted, (b) nor denied, (c) nor both asserted and denied, (d) nor neither asserted and denied. This statement is made troublesome by its last conjunct (d), because the negation of (c) should result in the negation of (d). Whether the difficulty lies in the logical form of negation or in the skeptic’s propositional attitudes that support (a) – (d), we impute the difference between Jaina logic and the Four-Fold Negation to a difference in their criteria of belief-assignment: when an agent is qualified to believe a sentence (to be true or false) varies between epistemological doctrines like skepticism and pluralism, and the skeptic position states that no one is qualified to believe some sentences. As claimed by Matilal: Roughly, the difference between Buddhism and Jainism in this respect lies in the fact that the former avoids by rejecting the extremes altogether, while the latter does it by accepting both with qualifications and also by reconciling them [4, p. 129].

The main challenge of the Catuskoti remains to see how no meaningful sentence could be ever affirmed and taken to be true, unlike the Jaina logic that makes a clear sense of truth and falsity and doesn’t amount to an acceptance of everything. But whether Nagarjuna’s logic corresponds to a rejection of everything will be reviewed elsewhere. Acknowledgment I wish to thank Jonardon Ganeri, for the gentle and helpful correspondence we had through emails; as well as Manuel Rebuschi, for the several suggestions he made during my talk about Jaina Logic in Geneva (SOPHA, 2 – 5 September 2009). Appendix: characterization of the Jaina connectives Jaina logic is to be described by its two main features, namely: its classical metalanguage, and its partial valuations.

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JL is classical, in the sense that: for every p, a(p) = 1 if and only if (thereafter: iff) a(p) ≠ 0; and the existential statements that translate the Jaina sentences follow the rules of classical sentential logic, including: double negation, commutativity, and associativity. The logical connectives are also defined in the classical way, so that (p → q) ≡ (~p ∨ q). Let ∪ and ∩ be the quasi-Boolean operations of join and meet on two answers in x = {1, 0}, such that x ∪ 1 = 1, x ∪ 0 = x, x ∩ 1 = x, and x ∩ 0 = 0. Then JL is partial in the sense that, for some of its valuations, x ∩ 1 = x or 0 whenever x = 1. Let x be a given answer in {0, 1}: either ‘yes’ (x = 1), or ‘no’ (x = 0). Then for every answer a in {a1, a2, a3} and every sentential metavariables α and β, a(α) = 1 and a(β) = 1 means that an agent believes ‘α’ and an agent believes ‘β.’ But those agents needn’t be the same: the agent who believes ‘α’ may not be the same as the agent who believes ‘β,’ and conversely. This entails that a(α ∩ β) is partial: a(α ∩ β) = 1 or 0 whenever a(α) = a(β) = 1. A characterization of the logical connectives will be followed by their corresponding matrix, where each logical value stands for an ordered combination of answers 〈a1; a2; a3〉. As an example of partial value, (7) – (6) means that the value of the corresponding sentence is either (7) or (6). Theorem 1. CG and CM are identical. Proof. By induction upon the number of logical connectives in CG and CM. CG and CM are identical iff, for every connective c∈{~, ∧, ∨, →} in CG, its counterpart c* in CG has the same characterization of its compound answers a1, a2, and a3. Given that the sole difference between c and c* lies in their third answer a3 (whether beliefs are either incomplete or inconsistent), only the valueconditions of a3 will count in JLM. Negation: ~G a1(~G(p)) = 1 iff ∃xBx(~p), i.e. a2(p) = 1; a1(~p) = 0, otherwise; a2(~G(p)) = 1 iff ∃xBx~(~p) = ∃xBx(p), i.e. a1(p) = 1; a2(~p) = 0, otherwise; a3(~G(p)) = 1 iff ∃x(~Bx(~p) ∧ ~Bx~(~p)) = ∃x(~Bx(~p) ∧ ~Bx(p)), i.e. a3(p) = 1; a3(~p) = 0, otherwise.

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Therefore: A(~G(p)) = 〈a2(p);a1(p);a3(p)〉. Negation: ~M a3(~M(p)) = 1 iff ∃x(Bx(~p) ∧ Bx~(~p)) = ∃x(Bx(~p) ∧ Bx(p)), i.e. a3(p) = 1; a3(~p) = 0, otherwise. Therefore: A(~M(p)) = 〈a2(p);a1(p);a3(p)〉. Thus, ~G and ~M are identical. ~ (1) (2) (3) (4) (5) (6) (7)

(2) (1) (3) (4) (6) (5) (7)

Conjunction: ∧G a1(p ∧G q) = 1 iff ∃xBx(p ∧ q). By first-order logic, for every sentence α and β we have: (E∧)

∃xBx(α ∧ β) ⇒ (∃xBx(α) ∧ ∃xBx(β)),

but the converse needn’t hold. That is: if a1(α ∧ β) = 1, then a1(α) = a1(β) = 1; but if a1(α) = a1(β) = 1, then a1(α ∧ β) = 1 or 0. Hence a1(p ∧G q) = 1 or 0 iff a1(p) = a1(q) = 1; a1(p ∧G q) = 0, otherwise. a2(p ∧G q) = 1 iff ∃xBx~(p ∧ q), i.e. ∃xBx(~p ∨ ~q). By first-order logic, we have: (I∨)

(∃xBx(α) ∨ ∃xBx(β)) ⇒ ∃xBx(α ∨ β)

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That is: a1(α ∨ β) = 1 if a1(α) = 1 or a1(β) = 1; a1(α ∨ β) = 0, otherwise. Thus a2(p ∧G q) = 1 iff a1(~p) = 1 or a1(~q) = 1, i.e. a2(p) = a2(q) = 1; a2(p ∨G q) = 0, otherwise. a3(p ∧G q) =

1 iff ∃x(~Bx(p ∧ q) ∧ ~Bx~(p ∧ q)), i.e. ∃x((~Bx(p) ∨ ~Bx(q)) ∧ ~Bx(~p ∨ ~q)) = ∃x((~Bx(p) ∨ ~Bx(q)) ∧ (~Bx(~p) ∧ ~Bx(~q)))

Let α= ~Bx(p), β = ~Bx(q), and γ = ~Bx(~p) ∧ ~Bx(~q). By first-order logic, we have: (Ass)

((α ∨ β) ∧ γ) ⇔ ((α ∧ γ) ∨ (β ∧ γ))

Hence a3(p ∧G q) = 1 iff ∃x((~Bx(p) ∧ ~Bx(~p) ∧ ~Bx(~q)) ∨ (~Bx(q) ∧ ~Bx(~p) ∧ ~Bx(~q))). For every sentence α: if ∃x(~Bx(~α)), then a2(α) ≠ 1, i.e. a1(α) = 1 or a3(α) = 1. Hence a3(p ∧G q) = 1 iff a3(p) = a1(q) = 1, or a3(p) = a3(q) = 1, or a1(q) = a3(p) = 1; a3(p ∧G q) = 0, otherwise. Let the partial values be marked in gray, when a(α) = 1 or 0. Therefore: A(p ∧G q) = 〈a1(p) ∩ a1(q);a2(p) ∪ a2(q); (a3(p) ∩ a1(q)) ∪ (a3(p) ∩ a3(q)) ∪ (a1(p) ∩ a3(q))〉 Conjunction: ∧M a3(p ∧M q) = 1 iff ∃x(Bx(p ∧ q) ∧ Bx~(p ∧ q)) = ∃x(Bx(p) ∧ Bx(q) ∧ Bx(~p ∨ ~q)). By (I∨), we have: Bx(~p) ⇒ Bx(~p ∨ ~q)), and Bx(~q) ⇒ Bx(~p ∨ ~q). Let α = Bx(p) ∧ Bx(q), β = Bx(~p), and γ = Bx(~p). By (Ass.), a3(p ∧M q) = 1 iff ∃x(Bx(p) ∧ Bx(q) ∧ Bx(~p)) or ∃x(Bx(p) ∧ Bx(q) ∧ Bx(~q)).

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Hence a3(p ∧M q) = 1 iff a3(p) = a1(q) = 1, or a3(p) = a3(q), a1(p) = a3(q); a3(p ∧M q) = 0, otherwise. Therefore: A(p ∧M q) = 〈a1(p) ∩ a1(q); a2(p) ∪ a2(q); (a3(p) ∩ a1(q)) ∪ (a3(p) ∩ a3(q)) ∪ (a1(p) ∩ a3(q))〉. Thus, ∧G and ∧M are identical. ∧ (1) (2) (3) (4) (5) (6) (7)

(1) (1) (2) (3)-(2) (4) (5)-(4) (6) (7)-(6)

(2) (2) (2) (2) (2) (2) (2) (2)

(3) (3)-(2) (2) (3)-(2) (6) (7)-(6) (6) (7)-(6)

(4) (4) (2) (6) (4) (4) (6) (6)

(5) (5)-(4) (2) (7)-(6) (4) (5)-(4) (6) (7)-(6)

(6) (6) (2) (6) (6) (6) (6) (6)

(7) (7)-(6) (2) (7)-(6) (6) (7)-(6) (6) (7)-(6)

Note: The bold italic value (in the first line of the first column) is actually a “degenerate” partial value, i.e. (1) – (8). While (8) is an impossible value, it can be easily proved that (1) ∧ (1) = (1): if A(p) = A(q) = (1), then everyone believes p to be true and everyone believes q to be true. If so, there is no agent who believes p and disbelieves q. Therefore, everyone believes p ∧ q to be true and, a fortiori, someone does. Hence A(p ∧ q) = (1). Disjunction: ∨G a1(p ∨G q) = 1 iff ∃xBx(p ∨ q). By (I∨), (∃xBx(p) ∨ ∃xBx(q)) ⇒ ∃xBx(p ∨ q). Hence a1(p ∨G q) = 1 if a1(p) = 1 or a1(q) = 1; a1(p ∨G q) = 0, otherwise. a2(p ∨G q) = 1 iff ∃xBx~(p ∨ q), i.e. ∃xBx(~p ∧ ~q). By (E∧), (∃xBx(~p) ∧ ∃xBx(~q)) ⇒ ∃xBx(~p ∧ ~q) but the converse needn’t hold.

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Hence a2(p ∨G q) = 1 or 0 iff a1(~p) = a1(~q) = 1, i.e. a2(p) = a2(q) = 1; a2(p ∨G q) = 0, otherwise. a3(p ∨G q) =

1 iff ∃x(~Bx(p ∨ q) ∧ ~Bx~(p ∨ q)), i.e. ∃x(~Bx(p) ∧ ~Bx(q) ∧ ~Bx(~p ∧ ~q)) = ∃x(~Bx(p) ∧ ~Bx(q) ∧ (~Bx(~p) ∨ ~Bx(~q))) By (Ass), a3(p ∨ q) = 1 iff ∃x(~Bx(p) ∧ ~Bx(q) ∧ ~Bx(~p)) ∨ (~Bx(p) ∧ ~Bx(q) ∧ ~Bx(~q)). For every sentence α: if ∃x(~Bx(α)), then a1(α) ≠ 1, i.e. a2(α) = 1 or a3(α) = 1. Hence a3(p ∨G q) = 1 iff a3(p) = a2(q) = 1, or a3(p) = a3(q) = 1, or a2(p) = a3(p) = 1; a3(p ∨G q) = 0, otherwise. Therefore: A(p ∨G q) = 〈a1(p) ∪ a1(q); a2(p) ∩ a2(q); (a3(p) ∩ a2(q)) ∪ (a3(p) ∩ a3(q)) ∪ (a2(p) ∩ a3(q))〉. Disjunction: ∨M a3(p ∨M q) = 1 iff ∃x(Bx(p ∨ q) ∧ Bx~(p ∨ q)) = ∃x(Bx(p ∨ q) ∧ (Bx(~p) ∧ Bx(~q))). By (I∨), we have: Bx(p) ⇒ Bx(p ∨ q)), and Bx(q) ⇒ Bx(p ∨ q). Let α = Bx(~p) ∧ Bx(~q), β = Bx(p), and γ = Bx(q). By (Ass), a3(p ∨M q) = 1 iff ∃x(Bx(~p) ∧ Bx(~q) ∧ ~Bx(p)) ∨ (Bx(~p) ∧ Bx(~q) ∧ Bx(q)). Hence a3(p ∨M q) = 1 iff a3(p) = a2(q) = 1, or a3(p) = a3(q) = 1, or a2(p) = a3(q) = 1; a3(p ∨G q) = 0, otherwise. Therefore: A(p ∨M q) = 〈a1(p) ∪ a1(q); a2(p) ∩ a2(q); (a3(p) ∩ a2(q)) ∪ (a3(p) ∩ a3(q)) ∪ (a2(p) ∩ a3(q))〉. Thus, ∨G and ∨M are identical. ∨ (1)

(1) (1)

(2) (1)

(3) (1)

(4) (1)

(5) (1)

(6) (1)

(7) (1)

80 (2) (3) (4) (5) (6) (7)

(1) (1) (1) (1) (1) (1)

(2) (3)-(1) (4) (5) (6)-(4) (7)-(5)

(3)-(1) (3)-(1) (5) (5) (7)-(5) (7)-(5)

(4) (5) (4) (5) (4) (5)

(5) (5) (5) (5) (4) (5)

(6)-(4) (7)-(5) (4) (5) (6)-(4) (7)-(5)

(7)-(5) (7)-(5) (5) (5) (7)-(5) (7)-(5)

Note: Just as with conjunction (see above), the above bold italic value is a partial one: (2) – (8). It can be equally proved that (1) ∨ (2) = (1). If A(p) = (1), then everyone believes p to be true. Therefore everyone believes p ∨ q to be true, in accordance to I∨: A(p ∨ q) = (1). Conditional: →G a1(p →G q) = 1 iff ∃xBx(p → q), i.e. ∃xBx(~p ∨ q) Hence a1(p →G q) = 1 iff a1(~p) = 1, i.e. a2(p) = 1, or a1(q) = 1; a1(p →G q) = 0, otherwise. a2(p →G q) = 1 iff ∃xBx~(p → q), i.e. ∃xBx(p ∧ ~q) Hence a2(p →G q) = 1 or 0 iff a1(p) = a1(~q), i.e. a2(q) = 1; a2(p →G q) = 0, otherwise. a3(p →G q) = 1 iff ∃x(~Bx(p → q) ∧ ~Bx~(p → q)) = ∃x(~Bx(~p ∨ q) ∧ ~Bx(p ∧ ~q)), i.e. ∃x(~Bx(~p) ∧ ~Bx(q) ∧ (~Bx(p) ∨ ~Bx(~q))) a3(p →G q) = 1 iff ∃x(~Bx(~p) ∧ ~Bx(q) ∧ ~Bx(p)) ∨ ∃x(~Bx(~p) ∧ ~Bx(q) ∧ ~Bx(~q))) Hence a3(p →G q) = 1 iff a3(p) = a2(q) = 1, or a3(p) = a3(q) = 1, or a1(p) = a3(q) = 1; a3(p →G q) = 0, otherwise. Therefore: A(p →G q) = 〈a2(p) ∪ a1(q); a1(p) ∩ a2(q); (a3(p) ∩ a2(q)) ∪ (a3(p) ∩ a3(q)) ∪ (a1(p) ∩ a3(q))〉. Conditional: →M

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a3(p →M q) = 1 iff ∃x(Bx(p → q) ∧ Bx~(p → q)) = ∃x(Bx(~p ∨ q) ∧ Bx(p ∧ ~q)), i.e. ∃x(Bx(~p ∨ q) ∧ (Bx(p) ∧ Bx(~q))) By (I∨), we have: Bx(~p) ⇒ Bx(~p ∨ q)), and Bx(q) ⇒ Bx(~p ∨ q). Let α = Bx(p) ∧ Bx(~q), β = Bx(~p), and γ = Bx(q). By (Ass), a3(p →M q) = 1 iff ∃x(Bx(p) ∧ Bx(~q) ∧ Bx(~p)) ∨ (Bx(p) ∧ Bx(~q) ∧ Bx(q)) Hence a3(p →M q) = 1 iff a3(p) = a2(q) = 1, or a3(p) = a3(q) = 1, or a1(p) = a3(q) = 1; a3(p →M q) = 0, otherwise. Therefore: A(p →M q) = 〈a2(p) ∪ a1(q); a1(p) ∩ a2(q); (a3(p) ∩ a2(q)) ∪ (a3(p) ∩ a3(q)) ∪ (a1(p) ∩ a3(q))〉. Thus, →G and →M are identical. → (1) (2) (3) (4) (5) (6) (7)

(1) (1) (1) (1) (1) (1) (1) (1)

(2) (2) (1) (3) (4) (6)-(4) (5) (7)-(5)

(3) (3) (1) (3) (5) (7)-(5) (5) (7)-(5)

(4) (4) (1) (5) (4) (4) (5) (5)

(5) (5) (1) (5) (5) (5) (5) (5)

(6) (6)-(4) (1) (7)-(5) (4) (6)-(4) (5) (7)-(5)

(7) (7)-(5) (1) (7)-(5) (5) (7)-(5) (5) (7)-(5)

Note: the bold italic value is a partial value, i.e. (2) – (8). But, again, it can be proved that (1) → (2) = (2). For if A(p) = (1) and A(q) = (2), then everyone believes p to be true and everyone believes q to be false (or, equivalently, ~q to be true). If so, then everyone believes p ∧ ~q to be true, hence everyone believes ~(p → q) to be true: A(p → q) = (2). Theorem 2. JLG is equivalent to K3. Proof. A logical system L1 is equivalent to another logical system L2 iff their subset of theorems T1 ⊂ L1 and T2 ⊂ L2 are identical: T1 = T2.

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K3 doesn’t have logical truths and neither does JLG, for no such sentence is designated under every interpretation of its components (the verification is left to the reader.) Therefore: JLG and K3 are equivalent to each other. Theorem 3. JLM is equivalent to LP. Proof. The same demonstration as for theorem 2. LP is classical logic minus the following classical theorems: (a) (b) (c)

(p ∧ ~p) |= q; p, ~p ∨ q |= q; p, p → q |= q;

(d) (e) (f)

p → q, q → r |= p → r; p → q, ~q |= ~p; p → (q ∧ ~q) |= ~p.

Each of these classical theorems has at least one counter-model in JLM, namely: A(p) = (3) and A(q) = (2) for (a), (b), (c), (e); A(p) = (1), A(q) = (3) and A(r) = (2) for (d); A(p) = (1) and A(q) = (3) for (f). Therefore: JLG and LP are equivalent to each other. References [1] Bharucha, F. & Kamat, R.V., Syādvada’s Theory of Jainism in Terms of Deviant Logic, Indian Philosophical Quaterly 9, 1984, 181 – 187. [2] Ganeri, J., Jaina Logic and the Philosophical Basis of Pluralism, History and Philosophy of Logic 23, 2002, 267 – 281. [3] Jaśkowski, S., A Propositional Calculus for Contradictory Deductive Systems, Studia Logica 24, 1969, 143 – 157. [4] Matilal, B.K., The Jaina Contribution to Logic, The Character of Logic in India, New York, 1998, 127 – 139. [5] Gokhale, P.P., The Logical Structure of Syādvada, Journal of the Indian Council of Philosophical Research 8, 1991, 73 – 81. [6] Priest, G., Jaina Logic: A Contemporary Perspective, History and Philosophy of Logic 29, 2008, 263 – 278. [7] Rescher, N., Quasi-truth-functional Systems of Propositional Logic, Journal of Symbolic Logic 27, 1962, 1 – 10.

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[8] Sylvan, R., A Generous Jainist Interpretation of Core Relevant Logics, Bulletin of the Section of Logic 16, 1987, 58 – 66.

REMARKS ON ANCIENT CHINESE LOGIC Jerzy Pogonowski Department of Applied Logic Adam Mickiewicz University www.logic.amu.edu.pl Some works by a Polish sinologist Janusz Chmielewski (1916–1998) on logical aspects of argumentation in ancient Chinese philosophy are discussed. In the paper the Wade-Giles transcription of Chinese characters is used.

1. Introductory remarks The philosophical tradition of the West (Europe) and the East (India and China) have developed in separation. However, there are some interconnections and, consequently, there has been some influence of one tradition on the other. In this paper the reconstruction of argumentation styles present in ancient Chinese philosophy in terms of Western formal logic is in focus. And it was Janusz Chmielewski who contributed considerably to this task: It was during the 1960s, when some scholars with a solid logical background began to write about logic in China, that Chinese logical tradition began to be interpreted in a way that could make sense to historians of logic in general. It is the considerable merit of Janusz Chmielewski to have acted on the insight that in order to study Chinese logic it is useful to know about Western formal logic. Janusz Chmielewski’s Notes on Early Chinese Logic I to VIII (1962 to 1969) are the first sustained attempt to apply formal logic to Classical Chinese texts [7, p. 21 – 22].

Below, we refer to the above mentioned Notes on Early Chinese Logic published in Rocznik Orientalistyczny: • Part I: 26, no. 1 (1962), 7 – 22; • Part II: 26, no. 2 (1963), 91 – 105;

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• Part III: 27, no. 1 (1963), 103 – 121; • Part IV: 28, no. 2 (1965), 87 – 111; • Part V: 29, no. 2 (1965), 117 – 138; • Part VI: 30, no. 1 (1966), 31 – 52; • Part VII: 31, no. 1 (1968), 117 – 136; • Part VIII: 32, no. 2 (1969), 83 – 103. We also take into account two more papers by Chmielewski: • J˛ezyk starochi´nski jako narz˛edzie rozumowania (Archaic Chinese as a Tool of Argumentation), Sprawozdania z prac naukowych Wydziału I Polskiej Akademii Nauk 8, 2, 1964, 108 – 133. • Zasada redukcji do absurdu na tle porównawczym (Reductio ad Absurdum Principle in a Comparative Perspective), Studia Semiotyczne XI, 1981, 21 – 106. The Polish Academy of Sciences prepares for publication Chmielewski works: Janusz Chmielewski Language and Logic in Ancient China. Collected Papers on the Chinese Language and Logic., M. Mejor (Ed.) Komitet Nauk Orientalistycznych Polskiej Akademii Nauk, Warszawa 2009. Below, we occasionally refer to our preface to that edition. By Ancient Chinese Logic we mean considerations present in chu-tzu pai-chia (100 Schools, (from 770 BCE to 221 BCE)), in the Springs and Autumns Period (from 722 BCE to 481 BCE) and the Warring States Period, (from the beginning of the 5th century BCE to 221 BCE): • Lao Tzu (from 604? BCE to 500? BCE), Chuang Tzu (from 370 BCE to 301 BCE), the Taoists. • Confucius (from 551 BCE to 479 BCE), Hsün Tzu (from 312 BCE to 230 BCE), classical Confucianism. • Mo Tzu (from 470 BCE to 391 BCE), the Later Mohists. • Hui Shi (the 4th century BCE), Kungsun Lung (from 325 BCE to 250 BCE), Sophists (or Dialecticians).

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• Han Fei (from 280 BCE to 233 BCE), the Legalists. Thus, we exclude from our considerations the Chinese Buddhist Logic, developed much more later. We are not going to add a separate list of references to the paper. The reader can find comprehensive bibliographies in e.g.: • Graham, A.C. 1978. Later Mohist Logic, Ethics and Science. The Chinese University Press and the School of Oriental and African Studies, London and Hong Kong. • Graham, A.C. 1989. Disputers of the Tao. Philosophical Argument in Ancient China. Open Court, La Salle, Illinois. • Hansen, C. 1983. Language and Logic in Ancient China. University of Michigan Press, Ann Arbor. • Christoph Harbsmeier, Language and Logic, Part I, [in:] Joseph Needham, Science and Civilisation in China, vol. 7, Cambridge University Press, Cambridge 1998.

2. On “Notes on early Chinese logic” Janusz Chmielewski himself was not a professional logician. Nevertheless, he acquired logical knowledge far beyond the elementary level. He was able, then, to make a successful use of it in his sinological works. Chmielewski defines his methodology as follows (Notes, I, 8): • to single out some more or less typical forms of reasoning occurring in the texts of early Chinese philosophers; • to define them from the standpoint and in terms of elementary formal logic; • to find out general logical laws and notions underlying them; and • as far as possible, to compare them with the ancient logical theory of the West.

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The reader can see a close similarity of this program to the one developed by Jan Łukasiewicz in his study of the Ancient Greek Logic. In parts I and II of the Notes Chmielewski uses the classical propositional calculus, in parts III – VIII he uses — and this is in his times a real novelty in sinological works — the first order predicate calculus. Actually, in both cases he uses quantification over propositional variables as well as over propositional functions which might suggest that he is working in higher order logic, or in the full type theory. This manner of symbolization, however, should be understood in most places as belonging to the level of metalanguage and not the object language itself. It is only in part IV of the Notes (Notes, IV, 93ff), where the reference to higher order logic is really essential, when Chmielewski postulates some rules of translation between archaic Chinese language and the language of higher order logic. Similar remarks apply to his paper “Reductio ad Absurdum Principle in a Comparative Perspective.” There are several places in the Notes where Chmielewski — always very cautiously — compares his interpretations against the contemporary logical theory and thus he goes far beyond the ancient state of logic in the West, e.g.: • He notices the importance of the λ-calculus developed by Alonzo Church which recently has been commonly used not only in logic, but also in the construction of mathematical models of ethnic languages and in computation theory, as well as in general in theoretical information science (Notes, IV, 107). • The same concerns combinatory logic, a system introduced by Moses Schönfinkel and being of special importance in the information science (Notes, IV, 108). • He is aware of the special role played by the most important logic different from the classical one, i.e. by the intuitionistic logic (Notes, V, 118). • When discussing the possibility of definition of general quantifier in terms of the so called zero quantifier Chmielewski enters (however without any reference to the classical papers by Andrzej Mostowski and Leon Henkin) the domain of generalized quantifiers which was advanced very vividly some years later. (Notes, V, 129 – 130).

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• He notices the system of positive logic, without the functor of negation (Notes, V, 133). • In part VII of the Notes Chmielewski refers to modal logic, when writing about a possible logical reconstruction of some of the Mohist Canons. Parts I and II of the Notes deal with the so called Chinese soriteses. Here Chmielewski proposes a completely new, from the logical point of view, interpretation of those kinds of reasoning. He rejects the interpretation in syllogistic terms and claims, providing a convincing justification, that one should analyze the arguments in question in terms of the classical propositional calculus. The interpretation becomes simple and adequate. Let us consider one example: 1. In order to obtain the kingdom there is a way: 2. If one obtains the people, one obtains the kingdom; 3. In order to obtain the people there is a way: 4. If one obtains the hearts of the people, one obtains the people; 5. In order to obtain the hearts of the people there is a way: 6. If one collects for the people what they like and does not impose on them what they dislike, [one obtains the hearts of the people]. The logical structure of this argument is transparent — only 2, 4 and 6 are premisses, and the conclusion is implicit (a usual situation in Ancient Chinese argumentation): Premiss 2: γ→δ Premiss 4: β→γ Premiss 6: α[→ β] (Implicit) conclusion: α → δ Chmielewski is perfectly right using propositional calculus for the reconstruction of this argument. Moreover, in Part I and Part II of the Notes one can find some general remarks illustrating Chmielewski’s view on logic and its connection with language, which play an important role also in the remaining parts e.g.:

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• Several types of reasoning found in the texts of ancient Chinese philosophers much more are examples of persuasion procedures (and thus a special kind of argumentation) than examples of (more or less formal) proofs. Chinese logic, in full accordance with the spirit of (several branches of) Chinese philosophy has much more pragmatic character than Western logic. • Reconstruction of forms of reasoning (arguments) should be conducted in a reasonable way: sometimes an argument contains pleonastic premisses which should be excluded from the reconstruction. The same concerns conclusions — it often happens that an argument is elliptic: the conclusion is omitted as obvious. How to deal with enthymemes is, of course, well known. One should also not forget that the arguments of ancient Chinese philosophers were often based on metaphor — it was not the literal meaning of the words used in the argument which accounted towards the proper understanding of it. To sum up, the linguistic shape of an argument is often not sufficient to its logical reconstruction, sometimes also several pragmatic factors should be taken into account. • The concept of truth is much more epistemological than logical (Notes, II, 97). Actually, in the logical reconstructions in question sometimes it is much more proper to talk about admissibility of the premisses or conclusion than about the problem whether they are true or false. In general, Chmielewski observes that ancient Chinese philosophers do not commit many drastic logical mistakes (from the point of view of modern logic). Sometimes, however, they are committing epistemological fallacies. Of course, there are also some purely logical mistakes in the analyzed texts: e.g. Mencius does not observe the asymmetry of implication. • Chmielewski distinguishes between implicit and explicit logic. As far as I can follow him, this distinction corresponds to some hidden, yet to be recovered, rules of argument in contraposition to transparent cases, where the underlying rule is well recognized. My impression is that Chmielewski did believe in the existence of the logic: most likely, the first order predicate logic. Hence he could be counted, if I am not mistaken, as adhering to the First Order Thesis, which is currently so vividly discussed by philosophers of logic.

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• I can bet that a very few logicians (if any) are aware of the fact that not all Boolean propositional functors have lexical exponents in any ethnic language. This is the case with archaic Chinese, as Chmielewski demonstrates (Notes, II, 104). In archaic Chinese there was no direct way to express the functor of alternative. Thus, instead of the direct construction α ∨ β in archaic Chinese we find equivalent to it, but indirect constructions of the form ¬α → β. Parts III and IV of the Notes are devoted to the analysis of the Mohist logic. Chmielewski proposes a logical structure of the Mohist hiao: it is a generally quantified implication, thus a formuła of the shape ∀x (ϕ(x) → ψ(x)). This suggestion is not without some reservations: implication in archaic Chinese logic might have an intensional character. Hence may be in the reconstruction one should rather use a strict implication (belonging to modal logic) than material implication (belonging to classical logic). Moreover, the arguments involving hiao refer to causal relationships (with the word ku, “reason, cause”). The formula: ∀x (ϕ(x) → ψ(x)) → (ϕ(a) → ψ(a)), where a is an individual constant, is a tautology of the predicate calculus. It corresponds, says Chmielewski, to certain Mohist arguments. But it is only an approximation of them: the first occurrence of an implication sign → should be perhaps replaced by strict implication, while the second occurrence of the same sign should be replaced by a functor corresponding to a causal relationship. Needless to say, such pedantry would result in an extremely complicated logical formula. But already the first approximation — done in the first order classical predicate calculus — explains a lot, as far as Mohist arguments are concerned. Consequently, Chmielewski’s interpretation shows that the Mohists did operate with a sophisticated logical machinery. Obviously, there was nothing close to a logical system comparable with that of the Stoics. However, one can find some cases of an appropriate (!) use of complicated logical patterns. In the Parts referred to one can also find some original and justified observations about connections between languages of logic and ethnic languages. We can appreciate Chmielewski’s overwhelming knowledge of the archaic Chinese language. He points out that such factors as e.g. the lack of inflection, monosyllabic structure of words, the lack of “parts of speech,” the system of writing, etc.

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surely influenced the implicit logic of the disputes of ancient Chinese philosophers. It follows without doubt that these features influence “the logic of the man on the street” as well. Chmielewski offers some remarks concerning “logical thinking” (Notes, IV, 108). Let me add that one can find more his remarks on this topic in the two his Polish papers mentioned above. Also in parts V and VI of the Notes, devoted to selected logical laws (the principle of double negation, the law of contradiction, the law of excluded middle, etc.) we find an interplay between language and logic. Chmielewski pinpoints places (e.g. Mohist Canons) where ancient Chinese philosophers made use of these laws. He also discusses different roles played by several (!) negations in archaic Chinese as well as by universal and restricted quantifiers and the zero quantifier. The logical heritage of the Later Mohists is, undoubtedly, the most developed piece of the Ancient Chinese Logic. The Dialectical Chapters of Mo Tzu contain not only logical but many methodological and epistemological issues as well. We find there definitions and statements arranged in a well designed order. Thus, we find, e.g.: • six definitions concerning descriptions; • thirty three definitions concerning action; • twelve definitions concerning knowledge and change; • eighteen definitions concerning geometry; • six definitions concerning debating. The purpose of logical analysis is characterized in Mo Tzu [6, p. 330] as follows: The purpose of logical analysis is (1), by clarifying the distinction between right and wrong (shih fei), to inquire into the principles of order and misrule; (2), by clarifying points of sameness and difference, to discern the patterns of names and objects; (3), by settling the beneficial and the harmful, to resolve confusion and doubts. Only after that may one by describing summarize what is so of the myriad things, by sorting seek out comparables in the multitude of sayings.

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Let us quote, after Harbsmeier [6, p. 330 – 345], a few examples of Canons (with explanations). We are not going to comment on them; the interested reader can find a detailed discussion of the matter in the monographs by Graham (Later Mohist Logic, Ethics and Science) and Harbsmeier quoted at the beginning of the paper. • Canon: The ‘reason’ (ku) is (such that) if and only if something has got it, it will come about. Explanation: Minor reason: having this, it will not necessarily (pi) be so. Lacking this, it will necessarily (pi) be so. . . Major reason: having this, it will necessarily be so. Lacking this, necessarily (pi) it will not be so. . . (A1) • Canon: Names (ming) are unrestricted, classifying or privative. Explanation: ‘Thing’ is unrestricted. Any object necessarily (pi) requires this name. Naming something ‘horse’ is classifying. For ‘like the object’ we necessarily (pi) use this name. Naming something ‘Jack’ is privative. This name stays confined in this object. . . (A78) • Canon: If you know kou (dogs), to say of yourself that you do not know chhüan (dogs) is a factual mistake (kuo). Explained by: identity of objects. Explanation: If the knowing of kou (dogs) is identical with the knowing of chhüan (dogs), then there is a mistake. If there is no identity, there is no mistake. (B40) • Canon: The standard (fa) is that in being like which something is so. Explanation: The mental picture (i), the compass, a circle, all these may serve as standard (fa). (A70) • Canon: By learning we add something (to our knowledge). Explained by: the objector himself. Explanation: He considers that learning does not add anything and accordingly informs the other. This amounts to causing the other to know that learning does not add anything, i.e., it amounts to teaching. If one believes that by learning one does not add anything, then it is inconsistent to teach. (B77) • Canon: Non-existence does not necessarily (pi) presuppose existence. Explained by: What is referred to. Explanation: In the case of non-existence of something, the thing has to exist before it is in this way non-existent. In the case of the non-existence of the sky’s falling down, it is non-existent without ever having existed. (B49)

93 • Canon: ‘All’ (chin) is none not being so. Explanation: Something is fixed of all (chü) of them. (A43) As for ‘some’ (huo), it is ‘not all.’ (No 5)

In parts VII and VIII Chmielewski is occupied with the concepts of similarity and difference. He stresses some difficulties in search of purely logical definitions of these concepts. In this critical approach he is obviously right. But the following remark should perhaps be added. Similarity and difference (or better, similarity and opposition) are not logical terms. Identity (and its negation, i.e. diversity) may be considered as logical constants, in Alfred Tarski’s sense (cf. [11]). But similarity and difference (or opposition) are much more ontological (and to some extent also epistemological) terms. It follows that one should first built mathematical representations of similarity (sometimes also called tolerance) and difference (opposition) and only then investigate logical properties of these representations. This is my conjecture, open to discussion and criticism. (Some modest proposals concerning mathematical models of similarity and opposition can be found e.g. in: [8], [9].) In his Notes Chmielewski proposes his own explication of the famous White Horse Paradox by Kungsun Lung: • A. Is it admissible that “a white horse is not a horse (Pai ma fei ma)?” • B. It is admissible. • A. Why should be so (ho tsai)? • B. ‘Horse’ is that by which we name the shape. ‘White’ is that by which we name the colour. Naming the colour is not (the same as) naming the shape. Therefore I say: “‘white horse’ is not (the same as) ‘horse’ (Pai ma fei ma).”

(translation of this fragment: [6, p. 304]). Chmielewski’s explication is done in terms of set theory, similarly to the solution proposed by: H. Greniewski and O. Wojtasiewicz [4, p. 241 – 243]. There already exists a huge literature on this paradox which, for the limitation of the space, will not be reported here. Let us only mention that the proposed solutions are connected e.g. with: • the problem of validity of synthetic statements (cf. e.g. Antisthenes views);

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• the problem of universals: White-horseness is not horseness.; • a possible consideration of mass-terms: White-horse-stuff is not horsestuff.

3. On “Archaic Chinese language as a tool of argumentation” This paper is devoted to a very subtle analysis of just one example of argumentation, from chapter 26 of Mo Tzu (translation from Chmielewski’s Polish original is mine): (1)(a) If there is righteousness in the world, there is life; if there is no righteousness, there is death. (b) If there is righteousness (in the world), there is richness; if there is no righteousness, there is poverty. (c) If there is righteousness (in the world), there is order; if there is no righteousness, there is disorder. (2)(a) [Now,] Heaven desires life in the world and abominates death. (b) (Heaven) desires richness in the world and abominates poverty. (c) (Heaven) desires order in the world and abominates disorder. (3) (Hence I know that) Heaven desires righteousness (in the world) and abominates unrighteousness.

Before the logical analysis of this argumentation, writes Chmielewski, one has to conduct a semantic analysis of it. The first step in the latter is an association of formal symbols to particular Chinese characters corresponding to lexical items:

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A B A-B C D E F G H I J K L M N O P R

Heaven below the world (= below Heaven) there is righteousness implication particle life there is not death richness poverty order disorder to desire possessive particle (3 singular) conjunction particle (adversative) to abominate negation particle

After such an association we obtain the following schema representing this argumentation: (1) (a) (b) (c) (2) (a) (b) (c) (3)

A-B C C C A

D D D M M M A

E E E N N N M

F I K F I K D

O O O O

G G G P P P P

D D D N N N R

E H E J E L H J L D

A commentary concerning this schema is in order: • The elements: A and A-B correspond to arguments (terms); all the remaining elements correspond either to logical constants or to predicates. • Complexes built in a symmetric and parallel way are separated in (1a) – (1c).

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• In (1a) – (1c) conjunction is expressed by mere juxtaposition; in (2a) – (2c) conjunction is expressed by the particle O. • It is syntactic structure of the text which enables us to distinguish premisses and conclusion in it. We can start the logical analysis proper now. Some more remarks, however, are necessary: • Adding C to any lexical item does not change the semantic value of the latter. Thus, e.g. CD and D mean the same. The element C is introduced for stylistic reasons only. • G expresses negation. A similar role is played by R (which occurs here only once). • Elements D, F, H, I, K and L correspond to one-place first order predicates. Elements M and P correspond to two-place predicates of higher order with different kinds of arguments. • One has to remember about a strong tendency in archaic Chinese (in the philosophical texts) to use antonyms. In particular, antonyms frequently replaced negations of terms and predicates. In the case just analyzed we have the following pairs of antonyms: F—H I—J K—L M—P Chmielewski observes that considering one element of such a pair as the negation of the other does not lead to wrong solutions and he accepts this point of view while interpreting antonyms in this example. • The possessive particle N points at the argument A-B. We can now begin a “translation” of the above argument into the language of (higher order) logic. First, we associate individual names (a and b), first order predicates (ψ1 , ψ2 , ψ3 ), second order predicate (Φ) and logical constants (negation  and implication →) with the symbols used in the former semantic analysis:

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A-B — a A — b N — a

(C)D — ϕ GD — ϕ RD — ϕ

F H I J K L

— — — — — —

ψ1 ψ1 ψ2 ψ2 ψ3 ψ3

M — Φ P — Φ

In result, we obtain the following representation of the argumentation in question (Chmielewski writes a one-argument first order predicate before its argument and a two-place second order predicate between its arguments, ∧ denotes conjunction): (1) (a) (b) (c) (2) (a) (b) (c) (3)

(ϕa → ψ1a) ∧ (ϕ a → ψ1 a) (ϕa → ψ2a) ∧ (ϕ a → ψ2 a) (ϕa → ψ3a) ∧ (ϕ a → ψ3 a) (bΦψ1a) ∧ (bΦ ψ1 a) (bΦψ2a) ∧ (bΦ ψ2 a) (bΦψ3a) ∧ (bΦ ψ3 a) (bΦϕa) ∧ (bΦ ϕ a)

We recall that antonyms are replaced by negations of the corresponding elements. It is evident that conditions (1a) – (1c) can be replaced by suitable equivalencies, and hence we obtain (here ≡ is the equivalence symbol): (1) (a) (b) (c) (2) (a) (b) (c) (3)

ϕa ≡ ψ1a ϕa ≡ ψ2a ϕa ≡ ψ3a (bΦψ1a) ∧ (bΦ ψ1 a) (bΦψ2a) ∧ (bΦ ψ2 a) (bΦψ3a) ∧ (bΦ ψ3 a) (bΦϕa) ∧ (bΦ ϕ a)

Now, it is clear that the argument contains elements which are redundant from the logical point of view. Each of the pairs: (1a) – (2a) (1b) – (2b) (1c) – (2c)

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has the same “deductive power” as all the premisses taken together. The redundancy is the result of stylistic effects chosen for persuasive reasons only. Thus, the above schema is equivalent to any of the schemes below, where i = 1, 2, 3: (1) (2) (3)

ϕa ≡ ψi a (bΦψi a) ∧ (bΦ ψi a) (bΦϕa) ∧ (bΦ ϕ a)

In the presence of standard assumptions concerning the logic used (e.g. that the substitutivity salva veritate does not destroy the validity of the rules of inference) the schema itself is certainly valid. Let us mention that Chmielewski was not interested in any metalogical problems concerning Ancient Chinese logic. This, of course, is justifiable: we do not deal here with a system of logic. The only thing we are analyzing are separate arguments. 4. On “Reductio ad absurdum principle in a comparative perspective” Chmielewski calls Reductio ad Absurdum Principle the following law of the classical propositional calculus: (p → ¬p) → ¬p. This law is also called the (weak) Clavius law. The name “Reductio ad Absurdum Principle” can be found in Principia Mathematica, where it is paraphrased (page 104): “if p implies its own falsehood, then p is false.” Let us recall that the law in a sense converse to it, i.e.: (¬p → p) → p, is also called the (strong) Clavius law. The reader should admire Janusz Chmielewski for his many-sided talents: a chosen logical principle is investigated in a wide comparative perspective, including the system of logic developed in Ancient Greece, logical and epistemological investigations in Indian Buddhist logic and logical and philosophical reflection in Ancient China. In addition to general remarks concerning antinomies and the principle in question in the texts of the mentioned three cultures we find also here some deep observations concerning the transfer of logical knowledge

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(in West and East) as well as some very interesting details, e.g. claims about quantification of the predicate in ancient Chinese, supported with evidence. We are not going to report on all Chmielewski’s achievements and claims contained in his paper. Let us only mention that he is mainly concerned with the analysis of the well known paradoxical statement that all statements are false. This paradox was known to Greek philosophers: one can find it already in Xeniades from Corinth, Plato, Aristotle and Sextus Empiricus. Chmielewski provides his own translation of the argument found in Mo Tzu which has exactly the same structure. We quote Harbsmeier translation of this argument (cf. [6, p. 345]): Canon: To claim that all saying contradicts itself is self-contradictory. Explained by: his saying (this). Explanation: To be self-contradictory is to be inadmissible. If these words of the man are admissible, then this is not self-contradictory, and consequently in some cases saying is acceptable. If this man’s words are not admissible, then to suppose that it fits the facts is necessarily illconsidered (B71).

Chmielewski analyzes this fragment carefully, discussing also some others translations of it. He also considers a possibility of a logical reconstruction of the argument with the use of quantifiers (because we are concerned here with all statements). Let us also add that the Later Mohists were fully aware of the self-reflexive paradoxes concerning truth and falsehood (cf. [6, p. 344]): Canon: To reject denial is inconsistent. Explained by: he does not reject it. Explanation: If he does not reject the denial (of his own thesis that the denial is to be rejected) then he does not reject the denial. No matter whether the rejection is to be rejected or not, this amounts to not rejecting the denial (B79).

Chmielewski discusses also explications of the above paradox present in some Buddhists’ texts, Indian as well as Tibetan. He stresses the connections of these explications with some epistemological assumptions accepted in Buddhists’ doctrines.

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5. Final remarks The paper is by no means a complete exposition neither of Janusz Chmielewski’s works nor of the Ancient Chinese Logic in general. The present author does not claim to be competent in Ancient Chinese Logic, he plainly reads about it from time to time. Some work on the subject has been done in his Department of Applied Logic, Adam Mickiewicz University in Pozna´n, Poland: • Anna Uryasz-Majewska, Wybrane podstawowe poj˛ecia staro˙zytnej logiki chi´nskiej (Selected basic notions of the Ancient Chinese Logic), Investigationes Linguisticae III, 1998, Institute of Linguistics, Adam Mickiewicz University, Pozna´n, 91 – 188. M.A. Thesis defended in 1991; supervisor: Jerzy Pogonowski. • Sławomir Sikora, Dialog o białym koniu (Bai Ma Lun) — metodologiczne problemy bada´n porównawczych logiki chi´nskiej i zachodniej (The White Horse Dialogue — the methodological problems of the comparative research of Chinese and Western logic), Investigationes Linguisticae XIV, 2006, Institute of Linguistics, Adam Mickiewicz University, Pozna´n, 97 – 110. I think that Chmielewski’s work is worth of further study by logicians interested in the history of logic. The reasons for that are numerous, e.g.: • Chmielewski’s Notes on Early Chinese Logic give us examples of a fruitful application of tools used in mathematical logic to the study of reasoning conducted in an ethnic language, in addition in a language from outside the sphere being in focus of interest of the modern mathematical logic. • Notes are waiting for the elaboration of some problems only marginally noticed by Chmielewski, problems which were in a sense anticipated by him, and which are of paramount importance in the construction of mathematical models of language. • One can hardly find anything comparable to the analysis given in “Archaic Chinese Language as a Tool of Argumentation.” It can be, without hesitation, recommended as a pattern of investigation in the logical analysis of ethnic languages.

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• Also Chmielewski’s analyses in “Reductio ad Absurdum Principle in a Comparative Perspective” are worth to be recommended. One may think of a research programme concerning the list of logical laws and rules of inference present in different cultures, conducted along the lines proposed by Chmielewski. Let us stress one point in connection with the last of the above remarks. I am personally strongly against any speculations about the so called logical relativism. Different cultures may have different epistemological attitudes. But as far as logic itself is concerned, there is no such diversity. First, let us make clear what we mean by logic here: Logic is a systemized set of valid schemes of inference [10, p. 222] All terms occurring in this definition have precise mathematical definitions. Thus, we can always decide whether a given argumentation (in a given ethnic language) is conducted according to some valid rule of inference. One may find a rich inventory of arguments in, say, philosophical texts originated in one culture compared with a relatively small amount of them in a distinct philosophical tradition. But this, by no means, is a proof that one culture uses a different logic than the other one. When it comes to an evaluation of arguments, we always rely on logic in the sense of the definition given above. One may think of a hypothetical culture which has developed, say, a system of intuitionistic logic and never “thought” of classical logic as the basis for arguments. The same may concern e.g. modal logic, many-valued logic or even some sort of infinitary logic. But still, all such allegedly culture-dependent logics are (in the strictly defined sense) comparable with classical logic. Such a discipline as ethnologic is, in my opinion, not only imaginary, but indeed without any real subject (in contraposition, of course, to such well-established disciplines as ethnolinguistics or anthropology of culture). The results obtained by Chmielewski strongly support the above thesis. Arguments found in the texts of Ancient Chinese philosophers can, without exception, be reconstructed in terms of modern formal logic. I think that Chmielewski’s results support also the opinion expressed by Pogorzelski [10, p. 223]: The question whether logical rules are not only true but, moreover, necessary, is far-reaching. It is connected with the question whether logic different from classical logic (not subclassical)

102 is possible. [. . . ] In other words, is a “different logic” conceivable? The fact that it does not seem to be possible (contrary to the fact that for instance “different physics” is conceivable) indicates the existence of compelling categories of interpreting the world. One can obviously wonder whether these categories are innate (compare Łukasiewicz: logical rules are made by the creator of the world) or if making propositional structures, which are used to describe the reality, imposes logic in one and compelling way.

References [1] Chmielewski, Janusz. Language and Logic in Ancient China. Collected Papers on the Chinese Language and Logic., [in:] M. Mejor (Ed.) Komitet Nauk Orientalistycznych Polskiej Akademii Nauk, Warszawa 2009. [2] Graham, A.C. Later Mohist Logic, Ethics and Science. The Chinese University Press and the School of Oriental and African Studies, London and Hong Kong, 1978. [3] Graham, A.C. Disputers of the Tao. Philosophical Argument in Ancient China. Open Court, La Salle, Illinois, 1989. [4] Greniewski, H., Wojtasiewicz, O., From the history of Chinese logic, Studia Logica IV, 1956. [5] Hansen, C. Language and Logic in Ancient China. University of Michigan Press, Ann Arbor, 1983. [6] Harbsmeier, Christoph. Language and Logic, Part I, [in:] Joseph Needham, Science and Civilisation in China, vol. 7, Cambridge University Press, Cambridge, 1998. [7] Needham, Joseph. Science and Civilisation in China, vol. 7, Cambridge University Press, Cambridge 1998. Part I: Christoph Harbsmeier, Language and Logic. [8] Pogonowski, Jerzy. Tolerance spaces with applications to linguistics, Adam Mickiewicz University Press, Pozna´n, 1981. [9] Pogonowski, Jerzy. Linguistics oppositions, Adam Mickiewicz University Press, Pozna´n, 1993.

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[10] Pogorzelski, Witold A. Notions and theorems of elementary formal logic, Białystok, 1994. [11] Tarski, Alfred. What are logical constants?, History and Philosophy of Logic 7, 1986, 143–154.

TALMUDIC HERMENEUTICS Avi Sion Geneva, Switzerland [email protected] This paper consists of excerpts from the author’s book Judaic Logic (Geneva, 1995), with a few slight modifications. The full original text may be found at www.TheLogician.net. © Avi Sion, 2009. Text reprinted here by kind permission of the Author, who still reserves all rights under Pan-American, European and International copyright conventions.

1. Traditional Teachings Talmudic law, known as Halakhah, was decided, with reference to the Torah, after much debate. In a first stage, the debate crystallized as the Mishnah (edited by R. Yehudah HaNassi in the 1st century CE); in a later stage, as the Gemara (meaning the Completion). The main, Babylonian, Talmud was redacted by R. Ashi in the 5th century CE; another, less authoritative one, known as the Jerusalem Talmud, was closed in the 4th century. The methods used in such discourse to interpret the Torah document are today known as ‘hermeneutic’ principles (or, insofar as they are prescribed, rules). In Hebrew, they are called midot (sing. midah), meaning, literally, ‘measures’ or ‘virtues.’ This Talmudic ‘logic,’ as we shall see, has certain specificities, both in comparison to generic logic and intramurally in the way of distinct tendencies in diverse schools of thought. Various Rabbis

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proposed diverse collections of such methodological guidelines, intending thereby to explain and justify legal decision-making.1 The earliest compilations were: the Seven Rules of Hillel haZaken (1st century BCE); the Thirteen Rules of Rabbi Ishmael ben Elisha (2nd century CE); and the Thirty-two Rules of Rabbi Eliezer ben Yose haGelili, of slightly later date.2 These lists are given as Baraitot, the first two in the introductory chapter to the Sifra (1:7), a Halakhic commentary to Leviticus, also known as Torat Kohanim, attributed to R. Yehudah b. Ilayi, a disciple of R. Akiba (2nd cent. CE), and the third within later works. Baraitot were legal rulings by Tanaim not included in the Mishnah; but they were regarded in the Gemara as of almost equal authority.3 According to Jewish tradition, at least since Geonic times (notably, Saadia Gaon) and still today, these rules all date from the Sinai revelation and were since then transmitted from teachers to pupils without interruption (orthodox commentators go to great lengths to explain the differences in listing). This claim is in part confirmed by statements in the Talmud and literature of that era, in which Rabbis claim to have received knowledge of certain rules from their teachers. For examples, klal uphrat, a R. Ishmael principle, is attributed to Nechunia b. Hakaneh (Tosefta Shevuot 1:7); ribui umiut, a R. Akiba principle, is attributed to Nachum Ish Gimzu (Shevuot 26a). But the historicity of the general claim has not so far been demonstrated by any pre-Talmudic evidence: in particular, there is no obvious mention of such interpretative principles anywhere in the Tanakh. It does not, in any case, seem likely that such rules would suddenly be ‘invented,’ as a conscious act, by their apparent authors or anyone else. 1

Readers may find it useful, in this context, to study: the articles on hermeneutics in the Jewish Encyclopedia (Vol. , pp. 30-33) and the Encyclopaedia Judaica (Vol. 8, pp. 366 – 372), as well as Bergman's Gateway to the Talmud (Ch. 13), and the Reference Guide to Steinsalz's English edition of the Talmud. 2 Note also that Malbim, in Ayelet haShachar, collects “all the hermeneutic rules scattered through the Talmudim and Midrashim” (Jewish Encyclopedia), which are reckoned as 613 in number. I did not look into this more modern source, which is likely to be rich. 3 As Scherman has pointed out, these Baraitot were different, in that they were not in themselves statements of law but explanations of how the laws were derived from the Torah source.

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The most likely scenario, from a secular point of view, is that they were for some time unconscious discursive practices by participants in legal debates; gradually, it occurred to some of these participants (most probably precisely those whose names have come down to us as formulators, or reporters and collectors, of hermeneutic rules) that they and their colleagues, and their predecessors, repeatedly appealed to this or that form of reasoning or argument, and such implicit premises would be made explicit (thereby reinforcing their utilization). Different such commentators would find some rules more convincing than others and thus compile selections; eventually, contending schools emerged. The following would be the natural stages of development of such a body of knowledge: first, unconscious practice (which might be correct or incorrect); second, dawning awareness of such practice (due to what we call ‘self-consciousness’); thirdly, verbalization, randomly to begin with (by exceptional individuals, focusing on the most outstanding practices), and then more and more widespread (more insights, by more people, as a cultural habit develops); fourthly, systematization (of the simplest kind, namely: listing) and dispute (as different lists are drawn up by different groups). In the case of Judaism, the next stage was merging results (by later generations, out of veneration making all the lists ‘kosher’ at once); and subsequently, there was a stage of commentary (trying to justify, explain – within certain parameters). However, sadly, as we shall see, the last natural stages of formalization and evaluation never occurred (until recently, outside orthodox circles). According to the Jewish Encyclopedia, these various lists were not, even in their own times, viewed as exhaustive. I am not sure how true that remark is, i.e. whether there is any statement in the Talmud or related literature which confirms the assumption that Hillel, R. Ishmael, R. Eliezer, or whoever compiled a list, did not consider himself as having succeeded in making a full enumeration of valid midot. At the other extreme, the view of traditionalists today, that these lists were all equally complete, is demonstrably just as conjectural, and based on anachronistic and circular arguments. What is in any event evident, is that the rules in each list were not in their own times uncontested. The school of Hillel was opposed by that of Shammai, and Rabbi Ishmael’s formulations were challenged by Rabbi

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Akiba ben Yossef. It is interesting to note that, at first, these opposing views were considered mutually exclusive; but, over time, they came to be used indiscriminately. (Still today, in Talmud study, people do not find it odd that a R. Ishmael rule might prevail in one context and a R. Akiba rule in another. Logically, one would have thought that just one of the systems would have to be adopted for the whole Talmud.) It apparently came to be considered that, although two dissonant rules may indeed lead to conflicting interpretations, the selection of one or the other of them as the finally applicable rule in any given single context, was a matter of tradition or majority decision; effectively, the correct conclusion was predetermined, and the rule selected only served as an ex-post-facto rationalization. Thus, the ontological status ascribed to the hermeneutic rules is that they were conditional on material factors – formalities activated or left dormant by textual content (which details were, one by one, designated by authorities, on the basis of transmissions or by vote). Although R. Akiba’s approach usually prevailed in practice, R. Ishmael’s thirteen midot are the most popularly known: they have become part of the daily liturgy and can be found in most Jewish prayer books. Since the above mentioned initial formulations, many attempts have been made to compile more complete lists (for instance, by Malbim). We will in the next sections analyze the main hermeneutic principles – at least, those of R. Ishamel – in some detail with examples; here, we will be content to only make some introductory comments. Broadly speaking, we refer to any thought process which tends to convince people as ‘logical.’ If such process continues to be convincing under perspicacious scrutiny, it is regarded as good logic; otherwise, as bad. More specifically, we consider only ‘good’ logic as at all logic; ‘bad’ logic is then simply illogical. In short, the term ‘logic’ may loosely refer to any form of discourse, but more strictly refers to forms capable of scientific validation. Logic, properly speaking, is both an art and a science. As an art, its purpose is the acquisition of knowledge; as a science, it is the validation of knowledge. Many people are quite strong in the art of logic, without being at all acquainted with the science of logic. Some people are rather weak in practice, though well-informed theoretically. In any case, study of the subject is bound to improve one’s skills.

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Logic is traditionally divided into two – induction and deduction. Induction is taken to refer to inference from particular data to general principles (often through the medium of prior generalities); whereas deduction is taken to refer to inference from general principles to special applications (or to other generalities). The processes ‘from the particular to general’ and ‘from the general to the particular’ are rarely if ever purely one way or the other. Knowledge does not grow linearly, up from raw data, down from generalities, but in a complex interplay of the two; the result at any given time being a thick web of mutual dependencies between the various items of one’s knowledge. Logic theory has succeeded in capturing and expressing in formal terms many of the specific logical processes we use in practice. Once properly validated, these processes, whether inductive or deductive in description, become formally certain. But it must always be kept in mind that, however impeccably these formalities have been adhered to – the result obtained is only as reliable as the data on which it is ultimately based. In a sense, the role of logic is to ponder information and assign it some probability rating between zero and one hundred. At the outset, it shall be pointed out that the Talmud’s hermeneutic rules are not all of a purely deductive nature, contrary to what may be thought at first glance. When the rules suggest a “derivation,” they do not necessarily refer to a mechanical relation between premises and conclusion. Most of the rules’ results are partly or entirely inductive; that is, they are, at best, a good working hypothesis within the given context of knowledge, which may possibly be replaced by another hypothesis or a deductive inference in an altered context of knowledge. Some of the rules, wholly or partly, represent deductive or inductive principles which can readily be justified by natural logic. Of these, some may be validated in formal terms (i.e. substituting symbols for specific contents); whereas others describe discursive acts which are rather intuitive – responses to material data without fixed patterns – and which can be approved with reference to broader epistemological considerations. However, some of the rules, wholly or partly, seem, from the point of view of natural logic, rather obscure and arbitrary, and remain acceptable only due to a claim that they are of Divine origin.

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The Talmud itself at least implicitly recognizes the inductive nature of many of the arguments in it. This is evidenced by the fact that when several alternative premises are given for a certain conclusion, it is viewed as being weak. The Rabbis argue: Nu, if each of the sets of reasons given was sufficient, why bother to adduce the others? From a deductive point of view, there is indeed no utility in giving many reasons; but nor is there any harm in it. It is only in inductive logic that giving more reasons increases the probability of the result, and therefore also suggests (incidentally) its relative weakness. Our remark that Talmudic (and indeed later rabbinic) reasoning is very often inductive, rather than purely and exclusively deductive, should be emphasized. It is contrary to popular belief (people are rather surprised when I suggest it), and so manifestly ignored by other writers that I would tend to claim it as original. If it is original, then it should be stressed as very important, among the most significant insights of the present work. In any case, it is evident and incontrovertible fact. The idea is disturbing, not to say devastating, to many people, because induction is thought of as inherently more fallible than deduction, and it is difficult to juggle with doubt and dogma. But in all fairness the truth of the matter is that deductive reasoning can also in principle and often does in practice err, and that inductive reasoning is not in principle necessarily weak nor does it always go wrong in practice. Each case must be considered on its own merits; one cannot make sweeping statements for or against such broad categories of mental process. 2. The Thirteen Midot Let us now briefly take a look at the tenor of the 13 Midot of R. Ishmael. We may distinguish three groups: a) Midot whose purpose is to infer information from the text, i.e. to make explicit what is implicit in it; this includes rules Nos. 1 – 3 and 12. b) Midot used to elucidate terms in the text, especially their extensional aspect; this includes rules Nos. 4 – 7. c) Midot serving to harmonize seeming or manifest incongruities in the text, including, as well as inconsistencies, mere redundancies,

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discrepancies, and other sources of perplexity; this includes rules 8 – 11 and 13. Admittedly, this grouping of the 13 Midot is a bit artificial. For, in a strict view, all inference of information is an eventual elucidation of terms and a prevention of inconsistency; and similarly, all elucidation of terms constitutes inference of information and harmonization; and likewise, all harmonization results in elucidation of terms and leads to inferences of information. Nevertheless, the immediate goals of these different movements of thought are sufficiently distinct to justify our subdivisions. A nice thing about these groupings is that they show a continuity of sorts in the approach of R. Ishmael, and explain and justify the sequence in which the midot were listed. The only misplaced midah in our view is No. 12, which should be closer to No. 2, or at least in the same group. Inferences of information Rule 1, qal vachomer (lit. lenient and stringent), refers to a-fortiori, a form of argument whose conclusion is essentially deductive, though there are in practice inductive aspects involved in establishing the premises, as we have seen. Within Judaic logic, this form of reasoning has in fact served as the paradigm of deduction, much as Aristotle’s syllogism (with which it is often confused) has had the honour within Western logic. The discovery of a-fortiori is, I would say, one of the most brilliant contributions of Jewish logicians to generic logic. It should be noted that afortiori has Biblical roots, as Jewish tradition has reported since Talmudic times if not earlier. Rule 2, gezerah shavah (lit. equal rulings), refers to arguments by analogy, or more specifically inferences based on homonymy (similarity of wording) or on synonymy (similarity of meaning). Reasoning by analogy was very common among the ancients, Jewish and otherwise, until the advent of the scientific method in relatively modern times; it could range from far-fetched comparisons to very credible equations. Of course, most arguments, including syllogism, are based on analogies, since conceptualization depends on our intuition of similarities between apprehended objects. However, not until recently was it fully understood that the legitimacy of an analogy rests on its treatment as a hypothesis to be tested, and repeatedly tolerated (i.e. not rejected) and even confirmed (if

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predictive) by evidence, more so than its alternative(s). So analogy is essentially an inductive mode of thought. While gezerah shavah is based on closeness of subject-matter, inferences from context appeal to the textual proximity of topics. Such logistical considerations are relatively incidental, but they lean on the fact that the text in question was written by an orderly mind. This form of reasoning includes: the rules known as heqesh (relating to two items in the same verse) and semukhim (relating to two items in adjacent verses), which are traditionally counted as aspects of rule No. 2 (though probably later inclusions under that heading); and rules classed as No. 12, meinyano (inference from immediate context) and misofo (inference from a later reference). Such reasoning has obviously got to be regarded as inductive, since however intentional the positioning of words, phrases or sentences, there have to be occasional changes of topic. A matter of related significance, note, is the assumption by R. Akiba that, in a Divine document such as the Torah, the choice and placement of words cannot be accidental; whence, no repeated word is superfluous and no missing word is insignificant, every letter counts, and so on. This view allows, indeed encourages, many an inference (or alleged inference). Be it said, R. Ishmael did not in principle agree on this issue, but considered that the Torah “speaks in the language of men.” The interpretations involved in analogical or contextual arguments may be intuitively reasonable enough, but they are not readily put in formal terms and are therefore difficult to validate systematically. In any case, applied indiscriminately, such arguments are bound to lead to difficulties – one line of reasoning may lead to one conclusion, and another to its opposite, there being no inherent logical protection against contradiction. And indeed, difficulties were often encountered. For this reason, many limitations were imposed on these rules; and ultimately, they were regarded as unusable without the support of an accepted tradition, or at least the approbation of the majority of the authorities. Rule 3, binyan av (lit. father construct), seems to refer to causal reasoning; that is, to finding the causes (in a large sense) of differences or changes, and thus predicting similar effects in other contexts. In a legal context, this means finding the underlying basis of known laws, so as to be able to make coherent laws in other areas. Here too, argument by analogy

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is involved, and the mode of thought is essentially inductive. The way the rule is traditionally worded (“a comprehensive principle derived from one text, or from two related texts”) gives a false impression that it refers to immediate or syllogistic inference; but we must look at its operation in actual practice to understand it, and in such event the role played in it by the process of generalization becomes evident. While such reasoning is relatively easy, nowadays, to express formally and control scientifically, the Rabbis (as we shall see) had a surprisingly hard time with it. Elucidation of terms Rules 4 – 7, labeled collectively as klalim uphratim, seem to concern class logic, to a large extent, as they involve the expressions klal (general) and prat (particular) in various combinations. Many arguments of this kind may be viewed as effectively proceeding from definable linguistic conventions – in the non-pejorative sense that they reflect certain uniformities of intent, in the style of Hebrew expression used by the Torah. For instances: the combination of a general term followed by a particular term, in close Torah verses or parts of a verse, yields a particular result (klal uphrat); whereas the reverse combination, of a particular term followed by a general term, yields a general result (prat ukhlal). As every writer or speaker knows, a maximum of information can be communicated in a minimum of words, through certain turns of phraseology. This seems to be the motive, here. Well and good, thus far – in theory. But in actual practice the expressions klal and prat cannot always be taken at their face value. Closer acquaintance with practical applications of the klalim uphratim rules reveals that their logic is not quite identical with that of Aristotle. In Western logic theory, inclusions or exclusions between broader concepts (genera, overclasses) and narrower ones (species, subclasses), or classes and their singular instances (individuals), are purely mechanical procedures, which presuppose clearly defined terms. Such subsumptive arguments can be readily represented pictorially by circles within or intersecting or outside other circles, known as Euler Diagrams, and are the domain of Aristotle’s syllogistic processes. But in the more Oriental logic of the Talmud, things are not so simple; terms are vaguer and may be taken to “imply” formally unrelated ones.

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On second thoughts, the truth is that in practice, even in Western thought, terms are not always at the outset clearly defined; rather, usually, the definition of a term is arrived at through a gradual, inductive process, as we focus on the subject matter more and more, and acquire a deeper knowledge of it. Sometimes we do decide by convention to name a phenomenon whose description we have already; but more often, we name a phenomenon before we are able to express its essence in words, and then work our way by trial and error to a satisfactory definition of it. This developmental aspect is not yet well accounted for in the classical theory of class-logic. Certain efforts at exegesis are rather contorted, and a great deal of fantasy and credulity are needed to accept them. R. Akiba’s methodology, where the terms used for the purposes of inclusion or exclusion are ribui (broad) and miut (narrow), seems especially weird to our minds. For instance, “sheep” may imply “birds” or even “garments,” without apparent rhyme or reason. This is why Maimonides regarded such arguments as having a mere mnemonic purpose. Their conclusions were foregone,4 received in the chain of oral tradition; nevertheless, the Rabbis made a determined effort to anchor them, however flimsily, in the written Torah. The best we can do to formalize such logic, then, would be to say that, given the tradition that the laws concerning a certain topic are X, Y, and Z; and that these laws are to be derived from a specified passage of the Torah, distinguished by the terms or phrases A, B and C; then, if X is related to A, and Y is related to B, it follows that Z is to be paired-off with C.5 The formal logic involved is therefore conjunctive and hypothetical: Granting that “if A and B and C, then X and Y and Z” 4

To illustrate this, a funny joke is circulated in Yeshivot: “How do you know you have to wear a yarmulke? Because it says Vayetse Yacov... Would Yacov go out without a kipah?” 5 We might cite as an example of such reasoning Rashi's “if it does not apply,” which Bergman clarifies as follows: “If the Torah indicates a halachah in a case or category where it is already known, then apply that halachah to another situation” (p. 120, my italics). Obviously, here, Rashi is appealing also to the R. Akiba principle that there is nothing repetitive or superfluous in the Torah. The problem remains, which other situation, and how is the choice to be justified!

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– and also knowing that “if A then X” and “if B then Y” – it follows that “if C then Z” (A and B being here, of course, tacitly with C). However, apart from this aspect, it is frequently difficult to honestly find formal justification for such argument; that is, how the connective relations of the major and minor premises were in the first place established. When in such contexts the Rabbis are found to argue between themselves at length, the discussion often does not revolve around such basic issues of proof, but is merely a controversy as to which of X, Y, Z is to be paired-off (seemingly arbitrarily) with which of A, B, C. The only way then left to us, to explain the unexplained, is to appeal to ‘tradition.’ Harmonization Rules 8 – 10, which start with the words kol davar shehayah bikhlal veyatsa (lit. whatever was in a general principle and came out), deal with sets of statements whose subjects are in a genus-species relation. Rule 8, although perhaps originally intended as one rule, has become traditionally viewed as having two variants, which we are calling lelamed oto hadavar and lelamed hefekh hadavar; these concerns cases where the predicates are also in a genus-species relation of sorts. Rule 9, liton toan acher shehu kheinyano, concerns predicates which are otherwise compatible; and rule 10, liton toan acher shelo kheinyano concerns incompatible predicates. Rule 11, which also starts with the words kol davar shehayah bikhlal veyatsa, and continues with the words lidon badavar hechadash, deals with situations where an individual changes classes and then returns to its original class. Rule 13, the last in R. Ishmael’s list, shnei ketuvim hamakhechishim, concerns other reconciliations of conflicting theses; note that this principle is to some extent reflected in the methodological concept of kushya and terutz (difficulty and its resolution). All these dialectical principles are quite capable of formal expression, and are demonstrably mainly inductive in nature, involving generalizations and particularizations. There are some deductive, logically necessary, aspects to them; but on the whole, as complexes of intellectual responses to given

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textual situations, they favour one course over another, which is logically equally possible, if not equally probable, and therefore they constitute inductive mental acts. One might well ask why God, the ultimate author of the Torah, expressed Himself in so tortuous and confusing a manner, that necessitated such complicated interpretative principles, instead of speaking plainly and straightly. The answer I received from teachers when I asked that question was that His purpose must have been to conceal the truth somewhat, so as to stimulate Torah study. Also, if everything was made clear in a systematic and explicit manner, the Torah could be studied fully in isolation; whereas, God wished it to be studied in a more social manner. Some also suggest as an answer, on the basis of qabalistic ideas, that if the Torah was perfectly explicit and unambiguous, then there would be no room for doubt in the world, and skeptics would have no opportunity to make the redemptive leap of faith, which is needed to safeguard human freedom of choice. If God was totally revealed, then humans would be forced, in fear and trembling, and out of infinite love, to surrender all personal will and identity. The diversity of the world was created and is maintained precisely through a concealment of some of the truth (for if the world is ultimately, in truth, unitary, then all appearance of plurality must be a sort of untruth). So much for the content, in brief, of R. Ishmael’s list of rules. Our analysis (here and elsewhere) somewhat justifies the order in which the rules appear in this list (except, as already stated, for rule 12). However, some of the groupings implied by this list are open to discussion. I would suggest that all inferences from context, including heqesh and semukhim (traditionally considered as subcategories of gezerah shavah) and meinyano and misofo, should have been grouped together under one heading (just as, for instance, gezerah shavah constitutes one heading with subdivisions). Especially, the klalim uphratim should, in my view, be reorganized, and counted as one heading, or as at most two (classifying each process according as its result is a klal or a prat),6 instead of four. Finally, in my opinion, the two variants 6

Including, appropriately separated, the two rules distinguished by the word hatsarikh, which are traditionally lumped together under No. 7. As elsewhere discussed, the treatment of complementarity as something distinct is an overreaction, in my view.

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of lelamed ought to be regarded as separate rules, comparable to the two rules liton toan acher. A comment worth making is that the arrangement and numbering of the midot may not be stipulations of R. Ishmael, but may be proposals of the compiler R. Yehudah. To my knowledge (without having researched the matter greatly), R. Ishmael did not systematically group and list his midot, but merely formulated them and referred to them individually in various contexts as the need arose; it is probably R. Yehudah (the author of the Sifra) who later brought them together in a list and organized them into 13 sentences in the given order. But the number 13 is not sacrosanct. According to Bergman, the Raavad noted the possibility of a count of 16 (counting rules Nos. 3, 7, 12 as two each); while others suggested counting rules 8 – 11 as one and thus supposedly arrived at a count of 10. My preferred manner of counting yields the number 13 – 2 + 1 = 12. In my view, this is wrong. Rule 11 is functionally radically distinct from rules 8 – 10, albeit the common opening phrase. And rules 8 – 10 are sufficiently different in their premises and conclusions to justify separate treatment, even though they are obviously a related series. This becomes clear in deeper analysis. It must be noted that, judging by actual Talmudic and rabbinic discourse, the inventory is incomplete.7 Orthodox commentators would not accept this last remark, and try to explain away every silence or disagreement of R. Ishmael (or R. Yehudah) concerning some rule or some detail of a rule mentioned by other authorities, earlier, contemporary or later. Since they regard the 13 rules as (an oral) part of the Revelation at Sinai, they must explain why Hillel listed only 7 rules, or R. Eliezer listed as many as 32. For this reason we find Bergman making statements like “Hillel certainly did not intend to dispute the teaching of R’ Yishmael,” even though Hillel lived a couple of centuries before R. Ishmael! Besides, how can one make 7

Other principles worth noting, which are in practice used for hermeneutic purposes, are rov (this statistical principle is usually associated with majority decision by judges, but it may also be applied to matters of judgment, as for instance in Avoda Zara 75b, where Num. 31:22-23 “every thing that may abide the fire” is understood by Rashi as referring to cooking utensils, since they are the metal implements habitually subjected to fire) and perhaps hazakah (which, again, is usually associated with the legal status quo, but in many contexts refers to empirical evidence).

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conjectures about a past person's “intentions,” without written record to support one's case, and say “certainly?!” Not only does Talmudic logic have specificities in comparison to generic logic, but there are different logical trends within the Talmud itself. That is already clear the well-known competitiveness between the schools of Hillel and Shammai, or between R. Ishmael and R. Akiba. But the differences embodied in explicit principles may not reflect all the underlying differences; there seems also to be unstated differences, which were not brought out into the open. This refers to the concept of the shitah: as is well known, there are leitmotifs which run through the legal rulings of individual Rabbis. What is very evident, upon closer inspection (which I will spare you here) is that it would be very difficult to claim that these various authorities based their work on a common blueprint. And this is of course a very unorthodox conclusion. The hermeneutic principles were intended to explain and justify the development of Jewish law from its Torah source. They were the methodological bridge between the Torah and the Mishnah and Gemara; the more or less logical techniques by means of which (to the extent that they are accurate renditions and exhaustively listed) the written foundation-document, together with the oral tradition, were transmuted into the Talmud. These are not to be confused with a further set of principles is traditionally transmitted in Judaism, which reflects more broadly the transition from Mishnah to Gemara, and then from Talmud to subsequent rabbinic Law, and finally the way Halakhah is actually taught and studied. These additional principles may be characterized as heuristic (practical rules of thumb), rather than hermeneutic (a priori methodologies), in that most of them constitute ex post facto summaries of certain uniformities in terminology, textual presentation and personal authority found in the Talmud. I say ‘most,’ because some of them though listed together with relatively incidental rules of thumb, are more or less objective logical forms and would have been more appropriately listed together with interpretative techniques. But the reason for their inclusion is usually to elucidate the terminology, rather than to deeply study their logical properties.

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Many of the heuristic principles were already made explicit in the Talmud itself, reflecting the intelligence, self-consciousness and unity of purpose of its protagonists, recorders and redactors (judging by a chart in Aiding Talmud Study, there were a couple of hundred named participants over a period of some 450 years). But some were evidently formulated in succeeding centuries, by Savoraim, Geonim, Rishonim and Acharonim. Current works in English describe such heuristics often in tandem with hermeneutics. 3. Exposition and Evaluation Traditional presentations of the principles and practice of rabbinic exegesis consist in listing the Thirteen Midot of R. Ishmael (at least, though other techniques may be mentioned, in contrast or additionally), describing roughly how they work, and illustrating them by means of examples found in the Talmud or other authoritative literature.8 Such an approach is inadequate, first of all, because the theoretical definitions of the rules are usually too vague for practical utility, and for purposes of clear distinction between similar rules. A simple test of practicality and clarity would be the following: if well defined, the rules should provide any intelligent person with a foolproof procedure, so that given the same database as the Rabbis, he or she would obtain the same conclusions as they did. The second important inadequacy in the traditional approach is the near total absence of evaluation; there are no validation procedures, no reductions to accepted standards of reasoning. There is no culture of frank critique; any criticism ventured is always kept well within traditional bounds and never allowed to get too radical. There is no denying the genius of R. Ishmael and others like him, in their ability to abstract rules of intellectual behaviour from the observation of 8

My analyses were made possible thanks mainly to Bergman's detailed presentation of the 13 Midot. Though I dislike that author's pompous tone and unquestioning fanaticism, and disagree with many of his specific positions, he is to be commended for his unusual efforts to clarify the hermeneutic principles. All too often, authors are simply content with listing examples with a minimum of reflection; he at least tries (if not always successfully) to sort out logical relations explicitly.

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their own and their colleagues’ thought-processes in various situations. Nevertheless, as closer scrutiny reveals, their failure to treat information systematically and their lack of logical tools, yielded imperfect results. In my work Judaic Logic, I propose some original ways to expose and evaluate rabbinic hermeneutics (mainly, the 13 Midot). The most important step in our modern method is formalization; this means, substituting variable-symbols (like ‘X’ and ‘Y’) for terms or theses of propositions. The formalization of relations is not technically valuable (apart from saving space), and tends to alienate and confuse readers; for these more abstract features of propositions, it is best to stick to ordinary language. Formalizing an argument, note, means: formalizing all explicit and tacit premises and conclusions. The value of this measure is that it helps us to clarify the situations concerned, the rabbinical responses to them, and the issues these raise. By this means, we move from a level akin to arithmetic, to one more like algebra. When we deal in symbols, we reduce immensely the possibility of warped judgment, due to personal attachment to some solution; all problems can be treated objectively. But it should be added that logical formalization is not always the most appropriate tool at our disposal; in some cases, epistemological and/or ontological analyses are more valuable. • We have two sets of data to thus formalize, or analyze in some manner: (a) the theoretical pronouncements of Rabbis (defining or explaining the rules, or guiding their utilization), and (b) the practical examples they give in support (illustrating or applying their statements). This work allows us to compare, and if need be contrast, rabbinic theory and practice. As we shall see, they do not always match. • Another utility of formalization or similar processes, is the possibility it gives us for comparing rabbinic conclusions to the conclusions obtained by syllogism or other such established logical techniques. This is the ultimate goal of our study, to determine without prejudice whether or to what extent rabbinic hermeneutics comply with deductive and inductive logic. As detailed analysis shows, they do not always parallel and even sometimes oppose the course taken or recommended by ordinary logic. In anticipation of such divergences, it is important to study the rabbinic hermeneutic principles carefully, and distinguish between their natural factors and their artificial factors. The natural aspects are immediately

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credible to, and capable of formal validation by, ordinary human logic, and thus belong to secular epistemology. The artificial aspects, for which Rabbis claim traditional and ultimately Divine sanction, are controversial and require very close examination, for purposes of evaluation or at least explanation. Our task with regard to such additives is to consider whether the rationales for them offered by the Rabbis are logical and convincing, or whether these factors ought to be regarded as human inventions and errors.9 It should be clear that we cannot allow ourselves the intention of masking any difficulties, but must engage in a “warts and all” exposé. The technicalities may be found hard-going by many people, but both secular and religious scholars, who endure through the ordeal, will be richly rewarded. They will find, not only an independent audit of rabbinic hermeneutics, but a methodological demonstration of universal value. By the latter remark, I mean that the same method of exposition (by formalization) and evaluation (with reference to formal logic) can be applied to other movements of thought in Judaism, or outside it, in other religions or other domains (philosophy, politics, or whatever). This detailed exposition and evaluation is, to repeat, found in the work called Judaic Logic. It is far too long-winded and intricate to be presented in so short a paper as the present one. Suffices for us here to review some of the conclusions of that study. We shall first express a verdict on rabbinic Hermeneutics. No doubt, certain doctrinaire defenders of Judaism are very upset for the devastating deconstruction of rabbinic hermeneutics in Judaic Logic. But facts are facts, logic is logic. There was no intention to discredit Jewish law; quite the opposite. Our method simply consisted in analyzing traditional data, examples and principles put-forward by Judaism itself, with reference to scientific logic. 9

We may regard the rabbinic principle ain mikra yotse miyedei pheshuto (quoted by Encyclopaedia Judaica, p. 371, with reference to Shab. 63a and Yev. 24a, and there translated as “a Scriptural verse never loses its plain meaning,” with the added comment “i.e., regardless of any additional interpretation”), as an implicit recognition that interpretations using the hermeneutic principles were not always natural. It may be asked how they managed to mentally accept conflicts between a midah-generated reading and a simple reading (pshat), given such a principle!

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Let us summarize, very briefly, the results of our research into the 13 Midot, with a view to distinguishing their natural and artificial aspects. Note first that all the rules suffer to some extent from vagueness and ambiguity, which means that they are bound to be applied with some amount of anarchy. • Qal vachomer is a natural and valid form of reasoning. It was reasonably well-understood and competently-practiced by the Rabbis (this is not of course intended as a blank-check statement, a blanket guarantee that all rabbinic a-fortiori arguments are faultless10), without weird embellishments. So, we can say that this first midah has essentially no artificial components; though rabbinic attempts to reserve and regulate use of this midah must be viewed as artificial add-ons. • Gezerah shavah is based on a natural thought-process, comparison and contrast, which applied to textual analysis pursues equations in meaning (synonymy) or wording (homonymy). Analogy is scientifically acceptable, though only insofar as it is controlled by adductive methods, namely ongoing observation of and adaptation to available data. While the Rabbis demonstrated some skill in such inference by analogy, they did not clearly grasp nor fully submit to the checks and balances such reasoning requires. Instead of referring to objective procedures, they tried to reserve and regulate use of this midah by authoritarian means; and moreover, they introduced logically irrelevant provisions, on the “freedom” of the terms or theses involved. Thus, this rule, though it has a considerable natural basis, eventually developed quite a large artificial protuberance, and should not in practice be trusted implicitly. • Inferences from context, including heqesh, semukhim, meinyano and misofo, are like arguments by analogy, in that the primitive mind accepts them immediately, just because they appear reasonable. But, upon reflection, we must admit the need for verification procedures; and, ultimately, the only scientific means we have is adduction (repeated testing, and confirmation or elimination, of hypotheses). In any event, 10

In particular, though the dayo principle was formulated by Rabbis, some other Rabbis resisted it; there were good reasons on both sides, meaning that it is sometimes imperative and sometimes avoidable, so that this theoretical controversy can be excused. However, there were in practice some inexcusable breaches of that principle – inexcusable, within the given context.

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proximity is not, even in theory, always significant; so one cannot formulate a hard and fast rule about it. It follows that the rabbinic attempt to do so is bound to be rather artificial, to the extent that it is presented as more than just a possibility among others. • Binyan av is a rabbinic attempt at causal logic. The induction of causes and effects is, of course, a natural and legitimate process, when properly performed, by observing the conjunction or separation of phenomena, tabulating the information and looking for behaviour patterns. The rabbinic attempt at such reasoning was, I am sorry to say, less than brilliant. The Rabbis seem to have grasped the positive aspect of causal reasoning, but apparently could not quite grasp the negative aspect. In practice, they may have often intuited causal relations correctly; but they had difficulty analyzing the relationship theoretically, in words. The outcome of such relative failure is that binyan av efforts must be viewed with suspicion, and classed among the artificial aspects of rabbinic exegesis. • The various klalim uphratim rules (including both R. Ishmael’s and R. Akiba’s variants) reflect a natural aspect of exegesis, but insofar as they rigidly impose interpretations which have conceivable alternatives, they must be judged as somewhat or occasionally artificial. This regards theory; regarding practice, we can go much further. In many cases, these rules are applied very artificially, being used as mere pretexts for contrived acts which have no real relation to them. If we regard every such false appeal to these principles as an effective instance of them (viewed more largely), then their artificial component is considerably enlarged. • With regard to the first few rules starting with the phrase kol davar shehayah bikhlal veyatsa, we found their common properties to be their concern with subalternative subjects (or antecedents) with variously opposed predicates (or consequents). Where the predicates are in a parallel relation compared to the subjects, the conclusion generalizes the minor predicate to the major subject (lelamed oto hadavar). Where the predicates are in an anti-parallel relation compared to the subjects, the conclusion renders the minor premise exclusive and particularizes the major premise (lelamed hefekh hadavar). Where the predicates are incompatible, the conclusion is similar in form to the preceding, though

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for different reasons; and perhaps additionally, it renders the minor subject and major predicate incompatible (liton toan acher shelo kheinyano). With regard to situations where the predicates are otherwise compatible (liton toan acher shehu kheinyano), our research has not determined the rabbinic conclusion and left the issue open. Now, in all these cases, except for the main conclusions of shelo kheinyano, which resolve significant inconsistencies in accord with natural logic, the rabbinic conclusions are deductively unnecessary: they are at best inductive preferences. However, since they are viewed by the Rabbis, not as tentative hypotheses open to testing, but as laws to be followed come what may, they must be considered as arbitrary and artificial. Furthermore, while we have attempted to determine the exact forms of these laws, the Rabbis themselves are not always clear on this issue, and occasionally misplace examples; this is an additional reason to regard their activities under these rubrics (except, to repeat, for legitimate harmonization) as suspect and artificial. • The rule lidon badavar hechadash, which the Rabbis were not sure how to distinguish, was found by formal methods with reference to examples to concern movements of individuals from one class to another and back; it was intended by R. Ishmael to raise a question with regard to corresponding changes in predication. While a literal approach to text would reject such a question, within a more open-minded exegetic system, it seems reasonable enough. Epistemologically, this rule instills exceptional caution in the situations concerned, making inferences conditional on reconfirmation. However, even if we do not classify this rule as overly artificial on theoretical grounds, we must regard some of its alleged applications with considerable suspicion, in view of the evidence that the Rabbis are unclear about it. • Lastly, the rule shnei khetuvim hamakhechishim, viewed as a wideranging harmonization principle, may be classed as an important aspect of natural logic. However, this essential validity does not automatically justify every dialectical act found in rabbinic literature; quite often, rabbinic interventions under this guise are rather forced. Furthermore, this rule may not, in fact, have been intended by R. Ishmael to cover every conflict resolution (or at least every conflict not resolved by preceding rules); its scope may have been intended to be premises with a

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common subject (or antecedent) and variously opposed predicates (or consequents). Such uncertainties in definition call for caution, too. In sum, this rule, as with most of the previous, in practice if not in theory, contains artificial factors. This summary makes clear that we cannot define in one sentence the distinctive features of rabbinic ‘logic,’ i.e. those aspects of it which are not granted universal validity by natural logic. Broadly speaking, the Rabbis developed distinct modes of thought due to lack of formal tools, consequent vagueness in theoretical definitions, and resulting uncertainties in practical applications. Their natural logic was gradually thickened by an agglutination of diverse artificial elements, which became more and more difficult to sort out, and more and more imposing. Being manifestly unjustifiable by natural means, these extra elements had to be defended by intimidation, with appeal to Divine sanction and the authority of Tradition. The verdict on most of rabbinic hermeneutics, emerging from our precise logical analysis has to be, crudely put, thumbs-down. In the last analysis, whatever it is, it is not a teaching of pure logic. There are, to be sure, many aspects of it which are perfectly natural and logical. But certain distinctive aspects of it, which we may refer to as peculiarly Judaic ‘logic,’ must be admitted to be, for the most part, either non-sequiturs or antinomial; in all evidence, products of very muddled thinking. We could, with an effort, make allowance for many of the latter processes, if they were viewed as ab-initio tentative hypotheses, inductive first-preferences, subject to further confirmation or at least to non-rejection by the remaining body of knowledge. But they are traditionally presented as irrevocable certainties, quasi-deductive processes, not subject to critical review (at least, without a special license granted to a privileged few). So we must evaluate them in that given framework. Whatever traditional claims, according to logic it is virtually inevitable that, in a large body of information, the adoption of unnecessary postulates and the arbitrary contradiction of given data will result in hidden, if not obvious, inconsistencies. All the more so, where the proof-text itself is rather ambiguous, disorderly and confusing, as is the Torah, so that one must proceed very carefully.

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To arrive at a consistent result, using artificial processes like R. Ishmael’s rules, it is essential to have a certain leeway, a possibility to retreat as well as advance. If each rule has to be applied rigidly and irreversibly, the endresult is bound to be untenable, and only capable of being sustained by lies and self-delusion. Even a simple, natural generalization of some Scriptural statement, through say a lelamed oto hadavar, may turn out to be in conflict with some other textual statement; how much more so with a complex, twisted paralogism, like say a lelamed hefekh hadavar. In such cases, we must either retract or modify the text: on what basis we are allowed to do the latter, without absolute logical need, I have no idea; it would seem much more justifiable to do the former. Surely, our primary axiom must be that the Torah is more reliable than rabbinic constructs. 4. The Sinai Connection Let us now, in the way of a conclusion, look into some aspects of the issue of the Sinaitic origin of Talmudic/rabbinic hermeneutics. The only conceivable defense against the results of the present research is to say that the rules of rabbinic exegesis constitute a secret code, by which instructions in the Torah are to be transformed into valid legal statements. This thesis suggests that God deliberately wrote the Torah in a misleading way, not wanting everyone to have access to His real intentions, but only a select few (the Jewish Rabbis), to whom a conversion table, the hermeneutic principles, was specially revealed for decoding purposes. Thus, according to this idea, God said (in effect) “when, for instance, I assign an implying predicate to a subordinate subject in the Torah, you must contradict the Torah statement where I assigned the implied predicate to the subaltern subject (lelamed hefekh hadavar).” Put in clear terms, this is effectively the defense proposed by the orthodox establishment. They put it more romantically, with reference to “allusions and hidden mysteries” which “defy literal interpretation,”11 but that is what they mean.

11

I quote Bergman again (p. 99), who uses this language with reference to Hagadic statements of the Rabbis; but I have seen similar language used with reference to the Torah.

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Thus, in that view, the Torah can, and often does, mean more or less than what it says. For this is what happens: when, without logical necessity, the Rabbis generalize a particular statement or read a statement exclusively (crediting the rest of, or the negation of, the subject with the negation of the predicate, beyond the license given by eductive logic), they add to the law; and when, likewise, they particularize a general statement, they subtract from the law. This thesis is not inconceivable, but it is rather far-fetched and difficult to believe. One may well wonder why God would want to engage in such shenanigans, and not speak clearly and straightly. If His purpose was to illuminate humankind in general, and the Jewish people in particular, with a perfect law, full of Divine wisdom and love, justice and mercy, purity and spirituality, why not say just what He means? Why would He need to mask His true intentions, and give the key to them only to the Rabbis? All this concerns, note well, especially situations which do not logically entail or call for the rabbinic responses. In situations where logic clearly demands a certain inference or resolution of conflict, there is no need of special revelations; everyone is (more or less) in principle naturally endowed with the required intellectual means. Rabbinic hermeneutics, as a Divinely-granted privilege, come into play, essentially, wherever logic is faced with a problematic issue, because Scripture, taken as a whole, does not answer some question, but leaves a gap. The gap may be an indefinite particular proposition: should we read it as general or contingent? In natural knowledge, the preferred course would be generalization. Alternatively, the gap may consist in total silence about some subject, without even a guiding particular proposition. In natural ethics, we might opt for permissiveness, or at best a conventional law. When dealing with a presumably Divinely revealed database, such as the Torah, instead of knowledge naturally developed in the minds of human beings, scientific logic cannot predict with certainty what the intent of the Law-Giver was, in the event of gaps. It is, arguably, more likely that an indefinite particular proposition be read as contingent, and it is conceivable that more radical gaps are to be filled by the decision of Divinelyappointed judges (as Deut. 17:8 – 13 suggests). The latter possibility would justify additions to the law (pronouncing an indefinite particular to be exclusive, or generalizing it, or formulating a completely new provision,

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are all references to previously unaddressed instances); but it would not justify subtractions from the law (other than particularizations called for by manifest contradictions, which cannot be resolved otherwise). Yet the Torah explicitly frowns on additions (tosafot) to, as well as subtractions (geronot) from, the Written Law, in passages like the following: Ye shall not add unto the word which I command you, neither shall ye diminish from it, that ye may keep the commandments of the Lord (Deut. 4:2). All this word which I command you, that shall ye observe to do; thou shalt not add thereto, nor diminish from it (Deut. 13:1).

It could be countered that ‘the word’ Moses here refers to, which may not be modified, includes not only the written law, but also the oral law. However, how can adherence to unwritten law be ensured? What, in such case, would addition or subtraction constitute? How would the boundaries be defined? Such passages could be interpreted literally, to imply that even where gaps are found, no human legislator or legislative body may presume to try and fill them. The very human, and particularly rabbinic, tendency to legislate about almost everything would seem to be illegal.12 In this perspective, when the written Divine law is obscure, albeit all efforts of pure logic made to clarify it, there is effectively no Divine law (on the subject at hand). The appointment of judges is then merely intended for the application of Divine law; that is, to decide in each case whether Divine law has been 12

Lewittes (p. 90), with reference to these two passages of Deut., comments: “Nevertheless, the masters of Jewish Law, in particular the Sages of the Talmud, did not hesitate to add new legislation to the corpus of Jewish Law. They interpreted the Biblical injunction quoted above to apply to each mitzvah in itself; i.e. not to add to a mitzvah a feature not prescribed for it by the Torah.... Furthermore, it was not considered a violation of this injunction if the additional legislation was clearly denoted as rabbinic and not Biblical in origin.” However, that explanation does not sincerely solve the problem; many laws in fact fall outside its scope one way or the other. It is just a smoke-screen: if we consider the final legislation point by point, we undeniably find many additions and subtractions.

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broken, or in whose favour Divine law leans, and impose the sentence, if any, required by those same laws. There is no delegation of powers to construct legislation with nearly Divine authority. Reason would suggest that all non-Divine legislation is subject to natural ethics or human convention, and thus possibly open to variation under appropriate circumstances. Judaism teaches that the written Torah is supreme, not open to doubt or review. By extension, this principle was later applied to the oral Torah. The authority of the Rabbis stemmed from the Torah itself; for instance, Deuteronomy 17:8 – 13: If there arise a matter too hard for thee in judgment (...); then shalt thou arise and get thee up, unto the place which the Lord thy God shall choose (...), unto the Levitical priests, and unto the judge that shall be in those days; and thou shalt inquire; and they shall declare unto thee the sentence of judgment (...); and thou shalt observe to do according to all that they shall teach thee.

The religious authorities were, first of all, the trustees of the oral transmission (many of these people, in the long line since Moses, are identified by name – for instance, in the Pirkei Avot, ch. 1). And secondly, it was foreseen that there would be gaps in knowledge, or changing circumstances, which would require wise and considered judgment by competent and recognized spiritual leaders. But there is clearly logical tension between the above quoted earlier statements of Moses and the later one. If the earlier statements are taken literally, then the later statement should not be taken as a license for new legislation but only as one for the strict application of existing law. Otherwise, where do we draw the line? We cannot clearly define the difference between permissible new interpretation and creative lawmaking. On the other hand, how else than by rational interference are we to deal with contradictions and gaps within the given text? They cannot just be ignored. In sum, the ‘secret code’ rationale is very fragile. It was intended, remember, as a last resort explanation of the illogic of the Midot (as above exposed). But this only holds together at best temporarily; since, as of the moment the code is broken and ceases to be secret, as done in Judaic Logic, the whole argument falls apart. One can, only so long as a mystery

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remains, argue that God wrote the Torah down differently than He intended it to be read, giving exclusively to Moses and his successors (the Rabbis) a codebook (the Midot) to translate His intentions. But, once the implied equations are made transparent and accessible to all, the idea that God expresses Himself in such uselessly tortuous ways becomes ridiculous. All esoteric claims are equally vain in the long run. Thus, similarly: the Oral Law as a whole stops to be a special privilege as soon as it is written down (as in Mishnah and all subsequent Halakhic works), and so one may well wonder why it was not handed down to us in writing to start with. It is thus easy to suppose that, from the first appearance of the midot (meaning near Talmudic times), they were simply the Jewish equivalent of Sophist argumentation (historically, we should perhaps rather make a comparison to Stoic preachers of Roman times), products of the logical incompetence and intellectual dishonesty of the speakers, and of the relative ignorance and gullibility of their listeners. The fact is that the artificial aspects of rabbinic hermeneutics give enough of an illusion of being complex logical arguments, to bamboozle into intellectual submission, anyone who feels unselfconfident in his or her logical abilities and/or who for emotional reasons is all too willing to be persuaded. The ‘secret code’ rationale plays only a supporting role, as eventual backup in debates with philosophers. In everyday practice, rabbinic hermeneutics ‘work,’ i.e. they are ‘convincing,’ because the defense against them demands a logical lucidity and expertise most people lack (be they Rabbis or laypersons). The power of persuasion of the midot was, of course, greater in the past than it is today; though some people, even educated people, continue to be moved by them. One non-negligible reason for the continuing credibility is the desire of Jews to hook up with the genuine, ages-old tradition of Judaism. They are not looking for absolute truth; they are looking for roots and wish to belong. They are willing to force their minds into the unnatural thought-processes of the Rabbis, because they regard their own current thought processes as equally artificially induced, by modern society and its media. But the pursuit of happiness must not be confused with that of truth. References

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[1] Bergman, Rabbi Meir Zvi. Gateway to the Talmud New York: Mesorah, 1985. [2] Carmel, Aryeh. Aiding Talmud Study. Rev. ed. Jerusalem, New York: Feldheim, 1991. [3] Encyclopaedia Judaica. Jerusalem: Keter, 1972. [4] Lewittes, Mendell. Principles and Development of Jewish Law. USA: Bloch, 1987. [5] Scherman, R. Nosson, ed. The Complete ArtScroll Siddur. 2nd. ed. New York: Mesorah, 1986. (With commentaries by the editor.) [6] Steinsaltz, R. Adin. The Talmud. The Steinsaltz Edition. A Reference Guide. Trans. and ed. R. Israel V. Berman. New York: Random, 1989. [7] The Jewish Encyclopedia. New York: Funk, 1968.

OCKHAM AND ORATIO MENTALIS Francesco Bottin Department of Philosophy Padua University Padua, Italy [email protected]

Ockham was the first to carry out a complete process of grammaticalization of mental language carefully outlining distinct specific rules for the written, spoken and mental language. From a formal point of view Ockham’s program appears to many contemporary scholars as incomplete and inadequate, particularly when he attempts to eliminate equivocity and synonymy in mental language. But, on the ground of the Augustinian epistemology, the Franciscan logician seems to be stating that mental language is not devoid of ambiguities because it is a perfect language, even if it is a language devoid of the unclearness due to the will to deceive, as it is the expression of the inner Self.

1. Augustine and Boethius In the first pages of his Summa logicae, before setting out the fundamental notions for the signifying function of terms, Ockham takes Boethius’s tripartite distinction of discourse into written, spoken, and mental, and immediately compares it to Augustine’s verbum cordis (something that Boethius had certainly not considered), stating that “these conceptual terms and the propositions composed of them are the mental words, according to St. Augustine …, belong to no language … and are incapable of being

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uttered aloud …”1 Unlike Augustine and Boethius, however, for Ockham these are three fully-developed and autonomous languages, in the sense that for each one of them it is possible to formulate specific norms of composition and formation: The written term is a part of a proposition which has been inscribed on something material and is capable of being seen by the bodily eye. The spoken term is a part of a proposition which has been uttered aloud and is capable of being heard with the bodily ear. The conceptual term is an intention or impression of the soul which signifies or consignifies something naturally and is capable of being a part of a mental proposition and of suppositing in such a proposition for the thing it signifies … [9], I, 1 (trans., p. 49).

Although Ockham is fully aware of the bitter debate that had taken place between the Franciscans and Dominicans over the verbum,2 he avoids taking part in it and deliberately aims to construct his doctrine on completely new bases, unsupported by either medieval or ancient authors. He merely finds a reference in Boethius to the names and verbs of mental language, as we have seen, and this is the only auctoritas, albeit vague, on which to base his concept:

1

See [9], I, 1 (English translation from: Ockham’s Theory of Terms, Part 1 of the Summa logicae, trans. and intro. by M. J. Loux, University of Notre Dame, Notre Dame-London 1974, pp. 49 – 50). 2 It is well-know that immediately after Aquinas’s death, Dominicans and Franciscans exchanged violent criticisms under the title of correctoria. In this context, even Aquinas’s doctrine of the verbum interius was severely criticised from many points of view. For an overall view see [7]. It should be noted, however, that for a certain period Ockham accepted the doctrine of verbum mentale as fictum and that he was fully aware, in admitting this, that he had come very close to the Thomists’ positions, even if they had been strongly criticised by the Franciscans. See, for example, the following passage of the Ordinatio: “Hoc … est mihi probabile, -- quamvis hoc non affirmem --, quod in intellectu, praeter ipsum actum intelligendi quando intelligitur aliquod commune ad plura, est aliquid in intellectu … quod est aliquo modo simile rei extra intellectae, quod a multis vocatur quasi quoddam idolum in quo aliquo modo ipsa res cognoscitur, quamvis rem singularem cognosci in illo non sit aliud quam ipsum cognosci, -- nisi forte praeter hoc cognoscatur ipsum esse conceptum alicuius alterius” (Ordinatio, I, d. 27, q. 2, pp. 205, 1—206, 4).

134 Nor should anyone be surprised that I speak of mental names and verbs … let him first read Boethius’ commentary on the De interpretatione; he will find the same thing here [9], I, 3 (trans. p. 54).

In reality, Ockham can find nothing more in Boethius than a general statement on the existence of three kinds of language. Accepting the interpretation that Porphyry had attempted to put forward to explain the controversial expressions used by Aristotle with his distinction between phonái and ta en tê phoné and graphómenoi and ta graphómena, that is to say, by introducing two neutral demonstratives that Boethius faithfully translates as ea quae sunt in place of their respective nouns, Boethius states that “the Aristotelians very correctly established that there are three kinds of language: one which can be written with letters, another that can be uttered with the voice, and a third that is built with thought … if there are three kinds of language (written, spoken, and mental) there is no doubt that there are also three parts of speech. Now, since the verb and the noun are the principal parts of speech there will be verbs and nouns which are written, others which are spoken, and still others which are thought with the silent mind.”3 These and other considerations by Boethius, however, had not led to any further development in his Aristotelian commentaries in elaborating a true mental language. Medieval writers, from Abelard onwards, were well aware of these statements by Boethius, but they had not been able to find in them the beginnings of a structuring of the three kinds of language that Ockham was to grasp [8, p. 192]. Indeed Boethius himself, in his brief reference to the triplex sermo, had limited himself to quoting his source Porphyry merely to resolve a hermeneutical problem in Aristotle’s text, and had not even felt it 3

See Boethius, In lib. De Interpretatione editio secunda, rec. C. Meiser, Teubner, Lipsiae 1880, pp. 29, 29—30, 10: “Quaerit ergo Porphyrius cur ita dixerit: Sunt ergo ea quae sunt in voce et non sic: sunt igitur voces; et rursus cur ita et ea quae scribuntur et non dixerit: et litterae, quod resolvit hoc modo. Dictum est tres esse apud Peripateticos orationes, unam quae litteris scriberetur, aliam quae proferretur in voce, tertiam quae coniungeretur in animo. Quod si tres orationes sunt, partes quoque orationis triplices esse nulla dubitatio est. Quare quoniam verbum et nomen principaliter orationis partes sunt, erunt alia verba et nomina quae scribantur, alia quae dicantur, alia quae tacita mente tractentur.”

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necessary to compare this source with what Augustine had said regarding the verbum cordis in his De Trinitate.4 Only when medieval thinkers began to compare Boethius’s various references to cogitabilis oratio or oratio quae in intellectibus est,5 which were presented as a genuine Aristotelian doctrine, with Augustine’s profound reflections on the verbum cordis was it possible to elaborate a complete semantic doctrine on oratio mentalis. In this perspective it was natural for medieval thinkers to move from a certain interpretation of the Aristotelian doctrine of the passiones animae as equal for all men, to the Augustinian doctrine of the verbum cordis, in which “thoughts are a kind of utterance of the heart … if anyone then can understand how a word can be, not only before it is spoken aloud but even before the images of its sounds are turned over in thought – this is the word that belongs to no language, that is to none of what are called the languages of the nations, of which ours is Latin …” [2, pp. 408 – 409]. In a speech in honour of St. John the Baptist, Augustine uses detailed analysis to explicitly state the difference between voice (vox) and word (verbum). He begins by establishing that both voice and word can be understood in two different senses: there is a voice which is “a certain indefinable sound which makes a noise and deafens the ears without any trace of intelligibility,” and a voice which, properly structured, is able to express a meaning and hence become word (verbum). Word in turn can be understood in two senses: on one hand is the verbum which “does not belong to any language … and which is of the greatest value even without a voice;” on the other is a word which “cannot be called word unless it has a means which expresses it,” that is to say, there is a sense of verbum which is only realised in its expression in a given language.6 4

In a second reference, Boethius states that it is a distinction put forward by Porphyry: “Porphyrius uero quoniam tres proposuit orationes, unam quae litteris contineretur, secundam quae uerbis ac nominibus personaret, tertiam quam mentis euolueret intellectus.” But no reference is made to Augustine here either. 5 In effect Boethius does not have any specific terminology to designate mental language. 6 See Augustine, Sermo 288: “Quaeramus quid intersit inter vocem et verbum: attenti quaeramus; non parva res est, nec parvam intentionem desiderat. Dabit Dominus, ut nec ego in explicando fatiger, nec vos in audiendo. Ecce duo quaedam, vox et verbum.

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Here Augustine uses expressions suitable for his public by speaking of a verbum cordis which “goes in search,” “tries to come out,” and “looks for a means of expression.” But in the De magistro this is given a theoretical definition in the concept of verbum imaginabile, which expresses the transition from verbum cordis, which in itself does not belong to any given language, to outward expression in a particular language, a concept which in the De Trinitate is further clarified as “the word which is neither uttered in sound nor thought of in the likeness of sound which necessarily belongs to some language, but which precedes all signs that signify it and is begotten of the knowledge abiding in the consciousness, when this knowledge is uttered inwardingly just exact as it is” [2, p. 410]. In the De doctrina Christiana, moreover, Augustine explicitly states that the verbum cordis in itself remains as it is and does not undergo any alteration, even if it transforms itself in order to find a means of outward expression. It is in any case in the De Trinitate that Augustine gives this consideration a systematic form when he distinguishes between the sense of verbum as a sensible image of the thing that is to be expressed, and that of verbum in corde strictly speaking. Finally, at the beginning of the 14th century, it was the concept of a mental language as autonomous, precisely because it is constituted of signa and not imagines, which was to allow Ockham to put forward a concept of language and signification capable of freeing itself from a complex, incoherent, and inefficient infrastructure which the Aristotelian tradition had been accumulating ever since it had first been put into Latin by the “Platonic” Boethius.7 Quid est vox? quid est verbum? Quid? Audite quod in vobis ipsis approbetis, et vobis ipsis a vobismetipsis interrogati respondeatis. Verbum, si non habeat rationem significantem, verbum non dicitur. Vox autem, etsi tantummodo sonet, et irrationabiliter perstrepat, tamquam sonus clamantis, non loquentis, vox dici potest, verbum dici non potest. Nescio quis ingemuit, vox est: eiulavit, vox est. Informis quidam sonus est, gestans vel inferens strepitum auribus sine aliqua ratione intellectus. Verbum autem, nisi aliquid significet, nisi aliud ad aures ferat, aliud menti inferat, verbum non dicitur.” 7 In the De consolatione philosophiae, Philosophy constantly refers to Plato as “noster,” something that is not true for Aristotle, even though his doctrines are used copiously .

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2. The autonomy of the oratio mentalis Ockham was the first to carry out the process known as the grammaticalization of mental language, in the sense that he reconstructs on a mental level all (or most) of the structures of speech which are found in spoken and written language. In effect, going far beyond Boethius’s isolated reference to the names and verbs of mental language, Ockham outlines a programme of correspondence between the structure of spoken language and the structure of mental language: There are, we have seen, three kinds of simple terms – spoken, written, and conceptual. In all three of spoken and written language terms are either names, verbs, or other parts of speech (i.e. pronouns, participles, adverbs, conjunctions, prepositions); likewise, the intentions of the soul are either names, verbs, or other parts of speech (i.e. pronouns, adverbs, conjunctions, prepositions) [9, p. 52].

For the other parts of speech, such as participles and pronouns, Ockham expresses some doubt as to whether a correspondence must be maintained with mental language. Indeed every utterance which contains a participle, such as Socrates est currens, can be transformed into an equivalent utterance with a verb, Socrates currit, since, Ockham affirms, “the participle of any verb, with the appropriate form of ‘to be,’ signifies precisely what the corresponding form of that verb by itself signifies.” In effect Ockham resolutely excludes from mental language only synonymous terms stating that “whatever is signified by an expression is signified equally by its synonymy” [9, p. 52]. Ockham presents two definitions of synonymy, but the one used in this context is the following: More broadly, expressions are synonymous which simply signify the same thing, so that nothing is in any way signified by one of the terms.

On the means of distinguishing between two terms which can be interpreted as synonyms, Ockham provides the following general criterion: those terms or those expressions are to be understood as synonyms which are used exclusively “to the embellishment of speech or something of that nature, so that the relevant multiplicity has no place at the conceptual

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level” and hence “the multiplicity of synonymous expressions in no way enhances the significative power of language; whatever is signified by an expression is signified equally by its synonymous” [9, p. 52]. There is a lively debate among contemporary scholars over the way in which Ockham might have understood synonymy in a concrete sense (see [4], [5]), and in particular how in the grammaticalization of mental language devised by Ockham – who attempts to maintain a correspondence between the parts of spoken and mental discourse solely in order to guarantee the truth or falsity of utterances – we often find ourselves up against clear incoherencies. Scholars agree on the other hand in recognising as fundamental to the relationship between spoken and mental language two innovations to traditional semantic doctrines which Ockham introduces for the first time and which are strictly correlated: ‘subordination’ and ‘sign.’ The relationship of ‘subordination,’ with which Ockham refers to the possibility of expressing with vocal signs the contents of a mental term is introduced as follows: I say that spoken words are signs subordinated to concepts or intentions of the soul not because in the strict sense of ‘signify’ they always signify the concepts of the soul primarily and properly. The point is rather that spoken words are used to signify the very things that are signified by concepts of the mind, so that a concept primarily and naturally signifies something and spoken word signifies the same thing secondarily [9, p. 50].

As we can see, this passage clearly establishes that the words of vocal language, though subordinated to those of mental language, do not signify “primarily and properly” the concepts of the soul, but rather things themselves. What is significant, however, is the overall justification of this conclusion. In reality Ockham reaches this conclusion by means of a series of passages which must be analysed in detail. In the first place he establishes a fundamental difference between the way of signifying of the two kinds of language: The concept or impression of the soul signifies naturally; whereas the spoken or written term signifies only conventionally.

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Even if this means of expression seems at first sight merely to reproduce the Aristotelian distinction between the conventional nature of spoken and written language and the naturalness and universality of the relationship between the passiones animae and things (which had been re-elaborated, in a variety of formulae including species, imagines, intentiones, conceptus, verba cordis, verba mentis etc., all considered as similitudines, that is to say, representations of things themselves), it is clear that Ockham strips all these entities of their content and takes them solely for their form, namely as signs: In general, whenever writers say that all spoken words signify or serve as signs of impressions, they only mean that spoken words secondarily signify the things impressions of the soul primarily signify [9, p. 50].

3. Subordination as conventionality Once we have determined that the three kinds of language have a common aspect as ‘signs,’ we must also determine the nature of the relationships that can be established between these different kinds of sign. In particular, the fundamental relationship, namely reference to the objects signified by these signs, will be placed on another plane. As signs, these three kinds of language can be characterised only by conventionality (as far as written and spoken language is concerned) or by naturality (as far as mental language is concerned), and by the concept of subordination which is introduced to explain the relation between signs. Even if the name of this relation is new, it must be said that it is an idea already present in John Duns Scotus, who had on various occasions observed that “regarding many signs co-ordinated with respect to the same signified thing we must admit that one is in some way the sign of the other because it allows us to include the other, since the more remote one could not exercise its function as a sign if it did not in some way first signify the nearer one, and yet we cannot for this reason say that one is the sign of the other.”8 8

See Scotus, Ordinatio I, d. 27, q. 1, vol. 6, p. 97, 15 – 19: “… potest concedi de multis signis eiusdem signati ordinatis, quod unum aliquo modo est signum alterius (quia dat intelligere ipsum), quia remotius non signaret nisi prius aliquo modo

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Naturally Ockham is much more specific than this. He establishes in the first place that the relation of subordination arises when we come across signs that are not of the same nature, in particular because mental language is made up of natural signs, while written and spoken languages are made up of conventional signs. It is not difficult to see in this differentiation the asymmetry that characterises the semantic triangle of Aristotelian origin, in which the left-hand side of the triangle (the relationship between words and concepts, that is) is conventional, while the right-hand side (the relationship between concepts and things) is characterised by a naturality. This asymmetry had become an insurmountable obstacle for any theory of the meaning of names which aimed to include these two irreconcilable aspects of the linguistic sign. Now, Ockham is able to overcome this difficulty because, by maintaining the particular nature of the different signs, he appropriately co-ordinates their signifying power: when a conventional sign signifies the same thing as a natural sign, its signifying function must be held to be subordinated to the former, in the sense that at a spoken level that relation is repeated conventionally which was natural in mental language. In this way it must be held that the relationship of subordination is none other than the giving to a conventional sign the meaning that naturally belongs to a concept. Now, this relation is asymmetrical because, as Ockham exemplifies, if a mental term, which naturally signifies its object, were to change meaning, the corresponding spoken term would also signify the new object, without any new linguistic convention. On the contrary, if by means of an agreement between two speakers, a certain spoken term was made to signify something different thing from what it had always signified, this would not lead to any change in mental language because the relationship between this language and the spoken one is purely conventional and hence can change at will.9 immediatius signaret, -- et tamen, propter hoc, unum proprie non est signum alterius, sicut ex alia parte de causa et causatis.” 9 See [9, p. 50]: “suppose a spoken word is used to signify something signified by a particular concept of the mind. If that concept were to change its signification, by that fact it would happen that the spoken word would change its signification, even in absence of any new linguistic convention … We can decide to alter the signification of a spoken or written term, but no decision or agreement on the part of anyone can have the effect of altering the signification of a conceptual term.”

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4. The nature and life of the sign The other notion that takes on a fundamental role to ensure that written, spoken, and mental terms all signify external objects, albeit in a different way, and not the mental objects which in various ways may be involved in the process of signifying, is that of sign. Ockham examines two conceptions of sign: … to silence hair-splitters it should be pointed out that the word ‘sign’ has two different senses. In one sense a sign is any thing which when apprehended brings something else to mind. Here, a sign need not, as has been shown elsewhere, enable us to grasp the thing signified for the first time, but only after we have some sort of habitual knowledge of the thing. In this sense of ‘sign’ the spoken word is a natural sign of a thing, the effect is a sign of its cause, and the barrel-hoop is a sign of wine in the tavern. However, I have not been using the term ‘sign’ in this wide sense. In another sense a sign is anything which (i) brings something to mind and can supposit for that thing; (ii) can be added to a sign of this sort in a proposition (e.g. syncategorematic expressions, verbs, and other parts of speech lacking a determinate signification); or (iii) can be composed of things that are signs of either sort (e.g. propositions) [9, p. 50].

In the first definition we can see the Augustinian concept of sign, understood as a vestigium or an imago10 which recalls to mind something whose relation with what it signifies we already know, either because it depends on previous knowledge which in this way is merely brought back to mind, or because it has been obtained thanks to a conventional institution of meaning, as is the case for many conventionally-established signs (such as the barrel-hoop to indicate where wine is sold, for example). Nevertheless, this Augustinian concept of sign has already been stripped in some way of the assumption that the sign must always be something that falls under the senses: in such a way it can also include concepts, as Roger Bacon had been the first to establish. But Ockham considers this concept of sign to be too general with respect to the purpose that he has in mind for it in his theory of language, precisely because, as Augustine had shown, it is not able to constitute the first knowledge of a thing, and for this reason Ockham restricts it to covering a particular kind of sign. Here he needs a specific definition of linguistic 10

For Ockham’s criticism of the concept of sign as vestigium and imago see [1].

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sign. Indeed, in his second definition, Ockham synthesizes two elements that had been absent from Augustine’s conception: (i) a sign in this sense is not only no longer a res, but an intellectus, and (ii) thanks precisely to this intelligible nature it can stand in mental language in the place of the object designated in a completely natural way and not by convention. In the observation that concludes this second definition of sign, Ockham specifies that “taking the term ‘sign’ in this sense the spoken word is not the natural sign of anything.” This is a curious observation that might appear to be redundant as a simple repetition of what we already know, namely that written signs and words are established by convention and hence have no relationship of natural resemblance with the things signified. But with this observation Ockham intends to call our attention to the fact that the impossibility of written signs and vocal names to be ‘natural signs’ of things depends or is expressed more rigorously by the second definition of sign. This definition specifies the general notion of sign with the clause that these are signs that are able to let us know things directly and that, as such, they can stand for the objects themselves in a proposition. Certainly we already know that voces had been divided into naturales and ab anima datae, in the sense that some voices signify by convention, while others, such as groans, naturally express a certain state of mind. But it is obvious that, as such, groans are not taken as words within a proposition. Voces, on the other hand, “conventionally instituted,” can stand for objects in spoken utterances precisely because they have received by institution not only the function of signifying their objects, but also the structuring that allows them to be parts of speech. The sense of Ockham’s statement then is the following: on the basis of the definition of sign, which involve the “standing for the object signified in a proposition,” no vocal sign can naturally signify its objects because vocal signs become parts of speech by a conventional structuring which changes from one language to another. Ockham’s observation, in any case, is aimed at highlighting the different situation of mental language, it too made up of terms, which are concepts themselves. The second definition of sign holds above all for these mental signs. As opposed to vocal signs ab anima data, like groans of pain, which however cannot enter the structuring of speech, there are therefore other signs ab anima data which go to constitute precisely mental language:

143 The act of understanding by which I grasp men is a natural sign of men in the same way that weeping is a natural sign of grief. It is a natural sign such that it can stand for men in mental propositions in the same way that a spoken word can stand for things in spoken propositions [9, p. 81].

Of the ancient discussion of the pathemata tes psyches – a concept of Aristotelian origin which led to interpreting the pathemata as opposed to the noemata and closer to sensible modifications of the soul [3] – Ockham merely retains the naturalness that characterises concepts and every immediate expression of joy or pain. But while the natural manifestations of the soul cannot be parts of speech, concepts, though they are natural expressions, constitute the terms of mental language and hence go to constitute a completely natural oratio mentalis. The relation of “stand(ing) for the signified thing” can obviously only be exercised by terms that have been recognised as significant and the nature of this relation depends on the nature of the signification. Hence, although both spoken and mental terms can exercise the function of “standing for,” this is expressed differently: in the case of spoken terms the conventional nature of the signification also entails the conventional nature of the supposition; in mental terms, the naturalness of the signification also entails the naturalness of the supposition. Ockham’s observation moreover aims at clarifying the difference which subsists between signs that are res (these, as spoken or written signs, have a physical reality which is overcome, to make them able to signify, only thanks to a convention), and those signs that are purely immaterial, which thanks to their immaterial nature can directly signify the very objects signified conventionally at a spoken level and can once again stand for objects naturally in the oratio mentalis. While the hermeneutical tradition on the passiones animae of Aristotelian origin showed itself to be unable to guarantee a form of realism and ended up with the ambiguities proper to all models based on representation, with these observations of a purely logico-semantic nature, Ockham indicates the only road open to direct realism, first on a semantic level, and then on a cognitive level: it is necessary to understand the passiones animae uniquely as signs of a particular kind. They are natural signs that are formed equally in all men who have the same experiences of the outside world. Hence the new notion of sign, applied to oratio mentalis, is able to

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resolve the controversies over the nature of the resemblance between image and object represented. Using therefore an adequate programme of subordination between the signs of language, Ockham is able to resolve the old problem of the meaning of names, in the literal sense of freeing it of a whole series of conceptual and lexical misunderstandings. Spoken words, although generated by the verbum mentis and subordinated to it, maintain the capacity to signify things directly without first having to pass through the signification of concepts or verba mentalia. In other words, we can say that Ockham understands subordination exclusively as a formal relation between signs, whose nature emerges differently from the different way it is a natural or a conventional sign with respect to the same object. In this way the purely formal aspect of subordination has no consequence on the relationship that conventional and natural signs have with objects, in such a way that, although they are subordinated, they can signify the same object directly, that is without having to repeat the subordination at the level of signification too. 5. Is a semantic triangle still possible? It is said that with Ockham the semantic triangle implicit in the Aristotelian concept of the relationship between words, passiones animae, and things, not only underwent a notable transformation, but also that the new conception of the relationship between words, verba mentalia, and things can probably no longer be expressed in the form of a triangle, however it is interpreted. If we wish to make a comparison, keeping the old framework, we must stress in the first place the fact that unlike in the semantic triangle, both words and concepts are able to signify their objects directly, even when these are extra-mental things. The main difference that can be seen in the new system clarifying the signifying function of names consists of the elimination of the mediation of concepts, understood in the most diverse ways as passiones animae, intentiones, intellectus, etc. and their substitution by the relation of subordination. This change enormously simplifies the complicated relationships implicit in the semantic triangle and allows direct access to things both by concepts, which are natural signs of the things themselves, and by names which no longer need

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intermediaries to carry out their signifying function. In this case words no longer pretend to describe the objects named by means of a mental representation that they should signify in the first place, but limit themselves simply to naming things. We can perhaps attempt to express this on a visual level not by a single diagram but by numerous diagrams, which could perhaps be assembled, even though this would end up by giving rise to considerable complication, and the diagram would no longer manage to express with intuitive clarity a conceptually complex doctrine. In any case, we can state the following points: (i) written words are subordinated to words uttered orally in the sense that there is a convention which associates the individual graphemes with the individual sounds of a voice, but taken in their entirety they are able to signify directly individual things, in the sense that anyone who sees these written signs and interprets these words refers them directly to the individual things in the outside world and not to mental concepts or elaborations; (ii) words spoken orally are in turn subordinated to concepts in the sense that they have been associated to certain concepts of the mind by convention; but anyone who listens to them and interprets their meaning understands that the signification of the concepts is conventionally subordinated to the signification of outward things and hence, although subordinate to the concepts of the mind, the words uttered refer conventionally to objects existing individually in the world outside the mind of he who utters them; (iii) concepts, finally, as immaterial signs, are able to signify directly and naturally the individual things of the outside world of which they are signs, and thanks to the relation of subordination they also transmit, but by means of a recognized convention and not by nature, the capacity to signify things directly to written and spoken words. To sum up, purely as an example, we could draw the following diagram:

146 conventional signification

written words (subordination) ↓ conventional signification

spoken words (subordination) ↓

individual objects of the outside world

natural signification

concepts

In this diagram, as we can see, the relationships no longer run exclusively along the sides of the triangle, but, on one hand we have three distinct languages (written, spoken, and mental) which have different relationships (conventional and natural) with the individual objects existing in the world. Furthermore, the relation of subordination regulates the relationships between the three different kinds of language. It is clear that this new situation can no longer be expressed uniquely on the basis of the relationship between words, concepts, and things as if it were a relationship between three elements, which the image of the semantic triangle had suggested. 6. Connotative terms In the Summa logicae Ockham states that “logicians use the term ‘signify’ in a number of ways” and he distinguishes four ways in which a term can signify its object: First, a sign is said to signify something when it supposits for or is capable of suppositing for that thing in such a way that the name can, with the verb ‘to be’ intervening, be predicated of a pronoun referring to that thing … In another sense we say that a sign signifies something when it is capable of suppositing for that thing in a true past, present, or future proposition or in a true modal proposition [9, p. 113].

These first two definitions, which put forward on a linguistic level the definition of signum already presented by Ockham as “anything which brings something to mind and can supposit for that thing,” have in common the condition that ‘signifying something’ involves a true

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predication of that thing, in the first case respectively when a given term signifies all and only those objects which exist at the time in which it is said to refer, and in the second case when it signifies all past, present, future or purely possible objects to which it is said to refer. Bearing in mind that by ‘signifying’ Ockham too understands, on the basis of Boethius’s translation, the “constituere intellectum” of something, which is equivalent to maintaining that our mind believes that a certain term has signified something and the mind of he who listens is satisfied when it is able to establish a true predication of an ostensive nature, by using a demonstrative pronoun and the copula, with the object which is to be signified. For example, the term ‘man’ correctly signifies its object if in a predication we can establish that ‘this is a man:’ in the first sense of “signifying,” this statement refers to all and only those men existing while they are pointed to by means of the senses; in the second sense, it refers to all men, past, present, future, or possible. Even if Ockham’s solution is put forward at a predicative level, that is at the level of enunciation, as only on this level can we have the truth or falsity of a proposition, the entire operation can also be thought on a pragmatic level: if the operation of pointing a finger (this is the function of the demonstrative) at a certain object succeeds, albeit under different conditions, the signifying process can also be said to succeed. But there are another two senses in which we say that a term signifies something: In another sense we say that a thing is signified by a word or concept which is taken from the expression or concept signifying that thing in the first mode, or when the thing is that on the basis of which the word or concept is imposed … In the broadest sense of all we say that a term signifies provided it is a sign which is capable of being a part of a proposition or a whole proposition and designates something, whether primarily or secondarily, whether in the nominative or one of the oblique cases, whether by actually expressing or merely connoting something, whether by signifying affirmatively or merely negatively [9, p. 114].

The third and the fourth mode of signifying are characterized by the following conditions:

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(a) with these modes of signifying, deixis by means of a demonstrative pronoun does not work as it is not possible to establish a correspondence between a demonstrative pronoun and a specific object; (b) as a consequence the condition of true predication between the pronoun and the object signified is not required either. In the third way in effect, although it can be considered analogous to the first, the presence of a secondary signification prevents the correct application of the two clauses indicated above. For example, if the term ‘white’ is used not to signify a particular white object but whiteness. In the fourth case, finally, we have a means of signifying so indirect as not to have to involve even the constituere intellectum clause, which as we have seen characterises signifying as such. In this fourth mode of signifying it is enough that “it involves something” (aliquid importat) in any way in which it involves it, or, as is specified, it connotes it. With this further division of signifying, therefore, Ockham intendes to introduce a distinction between absolute and connotative terms: Purely absolute names are those which do not signify something principally and another thing (or the same thing) secondarily. Rather, everything signified by an absolute name is signified primarily; A connotative name, on the other hand, is one that signifies one thing primarily and another thing secondarily [9, p. 69].

In the passage quoted, Ockham presents a list of absolute names, such as man, animal, goat, stone, tree, fire, water, sky, whiteness, blackness, heat, sweetness, smell, and taste. These names in the sense given by Ockham do not identify with their nominal definition, which for Ockham “is an expression explicitly indicating what is designated by an expression …so if someone who wants to teach another individual what is meant by the name ‘white’ says something like ‘something having whiteness.’” In effect, for Ockham, absolute terms have only one real definition, that is, a definition which in the strictest sense “is a complex expression signifying the whole nature of a thing without indicating anything extrinsic to the object defined” [9, p. 107]. Now, while absolute terms can have only one real definition, connotative terms can have many nominal definitions which are all synonymous expressions:

149 Strictly speaking, absolute names do not have nominal definitions, for a name with a nominal definition has only one such definition. Where a word has a nominal definition, the meaning of that word cannot be expressed by different sentences, such that terms from one sentence signify things not in any way designated by terms from other sentences. However, in the case of purely absolute terms the meaning of the name can be expressed by different sentences whose constitutive terms do not signify the same thing [9, p. 70].

Although he does not provide sufficiently long lists of absolute or connotative terms, but only a number of examples, Ockham nevertheless provides a general criterion for understanding when we are dealing with an absolute or a connotative term: Those who maintain that quantity is not an entity distinct from substance and quality must claim that all names from the genus of quantity are connotative [9, p. 71].

Naturally, as we know, it was Ockham himself who maintained precisely that there exist only objects represented by the categories of substance and quality, in the sense that for him there exist only individual objects and the real characteristics of these objects; hence, for him, absolute terms are terms which refer directly and principally to these types of object which can be catalogued in the only two categories admitted on an ontological level, manly that of substance and that of quality. Everything that is represented through other categories, on the other hand, which for Ockham do not have a true foundation, can be expressed by connotative terms, but not by absolute terms.11 In this way the distinction between absolute and connotative terms also ends up by playing a fundamental role in the ontological reduction which characterises the whole of Ockham’s thought.12 7. The significatio of names and suppositio In the above analysis of the meaning of terms by Ockham with his rigorous new use of the nature of the linguistic sign, characterised specifically as 11

See [9, p. 71]: “those who claim that every entity is either substance or a quality must hold that all the expressions in the categories other than substance and quality are connotative terms.” 12 On Ockham’s method of ontological reduction see [3].

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“that which can stand for something else,” it would see that Ockham attempted to connect two apparently distinct linguistic functions, significatio and suppositio. We find ourselves in fact in the strange situation of seeing Ockham use the notion of suppositio not only to define the sign in a strict sense, but also to establish the mode of signifying the various terms. The definition of suppositio, however, and its distinctions were to be introduced later by Ockham in his Summa logicae. We apparently find ourselves in the paradoxical situation of seeing this type of relation used systematically before it had been introduced. The discussion of the meaning of names, which arose from an analysis of Aristotle’s De interpretatione, referred to the object designated by these names without distinguishing between significatio and suppositio, but in practice embracing both these functions of language. It is a situation which is easily justified by the fact that, however we wish to understand them, these two functions of language have the purpose of establishing the type of relationship that exists between words and things. In reality the doctrines of significatio and suppositio had been progressively distinguishing themselves and in the 13th century Peter of Spain could straightforwardly establish that significatio was a “conventional representation of the thing by means of a voice” and suppositio consisted in turn of “taking a noun in place of something.”13 As we can see, the two definitions are distinguished solely by the use of the term ‘representation’ for significatio, which is introduced in this context precisely in order to establish that by means of voces we represent to ourselves on a cognitive level things themselves, and by the use of the term acceptio for suppositio, by which he attempts to establish a further and more specific relationship, still however within linguistic representation. Naturally if the different were based only on these two definitions it would be rather difficult to see an effective difference between significatio and suppositio, as it would in any case be a relation between words and things. In effect, Peter of Spain is quick to establish or rather to render explicit the difference between significatio and suppositio, something that these definitions were certainly not able to do: 13

See [11, pp. 79 – 80]: “Significatio termini … est rei per vocem secundum placitum repraesentatio … Suppositio vero est acceptio termini substantivi pro aliquo.”

151 However suppositio and significatio differ from one another because significatio takes place by means of the convention that a certain voice must signify a certain thing, while suppositio on the other hand consists of the taking of a term which already signifies a certain thing in place of an object.14

In further explanations, Peter of Spain not only establishes that significatio must come before suppositio, but sets out the following steps: (i) vox → (ii) vox significativa → (iii) nomen substantivum. On the first level we have all the voces, that is to say all the vocal articulations realized by means of the phonatory organs; on the second level we have some of these vocal articulations which have been conventionally chosen to represent objects on a spoken; on the third level, finally, we have meaningful voces which are admitted (acceptio) for certain representative uses of objects. Nevertheless, the attempt to properly distinguish between significatio and suppositio is highly problematic precisely in the context of the meaning of names. Even though it is distinct according to the criterion put forward by Peter of Spain, suppositio seems to continue to carry out a task which is not substantially different from that of significatio. Indeed in stating that suppositio is “the taking of a substantive in the place of something” Peter of Spain seems to make suppositio a form of significatio, albeit on a higher level, since the simple function established ad placitum whereby a certain vox signifies a certain type of object is further refined by the more specific function whereby a vox significativa, that is one that has already been subjected to impositio, stands for certain objects. This interpretation seems to be confirmed by the admission of a suppositio naturalis besides significatio. For Peter, suppositio naturalis takes place when “a common term is taken in place of all the objects which can by nature participate in it.” Even if this definition is expressed in terms of participation of a nature, which could make us think of something different from the function of assigning a meaning conventionally, the admission of 14

See [9, p. 80]: “Differunt autem suppositio et significatio, quia significatio est per impositionem vocis ad rem significandam, suppositio vero est acceptio ipsius termini iam significantis rem pro aliquo.”

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a suppositio naturalis besides significatio further reduces the possibility of distinguishing between the two linguistic functions, also because it is not specified whether suppositio naturalis at least doubles the function of including all possible meaningful objects in a different situation, that is exclusively within a proposition. Assumptions are certainly made more specific with Ockham, in the first place because he rigorously establishes the various meanings of significatio and then because he establishes an objective criterion to obtain suppositio. In effect, Ockham not only drops suppositio naturalis as ambiguous, but also explains the function of signifying in terms of suppositio. It is useful here to consider the definition of synonymous terms and equivocal terms. Synonymous terms are in effect defined in relation to their subordination to the concepts of the mind, whereby two terms are synonymous if they are subordinated to a single mental term, with the proviso that in doing this they must 1) signify the same thing and 2) signify it exactly in the same way. In an analogous way, Ockham holds a term to be equivocal which “in signifying different things ... is a sign subordinated to several rather than one concept or intention in the soul” [9, p. 75]. These definitions establish a strict correlation between “being subordinated to one or more concepts” and “signifying one or more objects.” In effect, as we have seen, for Ockham spoken or written terms do not signify concepts, but things directly. Now the concept of subordination is a concept which only belongs to the doctrine of signs, while signification expresses a first level of the relationship between words and things. Suppositio, in turn, expresses a higher more specific level of this relationship between words and things. 8. Oratio mentalis: ideal language and intimate self expression a) Searching for the perfect language In the perspective introduced by Ockham the meaning of the words we use to communicate is subordinated to the meaning of the terms of a purely mental language. At first sight the operation seems to respond perfectly to

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the need to render the common language we use every day in our interpersonal relations more rigorous. Ockham certainly agrees with Augustine in considering this language, made up of conventional signs, as the cause not only of ambiguity but also of the errors and lies that characterise human language. Anyone who has carefully read Augustine’s De magistro cannot but remain convinced that true communication must succeed in breaking the vicious circle of words which signify other words ad infinitum and allows us not only to deceive, but which also risks making effective interpersonal communication impossible. In Augustine, the need to make language unambiguous is strictly linked to a moral intention, at least in the broad sense of respect for one’s own identity. Taking an effective metaphor from Augustine, borrowed in turn from Plotinus, we could say that purely conventional language rules in the regio dissimilitudinis, while in intimior intimo meo there reigns the verbum cordis. But the interpretation of mental language in Ockham seems recently to have privileged exclusively the logical, formal aspect of mental language, in a strict correlation with the contemporary concepts of language itself. Peter Geach, for example, finds Ockham’s doctrine of mental language disappointing – in the context of contemporary theories of language obviously – because in practice Ockham limits himself to transferring the structures of Latin grammar to mental language, without offering any explanation of the way in which this language might have been translated into languages other than Latin, which often have a completely different grammatical structure [6, p. 102]. It is well known, for example, that Ockham intends to save the grammatical structure of the cases, since in his opinion the cases of the noun declinations also determine the logical structure of the truth or the falsity of the propositions in which they occur. Ockham exemplifies this by observing that the utterance “homo est homo,” namely “man is man” is always true, while the same utterance with the predicate in the genitive is false: “homo est hominis,” that is “man is of a man” is practically always false (unless we imagine a situation of servitude in which one man belongs to another man). The transition of a noun from nominative to genitive, therefore, determines the truth or falsity of the utterance and hence there must also be cases in mental language. But it has been rightly observed that

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the truth or falsity of utterances in other languages, which might not have declinations in cases, depends on the position of nouns in the proposition, as in the examples “Peter loves Julia” and “Julia loves Peter.” Hence grammatical structures, as such, in the historical languages depend on linguistic conventions in the same way as the meanings of words and hence it is unlikely that, as conventional, they can be part of a mental language that is characterised by naturalness. But in a short paper [13] Trentmann responded to these criticisms by maintaining that, in any case, with Ockham we acquire a number of fundamental elements which characterise an ideal language. In the first place Ockham clearly expresses the need to admit into mental language only those grammatical structures which determine the conditions of truth or falsity of propositions, even if he limits himself to giving some examples, without pretending to complete the entire project from a formal point of view. Since we cannot speak of a mental language without using a given historical language – at least not in the time in which Ockham lived – Ockham obviously only analysed the grammatical structures of Latin. By starting from a different historical language, he would probably have admitted different grammatical structures. This however should not lead us to conclude that in Ockham’s mental language there are incompatible grammatical structures, but merely that he was giving necessarily imperfect examples. Moreover, Trentmann maintains, this time decisively, that Ockham excluded from mental language two characteristic aspects of historical languages, namely synonymy and equivocation [13]. b) Synonymy and equivocity in mental language On these two aspects of mental language, in truth, Ockham’s works contain many indirect statements concerning the means of understanding synonymy and equivocation which tend in general to exclude that there can be synonymous and equivocal terms in mental language. But Ockham probably excludes these two grammatical structures explicitly only in a passage from the Quodlibeta: Every accident of a mental term is an accident of a spoken term, but not vice versa. For some things are accident of spoken terms because of requirement of signification and

155 expressiveness, and those belong to mental terms, whereas other things are accidents of spoken terms for the sake of the embellishment of speech, e. g. synonyms, and for the sake of grammaticality, and these do not belong to mental terms. 15

With a more detailed analysis of Ockham’s work, however, Paul Vincent Spade has noted at least a certain incongruity in Ockham’s statements. If his criterion of not making grammatical structures appear in mental language which do not determine the conditions of truth appears acceptable, it becomes decisively more problematic in Ockham’s attempt to identify in the historical languages the profound structures common to all languages. As for synonymy, moreover, Spade finds an incongruity in Ockham’s admission into mental language both of concrete terms and corresponding abstract terms, albus and albedo, for example. Indeed the concrete term albus is a typical example of a connotative term and its nominal definition is “something that has whiteness (albedo).” Now for Ockham connotative terms have the characteristic of being synonymous with respect to their nominal definitions, which as we have seen always imply the corresponding abstract term too (in the example used, the nominal definition of albus entails the presence of the abstract term albedo). In this way, after having denied the presence of synonymous terms, Ockham, ends up by accepting them, indirectly at least, with the admission of concrete terms and abstract terms into mental language [12]. Moreover, more recently, David Chalmers has attempted to demonstrate, prevalently from a theoretical point of view, that the complete elimination of synonymy from mental language would make it awkward and rigid and that Ockham therefore did not intend to eliminate it completely [5]. In effect, in his definition of synonymy, Ockham set out two conditions: 1) that two or more terms signify the same thing and 2) that they signify it in the same way (omnibus modis). Now, even though it is not clear what Ockham understood by “signify in the same way,” it is not difficult to identify various types of context in which two terms, apparently

15

See Ockham, Quodlibeta septem, ed. J.C. Wey, in Opera Theologica, S. Bonaventure, N. Y., 1980, V, q. 8, p. 513, 130 – 136 (English trans. by A. Freddoso and F.E. Kelley in William of Ockham, Quodlibetal Questions, Yale Univ. Press, New Haven-London 1991, pp. 428 – 429).

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synonymous, but which in reality signify differently from an epistemic point of view, also determine therefore different conditions of truth. An analogous reflection can be made for the equivocity of terms. Following tradition, Ockham distinguishes two types of equivocity: (i) In the first case a term is equivocal by chance. Here a term is subordinated to several concepts, but it would subordinated to one of these concepts even if it were not subordinated to the other(s); and similarly, it could signify one thing even if it did not signify the other(s) … (ii) but equivocality can also be intentional. Here, a word is first assigned to one thing or several things and is, thus, subordinated to one concept. But afterwards , because the things signified by the term are similar to or bear some other relation to other things, the term is used to signify something new. Its new use, however, is not merely accidental. If it had not been assigned to items of the first sort it would not be used in the second case [9, p. 76].

We can exemplify these first two types of equivocation with a) words which have two or more independent meanings, such as in Latin language “canis” if it refers to the constellation, to the animal, and to the fish or with b) words, which though they have received only one meaning, are used by analogy to mean different things, such as the word “man” which besides signifying an individual real man can signify a pictorial representation or a statue of a man. These first two modes of equivocation have in common the fact that they concern the conventional impositio of meaning on words. But in a subsequent passage from the Summa logicae Ockham explicitly states that equivocation in a strict sense does not take place so much at the level of the impositio of the meaning of words, but at the level of suppositio: We commonly affirm that equivocation consists of the different meaning of a term placed in a discourse ... but this is not said correctly ... whereby we must affirm more correctly that equivocation is to be defined as the calling (vocatio) many things by the same word or the same sign ... and I speak of “calling” so that we do not only think about significatio, but rather at suppositio or the standing for something …16 16

See Summa logicae, III, 4, cap. 2: “… dicitur quod aequivocatio est diversa significatio alicuius termini positi in oratione … sed istud non est bene dictum, nam non semper ubi est aequivocatio ibi est diversitas significationis … ideo dicendum est quod aequivocatio magis proprie definitur sic: aequivocatio est multorum vocatio sub eadem voce vel sub eodem signo. Ut vocatio non accipiatur hic pro significatione tantum sed magis pro suppositione seu pro alicuius positione.”

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Thus, although he has established that “only a word or another conventional sign can be equivocal or univocal and hence properly speaking the intention of the soul or the concept is neither univocal or equivocal,” Ockham now states that equivocation comes into play not so much with the conventional assigning of meaning to words, but when these are attributed with a certain suppositio in a specific linguistic context. Ockham held this proviso to be necessary because sometimes equivocation does not only arise because a double meaning has been imposed on a given term, either by chance or by analogy; in effect, equivocation also arises when we come up against a single act of impositio, as in the example, “man is a name,” where the term ‘man’ has received a single imposition of meaning and yet can signify different things, namely men in flesh and blood or the written or spoken term ‘man.’ In reality by specifying this, Ockham is distancing himself from the traditional way of explaining equivocation which made it a problem relating to the pure denomination of objects by words, but turns his attention to the fact that what is equivocal is precisely the use we make of words to refer to objects. Ockham’s clarification however constitutes a problem when we have to establish how the third type of equivocation is different from the first two. Ockham introduces it in this way: We have the third mode of equivocation when a certain expression is not taken with different meanings, but it is enough for its primary meaning not to correspond to its secondary meaning ... this type of equivocation does not happen because the word can mean different things, as happened in the first two types of equivocation, but it is enough for the same word to be able to supposit for different things. 17

The distinction between the first two modes of equivocation and this third mode seems to be extremely subtle: both the first two modes of equivocation, in Ockham’s clarification, and the third mode really take place at the level of suppositio, more than that of significatio. 17

See Summa logicae, III, 4, cap. 4: “… est tertius modus aequivocationis quando aliqua dictio non accipitur pro diversis significatis, sed ex hoc solum quod alicui comparatur quod non plus pertinet ad primarium significatum quam ad secundarium. Et iste modus accidit ex hoc quod vox potest significare diversa, sicut contingit in duobus primis modis, sed ex hoc quod eadem vox potest supponere pro diversis.”

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If we remember that for Ockham there is suppositio only when a term does not signify its objects independently of a certain linguistic context and hence only within a proposition, it becomes rather difficult to grasp the difference between the first two types of equivocation and the third, in the sense that all three, in as far as they depend on suppositio, take place only within a linguistic context and not in terms isolated from this context. This apparent contradiction in reality sheds light on Ockham’s correct attitude towards the nature of language: the fact that he has established the true nature of terms and now also their properties only within a linguistic context leads us to suppose that for him language is never a simple denominative game by means of which we attribute each thing with the name that signifies it. Even in the case of equivocation, as Spade has rightly observed, “the user of an equivocal term in one of the first two modes has adopted a certain set of subordination conventions that map the term into more than one concept … those conventions … are related to what is called ‘linguistic competence’ ” [12, p. 16]. Nevertheless, true equivocation arises when the language user chooses one of these linguistic competences in a specific proposition and in this sense Ockham was able to specify that equivocation of the first two types also only happens on the level of suppositio. In the third type of equivocation, namely that which depends on the contrast between the primary and the secondary meaning, there is no imposition of meaning and hence there is no previous elaboration of a linguistic competence and hence equivocation arises directly from the context. The distinction between the first two types of equivocation and the third, then, is to be sought in the fact that in the first two modes the terms are subordinated to mental terms, while in the third, which does not depend on impositio, there is no subordination to mental terms. An important consequence of these considerations should be that only the first two types of equivocation can be excluded from mental language, in so far as they are based on conventional subordination to mental terms. There is no reason on the other hand for excluding the third type of equivocation from mental language.

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In effect Ockham explicitly affirms that “this third type of equivocation can also be found in a purely mental proposition, while the first two types only take place in linguistic signs established conventionally.”18 It seems clear by now that Ockham did not intent to eliminate completely all forms of synonymy and equivocacy from mental language, even if he aimed to eliminate them in the case of concrete terms. But, as numerous recent studies have shown, “the language of thought, for Ockham, is not by itself semantically pure” [10, p. 61]. Therefore the typical Ockham clause “propositio est distinguenda” must be able to be applied to mental language too, in the sense that the language which arises strictly linked to things is also subject to analysis and does not always arise in its immediate purity, if only because it is not possible to separate the cognitive faculties of the subject, which at the level of mental language should always be perfect given the natural relationship that links them to their object, from the contents of these, which can become distorted on the basis of the sensible representation which we must make of them. This reinterpretation of the nature of the verbum mentis in Ockham has also led to a different assessment of the allegedly topical nature of his logical and linguistic doctrines. Claude Panaccio, for example, who has undertaken one of the most significant attempts “to constitute a dialogue between [Ockham’s] impressive theoretical system and certain parts of contemporary philosophy,” is convinced that it is possible “to translate Ockhamism into a contemporary philosophical idiom,” has limited himself more recently to observing that “Ockham, on the whole, is much more interested in nominalistic ontological economy than in the logical truth of human thought” [10, p. 73]. On the basis of this conviction he seems to want to re-assess the importance of Ockham’s logical and epistemological doctrines, which, as we know, have always been considered his greatest contribution not only to the cultural changes of his time, but have also inspired many thinkers, even in recent times.

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See Summa logicae, III, 4, p. 763: “Et est notandum quod iste tertius modus aequivocationis potest reperiri in propositione pure mentali, quamvis duo primi modi non habeant locum nisi in signis ad placitum institutis.”

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More specifically, Panaccio exemplifies his new perspective on Ockham when he observes that “the philosophical role of Ockham’s distinction between primary and secondary signification is to rarefy the ontology, not the mental apparatus for knowledge.” Furthermore he sees Ockham’s entire logical and linguistic project as aimed uniquely at establishing “relations between sign-token and singular objects of the world.” In his interpretation: “the duality of primary and secondary signification allows, in fact, for the admittance of a wide array of non-synonymous concepts, while avoiding the reification of universals and others abstract entities.” “To borrow Quine’s famous example,” Panaccio goes on, “‘renate’ (defined as ‘animal with kidney’) and ‘cordate’ (defined as ‘animal with heart’) can be admitted as two distinct, non-synonymous simple concepts even though their primary significates are the same … this can be done without recourse to special abstract properties of renateness or cordateness simply because the secondary significates of ‘renate’ and ‘cordate’ are not the same: ‘renate’ connotes kidneys; ‘cordate’ connotes hearts. Relational and quantitative concepts, being connotative, can be accepted in the same way without having to enrich the ontology with special entities such as relations or quantities … connotation in Ockham’s hands turns out to be a highly effective device in the service of ontological economy” [10, p. 58]. Even though these observations can be largely accepted, what gives rise to some perplexity is the attempt to pretend from Ockham something that he cannot give, ending up by misunderstanding the sense of his work from a historical and conceptual point of view. If for many years Ockham looked like a medieval Russell, which resulted in a forced interpretation of his logical and epistemological doctrines, now there is an attempt to consider him exclusively “as an aprioristic ontological reductionist,” without, that is, an adequate foundation to his logical doctrines. Even if Ockham’s doctrines might be found “inadequate” on an ontological level for a contemporary thinker with analytical tendencies, this does not remove the fact that Ockham was convinced he had based his criticism of traditional ontology precisely and exclusively on a logical and methodological plane. In order to evaluate his theories on the basis of the principles of the contemporary science of language, on the other hand, I believe that we should look at his attempts in the light of a more Augustinian assumption. Augustine’s work on the verbum cordis was carried out as a knowledge of

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self, one’s own inward acts, and one’s own identity. Only in this perspective is the effectiveness of interpersonal communication also guaranteed. The transparency of the language of the heart, however, does not exempt us from inquirere, from searching, and hence even the evidence of inner words is rather a conquest than an easy road on the search for truth. The same can be said for Ockham’s oratio mentalis. At a mental level too, in effect, on one hand the object may not result as evident and the faculty therefore necessarily limits itself to representing to itself, albeit in a totally natural way, an object which is not wholly evident in its real structure. On the basis of this we can conclude that the natural and evident character of mental language limits itself to grasping the nature of things, as this appears to us at any given moment. It is therefore a procedure which guarantees, more than the objectivity of knowledge, the transparency of the cognitive acts of the subject, who should be able, at any moment, to reach into the depths of his or her intimacy. Language therefore is not devoid of ambiguity because it is a perfect language, but it is only a language devoid of the ambiguities due to the will to deceive. Ambiguities, on the contrary, which depend on the epistemic state of the subject on a mental level should necessarily be dealt with by means of a logical procedure of analysis, at the end of which ambiguity can be dissolved. References [1] Andres T. De, El nominalismo de Guillermo de Ockham como filosofia del language. Madrid, 1969. [2] Augustine, De Trinitate, XV, 10, 18—19 (English trans. by E. Hill, in The Trinity, New City Press, New York, 1991). [3] Bottin F., Ockhams offene Rationalität, [in:] Die Gegenwart Ockhams, hrsg. von W. Vossenkühl und R. Schöngerber, VCH Verlagsgesellschaft, Weinheim, 1990, 51 – 62. [4] Brown D. J., The Puzzle of Names in Ockham’s Theory of Mental Language, The Review of Metaphysics, 50, 1996, 79 – 99. [5] Chalmers D. J., Is there synonymy in Ockham’s mental Language? [in:] The Cambridge Companion to Ockham, ed. P.V. Spade, Cambridge, 1999.

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[6] Geach P., Mental Acts: Their Content and Their Objects. London, 1957. [7] Glorieux P., Pro et contra Thomam. Un survol de cinquant années, [in:] Sapientiae procerum amore, ed. T.W. Köhler, Roma 1974, 255 – 287. [8] Nuchelmans G., Theories of the proposition. Ancient and medieval conceptions of the bearers of truth and falsity. North-Holland, Amsterdam, 1973. [9] Ockham, Summa logicae, eds Ph. Boehner, G. Gál, St. Brown, St. Bonaventure, N.Y. 1974. [10] Panaccio, C., Semantics and Mental Language [in:] The Cambridge Companion to Ockham, ed. P.V. Spade, Cambridge, 1999. [11] Petrus Hispanus, Tractatus, called afterwards Summulae logicales, ed. by L.M. De Rijk, Van Gorcum, Assen, 1972. [12] Spade P. V., Synonymy and Equivocation in Ockham’s Mental Language, Journal of the History of Philosophy, 18, 1980, 9 – 22. [13] Trentmann J., Ockham on mental, Mind, 79, 1970, 586 – 590.

ANALOGY IN THOMISM Petr Dvořák Institute of Philosophy Academy of Sciences of the Czech Republic Prague, Czech Republic [email protected] The paper introduces analogy in Thomism, focusing on the Thomist scholastic heritage. Two types of analogy are introduced: the analogy of attribution and the analogy of proportionality. These semantic phenomena of analogy are put to particular use in Thomism. This means that there are important classes of words whose meaning is expressed by means of analogy, i.e. the so-called transcendentals and divine attributes, and that antecedent metaphysical principles are true of the objects denoted by analogy. A question arises which type of analogy is more suitable to capture reality delimited by these metaphysical claims. The correct answer depends on the semantic properties of each type of analogy. It is argued that while the analogy of proportionality is employed to express similarity among objects, attribution captures their different ordering in the ontological realm.

This paper presents an explanation of the concept of analogy and its role in Thomism. The method of procedure is systematic rather than historical, or, more specifically, different views of historical authors within the Thomist tradition are treated in a systematic framework pertaining to the semantic phenomenon of analogy and its philosophical use. 1. Introduction From ancient times it has been a common practice to distinguish between two types of phenomena which have come to be treated as two types or forms of analogy in the scholastic tradition of the Middle Ages: the so-

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called analogy of attribution or proportion and that of proportionality. One can most certainly find this distinction in the writings of Thomas Aquinas (Thomas) and it amounts to no major simplification to say that the precise definition and nature, mutual relationship, and application to selected philosophical issues of the two aforementioned kinds of analogy is the sole interest of our present inquiry. There exist considerable differences among the followers of Thomas, the Thomists, on how they understand these types of analogy and on how they put them to use in philosophy. These differences are partly due to the fact that Thomas’ treatment of analogy is varied and, at times, even prima facie inconsistent. On the other hand, there are systematic reasons for treating and applying analogy in particular ways and not others. Thomists have become aware of some of these reasons only gradually in discussions with opponents. The result is a family of overlapping, yet at times to some extent opposing, theories concerning the two types of analogy and their application. Their proponents claim every one of them to be based and grounded in the thoughts and work of the Angelic Doctor. Hence, our task will be to explore and comprehend the various treatments of analogy in a systematic and orderly fashion. This goal calls for an appropriate order of procedure. First, we will outline systematically two types of analogy used in philosophy broadly conceived (including also the so-called rational or philosophical theology). Secondly, we shall specify the general metaphysical principles accepted in Thomism which give rise to different semantic requirements placed on analogy in the latter’s application. Thus we will avail ourselves of the necessary background in order to understand the relationship between both types of analogy in variant Thomist theories on the matter. Before our enquiry commences, however, a brief note appears to be called for. By “Thomists” we mean the intellectual followers of Thomas Aquinas from the heyday of scholasticism in the high Middle Ages through the Renaissance and early modern periods and up to the present. Different types of Thomism have evolved throughout the ages. In the late sixteenth century there appeared a distinct Jesuit version of scholasticism inspired by Thomas Aquinas, especially in theological matters (which was the official allegiance of the Jesuits in theology), yet far less concerned with strict adherence to all and every point Thomas had ever made. This appears to be

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in a marked contrast to Dominican theologians, for instance, who faithfully observed Thomas on all points and developed his teaching with much greater caution. Hence, the differences could be brought out as a matter of the degree of allegiance paid to Thomas’s teaching. Varied as these approaches might be, however, they all fall within the broad Aristotelian scholastic paradigm. In contrast, in the 20th century the scholastic paradigm, the scholastic way of doing philosophy, was to a lesser or greater degree enriched and reinterpreted in the light of other paradigms (e.g. Kantian and German idealism, phenomenology and existentialism, and analytic philosophy), resulting in transcendental, existentialist, analytic, and perhaps some other “Thomisms.” Fruitful as these current approaches might be, our focus will be on classical – by which we mean scholastic and neoscholastic – Thomism for the simple reason that it and no other stands as a departure point for rethinking in other paradigms using further conceptual tools. Consequently, the exploration of Thomist analogy in these extended frameworks would require a study of its own and is no part of our treatment. 2. Analogy of attribution Let us focus first on the two types of analogy and the shared ideas which would not be regarded as controversial by the Thomists as well as other scholastics of different schools. In our exposition we will take for granted the use of current general linguistic terminology and introduce scholastic conceptual apparatus rather than presuppose familiarity with the latter. Semantically speaking, the so-called analogy of attribution is really a type of polysemy, i.e. one word type is used with distinct yet related meanings. There exists a hierarchy among the meanings in that one is proper and the others are dependent on it. Hence we can call the proper meaning “primary” and the dependent ones “secondary.” The dependence is as follows. Let us take the concrete general term “F”. The proper meaning is that of “a possessor of the property F,” while the other meanings assume the general form of “something that bears the relation R to the property F” or “something that bears the relation R to a possessor of the property F.” Depending on the particular instantiation of “the relation R,” we obtain different secondary meanings all related to the primary meaning “a

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possessor of the property F” or “the property F” in that the primary meaning forms a constant part of the secondary ones. To take the notorious example used in virtually every discussion of the analogy of attribution (taken from Aristotle), the adjective – a general concrete term from the point of view of logical semantics – “healthy” means “a possessor of health” when applied to animals, “something that is a sign of health” when it stands for e.g. blood and urine, “something that restores health” denoting medicine, and “something that conserves health” when used of food. The meaning “health” forms a part of all the dependent secondary meanings.1 By “meaning” we have so far denoted the sense of an expression, e.g. the expression “F”. If we take “meaning” to stand for denotation, we can say that the secondary meanings are various things or objects related in different ways either to one thing which has the property F or to the property F itself in a thing that has it. In both cases this thing – possessor of the property F – is the primary meaning in the denotational understanding. Thus some animal, the bearer of health, is the primary meaning, and blood or urine, medicine, and food are secondary meanings exhibiting relations to the primary one. These things, i.e. meanings of F taken as denotations, are called the primary and secondary analogates. In contrast, the meanings understood as primary and secondary senses are dubbed primary and secondary analogues. Hence one can say that a word which is analogical by the analogy of attribution implies some order among the senses it expresses, i.e. among analogues, and among objects it denotes, i.e. analogates. This order is an order of priority and posteriority (per prius et posterius). Now it is important to recapitulate of what precisely the order consist as this ordering associated with the analogical 1

See Aristotle’s Metaphysics, book IV, 1003a. There might arise a question of whether only “health” or rather “a healthy animal” is a part of the dependent secondary meanings, for it seems that the relationship expressed by the latter meanings is to health in some animal rather than to health in general. That is to say, the relationship is to a healthy animal qua healthy. Aquinas himself is not clear on this point. If this were the case then the secondary meanings would be the following: “something that is a sign of a possessor of health” when it stands for e.g. blood and urine, “something that restores a possessor of health” denoting medicine, and “something that conserves a possessor of health” when used of food.

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usage of words is often put forth as the defining feature of analogy in general.2 Once again we have distinguished in word meaning (as is commonly the case in logical semantics) between the levels of sense and denotation which, philosophically (ontologically) speaking, is the level of rationally apprehended things, i.e. conceptualized things, concepts in short, and the level of things, properties, and relations. Obviously there is no analogy at all when a word is used only in the primary meaning, i.e. with primary sense denoting its primary object. In this situation there are no tokens of the word-type used with different (secondary) meanings, so one should not speak about primary senses and primary objects either. The “primary”– “secondary” distinction is applied only when there are tokens of the word used with non-standard meanings. On the level of sense these non-standard or, as we say, secondary, meanings require an understanding of the primary meaning as this is included in the secondary meanings as their constant part. As we have seen, there is sense inclusion, so a definition of any secondary sense contains the primary sense as its part. For example, the sense of “healthy” as in “healthy food” cannot be defined without the sense of “healthy” as in “healthy animal.” Hence, on this level the relationship of priority and posteriority implied is the one of definitional priority. On the level of denotation, these non-standard or secondary meanings require the understanding of the primary meaning as the secondary objects possess relations whose term is the primary thing. For example, healthy food bears a specific relation to an animal having health. Hence, on this level the relationship of priority and posteriority implied is the one of ontological priority: a relation of a particular thing, the secondary analogate (e.g. healthy food), to another one, the primary analogate (e.g. animal having health), which is the so-called term of the relation, presupposes that the term exists in some way. To sum up, there are two ordered relations in attributional analogy: there is the definitional priority and posteriority among the analogues on the conceptual level of sense and there is the ontological priority and posteriority among the analogates on the denotational level of things. 2

In Aquinas see for instance Summa Contra Gentiles, book I, chapter 32, Commentary on Metaphysics book 7, lesson 2, par. 7, On the Principles of Nature, chapter 6.

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Let us still dwell on the denotational level of things, the analogates: the relation of the secondary analogate to the primary analogate might be causal, as e.g. that of sustaining health in an animal, or otherwise, as e.g. being the sign of health in an animal. As any relation, it has a foundation which is some property present at least in one of the relata. For instance, the causal relation of healthy food to health in an animal is founded on the causal capacity of the food to sustain health. This founding property (capacity to sustain health) in the secondary analogate (healthy food) is not the same property as the one in the primary analogate (the property of health) which is conceptualized in the primary analogue in the primary sense and is the basis of the application of the word. In our example, the capacity to sustain health in food is not the property health which is found only in an animal. The application of the word is called denomination in the relevant scholastic texts (i.e. naming) because predicates are treated as general names of things. The denominating property, or form, as the scholastics say, inherent in the primary analogate, is called ratio. So what we are maintaining is that the ratio in the primary analogate and the foundational property in the secondary analogate are not the same, specifically speaking. That is, they are not of the same kind or type.3 Are we also saying that the ratio cannot be present in the secondary analogate at all? This point is trickier and has been a major matter of contention among the scholastics in general as well as Thomists in particular. In one sense we must necessarily deny its presence, in another not. In other words, the question is ambiguous and calls for differentiation or qualification. One thing appears to be clear: the point of the analogy of attribution is to apply a word whose sense expresses a certain property to a thing which does not have that property, on the grounds that there is a relation of the latter thing to the former’s having the property. So the secondary analogate does not have the property, or else, there would not be an analogical predication but a straightforward non-analogical one. For, clearly, if the secondary analogate had the property in question, the ratio, then it would be denominated directly and not in obliquo from something else, as it were. The secondary analogate would not be the secondary 3

Of course, they cannot be numerically the same as the analogates are numerically distinct entities.

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analogate but a primary one, which shows that the distinction would not be applicable at all and the application of the word would not constitute a case of analogy. So for there to be a specimen of the analogy of attribution, there must be at least one secondary analogate and for something to play the role of a secondary analogate it is required that a) the ratio is absent in it, and b) it has some kind of relationship to the primary analogate in which the ratio must be obviously present. This relationship is the basis of the analogical application of the word, the denomination. The scholastics would say that the denomination is not intrinsic but extrinsic, i.e. from outside. Requirements a) and b) appear to be necessary conditions of any attributional analogy. The semantic fact that the ratio is necessarily absent in the secondary analogate is further supported by the nature of the example used, i.e. by ontology: health can never be present in food, urine, blood, and medicine as the domain of the property of health includes only animals of which exclusively it is predicable. While condition b) is non-controversial, condition a) allows for some qualification. Let us raise the following question: is it not possible to name a thing based on a relationship to another thing regardless of whether it has the property in question? In other words, is it not possible to apprehend a thing as if it did not have the property regardless of the real ontological situation, i.e. whether it really has or does not have the property at issue? How could that be possible when we have just observed that the point of the analogy of attribution is to apply a word whose sense expresses a certain property to a thing which does not have that property, and that for something to play the role of a secondary analogate it cannot have the relevant property? Perhaps this might be an explanation: consider the situation that one is not interested in characterizing a thing as having the property F, but as having a specific relationship to something with the property F. The information one wishes to convey by such a naming is that the thing named has that relationship to some F, not that it is F itself. Not to say that something is F itself is not the same as saying that it is not F!

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Hence, one is entitled to claim that what is required for there to be a case of the analogy of attribution is not that the thing assuming the role of the secondary analogate actually does not have the relevant property, the ratio, but, rather, that it is not seen as having the property. Notice that this assertion is not the same as “seen not having the property.” So in applying the analogy of attribution to at least two things, one disregards that one or several of them have the property at issue, abstracts away from it and names the things now playing the role of the secondary analogates based on the relationship to the primary one. Hence we have to qualify the necessary conditions of the analogy of attribution. More precisely, we have to reformulate condition a), while b) stays the same. It is not required that the ratio be absent in the secondary analogate, but, rather, that a*) for a thing to play the role of the secondary analogate the ratio be not apprehended. So we propose to replace a) with a*). It seems then that the analogy of attribution represents the ontological situation in a certain way which might or might not differ from the real ontological situation. One could say that there is a semantic ontological model assumed by the analogy of attribution and the real ontological situation. In case these two differ, the point is not that the semantic ontological model presupposed or implied by the analogy of attribution falsely represents reality, the actual ontological situation, but that it is an incomplete model of reality as it really is. It would be a false model if it represented reality as if there was not a certain feature, namely, the property in the thing having the status of the secondary analogate. A prominent Renaissance Dominican commentator of Aquinas, Thomas de Vio, also known as cardinal Cajetan (1469 – 1534), appears to be making precisely this distinction. What we have called the semantic ontological model he dubs “formal understanding,” the real ontological situation is “material understanding:” It should be carefully observed that this first condition of such a mode of analogy, namely, that it is not according to genus of an inhering formal cause, but always according to

172 something extrinsic, should be formally and not materially understood. That is, it should not be understood in the sense that every name which is analogous by attribution is so common to the analogates that it pertains formally only to the first analogate and to the other analogates by extrinsic denomination, just as it happens in the case of health and medicine; for that universal is false, as it is evident concerning being and good [1, chap. 2, par. 11].

What if we dropped condition a) altogether? If we did this, then it would seem that the arguments above applied and there could not be any analogy of attribution: if the presence of the denominating property was recognized in both analogates, the analogy of attribution would be simply unintelligible. External denomination – naming from outside – would not be possible as this presupposes lacuna in the secondary analogate. As we have seen it is sufficient for the lacuna to be merely epistemological, a matter of apprehending the thing as criterion a*) maintains; it does not have to be real. Granted that there exists a theoretical possibility of naming something from outside even if it actually has the relevant property, and granted that the reason for doing so would be the aim of expressing the relevant relationship to the primary analogate, two further questions arise: first, would it not be the case that one and the same thing could in that situation be denoted by the same word yet with two different senses, and, second, are there any actual examples of such words? As for the first question, the answer is affirmative. A statement of the form “x is F” would be ambiguous. The first reading amounts to a nonanalogical predication “x has the property F” or, according to the scholastic identity understanding of predication, “x is something which is F.”4 The second reading is that of an analogical predication on account of the analogical predicate: “x is something which has a relationship R to some y which is F.” In our hypothetical situation both statements would be true. Let us turn our attention to the second question. We are now coming to the application of the analogy of attribution. In Thomism, this type of analogy has traditionally had two broad areas of application: the transcendental 4

We will assume this identity theory of predication as this seems to be required when concrete term predicates, such as “healthy,” are treated as names of things with the sense of “something which is healthy.” The analysis of attributional analogy is not dependent on this move, however. We make it because it is that presupposed in scholastic Thomism.

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concepts (and the associated terms) such as “being,” “one,” “true,” “good,” and possibly others as well as the so-called divine names, i.e. predicates applied to God and creatures alike such as “being,” “good,” “knowing,” “merciful,” etc. The first field of application is that of various branches of philosophy, especially metaphysics, ethics, and the philosophy of nature. The second set of words comes from the domain of theology, part of which overlaps with philosophy in the so-called rational or philosophical theology. In both of these sets of words it seems that in their application we are always faced with the situation in which some object denoted, x, has a certain property, say the property F, yet it has it on account of a different object y, which is F, and to which x is related. Hence, it is true that x is F, y is F, and x bears a relation R to y. This relation R is that of ontological dependence. x is F because y is F. To illustrate all this, let us take the word “being” as a representative of the first group. In Aristotelian ontology “substance” roughly denotes a thing, for instance a plant of some kind, while “accident” means a property of some type (possibly a relational property), such as the color green. Let x be some accident, e.g. green color, and y some substance, e.g. a plant. Now we can call the green color of the plant a being, for it has existence as such. We can also call it being because it has a relationship to the existing plant: the color would not exist if there were no plant, so the relationship is that of ontological dependence. We can distinguish the two predications of “being” of the color green: in the first application of “being” we ascribe to, say, “being 1” the meaning “has existence” or, rather, “something which has existence.” In the second application we predicate “being 2” as meaning “something which has a relationship of ontological dependence to the plant which has existence.” Clearly, “being 1” and “being 2” are predicates having different senses, even though they are affirmed of the same thing. 3. Divine attributes The same is true mutatis mutandis of the words from the second set, the socalled divine names, divina nomina, affirmed of creatures and God. Once again, let us have two predicates, “being 1” and “being 2.” While the former means “something which has existence,” the latter means

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“something which has a relationship of ontological dependence on God which has existence.” Now the relationship of ontological dependence between a creature and God is somewhat different from that existing between substance and an accident (i.e. a thing and its property).5 The relationship is a special type of causation called by the familiar term “participation,” taken from Platonic metaphysics. There is a metaphysical difference between the object having a certain property as participated and the object having that property fundamentally as the basis for participation by others. The latter is said to be the property rather than to have it as is the case with the former. Thus God is existence (being) rather than having it as any creature does. Now this metaphysical distinction perhaps points to some difficulty in our analysis which treats “God exists” and “creature exists” (being 1) in the same way as “God/creature has existence.” Our theory might not be fine-grained enough to capture the aforementioned nuances of participationist metaphysics, for it seems that this different mode of bearing or possessing the property should be also expressed somehow. We will return to this point later. At this point another difficulty is to be addressed first. Namely, the Thomists would unanimously claim that even “being 1” in both applications is analogical. So far our analysis is not able to accommodate this claim for the presupposition has been that the property of existence is also present in the secondary analogate and is only not apprehended when the secondary analogate is named from the primary one on the basis of a relationship it exhibits to it. The possibility that the denominating property is not present in the secondary analogate, as is the case with “healthy,” is ruled out in the case of “being” and the words in both sets – the philosophical and the theological – mentioned above. If this were the case, only “being 2” would be applicable to some property (accident) in relation to its substance (thing), and to some creature in relation to its creator. The property itself as well as the creature itself would not possess the relevant property and hence would not be “being 1.” But clearly, a property (accident) exists in its own right and – what is even more obvious – a 5

Both relationships are analogical, but in a different sense of the word “analogy,” that of proportionality discussed below.

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creature has its own being and other properties such as goodness, knowledge, etc. and is not said to have them only because God has them. If the real presence of properties in the secondary analogates were not allowed, then properties (accidents) would ontologically be just modes of a thing (substance) and, what is even worse, creatures and their properties would be only modes of the divine being. The former is in conflict with Aristotelian ontology, the latter with orthodox theism. So the properties at issue are bound to exist in the secondary analogates as well. The reader with some knowledge of the medieval discussion of divine attributes might object at this point. The problem of the so-called divine names concerns naming God, not creatures. So far we have discussed the situation of “naming” the creature: “being 1” applies to it in virtue of its own existence, “being 2” applies to it in virtue of its dependence relationship on the divine being. Only “being 2” is analogical in the attributional sense. But what is at stake is naming God, not creatures. Our terms used of creatures are univocal, i.e. used in the same (primary) sense. Once they are extended to name (characterize) God, they cannot be univocal any more, for this would annihilate divine transcendence (otherness from the world). On the other hand, they cannot be equivocal either (i.e. used in wholly different senses) as this would destroy the possibility of any meaningful positive discourse concerning God. Hence they are analogical – the different senses are distinct but related, or, what amounts to the same thing, their meaning is partly the same and partly different. Thus the primary analogate is not God, but the creature instead. For instance, “being 1” applied to a creature stays the same: “something which has existence.” Now God is the secondary analogate. When the word “being” is applied to Him based on the analogy of attribution, say “being 3,” the sense is the following “something which has a relationship of ontological foundation to the creature which has existence.” The relationship is precisely the converse of the one in “being 2.” There it was the relationship of the participating to the participated. Here it is the converse relation of the participated to the participating. Even here the secondary analogate, which is God in this application of the analogy, is supposed to contain the relevant property (existence). Thus again, it is not recognized in the application of the analogy. Yet, since we are ignorant of

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its precise character or mode, we “name” God indirectly from the property inherent in the creature. 4. Analogy of proportionality Now while all this is certainly true I will try to prove that there still remains the need for another kind of analogy. The situation is as follows. Of the creature we predicate (i) “being 1” meaning “something which has existence,” and as a secondary analogate of attributional analogy; and (ii) “being 2” meaning “something which has a relationship of ontological dependence on God which has existence.” Of God as a secondary analogate we predicate (iii) “being 3” meaning “something which has a relationship of ontological foundation to the creature which has existence.” The latter analogical predication about God presupposes predication (i) because the creature is the first analogate of which we predicate “being 1.” Similarly, analogical predication about the creature (ii) presupposes another predication, namely, that of the first analogate, which is now God. Of God we predicate (iv) “being 4.” What does it mean, however? The most natural answer would be: “the sense of ‘being 4’ is the same as ‘being 1,’ namely ‘something which has existence.’” This is because every analogical predication in the attributional sense presupposes a non-analogical predication (i.e. nonattributional) of the primary analogate which must possess intrinsically the property in question. But “being 4” cannot be the same as “being 1.” If it were the same, then the argument that we cannot use univocal predication

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of God, or else we lose the divine transcendence, could not be held at the same time. But so far we have no reason to deny that the predication of “being 1” is univocal, pace what the Thomists claim, for we have not found a way to accommodate the analogy in the attributional scheme. Hence we are now faced with two claims. The first is that “being 4” must be analogical in order to salvage divine transcendence. The second is that, according to the Thomists, “being 1” is analogical as well. It still remains to be seen why this should be so. It is clear that neither “being 4,” nor “being 1” can be analogical by the analogy of attribution, for this type of analogical predication presupposes a non-analogical one, i.e. univocal or analogical in a different sense. This different type of analogy must be like regular predication (univocity) in that it cannot abstract from the property at issue present in the thing of which the analogical word is predicated. The thing “analogically named” must have the property at stake, for the application of the word is based on its recognition. The predicate is applied in virtue of the thing having this property. In contrast, in attribution the application of the predicate is conditioned by the thing’s not having the property and the word is predicated in virtue of something different, yet related, having the property. Hence there cannot be a primary and secondary analogate in this new type of analogy in the same sense as in attribution. So now we will explain in what sense “being 4” is analogical in order to keep God transcendent, i.e. sufficiently different from creatures. By the same token this will allow us to see why “being 1” is similarly analogical as the Thomists observe. Recall the insight that the being of God and that of the creature are very different and that “God exists” and “a creature exists” ought to be treated differently, i.e. should differ in meaning. The point is that unlike in attribution where the meaning of the analogical word-token is different from and dependent on the meaning of the nonanalogical word-token of the same word-type (e.g. the meaning of “healthy” predicable of food is different from and dependent on the meaning of “healthy” predicable of an animal), in the new analogy both word-tokens have analogical meaning and there is a mutual dependence of sorts. This mutual dependence consists in the relationship of similarity. This similarity is essentially that between relational properties or

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“proportions” of the two analogates.6 Consequently, the analogy is designated as the analogy of proportionality. The relational properties of the distinct analogates are neither specifically, nor generically, identical. There is no way of saying in what feature or set of features precisely the similarity lies. But their similarity nevertheless allows for unification under one sense or concept. For instance, the military leader and the spiritual leader are both termed “leader” on account of the similarity of their roles or functions which could be brought out as relational properties. If “being” is predicated of a creature and God analogically by the analogy of proportionality, then the creature’s existential activities and God’s existential activities, although in many ways quite unlike, are nevertheless similar enough to give rise to some unitary sense expressed by the word “being.” Hence there is really no difference between “being 1” and “being 4.” Both express the same sense whose unity is, admittedly, looser than that of univocally predicated words whose sense expresses some specific or generic features. Thus the analogical sense or concept differs from the sense of any specific or generic terms. The analogy of proportionality appears to be more fundamental than that of attribution in that, semantically, the former in no way presupposes the latter. On the other hand, there are situations when attribution presupposes the use of proportionality as with the predication of “being” and other proportionally analogous terms such as “knowledge,” “good,” and others. It goes without saying that attribution can be based on regular, i.e. univocal predication, as are e.g. the analogous senses of “healthy.” 5. Metaphysical principles By now the reader might have noticed that there are different tasks assigned to analogy in Thomist philosophy; there are distinct but related roles it is supposed to fill. These give rise to certain semantic desiderata which might prima facie conflict with one another or at least exhibit some 6

These relational properties are typically complex rather than primitive, i.e. capable of being analyzed into other relational properties. Thus a relational property might stand for a set of relational properties. For example, the property of being a leader involves typically a set of relations to the subordinates. Thus although we treat them as singular properties, this does not preclude their further analysis.

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tension. What are these roles to be filled? Firstly there is the metaphysical claim concerning divine transcendence: God is very much unlike creatures (there is neither generic nor specific identity between God and creatures). Secondly, there are epistemological implications, i.e. the problem of “naming” (characterizing) God by words having roots in sense experience. Thirdly, the “otherness” of God is not absolute and there must be some commonality of a form weaker than generic or specific identity. This is because of the metaphysical principle according to which every cause resembles its effect. Fourthly, each “perfection,” i.e. property which does not imply some limitation on the part of its bearer inapplicable to God, is ontologically realized primarily in God and only derivatively – by participation – in creatures. So there is the relationship of ontological dependence alluded to above. Let us distil these claims into two principal ones only: 1. There is resemblance between God and creatures, not identity. 2. There is ontological dependence of participation between God and creatures. Let us simplify the matters a bit and observe that – semantically speaking – the former claim calls for the analogy of proportionality while the latter seems to find its semantic counterpart in attribution. Thus “being 4” (existence of God) was predicated analogically based on statement 1. As the dependence relationship can be viewed from two perspectives, i.e. from the angle of that in which the form is dependent and that in which it is not (leading to a different assignment of the roles of the primary and secondary analogates), there are two applications of attribution possible. Thus “being 2” is predicated of a created object and “being 3” of God based on these relationships. The above claims, however, having to do with the relationship of God to creatures are by no means all relevant philosophical claims related to the role of analogy in Thomism. One should not forget to add to these two “Platonic” claims their “Aristotelian” counterparts from the domain of metaphysics:

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1*. Existing (good…) things are not identical specifically or generically but only resemble one another in existence (goodness…). 2*. There is ontological dependence between substance and accidents. Again, it seems that while the first claim finds its semantic expression in proportionality, the second finds it in attribution. Do any of the philosophical claims exclude the employment of univocal predication? While the pair 1 and 1* do in the sense that there cannot be a univocal predication of the resembling objects at stake at the same time (God and creatures or existing and good entities), the pair 2 and 2* by themselves do not. In other words, proportionality excludes univocity to a certain degree, but attribution does not as is clear from the treatment of this type of analogy above. To what extent is univocal predication excluded in proportionality? First notice that the predication of “being 1” (the predication of existence of some created object) is analogical based on claim 1*, not univocal. Would it be possible to say that claim 1* also explains the predication of “being 4” of God? In other words, when we predicate existence of God based on proportionality, is it because of 1 or 1*? Above we maintained that the analogy is based on 1, but it seems that 1* is more general and would suffice. While claim 1 in effect states that for any F, there is no generic or specific identity between God and creatures in F, claim 1* asserts that for some G, there is no generic or specific identity between anything in G. Perhaps claim 1* would suffice in the case of predicating existence, but there are words that do not satisfy 1* and are predicated univocally of creatures based on underlying ontological generic or specific identities, yet when predicated of God, they are analogical based on 1. For instance, the application of the word “father” is grounded in a specific identity of objects in a particular relational property. Hence, fathers are not such things with which the principle 1* is concerned. Yet when predicated of God, human fathers and God the Father are not specifically identical but only resemble each other as principle 1 states. Hence the word predicated of human fathers and God is predicated in analogical sense, i.e. in that of the analogy of proportionality.

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So while it is true that in some sense 1* is more general because it speaks of anything, not only creatures in relation to God, in another sense it is not, because 1 applies to any F, while 1* applies to only some G. So 1* is more general compared to 1 only when a particular F is also a G, as is the case with “being” but is not the case always. Hence, in a different sense claim 1 concerning God and creatures is more general than 1*: while principle 1 excludes any identity between God and creatures whatsoever, principle 1* excludes identity among objects only in some cases and is compatible with there being identity in others. Our observations imply an important fact: it is not required for the analogy of proportionality to exclude all univocity. On the one hand, there are some words which cannot be predicated univocally and are predicated only analogically, such as “being.” Their associated meaning (sense) is always analogical. There is no corresponding generic identity between objects in being. On the other hand, there are other terms which allow of univocal predication based on proper non-analogical sense and which in some situations have analogical meaning. Let objects x and y be called F based on some generic identity, i.e. identity in some generic respect f. Then an object z which only resembles x and y in their having f, but in fact has only some similar respect g, can be also called F. Now the sense associated with “F” is analogical. The word “father” is of this sort as we have noticed above. Our list of metaphysical principles underlining the semantics of analogy is still not complete. We shall add one more “Platonic” principle related to claim 2. Recall that 2 states the following: 2. There is ontological dependence of participation between God and creatures. This implies the following claim: 3. There are ontological degrees to which a property is realized: while it is realized to the utmost degree in the object on which other objects participate, in the latter objects the realization is diminished.

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This claim in turn seems to imply statement 1, i.e. that there exists only similarity, not generic or specific identity. The property participated is not the same in the object of participation and in those participating on it. It is only similar. We know that this has to do with the “otherness” of God. In fact, claim 3 asserts that there exists an ontological hierarchy related to some property being realized; there is again a prior and a posterior, this time with regards to the realization of a property. This prior/posterior does not appear to be the same as the prior/posterior in ontological dependence, yet it is admittedly intimately related to it. We know already that the analogy of attribution serves to express the latter order of ontological dependence in Thomism. A key issue of contention among the followers of Thomas is the question of whether only attribution is capable of expressing order or whether proportionality could express it in some way too. It is our opinion that one can understand the principal divergences in teaching on analogy among various Thomists as resulting from different answers given to this question. 6. Relationship of the analogies If only attribution were capable of expressing order due to the hierarchy of prior and posterior imposed on the subjects of predication seen as the primary and secondary analogates, then it would have to express not only the order of dependence (claim 2), but also the order of realization of a property (claim 3). It is not clear, however, how attribution would be capable of expressing the order of realization as it presupposes that a relevant property is not present in the secondary analogate, or, more precisely, abstracts from its presence. Now there are two options open to the Thomist. Either he or she wishes to keep the expression of order as the sufficient condition of attribution. Then he is forced to reinterpret attribution in such a way that the abstraction from the relevant property in the secondary analogates is not a necessary condition of attribution. Alternatively, he keeps the standard view of attribution implying that the abstraction from the relevant property is its necessary condition – as Cajetan, his later fellow Dominican follower John Poinsot called John of St. Thomas (1589 – 1644), see [1, ch. 2], [2, question 8, article 3], and many others would hold – but then it seems difficult to maintain that the

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expression of order is a sufficient condition at the same time. Furthermore, another kind of analogy is capable of expressing order, i.e. proportionality. While the Jesuit Francesco Suarez (1548 – 1617) with his intrinsic attribution takes the first way [3, disputations 28, 32], most Thomists adhere to the second, denying that there is such a thing as “intrinsic attribution.” There appear to be difficulties in both options: while it is not at all clear how attribution is to work when the relevant property is predicated of both analogates, it is equally puzzling how proportionality is to accommodate order within the analogical sense of an analogical expression. As for the first difficulty, it seems that really the Suarezian reinterpretation of attribution dissolves the originally equivocal predication of attribution into a univocal predication, predicating of the analogates the same sense which implies some hierarchy in the objects denoted in relation to a greater and lesser realization of the property (see [5]). It is univocal because the property predicated is the same, pace its different degree of realization. In contrast, in proportionality the property is not the same but only similar. Once again, this keeps the otherness of God intact. If according to Suarez only attribution is capable of expressing order and, at the same time, is capable of expressing that both analogates have the same property, then there is no need for proportionality at all.7 Indeed, Suarez regards proportionality identical to metaphor and as such without much use in philosophy. The price to be paid seems clear: what is left is only analogy by name. As for the second problem, i.e. the issue of how proportionality can express order, Y. R. Simon, following Renaissance and early modern Thomists Cajetan and John of St. Thomas, claims that proportionality implies order by itself [9]. Perhaps the reason might be that claim 1 (and its counterpart 1*) are grounded in claim 3 by way of implication in Thomism (i.e. the uneven ontological realization of a property in objects implies similarity between them, not generic or specific identity in the property at stake). The fact that there are different modes of realization of a property would not be sufficient by itself. For instance, the followers of the 7

Recall that proportionality is invoked by those who regard attribution as unable to express the presence of a property in the secondary analogate.

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medieval theologian John Duns Scotus are prone to accept different modes, yet this does not by itself preclude there to be generic identity in that property among the objects and, consequently, univocal predication of that property. Claim 3 is to be read strongly in Thomism: it is not that there exists just any difference in realization of a property, but that this difference is much greater than that between different individuals of the same species or genus. That which possesses the property absolutely – which in fact is the property, i.e. is really identical with it rather than just has it – is so different from that which has it by way of participation only that one cannot speak of specifically or generically the same property in both cases. Analogical predication – that of proportionality – based on similarity does not by itself seem to imply any order among analogates. If we consider the term “leader” to be analogical, then there does not appear to be necessarily any hierarchy on the part of the objects denoted: e.g. a military leader, political party leader, mountain guide, spiritual leader, etc. No one particular leadership role seems to stand out clearly as some more essential embodiment of leadership. A Thomist philosopher keen to uphold that proportionality implies order would argue that if the similarity is the result of accepting claim 3, then there appears to be a hierarchy built into the analogical sense: the union of different, yet similar, relational properties of which some are more paradigmatic than others, as if some family members were better specimen of a particular resemblance than others, to put it in Wittgensteinian terms. This order results, however, from accepting some other ontological claim rather than being on account of the semantics of proportionality as such. Proportionality implies (as its necessary condition) that there is similarity between objects, not that this similarity necessarily exists due to uneven realization, as stated in principle 3. If every similarity were the result of this uneven realization of a property, then, of course, proportionality would imply that there is an ontological order of realization in the analogates. The assumption, however, that every similarity is due to uneven realization of some property remains to be proved.8 8

A prominent twentieth century neo-Thomist philosopher Yves R. Simon wrote: “In the language of St. Thomas, the expression “by priority and posteriority”… is synonymous with “analogically”… No doubt, this synonymity holds in proper proportionality, independently of any combination with another type of analogy,

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One might be tempted to see some order in proportionality based on examples where a word has analogical meaning on top of the proper, univocal one. In extending the word “memory” from its proper usage, denoting human mental capacity, via the analogy of proportionality to denote a certain analogical structure in computers, the word acquired a new analogical meaning. It is questionable, however, whether human memory somehow realizes the relational property of memory better or to a greater extent than computer memory and, consequently, whether there is a hierarchy of an ontological kind. Even if this were the case and the hierarchy or order of analogates were given by the semantic fact that the word applies to one analogate via its proper sense while the other because here also, despite intrinsicality in all cases, what is asserted of one analogate is denied of another. The irreducible togetherness of to be and not to be renders abstraction impossible and order conspicuous when it implies extrinsicality in all instances but one. It still implies order, though less conspicuously, when it admits of an abstraction that, once more, never uses ways of its own. We may not be able to determine, in all cases, what comes before and what comes after. We may even be uncertain as to whether a concept is analogical or not, but what cannot be doubted is that if the unity of a concept is analogical, its inferiors make up an ordered set…” [9, p. 164 – 165]. “The order comes from the peculiar nature of abstraction in analogy: So long as abstraction uses the ways of its own, it does not by itself, establish any kind of order. It effects a unity of universality and says nothing about priority and posteriority among the things of which the universal is predicable… But in analogy, abstraction uses ways that are not its own, for the obvious reason that the differentiating features exist in the common ground as actually as the common features. These differentiation features cannot be expressed except by assertion and negation of the common ground, and thus order is brought into logical existence, for assertion comes before negation and pure assertion comes before any complex in which negation plays a part. Analogical abstraction proceeds by “fusing together” the members of a set. But such “fusing together” involves assertions and negations that define priorities and posteriorities…” [9, p. 156]. Simon appears to be saying here that analogical sense implies assertions of a specific content of the analogates, say x and y, inseparable from denials of this same content. This is only to be expected where only resemblance exists between x and y, not generic or specific identity. Here sameness is fused with difference. How does this produce any ordering? The fact that there are degrees of difference, i.e. that the denial of the common content allows for degrees, implies that degrees of resemblance among x, y, and some z are possible. We could say that y resembles x more than z does. Does this imply that some property is realized in y more than in z? This does not seem to be evident.

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analogates are denoted by proportional analogy, such a picture would be inapplicable to divine attributes. The analogical extension is that from the creature to God so a particular divine characteristic would turn out to be the inferior member of the hierarchy of ontological realization. Perhaps one could argue that this is so only in the order of the imposition of names and that ideally the proper sense is the one through which divine property is denoted and the analogical sense – the result of the comparison of the creaturely property to the divine property – denotes the creature as having a similar property. There is, however, an epistemological lacuna in our knowledge of divine properties; there is no non-analogical sense of a word applicable to God in human language, so there does not seem to be any basis for analogical extension of meaning from God to creatures. Let us grant for the sake of argument that proportionality does express the order of realization. The one who would hold such a position (Y. Simon, Cajetan, John of St. Thomas) – let us call him a Proportionalist – might still have some need for attribution. It is questionable whether existence of the order of realization implies that there is an order of ontological dependence, i.e. it is debatable whether claim 3 implies statement 2. Perhaps it does (so both claims are therefore strictly equivalent, for we already know that 2 implies 3), but this is far from obvious. So, there might still be some need for expressing this order of dependence directly through the use of attribution and not only implicitly through the use of proportionality (if there is such an aforementioned implication of 2 by 3). By the Proportionalist understanding, however, proportionality is the basic analogy one cannot do without in philosophy (both metaphysics and philosophical theology) and attribution can be dispensed with. Evidently, this Proportionalist position is the direct opposite to the one adhered to by Suarez. In contrast, if proportionality does not imply order among the analogates as we maintain, attribution is definitely needed, for proportionality is not sufficient by itself. While the Suarezian position sees both orders (see claims 2 and 3) as being expressed by attribution and the Proportionalist position views them as expressed by proportionality (at least the order of realization), the middle of the road position we take requires both: proportionality for expressing the presence of the property in both analogates as well as attribution for expressing the orders between them;

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i.e. both the order of realization as well as the order of dependence. The latter stand has a clear affinity with the Suarezian stand that only attribution is capable of expressing order. It regards, then, both types of analogy to be necessary for use in philosophy. Granted that both are essential, the question now is which analogy is more basic, i.e. which takes precedence over the other. Above it has been assumed that the analogy of proportionality is primary in the sense that the relationship of ontological dependence expressed by attribution requires that a particular feature be present in the primary analogate. If the feature can be predicated of the primary analogate only analogically, i.e. by the analogy of proportionality, as is the case with “being” attributed to whatever plays the role of the primary analogate – whether it be God or a creature – then any attributional predication rests on this basic proportionality predication, i.e. presupposes it. On a closer look at the underlying principles: 1. There is resemblance between God and creatures, not identity. 2. There is ontological dependence of participation between God and creatures. It does not seem to be obvious that the second claim presupposes the first one. It is not clear that the claim which bases proportionality, i.e. (1), takes precedence over the one grounding attribution, i.e. claim (2). Ontologically it seems the other way around: because there is the relationship of participation between the objects, there is consequently similarity between them. The priority of analogies mentioned above is more of an epistemological or logical kind rather than ontological. One can find this difference also in Thomas Aquinas, who makes a distinction between the order of the imposition of names and the order of things, i.e. as they are. As we have seen, the question of priority of one analogy over another is another issue of contention among the Thomists which is even unsettled in the work of the master – Thomas – himself. Thomas prefers attribution but for two texts: On Truth, question 2, article 11 and his commentary on

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Aristotle’s Nicomachean Ethics, book I, lesson 7, par. 13 – 14.9 In the former the use of analogy concerns divine attributes. Thomas prefers proportionality over attribution because proportionality does not require there to be some determinate knowledge of a particular relationship between God and creation, as attribution would seem to require. In contrast, his preference for attribution in the other texts, especially in the two major Summae, is based on ontological considerations: the ontological primacy of God over creation and some types of being (e.g. substance) over others (accidents).10 It appears that the key to understanding the relationship between both analogies is that proportionality has a double role: it is basic in the sense that it enables one to predicate a certain property to both analogates (e.g. to a creature and to God). Yet we should not forget that it is, strictly speaking, not the same property. It is not one property but two distinct yet similar ones. Hence, attribution can pick up on this difference and express the fact that there exists an order between these properties and thus between their bearers, i.e. the analogates. It does so by denoting the secondary analogate which has the inferior of the properties as if this property were not there at all and the property, as such, existed only in the primary analogate. Thus, the inferiority of the property in the secondary analogate is expressed: the inferiority of both its realization compared to the similar property in the primary analogate as well as the inferiority based on its ontological dependence on the primary analogate’s having the similar property. So, to put it in a nutshell, while the analogy of proportionality expresses that both analogates possess similar properties, the analogy of attribution expresses that there is an order among them. Proportionality could be seen as semantically primary when the primary analogate of attribution is to be God: first we need to express that God has a certain property similar to the one in creatures. Attribution, however, can be seen as ontologically primary in that it expresses an order among analogates which is 9

For Thomas’ numerous texts on analogy in English translation see [6]. For Aquinas’ understanding of analogy in English see e.g. the classic study [7] and among the most recent contributions [8]. 10 See e.g. Summa Contra Gentiles, book I, chapter 34; Summa Theologica, part I, question 13, articles 5 and 6.

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constitutive of there being the properties in the analogates present at all; namely, in the secondary analogate which is the creature. By way of an example, let us take the word “wise” (sapiens). “Wise” normally has non-analogical (univocal) application to human beings. If applied to God, however, its sense is analogical (analogy of proportionality) and unites human and God as the bearers of human and divine wisdom respectively on the basis of their similarity. Then one can predicate “wise” of a human being in an attributionally analogical sense. Human as a secondary analogate is called wise because his wisdom is ontologically dependent on God (the order of dependence) and human wisdom is inferior to that of God (order of realization). This latter analogical use of “wise” is dependent on the former, i.e. on God being called wise. Ontologically speaking, however, there would not be any human wisdom if there were no perfect divine wisdom. So from this point of view one expresses a primary ontologically grounding relationship by attribution. One might object: would it not be possible to regard the first analogical application of “wise” also as a case of attribution where God would be the secondary analogate termed “wise” because he causes wisdom in a human? Besides the fact that Thomas explicitly rejects this thought (naming God on account of his causal role only) as we already know this would not express that there is something similar to human wisdom in God himself. Even if it did express that there is wisdom in God (perhaps nothing can be a cause of wisdom in the sense of participation unless it is wise itself, indeed, wisdom itself), this attributionally analogical concept associated with “wise” and the sense of “wise” applied to humans would be two different senses and there would be no unified philosophical treatment of “wise” possible. For as all scholastic Aristotelians would hold, any systematic philosophical treatment of a subject requires there to be a sufficiently unified concept of the subject. As Cajetan in his famous treatise On the Analogy of Names realized, only proportionality is able to provide such a unified concept, even though possessing a weaker unity than a concept or sense capable of univocal predication. Whether proportionality is able to guarantee a sufficiently unified sense is another question which forms a chief point of contention with the Scotist school, whose proponents would argue for the negative. The question boils down

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to the ontological issue of whether there is similarity between properties without there being either generic or specific identity at the same time. The solution advocated in our paper has the advantage of doing justice to Thomas’ writing for it does not force us to choose one analogy to the virtual exclusion of the other. As is clear, Thomas has a use for both types of analogy in his writings even if he might waver on which analogy is more important for philosophical purposes. This is roughly the solution of Sylvester of Ferrara (1474 – 1528) who, while accepting Cajetan’s arguments, was a more faithful follower of Thomas’ texts, especially the Summa Contra Gentiles, on which he commented [4, ch. 34]. By way of conclusion, we might summarize our position thus: if Thomists are right in claiming that there exists similarity among objects irreducible to generic or specific identity, then this gets expressed in language and thought by the analogy of proportionality. On the other hand, if there is some kind of hierarchy among objects, e.g. on account of ontological dependence or ontological grades among their properties, then this finds its linguistic realization in the analogy of attribution. Provided the background metaphysics is correct, both analogies play indispensable role in philosophical discourse concerning being, good and other transcendental notions as well as divine attributes. References Scholastic Thomist Works [1] Cajetan (Thomas de Vio), De nominum analogia. De conceptu entis (originally 1498). Pontificio Ateneo Angelicum, Rome, 1952 (English translation: Cajetan, The Analogy of Names and the Concept of Being. Duquesne University Press, Pittsburgh, 1953). [2] John of St. Thomas, Cursus philosophicus: Ars Logica (originally 1631–2), vol. I. Marietti, Turin, 1930 (Partial English translation: John of St. Thomas, The Material Logic of John of St. Thomas, University of Chicago Press, Chicago, 1955). [3] Suarez, F., Metaphysicarum disputationum, in quibus et universa theologia naturalis ordinate traditur et quaestiones omnes ad duodecim

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Aristotelis libros pertinentes accurate disputantur, Opera omnia (originally 1597), vols. 25, 26, Vivès, Paris, 1861. [4] Sylvester of Ferrara, Commentaria in Libros quatuor contra gentiles S. Thomae de Aquino (originally 1552), Orphanotrophii a S. Hieronymo Aemiliani, Rome, 1897. Other Works [5] Heider, D., Is Suárez's Concept of Being Analogical or Univocal?, American Catholic Philosophical Quarterly 1 (81), 2007, pp. 21—41. [6] Klubertanz, G. P., St. Thomas Aquinas on Analogy. Loyola University Press, Chicago, 1960. [7] Lyttkens, H., The Analogy Between God and the World: An Investigation of its Background and Interpretation of its Use by Thomas of Aquino. Almqvist and Wiksells Boktrycheri AB, Uppsala, 1952. [8] Mortensen, J. R., Understanding St. Thomas on Analogy, Dissertation.com, PhD thesis at Pontificia Università della Santa Croce, Rome, 2006. [9] Simon, Y. R., On Order in Analogical Sets (originally 1960), [in:] A. O. Simon (ed.), Philosopher at Work: Essays by Yves R. Simon. Rowman and Littlefield, Lanham, Maryland, 1999, pp. 135—171.

TOWARDS A LOGIC OF NEGATIVE THEOLOGY Paweł Rojek Department of Philosophy Institute of Philosophy Jagiellonian University Kraków, Poland [email protected] Negative theology developed by Pseudo-Dionysius the Areopagite is hardly considered as a consistent doctrine. The aim of this paper is to provide a coherent and extensive interpretation of its claims. I shall argue that the theory of negative theology consist in four theses concerning God’s positive and negative properties and his unknowability. Afterwards I will propose and discuss three consistent logical interpretations of that theory. Only one of these interpretations covers all the theses of negative theology and therefore is a preferable one.

1. Introduction To speak of logic of negative theology may sound distinctly odd. Many logicians, as well as some theologians, would not even agree on speaking about the logic of theology, let alone negative theology. They would argue that theology is based on the intuitive and perhaps irrational assumptions and that formal logic can hardly be useful in its clarification. Yet the negative theology seems to be the most mystical, illogical, and vague theory of God ever made. In contrast to positive theology, which simply says what God is, negative theology says only what God is not. In doing so, negative theology draws extreme conclusions. It denies not only all assertions given by positive theology, but also all positive assertions in general, and even denies its own denying. Pseudo-Dionysius the Areopagite, the founder of Christian negative theology wrote once:

193 It [supreme Cause, God − P. R.] is not soul or mind, nor does it possess imagination, conviction, speech, or understanding. [...] It is not immovable, moving, or at rest. [...] It is neither [...] divinity nor goodness. [...] It falls neither within the predicate of nonbeing nor of being. [...] There is no speaking of it, nor name nor knowledge of it. [...] It is beyond assertion and denial (MT V).1

It seems that Dionysius violates here not only some basic principles of positive theology, but also some basic principles of logic. Saying that God is neither being nor nonbeing he seems to reject the law of excluded middle. Holding that God is not divine, he seems to undermine the law of identity. Moreover, he plainly contradicts himself claiming that there is no speaking of God, whereas he actually does it. One may rends one’s garments − if this is not illogical, what is then? In this paper I shall nevertheless argue that negative theology (hereafter NT) might be interpreted in a consistent way. Further, I will provide no less than three logical interpretations, which I assume to be consistent. The object of analysis are writings of Pseudo-Dionysius the Areopagite, mainly his the most widely known treatise The Mystical Theology. This is a very short but extremely dense text, and these few pages are among the most influential pages in the history of philosophy and theology.2 Although the task I take here does not look very promising, some authors have tried to fulfill it. Fr. Józef M. Bocheński OP was, as far I know, the only logician who tried to clarify the doctrine of NT [4]. Although he rejected it after all, his analysis might be a good point of departure. John N. Jones [10] made some efforts to show the consistency of Dionysian doctrine, but he unfortunately did not use formal logic. However the vast majority of commentators willingly agree that Dionysius is incoherent in his speaking on God. They say that for Dionysius all our descriptions of God are inherently self-contradictory since God is transcendent, and any attempt to express the inexpressible leads to inconsistency. Moreover some scholars, e.g. Andrew Louth, Paul Rorem, and Vladimir Lossky, suggest that works of Dionysius are in general devoid of theoretical value. They 1

All citations of Pseudo-Dionysius are taken from [16]. I use the following standard abbreviations: MT = The Mystical Theology, DN = The Divine Names, CH = The Celestial Hierarchy, EH = The Ecclesiastical Hierarchy. The numbers after the abbreviation indicate chapter and subchapter. 2 For a short history of NT see [13].

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should be treated as a prayer, hymn of praise, or a way of unification with God, not as a philosophical treatise, and so any attempts to discover their logical structure are doomed to failure (see a review of approaches to Dionysian NT in [10, p. 356 – 357]). On the contrary, I assume that works of the Areopagite have theoretical, not only spiritual value, and I believe in general that holding God to be transcendent does not mean having to assume his inconsistency. I shall firstly (section 2) introduce four crucial theses of the NT, considered as a theory of God. These theses are problematic because they seem contradict each other. Secondly (section 3) I will propose a simple typology of possible interpretations of the NT. All of them take one of the theses as basic and try to reconcile the others with it. I distinguish three main interpretations, which would be called “agnostic,” “negative” and “positive” NT (I couldn’t find better labels). These interpretations will be discussed in the three following sections (4 – 6). I suggest that the last one is the most appropriate one and in principle agrees with St. Thomas Aquinas theory of analogy. 2. Four Theses of NT First, I would like to analyze two short passages from the Mystical Theology, namely the beginning and the end of that treatise, in order to extract some of the basic theses. These theses I call in general “NT,” although not all of them are strictly speaking negative. The first fragment is the following (numbers T1 & T2 inside text are mine): What has actually to be said about the Cause of everything is this. (T1) Since it is the Cause of all beings, we should posit and ascribe to it all the affirmations we make in regard to beings, and, more appropriately, we (T2) should negate all these affirmations, since it surpasses all being. Now we should not conclude that the negations are simply the opposites of the affirmations [...] (MT I, 2).

We find here two fundamental theses concerning God. The first one claims that God is a subject of all affirmation, the second, in contrast, that God is a subject of all negations. Affirmation means predicating positive property of something; what might be positive properties will be discussed soon. For the moment, the first thesis might be formulated as follows:

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(T1)

God has all positive properties.

Similar claims might be found in many other places in Corpus Areopagiticum. Dionysius writes for example that “he [God − P. R.] is all things since he is the Cause of all things” and that “every attribute may be predicated of him” (DN V, 8). Thesis (T1), no matter how strange it looks, has strong evidences in the texts of Dionysius. The second thesis seems to be just the opposition of (T1). Dionysius claims that (T2)

God has negations of all positive properties.

Again, this thesis is often repeated in the texts. Usually it follows immediately after the (T1). For example, after assertion that “every attribute may be predicated of him” Dionysius hastens to add that “yet he is not any one thing. He has every shape and structure, and yet is formless and beautyless” (DN V, 8). Our true cognition of God, as Areopagite often repeats, contains in “denying of all beings” (DN I, 5), but in many cases he is satisfied by denying only some of the properties commonly ascribed to God (e.g. MT V). (T1) and (T2), as they stand, are incoherent. This is the main reason why so many readers conclude that the Dionysian God is contradictory. As I indicated, some philosophers and theologians simply accept this alleged inconsistency of God. This path of thinking was consequently developed by Nicholas Cusanus, which in [12] put forward the idea of God as coinicdentia oppositorum, the synthesis of opposites. But if we look for a consistent interpretation of the NT, we need to find an alternative way to save logic. The first strategy is just to reject (T1). Dionysius in many passages openly says that (T2) is “more appropriate” than (T1), so the former might be taken as a basic statement of the NT (MT I, 2; CH II, 3). Indeed, two of three proposed below interpretations (“agnostic” and “negative”) accept (T2) and simply ignore (T1) in order to avoid contradiction. That move seems to be acceptable but surely is less sufficient than more extensive interpretations.

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The second option is to somehow interpret (T2) to reconcile it with (T1). Notice that Dionysius suggests that these theses are not necessary incoherent. He insists that “we should not conclude that the negations are simply the opposites of the affirmations” (MT I, 2). P. Rorem noticed [16, p. 135] that this passage directly contradicts a passage from Aristotle’s [1], in which he used identical terminology to argue that negations are the opposites of affirmations (17a, 31—33). Similarly, in another place Dionysius writes that the negative terms are applied by theologians to God “contrary to the usual sense of a deprivation” (DN VII, 1). It implies that Areopagite might use negations in a different way than Aristotle and contemporary logicians. What was exactly the meaning of Dionysian negation needs to be elucidated. However, what is important, it is possible with a suitable interpretation to hold both (T1) and (T2). The two of the proposed interpretations (“agnostic” and “positive”) seek an appropriate meaning of negation. Here is the second passage from the last part of the Mystical Theology (some parts of this chapter were cited at the beginning of the paper): It [supreme Cause − P. R.] is (T1) not soul or mind, nor does it possess imagination, conviction, speech, or understanding. [...] (T4) It cannot be spoken of and it cannot be grasped by understanding. [...] (T3) It is not immovable, moving, or at rest. [...] (T4) It cannot be grasped by the understanding since it is neither knowledge nor truth. [...] It is neither one nor oneness, (T2) divinity nor goodness. [...] (T3) It falls neither within the predicate of nonbeing nor of being. [...] (T4) There is no speaking of it, nor name nor knowledge of it. (T3) Darkness and light, error and truth − it is none of these. (T2) It is beyond assertion and (T3) denial. We make assertions and denials of what is next to it, but never of it, for it is both (T2) beyond every assertion, being the perfect and unique cause of all things, and, by virtue of its preeminently simple and absolute nature, free of every limitation, beyond every limitation; (T3) it is also beyond every denial (MT V).

It is easy to find in the above text some instances of (T2). Dionysius as usual denies that God has many properties usually predicated about him. He claims that God is not a mind, soul, and so on. Noteworthy, he consequently denies that God has a property of being God. It is a plain contradiction if he uses the word “God” in the same meaning, or if he understands negation in the usual way. However this need not be a case, as will be investigated in section 4.

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Dionysius however goes beyond (T2). The third thesis of the NT concerns the self-reference of Dionysian denial. He not only denies that God has positive properties, but also implicitly denies that he has negative ones. He claims that God is not “immovable,” not “nonbeing,” and that God is beyond “every denial.”3 Hence, (T3)

God has negations of all negations of positive properties.

The same thesis might be found in other places as well (MT I, 2; II). This thesis raises serious problems. On the one hand, (T3) contradicts (T2), since by the law of double negation, (T3) boils down to (T1). On the other hand, (T3) combining with (T2) give impression that Dionysius reject the law of excluded middle. Indeed, we are told that God “not immovable, nor moving,” “falls neither within the predicate of nonbeing nor of being,” or “it is beyond assertion and denial.” Again, Dionysius violation of laws of logic depends on his meaning of negation. If, as I suggested, he does not understand negation as an opposition, he might be reconciled with classical logic. In the last passage we find also the fourth and the last thesis of NT. Dionysius explicitly says that (T4)

God is unknowable.

We are often told that God cannot be “grasped,” “named,” and “known” (cf. DN XII, 3; EH VII, 10). In other place Dionysius claims that “we have no knowledge at all of incomprehensible and ineffable transcendence” of God (CH II, 3). In still another he repeats that One, that is God, is “ineffable and nameless” and “ungraspable and inscrutable” (DN VII, 1). Perhaps (T4) should be stated as “God is unspeakable,” since Dionysius makes no sharp distinction between unknowability and unspeakability.

3

Being “beyond every denial” is not however, as was noticed by J. J. Jones, de dicto negation of (T2), since Greek παν means here “each”, not “all together.” Dionysius denied all particular denials, but not the denial of all denials [10, p. 363]. The form of (T3) is therefore not “¬∀Q (¬Q(x))” but “∀Q (¬¬Q(x)).”

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The status of (T4) is not clear. There is something paradoxical that Dionysius, after developing a theory of God asserts that God cannot be grasped and named. We do know something, after all, even if our knowledge is exclusively negative. Perhaps (T4) is only an emphatic addition; however it is possible to construct interpretations, which takes it as a basic claim of NT. 3. Three interpretations of NT and Positive Theology The NT is therefore a complex of four theses which assert that (T1) God has all positive properties, (T2) God has negations of all positive properties, (T3) God has negations of all negations of positive properties and finally that (T4) God is unknowable. It is hard to find a more logically unfriendly doctrine. Nevertheless I suggest that there are some consistent interpretations of it. All these interpretations take however these theses with a different status. They accept one of them as basic and then interpret the other in the light of chosen principle. I distinguish four such interpretations. The first possible interpretation focuses on (T4) and claims that the basic idea of NT is the unknowability or unspeakability of God. Such an interpretation might be called “Agnostic Theology” since it claims that we have no knowledge of God, neither positive, nor negative. The other interpretations assume that we have some knowledge of God, even if it is only a negative one, and try to interpret (T4) as an additional expression of God’s transcendence, nothing more. These interpretations might be called “gnostic” (with no allusion to ancient gnosis movement). Now, gnostic theories divide into positive and negative. “Positive Theology” take (T1) as basic and simply rejects (T2) – (T5) as an exaggerated way of marking God’s transcendence. On the contrary, “Negative Theology” takes (T2) as fundamental and reject or interpret (T1). With no doubt Positive Theology cannot be a satisfactory interpretation of NT, for there is nothing negative in it. I have already suggested that interpretations of NT may differ in an accepted meaning of negation. I will discuss this later in details, but now I just want to indicate that negation might have usual “negative” or a special “positive” meaning. Accordingly, there are two interpretations of Gnostic

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Negative Theology, namely “Negative Negative Theology” and “Positive Negative Theology” (let me use these recursive expressions). All these interpretations might be ordered in a following way:

Theology

Agnostic

Gnostic

Positive

Negative Negative

Negative

Positive Negative

Of course there remains the tempting interpretation that the NT is just contradictory. However, as I have already noticed, I take into account only the consistent interpretations. Before I pass to the other interpretation, one problem must be touched upon. The difference between Positive and Negative Theology consist in predicating negative or positive properties of God. But what is the difference between them? Which properties are positive and which are negative? The syntactical criterion is plainly insufficient, since for instance blindness contains no negation, but refers to an absence of what might have been expected to be, and hence seems to be a negative property. Furthermore, according to St. Thomas such seemingly positive God’s properties as simplicity, perfection, or oneness are in fact negative [20, p. I, q. 3—11). Some of the proposals of defining positive properties would be discussed in section 5. I am not however going to provide a satisfactory criterion of positivity of properties. In the worst case, positive properties might be simply enumerated. I just want to stress that the following consideration abstract from the content of positive and negative properties. I shall focus on a form of NT, not on its matter.

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It must be noted that positive properties are sometimes understood in a very special way. Some authors take positive properties as “perfection”, that is “the highest degree of a given quality” [15, p. 626]. The standard examples of perfections are omnipotence (the highest degree of power), omniscience (the highest degree of knowledge), etc. Perfections play a crucial role in ontological arguments for the existence of God formulated by R. Descartes, G. W. Leibniz, K. Gödel, and J. Perzanowski (see [15] for details). In these arguments God is usually defined as a subject of all positive properties, that is perfections: (G1)

G(x) ≡ ∀Q (P(Q) ⊃ Q(x)). (I read “G(x)” as “x is God-like”4, and “P(Q)” as “the property Q is positive”).

Such ontological arguments presuppose a kind of Positive Theology. Definition (G1) is taken from [15] and will serve as a pattern for the definitions of God in the NT. I am not suggesting however that NT denies only perfections of God in this restricted sense. The range of negated properties is usually much wider. But, as I pointed out, I do not determine the answer to the question which properties matter. 4. “Agnostic NT” The first interpretation of NT takes (T4) as a primitive and modified or reject the remaining thesis to save a consistency of the doctrine. According to this interpretation NT stress the transcendence of God claiming that he is fundamentally unspeakable, inexpressible, or unknowable. God is, as it is sometimes said, “wholly other” or perhaps “the wholly other” (cf. [6, p. 71]). This interpretation is accepted by many commentators. Jerome I. Gellman interprets NT as a theory which claims that “for any predicate, P, in a 4

J. Bocheński takes “God” not as a predicate but as a definite description [4, p. 65 – 68]. For sake of simplicity I will follow J. Perzanowski [15] and adopt the predicate interpretation of the term “God.”

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language of finite beings, P cannot be truly predicated by God” [7, p. 158]. As it seems, he allows P stands both for negative and positive predicates. He continues that NT claims that God is not in the “sortal range” of any predicates of our language, where the sortal range is understood as a domain of objects of which Q or ¬Q can be predicated. Therefore if a negative theologian says for example that God is not wise, he wants to assert that it is not the case that Q is a predicate of God, but without implying that ¬Q is predicated of God. God is neither Q nor ¬Q. The motivation of this move is a conviction that all our predicates, belonging to the language of finite beings, denote imperfections and so cannot express perfect nature of God. The similar position occupies J. J. Jones which suggests that the denial of all properties, including negative ones, guarantees that God is not a particular thing. Since God is not a being, he cannot be known and spoken as beings are known or spoken of. “Negation relates to the (im)possibility of knowing and saying anything of God. It is, so to speak, a second-order rule for the employment of first-order names” [10, p. 368]. P. Rorem provides even more agnostic interpretation, since according to him Dionysius “denies and moves beyond all our concepts or ‘conceptual’ attributes of God and concludes by abandoning all speech and thought, even negations” [16, p. 99]. Thus in this “agnostic” interpretation the result of NT is not that God has negative properties, but that God is unspeakable, inexpressible and unknowable. The theory has at least two versions, based on the indicated ambiguity of (T4). The first claims that God is unspeakable, and therefore that religious discourse has no meaning at all.5 The second one claims only that God is unknowable, and although religious discourse has meaning, it says only what we do not know about God. After a brief consideration of the Theory of Unspeakable, I shall propose a logical analysis of the Theory of Unknowable. Many critics accused the Theory of Unspeakable of inconsistency. For example Michel Durrant claimed that

5

The “Theory of Unspeakable” was analyzed and criticized by Bocheński [4, p. 31— 36].

202 On this account in saying that God’s nature is fundamentally inexpressible we have already described God’s nature − namely that it is fundamentally inexpressible. In other words those who advocate this position cannot do so without contradicting themselves [6, p. 74].

On the contrary, Bocheński argued that the theory of unspeakable does not entail contradiction if only some usually accepted logical conventions are preserved. Let “Un(x, y)” stand for “x is an unspeakable object in language y.” The definition of God in the Theory of Unspeakable is then the following: (G2)

G(x) ≡ ∀y Un(x, y).

Does (G2) lead to contradiction? It seems plausible to hold that (G2) is formulated in meta-language, and that the domain of universal quantification is a class of all object languages. This definition allows attributing to God only the meta-linguistic property of being unspeakable and excludes every attribution of object-linguistic property. So, there is no self-reference in (G2) and hence no contradiction (see [4, p. 34]). If NT boils down to the Theory of Unspeakable, it is not however clear why it include also (T1) – (T3). If God is unspeakable then the religious discourse has no meaning and we cannot meaningfully neither affirm nor deny any properties of him. Perhaps (T4) should not be interpreted in a semantic way, but rather epistemologically or even ontologically. This is a motivation for the Theory of Unknowable. The Theory of Unknowable fits better to the NT since it not only accepts (T4) but also gives an interpretation of (T2) and (T3). To see this we need to introduce a concept of indeterminacy and to distinguish two different senses of negations. A logical theory of indeterminacy was developed for example by Alexander Zinov’ev [22, p. 92 – 95], and I shall adopt his considerations for the logical analysis of NT. Suppose that there exists object x, about which it cannot be stated if it has neither Q nor ¬Q. It might be for instance an elementary particle, which parameters cannot be determined, a theorem, which is neither provable nor disprovable, or an object in change. All such cases might be called “nonclassical,” since do not obey classical laws of excluded middle. Let “?Q(x)” stands for “it cannot be stated neither Q(x) nor ¬Q(x)” that is “x

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has Q indeterminately.” Since God is assumed to be fully unknowable object, he posses all positive properties indeterminately: (G3)

G(x) ≡ ∀Q (P(Q) ⊃ ?Q(x)).

This definition, unlike (G2) is not stated in meta-language. Therefore to avoid contradiction it must be assumed that at least G is not a positive property. What is important is that on the ground of the Theory of Unknowable it makes possible to deny in some sense all positive properties about God. Of course, not in the sense that God has ¬Q, since according to (G3) no such assertion is possible. But if we introduce the functor of indeterminacy, there appears the statement “x is not Q” is ambiguous. It may mean that “x has not Q” or “it cannot be stated if x has Q.” In the first case negation occurs in the de re position, in the second case stands de dicto. The first kind of negation Zinov’ev called “intrinsic negation,” since it relates to the operator of predication only, the second negation is “extrinsic,” since relates to all the statement. Let the sign “¬” stands for negation de re, and “~” for negation de dicto. The formula “¬Q(x)”means “x is not Q” or “x has an property ¬Q,” whereas the formula “~(Q(x))” stands for “it is not stated that x is Q”, or “it is not stated that x has a property Q” (I intentionally use the double parenthesis to mark an “extrinsic” character of negation).6 The logical relations between the functor of indeterminacy, negation de re and negation de dicto may be fixed by the following axioms: (D1) (D2) (D3)

~(Q(x)) ≡ ?Q(x) ∨ ¬Q(x), ~(¬Q(x)) ≡ Q(x) ∨ ?Q(x), ~(?Q(x)) ≡ Q(x) ∨ ¬Q(x).

From this it is provable that (1) 6

?Q(x) ≡ ~(Q(x)) ∧ ~(¬Q(x)).

Although A. A. Zinov’ev usually used different notation, he would accepted similar translation, cf. [22, p. 92].

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In the “classical” case, when indeterminacy is not allowed to, it is so that ¬Q(x) ≡ ~(Q(x)). However in a non-classical case, when indeterminate statements are acceptable, this is not true anymore. Furthermore, in this case the classical law of double negation ~(¬Q(x)) ⊃ Q(x) is not valid, as well as the formula ~(Q(x)) ⊃ ¬Q(x). It is so, since ~(¬Q(x)) implies either Q(x), or ?Q(x) and similarly ~(Q(x)) implies either ¬Q(x) or ?Q(x). Instead the following formulas are true: (2) (3)

¬Q(x) ⊃ ~(Q(x)), Q(x) ⊃ ~(¬Q(x)).

Moreover in a case with indeterminacy the classical law of excluded middle Q(x) ∨ ¬Q(x) is no valid for there exists the third possibility, namely ?Q(x). It is true however that (4)

Q(x) ∨ ¬Q(x) ∨ ?Q(x)

and (5) (6) (7)

Q(x) ∨ ~(Q(x)), ¬Q(x) ∨ ~(¬Q(x)), ?Q(x) ∨ ~(?Q(x)).

This simple theory of indeterminacy seems to be very useful in the analysis of the NT. The current interpretation provides an account for three theses, namely (T2) – (T4). (T1) must be simply rejected since nothing can be positively asserted about God. (T4) is explicated in terms of (G3). (T2) may be interpreted as involving an extrinsic negation. In this account (T2) claims that God has extrinsic negations of all positive properties, that is: (8)

G(x) ⊃ (∀Q (P(Q) ⊃ ~(Q(x))).

This formula follows from (G3) and (1). Every positive property is extrinsically negated about God. It does not mean however that God has

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intrinsically negative properties, since from (G3) and (1) it follows also that (9)

G(x) ⊃ (∀Q (P(Q) ⊃ ~(¬Q(x))).

This formula might be thought as an equivalent of (T3). There appears the double negation, but the first negation is extrinsic, and the second one intrinsic. The formula says that God has (extrinsically) no (intrinsic) negations of positive properties. Accordingly in this interpretation it make sense when Dionysius says that God is neither Q nor ¬Q, for example that he is not „greatness or smallness, equality or inequality, similarity or dissimilarity,” „neither nonbeing nor being” etc. (MT V). For it is true that (10)

G(x) ⊃ (∀Q (P(Q) ⊃ ~(Q(x)) ∧ ~(¬Q(x))).

In this way may be understood the Dionysian claim that God is “beyond every assertion and every denial” (MT V). The only exception is the claim that God is neither indeterminate nor determinate, neither unknowable nor knowable. But as it seems there is no such claim in Dionysius works. To sum up, both “agnostic” interpretations of NT, namely the theory of Unspeakable and the Theory of Unknowable, are consistent. The advantage of the latter theory is that it provides an account of three of four theses of NT. However, both theories are plainly inconsistent with (T1) and with general religious discourse and praxis. Every religious discourse contains sentences which ascribe to God some object-linguistic or determinate properties and if God has no object-language properties, or is totally unknown, there is absolutely no reason why it should be worshiped and praised (see [4, p. 36]). 5. “Negative NT” The starting point of the second interpretation of NT is (T2). This theory does not contend that language of religious discourse is meaningless or that God is unknowable. It asserts only that whatever might be predicated about

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God, God posses in a purely negative way. Hence we do know that God has no positive properties.7 In this interpretation a negation involved in (T2) is taken as a classical one, that is as obeying principle of double negation and the law of contradiction, therefore (T1) and (T3) must be rejected. In this account the God’s transcendence is highlighted by his pure negativity. Though it does not entail his inconsistency, it does imply in some sense his unknowability, and thus (T4) has its own place in this interpretation. In a familiar way God might be defined as an entity which possesses negations of all positive properties: (G4)

G(x) ≡ ∀Q (P(Q) ⊃ ¬Q(x)). (Negation here is used in classical way, with no distinction between de dicto and de re).

This means that God is a subject of all negated positive properties and fits closely to (T2). Formula (G4) needs however some restrictions in order to avoid contradiction. The set of positive properties, which is still very indefinite, must be limited in an appropriate way (see [4, p. 111 – 113]). Surely it cannot contain negative properties since denying of God negated property leads to affirmation, because ¬¬Q implies Q. Moreover, allowing for denying negative properties it directly leads to contradiction since God might be ¬Q and ¬¬Q that is Q. But it is not enough to grant that positive properties contains no negation, since, as was pointed in section 3, syntactical criterion is not satisfactory. One might try to amend this criterion on an epistemological basic, defining positive property as a directly perceived property or a property defined by formula containing only symbols of positive properties and terms of a positive logic [4, p. 112]. But even this formulation, as showed Bocheński, is not strict enough. However, as I indicated earlier, I am not going to propose any solution of this problem. 7

Such an interpretation of NT was also analyzed and criticized by J. Bocheński [4, p. 111 – 114].

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It is important to notice that surely the property of being God cannot be a positive one. If one wants to understand the claim that God is not divine, one must interpret “divinity” not in terms of (G4), but as a bundle of positive properties usually ascribed to God. Taking the distinction between positive and non-positive properties as granted, this interpretation has nevertheless an interesting feature. J. J. Jones argued that the main concern of Dionysius was to show that God is not a particular being [10, p. 357 – 358]. Many authors follow Stanisław Leśniewski and define “objects” or “beings” as subjects of properties (see [18]; [14, p. 71]). It is plausible to complete that classical definition by requirements that these properties must be positive. As a result, the definition of “object” or “being” is the following: (D4)

Ob(x) ≡ ∃Q P(Q) ∧ Q(x).

In this sense God is not being, since God has no positive properties: (11)

G(x) ⊃ ¬Ob(x).

If there is God, he exists only in a weak sense, in which something exists if there is something which is it. In other words, God would exist in Quine’s sense, but would not be an object in Leśniewski’s sense. Perhaps this result would be welcome by supporters of NT, since it shows the radical transcendence of God, which is not even an object in some important sense captured by (D4). Do we know God defined according to (G4)? Surely we can express somehow his exceptionally nature. But in some important sense we have no knowledge of God and (T4) finds its interpretation. Given the natural world, we understand the natures of things by contrast with what they are not. As Spinoza said, “all determinations are negation.” With no meanings opposed to and distinct from others, we could not use the language we do use. But God is a negation of all properties, and so he is beyond the whole system of oppositions and contrasts. Full negation yields a complete indetermination. In that sense God appears unknowable and unspeakable. To conclude, “negative” interpretation seems logically consistent and though does not cover all theses of NT, it develops in an interesting way

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some of its principles. However it is not compatible with (T1) and contradicts usual religious discourse and praxis. One cannot worship an entity of which one assumes only that has no positive properties (see again [4, p. 114]). 6. “Positive NT” The last interpretation of the NT accepts (T2) as basic and says that God has all negative properties, but interprets “negation” in a radically different way. (Yet it has nothing common with Zinov’ev distinction between intrinsic and extrinsic negations). This is why this interpretation reconciles (T2) with (T1) and (T3). Furthermore, it might provide an intelligible interpretation of (T4). The crucial point is to observe that the “negation” adopted in the theses of NT has a special meaning, far from a usual one. It has been pointed in section 2 that Areopagite realized that his “negations” are neither “deprivations” (DN VII, 1), nor “simply the oppositions of the affirmations” (MT I, 2). He explicitly admitted the possibility of holding both affirmation and “negation.” In many cases Dionysius claimed that God is “superior,” “beyond,” and “above” beings, or that God “surpasses” being. The following is a typical example of his use of “negation:” He [God − P. R.] is all things since he is the Cause of all things. [...] But he is also superior to them all because he precedes them and is transcendentally above them. Therefore every attribute may be predicated of him and yet he is not any one thing (DN V, 8; cf. I, 5).

It seems that to “negate” for Dionysius is to include something and at the same time to be beyond the “negated” thing. Hence Dionysian “negation” does not exclude affirmation. Somebody might object that if this is so it is not a negation at all. Nevertheless in such a case Dionysius would be charged of misusing some terms but not of inconsistency. I am not however sure if Dionysius really misuse here the word “negation.” Suppose that there are 100,000 dollars in cash on the table (it is not easy to imagine at this time of crisis). Suppose then that somebody asks if there are 10 cents. The answer “Yes” would be obviously right, but somehow misleading. It is in some sense acceptable to say “No.” “No,

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since there are much more then 10 cents on the table.” The word “No” does not stand for a classical negation, but express inadequacy between the supposition of this question on the one side, and the involved states of affairs on the other. The answer “Yes” for this question would imply that the sum on the table is comparable to 10 cents. The same situation obtains with questions such as “Is John an animal?” (“No! He is a man!”), “If Romeo likes Juliet?” (“No! He loves her!”), and so on. It seems therefore that in some context ordinary language allows “negations” which do not imply the logical negation. This “negation,” it might be said, has a positive, not a negative sense. It expresses excess, not a lack. I believe that it is a proper meaning of Areopagite’s “negation.” When he speaks negatively about God, he surely does not speak of emptiness, but of fullness. It might be argued that the answer “No” in all these cases violates H. P. Grice’s conversational Maxim of Quantity, which does not allow us to make contributions more informative than is required [8, p. 45 – 46]. However, I am not sure if it is really unnecessary information in some context that the object of question is something radically more than it was supposed to be. But even if it is not so, the charge of speaking too much is much less troublesome than the accusation of inconsistency. The same ambiguity of negation might be found in some philosophers. For instance a Polish phenomenologist Władysław Stróżewski [19] distinguished two meaning of negation. Besides the negation, which writes off the negated thing, there is a kind of negation which only marks the difference, and the result of the second kind of negation is something positive, not a pure lack. Similar sense is included in famous Hegelian “Aufhebung,” which is usually translated as “sublation” or “negation.” Hegel prizes this term for its ambiguity: We should note the double meaning of the German word Aufheben (to put by or set aside). We mean by it (1) to clear away, or annul: thus, we say, a law or regulation is set aside; (2) to keep, or preserve: in which sense we use it when we say: something is well put by. This double usage of language, which gives to the same word a positive and negative meaning, is not an accident, and gives no ground for reproaching language as a cause of confusion. We should rather recognize in it the speculative spirit of our language rising above the ‘either-or’ of understanding [9, § 96, A].

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That what was “Aufgehoben” does not disappear, but is “preserved,” “sublimated,” and “saved” in more perfect way. What results of sublating of something is something higher than the sublated item. Thus, I conclude that both ordinary and philosophical languages allow for positive negation which does not negate in a classical, negative way. To say that x positively has not Q means that x has Q in some particular, higher way. I shall write “!Q(x)” for “x positively has not Q,” “x has a positive negation of Q.” The meaning of positive negation might be partially fixed by the following axioms: (A1) (A2) (A3)

!Q(x) ⊃ Q(x), ¬(Q(x) ⊃!Q(x)), !!Q(x) ⊃ !Q(x).

From (A1) and (A2) it immediately follows that (12) (13)

¬Q(x) ⊃ ¬!Q(x), ¬(!Q(x) ⊃ ¬Q(x)).

Surely some further clarification of “!” is needed and possible, but all that I need for the moment is expressed in the above statements. In this interpretation NT claims that God has positive negations of all positive properties: (G5)

G(x) ≡ ∀Q (P(Q) ⊃ !Q(x))

This formula corresponds to the (T2). However form (G5) and (A1) it follows that (14)

G(x) ⊃ ∀Q (P(Q) ⊃ Q(x)),

God has all positive properties in a higher way but nevertheless he has them. God (positively) is not good, and is good; God is (positively) not wise, and is wise, and so one. There is no contradiction here. For the first time it is possible to reconcile (T2) and (T1) in one uniform interpretation.

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Moreover, since according to (A3) double positive negation yields single positive negation, it is possible to claim that God has positive negation of all positive negation of positive properties. (T3) has the following interpretation: (15)

G(x) ⊃ ∀Q (P(Q) ⊃ !!Q(x)).

Finally, when we assert that God has a positive negation of some property, we assert that he possesses that property in a higher way. We do not know however exactly what this means. Hence (T4) remains in some sense valid. Although we know that God is wise, we do not know what is for God to be wise. We know only that his wisdom is something more that mundane wisdom. So our ignorance does not affect the fact of being wise, but is restricted to the way in which God possess wisdom. We know that God is Q, we know that he is !Q, but we do not know what does it exactly mean. It is interesting to notice that Positive NT is very close to the Theory of Analogy developed by St. Thomas Aquinas and his followers. Some authors have even suggested that Thomistic theory is a version of NT ([5, p. 236]; [21, p. 118]). The Theory of Analogy concerns with the meaning of theological language. It asserts that the positive predicates applied to God are neither univocal, nor equivocal, but analogical. Univocal words have the same meaning; equivocal words have entirely different meanings (like in the case of homonyms). There are many interpretations of the traditional doctrine of analogical words. Some authors claim that the meaning of words predicated of God is somehow split into univocal and equivocal parts (see attempts of logical analysis of analogy [3], [4, p. 156 – 162], [6]). Other claim that the meaning of terms is the same, but the sense, that is the use, is different [2, p. 65 – 76]. Regardless of proper understanding of the theory, it says that we can use words not only to say what they mean but also to point beyond what we understand them to mean. We ascribe to God some properties without claiming to understand what these properties would be in God. As St. Thomas wrote, we do not know the “mode of signification” of these predicates, since denoted “perfections,” that is positive properties, are in God in “more eminent way than in creatures” ([20, p. I, q. 13, a. 3c]; cf.

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[11, p. 58]). When we say that God is wise we mean that the property we call wisdom in us exists in God in some higher and eminent way ([20, p. I, q. 13, a. 5). Therefore this is the same property, but possessed in a different, infinite mode. We do understand the property, but we do not understand that mode. The same conclusion is drawn in NT, but since Dionysius uses positive negation, it is expressed in a (positive) negative way. When Aquinas says that we know that God is Q but in some eminent mode, Dionysius simply says that God is (positively) not Q. Thus both great thinkers principally agree. 7. Conclusion I developed here three interpretations of NT. All of them express the transcendence of God, but in three different ways. The first one claims that God is fundamentally unknowable; the second one that God has no positive properties and in this sense is not an object; the third that God has all positive properties but in such perfect way, that it is more appropriate to say that he has no these properties at all. All these theories are, as I hope, consistent. Therefore, there is a logic of NT. There are even three such logics. These accounts differ in the way in which they succeed (or not) in interpreting the four theses of NT. Agnostic interpretation takes (T4) as basic, modifies (T2) and (T3) and rejects (T1). Negative interpretation starts with (T2), modifies (T4) and rejects (T1) and (T3). Finally Positive NT takes as basic (T2), agrees with (T3), and slightly modifies (T1) and (T4). So, I conclude that it is the best available interpretation of Dionysian NT. Jozef Wissink [21] has argued that there are two kinds of NT, namely “natural” and “Christian.” The first one appeared in many religious traditions and is motivated by a religious or an epistemological need of purification of theological concepts. It ends with a skeptical result that God is radically transcendent and totally unknowable. Christian NT has a different inspiration, since it is based not on a speculation, but on the revelation of God, which revealed himself as a Hidden One. NT is not in this case a result of skepticism, but the direct consequence of the revelation

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and must be consistent with some other revealed truth. As it seems, only the “positive negative” interpretation of NT fits the Christian model. This is, I believe, another reason to accept it. Acknowledgements I dedicate this paper to the memory of my teacher Professor Jerzy Perzanowski (1943 – 2009), an excellent philosopher and a virtuous man. The very early version of this paper was presented in Polish at a meeting of Student’s of Philosophy Scientific Circle at Jagiellonian University in December 2001. I profited then from discussions with Konrad Banicki. I am indebted to Walter Sisto for his help in brushing up my English. References [1] Aristotle’s Categories and Propositions (De interpretatione), transl. H. G. Apostle, Grinell, The Peripatetic Press, 1980. [2] Barrett SJ, C., The Logic of Mysticism (II) [in:] M. Warner (ed.), Religion and Philosophy (Royal Institute of Philosophy Supplement, vol. 31), Cambridge, Cambridge University Press, 1992, 61 – 69. [3] Bocheński, J. M., On Analogy [in:] A. Menne (ed.), LogicoPhilosophical Studies, Dordrecht, D. Reidel, 1962, 96 – 117. [4] Bocheński OP, J. M., The Logic of Religion, New York, New York University Press, 1965. [5] Davies, B., Aquinas on What God is Not [in:] B. Davies (ed.), Thomas Aquinas. Contemporary Philosophical Perspectives, Oxford, Oxford University Press, 2002, 227 – 242. [6] Durrant, M., The Meaning of ‘God’ (I) [in:] M. Warner (ed.), Religion and Philosophy (Royal Institute of Philosophy Supplement, vol. 31), Cambridge, Cambridge University Press, 1992, 71 – 84. [7] Gellman, J. I., The Meta-Philosophy of Religious Language, Noûs, 11, 1971, 151 – 161. [8] Grice, H. P., Logic and Conversation [in:] P. Cole and J. L. Morgan (ed.), Syntax and Semantics, vol. 3, Speech Acts, New York, Academic Press, 1975, 41 – 58.

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[9] Hegel, G. W. F., Logic. Being Part One of the Encyclopaedia of the Philosophical Sciences, transl. W. Wallace, Oxford, Oxford University Press, 1975. [10] Jones, J. N., Sculpting God: The Logic of Dionysian Negative Theology, Harvard Theological Review, 89, 1996, 355 – 371. [11] McCabe OP, H., The Logic of Mysticism (I) [in:] M. Warner (ed.), Religion and Philosophy (Royal Institute of Philosophy Supplement, vol. 31), Cambridge, Cambridge University Press, 1992, 45 – 59. [12] Nicholas Cusanus, Of Learned Ignorance, transl. G. Heron, London, Routledge & Kegan Paul, 1954. [13] Obolevitch, T., Platonizm w teologii wschodniochrześcijańskiej [Platonism in the East Christian Theology] [in:] W. Kowalski, T. Obolevitch (ed.), Metafizyka i religia [Metaphysics and Religion], Kraków, Wydawnictwo Naukowe PAT, 2006, 157 – 180. (In Polish). [14] Perzanowski, J., The Way of Truth [in:] R. Poli, P. Simons (ed.), Formal Ontology, Dordrecht, Kluwer Academic Publishers, 1996, 61 – 130. [15] Perzanowski, J., Ontological Arguments II: Cartesian and Leibnizian, [in:] H. Burkhardt, B. Smith (ed.), Handbook of Metaphysics and Ontology, vol. II, München, Philosophia Verlag, 1991, 625 – 633. [16] Pseudo-Dionysius: The Complete Work, trans. C. Luibheid, New York, Paulist Press, 1987. [17] Ross, J. F., Analogy as a Rule of Meaning for Religious Language, International Philosophical Quarterly, 1, 1961, 468 – 502. [18] Słupecki, J., St. Leśniewski's Calculus of Names, Studia Logica, 3, 1955. [19] Stróżewski, W., Z problematyki negacji [The Problem of Negation] [in:] W. Stróżewski, Istnienie i sens [Existence and Sense], Kraków, Znak, 1994, 373 – 395. (In Polish). [20] Thomas Aquinas, Summa Theologiae, transl. T. Gilby and others, London and New York, Blackfriars, 1964 – 1981, 60 vols. [21] Wissink, J., Two Forms of Negative Theology Explained Using Thomas Aquinas [in:] I. N. Bulhof, L. ten Kate (ed.), Flight of the Gods. Philosophical Perspectives on Negative Theology, Fordham, Fordham University Press, 2000, 100 – 120.

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[22] Zinov’ev, A. A., Foundations of the Logical Theory of Scientific Knowledge (Complex Logic), transl. T. J. Blakeley, Dordrecht, D. Reidel, 1973.

REASONING ABOUT THE TRINITY: A MODERN FORMALIZATION OF A MEDIEVAL SYSTEM OF TRINITARIAN LOGIC Sara L. Uckelman Institute for Logic, Language, and Computation Universiteit van Amsterdam [email protected] In Christian discourse, paralogisms arise when one blindly applies classical logic to the traditional definition of the trinity. Some people conclude, on the basis of these paralogisms, that logic cannot be coherently applied to the trinity. Others conclude that it can, but that the right logic is not classical logic, but an extension of it. An anonymous treatise from the late Middle Ages addresses the problems of paralogisms by introducing a semi-formal theory of predication and syllogistic reasoning which can be applied to both the trinity and to other subject matter. In this paper we formalize the theory presented in the medieval text, providing the contemporary philosopher and logician with a sound logic for reasoning about the trinity.

1. Introduction For many people, the phrase “logic in religious discourse” will bring to mind the paralogisms of the trinity, the various syllogistic arguments which have apparently true premises and valid structure, but which result in an apparently false conclusion. For example:

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The Father is God. God is the Son. ——————————————– Therefore, the Father is the Son. For both believers and non-believers, paralogisms like this pose a significant problem for the rational interpretation of Christian doctrine of the trinity. Many people draw the conclusion that the trinity is one part of religious discourse where there is no logic, where ordinary rules of inference simply do not apply. Others disagree and believe that the paralogisms are simply that — paralogisms — and that once the correct logic underlying trinitarian reasoning is isolated, the paralogisms will no longer appear to be valid. People in the latter camp can be divided into two types, those that believe that logic can be applied to the nature of God, but think that the appropriate logic is not the same logic as that used to reason about non-divine things, and those that believe that one and the same logic can be used for both reasoning about God and about non-divine things. An example of someone in the first camp is the 14th-century French logician Jean Buridan, who says in Book III, Part I, ch. 4 of his book on consequences: But it should be carefully noted that these rules do not hold in the case of God, [the terms for Whom] supposit for a simple thing one and triune at the same time. Whence although the Father is the same as the simple God and the Son is the same as the simple God, the Father is nevertheless not the Son; and although the same Father is God and not the Son, it is false nevertheless that the Son is not the same as God [3, 3.4.8, p. 265].1

More interesting are those in the other camp, and believe that not only can logic be applied to the trinity, but that it is the very same logic that we use in ordinary reasoning. This is the view of the anonymous author of a logical treatise De modo predicandi ac syllogizandi (DMPS) contained in ms. Munich, Bayerische Staatsbibliothek, lat. 17290, ff. 136r – 145v and edited in [7]2 , which discusses Sed diligenter aduertendum est quod hae regulae non tenent in terminis diuinis, qui supponunt pro re una simplicissima simul et trina. Vnde licet deo simplici sit idem pater et eidem deo sit idem filius, tamen filius non est pater; et licet idem pater sit deus et non filius, tamen falsum est quod filius deo non sit idem [2, p. 85]. 2 Few details about the authorship or localization of the text are known. Because Thomas Aquinas is referred to as a saint, the text was almost certainly written after his canonization 1

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modes of predication and syllogistic reasoning in the trinity. In this text, the author argues that the same logic can be used to reason about both divine and created things by making a distinction between different modes of being and modes of predication, distinctions which collapse when we talk about created things. He supports this conclusion by presenting a syllogistic logic which is adequate for reasoning about the trinity and from which ordinary, Aristotelian syllogistics can be recovered. (Interestingly, this author is often classed with people who fall in the former category, that is, the category of people who believe that logic can be applied to God but don’t accept a single logic (see, e.g., [4, p. 86]).) In this paper we give a formal reconstruction of the trinitarian syllogistic theory presented in the anonymous text, and show how it can be used to explain why traditional paralogisms appear to be valid but are in fact invalid. Chapter 7 of [10] is an expanded version of this paper. The technical details omitted from this paper can be found there. 2. The text The text can be divided into three main parts, each of which builds upon the previous one: 1. A discussion of modes of being. 2. A discussion of modes of predication. 3. A discussion of syllogistic reasoning. According to the author, the first of these is properly within the scope of philosophy (or, when it concerns the trinity, theology); the latter two make up the scope of logic. in 1323. On the basis of other textual and conceptual references, a composition date of the late 14th or early 15th century can be postulated, possibly in a Germanic setting. For further discussion of this, see Maier`u’s introduction [7, pp. 251, 255 – 257]. This manuscript is the only known manuscript containing this text. The text is, unfortunately, incomplete; in DMPS, par. 106, p. 286 an objection is introduced, and the text breaks off directly after, leaving the objection unaddressed. A translation of this text into English is given in Appendix C of [10]. Chapter 7 of [10] is an expanded version of present paper, and the technical details omitted from this paper can be found there. All Latin references are taken from [7], and the English translations from [10].

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The modes of being which can be found in the trinity are discussed in DMPS, paragraphs 6 – 24, p. 266 – 269; in DMPS, par. 25, p. 269, the author notes that, modes of being having been spoken of, we can now move to a discussion of modes of predication and syllogistic reasoning, for, as he says, “In truth logic, in so far as it suffices for the present purpose, consists in modes of predication and syllogistic reasoning.”3 Because predications are predications in some mode of being, before logic proper is discussed it is first required that the philosophical issues of modes of being be covered. Speaking anachronistically, we can say that the first 24 paragraphs were setting up the semantics of our system, explaining the underlying factors which will make certain predications true or false, and that starting in DMPS, par. 25 we are now being given syntax. Facts about generating modes of predication from the modes of being are discussed in DMPS, par. 25 – 32, pp. 269 – 270, and the discussion of syllogisms, which makes up the rest of the text, begins in DMPS, par. 33, p. 271. In presenting his syllogistic system, our author uses of two typically medieval developments in logic: supposition theory and expository syllogisms. From standardly accepted facts about the supposition of terms and the reduction of certain classes of general syllogisms to expository syllogisms, the author is able to isolate a class of divine syllogisms which are valid, and to justify their validity. Rules governing the validity of categorical syllogisms with mixed premises are given in DMPS, par. 51, p. 275 (for affirmative syllogisms) and DMPS, par. 57 – 60, p. 277 (for negative syllogisms). After a discussion of how these rules relate to expository syllogisms, the author summarizes the class of valid syllogisms which have two positive premises in DMPS, par. 93 – 96, pp. 284 – 295, and the class of valid syllogisms which have a negative premise in DMPS, par. 98 – 105, pp. 285 – 286. Unfortunately, DMPS, par. 106, p. 286 provides a counterexample to the system which has just been outlined, and as the text breaks off we are left with no indication as to how the author would have resolved this problem. The central argument of the text is that in order to properly reason about the nature of the trinity, we must distinguish three different modes of being and predication. When the author discusses the different modes of being of an object (divine or created), it should be understood that what he is speaking of is more properly called modes of identity, that is, different ways that two objects can be 3

[l]ogica vero, quantum ad propositum sufficit, in modis predicandi ac sylogizandi consistit.

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identical. He never speaks of an object simply existing in one of these modes of being, but rather of one object being the same as another object in one of these modes of being. The author distinguishes three modes of being, that is, three ways in which two things can be identical with each other: • Essentially (Essencialiter) • Personally or Identically (Personaliter/Ydemptice) • Formally or Properly (Formaliter/Proprie) Roughly speaking, two things are essentially the same if they share the same essence; but things which are essentially the same may still yet differ in the accidental properties that they share or in the definitions which define them. This distinction of types of identity can be found as early as Abelard. (For further discussion of Abelard’s views, see [5], especially p. 242.) In his Theologica ‘scholarium’ II, 95 – 99, Abelard distinguishes three ways that things can be the same [1, pp. 454 – 456]: • Essentially or in number (Essentialiter siue numero) • Properly or by definition (Proprietate seu diffinitione) • In likeness (Similitudine) Abelard’s three ways of being the same correspond to the three modes of being in the anonymous text we’re considering.4 Abelard’s essential identity is also called idem quod sameness, and Knuuttila glosses it as “[t]he sameness pertaining to the subject and predicate of a singular proposition in the sense that there is a third of which both are said.” This is distinguished from idem qui sameness, glossed as “the sameness between the meanings of terms.” This idem qui sameness covers both personal and formal (or proper) identity [6, p. 193]. Basically, if two things are essentially identical, then they share the same essence. If they are personally identical, then they share the same properties and definitions. Finally, if two things are formally identical, then they share sufficient similarity that they can be placed under the same genus, or form. (In DMPS, par. 32, p. 270 the author says that there “is a certain mode of being in which some things are formally the same, on the condition that in whatever way one As Knuuttila notes, “The originally Abelardian distinction between intensional (personal) and extensional (essential) identity was widely employed in later medieval Trinitarian theology and influenced late medieval logic” [6, p. 195]. 4

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is the other is also in the same way” (est quidam modus essendi quo aliqua sunt formaliter idem, ita quod in quocumque est unum in eodem est et alterum). It is not clear whether this condition is a sufficient or necessary condition for two things being formally identical.) The author’s system of divine syllogistics is based on distinguishing these three modes of being. In created beings, these distinctions collapse, which explains why ordinary, Aristotelian syllogistics works as well as it does, and why for so long no one realized that there was more to the story than that. (At the beginning of the text, in DMPS, par. 1, p. 265, the author apologizes for Aristotle, noting that because Aristotle’s focus was on the mode of being as it is found in created things, and hence his syllogistic system, which is based on predications expressing that mode of being, does not accommodate reasoning about non-created, i.e., divine things, we cannot fault him for not recognizing that his system could be extended to accommodate reasoning about the divine nature.) So what exactly do we mean when we speak of the trinity, or the divine nature, in the context of discussing this anonymous text? The author makes as few controversial assumptions about the nature of the trinity as possible. In DMPS, par. 4, pp. 265 – 266, the author says that: The mode of being in divinity is that three persons are one most simple essence and likewise the most simple essence [is] three persons and each of them.5

This view is essentially a compressed version of the Athanasian Creed, adopted in the 6th century: We worship one God in Trinity, and Trinity in Unity; neither confounding the Persons: nor dividing the Substance. . . But the Godhead of the Father, of the Son, and of the Holy Ghost, is all one: the Glory equal, the Majesty coeternal. . . The Father eternal: the Son eternal: and the Holy Ghost eternal. And yet they are not three eternals: but one eternal.6 Modus essendi in divinis est quod tres persone sunt una essencia simplicissima et eadem simplicissima essencia tres persone et quelibet earum (DMPS, par. 4, pp. 265 – 266). 6 Unum Deum in Trinitate, et Trinitatem in Unitate veneremur; neque confundentes personas: neque substantiam separantes. . . Sed Patris et Filii et Spiritus Sancti una est divinitas: aequalis gloria, coaeterna majestas. . . Aeternus Pater: aeternus Filius: aeternus [et] Spiritus Sanctus. Et tamen non tres aeterni: sed unus aeternus. 5

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The examples that the author uses when discussing the trinity mention eight different persons or properties of the trinity, for which we introduce notation now: E := Essence Su := Substance F := Father P := Fatherhood/Paternity S := Son Wi := Wisdom HS := Holy spirit C := Charity/Love P, Wi, and C are called by the author “personal properties” (DMPS, par. 23 – 24, pp. 268 – 269), following Peter Lombard. These properties are the distinguishing properties of the persons of the Father, the Son, and the Holy Spirit, respectively. (The essence also has essential attributes, namely sapiencia
et essencialiter dicte, iusticia, bonitas, etc. (DMPS, par. 19, p. 268). But we need not introduce new terms for these essential attributes, since they are all formally identical with the essence (DMPS, par. 19, 32, pp. 268, 270 – 271), and the author makes no further mention of them.) The comparison of the three modes of being used in this text with Abelard’s three modes of identity gives us some idea of what is meant when it is said that two objects are personally the same, or that they are formally distinct, but it does not give us information about the nature of the relationships ‘being essentially the same as,’ ‘being personally the same as,’ and ‘being formally the same as.’ No clear statement of the properties of these relations is given in the text, but we can extract some of them by looking at the examples of identities and distinctions that the author makes in DMPS, par. 5 – 24, pp. 266 – 269. A summary of these examples is given in Table 1 (note that some of the cells are not wholly filled in because the text is underspecific), where we let =e , =p , and =f be the relations of essentially identity, personal identity, and formal identity, respectively. Since essential identity is an equivalence class of which all parts of the trinity are members, we omit it from the table since it would appear in every cell. 3. The formal system The formal system we present here was developed in order to be able to model reasoning within a particular natural language, namely medieval Latin as it was used by logicians, specifically by the logician who is the author of the text under

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E F S HS P Wi C Su

E

F

S

HS

P

Wi

C

Su

=f =f ,=p =f ,=p =f ,=p =f =f =f =f

=f ,=p =p ,=f =p ,=f =p ,=f =p ,=f =f =f =f

=f ,=p =p ,=f =p ,=f =p ,=f =f =p ,=f =f =f

=f ,=p =p ,=f =p ,=f =p ,=f =f =f =p ,=f =f

 f = =p ,=f =f =f =p ,=f =p ,=f =p ,=f =f

=f  f = =p ,=f =f =p ,=f =p ,=f =p ,=f =f

=f  f =  f = =p ,=f =p ,=f =p ,=f =p ,=f =f

=f  f =  f =  f =  f =  f =  f = =f

Table 1: Formal and personal identity in the trinity

consideration. This is the cause of certain otherwise non-standard modeling choices that we make. In particular, we have designed our system to deal with ambiguous natural language statements such as Homo est animal. Because Latin does not have an indefinite or definite article, this sentence is ambiguous between the reading omnis homo est animal and quidam homo est animal. When the sentence is literally translated into English, this ambiguity manifests itself in questionable grammar: “Man is animal.” A more natural translation would add definite or indefinite articles or quantifiers, e.g., ‘the essence is the father’ for essencia est pater, which adds two definite articles which are not present in the Latin. Another way that features of our formal model will be determined by features of Latin is in the use of context-dependent indexicals like hoc (‘this’). When we say things such as haec tabula est viridis, we are saying something more than ‘some table is green’ but something less than ‘all tables are green.’ We will introduce specific operators into our language to be able to deal with issues surrounding the use of indexicals in this manner. 3.1. Language and models We use the language Ltrin consisting of a set of terms T; the relations =e , =p , =f and their negations =e , =p , =f ; the functions es and fs ; the quantifiers A, E, and !; and two punctuation symbols, [ and ]. T contains all of E, Su, F, S, HS, P, Wi, and C, and potentially other terms, e.g., ‘man,’ ‘cat,’ ‘Socrates.’ We use t as a variable ranging over T, and we use =∗ as a meta-variable over =e , =p ,

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=f when we need to make statements about all three relations. !t is to be read ‘this t’ (English) or hoc t (Latin). This operator will be used in formalizing ambiguous natural language sentences such as the ones just discussed. Traditional Aristotelian syllogistic logic is a term logic, not a predicate or propositional logic. This means that the formal system we develop will be neither a predicate nor a propositional logic, though, as we’ll see below, we will use predicate logic as a meta-logic when giving the truth conditions for formulas in models. Instead we will develop a logic whose basic constituent is the categorical proposition, though we will go a step beyond traditional medieval syllogistics by allowing boolean combinations of these categorical propositions. We begin by giving a definition of the set of basic terms and the set of quantified terms in our language: quant Definition 1 (Terms). The set Ttrin = Tbasic trin ∪ Ttrin is the set of terms of Ltrin where

• Tbasic trin is the set of basic terms of Ltrin , defined recursively as follows: – If t ∈ T, then t, tes , tfs ∈ Tbasic trin . basic – If t ∈ Tbasic trin , then [t]=∗ ∈ Ttrin .

– Nothing else is in Tbasic trin . We call terms of the form [t]=∗ equivalence terms. • Tquant trin is the set of quantified terms of Ltrin defined as follows: basic basic {At : t ∈ Tbasic trin } ∪ {Et : t ∈ Ttrin } ∪ {!t : t ∈ Ttrin } − If t ∈ Tquant trin then we let t be the result of removing the quantifiers from the front of t.

Definition 2 (Categorical Propositions). The set CATtrin of categorical propositions of Ltrin is defined as follows: • If t, t ∈ Ttrin , then t =e t , t =p t , t =f t ∈ CATtrin. We call categorical propositions of this type affirmative. • If t, t ∈ Ttrin , then t =e t , t =p t , t =f t ∈ CATtrin. We call categorical propositions of this type negative. • Nothing else is in CATtrin .

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Note that all categorical propositions are of the form Qt =∗ Q t for terms t, t and quantifiers (possibly null) Q, Q . If φ is a categorical proposition, then we indicate the type of identity in φ by φ∗ , and we call the term on the lefthand side of the identity sign the ‘subject’ and the term on the right-hand side the ‘predicate.’ Definition 3 (WFFs). The set WFFtrin of well-formed formulas of Ltrin is defined recursively: • If φ ∈ CATtrin , then φ ∈ WFFtrin. • If φ ∈ WFFtrin, then ¬φ ∈ WFFtrin . • If φ, ψ ∈ WFFtrin , then φ ∧ ψ, φ ∨ ψ, φ → ψ ∈ WFFtrin . • Nothing else is in WFFtrin . In order to prove some of the theorems in section 3.2, we need to isolate a special class of terms called divine terms; we’ll use the distinction between divine and created (non-divine) terms in our proof. Definition 4 (Divine Terms). The set Tdiv ⊆ Ttrin of divine terms of Ltrin is the set of all terms t ∈ Ttrin such that t only contains E, Su, F, S, HS, P, Wi, C and nothing else. We define the sets CATdiv and WFFdiv from Definitions 4, 2, and 3 by replacing trin with div throughout. Formulas gain meaning when they are interpreted in models. Definition 5 (Trinitarian Models). A structure ˙ e, = ˙ p, = ˙ f , es˙ , fs˙  Mtrin =
O, I, {t˙ : t ∈ T}, = is a trinitarian model iff: ˙ Su, ˙ F, ˙ S, ˙ HS, ˙ P, ˙ Wi, ˙ C˙ ∈ O. We use 1. O is a set of objects such that E, o, x, y, z, etc., as meta-variables ranging over O. 2. I : T → 2O associates a set of objects with each term of T, such that ˙ I(Su) = {Su}, ˙ ˙ I(S) = {S}, ˙ I(HS) = {HS}, ˙ I(E) = {E}, I(F) = {F}, ˙ I(Wi) = {Wi}, ˙ ˙ I(P) = {P}, and I(C) = {C}. I can be extended to ˙ =∗ ) = {x ∈ O : there is a y ∈ I  which covers equivalence terms: I  ([t] I(t) and x =∗ y}.

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3. = ˙ e is a partial equivalence relation on O such that if o ∈ / I(t) for all   /= ˙ e (that is, it is an equivalence t ∈ Tdiv , then for all o ∈ O,
o, o  ∈ relation on the interpretation of divine terms). 4. = ˙ p is a partial symmetric relation on O satisfying the conditions in Ta/ ble 1 such that if o ∈ / I(t) for all t ∈ Tdiv , then for all o ∈ O,
o, o  ∈ = ˙ p. 5. = ˙ f is a equivalence relation on O satisfying the conditions in Table 1. ˙ = ) then oes˙ = E; ˙ 6. es˙ , fs˙ are partial unary functions such that if o ∈ I  ([E] e ˙ if o ∈ [S] ˙ = then o ˙ = S; ˙ if o ∈ [HS] ˙ = then ˙ = then o ˙ = F; if o ∈ [F] f f f fs fs ˙ o ˙ = HS; and undefined otherwise. fs

Conditions 3, 4, and 5 of Definition 5 capture the fact that when we are reasoning about non-divine things, we can only make formal predications. When explaining why essential and identical predications do not show up in Aristotelian syllogistics, our author notes that though the terminists and the realists may disagree about whether there are only formal identities between created objects, or whether there are also personal identities, nevertheless they agree that all predications are predications of formal identity: And because in creation all predications are formal, because according to common opinion of the terminists all the things which are the same in creation are formally the same, therefore the mode of syllogizing through propositions concerning identical predications is not necessary in creation. However, according to the mode of the realists, according to which not all things in creation which are the same are formally [the same], still all predications are formal, which is clear because what is not formally the same according to the realists, according to they themselves must necessarily be denied of each other if indeed they are identically the same.7

The author does not specify whether, in the case of created objects, we are able to state non-identities of the essential and personal type (that is, whether we can Et quia in creaturis omnes predicaciones sunt formales, quia iuxta opinionem communem terministarum omnia que sunt idem in creaturis sunt formaliter idem, ideo non fuit necesse in creaturis modus sylogizandi per proposiciones de predicacione ydemptica. Secundum modum autem realistarum, secundum quem non omnia in creaturis que sunt idem sunt formaliter
idem, adhuc omnes predicaciones sunt formales, quod patet, quia que non sunt formaliter idem secundum realistas, secundum ipsos necessario negantur de semetipsis si eciam ydemptice sint idem (DMPS, par. 39 – 40, pp. 272 – 273). 7

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say of two created objects o and o that o =e o , and so forth). Because he does not say that it is possible to make statements of non-identities of these types when dealing with created objects (only that we cannot make predications of identities of these types), we have opted to not build into the system the ability to express these negated identities. We now give the truth conditions of the members of WFFtrin in a trinitarian model. Boolean combinations of categorical propositions are as expected: Definition 6 (Truth Conditions of Boolean Formulas). M
¬φ M
φ∧ψ M
φ∨ψ M
φ→ψ

iff iff iff iff

M
φ M
φ and M
ψ M
φ or M
ψ M
¬φ or M
ψ

For the categorical statements, we correlate the quantifiers of Ltrin with quantifiers in ordinary mathematical logic via an interpretation function int. Two of the quantifiers are standard — int(A) = ∀ and int(E) = ∃. As we noted earlier, indexical pronouns like ‘hoc,’ which we formalize with !, indicate something more than existence but something less than universality. Pronouns like ‘hoc’ are essentially context-dependent choice functions that, given a term, will pick out an appropriate witness for that term, given the context. We capture these two facts by interpreting ! with a generalized quantifier (cf. [8, 11]). For a term t, we indicate such a context-dependent choice function as χ! (t), which means 
we can define int(!) as {χ! (t)} for appropriate t. This leaves us with the empty quantifier, which shows up in formalizations of Latin sentences such as essencia est pater and homo est animal, which, as we noted above, are essentially ambiguous. Our author does not say how these sentences should be interpreted, but, given how his discussion of modes of being mirrors Abelard’s three ways of being identical, it’s reasonable that he would also subscribe to Abelard’s view of predication. Knuuttila summarizes Abelard’s view thus: In his Logica Ingredientibus Abelard argues that the simple affirmative statement ‘A human being is white’ [homo est albus] should be analysed as claiming that that which is a human being is the same as that which is white (idem quod est homo esse id quod album est) [6, p. 192].

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It is natural to read ‘that which is a human being’ universally, and ‘that which is white’ particularly. Thus, for statements of identity, involving =∗ , we stipulate that the int of the empty quantifier of a subject is ∀, and the int of the empty quantifier of a predicate is ∃. For statements of non-identity, involving =∗ , we stipulate that the int of the empty quantifier on either side of =∗ is ∀. The difference in how the empty quantifier is treated when it appears in a predicate is a result of the distributive force of negation; see Definition 8 below. Given these preliminaries, we can now give a uniform truth condition for categorical sentences: Definition 7 (Truth Conditions of Categorical Formulas). Let Q, Q be (perhaps empty) quantifiers, and t, t ∈ T. Then,   M
Qt =∗ Q t iff int(Q)x ∈ I(t) int(Q )y ∈ I(t ) (
x, y ∈ = ˙ ∗) We will see examples of these conditions in the next section when we discuss the formalization of natural language sentences concerning the trinity. Note that defining the truth conditions for the empty quantifiers in this way automatically deals with the issue of existential import, by allowing the inference, regularly accepted in the Middle Ages, from omnis homo est mortalis to quidam homo est mortalis, but not automatically allowing the inference, which is not so readily accepted by the medieval logicians (cf. [9, §1.2]), from nullus homo est immortalis to quidam homo non est immortalis, because M
At =f t when both I(t) = ∅ and I(t ) = ∅. 3.2. Properties of the system In this section we look at how the model presented in the previous section can be used to model the syllogistic theory presented in the anonymous text. First, note that it doesn’t really make sense to talk of axioms in the context of a syllogistic logic. This is because what is valid in a syllogistic logic is not sentences, but arguments, which means that the ‘axioms’ are simply rules for moving from two premises to a conclusion. In ordinary, non-divine, syllogistics, these rules are the perfect syllogisms, Barbara, Celarent, Darii, and Ferio. That / Tdiv : is, for t, t , t ∈

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Rule 1. Barbara: Celarent: Darii: Ferio:

If If If If

M
At M
At M
At M
At

=f =f =f  f =

t t t t

and and and and

M
At M
At M
Et M
Et

=f =f =f =f

t , t , t , t ,

then then then then

M
At M
At M
Et M
Et

=f =f =f  f =

t t t t

The validity of the affirmative syllogisms, Barbara and Darii, are governed by the rule called by the dici de omni by the medieval logicians, and the validity of the negative syllogisms, Celarent and Ferio, by the rule dici de nullo: Rule 2 (Dici de omni ). Whenever some predicate is said of some distributed subject, then of whatever is said to be of that distributed subject, of the same thing indeed it is said to be of that predicate.8 Rule 3 (Dici de nullo). Whenever some predicate is denied of some distributed subject, then of whatever is said to be of that distributed subject, of the same thing indeed it is denied to be of that predicate.9 The admissibility of the dici de omni et de nullo, and consequently of the four perfect syllogisms, follows straightforwardly from the fact that =f is an equivalence relation: Proof. Barbara Assume M
At =f t and M
At =f t . Then by Definition 7, the following two formulas hold: ˙ f )) ∀x ∈ I(t )(∃y ∈ I(t)(
x, y ∈ =

(1)

∀z ∈ I(t)(∃w ∈ I(t )(
z, w ∈ = ˙ f ))

(2)

Take arbitrary x ∈ I(t ). From (2) it follows that there is a y ∈ I(t ) such that
x, t ∈ = ˙ f . From (1), we know that there is some z ∈ I(t) such that
y, z ∈ = ˙ f . Since =f is transitive, we can conclude that
x, z ∈ = ˙ f . Since x was arbitrary, we have shown that the following holds: ˙ f )) (3) ∀x ∈ I(t )(∃z ∈ I(t)(
x, z ∈ = Quandocumque aliquod predicatum dicitur de aliquo subiecto distributo, tunc de quocumque dicitur tale subiectum distributum de eodem eciam dicitur tale predicatum (DMPS, par. 36, p. 272). 9 This rule is never explicitly stated by the author, but it would have been well-known to his audience. 8

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and hence that M
At =f t. Celarent Assume M
At =f t and M
At =f t . Then by Definition 7, (1) for every x ∈ I(t ) and y ∈ I(t),
x, y ∈ / = ˙ f , and (2) for every ˙ f . Take arbitrary z ∈ I(t) there is a w ∈ I(t ) such that
z, w ∈ = ˙ f. x ∈ I(t). By (2) there is some y ∈ I(t ) such that
x, y ∈ = By (1), for all z ∈ I(t),
y, z ∈ / = ˙ f . Now, suppose that there is a w ∈ I(t) such that
x, w ∈ = ˙ f . Since
x, w ∈ = ˙ f and
x, y ∈ = ˙ f , by ˙ f , which transitivity and symmetry of =f , this means that
y, w ∈ =  is a contradiction. Since x ∈ I(t ) was arbitrary, we can conclude that the following holds: /= ˙ f )) ∀x ∈ I(t)(∀y ∈ I(t)(
x, y ∈

(4)

and hence M
At =f t. Darii Assume M
At =f t and M
Et =f t . Then by Definition 7, ˙ f , and (1) for every x ∈ I(t ) there is a y ∈ I(t) such that
x, y ∈ = (2) there is a zˆ ∈ I(t) and w ∈ I(t ) such that
ˆ z , w ∈ = ˙ f . (1) and (2) together give immediately that there is a y ∈ I(t) such that
ˆ z , y ∈ = ˙ f , and hence there exists a z ∈ I(t ) and a y ∈ I(t) such that
z, y ∈ = ˙ f , which is the same as saying that M
Et =f t. Ferio Assume M
At =f t and M
Et =f t . Then by Definition 7, (1) for every x ∈ I(t ) and y ∈ I(t),
x, y ∈ /= ˙ f , and (2) there exists ˙ f . Suppose that there is z ∈ I(t) and w ∈ I(t ) such that
z, w ∈ = a y ∈ I(t) such that
z, y ∈ = ˙ f . Then by symmetry and transitivity, we would have
w, z ∈ = ˙ f and hence
w, y ∈ = ˙ f , which violates (1), and hence M
Et =f t.

A corollary of this is that Rules 2 and 3 are both sound. We are left with the cases where the terms do fall in Tdiv . The admissibility of the essential analog of Rule 1 follows immediately from the proof of the admissibility of that same rule, by substitution of =e for all occurrences of =f . For the other cases, the standard rule of validity for affirmative syllogisms only holds when the propositions in the premises and the conclusion are all of the same type. When they are not, we must use the following rules:

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Rule 4 (Dici de omni for mixed affirmative syllogisms). • Whenever some predicate is said formally of some distributed subject, then of whatever that subject is predicated identically, of the same that predicate is predicated identically.10 • Whenever some predicate is predicated identically of some distributed subject, then of whatever that subject is predicated formally, of the same that predicate is predicated identically.11 For mixed negative syllogisms — that is, ones with at least one negative premise — our rule is split into four parts: Rule 5 (Dici de nullo for mixed negative syllogisms). • When some predicate is formally denied of some distributed subject, then it is not necessary that of whatever that subject is predicated identically that of the same thing that predicate is denied identically or formally.12 • Whenever some predicate is denied identically of some distributed subject, then it is not necessary, if that subject is predicated identically of some term, that of the same that predicate is denied identically.13 • If some predicate is denied formally, that is in formal predication, of a distributed subject, of whatever that distributed subject is formally predicated, of the same that predicate is denied in formal predication.14 • Whenever some predicate is denied identically of some distributed subject, then of whatever that subject is said formally, of the same that Quandocumque aliquod predicatum dicitur formaliter de aliquo subiecto distributo, tunc de quocumque predicabitur tale subiectum ydemptice, de eodem predicabitur et tale predicatum ydemptice (DMPS, par. 51a, p. 275). 11 Quandocumque aliquod predicatum predicatur ydemptice de aliquo subiecto distributo, tunc de quocumque predicabitur tale subiectum formaliter, de eodem predicabitur tale predicatum ydemptice (DMPS, par. 51b, p. 275). 12 Quando aliquod predicatum negatur formaliter de aliquo subiecto distributo, tunc non oportet quod de quocumque predicatur ydemptice tale subiectum, quod de eodem negatur ydemptice vel formaliter tale predicatum (DMPS, par. 57, p. 277). 13 Quando aliquod predicatum negatur ydemptice de aliquo subiecto distributo, tunc non oportet, si tale subiectum predicatur ydemptice de aliquo termino, quod de eodem negatur ydemptice tale predicatum (DMPS, par. 58, p. 277). 14 Si aliquod predicatum negatur formaliter, idest in predicacione formali, de subiecto distributo, de quocumque predicatur formaliter tale subiectum distributum, de eodem negatur in predicacione formali tale predicatum (DMPS, par. 59, p. 277). 10

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predicate is denied identically.15 To formalize these, we define the notion of the distribution of a term within a formula: Definition 8 (Distribution). A term t is in the scope of ∀ iff one of the following holds:  1. t ∈ Tquant trin and is of the form At .

2. t ∈ / Tquant trin and is a subject. 3. t ∈ / Tquant trin and is a predicate of a negative categorical. If t is in the scope of ∀ in a categorical proposition φ, then we say that t is distributed in φ. With this definition, we can give the following formal statements of Rules 4 and 5. Rule 6 (Dici de omni for mixed affirmative syllogisms). If t and t are the terms of φ and t is distributed in φ, and Q is any quantifier, then • If φ = φf and M
φ, then if M
Qt =p t, then M
Qt =p t . • If φ = φp and M
φ, then if M
Qt =f t, then M
Qt =p t . Proving the admissibility of this rule is straightforward: Proof. Assume φ = φf , M
φ, and M
Qt =p t. Since t is distributed in φ and φ is affirmative, we know that φ is either of the form At =f Q t or t =f Q t , for some possibly empty quantifier Q . Looking at Table 1, the only formal identities (other than those which fall out of the reflexivity of =f ) are between the persons and their personal properties, and since the persons are personally identical with both themselves and their personal properties, it follows that M
Qt =p t . The other case follows similarly. Quandocumque aliquod predicatum negatur ydemptice de aliquo subiecto distributo, tunc de quocumque dicitur tale subiectum formaliter, de eodem negatur tale predicatum ydemptice (DMPS, par. 60, p. 277). 15

233

In Rule 1, there are only two syllogistic forms which have only affirmative premises, Barbara and Darii. For both of these, there are four possible ways to form a divine syllogism: either both premises are formal, both are personal, the major is personal and the minor formal, or the major is formal and the minor personal (DMPS, par. 54, p. 276). In the first case, the syllogism is valid, because: Secondly I say that if some predicate is said formally of a distributed subject, then of whatever thing that distributed subject is said formally, of the same indeed that predicate is said formally.16

Which is to say that the traditional dici de omni remains valid when considering categorical propositions with divine terms, not just ones containing only created terms. In the second case, the syllogism is not valid, because: When some predicate is predicated identically of a distributed subject, and if then that [subject] is said identically of some third term, then it is not necessary that that predicate indeed may be said of the same third term.17

The third and fourth cases are covered by Rule 6. Now for the negative syllogisms, Celarent and Ferio. Again we have four cases — the major premise is formal and the minor personal, the major premise is personal and the minor formal, both are personal, or both are formal. All four are expressed explicitly in the rule: Rule 7 (Dici de nullo for mixed negative syllogisms). If t is a distributed subject in φ and Q is any quantifier, then 1. If φ = φf and M
φ, then if M
Qt =p t, then neither M
Qt =p t nor M
Qt =f t follows necessarily. Secundo dico quod si aliquod predicatum dicitur formaliter de subiecto distributo, tunc de quocumque dicitur formaliter tale subiectum distributum de eodem eciam dicitur formaliter tale predicatum (DMPS, par. 53, p. 276). 17 Dico igitur primo. . . quod quando aliquod predicatum predicatur ydemptice de subiecto distributo, et si tunc tale dicitur ydemptice de aliquo tercio termino, tunc non oportet quod tale predicatum eciam dicatur ydemptice de eodem tercia termino (DMPS, par. 52, p. 275 – 276). 16

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2. If φ = φp and M
φ, then if M
Qt =p t, then M
Qt =f t . 3. If φ = φf and M
φ, then if M
Qt =f t, then M
Qt =f t . 4. If φ = φp and M
φ, then if M
Qt =f t, then M
Qt =p t . Again, proving the admissibility of these rules is straightforward: Proof. 1. We can prove this case by noting that M
E =f F and M
F =p E, but M
F =p F and M
F =f F. 2. This follows from the fact that, per Table 1, personal identities and non-identities only occur between the persons and their personal properties or between the persons and the essence, and that each person is formally distinct from both the essence and the personal properties which are not his characteristic property. 3. This valid case is identical with Celarent or Ferio (DMPS, par. 59, p. 277). 4. This case follows from (2) by contraposition.

With these tools to hand, it is possible to show that the four rules characterizing valid mixed affirmative syllogisms given in DMPS, par. 93 – 96, pp. 284 – 285 and the eight rules for mixed negative syllogisms given in DMPS, par. 98 – 105, pp. 285 – 286 are correct. These proofs are straightforward, and are left as exercises to the reader. More interesting is to see how this formal system can be applied to resolve the apparent paralogisms; we look to this in the next section. 4. Resolving the paradoxes In the previous section we introduced the ! quantifier but didn’t say much about its usage. The ! quantifier is used when we formalize natural language sentences about the trinity in order to make their import explicit. A paralogism arises when a syllogism appears to be sound but where the conclusion is intuitively false. These paralogisms can be blocked by recognizing the various ways that categorical propositions containing divine terms can be

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ambiguous. There are two main ways that categorical predications like this can be ambiguous. First, the type of identity being expressed by est is not made explicit. Paralogisms that arise from this type of ambiguity make up a large percentage of the fallacious arguments concerning the trinity: [M]any [fallacies] which are made in divinity, are made from identical conjunction of extremes with a middle, and because of this they are believed to be able to be connected with each other identically; or from identical and formal conjunctions, because of which conjunctions they are believed to be able to be connected with each other formally.18

As a result, to avoid paralogisms of this type we need to make explicit the type of identity (cf. DMPS, par. 81, p. 281). If we make explicit which type identity is being expressed by est (for purposes of examples we will take it to be =f ), then we still have a potential ambiguity, because there are two ways that we can interpret the sentence essencia est formaliter pater. By the default interpretation of the empty quantifiers that we introduced in the previous section, this sentence should be interpreted as omnis essencia est pater. But since in omnis essencia ˙ we could also interpret est pater, essencia stands for just one object, (namely E), the sentence as hoc essencia est pater without changing the truth conditions of the sentence (cf. DMPS, par. 34, p. 271). However, there is a second way that we could interpret omnis essencia est pater, namely by generalizing the subject term, e.g., omnis res que est essencia est pater (cf. DMPS, par. 56, 74, pp. 276, 280). The two interpretations are not equivalent, and they do not have the same signification: Briefly I say that these two propositions: every essence is the father, and: every thing which is the essence is the father, by the mode of signification and imposition do not have the same mentals (mentales) unless you want to abuse the term; and the subject of this: everything which is the essence is the father, taking the first ‘is’ identically, supposits formally for many things, namely for the three persons; however the subject of this: every essence is the father, supposits formally for one thing alone, namely for the essence, and only indistinctly and identically for [M]ulte
fallacie que fiunt in divinis, fiunt ex coniunctione ydemptica extremorum cum medio, et propter hoc creduntur inter se posse coniungi ydemptice; vel ex coniunctionibus ydemptica et formali, propter quas coniunctiones creduntur inter se posse formaliter coniungi (DMPS, par. 75, p. 280). 18

236 the three persons.19

The truth conditions for both versions are intuitive. Hoc essencia est essencialiter pater is a singular proposition, whose truth conditions are governed by Definition 7, that is, it is true if and only if the particular, singular thing which is the essence stands in the essential identity relation with [something that is] the father. Omnis res que est essencia est essencialiter pater is true if and only if everything which is the essence stands in the essential identity relation with [something that is] the father. Formally, the distinction is between: !E =f F/=f and E/=∗ =f F/=f Notice the introduction of =∗ into the first term; as the author notes in DMPS, par. 82, p. 281 – 282, if we want to expound essencia as omnis res que est essencia, we need to ask which type of identity is being expressed by this est. In DMPS, par. 84 – 88, pp. 282 – 283, the author argues in favor of interpreting omnis essencia est pater as only hec essencia est pater, and not as omnis res que est essencia est pater. While if we interpret it as omnis res que est essencia, then we can reason according to Rules 4 and 5, if we do so, then non salvabis omnes modos Aristotelis, ut patet de disamis (DMPS, par. 84, p. 282). Instead, if we singularize the subject terms and pay attention to the modification of the copulae introduced by essencialiter, personaliter, and formaliter, then “you will solve all paralogisms; you will even save all the modes of Aristotle.20 Taking this route, we will see that “many apparent distortions in the infidels themselves follow according to the mode of complete [distribution], of which nothing follows from the aforementioned modification of the copulae.”21 And thus we are Sed breviter dico quod iste due propociones: omnis essencia est pater, et: omnis res que est essencia est pater, ex modo significacionis et imposicionis non habent easdem mentales, nisi velis abuti terminis; et subiectum istius: omnis res que est essencia est pater, summendo primum ‘est’ ydemptice, supponit pro pluribus formaliter, scilicet pro tribus personis; subiectum autem illius: omnis essencia est pater, supponit pro uno solo formaliter, scilicet pro essencia, et indistincte vel ydemptice pro tribus personis (DMPS, par. 83, p. 282). 20 solves omnes paralogismos; salvabis eciam omnes modos Aristotelis (DMPS, par. 84, p. 282). 21 multa apparencia distorta ipsis infidelibus sequuntur ad modum de completa
distribucione, quorum nullum sequitur ad modificacione copularum predictam (DMPS, par. 89, p. 283). 19

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able to resolve the paralogisms. Furthermore, we can extract Aristotelian syllogistics from within the framework that we have provided. This allows us to say that reasoning about the trinity is not a “special case” which cannot be handled by regular syllogistic logic. Instead the situation is almost the other way around: Reasoning about creation is just a special case or a reduction of trinitarian syllogizing. We can do all of our logical reasoning within one formal system that handles propositions about divine and created things equally well. The fact that the predications used in syllogisms about the trinity can be formal, identical, or essential explains why we have paralogisms. The expository syllogism Hoc essencia divina est pater. Filius est essencia divina. Igitur, pater est filius. is valid and sound if the statements are all taken to be essential predications. The paralogism arises when we interpret the conclusion as making a personal or formal predication. Once this misinterpretation is cleared up, by making the type of predication explicit via our formal system and reasoning with expository syllogisms, then the paralogisms disappear. Acknowledgments The author was funded by the project “Dialogical Foundations of Semantics” (DiFoS) in the ESF EuroCoRes programme LogICCC (LogICCC-FP004; DN 231-80-002; CN 2008/08314/GW).

References [1] Abelard, P., Theologia ‘svmmi boni’ et theologia ‘scholarivm’, ed. by E.M. Buytaert & C.J. Mews, Turnholt: Typographi Brepols Editores Pontificii, 1987. [2] Buridan, J., Iohannis Buridani tractatus de consequentiis, ed. by H. Hubien, Louvain: Publications universitaires, 1976.

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[3] Buridan, J., Jean Buridan’s logic: the treatise on supposition, the treatise on consequences, ed. by P. King, Dordrecht: Reidel, 1985. [4] Hallamaa, O., Defending common rationality: Roger Roseth on trinitarian paralogisms, Vivarium 41(1), 2003, 84–119. [5] Knuuttila, S., Philosophy and theology in twelfth-century trinitarian discussions, [in:] S. Ebbesen & R.L. Friedman, eds., Medieval analyses in language and cognition, Copenhagen: Royal Danish Academy of Sciences and Letters, 1999. [6] Knuuttila, S., How theological problems influenced the development of medieval logic?, [in:] S. Caroti, R. Imbach, Z. Kaluza, G. Stabile, & L. Sturlese, eds., “Ad ingenii acuitionem”: studies in honour of Al´ ˆ fonso Maier`u, Textes et Etudes du Moyen Age 38, Louvain-la-Neuve: ´ F´ed´eration Internationale des Instituts d’Etudes M´edi´evales, 2007. [7] Maier`u, A., Logic and trinitarian theology De Modo Predicandi ac Sylogizandi in Divinis, [in:] N. Kretzmann, ed., Meaning and inference in medieval philosophy: studies in memory of Jan Pinborg, Dordrecht: Kluwer Academic Publishers, 1988. [8] Mostowski, A., On a generalization of quantifiers, Fundamenta mathematica 44, 1957, 12–36. [9] Parsons, T., The traditional square of opposition, [in:] E.N. Zalta, ed., The Stanford encyclopedia of philosophy, Fall 2008 edition, http://plato. stanford.edu/archives/fall2008/entries/square/. [10] Uckelman, S.L., Modalities in medieval logic, Ph.D. dissertation, Universiteit van Amsterdam, Amsterdam: ILLC Publications, 2009. [11] Westerst˚ahl, D., Generalized quantifiers, [in:] E.N. Zalta, ed., Stanford Encyclopedia of Philosophy Winter 2008 edition. http://plato.stanford.edu/archives/win2008/entries/ generalized-quantifiers/.

LATE MEDIEVAL TRINITARIAN SYLLOGISTICS: FROM THE THEOLOGICAL DEBATES TO A LOGICAL TEXTBOOK Paloma Pérez-Ilzarbe University of Navarra Pamplona, Spain [email protected] Jerónimo Pardo's analysis of the problems raised by some popular trinitarian paralogisms is studied in this paper. The purpose is to show how the notions employed by the theologians in order to solve theological problems were introduced into a textbook on logic to deal with some genuinely logical problems. First, the problem, common to all logical approaches, of achieving a fine-grained analysis of the logical form of syllogistical inferences. Second, the problem, typical of the terminist approach to logic, of guaranteeing that Latin is an adequate vehicle for logical analysis.

1. Terminist logic and the Trinity: some late-medieval theologians in a post-medieval textbook on logic Many studies have been devoted to showing the various aspects of the influence of medieval logic in Christian theology (see, for example, the works cited in [4, p. 183]). Thanks to these, we know how some medieval logical tools were applied to theological discussions. A widely studied topic is the application of the medieval theory of the ‘properties of terms’ (proprietates terminorum) to the specific problems generated by the complexity of trinitarian theology ([1], [2], [6] – [9], [10]).

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The Catholic doctrine of the Trinity (with a God who is one simple Essence and at the same time three Persons with different properties) is found to produce some paradoxes when it is arranged in accordance with the patterns of Aristotelian syllogistics. The terms used in trinitarian contexts (‘Essentia divina,’ ‘Pater,’ ‘Filius,’ ‘Trinitas,’ ‘generans,’ ‘spirans’...) allow us to construct some syllogisms which seem to follow the rules of Aristotelian logic but also seem to have true premises and a false conclusion. Among other responses, one very influential one lies in applying to the problematic cases the analytical tools provided by the theory of the proprietates terminorum. We can see this practice in the works of some brilliant theologians of the fourteenth century, such as Gregory of Rimini, Adam Wodeham and Peter of Ailly. These theologians saw the notions and distinctions belonging to the theory of the proprietates terminorum as a means to save the rationality of their theological discourse. But the connection logic-theology can be seen the other way round: not from the perspective of the theologians, but from the perspective of the logicians, who are confronted with a new domain of application for a well established and accepted logical doctrine. This is the perspective that I will examine. I will base my analyses on the work of a post-medieval logician, the Spaniard Jerónimo Pardo, who was a master of Arts and bachelor in Theology at the University of Paris at the very end of the fifteenth century. His Medulla Dyalectices (printed in Paris in 1500, and reprinted shortly after Pardo's death in 1505)1 offers a privileged viewpoint to look at the discussions that took place more than a century before. Pardo is transferring the logical discussions that were placed in a theological context to their proper context, that of logic. In this context, the doctrinal concerns are no longer present, so Pardo is able to assess each notion and distinction on its own, just in terms of its logical value.2 1

There is no modern edition of this work. I will quote from the 1505 edition, referring to it as MD. A provisional transcription of the relevant passages can be found at: http://www.unav.es/filosofia/pilzarbe1/medulla_dyalectices/medulla_dyalectices.html 2 Sometimes Pardo gives the name of the theologian who employed the notion he is discussing (and, for example, he quotes literally some passages from Gregory's and Peter's commentaries on the Sentences), but most of the time the author of the opinion is not identified but hidden under the expressions ‘aliqui dicunt,’ ‘alii dicunt,’

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Terminist logic was re-flourishig at that moment at the University of Paris, and Pardo was firmly convinced that it was the correct tool to carry out logical analyses. As far as syllogistical inference is concerned, the chapter De Syllogismis is fully devoted to show how the theory of the properties of terms can be used to dissolve any counterexample that might seem to threaten the Aristotelian doctrine. In this context, the different trinitarian paralogisms that were discussed in the theological works are just providing some examples of propositions that can be used to test the power of the terminist techniques of analysis.3 2. Jerónimo Pardo's theory of the syllogism: in search of the logical form Chapter nine of Pardo's Medulla Dyalectices focuses on the categorical syllogism, which is defined as a kind of formal consequence. Being a formal consequence, the validity of a syllogism should be characterized independently of any content. To this end, Pardo gives the Aristotelian rules dici de omni and dici de nullo a prominent role, as being the regulative principles of any categorical syllogism.4 In Pardo's words, this is the condition that a syllogism has to meet in order to be immediately governed by the principle dici de omni: Dici de omni: A syllogism is immediately governed by the dici de omni when nothing can be subsumed under the distributed subject of the major ‘communiter solet poni,’ etc. A further task to be done, which lies beyond the scope of this article, is that of identifying the supporters of every notion and distinction. 3 Pardo's Medulla can thus be added to the list given in [5, p. 70] of post-medieval textbooks which use some trinitarian paralogisms in discussing the syllogism. 4 Omnis autem talis sillogismus altera duarum regularum necesse est regulari: aut enim est affirmativus, et sic regulandus est per dici de omni, aut negativus, et sic regulandus est per dici de nullo. Que dicuntur principia regulativa sillogismorum tanta et talia, ut omnis sillogismus qui altera duarum regularum regulantur bonus et regularis dicatur et econverso, omnis autem qui altera harum regularum non regulatur, irregularis et inordinatus est dicendus et econverso. Ideo, omnem bonum sillogismum necesse est alteri harum regularum se conformare et omnem malum alicui harum se difformare; et ut unico complectatur verbo, cuiuslibet mali sillogismi solutio hec debet esse: quia non regulatur per dici de omni aut per dici de nullo (MD, fo. 126va).

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premise, of which the predicate of the major premise is not also affirmed in the conclusion. This happens when a) the major premise is a universal affirmative proposition, by means of which the predicate is affirmed of everything that the subject is said of; b) in the minor premise the subject of the major one is said of something, these things being then ‘subsumed’ under that term; and c) in the conclusion the predicate of the major premise is said of (at least some of) the things that were subsumed in the minor premise. For example, if we affirm in the major premise that every human being is an animal, and then in the second premise we subsume something, say Socrates, under ‘human being,’ then we are allowed in the conclusion to say of Socrates that he is an animal.5 In order for a syllogism to be immediately governed by the principle dici de omni, the major premise has to be de omni, that is, it has to affirm the predicate of whatever the subject is said of. This is usually accounted for in terms of distributio, which is the kind of suppositio that corresponds to the terms in the scope of a universal syncategorem, such as ‘omnis.’ The principle dici de nullo is analogously formulated and explained for the case of universal negative propositions. Pardo affirms that for any good syllogism we should be able to show that it is governed (either immediately or mediately) by one of these principles, and that for any bad syllogism we should be able to show how it is not governed by any of them. Let me summarise Pardo's proposal by the following General Rule: General Rule: A necessary and sufficient condition for the validity of a syllogism is that it is governed (either immediately or mediately) by the dici de omni or dici de nullo. 5

Unde sillogismus regulatur immediate per dici de omni quando nichil est sumere sub subiecto distributo maioris de quo non denotetur dici predicatum maioris in conclusione; ita quod in maiori propositione denotatur predicatum dici affirmative de quocunque dicitur subiectum, et in minore denotatur subiectum maioris dici de aliquo assumpto in minore, et in conclusione denotatur predicatum maioris dici de aliquo quod assumebatur in minore. Ut si dicam: omnis homo est animal, denotatur quod de quocunque verum est dicere quod est homo verum est dicere quod est animal; in hac minore: sed Sortes est homo, ly homo denotatur dici de aliquo; inferendo: ergo Sortes est animal, ly animal denotatur dici de illo assumpto in minori (MD, fo. 126va).

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Pardo devotes almost a half of the chapter De Sillogismis to discussing the moods Barbara and Darii, as they are immediately governed by the principle dici de omni, so they have (along with Celarent and Ferio, immediately governed by the dici de nullo) a founding character with respect to the remaining valid moods. His discussion always takes the same path: he starts by presenting some counterexamples (that is, some syllogisms that seem to have the form of one or other valid mood, but that seem to have true premises and a false conclusion, which is prohibited in the case of a valid inference); he then defines the logical notions and introduces the distinctions that he deems relevant for the solution; and he finally solves the counterexample by showing, with the help of the previously established notions and distinctions, either that the alleged syllogism is defective in its form or that, though it has the correct form, one of the premises is not true. The solutions proposed for the basic moods are taken as a model for the remaining moods, which Pardo does not analyse in such detail. As is stated above, Pardo entrusts the solution of the different paralogisms to various terminist theories (namely, the ones of suppositio, appellatio and ampliatio). The terminist notions and distinctions are thus used as a tool to safeguard the formal character of the syllogistical inference. In this context, many cases of trinitarian syllogisms are presented, as the peculiar semantics of the divine terms raises some special problems for the terminist approach to the Aristotelian theory of the syllogism. As a matter of fact, the first counterexample that Pardo offers is a syllogism with trinitarian terms, which seems to have a Barbara-form but where the conclusion does not seem to follow from the premises (since, according to orthodoxy, the premises are true and the conclusion is false): (1)

Omnis Essentia divina est Pater, omnis Filius est Essentia divina, ergo omnis Filius est Pater. Every divine Essence is the Father, every Son is the divine Essence, therefore every Son is the Father.

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Should this syllogism indeed have a Barbara-form, it would be governed by the principle dici de omni. That is, in the major premise one would be denoting that of whatever the subject is said, of the same thing the predicate is said; and in the minor premise one would be denoting that the subject of the major one is said of something; then, in the conclusion one would be denoting that the predicate of the major premise is said of the same thing that has been subsumed in the minor one. So, the application of the General Rule simply dissolves the paralogism, as we have the following two possibilities: either ‘omnis Essentia divina est Pater’ is de omni, and then the syllogism is a valid one, but the major premise is false (and thus it is not a problem that the conclusion is also false); or ‘omnis Essentia divina est Pater’ is not de omni, and then the syllogism is an invalid one (and thus it is not a problem that the premises are true and the conclusion is false).6 The possibility that the universal proposition ‘omnis Essentia divina est Pater’ is not de omni is allowed by the peculiar semantics of the divine terms. The term ‘Essentia divina’ can be said not only of the very divine Essence, but also of each of the divine Persons. If the predicate ‘Pater’ in ‘omnis Essentia divina est Pater’ is predicated of the divine Essence but not of each divine Person, this proposition cannot be said to be de omni. This means that, as far as trinitarian propositions are concerned, the syncategorem ‘omnis,’ which was usually considered to suffice to effect the required distribution in order for a syllogism to be governed by the dici de omni, seems to be no longer sufficient. The schema: B1. Omne A est B, omne C est A, ergo omne C est B, 6

Dicendum enim est illos sillogismos esse bonos si regulantur per dici de omni, ita quod in maiori denotetur quod de quocunque dicitur subiectum dicitur predicatum et in minori de aliquo denotetur dici subiectum, in conclusione vero de eodem denotetur dici predicatum. Sed maior est falsa in qua dicitur omnis essentia divina est pater, quia non de quocumque verum est dicere quod est essentia divina de eodem verum est dicere quod est pater, nam filius est essentia divina et tamen filius non est pater. Ideo, dicentes illum sillogismum bonum esse si maior sit de omni recte solvunt, si enim maior non est de omni non regulatur mediate per dici de omni, maior enim debet esse regula de omni secundum quam maior et conclusio sunt regulande (MD, fo. 127ra).

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Every A is B, every C is A, therefore every C is B, which could at first sight be considered as capturing the structure of the mood Barbara, is shown by the trinitarian counterexample to be an inadequate anlysis of the Barbara-form. The trinitarian domain is putting a high level of demand on the formality of consequences. If Pardo wants the syllogistic mood Barbara to be a formal consequence, he needs to go deeper into specifying its logical form. The notions and distinctions that Pardo is importing from the theological debates find in the Medulla their natural place, as they are introduced in successive steps to achieve a better grasp of the logical form. 3. Proprietates terminorum and syllogistical form: the notion of distributio completa Let us go back to example (1). In the minor premise, the peculiar semantics of the divine terms allows one to predicate ‘Essentia divina’ of ‘Filius.’ But in order for the syllogism to be governed by the dici de omni, the major premise must affirm the predicate of whatever the subject is said of. This raises a question concerning the universality of the major premise: when we say ‘omnis Essentia divina est Pater,’ are we affirming ‘est Pater’ of the Son? The theory of the proprietates terminorum allowed the theologians to put the question in semantic terms, either in terms of distribution, or in terms of supposition. That is, one can ask: is the term ‘Essentia divina’ distributed to the divine Son? Or: does the term ‘Essentia divina’ supposit for the divine Son? The answers are used by Pardo to elaborate a more accurate notion of syllogistical form. In order to find an answer that dissolves the paralogism, the theologians introduced a distinction between two forms of distribution of the divine terms: complete and incomplete. A universal proposition with complete distribution has stronger truth-conditions than a universal proposition with incomplete distribution. Pardo characterises these notions in terms of the notion of inferior, which is part of the terminist technique for specifying the truth-conditions of quantified propositions, the descensus to singulars:

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from a quantified proposition one can descend to a set of singular propositions, in which the predicate is said of the inferiora of the common term. But, again, the setting of the inferiora of the trinitarian terms is affected by their particular semantic behaviour. Pardo distinguishes two kinds of inferiora for the divine terms: per se and non-per se. For example, he explains that the term ‘Essentia divina’ has only one inferior per se, namely ‘haec Essentia divina que est Pater et Filius et Spiritus Sanctus,’ although in a non-per se sense, ‘Filius’ can also be considered to be an inferior of that term. Pardo thus formulates the distinction between complete distribution and incomplete distribution in the following terms: • For a universal proposition in which there is a complete distribution to be true, it is required that the predicate is truly said of everything that the subject is said of, whether it is per se under the subject or it is not. • For a universal proposition in which there is an incomplete distribution to be true, it is sufficient that the predicate is truly said of everything that is per se under the subject.7 This means that, in order for the syllogism to be governed by the dici de omni, the subject of the major premise has to be distributed with complete distribution. The domain of the Trinity has thus shown that there are (at least) two kinds of universality, one that suffices for the mood Barbara to be a formal consequence and one that does not suffice. Thus, the General Rule can be specified for the case of the mood Barbara by means of the following formulation: Rule 1: A necessary condition for the validity of a Barbara syllogism is that the subject of the major premise ‘A’ is distributed to everything that is A (omne quod est A).

7

Ad veritatem enim universalis in qua est distributio completa requiritur quod de quocunque dicitur subiectum de illo dicatur predicatum sive illud sit per se contentum sub subiecto sive non. Sed ad veritatem propositionis universalis cuius subiectum distribuitur distributione incompleta sufficit predicatum dici de omni per se contento sub subiecto (MD, fo. 127ra-b).

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The structure of the mood Barbara could then be better expressed by the schema: B2. Omne quod est A est B, omne C est A, ergo omne C est B. Everything that is A is B, every C is A, therefore every C is B. Of course, this distinction does not have any effect in the domain of the creatures (unless, says Pardo, one accepts Scotus' proposal of some naturas communes per indifferentiam), but it has to be posited if one wants the syllogistical consequence to hold independently of any matter. However, a further counterexample is going to require a finer analysis of the logical form. 4. The correct reading of the universal proposition Pardo also considers the following example of a syllogism with trinitarian terms: (2)

Omnis Essentia divina est Persona, Trinitas est Essentia divina, ergo Trinitas est Persona. Every divine Essence is a Person, the Trinity is the divine Essence, therefore the Trinity is a Person.

Even if we apply the previous analysis, the paralogism does not seem to dissolve. If we read the major premise as ‘omne quod est Essentia divina est Persona’ according to its de omni sense, it is still a true premise, and, when joined to the minor premise which is also true, it allows us to draw a false conclusion in accordance with the Darii-form.

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Pardo reports that this problem has led some theologians to say that a special kind of supposition is present here, namely a supposition extra numerum, as opposed to the supposition in numero,8 grounded on the fact that although whatever is the divine Essence is a Person (omne quod est Essentia divina est Persona), nevertheless not all things that are the divine Essence are a Person (non omnia que sunt Essentia divina sunt Persona), because three Persons are the divine Essence but they are not a Person. The solution of the paralogism involves the contrast between ‘omne quod est Essentia divina est Persona,’ which expresses the in numero sense of the major premise, and ‘omnia que sunt Essentia divina sunt Persona,’ which expresses the extra numerum sense. But Pardo does not consider that this distinction is necessary to dissolve the paralogism. All that is needed is a better reading of the de omni sense of the universal proposition ‘omnis Essentia divina est Persona.’ In the light of example (2), we can see that the expression ‘omne quod est’ does not provide a correct reading of the complete distribution. Instead, the expression ‘omne ens quod est’ has to be used. This allows us to solve the paralogism in analogy with (1), by showing that, when taken in its de omni sense, the major premise is false. ‘Omne ens quod est Essentia divina est Persona’ is a false proposition, because the Trinity is in fact a being which is the divine Essence (as the Trinity is the very divine Essence, which is of course a being which is the divine Essence), but the Trinity is not a Person.9 The ‘ens’ has the effect of fully extensionalising the analysis, thus avoiding the problems derived from the peculiar semantics of the trinitarian terms.10 This allows us to give a finer analysis of the Barbara-form: 8

A similar distinction can be found in Henry Totting of Oyta (see [6], [10]). Hiis tamen non obstantibus dico quod nullo sillogismo quis cogere potest ponere distributionem extra numerum que predicto argumento credebatur concludi. Dico enim illam propositionem: omnis essentia divina est persona in hoc sensu: omne ens quod est essentia divina est persona esse falsam, quia concedendum est quod trinitas est ens quod est essentia divina (bene enim sequitur: trinitas est essentia divina, ergo trinitas est ens quod est essentia divina) et tamen trinitas non est persona (MD, fo. 128ra). 10 Simo Knuuttila has studied the Abelardian origin of the extensional approach [4, pp. 192 – 198]. The ‘omne ens quod est’ is playing a similar role as the ‘idem quod’ device in Abelard's proposal. See also [3, p. 130] and [5, p. 73]. 9

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B3. Omne ens quod est A est B, omne C est A, ergo omne C est B. Every being that is A is B, every C is A, therefore every C is B. And also a new formulation of the rule governing the validity of the mood Barbara: Rule 2: A necessary condition for the validity of a Barbara syllogism is that the subject of the major premise ‘A’ is distributed to every being which is A (omne ens quod est A). 5. Proprietates terminorum and syllogistical form: the notion of supponere mediate The distinction between two kinds of universality can also be formulated in terms of suppositio, the property of terms by virtue of which they stand for something when they are used in a proposition. Pardo quotes Peter of Ailly,11 who distinguishes, as far as divine terms are concerned, between suppositing immediate and suppositing mediate. It should be noted that he is not introducing some new non-standard species of supposition, he is just distinguishing two ways in which a divine term, owing to the peculiarities of the trinitarian domain, can supposit for its supposita. The test to distinguish whether a term is suppositing in one or the other way is one of convertibility: • A term supposits mediate for something, say A, when the term supposits for A but it is not convertible with the corresponding term ‘A’ by means of the expression ‘omnis res que est A.’ • A term supposits immediate for something, say A, when the term supposits for A and it is convertible with the corresponding term ‘A’ by means of the expression ‘omnis res que est A.’ 11

From book I, question 5 of Peter's commentary on the Sentences.

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For example, the term ‘Essentia divina’ supposits for the Father, but it is not convertible with the term ‘Father’ in the terms ‘omnis res que est Essentia divina est Pater.’ Thus, ‘Essentia divina’ supposits just in a mediated way for the Father. In contrast, the term ‘Pater’ does supposit immediately for the Father, as the testing proposition ‘omnis res que est Pater est Pater’ is true. Pardo grounds this distinction in a parallel distinction between two ways of signifying that are peculiar to the trinitarian terms. On the one hand, each divine term has been imposed to signify something. This original imposition determines what he calls the ‘primary signification’ of the term. But, on the other hand, when we are talking about the Trinity, the thing which is primarily signified has some ontological connections with some other things that the term has not been imposed to signify. For example, the divine Essence, which is primarily signified by the term ‘Essentia divina,’ is identical with the Father. In virtue of this ontological connection, we can say that the term ‘Essentia divina’ also signifies the Father. But this is not by imposition, but rather by the very nature of the thing which is primarily signified. Thus, Pardo calls this a ‘secondary signification.’ As supposition is derived from signification, each divine term has two ways of suppositing: it supposits immediately for the things that are primarily signified, and it supposits in a mediated way for the things that are secondarily signified.12 The two kinds of universality, the one that suffices for the mood Barbara to be a formal consequence and the one that does not suffice, can thus be formulated in terms of supposition. If the subject of the major premise is taken as standing for just the things that it immediately supposits for, then the universality will not be sufficient for the requirements of formality. Instead, in order for the syllogism to be governed by the dici de omni, the 12

Et si dicas secundum istum modum dicendi quomodo declaranda est illa duplex suppositio premissa, mediata scilicet et immediata. Respondeo: suppositio ex significatione cognoscenda est, ideo cum ponatur talis duplex suppositio conformiter ponenda est duplex significatio: una primaria, secundum quam terminus ab impositore habet ut aliquid det intelligere, et cum iste terminus pater ab impositore non habet nisi quod significet patrem, ideo cum supponat pro patre immediatam suppositionem habet. Sed secundario significat rem idemptificatam cum illa re ad quam primo est impositus ad significandum, et hoc non est propter impositionem sed propter naturam rei significate que talis est ut habeat talem idemptitatem (MD, fo. 127vb).

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subject of the major premise has to be taken as standing for both the things that it immediately and mediately supposits for. This is how Peter of Ailly's distinctions, which he had used to solve his theological problems, are used by Pardo in his attempts to solve a logical problem: the sharpening of the notion of syllogistical form. Again, the General Rule can be specified for the case of the mood Barbara by means of the following formulation: Rule 3: A necessary condition for the validity of a Barbara syllogism is that the subject of the major premise is taken as standing for both the supposita that it immediately and mediately supposits for. Both the notion of complete distribution and the notion of mediately suppositing are formulated with the help of the complex expression ‘omne ens quod est’ (or the equivalent ‘omnis res que est’), which extensionalises the analysis in a way that is comparable with the modern quantificational approach. With this tool, one can provide an analysis of the trinitarian examples which dissolves their paradoxical character. But, before the logical value of this expression is definitely established, it has to overcome some difficulties. 6. Latin as a tool for analysis. The supposition of the term ‘Pater’ In fact, Pardo does not accept the notion of completa distributio without discussion (along with the corresponding notion of supponere mediate), when it is spelled out by means of the expression ‘omne ens quod est.’ These notions, useful as they have proved to solve the theological problems, both raise some logical problems to which the logician must pay attention. First, Pardo poses an objection against the notion of supponere mediate: it seems that it could be said that the term ‘Pater’ supposits for the Son, once it has been admitted that the term 'Pater' supposits for the divine Essence.13 13

Sed probaretur apparenter contra istum doctorem quod iste terminus pater mediate supponit pro filio si iste terminus pater mediate supponat pro essentia. Sic enim arguo: pater supponit pro omni ente quod est pater et pro eo distribuitur si completa distributione distribuatur, sed filius est ens quod est pater (sequitur enim: est essentia,

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The point is how far we can go with the possibility that ontological connections would extend the supposition of the terms beyond that which they have been imposed to signify. If the identity of the Father with the Essence leads one to admit that the term ‘Pater’ mediately supposits for the Essence, then it could be thought that the identity of the Essence with the Son would further lead one to admit that the the term ‘Pater’ mediately supposits for the Son. Pardo reports that some theologians were overcome by this objection, and that they therefore added a new distinction within the notion of mediate supposition: ‘primary’ mediate supposition and ‘secondary’ mediate supposition. When a term is taken in primary mediate supposition, it supposits for the thing which is identical to the immediate suppositum. When a term is taken in secondary mediate supposition, it supposits for the thing which is identical to the thing which is identical to the immediate suppositum. It seems as if the ontological connections would allow for limitless new supposita (that is, if ‘Pater’ supposits for the Essence, the identity of the Son with the Essence forces one to admit that ‘Pater’ also supposits for the Son), although the logical distinction between ‘primary’ and ‘secondary’ seems to safeguard the idea that the Father is not the Son. But Pardo does not agree with this limitless extension of the supposita of a divine term. This problem extends to the notion of complete distribution, as a term that is taken with complete distribution is taken as standing for both its mediate and immediate supposita. Does the term ‘Pater’ stands for the Son when it is taken with complete distribution, that is, the one required for the mood Barbara to be a formal consequence? Pardo does not want it to do that, but the problem will be very hard to solve, as what is involved here is no less than the limits of a logical analysis that is carried out by means of a natural language. The supposita of the term ‘Pater,’ when it is taken with complete distribution, for example in the proposition ‘omnis Pater est Pater,’ are determined by means of the Latin expression ‘ens quod est Pater.’ But the peculiar semantics of the trinitarian terms raises doubts about the adequacy

que essentia est pater), ergo est ens quod est pater, ergo iste terminus pater supponit pro filio (MD, fo. 127va).

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of this natural language expression to serve the purposes it has been designed for. In order to determine the supposita of the term ‘Pater’ when it is taken with complete distribution, one has to find the things of which ‘this is a being which is Father’ can be truly said. But in order to find this, one has to analyse the complex term ‘ens quod est Pater.’ Pardo uses in the analysis another terminist notion: the property of restrictio, which modifies the ‘normal’ supposition of a term, making it stand for fewer things than it normally stands for. Different kinds of restricting terms affect the restricted term in different ways. The question here is how the clause ‘quod est Pater’ is restricting the term ‘ens’ in the expression ‘ens quod est Pater.’ Pardo reports the opinion of those who think that in the complex expression ‘ens quod est Pater’ the term ‘ens’ is restricted to standing for the same things that the term ‘Pater’ stands for, just as in ‘ens quod est Sortes currit’ the term ‘ens’ does not supposit for any being, but it is restricted to stand just for Socrates. But Pardo's opinion is different, again motivated by the peculiar semantics of the trinitarian terms: when the restricting term is following a copula implicationis (that is, the copula which accompanies the relative ‘qui’), it does not restrict the term to stand for the thing that the restricting term stands for (in this case, for the Father), but for the thing which is the suppositum of the restricting term (in this case, for the thing which is Father), and this allows the term ‘Pater’ to supposit for the divine Essence. Again, a device for extensionalising the analysis (‘pro ente quod est Pater’) is operating.14 But the device demands that Pardo should put a limit on it. The problem is that, in order to determine the supposition of the term ‘Pater’ when it is used in a proposition, another supposition is involved, namely the 14

Respondeo: quando aliquis terminus restringitur per terminum sequentem copulam implicationis non habet ille terminus ex vi restrictionis quod stet precise pro eodem pro quo stat precise ille terminus restringens, sed habet tantum ille terminus ex vi restrictionis ut stet pro eo quod est illud pro quo supponit ille terminus restringens, sive illud sit alia res sive non. Exemplum: ut si per impossibile ponerentur due essentie divine quarum una esset pater et filius alia vero precise esse spiritus sanctus, tunc sic dicendo: essentia divina que est pater, ly essentia divina non staret precise pro esentia que est spiritus sanctus, staret tamen pro essentia que est filius, quia essentia que est filius est essentia que est pater; nec tamen debet dici quod ly essentia stet precise pro eodem pro quo ly pater (MD, fo. 128ra).

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supposition of the term ‘Pater’ which appears in the complex expression ‘ens quod est Pater.’ The peculiar semantics of the trinitarian terms would allow the first ‘Pater’ to supposit for the Son unless some limit is put on the supposition of the second ‘Pater.’ Pardo points out that the restricting term ‘Pater’ has to be taken as suppositing immediately. If it were taken as suppositing mediately, it would restrict the term ‘ens’ to stand for the thing which is the being which is Father (ens quod est ens quod est Pater), and this would allow the term ‘Pater’ to supposit for the Son (which is actually a being which is Father, namely, the Essence). The appropriate use of the notion of supponere immediate thus serves to dispel any doubts as to the adequacy of the expression ‘ens quod est Pater’ as a tool for logical analysis. 7. Latin as a tool for analysis. The universality of ‘omnis Pater est Pater’ When the problem is approached in terms of the notion of distributio completa, we encounter analogous difficulties. Pardo does not want the term ‘Pater’ to be able to supposit for the Son, but how can he prevent the term ‘Pater’ from being distributed to the Son in the proposition ‘omnis Pater est Pater?’ The complete universal sense of this proposition will be expresed by the following: ‘omne ens quod est Pater est Pater.’ How should this new proposition be interpreted? Pardo finds the elements he needs for an adequate answer in the discussions of the theologians concerning the following paralogism: (3)

Omne ens quod est Pater est Pater, Filius est ens quod est Pater, ergo Filius est Pater, Every being that is the Father is the Father, The Son is a being that is the Father, Therefore the Son is the Father,

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which seems to have the form of a Darii mood, and prima facie true premises.15 The theologians armed themselves with different logical tools in order to avoid the unwanted conclusion that the Son is the Father. Pardo examines in the first place the tools developed by Peter of Ailly, which he finds unsatisfactory. Trinitarian paralogisms had convinced Peter of Ailly that plain universality did not suffice in order for a syllogism to be governed by the dici de omni. What is needed is that the major premise should say that ‘of whatever the subject is said, of the same thing the predicate is said.’ But, while in the domain of the creatures the sign ‘omnis’ was thought to be enough for this task, the domain of the Trinity has shown that it is not enough, and consequently that the schema ‘omne A est B’ does not sufficiently expresses the universal form which is required for the syllogism to be valid. Thus, Peter made a distinction between being universal simpliciter and being universal secundum quid. A proposition is universal simpliciter when the subject is distributed as a matter of form (de forma), for example, when one says ‘omnis res que es A est B’ or ‘quicquid est A est B.’ In contrast, a proposition can be said to be universal secundum quid, that is, in an improper sense, when the distribution is not formally stated, and this happens whenever one just says ‘omne A est B.’ This means that, according to Peter, the universal syncategorem ‘omnis’ does not suffice to make a proposition universal enough (sufficienter universalis) in order to be the premise of a Barbara syllogism. Thus, he proposes the introduction of some complex syncategorems, such as ‘omnis res que est,’ ‘nulla res que est,’ as needed for the complete distribution of the subject in a simpliciter universal proposition. Once this kind of complex expression has been admitted as a universal syncategorem, Peter of Ailly is able to offer a double reading of the proposition ‘omne ens quod est Pater est Pater’ which is the major premise 15

Et confirmatur, quia quero an ista propositio: omnis pater est pater, aut est vera in sensu de omni aut non. Si dicas quod non, ergo pro aliquo supponit subiectum pro quo non supponit predicatum et illud non potest esse nisi filius aut spiritus sanctus. Si dicas quod est vera in sensu de omni, contra: sensus de omni est iste: omne ens quod est pater est pater, sed ipsum ostendo esse falsum, quia bene sequitur: omne ens quod est pater est pater, filius est ens quod est pater, ergo filius est pater. Premisse sunt vere, ergo et conclusio, et consequentia tenet in darii (MD, fo. 127va).

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of the example (3), depending on which part of it is acting as a universal syncategorem: a) If the syncategorem is the simple term ‘omne,’ then it is distributing the whole complex ‘ens quod est Pater,’ but then the syllogism will not be governed by the dici de omni, as the sign ‘omnis’ is not sufficient to make the complete distribution that is required. b) If the syncategorem is the complex ‘omne ens quod est,’ then it is distributing only the term ‘Pater,’ and then the syllogism will not be valid either, even if the distribution is indeed complete in virtue of the special syncategorem, because the subsumption in the minor premise has not been made under ‘Pater’ but under ‘ens quod est Pater.’ That is, according to Peter, the example (3) cannot be governed at all by the principle dici de omni (the problem being either one of distribution or one of subsumption). Pardo rejects the solution of putting a special syncategorem in order to make the complete distribution of the term ‘Pater,’ as the effect would be too strong. A sign distributing the term ‘Pater’ to any being which is Father would make it stand for the Son, as the Son is a being which is Father. He prefers to think that the syncategorem in ‘omne ens quod est Pater’ is the usual ‘omnis,’ which is distributing the term ‘ens quod est Pater’ (where the ‘ens’ is restricted by the ‘quod est Pater’ in the way explained before).16 The solution finally accepted by Pardo is that of Gregory of Rimini, who adopts the same strategy that proved useful for solving example (1). In order for the syllogism (3) to be good, the major premise has to be de omni, and this requires the distribution to be complete. That is, the subject ‘ens quod est Pater’ must be distributed to everything which is a being which is Father (omne ens quod est ens quod est Pater). Read in this way, the major premise ‘omne ens quod est Pater est Pater’ is false and the paralogism is solved.17 16

Si vero dicatur quod illa complexa: omnis res que est, nulla res que est, licet non sint sincathegoreumata supplent tamen quedam sincathegoreumata incomplexa distribuentia distributione completa. Arguitur sic: ex hoc sequitur quod ille terminus pater per illa signa quecunque sint illa distribuitur pro omni ente quod est pater et distribuitur pro filio, quia filius est ens quod est pater. Et sic habetur intentum quod iste terminus pater mediate supponit pro filio (MD, fo. 127vb). 17 Et ad sillogismum illud pretactum, cum arguitur: bene sequitur omne ens quod est pater est pater, filius est ens quod est pater, ergo filius est pater, dicitur quemadmodum dicit ille doctor qui prima facie videbatur reprobandus: quod illa

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But what is interesting is how the distinction between the two readings of ‘omne ens quod est Pater est Pater’ (as either de omni or not de omni) allows Pardo to save the truth of ‘omnis Pater est Pater.’ Again, Pardo has to put a limit on the application of the logical tools that the theologians employed within the domain of the Trinity. Although the proposition which is used to formulate the de omni sense of ‘omnis Pater est Pater’ (that is, ‘omne ens quod est Pater est Pater’) can be interpreted in a de omni sense when it is put as a premise of a syllogism, when it is employed as a tool for logical analysis it has to be interpreted according to its non-de omni sense, as this is the only way for the proposition ‘omnis Pater est Pater’ to be true in its de omni sense.18 We can see how the tools that the theologians had employed to solve the trinitarian paralogisms are being used by a logician to give the specific sense of the Latin expressions which are operating as metalinguistic devices for logical analysis. Pardo discovers that the same logical distinctions that are applied to the problems raised in the trinitarian domain by the object-language expressions can be applied to the problems raised by the metalinguistic expressions which spell out the sense of the objectlanguage expressions. With the help of these tools, Pardo is able to establish the logical value of the complex terms ‘ens quod est Pater’ and ‘omne ens quod est Pater’ as a tool for the analyis of the supposition and of the complete distribution of the object-language terms, respectively. Thus, the distinctions that Pardo has found in the theological discussions can be employed by the logician not only as a tool for the logical analysis of the syllogistical form, but also for the assessment of the very logical devices that are being used in this analysis. 8. Conclusion: What has to be said about syllogistical form? maior est distinguenda. Aut est de omni seu completa distributione distribuitur, et tunc maior est falsa, quia sensus est: omne ens quod est ens quod est pater est pater, et hoc est falsum. Si vero subiectum distribuatur incompleta distributione maior est vera, sed non sequetur quod non regulatur per dici de omni (MD, fo. 128ra-b). 18 Est igitur talis ordo: nam ista propositio omnis pater est pater est vera sive sit distributio completa sive incompleta. Et iterum, sensus que habet ut in ea est distributio completa est distinguendus, pro eo quod subiectum talis sensus potest distribui complete et incomplete, et ibi est status (MD, fo. 128rb).

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As a final objection to this whole approach, Pardo raises the question: is the mood Barbara a formal consequence? The naive objector would say that it is not, as we can construct a syllogism with a Barbara-form with true premises and false conclusion. In fact, the distinctions of the theologians have shown that there is a reading of the premise ‘omnis essentia divina est Pater’ that makes syllogism (1) to have true premises and false conclusion. The naive objector would say that syllogism (1) has the same form as the following, an impeccable example of the Barbara-form: (4)

Omne animal est substantia, omnis homo est animal, ergo omnis homo est substantia. Every animal is a substance, every human being is an animal, therefore every human being is a substance.

Thus, either the tools used by the theologians are defective, or syllogistical inferences are not actually formal consequences, but they hold in virtue of the material elements of discourse. But, needless to say, Pardo thinks that there is more to logical form than meets the eye, and therefore a better understanding of formality is needed to escape from the dilemma. The elements of the logical form that are usually taken into account are not sufficient to overcome every objection against the formal character of the syllogism. In contrast, Pardo will take considerable time specifying the elements which are relevant to the form. For the moment, in the light of the trinitarian paralogisms and as far as the Barbara-form is concerned, Pardo has made it clear that the kind of distribution (complete / incomplete) has to be taken as an ingredient of the syllogistical form.19 If the kind of distribution belongs to the syllogistical form, syllogism (1) can only be said to have the same form as syllogism (4) in the case where 19

Pardo deals separately with the expository syllogism, in which the middle is a singular term. The peculiarity of the trinitarian semantics (a singular term can supposit for more than one thing) will eventually call into question the very possibility of constructing a valid expository syllogism with trinitarian terms (MD, fo. 139vb-143ra).

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both major premises are interpreted with complete distribution.20 According to Pardo, the actual form of the mood Barbara is not B1, as logicians were inclined to think before they were confronted with the triniarian domain, but B3, which is the one that includes the complete distribution as a further component of the syllogistical form. This is an example of how the discussions of the theologians have helped the logician with the (endless?) task of revealing the logical form. Acknowledgments I would like to thank Ángel d'Ors and Heikki Kirjavainen for helpful comments on the draft of this article. I am also grateful to Hester Gelber, Alfonso Maierù, Olli Hallamaa and Simo Knuuttila for kindly having provided me with some material to complete my research. References [1] Ebbesen, S., The Semantics of the Trinity According to Stephen Langton and Andrew Sunesen, [in:] J. Jolivet & A. de Libera, eds., Gilbert de Poitiers et ses contemporains. Aux origines de la 'Logica Modernorum', Bibliopolis, Napoli, 1987, 401 – 435. [2] Hallamaa, O., Defending Common Rationality: Roger Roseth on Trinitarian Paralogisms, Vivarium 41/1, 2003, 84 – 119. [3] Knuuttila, S., The Question of the Validity of Logic in Late Medieval Thought, [in:] R. Friedman & L. O. Nielsen, eds., The Medieval Heritage

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Respondeo: de formalitate consequentie respondere valde tediosum est quemadmodum in materia de descensu pertractabitur. Si enim que communiter ponuntur ad formalitatem consequentie sufficerent, difficile esset salvare multas cavillationes quibus probaretur sillogismum non esse formalem consequentiam, sed de hoc alias. Sed ad predictam instantiam aliqui [Buridanus] respondent dicentes sillogismum non esse formalem consequentiam medio sumpto cum hac implicatione: ens quod est. Aliter forsitan posset dici quod tales sillogismi non sunt eiusdem forme nisi in utroque sit completa distributio. Si autem in neutra eorum sit completa distributio neuter eorum est bonus. Et quod variatio distributionis variet formalitatem consequentie ostenditur, nam suppositio videtur se tenere ex parte forme consequentie (MD, fo. 128rb).

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in Early Modern Metaphysics and Modal Theory, 1400–1700, Kluwer Academic Publishers, Dordrecht, 2003, 121 – 142. [4] Knuuttila, S., How Theological Problems Influenced the Development of Medieval Logic, [in:] S. Caroti et al., eds., ‘Ad ingenii acuitionem’: Essays in Honour of Alfonso Maierù, F.I.D.E.M., Louvain-la-Neuve, 2006, 183 – 198. [5] Knuuttila, S., Trinitarian Fallacies, Identity and Predication, [in:] P. Kärkkäinen, ed., Trinitarian Theology in the Medieval West, LutherAgricola-Society, Helsinki, 2007, 69 – 87. [6] Maierù, A., Logica aristotelica e teologia trinitaria. Enrico Totting da Oyta, [in:] A. Maierù & A. Paravicini Bagliani, eds., Studi sul XIV secolo in memoria di Anneliese Maier, Edizioni di Storia e Letteratura, Roma, 1981, 481 – 512. [7] Maierù, A., Logique et théologie trinitaire: Pierre d'Ailly, in Preuve et raisons à l'Université de Paris. Logique, ontologie et théologie au XIVe siècle, Z. Kaluza & P. Vignaux, eds., Vrin, Paris, 1984, 253 – 268. [8] Maierù, A., A propos de la doctrine de la supposition en théologie trinitaire au XIVe siècle, [in:] E. P. Bos, ed., Medieval Semantics and Metaphysics. Studies Dedicated to L. M. de Rijk on the Occasion of His 60 Birthday, Ingenium, Nijmegen, 1985, 221 – 238. [9] Maierù, A., Logique et théologie trinitaire dans le moyen-âge tardif: deux solutions en présence, [in:] M. Asztalos, ed., The Edition of Philosophical and Theological Texts from the Middle Ages, Almqvist & Wiksell, Stockholm, 1986, 185 – 212. [10] Shank, M. H., ‘Unless You Believe, You Shall Not Understand’: Logic, University and Society in Late Medieval Vienna, University Press, Princeton, 1988.

INEFFABILITY PERFORMANCE: CRITIQUE AND CALL Timothy Knepper Drake University, USA [email protected] Although Michael Sells’s Mystical Languages of Unsaying (1994) seemed poised to redirect the future of mysticism studies, fifteen years after its publication it has yet to receive an extensive critical analysis or positive structural implementation. This essay undertakes the former in order to prepare for the latter. After clarifying Sells’s basic thesis that apophatic discourse performs ineffability through linguistic-rule violation, it offers two critiques of Sells’s specific claims: apophatic discourse cannot violate (all) linguistic rules if it is to succeed at showing the ineffability of something; apophatic discourse does not therefore reenact an (absolutely) ineffable mystical experience. It then calls for an implementation of Sells’s general method that turns from the overtired and intractable question about the ineffability of mystical experience to the analysis of both the grammatical techniques and rules by which ineffability is expressed and the intra- and inter-religious patterns that show up in these techniques and rules.

The year 1994 witnessed the publication of a work that promised to transform the future of mysticism studies: Michael Sells’s Mystical Languages of Unsaying [13]. Rejecting the category of experience, at least in name, Mystical Languages of Unsaying turned from the hackneyed question of whether mystical experience was really ineffable to the novel issue of how claims of ineffability actually get expressed. Here, Sells argued that certain mystics, recognizing the logical problems pertaining to flat assertions of ineffability, opted instead to “perform ineffability” through the violation of linguistic rules, thereby showing how some language in fact cannot articulate the “object” in question. Then, Sells

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sought to establish the logic and conventions of such a discourse through a set of historical-literary analyses of the writings of five mystics in the Platonic-Abrahamic tradition. Thus, Sells’s Mystical Languages of Unsaying seemed poised not only to redirect the efforts of scholars of mysticism to a more fruitful and tractable area of inquiry but also to lead the way forward into that area of inquiry with a first attempt at interreligious textual comparison. Fifteen years later, however, Mystical Languages of Unsaying has yet to receive an extensive analysis, let alone a structural implementation.1 This essay aims to fill the first of these lacunae, undertaking a critical analysis of Sells’s project, in order to call for a positive implementation of Sells’s project. Note that this analysis will be critical, for it is my opinion that Sells’s project is deeply flawed as it stands. But it is also my opinion that, with proper modification and emendation, Sells’s project carries great promise for the future of not only mysticism studies but also comparative philosophy of religion. 1. The techniques and rules of apophasis: exposition and emendation Sells begins Mystical Languages of Unsaying by putting very succinctly what he considers to be the primary dilemma with transcendence: “X is beyond names” must name that which it claims to be beyond names. This is what Sells calls “the aporia – the unresolvable dilemma – of transcendence” [13, p. 2]. And it produces, according to Sells, three basic responses: silence, a distinguishing of the ways in which X is and is not beyond names, and the generation of a new mode of discourse that accepts this aporia as irresolvable [13].2 This new mode of discourse is apophasis, an attempt to “state” the aporia of transcendence by disrupting the rules and conventions of ordinary language. Apophatic discourse therefore goes beyond “apophatic theory,” performing the ineffability of some 1

While Mystical Languages of Unsaying has received a number of book reviews, a May 2009 search with the databases Academic Search Complete, ATLA Religion Database, JSOTR, and Philosopher’s Index reveals no extensive (i.e., at least article length) critical analyses of it. 2 For the argument that Sells’s description of this second response is “unnuanced and oversimplified” as well as the claim that Sells’s portrayal of this third response is extreme, see [10]. I am in agreement with Rocca on both points.

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transcendent in lieu of or addition to merely asserting its ineffability. In fact, apophatic discourse goes beyond even this: it reenacts a nonexperiential realization of or union with a non-substantial deity. But the expressed goal of Mystical Languages of Unsaying is not that of developing a general theory about apophatic discourse; it is that of establishing “the implicit logic and conventions of apophasis as a mode of discourse” [13, p. 6]. To this end, Sells undertakes a syntactic-semantic analysis of the writings of five mystics who share the cultural heritage of the West: Plotinus, Johns Scotus Eriugena, Ibn Arabi, Marguerite Porete, and Meister Eckhart. Although the vast majority of Sells’s work is in this respect avowedly historical,3 his introduction prepares the way for these historical-literary analyses by generalizing the following “three key features of apophasis:” (I1) A “creative tension” between metaphors of emanation and intentional creation by which certain key distinctions get “fused” (e.g., the distinction between that which flows out and that which receives the flow); (I2) A dis-ontological discursive effort to avoid reifying the transcendent as an entity by continually “un-saying” it; (I3) A dialectic of transcendence and immanence by which conventional logical and semantic structures are violated (e.g., reflexive/non-reflexive, perfect/imperfect, univocal antecedents of pronoun) [13, pp. 6 – 7]. And his epilogue follows up these historical-literary analyses by abstracting the following seven “formal principles of apophasis:”

3

Sells does not deal with all those texts that fit a formal definition of apophasis (e.g., Tao Te Ching, Vimalakirti Sutra), sticking to apophatic texts of a certain historical tradition (Neoplatonic and Abrahamic).

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(E1) The Aporia of Transcendence: the “open-ended process by which the original assertion of transcendence continually turns back critically upon itself;” (E2) A Language of Ephemeral, Double Propositions: the means by which “[n]o statement about X can rest as a valid statement but must be corrected by a further statement, which itself must be corrected in a discourse without closure;” (E3) The Dialectic of Transcendence and Immanence: the means by which “subject-object, self-other dichotomies are undone;” (E4) Disontology and Nonsubstantialist Deity: the means by which language “continually turns back upon the spatial, temporal, and ontological reifications it has posed” with respect to the transcendent; (E5) Metaphors of Emanation, Procession, Return: the means by which “causal explanation is displaced” through “a rich set of grammatical transformations” in which emanation is identified with return; (E6) Semantic Transformations: “the undoing of self-other, beforeafter, and here-there distinctions” through, for example, the fusion of pronominal antecedents, the undoing of reflexive and nonreflexive pronouns, and the destabilization of prepositions upon which temporal and spatial dualisms are based; (E7) Meaning Event: a literary mode constituted by the other six formal principles, which “is a reenactment (within grammar, syntax, and metaphor) of the fusion of self and other within mystical union” [13, pp. 207 – 209].4

4

Note that Sells presents these principles somewhat cautiously, saying that they are neither a “mechanism” nor a “formula than [sic] can be applied.”

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With these features and principles Sells does far more than simply offer a new approach for the study of ineffability; he proposes the implicit logic and conventions of apophatic discourse, the actual means by which apophatic writers violate conventional grammatical rules. That said, there are a number of organizational issues with Sells’s features and principles, most notably a good deal of overlap between them as well as a failure to distinguish clearly between actual grammatical techniques and their performative effects. For example, E1, E2, and E4/I2 overlap considerably, pertaining in different ways to the problem of naming the transcendent: because a non-substantial (E4/I2), transcendent (E1) deity cannot in principle be named, propositions containing such names or pronouns must be continually negated by subsequent propositions (E2/I2). And E3/I3 and E6 also overlap, as the performative effects of E3/I3 find their grammatical causes in E6: the undoing of self-other, subject-object, transcendent-immanent, and before-after dichotomies (E3/I3) are caused by the grammatical techniques fusion of pronoun antecedents, undoing of grammatical distinction between reflexive and non-reflexive action, destabilization of spatial and temporal prepositions, and fusion of perfect and imperfect action (E6).5 A proper evaluation of Sells’s claim that apophatic discourse is antinomian therefore first requires a reorganization of his formal principles, one that differentiates grammatical techniques from performative effects. Here is one such attempt:

5

Although Sells neglects to list this last technique under his “formal principles,” it figures prominently elsewhere in Mystical Languages of Unsaying.

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TABLE 1: Sells’s features and principles of apophasis: a reorganization Grammatical Technique Syntactic-Semantic Event Pronoun Anaphora Syntactic ambiguity between Ambiguity (I3, E3, E6) reflexive and non-reflexive senses of pronouns (such that pronouns denote two different antecedents) Verb Tense Ambiguity Syntactic ambiguity between (I3, E6) perfect and imperfect senses of verbs (such that verbs connote completed and ongoing action) Spatial-Temporal Semantic tension caused by Preposition Semantic identification of contrary Tension (I1, E3, E6) spatial-temporal prepositions (e.g., in and out, before and after) Emanation Semantic Semantic tension caused by Tension (I1, E5) identification of cause and effect of emanation, “stuff” and receptacle of emanation, processes of emanation and return Illocutionary Negation Violations of the laws of non(I2, E1, E2, E4) contradiction and excluded middle in successive illocutionary acts

Performative Effect Breaks down self-other, subject-object, transcendentimmanent dichotomies

Breaks down dichotomy

before-after

Breaks down self-other, subject-object, transcendentimmanent, before-after dichotomies Frustrates ordinary spatialtemporal understanding of causation

Negates or unsays referential expressions used to name the transcendent

While such a reorganization of Sells’s formal principles helps to make possible an evaluation of the claim that these grammatical techniques are rule-violating, also needed is an account of the rules that these techniques supposedly violate. Unfortunately, though, such an account is nowhere present in Mystical Languages of Unsaying. And this is most perplexing, not only since Sells’s thesis requires it but also since Sells seems to say as much on a number of occasions: It is only upon a foundation of conventional logic and semantics that the apophatic texts, at the critical moment, can perform (rather than assert) a referential openness — by fusing the various antecendents of the pronoun, or the perfect and imperfect tenses, or by transforming the spatial and temporal structures of language at the level of article, pronoun and preposition [13, p .8].

268 When the transcendent realizes itself as the immanent, the subject of the act is neither divine nor human, neither self nor other. Conventional logical and semantic structures — the distinction between reflexive and nonreflexive action, the distinction between perfect and imperfect tense, the univocal antecedent of a pronoun — are broken down [13, p. 7]. The paradoxes of transcendence and immanence, the coincidences of opposites, the displacement of the grammatical object — all are in violation of the conventional logic that functions for delimited entities. It is when language encounters the notion of the unlimited that conventional logic, not illogically, is transformed. The apophatic paradoxes are constructed upon a foundation of conventional logical distinctions; the more highly tuned the rationality of the kataphatic context, the more successful will be the apophatic paradox [13, p. 212]. The following seven readings will examine how in apophatic mystical union reference to the transcendent is undone, and how that undoing is reflected within language by a disorienting — at certain key points — of standard rules of reference and transcendence [13, p. 12].

Still, given Sells’s formal principles, the reader can fairly easily fill in the rules of ordinary language that these principles putatively violate. Here, again, is one such attempt: TABLE 2: Linguistic rules putatively violated by Sells’s features and principles of apophasis: an emendation (R1) Rule of reference in general: reference cannot succeed if the referent cannot be clearly identified as something. (R1a) Rule of reference applied to cases of pronoun anaphora ambiguity: reference cannot succeed if pronouns denote multiple antecedents. (R2) Rule of prediction in general: contradictory and contrary properties cannot be predicated of some referent. (R2a) Rule of predication applied to cases of verb tense ambiguity: predication cannot succeed if verbs connote both completed and ongoing action.

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(R2b) Rule of predication applied to spatial-temporal preposition semantic tension: predication cannot succeed if prepositions connote contradictory or contrary spatial-temporal meanings. (R2c) Rule of predication applied to emanation semantic tension: predication cannot succeed if the properties or actions of emanation connote contradictory or contrary properties. (R3) Rule of illocutionary acts in general: illocutionary acts cannot be achieved on contradictory propositional content. (R3a) Rule of illocutionary acts applied to cases of illocutionary negation: if the propositional content of any two illocutionary acts is in violation of either the law of noncontradiction or the law of the excluded middle, neither illocutionary act can succeed. Given these reorganized techniques and explicit rules, we are now in a better position to assess Sells’s claim that the former violate the latter. 2. The rules of ineffability performance: a first critique Do the grammatical techniques of apophatic discourse violate the grammatical conventions of ordinary discourse? Here, I’d like to begin by observing that even if apophatic techniques violate rules R1a, R2a, R2b, R2c, and R3a above, they do not and cannot violate basic rules of reference (R1), predication (R2), and illocutionary acts (R3). The writer can only show and the reader can only understand that something in particular is ineffable if they have first identified what that thing is and established that the category of ineffability is a coherent concept under which that thing can and does fall. (Of course, none of this needs to be done explicitly.) And so even if God is not a thing that can be picked out among things, God still needs to be identified as distinct from other things if one is to go about

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performing God’s ineffability.6 And even if God possesses supposedly contradictory and contrary properties, God must clearly fall under the category of ineffability in some respect if one is to go about performing God’s ineffability. In general, then, insofar as performances of ineffability communicate something about something, they require basic rules of reference, predication, and illocutionary force that cannot be violated. But what about Sells’s individual features and principles? In this case, I’d like to begin by observing the commonplace: readers make sense of the texts they read. When they encounter a stretch of ambiguous discourse, they therefore “disambiguate” it, employing what Umberto Eco has called “topics” (interpretive schemas) to actualize “isotopies” (semantic properties) of the text.7 Above all, two things should be emphasized here: first, isotopies are actual semantic properties of texts that make possible their disambiguation; second, isotopy is an “umbrella term” that encompasses “coherence at the various textual levels” [2, pp. 189, 190]. And this is what I will demonstrate below: the texts Sells reads possess isotopies that enable their disambiguation, and these isotopies are present at various levels in these texts. Due to a limitation of space, though, I’ll focus my attention below on the techniques that show up in the first chapter of Sells’s work (on Plotinus8) — pronominal anaphora ambiguity, emanation semantic tension, illocutionary negation — and the isotopy types that disambiguate them — discursive isotopies within sentences with syntagmatic disjunction, narrative isotopies connected with isotopic discursive disjunctions generating mutually exclusive stories, and 6

The premise of Sells’s chapter on Plotinus (ch. 1) — which will be examined in depth below — is that “any ‘static felicity’ or ‘superessential presence’ of an entity called ‘the one’ is undone within the Enneads, and that it is undone precisely in those passages where the expressions of Neoplatonic faith are most intense” [13, p. 15]. Even if so, this can only happen if the reader knows not only what it is that is being referred to here but also what it is that is being “said” of this referent. 7 Writes Eco: “the topic as question is an abductive schema that helps the reader to decide which semantic properties have to be actualized, whereas isotopies are the actual textual verification of that tentative hypothesis” [3, p. 27]. 8 Sells believes that an overview of Western apophasis begins with Plotinus — even though elements of it existed earlier in Heraclitus, Plato, and Aristotle — for “it was Plotinus who wove these elements [of apophasis] and his own original philosophical and mystical insights into a discourse of sustained apophatic intensity” [13, p. 5].

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discursive isotopies between sentences with paradigmatic disjunction, respectively. Nevertheless, I provide all of Eco’s isotopies here: FIGURE 1: Umberto Eco’s eight isotopy types [2, p. 193] with paradigmatic disjunction within sentences with syntagmatic disjunction discursive isotopies with paradigmatic disjunction between sentences with syntagmatic disjunction

INTENSIONS connected with discursive isotopic disjunctions

exclusive complementary

narrative isotopies unconnected with discursive isotopic disjunctions EXTENSIONS – extensional isotopies concerning possible worlds

Sells builds his case for the first of his Plotinian grammatical techniques — pronominal anaphora ambiguity — throughout his chapter on Plotinus, suggesting that some variation of the phrase it is as it makes it(self) to be is present in many of the passages that he translates from Enneads VI.8. In fact, though, there are actually only three occurrences of autos that Sells translates as him(self)/it(self), only one of which is at all referentially ambiguous.9 I provide it here in Sells’ translation, though with some crucial omitted passages restored (and the passage in question italicized): He is everywhere and nowhere … if nowhere he has happened to be, and if everywhere, then just as he is, he is everywhere so that he is the everywhere and the everyway. [But he, since he has the highest place, or rather does not have it, but is himself the highest, has all things as slaves; he does not happen to them, but they to him, or rather they happen around him; he does not look to them, but they to him; but he is, if we may say so, borne to his own interior, 9

The other two occurrences — which fall in one and the same sentence — will be noted below.

272 as it were well pleased with himself, the “pure radiance,” being himself this with which he is well pleased; but this means that he gives himself existence, supposing him to be an abiding active actuality and the most pleasing of things in a way rather like Intellect. But Intellect is an actualisation; so that he is an actualisation. But not of anything else; he is then an actualisation of himself. He is not therefore as he happens to be, but as he acts.] If then he exists in view of holding fast toward himself and gazing () toward himself, and the being () is for him that very gazing toward himself, he would then make him(self) () [oi/on poioi/ a'n au`to,n], and then he is not as he happened to be but as he wished to be, [and his willing is not random nor as it happened; for since it is willing of the best it is not random] [13, p. 25].10

Sells’s argues that the antecedent of the Greek term auton (him/self) is here unclear, split between the One and Nous, and therefore in fact refers to both antecedents. As Sells puts it, From the point of view of the pronoun, the reference is split, referring back to two different possible antecedents. From the perspective of the antecedents which are semantically fused at such a moment, the reference can be fused [13, p. 26].

So what we have here is an interpretive decision regarding a discursive isotopy within a sentence with syntagmatic disjunction: either auton functions reflexively, coreferring with the subject of poioi to One, or auton functions non-reflexively, referring to Nous, or auton functions both reflexively and non-reflexively, referring to both One and Nous.11 But which of these interpretive topics is best supported by an actual textual isotopy? Well, we can first say that, since auton is not in the nominative, does not intensify a substantive or pronoun, and is not preceded by an article, auton functions here as the simple personal pronoun him/it [14], §1204.12 But as Smyth’s Greek Grammar says, such oblique cases of autos 10

See Enneads VI.8.16.1—24. A few notes are in order about this passage. First, the restored/bracketed text is from the Armstrong translation of the Enneads. Second, Sells uses the convention () in lieu of the Greek term hoion since “it appears with such frequency that to translate it each time as ‘as it were’ would be cumbersome” [13, p. 25]. Third, Sells underlines the term is for reasons that are not identified but seem to be related to an effort to combat the misinterpretation of the One as a static entity. (Interestingly, Sells begins Mystical Languages of Unsaying by accusing modern translators of Meister Eckhart’s writings of selective interpolation [13, p. 1].) 11 Eco’s example of this isotopy is establishing the “co-reference” of they in “They are flying planes” [2, p. 194]. 12 I thank John Finamore for calling my attention to this.

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“usually take up a preceding noun (the anaphoric use)” [14], §1214. And the only proximate and prominent preceding noun is the implied subject of all the verbs in this sentence, the One.13 (Nous is mentioned three and four sentences earlier, but there only as a point of comparison for the actualizing activity of the One.) So, there is nothing for auton to refer to here except the One. So, only the first of our interpretive topics is verified by a textual isotopy: auton functions reflexively, co-referring with the subject of poioi to the One. (Note: both universal grammar and cognitive grammar confirm this.14) Of course, this doesn’t solve the problem of what it means that the One holds to himself, looks to himself, and makes

13

Compare Sells’ other two examples of pronominal antecedent ambiguity (which occur in one and the same sentence): “Perfect, seeking nothing, having nothing, needing nothing, [the one] overflowed, as it were, and its overflowing made its other. This begotten turned back toward it(self) and was filled and became the contemplator of it(self) and became nous” (to. de. geno,menon eivj auvto evpestra,fh kai. evplhfw,qh kai. ege,neto pro.j auvto. ble,pon kai. no/uj ou/toj) [13, p. 28], Enneads V.2.1. Here, since to hen is the only prominent noun prior to the introduction of “the begotten other,” these two occurrences of auto must refer to the One. Nous (the begotten) turns back toward the One and is filled and, by looking toward the One, becomes Nous. 14 In terms of the theory of government and binding: reflexives and pronouns are mutually exclusive categories, the former of which is bound by a local antecedent (within its governing category, in this case the clause oi/on poioi/ a'n au`to,n), the latter of which is not bound by a local antecedent [4, p. 240]; if auton is a pronoun, then it cannot be bound by a local antecedent, and must instead have some non-local antecedent; but the closest possible non-local antecedent, nous, is three sentences away; so it is very unlikely that the antecedent of auton is nous and therefore that auton is a pronoun; so the antecedent of auton is the subject of its governing category, one, and auton is a reflexive. In terms of cognitive grammar (where “anaphora constraints are not independently stated syntactic principles” but rather are “fundamentally semantic/pragmatic in nature” [5, p. 5]: since “reference points” anchor conceptual structures (dominions) and define contexts for anaphora constraints [5, pp. 53, 54], “[t]he antecedent for a pronoun must be sufficiently salient (i.e., distinct and prominent) within the context in which the pronoun appears that it can plausibly be construed as a reference point with the pronoun in its dominion” [5, p. 57]; while nous is not sufficiently prominent within the context in which auton appears, one is, functioning as the “trajector” not only in this sentence but also throughout this section; so, one is reference point of auton.

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himself.15 And more broadly, it doesn’t solve the problem of how an undifferentiated One can generate Nous (which is both same as and different from the One).16 But these might best be understood as problems that are endemic to any attempt to say how that which is supposedly absent of all duality gives rise to something else (which must be contained within it before hand and yet is externalized out of it as something different). To this we now turn. Given the problems involved in saying how the One emanates that which is both the same as and different from itself, it shouldn’t at all be surprising that the Enneads passage above is rife full of the adverb hoion (“as it were”). Sells calls hoion an “apophatic marker” that indicates that a name or predicate should not be taken at face alue. But he also claims that in the case of Enneads VI.8.16 hoion occurs with such frequency that its impact gets diluted, thereby resulting in an “unreadable text” [13, pp. 16 – 17]. (This is also the reason why Sells decides to “translate” hoion as () [13, p. 25].) Thus, what at first glance appears to be a decision concerning a discursive isotopy within a sentence with paradigmatic disjunction — i.e., which semantic value to assign to each hoion-marked term — turns out to be a decision concerning a narrative isotopy connected with isotopic discursive disjunctions generating mutually exclusive stories — i.e., whether or not to “read” passages containing clusters of hoion-marked terms as unreadable.17 And since all the hoion-marked terms above involve the dynamics of emanation,18 this narrative isotopy concerns more 15

Here, David Bradshaw says that Plotinus embraces the notion that the One is “supreme energeia,” though at the apparent cost of importing “minimal duality” into the One [1, p. 87]. Bradshaw’s believes that we need to understand this state as one that is absent of both experiential and ontological duality [1]. 16 Here, Bradshaw believes that the One generates Nous through its “supreme energeia,” which is epekeina energeias qua prōtē energeias [1, pp. 90 – 91], and which comes in two different modes, an external mode (that which imparts) and an internal mode (that which is imparted) [1, p. 76]. 17 Eco’s example of this isotopy is an extract from a French translation of Machiavelli that can be read as either a story about a friend of Domitian’s giving him an argument about power or a story of a friend of Nerva’s making Domitian the victim of a courtier’s wiles [2, p. 197]. 18 Curiously, Sells fails to mark one term (holding fast) and adds marks to three unmarked terms (awakener, rational life, these [being, nous, rational life]).

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specifically the way in which Plotinian emanation imagery should be interpreted in general — either as attempts to metaphorize emanation in partially adequate ways or as attempts to undermine such metaphors, showing their complete inadequacy. That Sells not only implies that the excerpt above from Enneads VI.8.16 is an “unreadable text” but also goes on to itemize the following “dilemmas” of emanation, shows, I believe, how he resolves this isotopy. The dilemma that the vessel is the content: “However, if the one produced the other by overflowing, and if that other is what flows from it, then to state that the other is filled by the same perpetual overflowing is to fuse together the vessel-content dualism on which the metaphor is based, or to first pose the dualism and then withdraw one element. This is the dilemma that the vessel is the content” [13, pp. 28 – 29]. The dilemma of emanation: “What flows out is identical to the act of flowing out, the result of the process is the process” [13, p. 29]. The dilemma of procession and return: “the procession cannot be said to have occurred until the return” [13, p. 29].

Now, I think we can first freely admit that no spatial-temporal imagery could ever completely adequately picture the emanation of every-thing from, and return of every-thing to, a principle that is no-thing, especially when a good deal of this emanation and return occurs in an intelligible, non-corporeal realm. And, so, the adverb hoion stands as an all-important signal not to take this imagery too literally.19 But let us not overlook the fact that Plotinus does a fine job of finding metaphors that are apt for showing how one “thing” might cause another “thing” that is both the same as and different from it. Here, flowing water and radiating light usually get the attention. But I’d like to focus instead on what may be a better metaphor, vegetative growth — the way in which a tree “causes” leaves to 19

Here Eric Perl reminds us that Plotinian causation is not an event or process at all; rather it is “nothing other than Platonic participation” [7, p. 19]. Later Perl adds, “When Neoplatonic vertical causation, or ‘procession,’ is understood as the dependence of the determined on its determination and hence as the differentiated appearance of the unitary determination, it becomes clear that the production of the effect is not an activity on the part of the cause, distinct from the cause itself” [7, p. 26].

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grow out of it that just are the tree and yet are differentiated aspects of the tree. This is an example of George Lakoff and Mark Johnson’s primary metaphor causation is movement out, which, qua primary, is a widespread, if not universal, means of understanding and speaking about causality in accordance with concrete bodily experience [6, pp. 45 – 59, 170 –234]. And I believe it provides a rather useful and adequate means of conceptualizing the relationship between the One and its emanations. More importantly, Plotinus seems also to believe so: for it is this very metaphor that Plotinus draws on in the closing sentences of the preceding section of the Enneads (VI.8.15). All this is to say that there is an isotopy at the level of the narrative or text that allows for the disambiguation of these hoion-marked terms in particular and the “dilemmas” of emanation in general. And this isotopy can be actualized through the employment of a topic that takes these occurrences of hoion as warnings not to take too literally, yet also not to take as completely inadequate, the terms and images that Plotinus uses in speaking about the dynamics of emanationreturn. The passages in which clusters of hoion-marked terms appear should not therefore be read as unreadable. Sells initially presents his last Plotinian technique in the form of the Plotinian image of removing a glowing mass from the center of a hollow sphere. This, says Sells, is an image of “apophatic abstraction: to reach into a reference and withdraw the delimited referent, to reach into the notion of contemplating something and withdraw the ‘some-thing’” [13, p. 18]. This is also nothing less than “unsaying” or “apophasis” itself (which is also associated here with the techniques of “double propositions,” “dialectic of transcendence and immanence,” and “paradox”). Sells attempts to establish this grammatical technique in a passage from Enneads VI.8.9, with respect to which he argues that delimiting statements such as “it is beyond all things” and “it is within all things” cannot be taken independently of one another. Rather: Meaning is generated between the two propositions: it is within all things—it is beyond all things. In effect, the smallest semantic unit is not the sentence or the proposition, but the double sentence or dual proposition [13, p. 21].

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Thus Sells credits Plotinus with structuring his propositions in Enneads VI.8.9 and elsewhere such as “to counter the tendency to give independent propositional status to the last sentence” [13, p. 22]. But, regarding this passage, he says more: When Plotinus writes that “it is neither X nor not X,” he violates the logical rule of the excluded middle. When he writes that “it is both X and not X,” he violates the law of noncontradiction” [13, pp. 20, 21].

Now note the obvious: these could be two very different claims. On the one hand, it might just be the case that it takes Plotinus a number of sentences to put the point as precisely as possible, and therefore that, if we want to understand his point as well as possible, we shouldn’t take any one of these sentences out of context. On the other hand, though, Sells seems to be arguing (at times) that Plotinus’ point is that there just isn’t a way to put the matter at all, and therefore that, if we want to understand this, we need to read these sentences as contradicting one another. Thus, we have here an interpretive decision regarding a discursive isotopy between sentences with paradigmatic disjunction.20 Should we read contiguous sentences that appear to affirm and deny the same predicate of the same subject as qualifications or contradictions of one another? Now, first note that the passage that Sells’ translates from Enneads VI.8.9 contains neither any violations of the law of non-contradiction nor, if all predicates are category mistakes of the One, any violates of the law of the excluded middle: Then there can be no “thus.” It would be a delimitation and a some-thing. One who sees, knows that it is possible to assert neither a thus nor a not-thus. How can you say that it is a being among beings, something to which a thus can be applied? It is other than all things that are “thus.” But seeing the unlimited you will say that all things are below it, affirming that it is none of them, but, if you will, a power of absolute self-mastery. It is that which it wills to be; or rather, the being that it wills to be it projects out into beings, while it remains greater than all its will, all will being below it. So neither did it will to be a thus, so that it would have to conform (to its thus), nor did another make it so [13, pp. 19 – 20], Enneads VI.8.9.38 – 48. 20

Here Eco’s example is that of the party-goer who praises the food, the service, the hospitality, the beauty of the women, and finally the excellence of the “toilettes” [2, p. 195].

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Note also that Sells himself both initially presents this passage as containing Plotinus’ struggle to avoid two unacceptable interpretations of the One’s causation — to say that the One wills emanation would imply some kind of need or lack, while to say that the One emanates by necessity entails that the One is compelled by something [13, p. 19] — and immediately follows this passage with the following diagram of Plotinus’ argument: (A) Because it is free, no other makes it what it is. It is what it wills itself to be. (B) But it cannot even be said to be limited to what it wills itself to be. (C) We should say that it projects the being, the quiddity, the ‘what’ that it wills itself to be, out into the realm of beings. It always remains beyond its own being and its own willing [13, p. 21].

Thus Sells himself seems to be giving the following interpretation of this passage: since the One is not a being that can be predicated, it is false to say either that the One freely wills emanation or that the One is predetermined to emanate;21 but since the One is determined by nothing, it is especially wrong to think that One emanates by necessity; therefore, it is better to think that the One freely wills itself to be; but since the One is prior even to will, this understanding of the One is not one of what the One actually is but rather of what the One becomes after and through emanation. This isn’t a bad interpretation [7, pp. 49 – 51]. But it also isn’t an interpretation that supports the extreme claim that Plotinus is here violating the laws of non-contradiction and the excluded middle. Rather, Sells carefully shows how Plotinus carefully specifies the respects in which these predicates do and do not apply. In support of this standard Aristotelian position on the law of non-contradiction, we can reiterate the standard Aristotelian position on the law of the excluded middle: in cases where predicates are category mistakes of their referent, opposite

21

Consider here the passage that Sells excerpts from Enneads VI.8.13 (wherein Plotinus makes it clear that the language of willing is not to be taken literally): “We use these names now for the sake of persuasion and in doing so we depart from strict accuracy” [13, p. 24].

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predicates may both be false.22 And to this, we can add a standard Searlean point about illocutionary force negation: when Plotinus says that “it is possible to assert neither a thus nor a not-thus,” Plotinus is negating the illocutionary force of assertion, not asserting a negative propositional content [12, p. 32], cf. [11]. Altogether then we have a picture that is not all that exceptional: about the One itself, it is not possible to assert; however, if one asserts of the One itself, one must do so by denying both will and necessity since all categories are mistakes of the One itself; and when it comes to speaking about the being that the One projects out into beings, it is better to think of the One as freely willing since necessity connotes determination by something else. This isotopy makes possible the disambiguation of this passage through the implementation of a topic that takes different uses of these predicates differently. All three of these Plotinian grammatical techniques are therefore easily disambiguated by textual isotopies that are readily accessible to the reader. Moreover, all three of these Plotinian grammatical techniques are governed by quite ordinary rules and conventions — the ordinary conventions of pronominal anaphora; the primary metaphor causation is emergence out; the standard Aristotelian positions on the laws of non-contradiction and the excluded middle; the common speech act distinction between illocutionary force negation and propositional content negation — not to mention basic rules of reference, predication, and illocutionary force. So, if it is the case that apophatic discourse has a distinctive set of rules and conventions, these rules and conventions absolutely require and therefore cannot contravene the rules and conventions of ordinary discourse. And if it is the case that apophatic discourse performs ineffability, it does so only insofar as it’s rule-governed. (Indeed, even if one chooses to read the Enneads “antinomianly” — and since this author is in more than one way “dead,” one is certainly able to do so! — one still reads the Enneads as rulegoverned.) Another way of putting all this is as follows: if apophatic discourse performs ineffability, it only ever does so relatively since it can never avoid trafficking in the rules and conventions of ordinary discourse. Now, given some of Sells’s comments in Mystical Languages of Unsaying, 22

Aristotle’s Categories 11b17ff, 11b38ff, 13b12ff; On Interpretation 19b20ff, 20a31ff; Prior Analytics 51b5ff.

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one might expect Sells to agree with this. The achievement of unsaying is, for him, “unstable and fleeting;” the tension between unsaying and saying, “momentary” [13, pp. 217, 3, 61]. Apophasis, for him, “seeks a momentary liberation” from the limitations of reference, but “cannot dispense with reference” [13, pp. 8 – 9; cf., 20, 31, 61]. And the performance of ineffability has, according to him, a range of intensity which “is a function of the frequency and seriousness with which the language turns back upon its own propositions” [13, p. 3]. But this, for Sells, indicates above all, not the limited success of apophatic discourse, not the dependence of apophatic discourse on ordinary rules, and not the way in which apophatic discourse can only ever show the relative ineffability of some “thing” — rather, it shows that apophatic discourse reenacts an underlying experience or object that is absolutely immediate and ineffable. 3. The experience of ineffability performance: a second critique Sells wants his performances of ineffability to be more than relatively antinomian grammatical techniques. He wants them to reveal the ineffability of their underlying mystical “experiences” or “objects.” To do so he draws upon Paul Ricoeur’s notion of a “meaning event,” though not without mangling it in the process. For Ricoeur, the meaning event is simply the dialectic of illocutionary act (event) and semantic content (meaning) [9, p. 11]. All discourse is necessarily constituted by both event and meaning; to consider one apart from the other is always an “abstraction” [9, p. 11]. For Sells, though, the meaning event is something entirely different. As the seventh of Sells’s seven formal principles of apophasis, it is not only a literary mode constituted by the other six principles of apophasis but also the “semantic reenactment” of mystical union: “The meaning event is a reenactment (within grammar, syntax and metaphor) of the fusion of self and other within mystical union” [13, p. 209]. As such the meaning event fuses or identifies or realizes event and meaning:

281 Meaning event indicates that moment when the meaning has become identical or fused with the act of predication. In metaphysical terms, essence is identical with existence, but such identity is not only asserted, it is performed [13, p. 9]. The mystical writers discussed below claim a moment of ‘realization’ — a moment in which, again, the sense and reference [i.e., meaning] are fused into identity with event [13, p. 9]. In other kinds of discourse, or other moments of apophatic discourse, we can distinguish the event (the act of predication) from the meaning (as sense and reference, the “what” and the “what about”). In mystical dialectic, when predication and reference become realization, there can be no distinction between meaning and event [13, p. 88].

Given the elementary nature of Ricoeur’s meaning event, it’s hard to know how to take these extravagant claims. If they say that, in the case of apophatic discourse, propositional contents always occur within or by way of illocutionary acts, then these claims are in fact trivial, saying nothing about apophatic discourse that sets it apart from non-apophatic discourse. If they say that, in the case of apophatic discourse, propositional contents are identical to illocutionary acts, then these claims are patently false, collapsing of the basic distinction between illocutionary force (doing) and propositional content (saying) that constitutes all discourse. And if they say that, in the case of apophatic discourse, propositional contents are somehow mystically unified with illocutionary acts, then, well, it’s not clear exactly what to say other than that Sells seems to have made a category mistake. In fact, though, this last reading of Sells’s meaning event appears to be the correct one, since Sells explicitly says in the introduction that he uses the notion of meaning event in contrast to the modern concept of experience [13, p. 10], and in general moves between the realms of experience and language as if they were cut from the same cloth. Sells claims that the meaning event continually shifts back and forth between mystical realization (experience) and linguistic predication: The meaning event is constantly being repeated. It cannot be possessed. The continual shift from predication to realization keeps the mind in constant activity, continually displacing the “object,” as the infinite regresses built into the discourse lead the reader deeper into the aporetic meditation. The meaning of narrative, philosophical, mythological, and poetic language (the kataphatic aspect) is not negated, but “what” is meant becomes one with the event of the perspective shift [13, p. 215].

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He asserts that mystical union (experience) breaks down the logical and semantic structures pertaining to reflexive and nonreflexive action, perfect and imperfect tense, and pronominal anaphora: When the transcendent realizes itself as the immanent, the subject of the act is neither divine nor human, neither self nor other. Conventional logical and semantic structures — the distinction between reflexive and nonreflexive action, the distinction between perfect and imperfect tense, the univocal antecedent of a pronoun — are broken down. This moment in which the boundaries between divine and human, self and other, melt away is commonly called mystical union [13, p. 7].

And he says that experience in general — which does not include mystical “experience” (more later) — possesses a grammatical object: As defined by many, experience presupposes (1) a grammatical object of experience; (2) mediation; (3) constructedness [13, p. 10]. If the non-intentionality claims of apophatic mystics are taken seriously, and if experience is, by definition, intentional, it necessarily follows that mystical union is not an experience. All experience must have a grammatical object, but the prime motivation of apophatic language is to subvert or displace the grammatical object [13, p. 214].

Why this conflation of experience and language? — Sells wants the antinomian nature of apophatic discourse to count as evidence for the nonintentional, unmediated, ineffable nature of mystical union. Thus in some cases Sells speaks of the meaning event as a recreation or imitation or reenactment of mystical union. The meaning event is the semantic analogue to the experience of mystical union. It does not describe or refer to mystical union but effects a semantic union that re-creates or imitates the mystical union [13, p. 9]. While the mystical experience of the self-revelation of the transcendent in the annihilated heart is called realization, the semantic re-enactment of that realization — what the writer and reader encounter in the act of writing and reading — is a meaning event [13, p. 208].

In other cases he refers to the meaning event as a reflection or effect of mystical union:

283 At the moment of mystical union, the undoing of self-other, before-after, and here-there distinctions is reflected in radical grammatical and semantic transformations [13, p. 208]. This moment in which the transcendent reveals itself as the immanent is the moment of mystical union. At this moment, the standard referential structures of language are transformed [13, p. 212].

And in still other cases he speaks of the meaning event as evidence for the immediacy or non-intentionality of mystical union: The readings of apophatic language of mystical union that follow focus upon how such language displaces the grammatical object, affirms a moment of immediacy, and affirms a moment of ontological pre-construction [13, p. 10]. If the non-intentionality claims of apophatic mystics are taken seriously, and if experience is, by definition, intentional, it necessarily follows that mystical union is not an experience. All experience must have a grammatical object, but the prime motivation of apophatic language is to subvert or displace the grammatical object. Similarly, the notion of the unmediated at the heart of apophatic mysticism (particularly that of Eckhart and Porete) contradicts the common opinion that all experience is mediated [13, p. 214].

It is therefore difficult to know what to make of Sells’ repeated claims that he makes no presuppositions about the relationship between mystical experience and apophatic discourse: This study makes no presuppositions about the exact relationship of the meaning event to the mystical experience of realization (such a relation is defined by each reader) [13, p. 208]. This study makes no presuppositions concerning mystical experience on the part of the writer or reader. The goal is to identify the distinctive semantic event within the language of unsaying, what I will be calling the “meaning event” [13, p. 9]. No attempt has been made here to discuss a common religious experience, or a common mystical experience. The goal of this study has been an understanding of a similarly structured semantic event that takes place within various versions of the apophatic mode of discourse. I have tried, apophatically, to refrain from defining this event [13, p. 216].

More to the point, though, Sells’s use of antinomian apophatic discourse as evidence for non-intentional mystical union just isn’t sound. Even if some writer’s apophatic discourse about some mystical experience is viciously

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antinomian, this need not mean anything about the content of this experience (not even that there actually was an experience). Moreover, the fact that even viciously antinomian apophatic discourse is about something suggests that, if there were underlying corresponding experiences, they too were about something and therefore were not non-intentional and immediate. And to say this is not, as Sells thinks, to dismiss apophatic discourse [13, p. 214]; rather, it is to enable apophasis as discourse, to allow apophasis to say (and unsay) something about something. And if this were not the case, Sells would have nothing to write about. Indeed, if Mystical Languages of Unsaying says anything about the experiences that are reflected in apophatic discourse, it suggests that these experiences are not ineffable, immediate, and non-intentional. Finally, it is just question begging to say that, if the modern notion of experience presupposes intentionality and mediation, then these “experiences” are not really experiences [13, p. 214].23 4. Comparative discourses of ineffability: a call Despite these critiques of the particular claims of Sells’s project — apophatic discourse momentarily violates (all) ordinary linguistic rules, in so doing it reenacts an (absolutely) ineffable experience of mystical union — Sells’s general method holds great potential for the study of expressions of inexpressibility. Some of this potential is merely negative: Sells’s general method frees us from the overtired and intractable question about the ineffability of mystical experience. But most of this potential is positive: Sells’s general method frees us for the analysis of the actual grammatical means by which authors attempt to speak about that which they claim they cannot speak about, the rules that govern those techniques, and the intra- and inter-religious patterns that show up with respect to such techniques and rules. While analyses of expressions of inexpressibility 23

And just because the category of experience is a “modern construct” doesn’t mean that pre-modern peoples didn’t have experiences — indeed, it’s hard to imagine things any other way. For all these reasons, I am in agreement with John Bussanich’s charge that Mystical Languages of Unsaying does not provide sound reasons “for doubting the validity of the influential Katzian constructivist approach” to mysticism [8, p. 97].

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can’t tell us whether the objects of such discourses actually are ineffable, they can tell us a great deal about the ways in which different authors from different religious cultures speak about different ultimate realities in similar and different ways by showing us the precise respects in which different ultimate realities are said or showed to be ineffable.24 They reveal how different authors conceptualize ultimate realities by showing us how different authors try to make language fail ultimate realities. I would therefore like to end this essay by calling for concerted and coordinated scholarly research in comparative discourses of ineffability.25 References [1] Bradshaw, David. Aristotle East and West. New York: Cambridge University Press, 2004. [2] Eco, Umberto. Semiotics and the Philosophy of Language. Bloomington: Indiana University Press, 1984. [3] Eco, Umberto. The Role of the Reader. Bloomington: Indiana University Press, 1979. [4] Haegeman, Liliane. Introduction to Government and Binding, 2nd ed. Cambridge, Mass: Blackwell Publishers, 1994. 24

My own work here has focused only on the writings of Pseudo-Dionysius the Areopagite, wherein I have identified grammatical techniques and rules pertaining to hyperpredication, negation (which includes aphairesis and apophasis), ineffability and unknowability assertion, ineffability and unknowability direction, metaphors of unknowable darkness and metaphors of unknowable height. What I find interesting about these techniques is the fine-tuned ways in which they attempt to articulate both the logical inapplicability of properties to God and the preeminent possession of properties by God, both the utter unknowability and ineffability of God and the preeminent knowability and effability of God. But what would be really interesting and important would be to see whether and how these techniques and patterns show up in other religious-philosophical texts, and thereby to learn something not only about the similarities and differences that show up among interreligious expressions of inexpressibility but also interreligious conceptions of transcendence. 25 This may be the “formal approach” to apophatic discourse that Sells distinguishes from his own “historical approach.” Dauntingly, Sells says that such a formal approach “would be a massive enterprise” [13, p. 4]. I agree. And this is precisely why concerted and coordinated scholarly activity is required.

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[5] Hoek, Karen van. Anaphora and Conceptual Structure. Chicago: The University of Chicago Press, 1997. [6] Lakoff, George, and Johnson, Mark. Philosophy in the Flesh. New York: Basic Books, 1999. [7] Perl, Eric. Theophany. Albany: State University of New York Press, 2007. [8] Bussanich, John. Review of Mystical Languages of Unsaying, History of Religions 35/1, 1995, 96 – 97. [9] Ricoeur, Paul. Interpretation Theory. Fort Worth: Texas Christian University Press, 1976. [10] Rocca, Gregory. Review of Mystical Languages of Unsaying, Theological Studies 55/4, 1994, 760 – 762. [11] Searle, John, and Vanderveken, Daniel, Foundations of Illocutionary Logic. New York: Cambridge University Press, 1985. [12] Searle, John. Speech Acts. New York: Cambridge University Press, 1969. [13] Sells, Michael. Mystical Languages of Unsaying. Chicago: The University of Chicago Press, 1994. [14] Smyth, Herbert Weir. A Greek Grammar. New York: American Book Company, 1920.