Logic, Language and Method - On Polarities in Human Experience: Philosophical Papers

Kuno Lorenz Logic, Language and Method − On Polarities in Human Experience

Logic, Language and Method − On Polarities in Human Experience Philosophical Papers

by

Kuno Lorenz

De Gruyter

The publication of this book was funded by the Laboratoire d’Histoire des Sciences et de Philosophie (CNRS), Archives Henri-Poncare´, Nancy, and by the Department of Philosophy at the Universität des Saarlandes, Saarbrücken.

ISBN 978-3-11-020312-7 e-ISBN 978-3-11-021679-0 Library of Congress Cataloging-in-Publication Data A CIP catalogue record for this book is available from the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. 쑔 2010 Walter de Gruyter GmbH & Co. KG, Berlin/New York Printing: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Contents Preface ......................................................................................................VII Part I Philosophical Logic and Philosophy of Language 1. Rules versus Theorems A new approach for mediation between intuitionistic and two-valued logic (1973)............................................. 3 2. On the Relation between the Partition of a Whole into Parts and the Attribution of Properties to an Object (1977)................................. 20 3. Basic Objectives of Dialogic Logic in Historical Perspective (2001) ................................................................................................... 33 4. Pragmatic and Semiotic Prerequisites for Predication. A dialogue model (2005) ..................................................................... 42 5. Pragmatics and Semiotics: The Peircean Version of Ontology and Epistemology (1994) ..................................................................... 56 6. Intentionality and its Language-Dependency (1985) ........................... 62 7. Meaning Postulates and Rules of Argumentation. Remarks concerning the pragmatic tie between meaning (of terms) and truth (of propositions) (1987) ........................................................ 71 8. What Do Language Games Measure? (1989) ...................................... 81 9. Features of Indian Logic (2008)........................................................... 92 Part II Methods in Philosophy, in Art, and in Science 1. The Concept of Science. Some remarks on the methodological issue ‘construction’ versus ‘description’ in the philosophy of science (1979) .................................................. 109 2. Is and Ought Revisited (1987) ........................................................... 124

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3. Competition and Cooperation: Are They Antagonistic or Complementary? (1994)..................................................................... 140 4. Another Version of Methodological Dualism (1997) ........................ 148 5. The Pre-Established Harmony Between the Two Adams (1993) ...... 162 6. On the Way to Conceptual and Perceptual Knowledge (1993).......... 171 7. Self and Other: Remarks on Human Nature and Human Culture (2002) ................................................................................................. 186 8. On the Concept of Symmetry (2005) ................................................. 198 9. Procedural Principles of the Erlangen School. On the interrelation between the principles of method, of dialogue, and of reason (2008) .......................................................................... 207 Acknowledgments ................................................................................... 219 Index........................................................................................................ 223

Preface It is delightful to see a representative sample of one’s papers on various subjects collected in a volume in one’s own lifetime, let alone when it is a collection of papers written in a language that is not one’s mother tongue. I am particularly glad that with the present volume which accompanies two volumes of papers in German – the one preceding and the other succeeding this publication – readers who read English but not German and wish to become acquainted with my treatment of current philosophical issues will have a chance to uncover ‘family resemblances’ that relate the papers with one another. In this process the key term ‘polarity’ in the subtitle may guide the reader’s attention. Of course, as the papers were written in a period extending over more than thirty years and owe their origin to very divergent occasions, partly being commissioned and partly offered, they exhibit different stages of awareness with respect to both the procedures applied and the terminology used. If there are any inconsistencies, they may be explained by developmental change. I have deliberately refrained from trying to produce up-todate versions as ‘final’ ones. This includes retaining occasional overlap at places where maintaining self-containedness makes it necessary. Work in progress should not hide the signs that it is in need of further improvement or refinement. The reader who wants to know how the latest and most advanced version up to now looks like, is advised to turn to the last paper (II.9) and read it first. In addition he will also find there a short exposition of how my treatment of issues as Dialogical Philosophy is related to the tenets of the Erlangen School and its philosophical ancestry. In each of the two parts the papers are divided into two blocks of four each by an additional paper that occupies the central position. Whereas within the four blocks chronological order is maintained, the two bisecting papers (I.5, II.5) are singled out, because they treat Peircean and Leibnizian issues, respectively, and may thus, through views of two philosopherscientists of the past, be particularly well suited to visualize the two main interests that underly the argumentations in the papers of the first and the second part. These interests are indicated by the two subtitles, one referring to questions of logic and language, and the other to questions of method. Notwithstanding that division, in some way each paper touches at least one of the countless oppositions that permeate the ways of coping with what

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happens and has happened around us and with us. Be it Doing and Suffering, Subject and Object, Truth and Meaning, Body and Soul, Particular and General, Practical and Theoretical, Knowing-how and Knowing-that, Thought and Action, and a multitude of others. Thus, procedural questions are not out of purview in the first part, and substantial questions do enter the papers of the second part. The reader will often encounter direct investigations into some of the oppositions, some others being touched alongside (I.1, I.2, I.3, I.7, I.9, II.1, II.2, II.3, II.4, II.6), and he equally should cast an eye on how the use of oppositions in the course of arguing interrelates with their status of being an object of argumentation as well. Looking back it appears now that in a way most of the papers may basically be considered as different steps en route of reducing the conceptual oppositions that permeate content and set-up of the papers to the fundamental one of looking at actions as ‘objects’ and as ‘a means’ in statu operandi, i.e., ‘from a distance’ and ‘while engaged in doing it’. This is true, already, of my billet d'entrée into professional philosophy: the dissertation on arithmetic and logic as dialogue games under the supervision of my late teacher Paul Lorenzen. This way of looking, I come to argue eventually, arises due to a basic anthropological feature: Humans are organized dialogically, as I and You. When A does something – A in the role of I – B (being identical with A in but a special case) ‘sees’ A doing that, B in the role of You. This simultaneity of two dialogical roles extends to the level of sign actions: When, in the role of I, A, besides e.g. speaking, ‘says’ something, then B, in the role of You, besides e.g. hearing A, ‘understands’ something (I.4, II.7, II.8). Philosophers are particularly prone to the danger of presenting their thoughts as if they were elected to represent mankind each for herself alone in telling how things are ‘really’ and often tend to forget about their dependence on others, if not their fellow philosophers. Usually, in science and philosophy we are accustomed to treat issues from a distance. This is in tune with considering the very verbalization as, already, a kind of detachment from its subject matter: words and things fall apart. On the other hand, we are well aware of how verbal activity anywhere and without restriction is the most prominent medium of being engaged with something: thinking and doing coincide. But when activities themselves, plain or verbal, are articulated as in papers like the present ones, a stage is reached where matters become quite complicated. What is at stake, is the relation of one’s individual activity, both plain and verbal, with the activity of others and with the particular objectives of activities that are supposed to be intersubjectively identifiable (I.6, I.8). It is not surprising, therefore, that many of the conceptual oppositions nourish competing philosophical doctrines and have done that since their inception with the consequence that scientific theories that avail them-

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selves of such doctrines as their conceptual frame inherit competitiveness with respect to each other. Sometimes this happens to such a degree that it becomes difficult to compare such theories adequately, because a common language of comparison is itself a matter of dispute. As an example, think of how in the last decades the behavioral sciences have lost ground to the cognitive sciences while there is no clear account of the reasons why this has happened. Even looking for reasons has become more difficult as their conceptual distinction from causes seems to evaporate. This again may be understood as an offshot of the gradual rise of biology as the paradigm of science and the decline of physics in that role. This shift has fuelled anew the age-old attempts to set up a unified account of the social and the natural sciences aiming to treat cultural features as a more advanced state of natural features. Language and reasoning, for example, are thus considered to be a more developed form of animal communication.The boundaries between the mental and the physical have become permeable, so much so, that now one often finds philosophy of mind being treated as just a version of neurobiology. Theory-building is encroaching upon more and more territory formerly held by mostly non-verbal and often plain activities. This trend is influential now, for example, in education, but in the arts as well. Theorizing gradually supplants practical expertise by having corpora of theoretical knowledge at one’s disposal that purports to rule over the practice in question. In ever more fields ‘knowing-that’ acquires priority to ‘knowing-how’ seemingly in line with the classical heritage that ars (τέχνη) is inferior to scientia (ἐπιστήµη). However, in antiquity it was justification that characterized ‘epistemic’ knowledge, a ‘knowing-why’, in contrast to a ‘knowing-that’, a merely true opinion. Only in modern times up to the present true opinion is held to be sufficient, if it serves successful guidance of individual and common practice. Practical knowledge or ‘knowing-how’ as represented by skills of various kinds and relegated to the sphere of mere ‘technic’ knowledge (artes) in antiquity, is still now, in a world quite different from its ancient predecessor, waiting to be acknowledged as on a par with theoretical knowledge rather than being treated as dependent on theories that explain how it works. The growing presence of theories of any kind, or, more generally, of theorizing in the loose sense of speaking about something rather than experiencing it in practice, is the result of a well-known modern development. We encounter a process where practical experience recedes from reach in more and more cases for ever more people and where growing world-wide communication induces apparent knowledge of a rapidly increasing number of matters beyond any chance of checking it individually. Thus, in the world we live, it becomes difficult to realize the mutual

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dependence of verbal and plain activities and likewise of semiotics and pragmatics, if the two kinds of activities are turned into objects of investigation. It is a world of particulars, subdivided into various categories depending on the structure of the language used (a fertile source of oppositions!), such that dealing with them is increasingly supported by numerous artificial devices, and especially persons including theorists play roles that disappear when speaking about them instead of with them. The difficulty of discerning verbal and plain activity in order to get an idea of their mutual dependency is aggravated even further, because the artificial devices that support plain activity take on various sign functions by themselves; just consider the case of doing something with the aid of computers even outside their use as a means of communication, if I may disregard the mere metaphor of ‘communicating with computers’. Philosophical investigations should make these features visible. They can do this by means of their reflexive nature that distinguishes them from scientific investigations, particularly with regard to the question of how the procedures applied relate to the subject matter treated. If I should have succeeded in bringing this to the fore at least in some of the papers – and the reader is the only one who can judge it – the present volume will have served its purpose. I should add, this publication would not have been possible without the efforts of quite a number of people to whom I am deeply grateful. Particularly, I want to thank my friend and colleague Professor Jürgen Mittelstraß (Universität Konstanz) and my friends and former students, Professor Gerhard Heinzmann (Université Nancy II) and Dr. Bernd Michael Scherer (Haus der Kulturen der Welt/House of World Cultures, Berlin) who encouraged me not to give up the whole plan. They took pains to raise financial support without which a book of this kind would be an unbearable risk to publish under present economic conditions. For generous financial aid I am indebted to my department at the Universität des Saarlandes as well as to the Centre National de la Recherche Scientifique Lorraine (CNRS, UMR 7117) which supported the present publication at Walter de Gruyter Publishing House. Special thanks go to Dr. Gertrud Grünkorn at de Gruyter’s who did everything to edit this volume in a form that shows all the features for which books published at de Gruyter’s are justly wellknown everywhere.

Part I Philosophical Logic and Philosophy of Language Philosophical Logic and Philosophy of Language

Rules versus Theorems A new approach for mediation between intuitionistic and two-valued logic I Contemporary critics of two-valued logic concentrate on the reasons for

accepting the tertium non datur A ∨ ¬A as a valid propositional schema. Brouwer explicitly states1 that only by unjustified extrapolation of logical principles from those which correctly describe the general relations among propositions on finite domains to those that allegedly regulate propositions on infinite domains, could it happen that A ∨ ¬A is accepted as valid. He was the first to observe that value-definite (decidably true or false) propositions do not generally transfer value-definiteness to their logical compounds. No better support could be found for the claim that the classical characterization of propositions as entities that are either true or false is inadequate. The union of the class of all true propositions and the class of all false propositions does not contain all logical compounds out of either true or false propositions; it does not contain, for example, certain as yet neither proven nor disproven universal propositions of elementary arithmetic. But nobody has seriously advanced the thesis that such propositions should not count as propositions at all.2 In fact, it is generally conceded that the usual way to form finite and infinite logical compounds makes sense even if nothing can be said about their truth-value. It is obligatory, then, to look for a better introduction of the term ‘proposition’ than the classical one and, of course, not only a syntactical introduction, which is trivial, but a semantical one. The validity concept of two-valued logic being dependent on the value-definiteness of propositions will consequently have to be given up and replaced by a concept of validity that works without recourse to the truth-value of the propositions in question. It follows that the classical introduction of logical particles by the (finite or infinite) truth table method has to be given up as well, or, rather, it has to ____________ 1 2

Brouwer 1925, p. 2. Skolem’s proposal of a strictly finite mathematics without any use of quantifiers is an exception; its radical implications would deserve special discussion, cf. Skolem 1923.

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be amended in such a way that the general definition of logical particles restricted to value-definite propositions yields the well-known classical ones. Various attempts in this direction have been made, most prominent, among others, the proof-theoretic interpretation of intuitionistic logic by Kolmogorov and the operationist interpretation by Lorenzen.3 In these attempts, the introduction of the junctor ‘if- then’ marks the starting point of a deviation from classical procedures. Kolmogorov replaces the concept of truth by the concept of provability and the provability of A → B correspondingly by the provability of B relative to a proof of A that is hypothetically assumed. The concept of proof and a fortiori of provability has to be taken over from existing unformalized mathematics. Lorenzen replaces the concept of truth by the concept of (generalized) derivability within some calculus K, such that derivability of A → B has to be read as admissibility of the corresponding rule α ⇒ β (with A K α and B K β) relative to K. Here, the concept of admissibility has to be accepted as intuitively clear. Actual difficulties of interpretation occur in both cases after iterating the logical composition, e.g., with ‘if-then’, and no way out is visible if propositions other than mathematical ones are candidates for logical composition. Yet, these attemps have cleared the way to the additional insight that not only is value-definiteness not hereditary generally to logical compounds, but that it is possible to ascertain the (non-logical) truth of logically compound propositions without recourse to the truth-value of the subpropositions. There are, for example, true (i.e., provable resp. derivable) and not logically true subjunctions A → B without any knowledge even about the value-definiteness of A or B: one may take a suitably chosen admissible rule α ⇒ β within an undecidable calculus. A successful criticism of two-valued logic has to be able, therefore, to balance a wider concept of proposition with a correspondingly wider concept of logical composition and to add an adequate concept of validity for propositional schemata, or, alternatively, an adequate concept of (logical) implication by keeping the meta-equivalence ‘A implies B’ if and only if ‘(A → B) ε valid’. The usual method of devising formal systems, i.e., of introducing a syntactical concept of validity for syntactically defined well formed formulas is, of course, insufficient. Heyting's formalization of intuitionistic logic did permit precise comparison with other calculi, calculi

____________ 3

Cf. Kolmogorov 1932, and Lorenzen 1955. Gödel has shown that an axiomatization of the concept ‘beweisbar’ (provable) within classical logic, somewhat different from Kolmogorov’s nonformalized version, can be used as a representation of intuitionistic logic, cf. Gödel 1933.

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of classical logic as well as of modal logic and others,4 but it could not answer the initial question of what kind of theory or rather ‘action’ (Denktätigkeit) 5 it actually is that is formalized by this or that formal system of logic. And intuitionism (in the spirit of Brouwer) consequently never claimed to be able to represent its logic fully by a formal system. In order now to gain a better understanding of the actual conflict, it is necessary to go beyond these introductory remarks by stressing a difference of points of view between the proponents of classical logic and the proponents of effective logic, which is so much taken for granted that it is hardly ever explicitly disputed. Since the time of Leibniz, classical logic is often referred to as a system of ‘truths’ which hold ‘universally’, in ‘all possible worlds’, and, therefore, independently of the special facts of the ‘actual world’, i.e., of the natural sciences. And mathematics is, following the programme of logicism, to be constructed as a special part of this system of logical truths. In precisely this sense logic, and with it mathematics, came to be considered as a system of tautologies without factual content. Classical logic is the formal frame for any scientific investigation, the a priori basis of empirical science. It makes no essential difference when Quine, expanding ideas of C. I. Lewis, disputes the distinction ‘a priori-empirical’ and stresses the uniformity of the whole corpus of scientific truths instead. 6 This corpus is not uniquely determined by observational facts, it is in need of conceptual and other theoretical constructions, e.g., mathematical ones, which are chosen by intrinsic criteria of perspicuity, economy, connectedness et alii of the system of science as a whole. Thus, it may well obtain that even logic, one of the central parts of the corpus of truths, has to be changed due to new observational facts in order to satisfy the aforementioned criteria. Yet, even then, logic, the system of accepted logical truths, be it in its formalized version derivable by a classical or by some other calculus, remains the formal frame of science. Logic may be called ‘relatively universal’, i.e., a system of universal truths relative to the actual state of science. On the other hand, effective logic (explicitly in its operationist interpretation), must be looked at as a system of ‘universal’ rules that are accepted whenever a system of rules of action, e.g., rules for producing proofs or rules for producing arbitrary strings of signs, has been laid down. In this case, the field of application for the rules of logic is not the world as the totality of facts, but rather the world as seen in terms of specific kinds of scientific human activities. Within mathematics, for example, the rules of ____________ 4 5 6

Cf. the review of the main results in Kleene 1952, § 81; especially important is the paper McKinsey/Tarski 1948. Cf. Heyting 1930, pp. 45–46. Cf., e.g., the paper Two dogmas of empiricism in: Quine 1953, pp. 20–46.

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effective logic may be used without restriction. And this obtains because mathematics is not viewed as a system of truths, even less logical ones, but is treated as an independent scientific activity which, together with its intrinsic rules, may use the rules of logic as additional ‘admissible’ ones. Effective logic is the material content of any scientific investigation, the ‘empirical core’ even within mathematics. It goes without saying that no uniqueness claim is added. The set of rules of effective logic may vary from one scientific activity to another, and are not even strictly determined by any one of those. It coincides even with the set of rules of classical logic in the case of strictly finite mathematics, as Brouwer explicitly observed.7 Thus, logic may again be called ‘relatively universal’, i.e., a system of accepted universal rules relative to the field of investigation. In the light of these considerations, the basic conflict is a question rather of the set-up of formal logic itself than of accepting this or that propositional schema as valid. And, indeed, the competing views, to treat logic either as a set of rules (‘for correct thinking’) or as a set of theorems (‘on the general behavior of thought’), trace back to the very beginning of formal logic, to Aristotle and his interpretation by posterity. The conflict is known under the rubric: logic – art or science?, the respective Greek terms being ‘τέχνη’ and ‘ἐπιστήµη’. The problem, at the beginning of logic in the Greek period, was to set up a discipline that realizes the possibility of well-founded argumentation without using these very means of argumentation under pain of begging the question. If logic, and in the case of Aristotle this means his syllogistic, would have to count as a science, it should obey the conditions laid upon a system of truths by Aristotle in order to have it represent an ἀποδεικτικὴ ἐπιστήµη. That is, there should exist a set of first true premisses8 out of which all further truths may be inferred by (apodeictic) syllogisms. But syllogisms never count as propositions (λόγοι ἀποφαντικοί), nor do perfect syllogisms count as axioms (ἀρχαί), nor are the reductions of the syllogisms to perfect ones called ‘proofs’ by Aristotle.9 Aristotle does not treat his syllogistic as a science. On the other hand, if the set-up of syllogistic would represent an art in the strong sense of a διαλεκτικὴ τέχνη, there should exist first premisses accepted for the sake of argument (τόποι), from which those propositions ____________ 7 8 9

Cf. Brouwer 1925. The other conditions that Aristotle imposes on these first true premisses, i.e., the axioms or, rather, ‘principles’ (ἀρχαί), as they are now called, are of no concern for our purposes; cf. An. post. 71b. This has been made a point in the convincing operationist interpretation of Aristotle’s syllogistic against the arguments of Łukasiewicz in: Ebbinghaus 1964.

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about which the argument is concerned follow by (dialectic) syllogisms. It is obvious that the apparent axiomatic treatment of Aristotle’s syllogistic does not comply with these specifications, either. Syllogisms are used both for the sciences and for the arts, but they cannot themselves belong to either of them.10 Consequently, a syllogism which is defined twice, in the Prior Analytics and in the Topics, as a “linguistic expression (λόγος) in which, something having been posited, something other than the underlying results necessarily through the underlying”11 should neither be read as a theorem nor as a rule, though both possibilities have been adopted alternatively through the centuries.12 This view gets further support by observing Aristotle’s own argumentation on behalf of his choice for dealing with the objects of the Analytics. Instead of using the terminology of apodeictic or dialectic reasoning that would throw some light on Aristotle’s own opinion as to where to place the Analytics, he uses ‘analytically’ (ἀναλυτικῶς) instead of ‘apodeictically’, and ‘logically’ (λογικῶς) instead of ‘dialectically’ together with the interesting feature that most of his arguments on a certain point appear twice, once framed as an analytical one, and then as a logical one.13 This indefiniteness on the status of the arguments for the set-up of argumentation itself should not really give rise to surprise. A far more detailed investigation is needed to free the beginnings of logic from the air of circularity. The reason syllogisms are treated neither as theorems nor as rules is simply that in a way they are indeed both theorems and rules, depending on the level of argumentation. They can, any one of them, be considered as rules of inference14 – the syllogistic method in use is justly called a συλλογιστικὴ τέχνη by Aristotle15 – but as soon as the syllogisms are not considered with respect to their producing something out of something, but ____________ 10 Cf. Aristotle Met. 995a. 11 Top. 100a25–26; cf. An. pr. 24b19–20. 12 For two modern proponents of either possibility, cf. Łukasiewicz 21957, and Scholz 2 1959. Łukasiewicz interprets syllogisms as generalized subjunctions, Scholz reads them as rules of inference. For example, PaQ, QaR≺PaR (modus barbara) becomes ∧P,Q,R. a(P,Q)∧a(Q,R)→a(P,R). in Łukasiewicz, and PaQ; QaR ⇒ PaR in Scholz. 13 Cf., e.g., An. post. 83b–84b, where there are two ‘proofs’ for the claim that to each science there must exist first true and undemonstrable principles. 14 This is done successfully in the operationist interpretation of Aristotle’s syllogistic by Ebbinghaus as mentioned in note 9. In addition, the interchange of terms in PeQ (ἀντιστροφή), the contradictoriness of PaQ and PoQ, and of PeQ and PiQ, and the contrariness of PaQ and PeQ, all these formulated verbally by Aristotle, are given the form of rules; then all other valid syllogisms are ‘provable’ as admissible rules relative to the initial set of valid rules. 15 Cf. Soph. Elench. 172a36.

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as entities sui generis, those rules of inference may be transferred into (logical) implications, i.e. three-place (meta)propositions on propositions, and thus theorems.16 Hence, syllogistic in the sense of a theory of the valid rules of inference may be taken as an early anticipation of the position held by the proponents of effective logic, now in a refined version: effective logic is to be considered as a theory, i.e., a system of truths, about the universally admissible rules within arbitrary systems of rules of action. Naturally, in the course of history, syllogistic has been treated the other way round, too. For example, according to the most influential diplomatic vote of scholasticism, which can be found in the Summulae Logicales (ca. 1250 A.D.) of Petrus Hispanus, who later became Pope John XXI, the definition of logic runs like the following: dialectica (i.e., logic) est ars artium et scientia scientiarum ad omnium methodorum principia viam habens;17 and Duns Scotus gives an interpretation of this twofold determination: logic is a science respectu materiae ex qua constat, and logic is an art respectu materiae in qua versatur. This distinction may now be taken as an anticipation of the position held by the proponents of classical logic, here again in a refined version, insofar as the system of tautologies can be enumerated by a calculus, i.e., a system of rules. More in the line of Aristotle, the more radical schoolmen such as Buridan in his Summa de Dialectica just dropped any mention of logic as a science and kept only its characterization as ars artium which, therefore, leads again to the position of the proponents of effective logic. Hence, classical logic is the result of starting with arbitrary theories that obey the axiomatic method by concentrating on the forms of truths within arbitrary domains, and then formalizing this system of formal truths by means of some calculus, thus getting a praxis on top of the theories. Effective logic, on the other hand, starts with arbitrary calculi built up by the constructive, i.e., genetic, method, and proceeds to a theory about the generally admissible rules within the calculi – a theory which can afterwards likewise be formalized; here we have a theory on top of the praxis. Now it looks almost like a matter of taste how one is going to choose the level for a reasonable beginning of formal logic. Yet, the following constructions claim that there is an adequate solution of the conflict ____________ 16 This is an accord with the characterization of arts and sciences in An. post. 100a8– 10. Arts are concerned with the world of coming-to-be and passing-away, sciences are concerned with the world of being. 17 There are references to the art-science dispute and its medieval background in the course of discussing modern operationist logic (Brouwer, Wittgenstein, Kolmogorov, Lorenzen) in: Richter 1965.

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between a logic of rules and a logic of theorems and, more generally, a proper approach to the problem how praxis and theory interact in the case of logic.

II The starting point is again very close to the actual origin of logic in antiquity. With Aristotle, and even more with Plato, logic – or rather dialectic, the term being a strong hint by itself – had to provide the means by which sound argumentation could be distinguished from unsound argumentation.18 This has been a practical necessity in face of the highly developed sophistic technique to provide proofs for arbitrary theses on demand. And indeed, if it is granted that any scientific activity, be it on practical or on theoretical matters, is characterized as scientific by the fact that there is a justification available for each and every assertion put forth in the course of this activity (the possible linguistic articulations of nonlinguistic acts included!), there is no other basis for the construction of logic than to look for a methodical introduction of the linguistic elements of assertions and from there to proceed to the use of assertions within argumentations. Such an introduction of elementary linguistic elements shall be called primary praxis and will be executed within properly stylized teach-and-learn situations for these elements. As far as simple singular and simple general terms are concerned, the details of the procedure do not bear upon the set-up of formal logic. They have been discussed extensively elsewhere.19 For our purposes, it is sufficient to remark that introducing words by means of teach-and-learn situations guarantees their public understandability. Furthermore, it should be clear that the determination of a primary praxis in the given sense is a process post hoc, something man does in order to gain precise knowledge concerning his abilities and their limits after he has used speech and other acts meaningfully in the context of life. ____________ 18 Cf. the first sentence of the Topics (100a18–20), where the purpose of the treatise is characterized as “finding a method, by which we shall be able to argue (συλλογίζεσθαι) on any problem set before us starting from accepted premises (ἔνδοξαι) such that, when sustaining an argument (λόγος), we shall avoid saying anything self-contradictory”. It was E. Kapp who showed convincingly that the origin of Aristotle’s syllogistic (still taken to be a theory) is situated in the actual sophistic discussions on public affairs, cf. Kapp 1942. 19 Cf. the second part in: Lorenz 1970. There will be found special references to competing proposals in: Quine 1960, and in: Strawson 1959.

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The introductions in question do not each constitute a creatio ex nihilo, they are rather ‘recreationes’, that is, a system of methodical reconstructions of that which has already been said and done. Another feature of the primary praxis is important: Due to the teach-and-learn situations connected with the introduction of terms, there is no difference between the situation articulated by means of the terms, and the situations in which those terms are used. No use of terms other than introducing terms has as yet been the object of consideration. But this, of course, is a trivial part of human speech. The special power of linguistic communication becomes apparent only, if the situations which underlie words, phrases, or sentences are different from the situations in which these words, phrases, or sentences are used. In that case, understandability of the linguistic expressions is not enough, a special link between the two situations is needed to secure the proper function of language. This link is provided by the detailed reconstruction – again through teachand-learn-situations – of possible uses of linguistic expressions after their introduction. Any such introduction of a use of linguistic expressions different from the introduction itself shall belong to the secondary praxis, e.g., the use of terms as wishes, questions, or propositions. The way this is done guarantees the public justifiability of linguistic expressions in addition to their understandability. The special act of asserting propositions (as distinguished from their use, e.g., in story-telling) involves a justifying procedure within the secondary praxis – a procedure that has to be introduced together with the use of terms as assertions – such that the validity of assertions can be defined by means of this procedure. It is even possible to distinguish words and sentences along these lines. If the situation articulated by a linguistic expression coincides with, or is at least part of the situation in which that expression is used, only its understandability is of concern, and the linguistic expression shall count as a word; but if those two situations are wholly different, both understandability and justifiability have to be secured and the linguistic expression shall count as a sentence. This now is the exact point for characterizing the justifying procedure of assertions as a dialogue, or an argumentation between two partners. To assert a proposition makes sense only, if there is someone on the other side, albeit fictitiously, who either denies or at least doubts the asserted proposition. But it is not enough merely to argue about propositions, there must exist precise stipulations on the rules of argumentation, rules which, in a way, define the exact meaning of the proposition in question. A proposition shall be called ‘dialogue-definite’ under the condition that the possible dialogues on this proposition are finished after finitely many steps according to some previously stipulated and effectively applicable rules of argumentation, such that, at the end, it can be decided who has

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won and who has lost. Hence, dialogue-definiteness of propositions means that the relevant concept of a dialogue is decidable. And it is this concept of dialogue-definiteness that is to replace the age-old value-definiteness as the characterizing feature for linguistic expressions to be propositions.20 Further considerations will show that the class of value-definite propositions is indeed a proper subclass of the class of dialogue-definite propositions and that, therefore, the justifiability of propositions as introduced by the dialogue procedure does not coincide with their verifiability. Yet, ‘truth’ and ‘falsehood’ for propositions can now be defined on the basis of the dialogue game associated with each proposition. Such a definition marks the beginning of a theory about (primary and secondary) praxis, insofar as (meta)propositions on the actions within the praxis get introduced. In a certain sense, even the secondary praxis itself contains a theoretical element, namely the propositions themselves, which get their meaning by the rules of argumentation about them. Hence, it might be appropriate, at least for the systematic purposes of the whole set-up, to distinguish an object-theory (the class of propositions introduced within the secondary praxis) from a metatheory (about primary and secondary praxis), the propositions of which cannot, of course, exist without the same pragmatic foundation as the propositions on the ground level. At this early stage, already, the interaction of praxis and theory is far more complicated than the usual presentation of logical theories permits us to suppose. As a preparation for defining ‘truth’ and ‘falsehood’ for propositions, it is useful to observe that win and loss of a dialogue about a given proposition will in general depend upon an individual play of the game and will not be a function of the proposition alone. But the strategies of either player of the game are invariant against the choice of arguments of the other player. Hence, a proposition A shall be called ‘true’, iff there is a winning strategy for A; this means that the player who is asserting A – the proponent P – will be able to win a dialogue on A independently of the choice of arguments of the opponent O. Accordingly, a proposition A shall be called ‘false’ iff there is a winning strategy against A, i.e., the opponent can win a dialogue on A independently of the moves of the proponent.21 I have shown elsewhere22 that ____________ 20 The concept of dialogue within such a context has originally been introduced in: Lorenzen 1960; 1961, for the purpose of a better understanding of operationist logic. Its further explication, especially with respect to a pragmatic foundation of the calculi of intuitionistic and two-valued logic is due to K. Lorenz; cf. Lorenz 1968; 1978. 21 If, as usual, the validity of the logical principles is presupposed on the metatheoretic level, it would be possible from the validity of the saddle-point theorem for finitary two-person zero-sum-games to infer that propositions are either true or

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the dialogue on the metaproposition ‘A is true’ coincides with the dialogue on A itself, which means that the traditional condition of adequacy for any definition of truth is satisfied: ‘A is true iff A’.23 The next step of the theory about the (primary and secondary) praxis is concerned with the justification of the rules of argumentation that constitute the secondary praxis. Again, this is done with a few accompanying remarks to the following proposal of a structural rule for dialogues, because space does not permit extensive elaboration on that point here.24 (Dl) Dialogues about propositions consist of arguments which are put forth alternatively by an opponent O and a proponent P. The arguments follow certain rules of argumentation that belong to the game such that each play ends up with win or loss for either player. (D2) With the exception of the improper initial argument, each argument either attacks prior ones of the partner or defends those of one's own upon such an attack, but does not act simultaneously in both ways: the proper arguments split into attacks and defenses. (D3) Attacks may be put forth at any time during a play of the game (rights!). (D4) Defenses must be put forth in the order of the corresponding attacks (upon which the defense answers), yet may be postponed as long as attacks can still be put forth: always that argument which has been attacked last without having been defended yet, has to be defended first (duties!). (D5) Whoever cannot – or will not – put forth an argument any longer, has lost that play of game; the other one has won it.

(Dl) is obviously not in need of further explanation; (D2) may be accepted as defining the special dialogue character of the game; and (D5) codifies equally current rules of win and loss. The only items in need of some further comments are (D3) and (D4) that regulate rank and order of attacks and defenses. With respect to the generality of rules, the right to attack shall not depend on a special position reached during a play of the game and, hence, shall not become void until the end of each play. On the other hand, the given order of defenses is a consequence of the stipulation ____________ false, cf. Berge 1957. Now, without begging the question, there is only a practical meaning of ‘either-or’ on the metalevel available, i.e., decidability of choice, but that cannot happen, because it is not generally decidable which part of the alternative holds; it is only decidable who has won a particular play of the game and who has lost it. 22 Cf. Lorenz 1968, pp. 35–36. 23 Cf. Tarski 1956, pp. 187–188. For a discussion about the danger of semantic antinomies, if this condition of adequacy is used as a schematic definition of truth, cf. Lorenz 1970, pp. 44–46. 24 For further details consult again Lorenz 1968, pp. 37–39.

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in (Dl) to argue alternatively together with the rule of win and loss in (D5), if to both players is guaranteed that neither must defend upon an attack unless this attack has been defended first upon a counter-attack. Now, (D1) – (D5) are not sufficient to secure finiteness of the individual plays of the game. There is lacking a regulation on the number of attacks permitted against a single argument during a given play. Yet, since any choice of bounds would be arbitrary, it might be accepted as reasonable that this choice should become part of the dialogue game itself. After the initial argument has been laid down by P, first O shall choose a natural number n as the maximal number of attacks to be directed against a single argument of P, then P shall choose a natural number m analogously. Only now the proper dialogue about the initial argument may start obeying the following additional stipulation of the structural rule. (D6 n,m) During a play of the game, any argument may be attacked by the opponent at most n-times, by the proponent at most m-times.

In order actually to play a dialogue game according to the given rules, the rules of argumentation in (D1) have to be specified. This can be done by laying down a schema of attacks and defenses, which shows all possible attacks against an argument as well as all possible defenses of this argument upon each of these attacks. And, in general, this specification is possible only by special reference to the internal structure of the propositions concerned. The structural rule is purely ‘formal’ in the sense that no special knowledge about the proposition is needed, whereas the rules of argumentation are ‘material’ in so far as they have to make use of the actual set-up of the propositions, their ‘content’ in the terminology of traditional philosophy. Yet, there is a possibility of determining special rules of argumentation that are, in a way, formal, too, namely, those that make use only of the fact that propositions may be composed out of subpropositions. This leads to the concept of logical composition that in turn affects the introduction of further terms into the primary praxis, the so-called logical particles. A proposition A shall be called ‘logically composed’ out of propositions from a class K of given dialogue-definite propositions, if the schema of attacks and defenses associated with A contains only propositions from the class K. By means of such special rules of argumentation, the particle-rules, any dialogue about A is reduced to dialogues about the subpropositions of A: A is dialogue-definite, too. Easy combinatorial considerations show 25 that for a complete survey it is sufficient to discuss unary, binary and infinitary logical particles only, ____________ 25 Cf. Lorenz 1968, pp. 41–43.

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under the condition to restrict the schemata in question to those that contain each subproposition just once and that use as further attacks certain non-assailable orders for defense or doubts, symbolized by ‘?’ with added indices. Scheme 1 will be self-explanatory. (As notation for plays of the dialogue game it seemed to be sufficiently suggestive to use two columns such that the rows are reserved for the consecutive attacks (from top to bottom with an index of the row number of that argument against which the attack is placed) together with an entry for the chosen defense – if any – upon that attack; to recover the order of moves one may enumerate the arguments, if necessary.) position negation (not)

*A ⌐A

attacks ?

¬A

A

A*B conjunction (and) adjunction (or) subjunction (if-then)

abjunction (but not) injunction (neither-nor) (all) (some) (no)

A∧ B

attacks 1? 2?

defenses A

defenses A B A B

A∨ B

?

A →B

A

B

A ←B

B A ? B ? A B

A

A

B

A

B

A

B

*xA(x) attacks ∧xA(x) ?n ∨xA(x) ? A(n) xA(x)

B A

defenses A(n) A(n)

scheme 1

As an example of a dialogue we will discuss the assertion A0 ((a→b)→b)→((b→a)→a) for dialogue-definite propositions a and b. Furthermore, we will make use of one of the main results of the theory of dialogue games, namely that the class of propositions for which there are

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winning strategies is not changed if O is limited to at most one attack against an argument of P, whereas P may choose any number of attacks against a single argument of O.26 In the position as given in scheme 2, P must either have a winning strategy for b in order to win by defending a→b with b upon the attack a or a winning strategy against a. If, on the other hand, there exists a winning strategy for a, P should have defended his second move upon the attack b→a instead of counter-attacking the first move of O with a→b. Should O have chosen the defense b upon the attack a→b as his fifth move, P would have attacked the third move with b, and any attempt of O to start now a subdialogue about this b of P would lead to an imitation of this sub-dialogue by P about the b of O. O 1) 3)

(a→b)→b b→a

(0) (1)

5)

a

(3)

(1)

P A0 (b→a)→a

2)

a→b

4)

scheme 2

O must finally defend upon the attack b of P with a, and P in turn defends his second move with a. Any further attempt of O to try a sub-dialogue about this a of P results in an imitated sub-dialogue of P about the a of O. Therefore, if a is value-definite, A0 is true independently of the truth or falsehood of b. And it can be seen that there is a chance of winningstrategies which are formal in the sense that nothing need be known about the truth or falsehood of the prime propositions, as is the case, e.g., for A1 (a ∨ b)→((b→a)→a) (scheme 3). O

P A1 1) a ∨ b (0) (b→a)→a 2) 3) b→a (1) a 6) 5) a (1) ? 4)

O

P A1 1) a ∨ b (0) (b→a)→a 3) b→a (1) a 5) b (1) ? 7) a (2) b

2) 8) 4) 6)

scheme 3

In these cases, the win of a play for P does not depend on the outcome of the dialogues about the prime propositions, the crucial point being only the ____________ 26 Cf. Lorenz 1968, pp. 85–87.

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possibility for P not to place a prime proposition as an argument until the same prime proposition has been placed as an argument by O. A special rule for formal playing can, hence, be formulated: (D7m) Prime propositions cannot be formally attacked; they may be put forth by the opponent without restrictions, whereas the proponent may only take over prime propositions from the opponent, each at most m-times, if m is the bound for the number of attacks against arguments of the opponent.

The restriction for P not to take over a prime proposition of O more than the maximal number of attacks he has chosen to place against a single argument of O during a play, is necessary in order to guarantee that the existence of a formal winning strategy, i.e., a winning strategy using formal playing only, for a proposition A is invariant against substitution of logically composed propositions for prime propositions within A. On the basis of the construction up to this point, it is now possible to introduce the concept of logical (or formal) truth of propositions by the existence of formal winning-strategies for them. And a propositional schema is valid iff propositions bearing that schema are logically true. This definition of validity for arbitrary dialogue-definite propositions works independently of any assumption on the truth or falsity of their prime propositions. It is, therefore, beyond the range of the classical theory of logical truth, even if the domain of propositions gets restricted both to value-definite ones and to quantifier-free ones. For, it is one thing to define the (classical) logical truth of a proposition A by the existence of (material) winning strategies for all propositions A* which result out of A, if the set of prime propositions of A runs through all combinations of truth and falsehood with respect to these prime propositions, and it is another thing to define the logical truth of a proposition A by the existence of a formal winning strategy for A. And, indeed, these two concepts do not coincide even within this restricted domain of propositions, as example A0 already shows. The propositional schema ((a→b)→b)→((b→a)→a) is valid classically (with the special point that only the value-definiteness of a is needed), though there is no formal winning strategy for it. The classical theory does not even permit the definition of this difference between general material truth and purely formal truth. It is the pragmatic approach to formal logic by means of dialogue games as it has been sketched here that leads to the definability of a concept of formal truth for propositions which do not generally satisfy the classical condition of value-definiteness. Hence, the dialogue concept makes it possible to define formal truth independently of material truth, whereas the classical theory is char-

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17

acterized by the reduction of formal truth to material truth, namely as general material truth. At this late stage now it is reasonable to formalize the theory of dialogue games, and with this method to return to a praxis at a higher level, which, as the praxis of the calculi of logic, has been the usual starting point for contemporary logical theory. With the help of this formalization, it is possible to prove the main theorem of the theory of dialogue games: The class of valid propositional schemata coincides with the class of intuitionistically valid propositional schemata.27 If, furthermore, all classically valid propositional schemata shall be gained by formal winning-strategies, this means fictitiously to assume the value-definiteness of suitable propositional sub-schemata of the propositional schema in question: Any logically true proposition in the classical sense is logically true in the effective sense, if only suitable tertium-non-datur hypotheses are added. As an instructive example it is easily checked that there is a formal winning strategy for the classical disjunction ∧xa(x) ∨∨x¬a(x) under the three tertium-non-datur hypotheses ∧x. a(x) ∨ ¬a(x). , ∧xa(x) ∨ ¬∧xa(x) and ∨x¬a(x) ∨ ¬∨x¬a(x). Intuitionistic or effective logic is the logic of dialogue-definite propositions, two-valued or classical logic is the logic of the subclass of value-definite propositions. And it is quantification theory which shows the necessity for transition from the one to the other. In any case, we can conclude that logic is primarily the theory of a structured praxis. The logic of antiquity has not in general been very conscious (at least in the eyes of its interpreters) of the pragmatic basis of logic taken as a science. As a substitute, one often uses the very misleading phrase of the ‘ontological background’ of ancient logic. Yet, logic is, secondarily (by means of formalization), again a praxis of a structured theory. On the other hand, now, modern logic since Leibniz has minimized the importance of the possible theoretical basis of a calculus of logic, that is, logic taken as an art. Presumably, one had doubts about the precision which could be imposed on a logic formulated only within ordinary language. In both cases, there has been no clarity about the details of a step-bystep procedure from a praxis via a theory again to a praxis which, as we have tried to show, is the necessary minimum to get the means for an adequate solution of the dispute on the true nature of logic.

____________ 27 Cf. for a proof: Lorenz 1968; another proof in: Stegmüller 1964.

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References Berge, Claude, 1957: Théorie générale des jeux à n personnes, Paris: Gauthier-Villars. Brouwer, Luitzen E. J., 1925: Über die Bedeutung des Satzes vom ausgeschlossenen Dritten in der Mathematik, insbesondere in der Funktionentheorie, in: Zeitschrift für reine und angewandte Mathematik 154, pp. 1–7. Ebbinghaus, Kurt, 1964: Ein formales Modell der Syllogistik des Aristoteles, Göttingen: Vandenhoeck und Ruprecht. Gödel, Kurt, 1933: Eine Interpretation des intuitionistischen Aussagenkalküls, in: Ergebnisse eines Mathematischen Kolloquiums, Heft 4, pp. 39–40. Heyting, Arend, 1930: Die formalen Regeln der intuitionistischen Logik, in: Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalischmathematische Klasse, pp. 42–56. Kapp, Ernst, 1942: Greek Foundations of Traditional Logic, New York: Columbia University Press. Kleene, Stephen C., 1952: Introduction to Metamathematics, Princeton/Toronto/ NewYork: D. van Nostrand Company. Kolmogorov, Andrej N., 1932: Zur Deutung der intuitionistischen Logik, in: Mathematische Zeitschrift 35, pp. 58–65. Lorenz, Kuno, 1968: Dialogspiele als semantische Grundlage von Logikkalkülen, in: Archiv für mathematische Logik und Grundlagenforschung 11, pp. 32–55, 73–100. Lorenz, Kuno, 1970: Elemente der Sprachkritik. Eine Alternative zum Dogmatismus und Skeptizismus in der Analytischen Philosophie, Frankfurt am Main: Suhrkamp. Lorenz, Kuno, 1978: Arithmetik und Logik als Spiele [1961] , (partially reprinted) in: Paul Lorenzen/Kuno Lorenz, Dialogische Logik, Darmstadt: Wissenschaftliche Buchgesellschaft, pp. 17–95. Lorenzen, Paul, 1955: Einführung in die operative Logik und Mathematik, Berlin/ Göttingen/Heidelberg: Springer. Lorenzen, Paul, 1960: Logik und Agon, in: Atti del XII Congresso di Filosofia (Venezia, 12–18 Settembre 1958). v 4, Firenze: Sansoni Editore, pp. 187–194. Lorenzen, Paul, 1961: Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods. Proceedings of the Symposium on Foundations of Mathematics (Warsaw, 2–9 September 1959), Oxford/London/New York/Paris: Pergamon Press, pp. 193–200. Łukasiewicz, Jan, 21957: Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic [1951], Oxford: Clarendon Press. McKinsey, J. C. C./Tarski, Alfred, 1948: Some theorems about the sentential calculi of Lewis and Heyting, in: The Journal of Symbolic Logic 13, pp. 1–15. Quine, Willard V. O., 1953: From a Logical Point of View. 9 Logico-Philosophical Essays, Cambridge Mass.: Harvard University Press. Quine, Willard V. O., 1960: Word and Object, New York/London: John Wiley & Sons. Richter, Vladimir, 1965: Untersuchungen zur operativen Logik der Gegenwart, Freiburg/München: Verlag Karl Alber. Scholz, Heinrich, 21959: Abriß der Geschichte der Logik, Freiburg/ München: Verlag Karl Alber. Skolem, Thoralf A., 1923: Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unend-

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lichem Ausdehnungsbereich, in: Skrifter utgit av Videnskapsselskapet i Kristiania, I. Matematisk-naturvidenskabelig klasse, No. 6. Stegmüller, Wolfgang, 1964: Remarks on the Completeness of Logical Systems Relative to the Validity Concepts of P. Lorenzen and K. Lorenz, in: Notre Dame Journal of Formal Logic 5, pp. 81–112. Strawson, Peter F., 1959: Individuals. An Essay in Descriptive Metaphysics, London: Methuen & Co. Tarski, Alfred, 1956: The Concept of Truth in Formalized Languages [German 1936], in: Alfred Tarski, Logic, Semantics, Metamathematics. Papers from 1923 to 1938, transl. by J. H. Woodger, Oxford: Clarendon Press, pp. 152–278.

On the Relation between the Partition of a Whole into Parts and the Attribution of Properties to an Object I Since the invention of the calculus of individuals, first 1916 by Leśniewski,1 there has been a continuous discussion about the question of how far it may serve as a substitute for the usual calculi of set theory. The numerous analogues of operations on sets using the terminology of parts and wholes within the theory of individuals gave rise to hopes that the whole hierarchy of logical types might – at least to a certain extent – be represented in a theory of one logical type only. It was Nelson Goodman together with Henry Leonard who, in their classical paper on the calculus of individuals,2 inferred explicitly from the interpretative power of a combined formal system using ‘is a member of’ together with ‘is part of’ as primitive notions that “the dispute between nominalist and realist as to what actual entities are individuals and what are classes is recognized as devolving upon matters of interpretative convenience rather than upon metaphysical necessity”.3 It is a matter of habit, they declare, to treat qualities in interpretations as entities of one type higher than things, one could equally proceed the other way round. What counts is that the concept of being an individual and that of being a class distinguish one segment of the total universe from the rest in a different manner: To conceive a segment as a whole or individual offers no suggestion as to what the potential subdivisions of that segment must be “whereas to conceive a segment as a class imposes”, so the authors say, “a definite scheme of subdivisions – into subclasses and members”.4 Apart from the obvious fact that to subdivide into members of a class is by no means more definite than to subdivide into parts of an individual, it does happen, though, that some interrelations of classes – the authors quote the example of windows and buildings where each window is a part ____________ 1 2 3 4

Leśniewski 1916. Goodman/Leonard 1940. Op. cit., p. 55. Op. cit., p. 45.

On the Relation

21

of some building – cannot be expressed using the class-membershiprelation alone, unless windows and buildings are reinterpreted as entities of some other logical type – here, e.g., specified classes of atoms. And it is this crucial feature of logico-linguistic world representation that calls at least for an amendment to set theory, if not for its, possibly only partial, substitute through a mereology. This argument sounds convincing, would not the usual consequence, the set-up of a formal system – with, e.g., the following well known postulates: 1. Transitivity of the relation ‘is part of’ (≤) 2. Unique existence of the composite κP (called ‘fusion’ by Goodman and Leonard) for each non-empty predicate P (κP ι x.∧y.x○y↔∨z.P(z) ∧ z○y… with x○y ∨z.z≤x∧z≤y. as definition of ‘overlapping’)

– as a companion to a formal system of set theory, have to make use of standard interpretations of the two primitives, ‘ ’ and ‘≤’, in the metalanguage. When we write, as usual, ‘P(n)’ to represent the object n as falling under the concept ϑP, i.e., the intension of the predicate ‘P’, or the object n as being a member of the class P, i.e., the extension of the predicate ‘P’, we do this in the relational cases, x y and x≤y, as well – at least with respect to potential interpretations –, and it is equally common, and now on the pure syntactic level irrespective of semantic considerations, to consider the symbols as parts of the formula that is made out of them by, usually, concatenation. So we should answer the question of how in ‘x y’ – for ‘x≤y’ the argument runs in an analogous fashion – the phrase ‘x and y are, in this order, relata of the class-membership-relation’ interferes, contrasts, or correlates with the phrase ‘ ‘x’, ‘ ’, ‘y’, and linear combinations thereof are parts of ‘x y’ ’. We can read ‘x y’ – and likewise ‘x≤y’ – either semantically or syntactically, only by either using the standard interpretation of ‘ ’ or by using the standard interpretation of ‘≤’, or variants thereof. For convenience of presentation from now on, I go back to ‘xεP’ with some predicate letter ‘P’ as the logical basis of the notation ‘x y’ (being a suggestive shorthand for ‘x,yε ’ with ‘y’ referring to the extension of ‘P’), and I read ‘xεP’ with the affirmative copula ‘ε’ as attribution of property P to object x. It seems clear, then, that the relation ‘is part of’ obtains only among objects (of whatever type), in contrast to attribution occurring on the (meta)level of stating that, e.g., a relation x≤y, i.e., x,yε≤, holds. Hence, attribution is not a relation, but a means to articulate that properties,

22

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relations included, hold of objects. The usual procedure to represent attribution seemingly on the object level by using the extensional classmembership-relation between objects of consecutive types hides this fact. In such a way, attribution has just been pushed one step higher, with x y instead of xεP, from xεP to x,yε . Thus, rather than claiming that ‘is a member of’ and ‘is part of’ establish relations among objects of different types and among objects of equal types, respectively, one should distinguish both notions more fundamentally as one of description and one of construction. Attribution is a means to describe (to express, to state, or whatever term is used in ordinary discourse) a certain (elementary) state of affairs, even if this ‘connection’ between the object level and the language level is transposed into the object level and represented by the ordinary settheoretical -relation. Partition is a means to construct (to analyse, to build up, to subdivide, or whatever term is used in ordinary discourse for the procedure to derive parts from a whole or a whole out of parts) a certain (non-elementary) object out of other objects that afterwards count as parts of a whole. It is worthwhile to note that within this framework there is as yet no decision, whether this construction works by synthesis (starting with the parts) or by analysis (starting with the whole). For example, to construct natural numbers through counting, that is, successively adding one (to be described afterwards as applying the successor function), should be called ‘construction by synthesis’ (yielding ‘additive’ parts for each natural number considered as a whole), whereas the representation of natural numbers as a product of prime numbers may count as an example of a ‘construction by analysis’ (yielding ‘multiplicative’ parts for each natural number considered as a whole). Corresponding examples from the sciences abound. In addition, we may observe that within the restriction of a given domain of objects the usual device of object forms or terms serves to represent forms of partition, whereas propositional forms or formulas – at least in the elementary case – represent forms of attribution. And the well known interrelation of terms with formulas give us the first hint of how to approach the more general relation between the language of parts and wholes and the language of properties. Yet, to get a fair account of the difference, it is useful first to treat an example simultaneously by description and by construction. Let us take a logically compound formula, e.g., A a ∧ ¬b. Then, it is possible to describe this formula by ι xA(x) using a predicate A on the domain of expressions with roughly the following meaning: A(x) x is derivable in one of the usual calculi for constructing wellformed formulas and x is linearly composed out of a, ∧, ¬, b, in that order. One could even

On the Relation

23

axiomatize these syntactic notions of derivability and concatenation such that it is a question of logical consequence from the axioms, whether ι xA(x) exists or not. On the other hand, one may simply construct the formula A according to the rules of a suitable calculus for deriving well-formed formulas. The way of description needs a proof of (unique) existence of the described object, the way of construction is by itself a proof of existence of the constructed object. As can be easily seen, the difference comes about essentially through the change from parts (of an object) to the property (of that object) of having those parts. And, furthermore, this simultaneous treatment shows that the natural way of comparison between part/whole and object/property (resp. class) does not follow the course of considering an object once a whole (or an individual) and once a property (or a class), though comparison of the respective formal properties of x≤y and x y in the calculi of the theory of individuals and the calculi of set theory is bound to concentrate on this line of investigation. It, rather, turns out that greater reward may be expected from looking more closely to the usual terminological mix-up when having parts are said to be properties of objects and when properties of objects are said to be partial determinations of objects: both parts and properties belong to objects. If we look at our simple example with this idea in mind, another observation leads to the following generalisation: To identify an object nεN, it is necessary either to have at least one characterizing property Q for determining that object, i.e., n = ι xQ(x), where N is to be taken as a set of values for the variable x, or to have at least one characterizing part ι R for constituting that object, i.e., ∨R(x) x