The Lvov-Warsaw School The New Generation

THE LVOV-WARSAW SCHOOL – THE NEW GENERATION

POZNAē STUDIES IN THE PHILOSOPHY OF THE SCIENCES AND THE HUMANITIES VOLUME 89

EDITORS Jerzy BrzeziĔski Andrzej Klawiter Krzysztof àastowski Leszek Nowak (editor-in-chief) Izabella Nowakowa Katarzyna Paprzycka (managing editor)

Marcin Paprzycki Piotr Przybysz (assistant editor) Mikoáaj SĊdek Michael J. Shaffer Piotr Ziemian

ADVISORY COMMITTEE Joseph Agassi (Tel-Aviv) Étienne Balibar (Paris) Wolfgang Balzer (München) Mario Bunge (Montreal) Nancy Cartwright (London) Robert S. Cohen (Boston) Francesco Coniglione (Catania) Andrzej Falkiewicz (Wrocáaw) Dagfinn Føllesdal (Oslo) Bert Hamminga (Tilburg) Jaakko Hintikka (Boston) Jacek J. Jadacki (Warszawa) Jerzy Kmita (PoznaĔ)

Leon Koj (Lublin) Wáadysáaw Krajewski (Warszawa) Theo A.F. Kuipers (Groningen) Witold Marciszewski (Warszawa) Ilkka Niiniluoto (Helsinki) Günter Patzig (Göttingen) Jerzy Perzanowski (ToruĔ) Marian PrzeáĊcki (Warszawa) Jan Such (PoznaĔ) Max Urchs (Konstanz) Jan WoleĔski (Kraków) Ryszard Wójcicki (Warszawa)

PoznaĔ Studies in the Philosophy of the Sciences and the Humanities is partly sponsored by SWPS and Adam Mickiewicz University

Address:

dr Katarzyna Paprzycka . Instytut Filozofii . SWPS . ul. Chodakowska 19/31 03-815 Warszawa . Poland . fax: ++48 22 517-9625 E-mail: [email protected] . Website: http://PoznanStudies.swps.edu.pl

POLISH ANALYTICAL PHILOSOPHY

Volume VI Editors: Jacek Juliusz Jadacki (editor-in-chief) Leszek Nowak Jan WoleĔski Jerzy Perzanowski Ryszard Wójcicki

THE LVOV-WARSAW SCHOOL – – THE NEW GENERATION

Edited by Jacek Jadacki and Jacek PaĞniczek

Amsterdam - New York, NY 2006

The paper on which this book is printed meets the requirements of "ISO 9706:1994, Information and documentation - Paper for documents Requirements for permanence". ISSN 0303-8157 ISBN-10: 90-420-2068-7 ISBN-13: 978-90-420-2068-9 ©Editions Rodopi B.V., Amsterdam - New York, NY 2006 Printed in The Netherlands

CONTENTS Jacek Jadacki, Jacek PaĞniczek, The Lvov-Warsaw School: Its Contemporary Inheritors and Investigators in Poland and Abroad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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PART I. THE SCHOOL: ITS ORIGINS AND SIGNIFICANCE

Barry Smith, Why Polish Philosophy Does Not Exist . . . . . . . . . . Jacek Jadacki, The Lvov-Warsaw School and Its Influence on Polish Philosophy of the Second Half of the 20th Century . . . .

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PART II. OBJECTS AND PROPERTIES

John T. Kearns, An Elementary System of Ontology . . . . Jacek PaĞniczek, Do We Need Complex Properties Ontology? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrzej Biáat, Objects, Properties and Russell’s Paradox . . Joanna OdrowąĪ-Sypniewska, On the Notion of Identity . .

.. in .. .. ..

. . . . 87 Our . . . . 113 . . . . 129 . . . . 143

PART III. PROGNOSES, NORMS AND QUESTIONS

Tomasz Placek, A Puzzle about Semantic Determinism: àukasiewicz’s “On Determinism” Years Later . . . . . . . . . . . Max Urchs, Causality in Chaotic Environment: Does Strong Causality Break Down in Deterministic Chaos? . . . . . . . . . . . Jan WoleĔski, Three Contributions to Logical Philosophy . . . . . . . Andrzej WiĞniewski, Reducibility of Safe Questions to Sets of Atomic Yes-No Questions . . . . . . . . . . . . . . . . . . . . . . . . .

171 187 195 215

PART IV. CATEGORIAL GRAMMAR

Peter Simons, Languages with Variable-Binding Operators: Categorial Syntax and Combinatorial Semantics . . . . . . . . . . 239 Urszula Wybraniec-Skardowska, On the Formalization of Classical Categorial Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

6 PART V. INTENTIONALITY, SENSE AND CONSEQUENCE

Liliana Albertazzi, Retrieving Intentionality: A Legacy from the Brentano School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kazimierz TrzĊsicki, Logical and Methodological Assumptions of the Ajdukiewicz’s and Kripke-Putnam’s Views of Meaning . . . Anna Jedynak, On Linguistic Relativism . . . . . . . . . . . . . . . . . . Dale Jacquette, Tarski’s Analysis of Logical Consequence and Etchemendy’s Criticism of Tarski’s Modal Fallacy . . . . . . . .

291 315 325 345

PART VI. TRUTHS AND FALSEHOODS

Arianna Betti, Sempiternal Truth. The Bolzano-TwardowskiLeĞniewski Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Artur Rojszczak, From the Act of Judging to the Sentence: The Truth-Bearer and the Objectivisation of Truth . . . . . . . . . . . . Wojciech ĩeáaniec, What does “Truth in Virtue of Meaning” Really Explain? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Józef Misiek, Do We Need a Definition of Truth? . . . . . . . . . . . .

371 401 421 445

PART VII. RATIONALITY: ITS CRITERIA AND DEFINITION

Ryszard Kleszcz, Criteria of Rationality . . . . . . . . . . . . . . . . . . . 469 Mieszko Taáasiewicz, On the Concept of Rationality . . . . . . . . . . . 485

THE LVOV-WARSAW SCHOOL: ITS CONTEMPORARY INHERITORS AND INVESTIGATORS IN POLAND AND ABROAD

Habent sua fata libelli. According to our original plans, this book was scheduled as the symbolic closure of the 20th century within the sphere of influence of the Lvov-Warsaw School. A serious illness suffered by one of us and the sudden violent death of our young friend, Artur Rojszczak, completely changed those plans. However, despite the various difficulties, this book is to finally appear in press. Let us treat it, therefore, in this altered situation, as our “tvardovskian,” symbolically tardy inauguration of the 21st century. We propose below a short survey of the content of this anthology.

Part I. The School: Its Origins and Significance The assumption in the title of Barry Smith’s article “Why Polish Philosophy Does Not Exist” is that there is no Polish philosophy. Through the process of exposition and commentary, this, initially provocative, assumption turns out to be a great compliment to the (scientific) style of philosophising in the Lvov-Warsaw School. There is no Polish philosophy in Poland because in Poland, especially since Twardowski’s times, it is philosophy per se that has been pursued, which is thus a constituent of world philosophy. Smith’s article gives a brilliant introduction to the extraordinary phenomenon of Twardowski’s circle, which brought Polish philosophers into the world stream of modernisation in philosophy. The significance of the School is sketched in Jacek Jadacki’s study “The Lvov-Warsaw School and Its Influence upon Polish Philosophy of the Second Half of the 20th Century.” The author advocates the view that, independently of the long ideological pressure of the communist regime, the School’s influence on Polish philosophy of the last fifty years turned out to be decisive. Jadacki proposes a certain division of the

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 7-15. Amsterdam/New York, NY: Rodopi, 2006.

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history of this influence and he gives an overview of the most important achievements of the School in the domains of formal logic, ontology, epistemology, methodology, praxiology, philosophy of science, and axiology.

Part II. Objects and Properties LeĞniewski’s system of logic still captivates logicians and philosophers because of its formal elegance and ingenuity and stimulates them to further developments and new interpretations. LeĞniewski’s Ontology is frequently criticised for lacking a category of referring expressions which makes it incapable of accommodating expressions for collections of individuals. In his paper “An Elementary System of Ontology,” John T. Kearns develops a natural deduction system which is a restricted version of LeĞniewski’s Ontology and proves its soundness and completeness. Although the semantic account he provides treats nouns as predicates (as expressions with no ontological commitment), yet there is a way to conceive the language of Ontology as a truncated version of a larger language which does contain referring expressions. A revised system which contains such expressions is sketched. Philosophers sometimes wonder whether, apart from simple properties like for example being a cat or being black, there exist complex properties like being a non-cat or being a black cat. Putting aside the metaphysical issue of the existence of such properties, Jacek PaĞniczek, in his paper “Do We Need Complex Properties in Our Ontology?”, argues that complex properties could be a very fruitful ontological concept when conjoined with a liberal ontology of objects. Roughly speaking, complex properties may be used as a means for the categorisation of objects. PaĞniczek considers an algebra of properties which consists of simple properties and which is closed under Boolean operations. The principles that express possible dependencies between simple and complex predication are then enumerated and several categories of objects that obey all or only selected principles are discussed (particularly important among them are the categories of individuals, incomplete, and inconsistent objects). Russell’s paradox has been a recalcitrant problem in philosophical logic and has provoked philosophers to a constant search for still another “solution” to it. The so-called Russell’s property, which appears in this paradox, is often considered to be an abnormal property which needs to be somehow dismissed. Andrzej Biáat claims in his paper “Objects,

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Properties and Russell’s Paradox” that there are properties that are not objects and that the Russellian property is one of them. His argumentation is based on a formal-ontological system that allows for at least some properties not to be objects. When one tries to display Russell’s paradox within this system one comes to the conclusion that the property that applies to those and only those properties that do not apply to themselves is not an object. Apart from the paradox itself, Biáat’s system offers an interesting unorthodox ontological approach to objects and properties (usually it is assumed either that the classes of objects and properties are disjoint or that every property is an object). The principal aim of Joanna OdrowąĪ-Sypniewska’s paper “About the Notion of Identity” is to analyse the notion of identity and to clarify the terminology relevant to this notion. OdrowąĪ-Sypniewska characterises two main types of identity relevant to objects: abstract identity and numerical identity. She introduces further subtypes and points out some of the relations between objects which are identical numerically, and those which are identical abstractly. After describing some of the qualities of identity, the author examines the relationships between identity and such relations as sameness, kinship, likeness, equivalence, equality and difference. The author also proposes a criterion that would make the identification of identity possible.

Part III. Prognoses, Norms and Questions The subject of Tomasz Placek’s considerations in “A Puzzle about Semantic Determinism” are the ontological aspects of the discussion of truth, conducted by Twardowski and his pupils. Starting with àukasiewicz’s remarks that the thesis of the eternity of truth implies determinism, Placek expresses the view that the consequence of this thesis is the semantic but not ontological (fully-fledged) determinism. He demonstrates that àukasiewicz’s incorrect conclusions are due to his interpretation and formal characterisation of the phrase ‘it is true at a given time’ (depending on whether it is an operator or rather a predicate that ascribes a logical value; depending further on whether negation commutes with it or does not). Placek’s considerations appeal to a variety of possible intuitions: from intuitive ones to those proposed by the proponents of the logic of vagueness. The historical moral of Placek’s paper is that the birth of many-valued logic was polluted by the sin of imprecision.

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The question whether it is possible to build a bridge between deterministic (in particular causal) and chaotic visions of the world is undertaken by Max Urchs in his article “Causality in Chaotic Environment. Does Strong Causality Break Down in Deterministic Chaos?” Difficulties appear because chaos theories deny the validity of the principle of invariance of similarity, which lies at the basis of determinism in science. In the author’s opinion the solution is to consider causal and chaotic perspectives only as perspectives that can not be used together in one description of our world. His conclusion is, “There is no such thing as chaotic causality nor causal chaos.” Jan WoleĔski’s text “Three Contributions to Logical Philosophy” contains three short analytical arguments. In the third argument, “Determinism and Sentences about the Future,” WoleĔski shows (referring to Twardowski, LeĞniewski and Jordan) that the thesis on sempiternality of truth has nothing to do with the controversy between traditional determinism and indeterminism or, strictly speaking, that this thesis is compatible with both positions. The second argument, “God, Foreknowledge and Freedom,” presents the author’s formal criticism of a contemporary version of theodicy. The main aim of the first argument, “Permissions, Prohibitions and Two Legalisms,” is logical analysis of British legalism (lack of prohibition implies permission) and German legalism (lack of permission implies prohibition). Although non-classical logics were not the main interest of the LvovWarsaw school, it is well known that important results in many-valued logic, (alethic) modal logic, intuitionistic logic and paraconsistent logic were achieved by the representatives of the school. It is less well known, however, that Ajdukiewicz was one of the pioneers of the contemporary logic of questions (and answers). His first paper on questions appeared in 1926 and his most influential contribution to the field was published in 1934. After the war, the logic of questions was still an area of interest for Polish logicians; names such as Henryk HiĪ, Leon Koj, and Tadeusz KubiĔski are the first to be mentioned in that context. This volume includes a paper by Andrzej WiĞniewski, a student of KubiĔski, entitled “Reducibility of Safe Questions to Sets of Atomic Yes/No Questions.” WiĞniewski’s paper gives an example of the Polish-style approach to the logic of questions: formal tools are applied in order to solve a problem that is important not only to the logic of questions itself but also to philosophy.

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Part IV. Categorial Grammar Kazimierz Ajdukiewicz has been recognised as the founder or cofounder (with Bar-Hillel) of the theory of categorial grammar whose relevance for contemporary linguistics and philosophy of language can hardly be overestimated. Two articles in our book refer to Ajdukiewicz’s ideas in this domain. Peter Simons in his article “Languages with Variable-Binding Operators: Categorial Syntax and Combinatory Semantics” proposes an original method of surmounting the difficulties connected with the description of languages containing variable-binding operators in terms of categorial grammar. The main idea of this method is to construct an extended categorial grammar based on distinguishing the role of place marking and place filling. Simons also presents an outline of combinatorial semantics for languages with operators. On the other hand, Urszula Wybraniec-Skardowska in her article “On the Formalisation of Classical Categorial Grammar” takes into consideration LeĞniewski’s and Ajdukiewicz’s views on language in discussing the latter philosopher’s conception of classical categorial grammar in a historical context. She aims to accommodate this conception in an axiomatic system. The system of categorial grammar developed in this paper represents the concretistic approach to language, according to which tokens are the basic elements of language as opposed to the more frequently considered Platonist approach, which assumes type-tokens as basic.

Part V. Intentionality, Sense and Consequence Liliana Albertazzi, in her paper “Retrieving Intentionality. A Legacy from the Brentano School” shows that the conception of intentionality developed by Kazimierz Twardowski and his successors can serve as a tool for resolving some important problems in contemporary cognitive science. One of them, according to her, is the question of how to explain our specific (scil. dynamic) perception of immobile things (like the Leaning Tower), various immobile depictions of one object or symbols of mobile things (like mural representations of galloping horse, or exit signs), or the apparent movement of static phenomenal figures (like a moving spiral). To explain such perceptual situations, in which we transform sense data into meaningful experience events, one needs to use the subtle conceptual apparatus of intentional acts. Thanks to this

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apparatus, traditional controversies concerning the objective/subjective and the perceptual/intellectual character of duration can be precisely formulated. The main focus of Kazimierz TrzĊsicki’s contribution is indicated by his title “Ajdukiewicz’s Logical and Methodological Assumptions and the Conceptions of Meaning in Kripke and Putnam.” In Ajdukiewicz’s conception of meaning, the following (reconstructivist) assumptions are the most important. Firstly, all the semantic functions of expressions (including natural functions) must be relativised to the respective language. Secondly, natural languages are considered as an extrapolation of formal (logical) language, which is the only language in the proper sense of the word. Thirdly, the subject of the logical theory of language is language understood as a product. The problem of the relation between languages used by us and our world perspectives – posed in the thirties by Ajdukiewicz – is analysed by Anna Jedynak in her paper “On Linguistic Relativism.” According to linguistic relativism, using different languages results in different visions of world, and sums of those visions are mutually inconsistent. In consequence, different languages cannot be used simultaneously. Jedynak argues for the thesis that it is possible to compile the conceptual apparatuses of languages that are believed to be incommensurable without the risk of cognitive conflict. The impression that it is not possible arises only for pragmatic reasons. Alfred Tarski’s groundbreaking classical semantical works are still in the centre of philosophical discussions within the philosophy of language. Recently, John Etchemendy in his book The Concept of Logical Consequence (Cambridge: Harvard University Press: 1990) objected against Tarski’s identification of the logical relation of consequence with its ordinary intuitive counterpart and diagnosed Tarski’s position as an instance of “modal fallacy.” Roughly, the identification depends on the extent of the logical and extralogical terms that can be involved in the consequence relation. Dale Jacquette, in his paper “Tarski’s Analysis of Logical Consequence and Etchemendy’s Criticism of Tarski’s Modal Fallacy,” carefully analyses Etchemendy’s argumentation comparing it with Tarski’s understanding of consequence. He shows that Etchemendy’s objections can be avoided by an appropriate restriction of the terms relative to which the satisfaction of the consequence relation is determined. He stresses that Tarski’s views concerning the concept of consequence need not require a defence so long as we carefully follow Tarski’s intentions.

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Part VI. Truths and Falsehoods The historical origin of the Lvov-Warsaw School’s famous discussion on the absoluteness and the eternity as well as the sempiternity of truth is elucidated in “Sempiternal Truth. The Bolzano-Twardowski-LeĞniewski Axis” by Arianna Betti. She finds the seed for Twardowski’s and LeĞniewski’s ideas in Bolzano’s writings. As it appears, the thesis of the sempiternity of truth serves as a basis for three different ontological systems that differ, among others, in what they consider to be the truth bearers: for Bolzano, a Platonist realist, the truth bearers are ideal propositions-in-themselves; for Twardowski, an Aristotelian realist, they are typical judgements (abstracted from particular utterances); for LeĞniewski, a nominalist, they are concrete (uttered or written) sentences. Thus, for Bolzano and Twardowski, truth is atemporal whereas, for LeĞniewski, it is omnitemporal. Betti’s analysis throws an interesting light on the controversy between KotarbiĔski and LeĞniewski concerning the role of free creation in determining the ontological status of logical values as well as of their bearers. The problem of truth bearers in the Bolzano-Tarski tradition is examined also by Artur Rojszczak in his work “From the Act of Judging to the Sentence. The Truth Bearer and the Objectivisation of Truth.” It is an instructive appendix to Betti’s analysis. According to Rojszczak, the problem’s background consist in the so-called objectivisation of knowledge, i.e., in regarding it (or rather its parts) as being absolute (timeless, non-spatial) and independent from knowers (or, rather, their cognitive acts). Similarly, the objectivisation of truth can be described as the absolutisation of truth bearers or truth makers – making them independent, in other words. Wojciech ĩeáaniec’s article “What does ‘Truth in Virtue of Meaning’ Really Explain” takes up the old issue of analytic propositions. The author attempts to show in what “technical” sense propositions judged to be analytic by the logical empiricist tradition, up to and including Quine, can actually be demonstrated to be analytic. He delineates the procedures for showing that a given proposition is analytic in virtue of its logical form and the meaning of its constituent expressions, thereby giving more distinct contours to the somewhat hazy concept. Yet, the final conclusion is meant to convince the reader not to acquiesce to the logical empiricist account of analytic propositions by showing that the “truth in virtue of meaning” of some analytic propositions can be established only at the price of accepting other non-empirical propositions, the analyticity of which would have to be shown first. Thereby, a distinction among

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analytic propositions is introduced which is remotely reminiscent of the old analytic-synthetic a priori distinction. It has been said that analytical philosophy has had an advantage over non-analytical philosophy because of its ability to . . . analyse its own mistakes and prejudices. The essay “Do We Need a Definition of Truth?” is a brilliant illustration of this point. It is commonly known that one of the sources of Tarski’s semantical achievements in the thirties was the liar antinomy in àukasiewicz’s formulation. Józef Misiek argues that the attitude of analytical philosophers to this antinomy was actually the cause of a chain of misunderstandings, which paralysed their inquiry for years. Misiek’s answer to the title question of his paper is negative. The view according to which clarity is to be identified with definability is a harmful superstition. A humorous, coincidental corollary of the paper is that “the liar antinomy [. . .] is [only] a problem for the liar himself.” The more serious aspect of Misiek’s considerations consists in an adroit explanation of some vital facts in the history of philosophy of the last few decades.

Part VII. Rationality: Its Criteria and Definition According to the standard model of rationality accepted in the LvovWarsaw School and expressed by Ajdukiewicz and then Szaniawski (among others), our beliefs are rational when they fulfil three criteria: when they are articulated precisely, when they fulfil logical requirements (i.e., when they are related deductively and when there are no contradictions between them), and when they are sufficiently justified. Because many difficulties have been encountered in connection with these criteria, Ryszard Kleszcz in his paper “Criteria of Rationality,” tries to evade the problems by modifying the criteria. His modifications are twofold. Firstly, he proposes to implement different sets of criteria in different belief domains. Secondly, he argues that particular sets of criteria be weakened depending on the degree to which a given domain is theoretically complex. Kleszcz discusses in detail various versions of the three criteria mentioned above; in addition he discusses another two criteria, namely: (self)criticism and solvability of examined problems. The aim of Kleszcz’s strategy is to avoid the destructiveness of methodological relativism. The problem of a non-criterial definition of ‘rationality’ is considered by Mieszko Taáasiewicz’s in “On the Concept of ‘Rationality’.” He starts by distinguishing ontical and non-ontical (epistemic or pragmatic)

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rationality. Then, he points out that the variability of the criteria of rationality does not exclude the invariability of the notion of rationality. The desired definition of ‘rationality’ should ensure that the question of possessing respective (non-graded) properties is decidable and that logically incompatible theories are suitable simultaneously. Taáasiewicz considers as a candidate the definition of rationality as identified with what stands to reason. *** We hope that the texts presented above provide sufficient evidence to judge the accuracy of the opinion expressed in Smith’s introductory essay that they meet the contemporary “international standards of training, rigour, professionalism and specialisation” in philosophy. Jacek Jadacki Jacek PaĞniczek

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PART I THE SCHOOL: ITS ORIGINS AND SIGNIFICANCE

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Barry Smith WHY POLISH PHILOSOPHY DOES NOT EXIST

1. The Scandal of “Continental Philosophy” There are many hundreds of courses taught under the title “Continental Philosophy” (C.P.) each year in North-American universities. Such courses deal not with philosophy on the continent of Europe as a whole, however, but rather with a highly selective portion of Franco-German philosophy, centred above all around the person of Martin Heidegger. Around him is gathered a rotating crew of currently fashionable, primarily French thinkers, each successive generation of which claims itself the “end” of philosophy (or of “man,” or of “reason,” of “the subject,” of “identity” etc.) as we know it. A sort of competition then exists to produce ever wilder and more dadaistic claims along these lines, a competition that bears comparison, in more than one respect, with the competition among Hollywood film directors to outdo each other in producing ever more shocking or brutal or inhuman films. The later Husserl, Heidegger’s teacher, is sometimes taken account of in courses of this “Continental Philosophy”; not, however, Husserl’s own teacher Brentano and not, for example, such important twentieth-century German philosophers as Ernst Cassirer or Nicolai Hartmann. French philosophers working in the tradition of Poincaré or Duhem or Bergson or Gilson are similarly ignored, as, of course, are Austrian or Scandinavian or Czech philosophers. What, then, is the moment of unity of this “Continental Philosophy”? What is it that Heidegger and Derrida and Luce Irigaray have in common, which distinguishes them from phenomenologists such as Reinach or Scheler or the famous Daubert? The answer, it seems, is: antipathy to science, or more generally, antipathy to learning and to scholarly activity, to all the normal bourgeois purposes of the Western university (and we note in passing that, as far as phenomenology is concerned, it was Heidegger who was responsible for terminating that previously

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 19-39. Amsterdam/New York, NY: Rodopi, 2006.

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healthy scientific line which had brought forth such masterpieces as Brentano’s Psychology form an Empirical Standpoint and Husserl’s Logical Investigations). This rejection of the values associated with normal scholarly activity is combined, further, at least in the case of those French thinkers accredited as “Continental Philosophers” – with a substitution of politics for science (where politics, too, is to be understood in a broad sense  a sense broad enough to include also the adolescent fringe). Philosophy thereby becomes transformed into a strange type of ideologically motivated social criticism. This transformation is sometimes defended, especially by American apologists for “Continental Philosophy” such as Richard Rorty, by appeal to an argument along the following lines: i. ii.

All scientific activity is in any case an exercise of social power (here the work of Kuhn is often called in aid). The putative distinctions between “knowledge” and “power” or between “descriptive” and “performative utterances” are therefore spurious  such distinctions must be “deconstructed” (in the manner of Foucault et al.).

Hence: iii.

Philosophers should cast aside the pretension that they are seeking knowledge and should instead engage exclusively in the struggle to shift the relations of power in society (and here we note that it is above all radical feminist groupings who have gained most from the widespread acceptance, in North America, of different versions of this argument).

(The problems with the argument are, of course, legion. To mention just one obvious stumbling block: if this is indeed an “argument” in defense of what might best be described as a grab for power on the part of certain groups, then this can only be because there is, after all, a distinction between descriptive and performative utterances, for if its premises did not themselves have validity as descriptive truths, then the argument would lose all force as justification.) That the discipline of philosophy has been subject  in certain circles  to a transformation of the sort described is at the same time masked by the use of new styles of writing which are designed to fool outsiders and to protect the circles of initiates from potentially damaging criticism. The most prominent mark of such styles of writing is the heavy use of pseudo-scientific jargonizing inspired by sociology and psychoanalysis. In addition, and especially in “post-modern” circles, they

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are marked by the utilization of various tricks of irony and self“quotation,” by means of which the authors of the new philosophy seek to distance themselves from the responsibility of making assertions which might be judged as true or false. Finally, however, the new writing style is often marked by the use of what can only be called pornographic devices. Consider the following characteristically pretentious passage, chosen at random from Derrida’s Spurs, in which the French Doctor Criminale undertakes to “deconstruct” the petit-bourgeois assumption according to which the two concepts of truth and castration would be somehow distinct: The feminine distance abstracts truth from itself in a suspension of the relation with castration. This relation is suspended much as one might tauten or stretch a canvas, or a relation, which nevertheless remains – suspended – in indecision. In the HSRFȒ. It is with castration that this relation is suspended, not with the truth of castration – in which the woman does [not 1] believe anyway – and not with the truth inasmuch as it might be castration. Nor is it the relation with truth-castration that is suspended, for that is precisely a man’s affair. That is the masculine concern, the concern of the male who has never come of age, who is never sufficiently sceptical or dissimulating. In such an affair the male, in his credulousness and naivety (which is always sexual, always pretending even at times to masterful expertise), castrates himself and from the secretion of his act fashions the snare of truth-castration. (Perhaps at this point one ought to interrogate – and “unboss” – the metaphorical fullblown sail of truth’s declamation, of the castration and phallocentrism, for example in Lacan’s discourse.) (Derrida 1978, pp. 59f )

Or consider this pudding of similar nonsense from Luce Irigaray: Gynecology, dioptrics, are no longer by right a part of metaphysics  that supposedly unsexed anthropos-logos whose actual sex is admitted only by its omission and exclusion from consciousness, and by what is said in its margins. And what if the “I” only thought the thought of woman? The thought (as it were) of femaleness? And could send back this thought in its reflection only because the mother has been incorporated? The mother  that all-powerful mother denied and neglected in the self-sufficiency of the (self) thinking subject, her “body” henceforward specularized through and through. (Irigaray 1985, p. 183)

Or again:

1

The ‘not’ is left out by the translator, to no apparent consequence.

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Inside Plato’s  or Socrates’  cave, an artificial wall curtain  reenactment, reprise, representation, of a hymen that has elsewhere been stealthily taken away, is never, ever crossed, opened, penetrated, pierced, or torn. (Irigaray 1985, p. 249)

As Ms. Irigaray explains: Any hint, even, of theory, pulls me away from myself by pulling open  and sewing up  unnaturally the lips of that slit where I recognize myself, by touching myself there (almost) directly. (Irigaray 1985, p. 200)

(It is, incidentally, one not inconsiderable victory of radical feminism in the Anglosaxophone countries that the C.P.-obsession with sex, as revealed in passages such as the above, has been introduced into the pages of even the most technical scientific journals via the banishment of the unmarked personal pronoun and its replacement with a pervasive and senseless switching back and forth of gendered “she’s” and “he’s.”)

2. Philosophy in Poland What, now, of the fate of philosophy in Poland? We note in passing how sad is the spectacle presented by the host of young students of philosophy in Poland currently devoting its energies to deconstructionist and to other non-serious and ultimately corrosive philosophical fashions. More important for our purposes, however, is the degree to which Poland’s own philosophers have fared so badly as concerns their admission into the pantheon of “Continental Philosophers.” Why should this be so? Why, to put the question from the other side, should there be so close an association in Poland – at least since 1894 – between philosophy and logic, or between philosophy and science? 2 One can distinguish a series of answers to this question, which I shall group together under the following headings: (a) (b) (c) (d) (e) 2

the role of socialism; the disciplinary association between philosophy and mathematics; the influence of Austrian philosophy in general and of Brentanian philosophy in particular; the serendipitous role of Twardowski; the role of Catholicism.

What other country – to mention just one symptom of the association I have in mind – would publish an encyclopedia entitled Philosophy and Science (Cackowski, Kmita and Szaniawski, eds. 1987)?

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3. Socialism and Scientific Philosophy Much of what needs to be said about the Polish case can be derived, with suitable modifications, from considering the case of Austria. Consider, in this light, the following passage from the autobiography of A. J. Ayer, who in 1932 spent a protracted honeymoon of just over three months in Vienna before returning to Oxford to write Language, Truth and Logic: The members of the Vienna Circle, with the notable exception of Otto Neurath, were not greatly interested in politics, but theirs was also a political movement. The war of ideas which they were waging against the Catholic church had its part in the perennial Viennese conflict between the socialistic and the clerical reaction. (Ayer 1977, p. 129)

A more explicit version of the same thesis put forward by Johannes Dvorak (also quoting Neurath): In light of the fact that the bourgeoisie – especially in Central Europe – had discharged itself of all enlightenment traditions and paid homage rather to the cults of irrationalism, while the proletariat struggled for a rational formation of society, the hope certainly prevailed that “It is precisely the proletariat which will become the carrier of a science without metaphysics.” (Dvorak 1985, p. 142)

Not only Neurath and Dvorak but also other scholars working on the background of the Vienna Circle have defended a view according to which the flowering of scientific philosophy in Central Europe between the wars is to be regarded precisely as part of a wider struggle between left and right, between science and reaction. I do not believe that we need spend too much time on this purported explanation as far as Poland – a land not of proletarians but of peasants and nobility – is concerned; but the reader is asked to hold her horses before rushing forward with objections to a political account of the rise of scientific philosophy in Poland along the lines suggested.

4. Safety in Numbers At the dawn of Polish independence in 1917, as part of a widespread campaign in favour of the conception of science as a laudable form of public service in the cause of the new Poland, the mathematician Zygmunt Janiszewski committed Polish mathematicians to a program designed to take advantage of the talents of the Polish mathematical community via systematic collaboration and concentration on specific

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problems and topics of research (see Janiszewski 1917). Janiszewski’s project, which proved to be of great success, is important for our present purposes for a number of distinct reasons: (i)

(ii)

(iii)

The topics chosen, above all in the area of set theory, were such as to lead to the possibility of fruitful collaboration between mathematicians and philosophers working in the area of logic. The foundations for such collaboration were laid already in 1894 when Twardowski began to teach philosophy at the university of Lvov; at that time Twardowski advised his students to study the science of mathematics in addition to philosophy, and some of his brightest students subsequently fell under the influence of the mathematician SierpiĔski, who taught in Lvov from 1910. Janiszewski was conscious of the comparative advantage possessed by smaller countries in those fields of scientific research not requiring significant expenditures on facilities and equipment. One can point, additionally, to a certain comparative advantage enjoyed by scholars in countries such as Poland in those fields – such as mathematics or music – where the issue of the native language of the scientist or artist is of secondary importance. (The achievements of Poland in the field of mathematics are matched, significantly, by similar achievements on the part of Hungary and Finland.) These comparative advantages of smaller nations on the world stage can be carried over also to other spheres, including philosophy, and especially to those areas of philosophy most remote from issues of politics and national culture. Mathematicians, and logicians and scientific philosophers, may also enjoy the advantages of relative personal and professional safety in turbulent political times, in a way which may not be possible for thinkers working in such fields as ethics or political theory or history. The relative superiority of work done in logic in Eastern as opposed to Western Germany is, I believe, to be explained in part in terms of this factor. On the other hand, however, given the role played by logical philosophers in the Solidarity movement and in the Polish underground university during the war, it is more difficult to gauge the significance of this factor in the case of Poland.

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5. Polish Philosophy Is Austrian Philosophy In his paper “Wittgenstein and Austrian Philosophy,” Rudolf Haller writes as follows: I wish [. . .] to defend two theses: first, that in the last 100 years there has taken place an independent development of a specifically Austrian philosophy, opposed to the philosophical currents of the remainder of the German-speaking world; and secondly that this development can sustain a genetic model which permits us to affirm an intrinsic homogeneity of Austrian philosophy up to the Vienna Circle and its descendants. (Haller 1981, p. 92)

The grain of truth in this passage can be seen already if we consider the degree to which the writings of such exemplary Austrians as Bolzano, Mach, Meinong, Twardowski, Popper and Gustav Bergmann, exhibit radical differences of style as compared to German philosophers such as Hegel, Heidegger, or Habermas who are standardly associated with Germany (see Smith 1991a; Mulligan 1993). Most simply put: the former employ a sober scientific style and shun pretensions. There are also associated differences pertaining to the differential role of science and logic, as opposed to that of politics in the two traditions  differences which, as we shall see, serve to explain why it is (certain selected) German and not Austrian philosophers who have been taken up into the bosom of “Continental Philosophy” in North America. These are differences which are rooted deeply in history, and they do much to explain why Germany – in spite of the fact that it has brought forth such giants of mathematical logic as Frege, Hilbert and Gentzen – has taken so long to develop a community of analytic philosophers on its home territory and why not a few of those most centrally responsible for this development – above all Wolfgang Stegmüller – have hailed from Austria (or more precisely, in Stegmüller’s case, from the South Tyrol). I have sought elsewhere (see Smith 1994) to demonstrate the degree to which philosophy in Austria in the period from 1890 to the Anschluss was influenced by the thinking of Franz Brentano, and as the manifesto of the Vienna Circle points out, many of the characteristic concerns even of the logical positivist movement were foreshadowed in his writings: As a Catholic priest Brentano had an understanding for scholasticism; he started directly from the scholastic logic and from Leibniz’s endeavours to reform logic, while leaving aside Kant and the idealist systemphilosophers. Brentano and his students showed time and again their understanding of men like Bolzano and others who were working towards a rigorous new foundation of logic. (Neurath, Carnap and Hahn 1929, p. 302)

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In support of this contention as to the importance of Brentano, it is remarkable to consider the fact that the most important centers of scientific philosophy in Continental Europe – Vienna, Prague, Graz, Berlin, Göttingen and Lvov – were precisely those cities in which Brentano’s most distinguished students had held chairs in philosophy from the 1890s onwards. Brentano was not only sympathetic to the idea of a rigorously scientific method in philosophy; he also shared with the British empiricists and with the Vienna positivists an anti-metaphysical orientation, manifesting an especially forceful antipathy to the “mystical paraphilosophy” of the German idealists and stressing in all his work the unity of scientific method. Brentano’s writings involve the use of methods of language analysis similar in some respects to those developed later by philosophers in England. The thesis of an internally coherent tradition of Austrian philosophy is not, however, without its problems. Thus, while it seems that the works of Brentano, like those of Meinong and Husserl, were mentioned in discussions of the Vienna Circle, in the case of Brentano, at least, these writings were discussed primarily because Brentanian ethics was chosen by Schlick as a special object of scorn. Schlick himself was of course a German, as also were the three thinkers  Carnap, Reichenbach and Hempel  who are held by many to have made the most important contributions to philosophy of the logical positivist sort.

6. The Serendipitous Role of Twardowski “Austrian Philosophy” is, in any event, by no means as unified a phenomenon as some might like to believe. Perhaps, though, we can maintain a parallel thesis in regard to Poland, where the role of Brentanian ideas is more easy to gauge in light of the singular dominance, in the history of philosophy in Poland, of one man: Kazimierz Twardowski. The influence of Twardowski on modern philosophy in Poland is all-pervasive. Twardowski instilled in his students a passion for clarity and rigour and seriousness. He taught them to regard philosophy as a collaborative effort, a matter of disciplined discussion and argument, and he encouraged them to train themselves thoroughly in at least one extra-philosophical discipline and to work together with scientists from other fields, both inside Poland and internationally. This led above all, as we have seen, to collaborations with mathematicians, so that the Lvov school of philosophy would

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gradually evolve into the Warsaw school of logic, as Polish scientific philosophers availed themselves of the new techniques of formal philosophy developed by Frege and Russell. 3 Twardowski taught his students, too, to respect and to pursue serious research in the history of philosophy, an aspect of the tradition of philosophy on Polish territory which is illustrated in such disparate works as àukasiewicz’s groundbreaking monograph on the law of non-contradiction in Aristotle and Tatarkiewicz’s highly influential multi-volume histories of philosophy and aesthetics. In 1895, at the age of 29, Twardowski was appointed professor of philosophy in Lvov, still at that time an Austrian town. This meant that, like the Jagellonian University in Cracow, its university enjoyed a rather liberal and tolerant atmosphere. Thus Poles were allowed to study and to be taught by their own lecturers and professors, where “in the other parts of partitioned Poland they were engaged in a most savage struggle for national and economic survival” (Jordan 1945, p. 39). Twardowski taught at the university of Lvov until 1930 and continued to hold seminars until his death in 1938. His success in establishing a modern tradition of exact and rigorous philosophy in Poland can be seen in the fact that more than 30 of his Ph.D. students acquired professorships, and by the inter-war period his students held chairs in philosophy departments in all Polish universities with the single exception of the Catholic University in Lublin.4 As Tarski expressed it in a letter to Neurath of 1930: “almost all the researchers, who pursue the philosophy of exact sciences in Poland, are indirectly or directly the disciples of Twardowski” (Tarski 1992, p. 20). It has been suggested that Twardowski’s teaching was in some sense philosophically neutral, that the unity of his school was rooted in a common training in methods and habits of work, rather than in the handing down of any shared doctrines. Jordan, for example, asserts that the members of Twardowski’s school were not linked by any “common body of philosophical assumptions and beliefs.” Twardowski led his students, rather, “to undertake painstaking analysis of specific problems which were rich in conceptual and terminological distinctions, and

3

WoleĔski (1989) is now the standard history of the Lvov-Warsaw School. On Twardowski’s teaching see Skolimowski (1967, p. 26f ), who refers to Twardowski’s “Spartan drill.” On Twardowski’s intellectual development see Dąmbska (1978) and Twardowski (1991). 4 On Twardowski’s influence, see, again, WoleĔski (1989, Ch. 1, part 2), and also Skolimowski (1967, ch. II).

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directed rather to the clarification than to the solution of the problems involved” (Jordan 1963, pp. 7f ). While Twardowski held no truck with the system-building “philosophical” philosophies of the past, his work was nonetheless marked by a certain metaphysical attitude, which reveals itself in the work of those philosophers who came under his influence. This applies even to those – like Twardowski’s son-in-law Ajdukiewicz, also in other respects a noted Austrophile (see Giedymin 1982) – who were at certain times attracted by the positivism or reductionism of the Vienna Circle (see, e.g., Ajdukiewicz 1978, p. 348; Küng 1989). It applies to àukasiewicz, to KotarbiĔski, and to philosophers such as Drewnowski and Zawirski, who developed a conception of metaphysics as a hypothetical-deductive science to which the axiomatic method should be applied.5 The metaphysics to which Twardowski subscribed is that of Brentano (see Twardowski 1991),6 and Twardowski’s influence upon the content of modern philosophy in Poland can accordingly best be understood in terms of certain Brentanian ideas and attitudes which Twardowski conveyed to his Polish disciples. This influence reveals itself, more precisely, in the fact that modern philosophy in Poland is marked, on the one hand, by an attitude of metaphysical realism and, on the other hand, by a concern with the notion of truth as correspondence, both of which Twardowski had inherited – with some Bolzanian admixtures – from the early Brentano. Thus while Meinong’s theory of objects is a more widely known example of a generalized ontology built up on the basis of descriptive psychological analyses of the different kinds of mental acts, it was in fact Twardowski, of all the Brentanians, who was the first to develop a generalized ontology in this sense. As Ingarden puts it, Twardowski’s Content and Object (1894) is, “so far as I know, the first consistently constructed theory of objects manifesting a certain theoretical unity since the times of scholasticism and of the ‘ontology’ of Christian Wolff” (Ingarden 1938, p. 258, quoted in Schnelle 1982, p. 99). In some cases, a direct interest in Brentano and his school was inherited from Twardowski by his students. This is especially true of Ingarden; but it holds also of LeĞniewski, and àukasiewicz was subject to 5

See e.g. Jordan (1945, p. 38). A similar conception is represented in the work of contemporary Polish philosophers such as J. Perzanowski and many others. 6 As àuszczewska-Rohmanowa puts it, “Twardowski saw as his exclusive task the realization of the ideas of Brentano on Polish soil, ideas with which he himself in a way grew up and which he held to be indubitably correct” (àuszczewska-Rohmanowa 1967, p. 155, as quoted in Schnelle 1982, p. 90).

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the influence of Brentano’s ideas, too. He studied not only with Twardowski but also with Stumpf in Berlin and with Meinong in Graz, and among his earliest papers are a number of short reviews of works by Husserl, Höfler, Stumpf and Meinong. It would be wrong to suggest that specifically Brentanian doctrines were taken over whole by Twardowski’s students. Yet the implicit or explicit concern with metaphysics, and especially with realistic metaphysics and with truth as correspondence, is a constantly recurring feature of their work. Investigations in the ontology of truth, or of those relations between sentences and objects which are constitutive of truth, have been quite peculiarly prominent features of Polish philosophical writings from Twardowski to the present day, and they have coloured especially the Polish reception of the philosophy of Wittgenstein. 7 Even the early work of Tarski can illuminatingly be viewed in this light, though Tarski did not himself study with Twardowski (see Tarski 1956, p. 155, n. 2; WoleĔski and Simons 1989). At all events, though, it cannot be denied that an interest in the philosophy of truth has been a highly conspicuous moment of modern philosophy in Poland. The idea of realism, on the other hand, may initially be thought to have played a less prominent role. On closer inspection, however, we see that the realist attitude which Twardowski promulgated has in fact been taken for granted by Polish philosophers as something almost universally shared. Realism, even Aristotelian realism, is an unquestioned presupposition of LeĞniewski’s work and of that of his principal successors. It governs the work of Ingarden, dictating even the latter’s interest in the phenomena of aesthetics.8 It has been of repeated concern to Ajdukiewicz, and it has coloured also the work on epistemology of KotarbiĔski and his pupils (see Jordan 1945). And in each case, Twardowski has played a role in determining both the terminology and the thinking of the philosophers in question.

7. Scientific Philosophy and Catholicism Can we, then, accept an explanation of the rise and entrenchment of scientific philosophy in Poland in terms of the uniquely powerful influence of Twardowski? Before we move to answer this question, let us 7

From a wide selection of more recent works one might mention: Borkowski (1985), Perzanowski (1985), Suszko (1968), and Wolniewicz (1985). 8 See the Preface to his (1931); see also Ingarden’s critical writings on Husserl’s idealism, above all in his (1929).

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consider one further factor, which turns on the fact that Poland, like Austria, is a Catholic country. For some have offered religious explanations as to why scientific philosophy should have taken root in these countries – but not in (Protestant, Northern) Germany. Here again we may turn to Neurath, who writes as follows: Catholics accept a compact body of dogma and place it at the beginning of their reflections, [thus] they are sometimes able to devote themselves to systematic logical analysis, unburdened by any metaphysical details. [. . .] Once someone in the Catholic camp begins to have doubts about a dogma, he can free himself with particular ease from the whole set of dogmas and is then left a very effective logical instrument in his possession. Not so in the Lutheran camp, where [. . .] many philosophers and scholars from all disciplines, while avoiding a commitment to a clear body of dogma, have retained half-metaphysical or quarter-metaphysical turns of speech, the last remnants of a theology which has not yet been completely superseded [. . .]. This may explain why the linguistic analysis of unified science prevailed least in countries where the Lutheran faith had dealt the hardest blows to the Catholic Church, despite the fact that technology and the sciences that go along with it are highly developed in these countries. (Neurath [1933] 1987, p. 277)

Hence, Neurath claims, the “revolt against the metaphysical tradition is succeeding outside Lutheran countries in Calvinistic as well as in Catholic ones” and he notes with pride that there are in Austria “no such metaphysical autocrats as Heidegger, Rickert or others” (Neurath [1933] 1987).

8. A Copernican Shift Neurath is certainly onto something when he points to differential features of the history of Germany and Austria/Poland in this way. There are a number of severe problems with his specific thesis, however. Thus Heidegger himself was steeped rather in Catholic than in Lutheran metaphysics as a young man; and there are many Catholic countries, in other respects comparable to Poland, where logical empiricism and analytic philosophy have failed to take substantial root, just as there are Lutheran countries (Finland is here the most striking example), and of course countries of Anglican-Episcopalian filiation – not mentioned at all by Neurath – which have served as veritable bastions of the analytic tradition. Similar objections can be raised even more forcefully against our first (political) class of explanations of the rise of scientific philosophy in

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Central Europe (“‘scientific philosophy is the philosophy of the workers’ movement”). Not only is it the case that socialist movements in France or Spain or Italy, otherwise comparable to those in Poland and Austria, gave rise to no comparable movements in philosophy, but it is also the case that many of the most important thinkers associated with the Vienna circle and with the rise of scientific philosophy in Poland treated socialism with disdain (see Smith 1996). What, then, of explanations of the rise of scientific philosophy in Poland which stress the role of Austria in general and of Brentano and Twardowski in particular? Here again there are problems, not least as a result of the fact that the available explanations do little to explain why the ideas of Brentano and Twardowski were able to plant such firm and lasting roots precisely on Polish soil, where the contemporary influence of Brentanism in Austria itself, and in other parts of the former AustroHungarian Empire, is almost vanishingly weak. These problems become all the greater when one reflects on the tremendous difficulties faced by philosophers in Poland in keeping alive their philosophical traditions through the course of this country. It may, indeed, be possible to adjust and amend the offered explanations in order successfully to confront this and the remaining difficulties and thus to provide a reasonable explanation of the rise of scientific philosophy in Poland. I would like, however, to look at the matter from another, quite different perspective, and to raise instead the question: why did scientific philosophy not take root in, say, Bulgaria or Tadjikistan? The answer to these questions is I hope rather clear: scientific philosophy, or in other words a philosophy that respects the values of clarity, precision, seriousness, rigour and technical competence, is the product of an advanced intellectual culture and of the Western university. But why, then, did scientific philosophy to such a marked degree fail to take root in Germany? Recall, again, Neurath’s explanation of the metaphysical character of German philosophy in terms of the historical experience of the German people. We had occasion to reject this explanation when formulated in religious terms. What happens, however, if we recast the account in terms of the history of the Germans in the more narrow political sense? This manifests, from our present (certainly, radically over-simplified) point of view certain peculiar features as contrasted to the political history of the English or the Austrians. For philosophy has played a role in the history of the German state that is quite unique. Just as England has its National Theatre, and America has its Constitution and its Declaration of Independence, so Germany has its National Philosophy: Kant, Fichte,

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Hegel, Schlegel, etc. are national monuments of the German people, whose memory is held sacred not least because they were so closely involved in creating that unified national consciousness which made possible Germany itself as a unified nation state. Philosophical thinkers were made to play a role in the history of Germany that is analogous to the role of Mickiewicz in the history of Poland, of Homer in the history of Greece, or of Shakespeare in the history of the English. From this point of view, it is no accident that the most impressive German contributions to philosophical scholarship have consisted precisely in great critical editions of the classical German idealist philosophers. (For a consideration of the role of these editions in German culture, see Smith 1992.) The size and power of Germany in the twentieth century has furthermore ensured that the Germans have been able to keep alive their national philosophical tradition and to protect it in a way that would have been impossible for the smaller nations of Central and Eastern Europe. Certainly outside influences have occasionally been absorbed into the large central mass of post-Kantian German philosophising. Elements of the thinking of Searle, for example, have been absorbed into the work of Habermas and Apel; but they have thereby been transformed into something that is, from the perspective of the analytic tradition in which Searle’s thinking has its origins, scarcely recognisable. Attempts were made in the nineteenth century, in part under the inspiration of Fichte and the German model, to found a (“Messianistic”) national philosophy in Poland also: as the “Christ among the Nations” Poland has a quite special mission in the history of European Christianity. (Compare, in this respect, the work of such figures as Cieszkowski, Trentowski, Hoene-WroĔski; as discussed in Kuderowicz 1988.) Such attempts proved to be of little lasting influence, however, and were, in fact already in the 1870s overwhelmed by the more forwardlooking so-called “positivists” in Warsaw, who accused the nationalistic philosophers of being too naively idealistic in their goals and of thereby promoting national tragedies. The positivists advocated instead the virtues of “small” or “organic” work, which consisted not least in the promotion of science among the people in the spirit of Mill, Comte and Spencer (see Jadacki 1994). Now there is, of course, no important national philosophy in Britain or Austria, either: the imperial, multinational tradition seems for such purposes to be both too broad and too loose. There is also, and correspondingly, no particular concern for the purity or authenticity of the language of philosophy in England or Austria, for such issues there

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do not go hand in hand with issues of national-cultural integrity. And that Poland, as the mother of many nationalities, is to be seen as being allied to England and Austria in this respect manifests itself further in the fact that the Polish intellectuals, too  from Twardowski, born in Vienna in 1866, to LeĞniewski, born in Petersburg in 1886  have often been multinational in their aspirations and have thus not shunned the use of Western languages in their work. How different is the case of Germany (and, by extension, France). The works of Kant, Fichte, Hegel, Heidegger are master texts of the German people, and like all such master texts, be they the master texts of a religion, a sect, a people or a culture, they manifest that type of density and obscurity which goes hand in hand with the tendency to spawn a commentary literature, with all that this implies by way of association with the commentary literatures on, for example, Aristotle, the Bible, or the writings of Marx and Engels (see Smith 1991b). German philosophers have in fact for centuries been schooled systematically in the habits of a philosophical culture in which the most important textual models are associated with a need for commentaries and with what one might call a hermeneutic intransigence. They grow up further in a philosophical culture which is sealed off by firm disciplinary boundaries from the empirical sciences and which places a high value not on consistency and clarity, but rather on “depth” and “authenticity.” Teamwork and the exercise of mutual criticism and persistent argument, international collaboration, and indeed the search for any sort of “truth” in philosophy, are from this perspective simply out of place (see Puntel 1991). Philosophy, rather, is seen as something that should come directly from the heart, as a direct expression of the author’s soul or “spirit.” Consider, in this context, the mind-deadeningly repetitive stream-of-consciousness rantings of Derrida who shows how, in this as in so much else, French philosophy (or more precisely, that part of French philosophy that has been approved as part of “Continental Philosophy”), has become little more than a parody of its German model (the result of applying to German philosophy a Nietzschean sceptical rejection of the very idea of seriousness). In the wider world, of course, it is not classical German idealism, with its national, textual and historical associations, but rather empirical, or at least scientifically oriented, philosophy that has come to constitute the contemporary mainstream. The latter is, for reasons not altogether accidental, a philosophy which values logic, argument and technical competence more highly than those literary, ideological and historical qualities which are at a premium in certain philosophical circles in

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Germany and France. Moreover, it seems likely to be the case that (whether for good or ill), as the discipline of philosophy becomes ever more a creature of the modern university, it will come to be marked to an increasing degree by the factor of professionalization, so that respect for technical competence and for the scientific method and the rejection of hagiography and the use of a mystificatory style will come increasingly to characterize the discipline of philosophy as a whole. The most prominent Polish philosophers have, when seen from this perspective, been speaking prose all along without knowing it.

9. Why Polish Philosophy Does Not Exist If, now, we return to our question as to how we are to explain the rise of scientific philosophy in Poland, then we can see that this question in fact needs no answer. In Poland, exactly as in Austria, and Scandinavia, and exactly as in England and the rest of the Anglosaxophone world, the rise of scientific philosophy is an inevitable concomitant of the simple process of modernization. Just as the term ‘Austrian Philosophy’ is a misnomer to the degree that it suggests that there is a corresponding national or regional or ethnic philosophy, or a special Austrian way of doing philosophy that is unavailable to those born (say) outside the borders of the former Habsburg Empire; and just as the term ‘women’s philosophy’ is a misnomer to the extent that it suggests that there is a special way of doing philosophy that is available only to those of feminine gender, so also the term ‘Polish philosophy’ is a misnomer  and for just the same reasons. For Polish philosophy is philosophy per se, it is part and parcel of the mainstream of world philosophy – simply because, in contrast to French or German philosophy, it meets international standards of training, rigour, professionalism and specialization.

10. The Law of Conservation of Spread Why, then, is “Continental Philosophy” so popular in certain circles? Why do young philosophers in Poland in such large numbers choose to read – or rather buy – the newly made translations of the ghastly works of Derrida and his ilk? Why, more generally, should intelligent people come to hold that it is not necessary, in doing philosophy, to meet the normal standards of clarity and precision? Why, indeed, should some have found it attractive to reject the very goal of clarity in philosophy,

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and to praise, instead, the putative virtues of obscurity and depth? To answer these questions I should like, appealing to an analogy with the physicist’s law of conservation of matter, to advance a law of conservation of the various branches of intellectual concern which have traditionally, in the West, been grouped together under the heading “philosophy.” If one or other of these branches is in one way or another suppressed, or so I want to suggest, then it will somehow find a way to force itself through in some new and unexpected territory, or in some new and bastardized form. (If Marxist philosophy, broadly conceived, is no longer able to be taken seriously in the fields of economics or political theory, then it will rise again in the field of, say, “comparative literature” or “critical legal theory” or in the transzendentale Sprachpragmatik of Habermas and Apel.) Something like this, I suggest, has been the fate of many of the classical philosophical concerns now customarily dealt with by those pleased to call themselves “Continental Philosophers,” many of whom are grouped together in American universities in departments of comparative literature, of film studies, of “woman studies,” and so forth. For the best philosophical minds in the Anglo-Saxon world – and in Poland – have in recent decades turned primarily to logic and to the related branches of our discipline. To put it plainly: really existing logical or scientific philosophy, philosophy as concretely practiced in alliance with logic and science, has been overly narrow in the scope of the problems with which it has deigned to concern itself and has been too often associated with metaphysical standpoints (of nominalism, inscriptionalism, reism, eliminativist materialism, and so on) on the basis of which it is difficult, to say the least, to do justice to what we might call the phenomenological aspects of human existence. Those who have sought answers to the broader philosophical questions have thus fallen into the arms of those “Continental Philosophers” whose knavish tricks have been described above. Part of the blame for the excesses of the “Continental Philosophers” is, accordingly, to be laid at the door of Ryle and Quine and Carnap, who have played something like the same role, in the turning away from a unified adequatist metaphysics in our own day, as was played by Galileo (or the Galileo of Husserl’s Crisis) in an earlier period. (It is not for nothing that Quine, together with Heidegger and Derrida, is one of the heroes of Richard Rorty.) The problems of narrowness of scope and of metaphysical reductionism in philosophy were, perhaps, of lesser significance in Poland  where Ingarden, above all, kept alive the flame of Grand Metaphysics, but even in Poland there were some influential members of the Lvov-Warsaw School who have

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muddied the waters by being less than wholly clear as to whether they were or were not engaged in “metaphysics.”

11. What Is to Be Done? How, now, should those – the contemporary heirs of Russell and àukasiewicz  be they in Cracow or Canberra, in Pittsburgh or PoznaĔ – who believe in truth in philosophy, react to these developments? Should they simply ignore “Continental Philosophy” and the text- and commentary-based traditions of philosophizing in Germany and France from out of which it grew? Can they justifiably embrace the hope that they will all simply go away? Or is it not much rather the case that a fashion economy, when once established, manifests a quite remarkable resilience? Should they, as is now all too customary, allow the inhabitants of the C.P.-ghetto of Heideggerians, Derridians and Irigarians to perform their antics undisturbed, whether in the spirit of pluralistic tolerance or in that of scornful disdain? To react in this fashion would, I believe, be a great mistake. This is not because I believe that the proper reaction to the cynicisms, relativisms and irrationalisms which predominate in so many corners of our “postmodern” world would be to form a new “movement” charged with agitation on behalf of the scientific conception of philosophy. For as Schlick, however dimly, saw, the formation of a movement of “scientific philosophy” – to be ranked alongside “women’s philosophy,” “Australian regional philosophy,” and the like – can only contribute to the widespread confusion of supposing that there are different sorts of truth (see Smith 1996): scientific truth, women’s truth, aboriginal truth, proletarian truth, aryan truth, Tadjik truth, German truth, Jewish truth, and so on. Rather, we should orient ourselves more steadfastly around the idea that it is the proper business of philosophy to search for truth (for truth simpliciter), including truth in the various fields of the history of philosophy. This must imply also a search for truth even in relation to those byways of philosophical history and of philosophical concern that do not fit well into the customary and rather narrow picture of philosophical history which has been favoured by analytic philosophers hitherto. It must imply, indeed, a search for truth in the history of German and even of French philosophy in all its breadth. We should shine light, if one will, upon the dark places of our discipline and seek out the monsters that are breeding in its mists.

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It would be one incidental benefit of the study of the history of philosophy along these lines that it would help to make clear to philosophers and others that in former times, too, which is to say in previous dark ages of philosophical development, generations of philosophers have repeatedly been wont to declare themselves as constituting the “end,” or the “death,” of philosophy as we know it and have thereby engaged in competition with their predecessors in the wildness of the antics with which they have set out to support such claims. On the other hand, however, it will become clear also to the student of this catholic history of philosophy that such dark periods in philosophical history were in each case succeeded by new and healthier phases, in which truth and reason were once more, and with renewed vigour, given their due. 9

University at Buffalo Department of Philosophy 135 Park Hall Buffalo, NY 14260-1010, USA e-mail: [email protected]

REFERENCES Ajdukiewicz, K. (1978). “The Scientific World-Perspective” and Other Essays. Dordrecht: D. Reidel. Ayer, A.J. (1977). Part of My Life. London: Collins. Borkowski, L. (1985). A Formulation of the Classical Definition of Truth. Studies in Logic and Theory of Knowledge 1, 33-44. Cackowski, Z., J. Kmita, and K. Szaniawski, eds. (1987). Filozofia a nauka: Zarys encyklopedyczny [Philosophy and Science. An Encyclopaedic Outline]. Warszawa: Ossolineum. Dąmbska, I. (1978). François Brentano et la pensée philosophique en Pologne: Casimir Twardowski et son ecole. In: R.M. Chisholm and R. Haller (eds.), Die Philosophie Franz Brentanos, pp. 117-130. Amsterdam: Rodopi. Derrida, J. (1978). Spurs: Nietzsche’s Styles. Translated by B. Harlow. Chicago & London: The University of Chicago Press. 9

I am grateful to Jacek Jadacki, Anna Kanik, Jerzy Perzanowski, Artur Rojszczak, Jan WoleĔski and to my esteemed translator, all of whom provided valuable help and inspiration during the preparation of this paper. Research for this publication was supported by a grant from the International Research and Exchanges Board, with funds provided by the US Department of State (Title VIII) and the National Endowment for the Humanities. None of these organizations is responsible for the views expressed.

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Dvorak, J. (1985). Wissenschaftliche Weltauffassung, Volkshochschule und Arbeiterbildung im Wien der Zwischenkriegszeit. Am Beispiel von Otto Neurath und Edgar Zilsel. In: H.J. Dahms (ed.), Philosophie, Wissenschaft, Aufklärung. Beiträge zur Geschichte und Wirkung des Wiener Kreises, pp. 129-143. Berlin & New York: Walter de Gruyter. Giedymin, J. (1982). Science and Convention: Essays on Henri Poincaré’s Philosophy of Science. Oxford: Pergamon Press. Haller, R. (1979). Studien zur österreichischen Philosophie. Amsterdam: Rodopi. Haller, R. ([1975] 1981). Wittgenstein and Austrian Philosophy. In: J.C. Nyíri, ed., (1986), pp. 91-112. Haller, R. (1986). Zur Historiographie der österreichischen Philosophie. In: J.C. Nyíri (ed.), (1986), pp. 41-53. Haller, R. (1986a). Fragen zu Wittgenstein und Aufsätze zur österreichischen Philosophie. Amsterdam: Rodopi. Haller, R. (1988). Questions on Wittgenstein. London: Routledge. Haller, R. (1993). Neopositivismus: Eine historische Einführung in die Philosophie des Wiener Kreises. Darmstadt: Wissenschaftliche Buchgesellschaft. Ingarden, R. (1929). Bemerkungen zum Problem Idealismus-Realismus. In: Festschrift Edmund Husserl zum 70. Geburtstag gewidmet (Jahrbuch für Philosophie und phänomenologische Forschung Ergänzungsband), pp. 159-190. Halle: Niemeyer. Ingarden, R. ([1931] 1973). Das literarische Kunstwerk. Halle: Niemeyer. English translation: The Literary Work of Art (Evanston: Northwestern University Press, 1973). Ingarden, R. (1938). DziaáalnoĞü naukowa Twardowskiego [Scientific Activity of Twardowski]. In: Kazimierz Twardowski: Nauczyciel – uczony – obywatel [Teacher  Scholar  Citizen], pp. 13-30. Lwów: PTF. Irigaray, L. (1985). Speculum of the Other Woman. Translated by G.C. Gill. Ithaca: Cornell University Press. Jadacki, J.J. (1994). Warsaw: The Rise and Decline of Modern Scientific Philosophy in the Capital City of Poland. Axiomathes 2-3, 225-241. Janiszewski, Z. (1917). O potrzebach matematyki w Polsce [On the Needs of Mathematics in Poland]. WiadomoĞci Matematyczne 7 (1963), 3-8. Jordan, Z. (1945). The Development of Mathematical Logic and of Logical Positivism in Poland between the Two Wars. Polish Science and Learning, vol. 6. Oxford: Oxford University Press. Partially reprinted in: McCall (1967), pp. 346-397. Jordan, Z. (1963). Philosophy and Ideology: The Development of Philosophy and Marxism-Leninism in Poland since the Second World War. Dordrecht: Reidel. Kuderowicz, Z. (1988). Das philosophische Ideengut Polens. Bonn: Bouvier. Küng, G. (1989). Ajdukiewicz’s Contribution to the Realism/Idealism Debate. In: Szaniawski, ed. (1989), pp. 67-85. àuszczewska-Rohmanowa, S. (1967). Program filozofii naukowej Kazimierza Twardowskiego [Kazimierza Twardowski’s Project of Scientific Philosophy]. Studia Filozoficzne 4, 154-168. McCall, S., ed. (1967). Polish Logic 1920–1939. Oxford: Clarendon Press. Mulligan, K. (1993). Post-Continental Philosophy: Nosological Notes. Stanford French Review 17, 133-150. Neurath, O., R. Carnap and H. Hahn (1929). Wissenschaftliche Weltauffassung: Der Wiener Kreis. Vienna: Wolf. English translation: The Scientific Conception of the World, in: O. Neurath (ed.), Empiricism and Sociology (Dordrecht: Reidel, 1973), pp. 299-318. (Page references are to the English translation.)

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Neurath, O. ([1933] 1987). Unified Science and Psychology. In: B. McGuinness (ed.), Unified Science (Dordrecht: Reidel), pp. 1-23, 274-278. (Page references are to the English translation.) Nyíri, J.C., ed. (1981). Studien zur Österreichischen Philosophie. Munich: Philosophia. Nyíri, J.C., ed. (1986). From Bolzano to Wittgenstein: The Tradition of Austrian Philosophy. Vienna: Hölder & Pichler-Tempsky. Perzanowski, J. (1985). Some Observations on Modal Logic and the Tractatus. In: R.M. Chisholm et al. (eds.), Philosophy of Mind – Philosophy of Psychology, pp. 544-550. Vienna: Hölder & Pichler-Tempsky. Puntel, L.B. (1991). The History of Philosophy in Contemporary Philosophy: The View from Germany. Topoi 10, 147-154. Schnelle, T. (1982). Ludwik Fleck – Leben und Denken: Zur Entstehung und Entwicklung des soziologischen Denkstils in der Wissenschaftsphilosophie. Freiburg i. Br.: Hochschulverlag. Skolimowski, H. (1978). Polish Analytic Philosophy: A Survey and a Comparison with British Analytic Philosophy. London: Routledge and Kegan Paul. Smith, B. (1991a). German Philosophy: Language and Style. Topoi 10, 155-161. Smith, B. (1991b). Textual Deference. American Philosophical Quarterly 28, 1-13. Smith, B. (1992). Thesen zur Nichtübersetzbarkeit der deutschen Philosophie. In: D. Papenfuss and O. Pöggeler (eds.), Zur philosophischen Aktualität Heideggers, vol. 3: Im Spiegel der Welt: Sprache, Übersetzung, Auseinandersetzung, pp. 125-147. Frankfurt: Klostermann. Smith, B. (1993). The New European Philosophy. In: B. Smith (ed.), Philosophy and Political Change in Eastern Europe, pp. 165-170, 191-192. La Salle: The Hegeler Institute. Smith, B. (1994). Austrian Philosophy: The Legacy of Franz Brentano. La Salle & Chicago: Open Court. Smith, B. (1996). Austria and the Rise of Scientific Philosophy: The Neurath-Haller Thesis. In: K. Lehrer and J.C. Marek (eds.), Austrian Philosophy Past and Present, pp. 1-20. Dordrecht: Kluwer. Smith, B., ed. (1994). European Philosophy and the American Academy. La Salle & Chicago: Open Court. Suszko, R. (1968). Ontology in the Tractatus of L. Wittgenstein. Notre Dame Journal of Formal Logic 9, 7-33. Szaniawski, K., ed. (1989). The Vienna Circle and the Lvov-Warsaw School. Dordrecht: Kluwer. Tarski, A. (1956). Logic, Semantics, Metamathematics. Oxford: Clarendon. Tarski, A. (1992). Drei Briefe an Otto Neurath. Grazer Philosophische Studien 43, 1-32. Twardowski, K. (1894). Zur Lehre vom Inhalt und Gegenstand der Vorstellungen: Eine psychologische Untersuchung. Vienna: Hölder. Reproduced: Munich & Vienna: Philosophia, 1982. English translation: On the Content and Object of Presentations, translated by by R. Grossmann (The Hague: Martinus Nijhoff, 1977). WoleĔski, J. (1989). Logic and Philosophy in the Lvov-Warsaw School. Dordrecht: Reidel. WoleĔski, J., P.M. Simons (1989). De Veritate: Austro-Polish Contributions to the Theory of Truth from Brentano to Tarski. In: Szaniawski, ed. (1989), pp. 391-442. Wolniewicz, B. (1985). Ontologia sytuacji: Podstawy i zastosowania [Ontology of Situations: Foundations and Applications]. Warszawa: PWN.

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Jacek Jadacki THE LVOV-WARSAW SCHOOL AND ITS INFLUENCE ON POLISH PHILOSOPHY OF THE SECOND HALF OF THE 20TH CENTURY

0. Thesis My thesis reads: Many factors influenced Polish philosophy of the second half of the 20th century; among those factors were philosophical traditions that had been present in Poland for a long time as well as traditions that appeared only after the Second World War with the influence of foreign ideas (mainly European and American analytical thought) and external political circumstances also playing a formative role. But it was the Lvov-Warsaw School whose influence proved decisive  in any case upon what was of the greatest value in Polish philosophy of this period Firstly, the majority of the most respected philosophers were students of the School’s representatives and many of them also declared their membership of it. Secondly, the School determined the program of scientific philosophy that held good in Poland  not without intervals of course – during the course of the entire 20th century. This was the program formulated by Twardowski at the beginning of his Lvov professorship and articulated emphatically by Jan àukasiewicz in Warsaw. In the second fifty-year period, the postulate of the scientific character of philosophy was strongly re-accentuated by Andrzej Grzegorczyk (1989). Thirdly, in the School two complementary methods of realising this program were present: the method of semantic analysis (Kazimierz Twardowski, Tadeusz CzeĪowski) and the method of formal reconstruction (Jan àukasiewicz, Stanisáaw LeĞniewski). Fourthly, problems which were exposed and elaborated by representatives of the first generation of the School have remained central in the research of their successors. Fifthly, proposals put forth by the main Polish

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 41-83. Amsterdam/New York, NY: Rodopi, 2006.

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philosophers of the last fifty-year period usually referred to the results achieved in the School. They were either improvements of those results or counterproposals based on reliable criticisms of those results. I shall concentrate on the last two matters, because their importance is not only of a historical or local character. A systematic examination of the theoretical problems and results reached in contemporary Polish philosophy may be of use for many philosopher-specialists in their current research and may make the actual state of Polish and, in consequence, Euro-American philosophy easier for a philosopheramateur to comprehend (and who of us is not just an amateur in the majority of philosophical disciplines?!). With this in mind, I include a comparatively rich bibliographical guide, focusing especially on texts available in English. Before discussing the influence of the Lvov-Warsaw School on the Polish philosophy of the second half of the 20th century, I shall briefly describe the stages 20th-century Polish philosophy went through and its institutional and publishing basis.

1. Division into Periods From the perspective of the Lvov-Warsaw School, twentieth-century Polish philosophy began . . . in the 19th century (in 1895 to be precise) when Twardowski, the founder of the School, took the chair of philosophy at Lvov University. The century which has passed since that memorable year falls naturally into five twenty-year phases, preceded by the five-year prologue (1895-1900). 1.1. The Phase of Crystallisation During the first twenty years (1900-1920)  let us call them the phase of crystallisation  the process of forming the creative personalities of the main representatives of the first generation of Twardowski’s students took place; the majority of them having been born in the 1880s: Wáadysáaw Witwicki (b. 1878), Jan àukasiewicz (b. 1878), Bronisáaw Bandrowski (b. 1879), Marian Borowski (b. 1879), Zygmunt Zawirski (b. 1882), Kazimierz SoĞnicki (b. 1883), Stanisáaw LeĞniewski (b. 1886), Tadeusz KotarbiĔski (b. 1886), Wáadysáaw Tatarkiewicz (b. 1886), Stanisáaw Kaczorowski (b. 1888), Daniela Tennerówna-Gromska (b. 1889), Tadeusz CzeĪowski (b. 1889) and Kazimierz Ajdukiewicz (b. 1890). This was a generation to which Czesáaw Znamierowski (b. 1888), a philosopher and a jurist and Zygmunt Janiszewski (b. 1888),

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a mathematician, also belonged. Notwithstanding the fact that they studied abroad (Znamierowski with Cornelius and Janiszewski with Poincaré), both of them found themselves in the orbit of the influences of Twardowski’s School and played an important part in it. Janiszewski was one of the main initiators of the co-operation between mathematicians and philosophers, which resulted in the Warsaw Logical School. Znamierowski brought about a fusion of Leon PetraĪycki’s philosophy of law with the paradigm of Twardowski’s School. In the first five years of this period, the majority of the representatives of the second generation of the School came into the world: Alfred Tarski (b. 1901), Dina-Janina Sztejnbarg-KotarbiĔska (b. 1901), Edward PoznaĔski (b. 1901), Aleksander Wundheiler (b. 1902), Mordchaj Wajsberg (b. 1902), Józef Maria BocheĔski (b. 1902), Jan Salamucha (b. 1903), Izydora Dąmbska (b. 1904), Seweryna àuszczewska-Romahnowa (b. 1904), Jerzy Sáupecki (b. 1904), Henryk Mehlberg (b. 1904), Adolf Lindenbaum (b. 1904), MojĪesz Presburger (b. 1904), Maria KokoszyĔska-Lutmanowa (b. 1905), Leopold Blaustein (b. 1905), Stanisáaw JaĞkowski (b. 1906), Bolesáaw SobociĔski (b. 1906) and Stefan SwieĪawski (b. 1907). Józef Iwanicki (b. 1902) was born at the same time. His path was similar to that taken by Twardowski’s students, although he studied not in Lvov but in Strasbourg. All of them would join the group of their earlier-born colleagues: Antoni Korcik (b. 1892), Antoni PaĔski (b. 1894), Mieczysáaw Walfisz-Wallis (b. 1895), Maria NiedĨwiecka-Ossowska (b. 1896), Jan Franciszek Drewnowski (b. 1896), Stanisáaw Ossowski (b. 1897) and Janina Hosiasson-Lindenbaumowa (b. 1899). This period was ended by the premature death of two uncommonly talented scholars, Bandrowski (d. 1914) and Janiszewski (d. 1920) mentioned above, and by the outbreak of the First World War (waged in great part on Polish territories), accompanied – in Poland’s case  by the Russian invasion (luckily victoriously repulsed), which interrupted normal scientific activities. 1.2. The Phase of Prosperity The second twenty-year period (1920-1940) was also closed by violent military and political events: in 1939, Poland was attacked by its allied neighbours  Nazi Germany and Bolshevik Russia; this time, it was defeated and lost its independence for fifty years. In philosophy  as well as in the whole Polish culture  this second period was a phase of splendid creative prosperity: talents that crystallised during the former period bore fruit in the form of original scientific work  ideas, conceptions and systems. At the same time, in the first part of the period,

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the future apostles and improvers of these results come into the world; they will belong to the third generation of the School: Jerzy àoĞ (b. 1920), Zygmunt ZiembiĔski (b. 1920), Leon GumaĔski (b. 1921), Halina Mortimer (b. 1921), Ija Lazari-Pawáowska (b. 1921), Jan Gregorowicz (b. 1921), Andrzej Grzegorczyk (b. 1922), Marian PrzeáĊcki (b. 1923), Tadeusz KubiĔski (b. 1923), Henryk Stonert (b. 1923), Jerzy Pelc (b. 1924), Tadeusz Pawáowski (b. 1924), Klemens Szaniawski (b. 1925), Jerzy Giedymin (b. 1925), Zdzisáaw Augustynek (b. 1925), Wojciech Pogorzelski (b. 1927), Bogusáaw Wolniewicz (b. 1927), Lech Dubikajtis (b. 1927), Zbigniew CzerwiĔski (b. 1927), Bogusáaw IwanuĞ (b. 1928), Leon Koj (b. 1929), Andrzej Malewski (b. 1929), Henryk Skolimowski (b. 1930), Tadeusz Kwiatkowski (b. 1930), Stanisáaw Surma (b. 1930), Ewa ĩarnecka-Biaáy (b. 1930), Witold Marciszewski (b. 1930), Zdzisáaw Ziemba (b. 1930) and, just after them, Ryszard Wójcicki (b. 1931), Jerzy Kmita (b. 1931) and Andrzej Siemianowski (b. 1932). They joined the people born in the previous ten years: Jordan (b. 1911), Czesáaw Lejewski (b. 1913), Andrzej Mostowski (b. 1913), Ludwik Borkowski (b. 1914), Stanisáaw Mazierski (b. 1915), Jerzy Kalinowski (b. 1916), Helena Rasiowa (b. 1917), Henryk HiĪ (b. 1917), Stanisáaw KamiĔski (b. 1919) and Roman Suszko (b. 1919). In his remarkable History of Philosophy, Tatarkiewicz characterised the situation of Polish philosophy at the threshold of the second half of our century: The second great war found it in blooming state [. . .]. It was destroyed by occupants between 1939 and 1944. The great part of the young generation perished in fights or was murdered in German [and Russian, let us add] camps. And the great part of scientific workshops, libraries and institutes, was devastated, robbed, razed to the ground. For Poland, much more than for other countries, these years closed an important and rampant, but short and unfinished epoch. (Tatarkiewicz 1981, p. 371)

1.3. The Phase of Destruction The external circumstances mentioned above meant that the third twenty years (1940-1960) deserve to be called the “phase of destruction.” Firstly, its beginning years witnessed the death of Twardowski (d. 1938), Borowski (d. 1938) and LeĞniewski (d. 1939). Then, the war devoured Wajsberg (d. 1939), Lindenbaum (d. 1941), PaĔski (d. 1942), HosiassonLindenbaumowa (d. 1942), Blaustein (d. 1944) and Salamucha (d. 1944). Soon after the war, Zawirski (d. 1948) and Witwicki (d. 1948) passed away. Afterwards, the country was exposed to the darkest five years of

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communist terror. Shortly after both àukasiewicz (d. 1956) and Wundheiler (d. 1957) died, both having emigrated earlier. It is hardly surprising that Polish analytical philosophy  which was an object of special pressure of the ideologists of the regime  went into the intellectual underground or, at best, was reduced to formal logic. At the time, this looked like the final destruction of the analytical tradition in Poland. Skolimowski, a historian of this tradition, wrote in 1967: The continuous development of the analytical movement [in Poland] led to its finest results in the late 1920s and in the 1930s. The war shattered this continuity. After the war, analytical philosophy never regained its previous strength; the 1950s saw its definitive decline. (Skolimowski 1967, p. 260) [In the early 1960s], the analytical movement become emasculated. [ . . .] Analytical philosophy is no longer a dominant trend in Poland; its strength has been diluted; its output drastically limited. (p. 235)

Skolimowski could not know, of course, that in this period the future representatives of the fourth generation of Twardowski’s successors were born and that they would give an analytical direction to Polish philosophy in the last ten years of the 20th century. Józef Herbut (b. 1933), Tadeusz Batóg (b. 1934), Barbara Stanosz (b. 1935), Stanisáaw MajdaĔski (b. 1935), Grzegorz Bryll (b. 1935), Zbigniew Zwinogrodzki (b. 1935), Stefan Zamecki (b. 1936), Adam Nowaczyk (b. 1936), Robert Leszko (b. 1937), Bohdan ChwedeĔczuk (b. 1938), Stanisáaw Kiczuk (b. 1938), Andrzej Bronk (b. 1938), ElĪbieta Pietruska-Madej (b. 1938), Edward NieznaĔski (b. 1938) and Tadeusz Prucnal (b. 1939) were born before the war. Then, successively: Krystyna Zamiara (b. 1940), Jan WoleĔski (b. 1940), Urszula Wybraniec-Skardowska (b. 1940), Eugeniusz ĩabski (b. 1940), ElĪbieta KaáuszyĔska (b. 1941), Mieczysáaw Omyáa (b. 1941), Zdzisáaw Kowalski (b. 1942), Leszek Nowak (b. 1943), Jerzy Perzanowski (b. 1943), Jacek Hoáówka (b. 1943), Ulrich Schrade (b. 1943), Joanna Górnicka-Kalinowska (b. 1943), Józef Wajszczyk (b. 1944), Teresa Hoáówka (b. 1944), Grzegorz Malinowski (b. 1945), Jan Zygmunt (b. 1945), Jacek Jadacki (b. 1946), Józef ĩyciĔski (b. 1948), Urszula ĩegleĔ (b. 1949), Wojciech Patryas (b. 1949), Adam Grobler (b. 1949), Roman Murawski (b. 1949), Witold StrawiĔski (b. 1949), Barbara TuchaĔska (b. 1949), Janusz Czelakowski (b. 1949), Michaá Krynicki (b. 1950), Kazimierz Jodkowski (b. 1950), Ryszard Kleszcz (b. 1950), Wojciech Buszkowski (b. 1950), Zbysáaw MuszyĔski (b. 1951), Marian Grabowski (b. 1951), Jacek PaĞniczek (b. 1951), Jerzy Pogonowski (b. 1951), Teresa Rzepa (b. 1952), Maágorzata Czarnocka

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(b. 1952), Wojciech Sady (b. 1952), Adam Jonkisz (b. 1953), Stanisáaw Judycki (b. 1954), Andrzej Pietruszczak (b. 1954), Cezary Gorzka (b. 1955), Marek Rosiak (b. 1956), Anna Jedynak (b. 1956), Jan Czerniawski (b. 1957), Tadeusz Szubka (b. 1958) and Andrzej WiĞniewski (b. 1958). 1.4. The Phase of Restoration The fourth twenty years (1960-1980) constitute a phase of the restoration of the Lvov-Warsaw School’s influence on Polish philosophy. This phenomenon occurred even though three important representatives of the first generation of Twardowski’s School passed away: Ajdukiewicz (d. 1963), SoĞnicki (d. 1971) and Tennerówna-Gromska (d. 1973), as well as Znamierowski (d. 1967). Also some members of the second generation died: Ossowski (d. 1963), JaĞkowski (d. 1965), Korcik (d. 1969), Kaczorowski (d. 1971), Ossowska (d. 1974), Drewnowski (d. 1978) and abroad  PoznaĔski (d. 1974) and Jordan (d. 1977). It was especially poignant that three pillars of the third generation died: young Malewski (d. 1963) together with Mostowski (d. 1975) and Suszko (d. 1979), both in the prime of their creative life. On the other hand, within this period fall the dates of birth of the representatives of the fifth generation of successors of the School tradition: Tomasz Placek (b. 1960), Andrzej Biáat (b. 1960), Andrzej Rogalski (b. 1961), Marek Lechniak (b. 1962), Wojciech Krysztofiak (b. 1963), Tomasz Bigaj (b. 1964), Jacek Wojtysiak (b. 1967), Krzysztof Wójtowicz (b. 1967), Andrzej Rojszczak (b. 1968), Kazimierz Dudkiewicz (b. 1970), Joanna OdrowąĪ-Sypniewska (b. 1971) and Mieszko Taáasiewicz (b. 1973). 1.5. The Phase of Expansion In the period of the fifth twenty years (1980-2000) we witnessed the phase of expansion. True enough, the last representatives of the first generation of the School passed away: Tatarkiewicz (d. 1980), CzeĪowski (d. 1981) and KotarbiĔski (d. 1981). After the death of SobociĔski (d. 1980), KokoszyĔska-Lutmanowa (d. 1981), Dąmbska (d. 1983), Tarski (d. 1983), BocheĔski (d. 1995) and KotarbiĔska (d. 1997), SwieĪawski is the only active member of the second generation, being the unquestionable senior of the School. The third generation was decimated as well, for the following philosophers died: Mortimer (d. 1984), KamiĔski (d. 1986), Sáupecki (d. 1987), Szaniawski (d. 1990), Stonert (d. 1992), Mazierski (d. 1993), Giedymin (d. 1993), Borkowski (d. 1993), Rasiowa (d. 1994), Lazari-Pawáowska (d. 1994), IwanuĞ (d. 1995),

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Pawáowski (d. 1996), ZiembiĔski (d. 1996), Gregorowicz (d. 1998), Pszczoáowski (d. 1999) and Augustynek (d. 2001). At the same time, however, the activity of their students and the students of their students increased dramatically.

2. Institutional and Publishing Basis The theoretical efforts of the philosophers would not have been fruitful without the great organisational work of the environment of Twardowski’s students and their successors. Let me present the most important organisational enterprises. In the phase of restoration: the Department of Praxiology at the Polish Academy of Science (1965) was established on KotarbiĔski’s initiative (1965). At KotarbiĔski’s, Suszko’s and Pelc’s suggestion the Polish Semiotical Society (1967) was founded. The yearly Conferences on the History of Logic initiated at the beginning of this period (1959) by CzeĪowski became a customary phenomenon in Polish philosophical life. The conference on the analysis of the notion of justification (1961) organised by Ajdukiewicz and the Winter Formal Logic School inaugurated ten years later (1970) were of the utmost significance. The pressure on the political regime was so great that during the second Congress of Polish Science (1953) the communists were forced to put into their ideologists’ mouths a declaration of the reintroduction of logic teaching in universities and other academic schools. The quarterly Philosophical Movement founded by Twardowski (which appeared in 1911-1914, 19181939 and 1948-1950) then revived in 1958, and the annual Studia Logica founded by Ajdukiewicz in 1953 (a quarterly in English since 1974) were joined by further periodicals: Praxiology (Polish version) (1962), Methodological Studies (1965), Semiotical Studies (1970), Bulletin of the Section of Logic [PAS ] (in English) (1972), Reports of Mathematical Logic (in English) (1973), Philosophical Problems in Science (1978), Problems of the Science of Science (1978) and Reports on Philosophy (in English) (1977). Important series appeared: PoznaĔ Studies in the Philosophy of the Sciences and the Humanities (in English) (1975, Amsterdam: Rodopi  the initiative came from Nowak) and PoznaĔ Studies in the Philosophy of Science (Polish version) (1976, PoznaĔ: UAM; since 1994 it appears as PoznaĔ Studies in the Philosophy of the Humanities). Small Encyclopaedia of Logic (1970) edited by Marciszewski and Small Encyclopaedia of Praxiology and the Theory of Organisation (1978) written by Pszczoáowski were published. Collected

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works of coryphaei of the Lvov-Warsaw School were published: KotarbiĔski’s writings, published during the previous period (1957), were joined by Ajdukiewicz’s writings (1960) as well as those of àukasiewicz (1961), Twardowski (1965), Ossowski (1966) and Tatarkiewicz (1971-1972). Their valuable handbooks were also re-edited: àukasiewicz’s Elements of Mathematical Logic (1958) and KotarbiĔski’s Elements of the Theory of Knowledge, Formal Logic and Methodology of Science (1961). The publications of Pawáowski’s anthology Logical Theory of Science (1966) and Pelc’s anthology Polish Semiotics 18941969 (1971) were important editorial events too. In the phase of expansion: the Polish Society of Logic and Methodology of Sciences (1993) was founded by Wójcicki and KabziĔski. Perzanowski inaugurated The Logical-Philosophical Workshops (1994). Conferences on the Application of Logic in Philosophy and Foundations of Mathematics started taking place. Wójcicki organised The Summer School for Theory of Knowledge (1998). During the sixth Polish Philosophical Congress (1995), Pelc contrasted literary philosophy with scientific philosophy and recognised only the latter as academic philosophy. Philosophical Quarterly (1990; it appeared in 1923-1938 and 1946-1950) and Philosophical Review (1992), founded by Weryho (it appeared in 1898-1939 and 1946-1949), were resumed. New periodicals were started: Studies in Logic, Grammar and Rhetoric (in English) (1982), Bulletin of the Commission of Logic, Warsaw Scientific Society (1991), Praxiology (English version) (1992) and Philosophy of Science (1993). Valuable series appeared: Realism  Rationality  Relativity (1984, Lublin: UMCS), Logic and Applications of Logic (1985, Warsaw: PWN), Studies in Logic and Theory of Knowledge (in English) (1985, Lublin: KUL), Sign  Language  Reality (1990, Warsaw: PTS), Logic and Logical Philosophy (in English) (1993, ToruĔ: UMK), Philosophy  Logic  Logical Philosophy (1995, ToruĔ: UMK), Dialogikon (1995, Kraków: UJ), Foundations of Science (in English) (1998, Dordrecht: Kluwer  series edited by Wójcicki), Polish Analytical Philosophy (in English) (1999, Amsterdam: Rodopi  subseries of PoznaĔ Studies in the Philosophy of the Sciences and the Humanities). Marciszewski edited the Dictionary of Logic as Applied in the Study of Language (English version) (1981) and Formal Logic: An Encyclopaedic Outline (Polish version) (1987); Cackowski, Kmita, Szaniawski and SmoczyĔski edited Philosophy and Science: An Encyclopaedic Outline (1987). The enterprise of publishing collected papers of the representatives of the first two generations of the Lvov-Warsaw School was continued: the writings of Ossowska (1983), CzeĪowski (1989), KamiĔski (1989-1998),

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KotarbiĔska (1990), KotarbiĔski (1990-2003), BocheĔski (1993), Tarski (1995), Drewnowski (1996), Salamucha (1997) and àukasiewicz (1998) saw the light of the day. Ajdukiewicz’s classic handbook Problems and Orientations of Philosophy (1983) and àukasiewicz’s famous monographs On the Principle of Non-Contradiction in Aristotle (1987) and Aristotle’s Syllogistic from the Standpoint of Contemporary Formal Logic (Polish version) (1988) were re-edited. It is worth adding that many representatives and sympathisers of the Lvov-Warsaw School (i.e. Dąmbska, Lazari-Pawáowska, KubiĔski, Szaniawski, IwanuĞ, Nowak, and Perzanowski) became involved in the Solidarity movement, though the slogan of non-intervention in political controversies belonged to the program of the School. The activity of Polish philosophers was accompanied by an increasing interest in Polish analytical philosophy abroad. At the beginning of the phase of expansion (1989), the yearly Philosophical Lectures devoted to Twardowski were inaugurated in Lvov (which since the Second World War has been within the borders of Ukraine). Then, conferences started: Stanisáaw LeĞniewski aujourd’hui (Grenoble 1992), The Lvov-Warsaw Philosophical School and Contemporary Philosophy (Lvov-Warsaw 1995), àukasiewicz in Dublin (Dublin 1996) and Alfred Tarski and the Vienna Circle (Vienna 1997). This growth in interest was certainly stimulated to a great extent by the promotional activities of the Poles themselves. Even before the war the state of Polish analytical philosophy was described in French by Zawirski (1935) and in German by KotarbiĔski (1933) and Ajdukiewicz (1935). After the war, new works were written: in English by Zbigniew Jordan (1945, 1973), Skolimowski (1967), Ingarden (1973), Pelc (1973), KamiĔski (1977). Szaniawski (1980), Jadacki (1980) and, most importantly, WoleĔski (1977); in French by BocheĔski (1947), KotarbiĔski (1958), Ostrowski (1971), and PrzeáĊcki and Jadacki (1993). Historical texts written by foreign analysts were also of great importance: in German by Franzke and Rautenberg (1972); in English by Simons (1992) and Smith (1996). Of course, publishing original texts in an English translation was crucial here. The fundamental works of KotarbiĔski (1966), àukasiewicz (1970), Tatarkiewicz (1970-1974, 1973, 1980), Twardowski (1977, 1999) and LeĞniewski (1992), CzeĪowski (2000), Salamucha (2003) as well as McCall’s (1967), Pelc’s (1979), Krajewski’s (1982, 2001), Szaniawski’s (1989), Coniglione, Poli and WoleĔski’s (1993), WoleĔski’s (1994), Kijania-Placek and WoleĔski’s (1998) and WoleĔski and Köhler’s (1999) anthologies began to appear.

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3. Formal Logic It is natural that the influence of the Lvov-Warsaw School on Polish philosophy of the second half of the 20th century manifests itself mainly in the domain of formal logic, which became a kind of international visiting card of the School as early as in the 1930s  thanks to a great German thinker  Scholz. 3.1. Sentential Calculus Sentential calculus, which had been an object of investigations under àukasiewicz’s leadership at Warsaw in the period of prosperity, in the next years continued to be studied mainly by logicians assembled around Sáupecki and Borkowski. Outside the group were GumaĔski (1981), who constructed equivalence (reversible) systems and Pietruszczak (1991), who was occupied with quantifierless calculi. Grzegorz Malinowski (1990) developed Chrisippian (two-valued), as well as non-Chrisippian (many-valued) logic initiated by àukasiewicz, including three-valued logic. Zawirski’s suggestions as to the possibility of using it in physics, have been taken up by Kiczuk (1995). But new ideas appeared as well. Firstly, àukasiewicz’s questioning Chrisippos’ principle of bivalence was followed by Suszko, who rejected Frege’s principle that the denotation of sentences is to be identified with their truth-value. Thus, the next non-classical logic – after the non-Chrisippian one – came into being: namely, a non-Fregean logic, which assumes that the denotation of sentences is to be identified with a corresponding situation. Omyáa (1986) systematised this new logic. Secondly, systems of nihilistic logic were constructed (ĩabski 1995). They were founded on a conception of truth, according to which a sentence of the form ‘The sentence p is true’ is synonymous with sentence p itself; such systems can be used to solve some antinomies. The genesis of research on «paradoxical» logics was analogous; they allow for a substitution of sentential variables with nonsensical (or sense-losing) expressions (Piróg-Rzepecka 1966, 1977, 1985). Thirdly, studies in logics which result from the weakening of classical logic by the elimination of some axioms (or rules of inference, respectively) were undertaken. Special attention, inspired by JaĞkowski, was devoted to intuitionistic logic, which rejects the principle tertium non datur and which was already an object of Zawirski’s interest, as well as to para-consistent logic, allowing for  or rather ignoring  contradiction by removing Duns Scotus’ law from among the axioms (JaĞkowski 1948; da Costa and Dubikajtis 1968, 1977).

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3.2. Nominal Calculus All three versions of the nominal calculus considered by Twardowski’s immediate students were pursued. Sáupecki (1955), Iwanicki (1965) and Kwiatkowski (1980) worked on an adequate interpretation of Aristotle’s syllogistic, NieznaĔski proposed his own axiomatisation for it and  contrary to àukasiewicz’s standpoint  IwanuĞ (1969) discussed the possibility of enriching it by introducing names that denote the empty and the full sets. In the area of functional calculus, codified by Borkowski (1958-1960), interesting results were obtained by the generalisation of the notion of quantifiers  by taking into account ramified quantifiers and by the construction of systems containing such «bifurcational» quantifiers (Krynicki, Mostowski and Szczerba 1995). Extending functional calculus (of the first order) to «fictional» (Meinongian) logic  pendant to the above mentioned enrichment of syllogistic  made by PaĞniczek (1988, 1998) is even more interesting. LeĞniewski’s ontology also continued to be an object of interest (Sáupecki 1955; Lejewski 1958; IwanuĞ 1969, 1973, 1980; Borkowski 1970). Elementary nominal calculi, close to ontology, were built by KubiĔski (1971a), while Rogalski (1995) adjusted ontology to the needs of the reconstruction of mediaeval metaphysics. 3.3. Erotetics The genesis of contemporary erotetics in Poland can be found in Twardowski’s incidental remarks and in Ajdukiewicz’s theory of questions developing these remarks. Their ideas were taken up and completed by Giedymin (1964), KubiĔski (1971), Koj and WiĞniewski (1989), WiĞniewski (1990) and Jadacki (2003), as well as by Leszko (1980, 1983), who used the theory of graphs and matrices for that purpose. 3.4. Mereology Attempts at completing LeĞniewski’s mereology have not ceased. Lately, Pietruszczak (1996) studied it intensively and Gorzka (1999)  on Tarski’s inspiration  has extended it (by introducing the notion of the diameter of a region) with a view to constructing an ontology without points. At the same time, attempts at constructing an alternative mereology have started, using the notion of founding (Rosiak 1995, 1996).

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3.5. Metalogic Metalogical research went in two directions. The first (metamathematical) direction, determined by Tarski, contained a generalisation of his theory of deductive systems (Sáupecki and Borkowski 1963; Rasiowa 1963: Rasiowa and Sikorski 1968; Sáupecki and Bryll 1965), theory of logical calculi (Wójcicki 1988) and the theory of proof (consequence), in particular (Zygmunt 1984). The second direction aimed at a satisfactory reconstruction of the logic of induction (Hosiasson-Lindenbaumowa 1941; Mortimerowa 1982). 3.6. Semantics The model-theoretical semantics initiated by Tarski turned out to be the dominant semantics in Poland. Pelc (1971) contrasted with it functional semantics – and, broadly, semiotics  as more suitable for the analysis of natural language. Starting from a similar motivation, Wybraniec-Skardowska (1985, 1991) chose categorial semantics (she constructed an original axiomatisation for it), while Pogonowski (1993) declared for combinatorial semantics. Moreover, specific semantics for languages of many-valued logics were examined (Lechniak 1999). Among particular semantic problems, the main attention was paid to the problem of empty, ambiguous and quotational subject terms as well as to self-referring expressions, probably because of their antinomiogeneity. After KotarbiĔski, semantic functions of empty names were discussed by Dąmbska (1949) and GumaĔski (1960, 1961) as well as by logicians interested in systems that allowed empty names in their vocabulary (IwanuĞ 1976; Wybraniec-Skardowska and Chuchro 1991). The problem of ambiguous names was analysed by KubiĔski (1958), by PrzeáĊcki (1964), who extended it to the problem of undetermined expressions and interpreted in the model-theoretical semantics, and by MuszyĔski (1988); recently, OdrowąĪ-Sypniewska has published a detailed monograph concerning this subject (2000). KubiĔski (1965) tried to find a remedy for the difficulties concerning the usage of quotational names, indicated already by LeĞniewski and Tarski. Indexical expressions were examined by Koj (1964). On self-evident grounds, semantic antinomies and paradoxes, which played such an important role in the logical research carried out by Twardowski’s students, also attracted a lot of interest. Among people who came back to them were: Suszko (1957), focusing on the liar antinomy (in àukasiewicz’s formulation); Koj (1963), linking semantic

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antinomies with the problem of transparency; Stanosz (1965), analysing the paradox of intensionality. 3.7. Pragmatics Two pragmatic relations were the main object of examination: asserting and understanding. The stimulus for examining the former came from Ajdukiewicz and for examining the latter from Dąmbska. Detailed reviews and systematisations of the problems of assertion were written by MajdaĔski (1974) and Patryas (1987). Koj (1969), Kmita (1971) and Jadacki (2003) analysed the nature and criteria of understanding (within the framework of the theory of interpretation).

4. Ontology In ontology as practised in Poland in the past fifty years, two (actually complementary) tendencies competed one with another: in the last ten years they have been emphatically articulated as Perzanowski’s formal ontology program (ontologic) (1996) and as Placek’s experimental ontology program (or metaphysics in general) (1997). Both programs refer to Augustynek’s ontological program. The program of experimental ontology is (consciously or not) a radicalisation of Augustynek’s postulate, which requires that an ontological system be compatible with contemporary physics (Augustynek and Jadacki 1993). 4.1. Theory of Being or Existence In the domain of the theory of being, two thematic spheres dominated: the analysis of the notion of existence and the program of unifying (the picture of) reality. It was Twardowski, LeĞniewski and Ajdukiewicz who initiated contemporary analysis of the notion of existence and non-existence in Poland. Then the subject was taken up by GumaĔski (1960), who formulated it in terms of existential assumptions, KubiĔski (1985a), Wojtysiak (2002), who drew subtle semantic distinctions in this area, Czarnocka (1986), who examined the nature and criteria of existence in the natural sciences, and PrzeáĊcki (1979, 1980), who focused on the ways of eliminating problems connected with sentences about non-being. The program of unifying reality took either the form of argumentation in favour of the structural unity of the world (Tempczyk 1978, 1981) or the form of realisation of the postulate of onto-categorial reduction.

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As far as reduction of ontic categories is concerned, the Lvov-Warsaw School entered the second half of the 20th century with KotarbiĔski’s reism seriously impaired by Borowski’s and Ajdukiewicz’s criticism. This criticism was continued by Szaniawski (1977), who indicated the unlikeliness of a satisfactory interpretation of the distributive notion of a set in this system, as well as by PrzeáĊcki (1984) and Wolniewicz (1990). However, reism also had its defenders (Czerniawski 1997). In the meantime, competitive conceptions have appeared. In those conceptions categories other than things were accepted as basic categories, namely: properties (attributivism), states of affairs/facts (situationism), events (eventism) and processes (processualism). ĩabski gave a formal shape to attributivism (1988) and Nowak developed the idea of a negative ontology opposed to (positivistic) attributivism (1998, 2004). Situationism was elaborated by Wolniewicz (1968, 1985) and Omyáa (1996). These attempts were accompanied by the analysis of the notion of a state of affairs covering negative (Kowalski, Krzysztofiak and Biáat 1998) and intentional (fictional) states of affairs (Pelc 1983; PaĞniczek 1991, 1998). The most perfect shape was given to eventism, considered by its authors (Augustynek and Jadacki 1993) as an ontology, which is compatible with relativistic physics. Processualism had its adherent in Tempczyk (1986). The structural basis for these new reductive formal ontologies  and most certainly for eventism  is set theory. Quite different  namely combinatory  character was given by Perzanowski (1995) to his refined ontological systems. 4.2. Theory of Necessity and Possibility The analysis of necessity and possibility has been carried out almost exclusively within the range of modal logic (see below). 4.3. Theory of Time and Space Augustynek has undertaken detailed studies on time  referring to LeĞniewski’s and KotarbiĔski’s polemics, Ajdukiewicz’s conception and Zawirski’s and Dąmbska’s considerations concerning the logical status of sentences about the future. Firstly, Augustynek proposed a definition of time (1970) compatible with relativistic physics; secondly, he analysed various properties of time (topological and symmetric, in particular) (1975); thirdly, he introduced relational notions of the past, the present and the future (1979). Snihur (1990) was his opponent regarding this last issue.

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Perzanowski’s analyses concerning space (within the framework of a more capacious system of locative ontology) (1993) match in rigor and elegance Augustynek’s analyses of time. 4.4. Theory of Change and Motion Theory of change should solve two difficulties: how to get over the paradoxes of motion and becoming and how to reconcile changes with identity of changing objects. Regarding the first matter, Ajdukiewicz proved that it is possible to describe change without infringing on the principle of non-contradiction or the principle of excluded middle (1948). In this matter, Placek’s subtle analyses turned out to be decisive (1989, 1995). Regarding the second matter, CzeĪowski wrote a short paper (1951a), while Augustynek gave it more attention, proposing his own definition of gen-identity (1981). 4.5. Theory of Determination and Causality The notions of determination and causality, and determinism and causalism, respectively, so absorbing for àukasiewicz and (later on) for KotarbiĔska, were reconstructed precisely by Zbigniew Jordan (1963a), Mazierski (1961), Augustynek (1962) and TrzĊsicki (1989).

5. Epistemology  Methodology  Praxiology During the last fifty years traditional epistemology has in fact not been cultivated in the Lvov-Warsaw School. Ajdukiewicz’s standpoint was decisive here: according to him, epistemological problems could be studied only after a suitable semantic paraphrase and after such a paraphrase they became indistinguishable from respective methodological problems. However, some people saw the necessity of distinguishing epistemology from methodology (Zamiara 1974). On the other hand, methodology itself could be recognised either as a fragment of praxiology, constructed in the second half of our century, or  on the ground of some assumptions  as a fragment of a theory of behaviour (Malewski 1964), or, finally, as a fragment of the cognitive sciences (Bobryk 1988) and the theory of artificial intelligence (LubaĔski 1975). In the youngest generation there is a tendency to return to traditional problems of epistemology in their original form, but with contemporary methods, e.g. the realism-idealism controversy (Krysztofiak 1999).

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5.1. Programs In Polish philosophy of the last half a century, four main (usually competing) methodological programs have functioned: apragmatical and pragmatical, on the one side and descriptive and normative, on the other. Moreover, they have been realised either by means of semantic analysis or by means of formal reconstruction. A clear differentiation between the first two programs  referring to the general distinction between acts and results made by Twardowski  appeared thanks to Ajdukiewicz (1948a). Thus, it was realised that (apragmatical) meta-science should be carefully separated from psychology and sociology of knowledge. Sociology of knowledge  with some elements of historiosophy  became an object of greater interest only in the last phase of the twentieth-century Polish philosophy (Pietruska-Madej 1980; Jodkowski 1909; Jonkisz 1990, 1998). In this trend they analysed such matters as the question of continuity (paradigms) and changeability (revolutions) in science. The meta-scientific attitude was dominant (Wójcicki 1974, 1982). The psychological and sociological analyses provoked, at once, serious objections of «apragmatists» (KaáuszyĔska 1994a). Apart from apragmatical or pragmatical interest, methodologists were divided with respect to descriptive and normative approaches. Some of them (Giedymin 1961; Kmita 1971, 1976, 1980; Nowak 1971, 1973; Sady 1990) wanted to restrict themselves to logical reconstruction of real procedures used by scholars to obtain these results. Others (Koj 1998; Hoáówka 1998) thought that their main task is the formulation of norms that determine the methodological duties of scientists. All these programs were criticised by Misiek (1979). 5.2. Knowledge-Generating Procedures Within both the pragmatical and apragmatical programs the knowledgegenerating procedures and their results were the object of detailed study. Firstly, analyses concerned observation and evidence (Rojszczak 1994), in general, as well as experience (Kalinowski 1991; Czarnocka 1992) and measurement (KaáuszyĔska 1983), in particular. Measurement was subjected to a deep analysis by Ajdukiewicz in the final period of his life (1961). In the School, a broad understanding of empirical knowledge dominated, which included not only knowledge acquired by introspection but also by axiological intuition (CzeĪowski 1989; PrzeáĊcki 1996). Problems of observation were formulated, in general, in terms of observational sentences; in such a context, the problem of the analytical

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components of factual sentences corresponded to the problem of the theoretical components of scientific facts (Jodkowski 1983). It was usually connected with the problem of the status of theoretical terms (Borkowski 1966; PrzeáĊcki 1969, 1993; Nowaczyk 1985, 1990; KaáuszyĔska 1994). ĩytkow (1979) identifies those terms with sets of operational procedures. Secondly, studies in inferential procedures (ways of reasoning), begun already by Twardowski and àukasiewicz, were creatively continued. Polish philosophers analysed both infallible (deductive) and fallible (inductive) inferences. As far as infallible inferences are concerned, the work focussed on the problems of mathematical proof (Sáupecki and Pogorzelski 1962) and its algorithmisation (Oráowska 1973; Zwinogrodzki 1976; Rasiowa and Banachowski 1977; Marciszewski and Murawski 1995), on the one hand, and the problems of verification (CzeĪowski 1951), on the other. Ajdukiewicz began the studies on the problem of the logical reconstruction of fallible inferences; Mortimer (1982) took it up in its full generality and Oráowska and Pawlak (1984) interpreted this kind of reasoning as inferences in systems with incomplete information. Other philosophers examined particular fragments of the logic of induction: the theory of inference by analogy (Dąmbska 1962) and the possibility of its mechanisation (Zwinogrodzki 1982); the theory of «historical» inferences (i.e., inferences on the basis of testimony) and the question of the reliability of informants (Giedymin 1961); finally, the theory of probabilistic (CzeĪowski 1952) and statistical inferences, in the light of which it appeared that the majority of fallible methods of inference have no degree of infallibility (Szaniawski 1994). A separate study was devoted to the notion of certitude (Sady 1993). Problems of deduction were seen, more and more commonly, as problems of justification; moreover deductive justification was opposed to deductive inference (Borkowski 1966). Analogously, induction was tied with the context of discovery, which was logically reconstructed in the spirit of the School (Zamecki 1988; Pietruska-Madej 1990; Sady 1990). Prognostics (WoleĔski 1984) and explanation, as well as the notions of scientific law (Pelc, PrzeáĊcki and Szaniawski 1957; Mazierski 1993) and hypothesis (Herbut 1978) respectively, were engaging a good deal of attention. Thirdly, procedures of formulating problems were not neglected. The theory of questions took the shape of erotetic logic. Among particular matters, the notion of the essence of a problem was at the centre of investigations.

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Fourthly, mereology and set theory were used to describe procedures of partition, classification and ordering. In particular, the theory of classification was developed by CzeĪowski (1950), Batóg and àuszczewska-Romahnowa (1965) and Batóg (1994), to whom we owe its generalisation. Fifthly, Polish philosophers gave a considerable amount of thought to the verbalisation and the interpretation of theories. Twardowski’s followers have always remembered his postulate of clarity and, even if they did not accept officially its theoretical foundation, arguing that sometimes a clear thought cannot be expressed clearly (Gorzka 1990), they have in practise struggled to observe this postulate in the highest degree. In this area, studies culminated in the analysis of definition, inaugurated in Poland in the contemporary manner by LeĞniewski and Ajdukiewicz. Afterwards, many people were occupied with the theory of definition: KokoszyĔska-Lutmanowa (1971, 1973), who favoured the view that there is only one notion of definition, contrary to Ajdukiewicz (1958), Gregorowicz (1962), Stonert (1959) (in the deductive sciences) and Pawáowski (1978) (in the humanities). 5.3. Rationality In the domain of epistemologico-methodological problems, the main subjects of inquiry were definition, typology and criteria of rationality. Pre-war Dąmbska’s research on irrationalism (1937) before the war constituted a background for later attempts to deal with this question. Her research was connected directly or indirectly with the work of Grzegorczyk (1993, 1997), who finds in rationality a distinctive feature of the European civilisation, of PrzeáĊcki (1996), who extends the notion of rationality beyond the limits of scientific knowledge, of with Marciszewski (1991), who concentrates his considerations mainly on the rationality of discussion, of ĩyciĔski (1993), who struggles to find the place of rationality within the compass of religion, and, finally, of Grobler (1993), who analyses especially the notion of deferred rationality. Apart from these considerations, rationality has been approached in decision theory (Szaniawski 1994). Finally, Kleszcz (1998) presented a review of the results of investigations in this area, distinguishing seven types of rationality (conceptual, logical, ontological, epistemological, methodological, practical and axiological); he contrasted the rationality of beliefs with the rationality of acts; described the difference between rationality, on the one hand, and irrationality or non-rationality, on the other; drew up a list of criteria

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for rationality, introducing into it: verbal precision, application of logical laws, criticism and resolvability of entertained problems. StrawiĔski (1991) also added simplicity to them and devoted a special study to it. An original conception of rationality was constructed by Taáasiewicz (2002). 5.4. Truth The question of truth has not stopped being a object of deep inquiries since Twardowski presented a persuasive criticism of alethic relativism and Tarski developed the semantic (model-theoretical) version of the correspondence conception of truth. Nobody in this environment – especially after KokoszyĔska’s additional explications  questioned alethic absolutism. Ajdukiewicz quickly retracted his radical conventionalism. On the other hand, Dąmbska (1962, 1975a) argued that radical conventionalism did not lead to alethic relativism because conventions do not have to be arbitrary. The acceptance of moderate conventionalism (Giedymin 1982; Siemianowski 1983, 1989) could be reconciled, a fortiori, with the absolutist conception of truth. Let us add that in general scepticism in this matter was not shared (WiĞniewski 1992). Similarly, the opinion that the correspondence conception of truth is correct, dominated. It was pointed out that both the coherentist and pragmatic conceptions are inadequate. This was done by either referring to common sense (ChwedeĔczuk 1984) or to philosophical interpretations of limiting theses (WoleĔski 1993). On the other hand, Grzegorczyk (1996) explicitly proved the accuracy of the correspondence conception. This does not mean, however, that the explanatory power of alternative conceptions of truth was not investigated. Thus, a «non-Fregean» version of the correspondence conception was constructed where states of affairs stated by sentences are considered to be extra-linguistic counterparts of sentences. Such a version was recognised as the most adequate interpretation of the classical (Aristotelian) solution (Borkowski 1995; NieznaĔski 1984; Biáat 1994, 1995; Jadacki 2003). Tomasz Jordan’s (1989) attempts tended in a similar direction  approaching the intuitions of natural languages. On the other hand, Grobler (1993)  not without sympathy  presented a version of the pragmatic conception, namely the dynamic (approximative) version, in which the property of being true is replaced by the relation of being-closer-to-the-truth-than. Jacek Malinowski (1995) studied the illocutionary version of the pragmatic conception with efficiency as the equivalent of truth. The consensual version of the pragmatic conception was reconstructed lately by Kijania-Placek (2000). ĩabski (1995) built an original logic for the nihilistic conception.

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5.5. Praxiology Although the problem of distinguishing and analysing actions (versus products) was formulated by Twardowski, it was KotarbiĔski (1956, 1966b) who was the real creator of the theory of action  i.e., praxiology. Afterwards, Podgórecki (1962), Pszczoáowski (1969) and ZiembiĔski (1972a) proposed important contributions and improved syntheses.

6. Philosophy of Science 6.1. Classification of the Sciences Twardowski made a penetrating analysis of the traditional classifications of sciences: the classification into a priori sciences and a posteriori sciences, in particular. The most universal inquiries into the notion of science and the classification of scientific disciplines – from various points of view – were undertaken by KamiĔski (1961). 6.2. Philosophy of Mathematics In philosophy of mathematics, Batóg (1996) and Murawski (1995, 1999) were active. In particular, the question of the philosophical significance of reverse mathematics held the attention of the latter. Now, two representatives of the younger generation have joined them; they have examined in great depth the arguments on account of mathematical realism (Bigaj 1997; Wójtowicz 1999) and intuitionism (Placek 1999). 6.3. Philosophy of Physics, Chemistry and Biology In philosophy of physics, the question of the philosophical interpretation of relativistic physics was continuously at issue (Czerniawski 1993). Lately, the theory of chaos and its implications for general philosophy have become an object of study (Tempczyk 1995, 1998). At the beginning of the period in question, Mehlberg (1951) considered the controversy between idealism and realism in contemporary physics; and at the end of it, Jodkowski (1996) studied the controversy between evolutionism and creationism in contemporary biology. Pietruska-Madej’s attention (1975) turned to the philosophy of chemistry.

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6.4. Philosophy of Psychology and Sociology Two philosophers from the circle of the Lvov-Warsaw School, Bobryk (1988) and Rzepa (2002), occupied themselves with the philosophical problems of psychology. 6.5. Philosophy of the Humanities: Linguistics, Jurisprudence and History The most serious results in the philosophy of linguistics were achieved by the program of formalising theoretical phonology, formulated and realised by Batóg (1967) and Pogonowski (1979, 1981). As a result, the reduction of the basic phonological categories was obtained (Batóg 1967). Relatively many philosophers worked creatively in the philosophy of law (ZiembiĔski 1963; Gregorowicz 1963; WoleĔski 1972, 1980, 1999; Nowak 1968, 1973). For instance, the status of juristic definitions (Gregorowicz 1962), modes of justifying juristic norms (ZiembiĔski 1972) and the relation between juristic and ethical norms (ZiembiĔski 1966) were investigated. Giedymin (1961, 1964), Nowak and Kmita (1968), and Kmita and Zamiara (1989) pursued the methodology of history – and the humanities, more broadly – concentrating their efforts especially on the reconstruction of interpretation procedures and also on the explaination of the status of theoretical terms in the theory of belles-lettres. 6.6. Reductionism and Holism The problem of integrating the sciences and the chances of reductionism in this field were investigated by StrawiĔski (1997) and Grobler (1993). Jedynak (1998) probed in detail the empiricist version of reductionism. She showed that it cannot be fully realised because of a disharmony of its particular components. Siemianowski (1988) indicated the consequences of radical empirism.

7. Axiology 7.1. Description, Estimation and Norm It was Twardowski’s Lvov lectures in ethics that proved to be the main impulse for inquiries in axiology (or, strictly speaking, ethics) among Polish philosophers influenced by the Lvov-Warsaw School. In these

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lectures, the founder of the School declared himself as a cognitivist and an axiologist regarding the question of the relations between description, estimation and norm. This standpoint  mainly thanks to Znamierowski (1957), CzeĪowski (1989) and Ossowska (1947, 1963)  has become the dominant paradigm in this context. Ethics  as Ossowska emphatically stated  could be a science and not just a set of moral norms, but it would have to be a real science of moral phenomena which would provide their scientific description and which would contain meta-ethics (1970) as well as psychology and sociology of morality (1963). Within the framework of the last of these, Ossowska, herself, reconstructed two examples of an ethos present in Polish society: the chivalry ethos (1973) and the middle-class ethos (1956). 7.2. Ontic Status and the Universality of Values In the controversy regarding the ontic status of values Twardowski  and other representatives of the School after him (Tatarkiewicz 1919, 1938; PrzeáĊcki 1989)  took the objectivist position: some objects are goods per se and not because somebody considers them as goods. The construction of a formal theory of goods has become a goal and it was CzeĪowski who gave the outline of such a theory first (1960). Objectivism was joined, in principle, with absolutism. At the same time, the analysis of scepticism (Dąmbska 1948) and relativism in relation to the question of the universality of values was engaging a good deal of attention. Lazari-Pawáowska (1975), following the slogan of the School  clara et distincta  contrasted axiological relativism with methodological, situational and cultural relativisms. Hoáówka presented a monograph on the problem (1981). 7.3. Motivation and Respecting Norms In the controversy regarding the sources of approval of moral norms, Twardowski and his followers were adherents of autonomism: moral norms do not need external justification, in general, and religious justification, in particular. For that reason, systems of independent ethics were developed (KotarbiĔski 1958a); they assumed the shape of atheistic Christianity (PrzeáĊcki 1989), i.e., hic et nunc catholic ethics but without theistic theses. Usually, autonomism was connected with intuitionism (CzeĪowski 1989). According to axiologism, good is what is commanded; and, what is good in individual situations is  in the intuitionists’ opinion  simply

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«visible». Thus, it is not surprising that the «organ» of moral cognition, conscience, was analysed in detail (Górnicka-Kalinowska 1992). Twardowski was rigorist as to respecting moral norms. None of his followers proclaimed (or respected) such a radical rigor and some people (KotarbiĔski 1966a; Lazari-Pawáowska 1992) tended rather towards a «soft» utilitarianism, whose praxiological version was presented by Pszczoáowski (1982). Anyway, philosophers referring to Twardowski’s program of scientific ethics offered deep analyses of the notions of responsibility and fault (Znamierowski 1957a), liberty (LazariPawáowska 1992) and justice (Ajdukiewicz 1939), including just distribution of goods (Szaniawski 1994). 7.4. Ethical Systems Separating, following Twardowski and Ossowska, ethical standpoints from the science of moral phenomena (as science ex definitione) Polish philosophers working under the sign of the Lvov-Warsaw School tried to put their own «unscientific» ethical views into the shape of possible, rationally reconstructed systems. Altruism, humanitarianism and perfectionism were stable reference points here. It is necessary (according to altruism) to take care not only of our own welfare but also  and maybe first of all (PrzeáĊcki 1989)  of others’ welfare. The dominant position was occupied by the ethics of favour towards others (Znamierowski 1957c), of good relations with others (Ossowska 1949) or at least of esteem in relation to others (Witwicki 1957). Fair life consists just of taking care of others’ welfare (KotarbiĔski 1966a). In connection with the altruistic attitude, semantic analyses of such notions as tolerance (Lazari-Pawáowska 1992) were put forth. In the controversy between maximalism and minimalism, the majority opinion was on the side of minimalism. Taking care of others’ welfare should manifest itself, in particular, in trying to minimalise the pains that others experience. This was the position of humanitarianism (LazariPawáowska 1992). As far as our own good is concerned, the perfectionistic-ascetic standpoint was dominant: we ought to perfect our virtues but the number of perfected virtues  if the enterprise is to be successful  should be radically limited to civic virtues in particular (Ossowska 1970). Hedonism was in fact only an object of theoretical interest. Tatarkiewicz (1947) offered a splendid analysis of the notion of happiness, separating the happiness of an ethical character from vital, psychological and dispositional happiness. One of the results of this analysis was a

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justification of the thesis that hedonist happiness cannot be a rational aim of human activity. 7.5. Aesthetics In aesthetics, studies were focused on the psychology of creation and perception of art (Tatarkiewicz 1951; Wallis 1968). Pawáowski (1989) initiated an inquiry into aesthetic values, trying to make use of Ockham’s razor also in this sphere.

8. Formalisation and Axiomatisation of Various Domains of Knowledge According to the paradigm of the Lvov-Warsaw School  at least in the version which owes its shape to àukasiewicz (1927) and LeĞniewski (1927)  the final form of philosophical disciplines should be their logical reconstruction and presentation in the shape of axiomatised formal systems. In the second half of our century, Poles constructed a few important systems of such «regional» logics, mainly of intensional character (Malinowski 1989). 8.1. Natural Deduction As far as a fragment of methodology is concerned, such a system was presented as early as 1934 by JaĞkowski as a system of natural deduction (assumptional logic). It was a realisation of àukasiewicz’s postulate (1927) to reconstruct the real modes of reasoning used in mathematics in logical terms. Afterwards, many philosophers worked on developing and perfecting this system: Iwanicki (1949), Sáupecki and Borkowski (1963), and lately Dudkiewicz (1988), who concentrated his efforts on using the method of semantic matrices. 8.2. Deontic Logic Deontic logic was a regional logic for ethics and jurisprudence. It was developed by Kalinowski (1965, 1972, 1996), GumaĔski (1980, 1981), Ziemba (1969, 1983), Ziemba and ĝwirydowicz (1988) and WoleĔski (1990).

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8.3. Diachronical Logic With historical inquiries in mind, Suszko (1957a) built a system of diachronic logic. 8.4. Relevant Logic and Categorial Grammar In the domain of natural languages, LeĞniewski’s and Ajdukiewicz’s ideas were developed; they were put into the mature form of categorial grammars (Suszko 1958-1960; Stanosz and Nowaczyk 1976; Buszkowski 1989). On the other hand, problems with using «normal» logic to examine natural languages  signalised by Tarski  resulted in the development of relevant logic (Tokarz 1993). 8.5. «Creational» Logic Praxiology «obtained» a «creational» logic, i.e., logic of action (efficiency) (KubiĔski 1985). 8.6. Doxastic (Epistemic) Logic It was àukasiewicz who set the framework for inquiries into the logic of convictions in Poland with his analysis of systems with functors of acceptance and rejection, written in connection with his reconstruction of Aristotle’s logic. àukasiewicz’s work concerning the logic of rejection was continued by Wybraniec-Skardowska and Bryll (1969), as well as by Sáupecki, Bryll and Wybraniec-Skardowska (1971-1972). Marciszewski devoted a monograph to the general theory of beliefs (1972). 8.7. Modal, Temporal, Transformational and Causal Logic Polish philosophers were also interested in the logical reconstruction of problems in the ontology of physics. Various modal systems for the notions of necessity and possibility were built (JaĞkowski 1951; ĩarnecka-Biaáy 1973; Perzanowski 1989). A review of various attitudes towards problems of modality in logic and philosophy was elaborated (ĩegleĔ 1990). Great efforts were made to construct a temporal, «transformational» logic that could help to avoid the problem of change, which threatened the principle of non-contradiction. Using àukasiewicz’s, Zawirski’s, Sáupecki’s and àoĞ’s ideas, Rogowski (1964), Kiczuk (1984a, 1991), and Wajszczyk (1989, 1995) engaged in research in this domain. Wajszczyk proposed detailed systems both for dichotomous (being  non-being and vice versa) and continuous changes. In the case of causal

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logic, as in the case of modal and temporal logic, the impulse came from àukasiewicz and JaĞkowski (1951); later philosophers from a younger generation (Kiczuk 1984, 1995) have joined them. 8.8. «Theological» Logic The program of scientific philosophy, formulated by Twardowski (1921), was later extended  by àukasiewicz, Drewnowski, BocheĔski and Salamucha  to cover also theology. The extension was undertaken against the opinions of, e.g. Witwicki (1939), who claimed that theological problems (and religion in general) belonged to an irrational sphere. Witwicki’s view was endorsed by ChwedeĔczuk (1997, 2000) and  it seems  by the majority of the School’s sympathisers. Nevertheless, the opposite view has never lacked defenders (ĩyciĔski 1985-1986; Bronk 1996). The main efforts were put into the logical analysis of traditional justifications (proofs) of the basic theistic thesis that God exists (Salamucha 1934; BocheĔski 1965; NieznaĔski 1980).

9. Final Remark I end here my review of the theoretical problems and the results achieved by contemporary Polish philosophy that acknowledges its links with the Lvov-Warsaw tradition. To avoid possible misunderstandings, I must stress that my picture is necessarily of a sketchy character. One can hardly expect more than an introductory diagnosis, if one realises that now, in Poland, there are circa two thousand active philosophers and that two hundred of them are full titular professors!

Uniwersytet Warszawski Department of Philosophy ul. Krakowskie PrzedmieĞcie 3 00-047 Warszawa, Poland e-mail: [email protected]

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PART II OBJECTS AND PROPERTIES

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John T. Kearns AN ELEMENTARY SYSTEM OF ONTOLOGY

1. Ontology According to SobociĔski (1949), the Polish logician Stanisáaw LeĞniewski devised his logical system Ontology in order to capture or express the notion of a distributive class. However, it isn’t clear to me that Ontology involves any sort of classes. I think LeĞniewski’s system is best understood as a theory concerned with some features of common nouns – in contrast to first-order theories, which focus on referring expressions and predicates of individuals. In this paper I will explain my understanding by developing elementary systems of Ontology in which the semantic account makes no provision for distributive classes. After developing these systems of Ontology, I will discuss collections, which I think are close to what LeĞniewski understood distributive classes to be. As it turns out, the elementary systems of Ontology are not suited for making statements about collections. I will finish by sketching changes in one system of elementary Ontology which allow it to incorporate statements about collections. In order to understand how logical systems can shed light on natural languages, I adopt a speech act approach to language and logic. I won’t try to defend this here (I have made such an attempt in Kearns 1996), but will simply state certain principles and use them to motivate my analyses. On my view, the primary linguistic reality is constituted by the linguistic skills and the language acts of human beings. A language act, or speech act, is a meaningful act performed with an expression. Such an act can be performed by speaking out loud, or writing, or (merely) thinking. And it isn’t just persons producing expressions who perform language acts. The person who listens with understanding, or who reads, is using the expressions she hears or reads to perform language acts. It is language acts, not expressions, which are the primary bearers of meaning. These acts have the meanings that the language user intends.

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 87-112. Amsterdam/New York, NY: Rodopi, 2006.

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But various expressions are conventionally used to perform acts with specified meanings, and it is customary for a person to intend a meaning conventionally associated with the expression she uses. From this perspective, there is a sharp distinction that must be drawn between syntax and semantics. Syntactic features characterize expressions. Indeed, expressions can be considered to be syntactic objects. But expressions don’t have semantic features; these features are instantiated by linguistic acts. To illustrate, consider the situation where Eileen looks at the vase on her desk, and says “That flower is a carnation.” We can analyze her statement as follows: (1)

(2)

The phrase ‘That flower’ is used to direct her attention to a particular flower, or to express her attending if she was already looking at the flower. The referring act “sets up” Eileen’s act of using ‘is a carnation’ to acknowledge the flower to be a carnation.

The referring act, the acknowledging act, and the enabling relation between these acts constitute Eileen’s statement. Her statement has both a syntactic character and a semantic structure. The syntactic character is supplied by the expressions used, their syntactic categories, and the order in which the expressions occur. The semantic structure of the statement is independent of the expressions used. We give semantic structure by indicating the kinds of component acts performed and the way in which they are organized. The analysis above of Eileen’s statement presents the semantic structure of her statement. The artificial languages of modern logic are not used by people to perform language acts. No one speaks these languages, or uses them for thinking. We might regard the logical languages as languages that can, in principle, be used to perform speech acts, even though we don’t actually speak or write these languages. I find it more useful to consider the sentences of artificial languages as representations. They are representations of language acts, ordinarily of propositional acts. (A propositional act is one that can appropriately be evaluated in terms of truth and falsity.) However, instead of describing logical-language sentences as representations of propositional acts, I will more often speak of the sentences as representing natural-language statements. The syntactic account for an artificial language gives rules for constructing sentences of the artificial language. These rules need have no relation to principles governing expressions in natural languages. But the truth conditions for logical-language sentences are the conditions of

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the statements these represent. Deductive systems codify artificiallanguage expressions. 2. The Language L LeĞniewski’s system Ontology, which is described in Luschei (1962), is formulated in a very different way than are standard systems of firstorder and higher-order logic. The practice now is to completely specify an artificial language by indicating all of its basic symbols and the manner in which these are combined to constitute formulas or sentences. In an intuitive sense, these artificial languages are “finished products.” In contrast, LeĞniewski conceived of Ontology as a “growing” system. Whatever symbols the artificial language possesses at a given time can always be augmented by the introduction of new symbols. It is also always possible to augment the language as it is at a time with new syntactic categories. In LeĞniewski’s formulation, there are rules for adding definitions to Ontology, which definitions introduce both new symbols and new categories. According to Lejewski (1995), LeĞniewski regarded creative definitions as a philosophically important feature of his systems, but the unfinished character of Ontology is a drawback for some purposes. And this character isn’t really essential for carrying out the normal types of logical investigation. The language L of Elementary Ontology will be specified at the start. The fundamental category of expressions in L is common nouns, or noun constants: A, B, C, A1 , . . . A noun constant is used to represent some acts performed with common nouns or noun phrases in natural languages. A noun constant will label zero, one, or more than one individuals (like ‘square circle’, ‘capital of New York State’, and ‘dog’). Capital letters from the end of the alphabet are used as noun variables: X, Y, Z, X1 , . . . The language L contains one sentence-forming functor ‘H’ for noun constants; it takes two noun constants as arguments. An atomic sentence of L is written: [A H B]

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which means that the one and only A is a B. It is unfortunate that epsilon is also commonly used for set membership, because LeĞniewski’s epsilon is intended to capture a common use of ‘is’. I will follow Simons (1987) in employing two styles of epsilon. The epsilon ‘’ is for set membership, while ‘H’ is used in Ontology. A sentence: [A H B] is true if, and only if (from now on: iff), there is exactly one A and this A is also a B. It is characteristic of what I am calling elementary Ontology that there is only this one functor which combines with common nouns to yield sentences. Other sentence-forming functors can be introduced by definitions, but defined expressions are regarded as mere abbreviations. Elementary Ontology does not contain variables in the categories of sentence-forming functors for noun constant arguments. It isn’t clear whether LeĞniewski failed to recognize that proper names and other referring expressions belong to a different syntactic category than common nouns, or whether he simply thought the distinction between singular terms and common nouns is of no great importance. Some terms label no individuals, some label exactly one, and some label more than one, but all these terms are adequately accommodated by a single category of Ontological expressions. Given LeĞniewski’s perspective, a good reading for the functor of elementary Ontology might be ‘is a’. Although it makes sense to have a sentence: [B H C] where B labels many individuals (e.g. ‘Dog is an animal’), the sentences are only true when the subject labels exactly one individual. LeĞniewski’s attitude toward singular terms and common nouns appears to be endorsed in Simons (1987), but this attitude/understanding is incompatible with the present speech-act approach to language. Referring acts are the fundamental linguistic acts by which language users “pick out” objects in the world, and so “fasten” their statements to the world. Referring acts are fundamentally different from characteristic acts performed with common nouns, and it is quite misleading for expressions in a single logical-language category to be used to represent both referring acts and common-noun acts. In the present paper, noun constants are not referring expressions. As a reading for a sentence ‘[A H B]’, we could say “There is exactly one A, and it is a B” or “The one and only A is a B,” but these are both cumbersome. I prefer to treat noun constants as plural nouns, and will

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provide a reading that is grammatically appropriate for this treatment. I propose to introduce a new, quasi-technical, use for ‘singly’, and to read ‘[A H B]’ as “A’s are, singly, B’s.” A’s are, singly, B’s just in case there is exactly one A, and this A is a B. It is understood that the plural reading carries no indication of a plurality of objects. The language L contains the connectives ‘~’ and ‘›’, and the universal quantifier ‘’. The connectives ‘&’, ‘Š’, and ‘{’ are defined in standard ways. The quantifier ‘’ is combined with a noun variable D to constitute a quantified phrase (D). The particular quantifier ‘’ is defined in a standard way. The language L does not permit vacuous quantification. The connectives have the standard, truth-table, truth conditions. (They are used to represent statements having these truth conditions.) But the quantifiers are not understood in the standard way. The quantified phrase ‘(X)’ doesn’t mean for every individual X, because ‘X ’ isn’t an individual variable. To make use of quantifiers and make sense of quantification in L , we are forced to construe quantifiers substitutionally. To understand how to do this, let us notice that common nouns in English can be used in different ways. A common noun can be predicated of an individual that has been identified by a referring act, as in: Mary is a gymnast. Common nouns can be used nonpredicatively as subjects of sentences: Gymnasts are athletes. The use as a predicate of individuals seems to be prior to the second use, for in order to speak meaningfully of gymnasts, one need to know what it is to be a gymnast. But to use common nouns predicatively, we must have referring devices for picking out particular individuals. The language L contains no referring expressions (no syntactically-signalled singular terms). However, we shall construe noun constants of L as expressions that can in other circumstances be predicated of individuals. A natural language predicate like ‘(is a) dog’ can be conceived as expressing a function from objects in the world to truth and falsity. So to interpret L , we begin with an empty or nonempty domain E of individuals. An interpreting function f of L for a nonempty E takes each common noun D to a truth-valued function on E . So, for example, if U is an individual in E , then f (A) applied to U – f (A)(U) – yields the value truth or the value falsity. When E is empty, an interpreting function f of

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L for E will take each common noun D to a truth-valued function on some other, nonempty, domain. The “idea” for the universal quantifier is that a sentence: (X)M should be true if M becomes true no matter which common noun is substituted for the free occurrences of ‘X ’ in M. But there is a problem here. We want a universally quantified sentence (the statement represented by such a sentence) to have a truly “unlimited” force. But a particular interpreting function f might assign values to the common nouns of L in such a way that M becomes true for all the interpreted nouns that happen to be in L , but there is some noun that might have been in L for which M would be false. For example, consider the domain of human beings, and let all noun constants in L be assigned the truth-valued function which yields truth for females and falsity for everyone else. Then if we understand the quantified phrase with respect to (interpreted) noun constants of L , the sentence: (X)~[X H A] which says that no noun constant can replace ‘X ’ in ‘[X H A]’ to yield a true sentence, is true. But there are lots of females in our domain. If we had the right nouns in L , we would have true sentences of that form. (All we need is a noun which labels exactly one female human being.) To overcome this difficulty, we don’t understand universally quantified phrases just with respect to nouns as interpreted by one function f. We consider them with respect to nouns as interpreted by f and all “alternatives” to f which differ from f in their treatment of some one noun constant. That way a sentence (D)M is true if M becomes true no matter what noun is substituted for D in M, including nouns found in extensions of L . A sentence (D)M does not make (represent) a statement about all objects in the domain E . For among the nouns “covered” by the universally quantified phrase will be nouns which are true of nothing (nouns like ‘square circle’). To make a statement about all objects in the domain, we need something like this: (X)[[X H X] Š . . . X . . . ] LeĞniewski didn’t regard sentences of Ontology as representations of ordinary statements, for that construal of artificial-language sentences depends on a speech-act/language-act perspective. But LeĞniewski may

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have thought that Ontology is adequate for analyzing ordinary statements, and that Ontology “captures” ordinary language in the sense that Ontology possesses the resources for giving faithful translations of ordinary statements (once Ontology is supplied with a suitable vocabulary). However, the use of nouns for talking about all the objects that the noun labels presupposes the use of nouns as predicates of individuals picked out by using referring expressions. An adequate language must contain expressions for referring to single individuals, and predicate (and relational) expressions which can be combined with referring expressions to characterize the objects referred to. LeĞniewski’s system Ontology fails to provide a category of singular referring terms (a category of singular terms for representing referring acts). We have construed noun constants as predicates of individuals, but Ontology provides no “place” for sentences predicating these constants of individuals. With respect to the goal of representing and investigating naturallanguage statements, Ontology is inadequate in the sense that it is incomplete. Singular terms are not, as LeĞniewski may have thought, just general terms that happen to apply to a single object. A statement made with this sentence: Denver is a large city. is not claiming that Denvers are, singly, large cities. The statement identifies the city Denver, and acknowledges this to be a large city. Ontology, especially a system which goes beyond elementary Ontology, possesses the resources to represent some natural-language statements, but many ordinary statements can be represented only after Ontology is supplemented with a category of referring expressions. Although a statement which characterizes Denver as a large city is not adequately represented by a sentence: [D H C] this Ontological sentence is a kind of “counterpart” to the naturallanguage statement. For every genuine singular term like ‘Denver’, we can introduce a corresponding common noun, say ‘denver’, which is true of exactly the object associated with the singular term. And whenever we have a true statement made with a sentence like ‘Denver is a large city’, we can make a corresponding true statement to the effect that denvers are, singly, large cities.

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3. A Careful Formulation of Elementary Ontology The well-formed formulas (wffs) of L are given by: (1) (2) (3) (4)

If D, E are noun constants or noun variables of L , then [D H E ] is a wff, an atomic wff of L . If M, N are wffs of L then so are ~M, [M › N ]. If M is a wff containing free occurrences of noun variable D, then (D)M is a wff. All wffs of L are obtained by (1)-(3).

Let E be a nonempty domain of individuals. A function I on L is a truthvalued function iff for each U in E , I(U) = t or I(U) = f, where t and f are the truth values. Let E be a domain of individuals. If E z ‡, an interpreting function f of L for E assigns a truth-valued function on E to each noun constant of L . If E = ‡, then an interpreting function f of L for E assigns a truthvalued function on an arbitrarily selected nonempty domain to each noun constant of L . Let f be an interpreting function of L for E . Let E be a noun constant of L . Then an interpreting function f * of L for E is a E-variant of f iff f * agrees with f on the value assigned to every noun constant with the possible exception of E. Let f be an interpreting function of L for a domain E . The valuation of L determined by f is as follows: (1)

(2) (3)

An atomic sentence [D H E ] is true iff f (D) is a function which is true for exactly one individual U in E , and f (E )(U) = t. The sentence is false otherwise. Sentences ~M, [M › N ] have values determined by the truth-tables for ‘~’ and ‘›’. Let (D)M be a sentence of L , and let E be the first noun constant in alphabetical order not occurring in M. Then (D)M is true for the valuation determined by f iff the result of replacing all free occurrences of D in M by E is true for every E-variant of f.

The deductive system EO is a natural deduction system which uses tree proofs to establish that argument sequences M1 , . . . , Mr / N (r t 0) are theorems. Here, the Mi are premisses and N is the conclusion. In tree proofs of EO, only sentences occur as steps. For the propositional part of EO, the elementary rules are:

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› Introduction M N [M › N ] [M › N ]

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Contradiction Elimination M ~M N

The following proof establishes the logical validity of argument sequence ‘M, ~[M › N] / ~M ’: M ›I ~ [M › N ] [M › N ] CE ~M A nonelementary rule of inference cancels, or discharges, one or more of the hypotheses used to reach premisses for the rule. In illustrating nonelementary rules, the hypotheses that are cancelled are enclosed in braces. › Elimination {M }{M } [M › N ] P P P

~ Elimination {~ M} M M

The following proof establishes ‘~[M › N] / ~M ’ to be a theorem: x

x

~~M ~M CE M ~ E , canceled ~ M M ›I ~ [M › N ] [M › N ] CE ~M ~ E , canceled ~ ~ M ~M Hypotheses which are cancelled have an ‘x’ placed above them. The rule for establishing theorems of EO is as follows: An argument sequence M1 , . . . , Mr / N is a theorem of EO iff there is a tree proof whose uncancelled hypotheses are among M1 , . . . , Mr and whose conclusion is N. ~ To illustrate the rules for the universal quantifier, I will use ‘ S ’ to designate the result of substitution. If M is a wff, D is a noun variable, E is a noun variable or a noun constant, then

~D S M/ E

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is the result of replacing all free occurrences of D in M by E. The quantifier rules for EO are the following:  Elimination  Introduction (D) M ȕ is a noun constant which ȕ is a noun ~ D S M/ does not occur in M or in any constant ~D E S M/ other uncancelled hypothesis E (D) M leading to the occurrence of ~D S M/ E

The rules for ‘H’ are as follows: RX [D H E] [D H D ]

SY [D H E][E H E] [E H D]

TR [D H E][E H J ] [D H J ]

H Introduction [G H J ] (D)(E)[[[D H J ] & [E H J ]] Š [D H E]] [J H J] Given a domain E of individuals, an argument sequence M1, . . . , Mr / N of L is logically valid with respect to E iff there is no interpreting function f of L for E such that each of M1, . . . , Mr is true for the valuation determined by f , but N is false for this valuation. (If N is true for the valuation determined by f , we indicate this: f (N ) = t.) An argument sequence M1, . . . , Mr / N is logically valid iff it is logically valid with respect to every domain of individuals. It is fairly obvious that EO is sound  i.e., that every argument sequence which is a theorem of EO is logically valid. I will sketch a proof of this result without filling in the details, for the argument is entirely straightforward. An inference figure is a “move” in a tree proof, an instance of one of the rules of EO. The rank of a tree proof * is the number of inference figures in *. The minimum rank is 0; a sentence M standing alone is a tree proof having rank 0  this proof establishes the argument sequence M / M. LEMMA. Let * be a tree proof of EO having uncancelled hypotheses M1, . . . , Mr and conclusion N. Let E be a domain of individuals and f be an interpreting function of L for E such that f (M1 ) = . . . = f (Mr) = t. Then f (N ) = t.

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This is proved by induction on the rank of *. THEOREM 1. Every theorem of EO is a logically valid argument sequence. This is an easy consequence of the lemma. We cannot prove that EO is complete. For given a domain E of individuals, an interpreting function f of L for E , a universally quantified sentence (D)M is true if M is true for every subset ] of E as value of D. (This amounts to the same as saying that ~D S M/ E is true for every E-variant of f.) If E is denumerable, then it has nondenumerably many subsets. We can establish a version of Gödel’s Incompleteness Theorem for the system EO. Elementary Ontology is stronger in expressive power than first-order logic, for first-order logic is complete, while elementary ontology is not. It is an advantage of standard logic as compared to Ontology that firstorder logic is a “natural” subsystem of higher-order logic, one which is sound and complete. Elementary Ontology is a natural subsystem of Ontology, but it is sound without being complete.

4. First-Order Ontology

We shall characterize a subsystem of Elementary Ontology which is fairly natural, and which is sound and complete. First-order Ontology is obtained from Elementary Ontology by dropping the quantifier ‘’, and replacing it by the restricted quantifier ‘i’. A sentence (iD)M has the significance of: (D)[[D H D] Š M]. And the (defined) existential quantifier ‘i’ gives sentences (iD)M which have the significance: (iD)[[D H D] & M]]. The language L i is obtained from L by replacing the (unrestricted) quantifier ‘’ by ‘i’. The restricted quantifier ‘i’ is defined: (iD)M = (def ) ~(iD) ~M. Given a domain E and an interpreting function f of L i (or L ) for E , the value of the restricted quantifier is determined as follows: Let (D)M be a sentence of L , and let E be the first noun constant in alphabetical order not occurring in M. Then (iD)M is true for the valuation determined by f iff the result of replacing all free occurrences of D in M by E is true for every E-variant of f in which E is assigned a truth-valued

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function which yields the value true for exactly one individual in E . If E = ‡, the sentence (iD)M is (vacuously) true. The deductive system 1O has the same rules for connectives as EO. The rules for the restricted quantifiers are the following: i Elimination ( i D) M [E H E] ~D S M/ E

i Introduction ȕ is a noun constant which {[E H E]} does not occur in M or in any ~D uncancelled hypothesis other S M/ than the one in braces in the E proof of ( i D) M ~D S M/ E

The rules for ‘H’ are the same for 1O as for EO except for the rule H Introduction: H Introduction [G H J ] ( i D)( i E)[[[D H J ] & [E H J ]] Š [D H E]] [J H J] It is a straightforward matter to prove the following: THEOREM 2. Every theorem of 1O is logically valid. Let X be a set of sentences of L i and let M be a sentence of L i. Then M is a logical consequence of X, and X logically implies M (X ş M) iff there are no domain of individuals E and interpreting function f of L i for E such that each member of X has value t for (the valuation determined by) f , but f (M) = f. Let X be a set of sentences of L i and let N be a sentence of L i. Then N is deducible from X (X Ō N) in 1O iff there are sentences M1, . . . , Mr which are members of X such that M1, . . . , Mr / N is a theorem of 1O. Let X be a set of sentences of L i. Then X is consistent with respect to 1O iff there is no sentence N of L i such that both X Ō N and X Ō ~N. Let X be a set of sentences of L i. Then X is maximal consistent with respect to 1O iff X is consistent with respect to 1O, and for every sentence M of L i, either M is a member of X or X ‰ {M} is not consistent. Let X be a set of sentences of L i. Then X is instantially sufficient iff for every sentence ~(iD)M of L , if ~(iD)M is a member of X, then for some noun constant E,

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~D [[E H E]& ~ S M /] E is a member of X. LEMMA. Let X be a maximal consistent, instantially sufficient set of sentences of L i. Then (1) ~M  X iff M  X; (2) [M › N]  X iff M  X or N  X; (3) (iD )M  X iff for every noun constant E, ~D [[E H E]& ~ S M /] E is a member of X. THEOREM 3. Let X be a maximal consistent, instantially sufficient set of sentences of L i. Then there is a domain of individuals E and an interpreting function f of L i for E such that for every sentence M of L i, f (M) = t iff M  X. Proof. Let E be a noun constant of L i such that [E H E ]  X. Then C(E ) = {J _ J is a noun constant of L i such that [J H E ]  X }. Let E = {C(E ) _ E is a noun constant such that [E H E ]  X }. If E z ‡, then let f be the function defined on the noun constants of L i such that if J is a noun constant, then f (J ) is the function I such that I[C(E )] = t iff [E H J ]  X, and I[C(E )] = f otherwise. If E = ‡, then f(J ) is a function on some arbitrarily selected non-empty domain (say {1, 2, 3, 4}) that is true for 0 or more elements and false for the rest. Now we can establish the theorem by arguing by induction on the length of M. Let L i+ be the extension of L i obtained by adding the following noun constants: D, D1, . . . LEMMA 1. Let X be a consistent set of sentences of L i. Then X can be extended to a maximal consistent set of sentences of L i. LEMMA 2. A maximal consistent set of sentences of L i can be extended to a maximal consistent, instantially sufficient set of sentences of L i+. THEOREM 4. Let X be a consistent set of sentences of L i. Then there is a domain of individuals E and an interpreting function f of L i for E such that each member of X has value t for the valuation determined by f . THEOREM 5. Let X be a set of sentences of L i and M be a sentence of L i such that X ş M. Then X Ō M.

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5. Referring

In the following sections we will consider Elementary Ontology. It is straightforward to adapt our remarks to the system First-order Ontology. Before I consider collections of objects, and determine whether sentences of L are suited to represent statements about collections, I will discuss referring. Referring takes place when a person uses an expression to direct her attention to a particular object, and there really is an object of her attention. A person can use a name to refer: Napoleon was poisoned. Several of our friends live in Paris. But she can also refer with other expressions: That girl is quite tall. The author of Paradise Lost was English. A person who uses an expression to direct her attention to a genuine object has identified the object; referring is a kind of identifying. If someone is to use an expression to refer to an object, she must be connected to the object in some way that she knows about. If she experiences or has experienced the object, she has a direct connection to the object (so long as she doesn’t forget). If she merely learns about the object, she is connected to the object via her sources of information and their connections to the object. But there are other connections that a person can exploit in referring to an object. For example, I have never been to El Salvador, but I have read and heard about El Salvador. I know (roughly) how I am situated with respect to El Salvador, and I know how to get there from where I am now. In referring to El Salvador, I intend the country to which I have this geographical connection. To refer to an object in the unqualified sense, there must really be an object, and I must succeed in identifying it. But we also talk and think about fictional characters. Different people can discuss the same fictional character. It is useful to recognize fictional forms of identifying and referring so that we can accommodate talk about Sherlock Holmes or Lady Macbeth. I follow Searle (1974) in holding that the person who tells a story is pretending to refer to characters and make assertions about them. But the storyteller creates an objective situation which provides targets for fictional referring acts. It doesn’t very often happen that a person directs her attention, intending to refer to an object, when there is no object. Even when a person uses an expression which is inappropriate or incorrect for an object, she can successfully refer to the object. The target of the referring

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act is determined by the language user’s intentions, not by the expression she uses. But if it should happen that an attempted referring act has no target, then any characterization of the “referent” will be incorrect. Suppose that in Mary’s statement: The man who did this is inconsiderate. the act performed with the subject expression has no referent. Then her statement isn’t correct, no matter whether or not we decide to call it false. But while we can deny her statement, Mary can’t do this herself by saying something like: It isn’t true that the man who did this is inconsiderate. For Mary’s use of the referring expression marks a commitment to acknowledge the object of attention to exist. This is a presupposition of her statement, even when she uses a negative prefix. I have discussed referring acts, because I want to compare our talk about collections to referring acts. Sentences of Ontology, of Elementary Ontology, and of First-order Ontology do not represent referring acts. But perhaps such sentences (or their components) do represent acts of talking about collections.

6. Collections

When a speaker says something like: George is smarter than he looks. she uses the word ‘George’ to identify the person George. In a similar way, I think, the person who says: Dogs make better pets than cats. is using the word ‘dogs’ to direct her attention to something. That something is the collection of dogs. Unlike the words ‘set’ and ‘class’, the word ‘collection’ has not got an established technical use. So I am pressing this word into service for my own ends. My use of ‘collection’ has features in common with ordinary uses of this word, but I am not trying to capture ordinary usage. My collections are just what I say they are. However, I do intend by my use of ‘collection’ to “pick out” entities that play an important role with respect to thought and language.

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A collection of objects is an aggregate of those objects. It is made up of those objects, considered together. The collection has no internal structure, but is unified by being the object of someone’s attention. A real collection, as opposed to a fictional one, contains only existing real objects. There is no collection at present which contains the city Troy, or Carthage. A collection can gain and lose members. The collection of dogs at one time contained Rin Tin Tin, but no longer does, since the Rin Tin Tin of the movies no longer exists. The collection of dogs will have new members next year, and will have lost some of its current members. However, we can also consider the collection of exactly those dogs alive at this moment. That collection will not gain new members, and will cease to exist once one of those dogs dies. There are also collections which go out of existence, and are later reconstituted. The collection of Farmer Jones’ sheep will go out of existence if that farmer sells them all, and be reborn when he subsequently buys new sheep. A collection is constituted by its members; there is no empty collection. When the members of a collection cease to exist, so does the collection. And a collection of a single object consists of that object. On my analysis of linguistic acts, collections are important in connection with some uses of plural nouns. A single sentence can be used (on different occasions) to perform acts with different semantic structures, but a speaker would commonly use these sentences: All dogs are animals. No dogs are birds. Some dogs are collies. Some dogs are not collies. to direct her attention to the collection of dogs, and to acknowledge the collection to all be animals, never be birds, etc. The statements made might be concerned only with the collection of dogs at present, but in making the universal statements, it would be natural to intend the collection of dogs at all times. As customarily used, the first word of each categorical sentence qualifies the acknowledging act. The dogs are all animals, in no case birds, sometimes collies, etc. The common use of the categorical sentences is different in this respect from common uses of sentences like these: Every dog is a mammal. Some dog has rabies.

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Characteristic acts of using these last sentences involve the quantificational use of noun phrases, and will not pick out the collection of dogs. A collection must be distinguished from what I will call a group. A group is a structured whole of objects. Its internal structure is what unifies the elements making up the group. A group is also a concrete object, while a collection is slightly abstract, for the collection is constituted by its elements considered apart (in abstraction) from any relations they might have to one another. (But collections are not abstract entities of the sort that numbers, concepts, and propositions have traditionally been thought to be.) A wolf pack is a group as opposed to a collection. The pack is constituted by its component wolves linked by “kinship” and hierarchical relations. The collection of wolves in that pack is just the wolves, and not the relations. If the relations between wolves in the pack changes, the group/pack is changed, but the collection will not be affected. The Solar System is a group in my sense, constituted by the Sun and its planets (and their moons, and smaller bodies, etc.). When someone uses an expression to direct her attention to a particular real object, she has referred to/identified that object. In referring, she makes use of (exploits) relations that she knows about linking her to the object. She is situated by these relations with respect to the object, so that the object has a location prior to and independent of her referring act. Groups are concrete objects found in the world, and are qualified to be the objects of referring acts. However, collections are not. Someone who uses an expression to direct her attention to a collection has not referred to that collection. For collections do not have an independent “standing” apart from acts of attending to (thinking of) them. There are no antecedent relations linking language users to collections that the language users can know about so as to intend the object reached by the relations. Although a collection is not suited to be the object of someone’s referring act, a language user can direct her attention to a collection. How does she do this? The language user attends to a collection with respect to a linguistic practice. If there is a predicate I which is used to acknowledge objects to be \, then there are criteria associated with the predicate. The language user employs these criteria to correctly acknowledge an object to be \. In using the predicate to attend to the collection of \’s, the language user is attending to whatever objects satisfy I’s criteria. This collection does not constitute an object apart from the language user’s linguistic practice and her attending act. Given

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the linguistic practice, the collection is merely potential until someone actually attends to it. A collection which is constituted by someone’s attending to it is not a unified whole. It hasn’t got sufficient unity to be an indivisible element of a further collection  unless it is a collection consisting of a single object. So if X is a collection and Y is a collection, the collection of X and Y is no different from the collection of X ’s and Y ’s. Using a set theoretic analogy, if X and Y are collections, we can consider the collection X ‰ Y but there is no collection {X, Y}  alternatively, we might have it that the collection {X, Y} “reduces” to X ‰ Y.

7. Sets and Collections

Collections are, according to me, genuine elements of the world, though they depend for their existence on being the objects of people’s attention. One common reason for attending to a collection is to predicate something of its members. There is affirmative universal (distributive) predication: All cats are felines. And negative universal predication: No tigers make good pets. And “typical” distributive predication: Birds fly. Dogs bark. We also attend to collections to provide targets for collective predication: Dinosaurs are extinct. Bats are numerous. Collections are important for many of the things we say (and think) about the world. In contrast to collections, sets are not important for our ordinary linguistic practice. Sets do play important roles in mathematics and mathematically-related disciplines, but the ontological status of sets has been controversial since Cantor founded set theory a century ago. It is my opinion that logicians and philosophers who regard sets as fictions are correct. Although no one can refer in the unqualified sense to a set, fictional-referring to sets is possible. The set-theoretic story is consistent

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and coherent, though not yet complete, and this story can profitably be investigated and developed. In giving the truth conditions of sentences of L and L i, and in proving soundness results for EO and 1O and a completeness result for 1O, I have made free use of set-theoretic concepts and techniques. This is entirely appropriate, for set-theoretic models are legitimate devices to use to investigate theories of various kinds. That we can use set-theoretic techniques to establish the strong completeness of 1O is convincing evidence that 1O is adequate to the arguments made with sentences of L i (made with statements represented by those sentences). Our proofs are probably the best we can do by way of supporting the adequacy of 1O. However, the sentences of L and L i are intended to represent statements of kinds that people might actually make. A set-theoretic interpreting function is merely an abstract model of what I will call a fully specific interpretation of a logical language. A fully-specific interpretation ties logical-language expressions to specific kinds of linguistic acts, performed with specified expressions. A fully-specific interpretation of either L or a first-order language wouldn’t be based on a domain of individuals which is a set, but would instead have a domain which is a collection given by using a noun or noun phrase. A genuine collection is nonempty, but we could also use some expression like ‘square circle’ or ‘healthful cigarette’ to specify an empty domain. Since all collections have members, it is a façon de parler to speak of an empty domain. A fully specific interpretation determines which kinds of acts are represented by the various expressions. For a first-order language, one might do this by saying something like: Let ‘F(x)’ mean x is a fox Let ‘P(x, y)’ mean x is a parent of y Etc. For Ontology, we might have this: Let ‘A’ be a noun constant for anteaters Let ‘B’ be a noun constant for bananas Etc. The interpreting functions we have employed above assign truth-valued functions to noun constants. Such functions are extensional mathematical counterparts to predicative acts, but they don’t “do justice” to these acts. For one function might correspond to different predicates (predicative

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acts). And the function can only capture the extension of the predicate at some particular moment.

8. Capturing Talk about Collections

If someone attempts to refer to an object and, further, say something about the object, when there is no object  perhaps with this sentence: The person who did this is deranged. then her statement is not correct. I prefer to classify such statements as false rather than making a special category for them. These false statements have an existential presupposition that might also characterize a speaker’s denial of the statement. If both of these statements: The person who did this is deranged. It isn’t true that the person who did this is deranged. presuppose that there is a person who did whatever this is, and no person did it, then both statements are false. (Even someone who holds out for calling them neither true nor false must admit that neither is correct.) I think it is also the case that the person who directs his attention to a collection, and goes on to say something about this collection, presupposes the existence of the collection  his attending act commits him to acknowledge the collection to exist, which “amounts” to acknowledging the collection to have members. A person who, by mistake, directs his attention to a nonexistent collection, and goes on to make a statement “about” this collection, has said something false (incorrect) no matter how he characterizes the collection. For the “success” and correctness of his statement depends on his attentiondirecting act having a genuine target. (But we can say true things about collections, like the collection of dinosaurs, that once existed and no longer do.) If there are no I’s, then someone who uses these sentences: All I’s are \ No I’s are 4 to claim that the (collection of) I’s are all \, and never 4, has made false statements. But if she uses these sentences: It is false that all I’s are \ It is false that no I’s are 4

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and still directs her attention to the collection of I’s, her statements are still false. When someone uses a predicate or noun I to direct her attention to the collection of I’s, she has used the expression directedly. The directed use of the expression involves an existential commitment (or presupposition). But when the same (or a derivative) expression is predicated of an object, there is no existential presupposition. The speaker who uses a sentence: D is a I to claim that D is a I is making/reflecting a commitment to accept statements to the effect that something is a I and that I’s exist. But if she denies the statement, using one of these: D is not a I It is false that D is a I she is not committed by what she says to accept a statement that I’s exist. The predicative use of an expression is not characterized/ constituted by existential commitment. As the language L (and also L i) has been developed to this point, no noun constants are used directedly. A statement represented by: [D H I] to the effect that D’s are, singly, I’s is false if either there are no D’s or no I’s. (It is also false if there are D’s and I’s, but either there is more than one D or the single D is not I.) And if this statement is false, then the following will represent a true statement: ~[D H I] The language L does not contain a category of referring expressions, so in L there are no sentences which predicate a noun constant of an individual picked out by a referring expression. But we think of L as a fragment of a larger language in which the noun constants are predicated of individuals (in which the noun constants represent acts predicating something of individuals). However, an expression can be used predicatively without being predicated (or denied) of an individual. What is characteristic of the predicative use of a predicate expression is the absence of existential commitment/presupposition. (I am stipulating my own usage here, not reporting conventions governing the use of English.) A sentence: [D H I]

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has the sense: Exactly one thing is D, and that thing is I. In this sentence (phrase) I have used to give the sense, both ‘D ’ and ‘I ’ have been used as predicates. If Ontology is intended to capture or represent statements about collections, then it is unsuccessful. Ontology captures instead (some) predicative uses of common nouns. I will expand the language L to L which contains expressions for collections (expressions representing acts of using words to direct one’s attention to collections). In L , noun constants which have underlining are used to represent the directed use of nouns. A plain noun constant is one of: A, B, C, A1 , . . . A directed noun constant is obtained by underlining a plain noun constant. The wffs of L are wffs of L . In addition, we have: If M is a wff of L , and contains occurrences of plain noun constant D, then the result of replacing some or all occurrences of D by D in M is a wff of L . As for truth conditions, we treat a sentence M containing directed constants D1 , . . . , Dn (and no others) as if it were the conjunction: M* & (X)[X H D1 ] & . . . & (X)[X H Dn ] where M* is obtained from M by replacing all occurrences of directed noun constants by their plain counterparts. So, for example, both a sentence: [A H B] and: ~[A H B] imply that something is a B. If there are no B’s, then both of the sentences are false. A single English sentence can be used (on different occasions) to make statements having different semantic structures. But someone making a statement with this sentence: All dogs are animals. would be very unlikely to use ‘dogs’ predicatively. In sentences of L , any place that can be occupied by a constant D can also be occupied by the plain version D, and conversely. However, while sentences of L and L (and also L i) represent statements which someone can make, not all of the artificial-language sentences represent kinds of statements that are common.

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To provide a completely adequate deductive system for L , it would be necessary to make use of elements of illocutionary logic, as explained in Kearns (1997, 2000, and 2002). But doing this would make the present paper much too long. I will briefly sketch the deductive system EO, and won’t prove results about the system. EO is obtained from EO by adding rules for directed constants. To illustrate these rules, if M is a wff and D is a directed constant, I will write ‘MD ’ to indicate that M contains one or more occurrences of D. To understand the significance of the new rules of EO, we must regard the system as being used with respect to some established body of knowledge. We use proofs/arguments in the deductive system to establish consequences of this knowledge. If M is a sentence representing a statement of what is known, M can be the top node of a branch in a tree proof without being a hypothesis of the proof. In such a case, M is an initial assertion of the tree proof. A tree proof from initial assertions M1, . . . , Mr and hypotheses N1, . . . , Ns to conclusion P establishes that the argument sequence N1, . . . , Ns / P is correct or valid with respect to the body of knowledge K. In EO, the restrictions on the constant generalized by the rule  Introduction require that this constant occur in neither an initial assertion nor a hypothesis leading to the sentence which is the premiss for  Introduction. No hypothesis in a tree proof of EO can contain occurrences of a directed constant D. The additional rules of EO are the following: Directed Introduction [ D H E] [ D H E]

Directed Transmission MD N ND

The premiss is the conclusion of a tree proof with no uncancelled hypotheses; thus it is an addition to the body of knowledge.

N contains occurrences of the plain constant D, and some or all of these are replaced by D in ND.

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Directed Elimination MD M

Some or all occurrences of D in MD are replaced by D in the conclusion.

It is clear that there are no logical truths of L which contain directed constants. It should also be noted that an argument sequence: [D H E ] / [D H E ] is not inferentially correct in the sense that an inference from its premiss to its conclusion is unconditionally legitimate. Whenever the premiss is true, so is the conclusion. Asserting the premiss commits a person to asserting/accepting the conclusion. But supposing the premiss does not commit a person to a supposition of the conclusion. (If we suppose some statements to be true, and then derive a conclusion from these statements, our conclusion has the force of a supposition even though we don’t prefix the conclusion with the word ‘suppose’.) The directed use of a noun constant has the force of an assertion, so that supposing: [D H E ] is like saying: I assert that E ’s exist, and I will suppose that the D ’s are, singly, E ’s.

9. Some Conclusions

The plain system Elementary Ontology consists of the language L , the account of truth conditions for sentences of L , and the deductive system EO. This system is a truncated version of LeĞniewski’s system Ontology. The system Elementary Ontology isn’t exactly like LeĞniewski’s system, for it doesn’t have the “growing” character of LeĞniewski’s system, and it contains a semantic account  which LeĞniewski did not provide. However, the open character of LeĞniewski’s system isn’t essential, and the account of truth conditions seems pretty much what LeĞniewski must have had in mind, except for the fact that noun constants are construed as predicates of individuals. The system First-order Ontology is a restricted version of Elementary Ontology in which the only quantifier is restricted to one-individual noun constants. This makes a sentence ‘( iD)M ’ equivalent to a sentence about all individuals in the domain. The system 1 O is sound and

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complete. LeĞniewski himself probably wouldn’t have had any use for First-order Ontology, but I think it is useful to call attention to a fundamental system of Ontology which is complete as well as sound. LeĞniewski failed to appreciate the essential difference between referring expressions and common nouns, for, according to SobociĔski (in his classroom lectures), LeĞniewski thought that a statement to the effect that Socrates is a human being is captured by an Ontological sentence like this: [Socrates H human being] This failure kept LeĞniewski from recognizing that Ontology provides an inadequate analysis for the whole of a natural language. But Ontology is suited for analyzing some uses of common nouns. However, Ontology isn’t suited for analyzing statements about collections (as I understand them), and I think that collections are those entities which are genuinely distributive classes. We say things about these collections in order to predicate properties of each of their members. The directed system Elementary Ontology, which consists of L , its semantic account, and EO, does allow us to analyze statements about collections. However, the directed system would be more satisfactory if it incorporated principles and notation proper to illocutionary logic, for the notation characteristic of directed constants marks a force analogous to illocutionary force, and would be more conveniently treated in a system with symbols for illocutionary force.1

University at Buffalo Department of Philosophy 607 Baldy Hall Buffalo, NY 14260-1010, USA email: [email protected]

REFERENCES Kearns, J.T. (1996). Reconceiving Experience: A Solution to a Problem Inherited from Descartes. Albany/New York: State University of New York Press.

1

In writing and revising this paper, I have received many helpful suggestions from my colleague John Corcoran. They have substantially improved the paper in both style and substance. Comments from Barry Smith were also extremely useful.

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Kearns, J.T. (1997). Propositional Logic of Supposition and Assertion. Notre Dame Journal of Formal Logic 38, 325-349. Kearns, J.T. (2000). An Illocutionary Logical Explanation of the Surprise Execution. History and Philosophy of Logic 20, 195-214. Kearns, J.T. (2002). Logic Is the Study of a Human Activity. In: T. Childers and O. Majer (eds.), The Logica Yearbook 2001, pp. 101-110. Prague: Filosofia. Lejewski, Cz. (1995). Remembering Stanisáaw LeĞniewski. In: D. Miéville and D. Vernant (eds.), Stanisáaw LeĞniewski Aujourd’hui, pp. 25-66. Paris, Grenoble & Neuchatel: Groupe de Recherches sur la philosophie et le langage & Centre de Recherches Sémiologiques. LeĞniewski, S. (1927). O podstawach matematyki [On Foundations of Mathematics]. Przegląd Filozoficzny 30, 164-206; 31, 261-291; 32, 60-101; 33, 77-105; 34, 142-170. LeĞniewski, S. (1929). Grunzüge eines neuen Systems der Grundlagen der Mathematik. Fundamenta Mathematicae 14, 1-81. LeĞniewski, S. (1930). Über die Grundlagen der Ontologie. Sprawozdania z posiedzeĔ Towarzystwa Naukowego Warszawskiego, Wydziaá III (Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III) 23, 111-132. LeĞniewski, S. (1932). Über Definitionen in der sogenannten Theorie der Deduktion. Sprawozdania z posiedzeĔ Towarzystwa Naukowego Warszawskiego, Wydziaá III (Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III) 24, 289-309. Luschei, E.C. (1962). The Logical Systems of LeĞniewski. Amsterdam: North Holland. Searle, J.R. (1974). The Logical Status of Fictional Discourse. New Literary History 6, 319-322. Reprinted in: Expression and Meaning (Cambridge: Cambridge University Press, 1979), pp. 58-75. Simons, P. (1987). Parts: A Study in Ontology. Oxford: Oxford University Press. SobociĔski, B. (1949). L’Analyse de l’Antinomie russellienne par LeĞniewski. Methodos 1, 94-107, 220-228, 308-316; 2, 237-257.

Jacek PaĞniczek DO WE NEED COMPLEX PROPERTIES IN OUR ONTOLOGY?

In discussing ontological issues philosophers sometimes mention complex properties. However, they usually dismiss the concept as theoretically useless and even dangerous (see Grossmann 1972). Only some of them try to argue in favour of complex properties and even then their arguments are not sufficiently convincing to give rise to more serious consideration of this concept (see, for example, Meixner 1991). In the present paper, we aim to show the ontological usefulness of complex properties and to suggest that these properties can be treated as a formal-ontological means of categorial differentiation of objects. The main idea is that objects may differ not only with respect to simple properties but also with respect to complex properties. In particular, two objects may possess the same simple properties and differ in complex ones. We believe, however, that the configuration of simple and complex properties is neither accidental nor arbitrary but that it is amenable to certain general ontological rules. We adopt here a liberal ontological approach and do not restrict our considerations to any particular category of objects (like individuals, for example). The problem of complex properties is very often considered parallel to the problem of complex predicates (the latter is discussed usually by logicians or linguists) and the two problems are in fact inseparable (see especially: Swoyer 1997, 1998). Quite often acceptance of complex properties is motivated by linguistic reasons. In semantic analyses of language, we need complex predicates and these should be equipped with interpretations. Here we will prefer the “ontological” approach but, for obvious reasons, we will talk, explicitly or implicitly, also about complex predicates.

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 113-128. Amsterdam/New York, NY: Rodopi, 2006.

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1. Introducing Complex Properties: A Formal Theory of Complex Properties What can be considered a complex property? Given any (simple) properties P and Q, we talk about the negation (or: complement) of P: ¬P, the conjunction of P and Q: (P š Q), or the disjunction of P and Q: (P › Q). One can take into account kinds of complex properties other than those built up by means of connectives such as ¬, š, ›, for example properties built up by means of modal operators or relations (the socalled relational properties). Here we focus only on complex properties composed of simple properties and connectives. 1 The problem whether complex properties exist somehow apart from simple properties belongs to metaphysics. However, this problem, when considered exclusively within a metaphysics of properties, is not particularly interesting and we are not going to discuss it in this paper. Our concern is primarily ontological. This means that we consider various categories of entities in order to discover dependencies between simple and complex properties, as if they were existent, alongside dependencies between properties and objects. We then consider various categories of objects relative to these dependencies. Any discussion of properties should, of course, be accompanied by a discussion of objects qua subjects of predication. Only in this context may we reasonably investigate ontological relations between simple and complex properties by asking whether the predication that involves complex properties implies, is implied by, or is equivalent to, the predication that involves respective simple properties (e.g. whether the fact of possessing P is equivalent to the fact of not possessing P). This possibility would make the concept of complex properties ontologically relevant. Let a represent an object and aP express the predication: a possesses P. We may here distinguish the following principles of dependency between simple and complex predication, i.e., between possessing simple properties and possessing complex properties. P1. P2. P3. P3.* P4. P5. 1

a¬P Š ¬aP (consistency) ¬aP Š a¬P (completeness) a(P š Q) Š aP š aQ ¬a(P š ¬P) Š (a(P š Q) Š aP š aQ) aP š aQ Š a(P š Q) (closeness under conjunction) aP › aQ Š a(P › Q)

The case of relational properties is more complicated from a formal point of view but at the same time it is not much more interesting from a philosophical point of view.

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P6. P7.

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a(P › Q) Š aP › aQ a(P Š Q) Š (aP Š aQ)

We assume that connectives are mutually definable in the usual way when they are applied to properties: R1.

If Į { ȕ is an instance of a tautology expressed by means of property variables then for every object a: aĮ { aȕ holds.

E.g. an object a possesses a complex property (¬P › Q) iff it possesses the property (P Š Q). Additionally, we will consider the following two rules: R2.

If Į is an instance of a tautology expressed by means of property variables then for every object a: aĮ holds (R2ƍ: ¬a¬Į holds also).

If R2 is assumed then every object possesses the property (P Š P) in particular. R3.

If Į Š ȕ is an instance of a tautology expressed by means of property variables then for every object a: aĮ Š aȕ holds.

According to R3 objects are closed with respect to logical entailment, i.e. logically closed. E.g. if a(P š Q) then aP and aQ as well. Obviously, R3 is stronger than R1, i.e. R1 follows from R3. It should also be noticed that if we additionally assume, and this seems quite reasonable, that every object possesses at least one property then R2 will also follow from R3.2 Let us indicate some other logical interdependencies between P1-P7 and R2, R3 if R1 is assumed. When all principles and rules listed above hold, this situation will be called individuality. FACT FACT FACT FACT FACT FACT FACT FACT

2

1. 2. 3. 4. 5. 6. 7. 8.

P1-P7 are independent from each other. P1-P4 entail R2 and R3. R3 entails P3. P7 and R2 entail R3. P2 and P4 entail P3*; P2, P4, and R2ƍ entail P3. R2 and P7 entail P4. P3 and P4 entail P7. The following groups of principles and rules are equivalent and each of them entails individuality: (a) P1-P4; (b) P1, P2, P5, P6; (c) P1, P2, P7, R2; (d) P1, P2, P4, R2ƍ; (e) P1, P2, P4, R3.

Suppose that D is a tautological property and that for some E, aE. In this case E Š D is also a tautological property and, according to R3, aE Š aD and hence aD.

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FACT 9. P1 and P2 do not entail any of P3-P7.3 Since R1 is assumed we may model (interpret) our basic ontology of properties in a Boolean algebra: P = , where P is a set of properties, ˆ, ‰ are distributive lattice operations of join and meet respectively, * is the operation of complement, 0 and 1 are the minimum and maximum elements respectively.4 This modelling is also treated as a semantics for a language in which we talk about complex properties by means of predicate variables and connectives (the language can be successively enriched). Thus, let I be an interpretation function which assigns to P, Q, etc. the elements of P and assigns to complex properties their Boolean counterparts: I(¬P) = I(P)*, I(P š Q) = I(P) ˆ I(Q), I(P › Q) = I(P) ‰ I(Q), I(P Š Q) = I(P)* ‰ I(Q). Thus, I(P › ¬P) = 1 and I(P š ¬P) = 0. Let I assign to object variables a, b, . . . subsets of the power set of P, i.e. sets of properties. We then say that these sets constitute respective objects and that the proposition aP is true if and only if I(P)  I(a). Thus, objects in this semantics are represented by sets of properties which they are supposed to possess. Yet, this is just a representation which does not commit us to a bundle theory of objects. As can be expected, R1 holds in the algebra P but neither R2 nor R3 does.5 Also, none of P1-P7 are valid in this semantics. In order to be able to express all facts about the properties in P, we obviously need a much richer formal language than the one adopted above. Most ontologists would presumably agree that all principles P1-P7, R1, R2, R3 are true. As, certainly, would all philosophers who admit individuals as the only category of entities (this motivates our choice of the term individuality in this context). The truth of these principles means two things. First, it means that complex predication is always reducible to simple predication (complex predicates can then be viewed as merely a trivial notational variant of the simple predicates). Second, we can accept an ontology of complex properties and treat P1-P4 as the metaphysical theses stating jointly that there exist only individuals, i.e., objects insensible to a distinction between simple and complex properties. However, even if we do not admit entities other than individuals, the concept of complex properties and its related principles allow us to gain an insight into the ontological nature of individuals. Generally, when we 3

We omit the proofs of theorems included in this paper. Most of those proofs are quite elementary or they follow from well-known facts concerning Boolean algebras. 4 A reader not acquainted with Boolean algebras may consult any elementary textbook on this topic. 5 Essentially the same ontological framework for semantics of natural language is adopted in Keenan and Faltz (1985).

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choose some of the principles P1-P7 and R1, R2, R3 as the axioms of our theory, this decision is metaphysical and means that there only exist objects that obey these principles. Yet, if we discuss objects that fulfil some of these principles then this is merely an ontological approach, which does not entail any metaphysical theses.6

2. Ontological Interdependencies between Objects and Properties If we assume that properties are more basic entities than objects then the ontology of properties can induce an interesting ontology of objects (but not a metaphysics thereof). If we assume the converse ontological dependence then the possible relations between properties such as entailment and possible operations on them such as negation, conjunction, etc., can be entirely determined by the ontology of objects. The second ontological view is certainly much more popular among philosophers. The extensional approach to properties is a paradigmatic example of this view. 7 We simply conceive of properties as sets of individuals and the relations between those sets determine all the possible relations between properties. The relations in question are usually established, especially in formal ontologies, by identity and definition axioms (for properties and objects, respectively): IdPR IdOB

a (aP { aQ) Š P = Q P (aP { bP) Š a = b

IdPR states that coextensive properties are identical and this is the most straightforward formulation of extensionalism. IdOB is the strong version of Leibniz’s principle. Although in these axioms we quantify over objects and properties, respectively, the axioms carry no metaphysical

6

Here we distinguish between the meaning of ‘ontological’ and ‘metaphysical’. By the latter term we mean those theses that state or entail the existence of such and such objects or objects of a such and such kind. Ontological theses also involve existence but they state dependencies between different categories of entities, forms of entities, structures of predication, etc. and basically they do not entail any existential theses. I say ‘basically’ since the issue depends on what criterion of ontological commitment we accept. The ‘ontological-metaphysical’ distinction has been precisely defined by a Polish phenomenologist, Roman Ingarden (see Ingarden 1964). 7 Given a set of individuals D, the structure is a Boolean algebra with inclusion as its ordering. See section 5.

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commitments in the sense of entailing the existence of objects and properties.8 DfPR DfOB

 P a (aP { Į), where P does not appear in Į.  a P (aP { Į), where a does not appear in Į.

These two definition axioms are strictly metaphysical – they explicitly state the existence of properties and objects. Of course, since the content of the formula Į determines a particular property or a particular object, the ontology of properties and objects directly hinges on the expressive power of the language in which Į is phrased. It is possible to assume both DfPR and DfOB, i.e., to allow mutual ontological definability of objects and properties. However, one should be aware that this gives rise to a circularity which may result in a contradiction. According to our approach properties are ontologically basic. This seems to make objects depend on properties since objects are constituted by properties. However, such an interpretation is only partially correct, for although objects that possess the same properties are indistinguishable, nothing guarantees that there exists an object that possesses such and such properties (including complex ones). Even if we introduce the definition axiom DfOB into our theory it will have no significant creative power. We will be able to define new objects only relative to those already present. In particular, we may define the negation, the inversion, the conjunction, and the disjunction of objects: [¬a]P { ¬aP, [~a]P { ¬a¬P, [a š b]P { aP š bP, [a › b]P { aP › bP.9 Yet, the axiom DfOB, as it stands, does not allow us to “create” objects from any arbitrary properties. This goal of “creativity” can be at least partially achieved if we introduce the symbol of identity into the language of our ontology. We can then define (bring into existence) every object constituted by a finite number of selected properties, e.g.  a P (aP { P = Q1 š P = Q2 š . . . š P = Qn ) states the existence of an object constituted by properties: Q1 , Q2 , . . . , Qn . Of course, the cardinality of objects that can be defined depends strictly on the number of properties in P. It should be stressed that we cannot adopt DfPR instead of introducing complex properties in the way we have done. If we did that, complex properties would be dependent on objects. In particular, the only 8 According to Quine’s criterion, they do in the sense that there must exist at least one object or one property when quantification is based on classical logic. 9 Notice that when we use a tautology or contradiction for Į, we may define the object that possesses every property or the object that possesses no property. These are certainly very bizarre objects and hardly anyone can accept them.

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way of defining the negation of a property is:  P a (a¬P { ¬aP). If we insisted on the uniqueness of P then this would force us to treat this property extensionally – for every two coextensive but distinct properties their negations would be identical. 10 As such, the algebra of properties does not bring into being any object. All we can say is that objects may be represented by sets of properties and this fixes the upper boundary of the class of all objects that could be. We may adopt the Meinongian approach and allow every set of properties to represent an object, perhaps with the exception of the empty set (see below). We can distinguish many categories of objects (although clearly a finite number of them) which fulfil various combinatorically possible systems of principles and rules listed above. However, as we already noticed, this does not mean that we are ontologically committed to the existence of those objects, i.e., that we must accept some metaphysical theses. Moreover, from an ontological point of view, most of those categories seem quite uninteresting and one could hardly imagine objects that exemplify them.

3. Negation and Conjunction of Properties. Completeness, Consistency and the Conjunctive Closure of Objects Perhaps the negation (or the complement) of a property is the most often mentioned type of complex property. In a natural language, we usually refer to the negation of a property by means of prefix like non- or not inserted in front of the respective predicate expression, e.g. ‘non-human’, ‘not a cat’, etc. Two properties are called contradictory if one is the negation of the other. The concept of negation enables us to express two formal-ontological features of objects – their consistency and completeness. An object a is said to be consistent if and only if P1 holds for all properties, i.e., an object cannot possess contradictory properties.11 An object is said to be complete if and only if P2 holds for every property, i.e., an object possesses at least one of every two contradictory properties. Usually consistency and completeness – if considered by traditional philosophers 10 This is the problem of every intensional theory of properties in which DfPR appears as a theorem. The problem can, to a certain extent, be evaded only in theories with two modes of predication (see Zalta 1983 and 1988). 11 There is a weaker concept of contradictory properties according to which being round and being a square are contradictory, see below.

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at all  are attributed to all individuals or to all objects. But then these two concepts turn out to be universal in the domain of objects and so not ontologically relevant. On the other hand, they are quite trivial if one does not acknowledge complex properties or treat those properties as reducible to simple ones. Thus, all objects turn out to be ex definitione complete and consistent. Let us emphasize that consistency and completeness have a formal-ontological character and as such the notions do not need to entail any metaphysical existential theses, in particular, they do not need to entail any theses about the existence of inconsistent or incomplete objects. However, there are some metaphysical theories that allow for inconsistent and incomplete objects. Here we should mention perhaps the most characteristic theory of this kind, i.e. the Meinongian theory of objects. The Meinongian ontology comprises, apart from existing objects which are consistent and complete, also non-existing objects which are inconsistent or incomplete (an example is the non-round circle). It is worth mentioning that, in this theory, the concepts of consistency and completeness play an essential role in distinguishing new ontological categories of objects different from existing individuals. Since consistency and completeness are independent  neither entails the other  their intersection gives rise to four categories of entities: consistent and complete, inconsistent and complete, consistent and incomplete, inconsistent and incomplete. It is a commonly shared view that existing objects (individuals) are consistent and complete, although not every consistent and complete object must exist. 12 There is also another notion of inconsistency, different from the one given above. Let us call it impossibility. An object is said to be impossible if it possesses a contradictory property, e.g. a(P š ¬P). R2ƍ states that every object is possible (is not impossible). If P3 or R3 is assumed then every impossible object is inconsistent; if P4 is assumed then every inconsistent object is impossible. The conjunction of properties is still another kind of complex property frequently discussed. Some philosophers accept conjunctions of properties as the only sort of complex properties.13 However, there seem to be no good reasons for rejecting either P3 or P4 as a principle true for all objects and properties. In particular, one can treat a conjunction P š Q as a property “stronger” than P and Q alone in the sense that generally, i.e., P3 holds for every object while P4 does not need to hold for every 12

It is easy to construe a counterexample (to this view) on the ground of Boolean algebra. Armstrong admits the conjunction of properties to be a property but he deprives of this status the negation and disjunction of properties (see Armstrong 1997). 13

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object. Let us notice here that, in a certain sense, P3 is in fact weaker than P4. For, on the one hand, P3 follows from the logical closure expressed by R3 (which is a very natural condition supposed to be obeyed by all objects as explained below); on the other hand, P3 follows from P4 for complete objects that do not possess the empty property (see FACT 4). We see also that complete, consistent, possible objects that are closed under conjunction are individuals (see FACT 7d). It is worth considering another class of objects that embraces the class of individuals, viz. the class of consistent objects, i.e. objects that fulfil P1. Of course, consistent objects do not need to be incomplete but the category of incomplete and consistent objects is particularly interesting. Intuitively, objects that are referred to by definite descriptions such as the wooden table, the black cat, the rectangle with equal sides are consistent but not complete objects. In particular, the wooden table is supposed to be wooden and to be a table but neither to be square nor to be round (non-square). Another example of this kind of incomplete entity is provided by fictional objects. Those objects are to a large degree undetermined with respect to their properties. E.g. Sherlock Holmes is neither an orphan nor a non-orphan. This underdetermination follows from the obvious fact that it is impossible to invent a complete object, i.e., for every property to ascribe that property or its complement to a fictional object. It is worth noting that incomplete objects play an important role in the Meinongian ontology as entities that mediate intentional relation or reference to complete, in particular, to existent objects. Incomplete objects as determined merely by P1 can be subjected to further ontological refinement. Let us consider the rectangle with equal sides. This object can be thought of as constituted by two properties: being a rectangle and being a geometrical figure with equal sides. As such the rectangle with equal sides is a rectangle and it is a geometrical figure with equal sides but that object does not possess the conjunction of the two properties, i.e., it is not a rectangle with equal sides. This certainly may be seen as a counterintuitive result: why is the rectangle with equal sides not a rectangle with equal sides? This will not be the case if objects distribute the conjunction of properties, i.e. if they obey P4. Such incomplete objects closed under conjunction of properties can be called general objects.14 14

One is not supposed to associate any special meaning with this term. However, it may be mentioned that this meaning of “general objects” is close to the notion of a general object assumed by some philosophers of the Lvov-Warsaw School (see àukasiewicz 1987, ch. XVIII; KotarbiĔski 1920; LeĞniewski 1992).

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Let us now assume that the rhomb with equal diagonals is constituted of the following properties: being a rhomb, being a figure with equal diagonals, being a rhomb with equal diagonals. Intuitively the rectangle with equal sides is the same as the rhomb with equal diagonals and the square (constituted of a single property: being a square). So how can one render formally the fact that, in one sense, these objects are identical while, on the other hand, they are different because they are constituted by different properties? The following solution to this question may be suggested. First, we need to assume that complex properties: being a rectangle with equal sides and being a rhomb with equal diagonals are identical as well as identical with the simple property of being a square. The three objects mentioned above turn out then to have the same infimum: the property of being a square. This feature may be taken generally as a mark of a special sense of object identity, say virtual identity: two objects are virtually identical if and only if they have the same infimum. In this sense the rectangle with equal sides, the rhomb with equal diagonals, and the square are virtually identical; analogously the Polish Pope, the Cardinal born in Wadowice, John Paul II are virtually identical objects (although they do not need be strongly identical).15 Notice, that for objects which are logically closed (R3) and closed under conjunction virtual identity implies strong identity.

4. Property Entailment. Objects Closed under Entailment An important problem connected with properties is whether there is any entailment between them.16 It seems to be true that redness entails colouredness, roundness entails non-squareness. One could simply answer that a property P entails a property Q if and only if (necessarily) every object that possesses P possesses (or possesses necessarily) Q as well, i.e. a (aP Š aQ) or: a (aP Š aQ). However, this answer would mean that property entailment is basically dependent on actually existing objects (the range of individual variables) or merely possible objects and in this sense the entailment is accidental from a metaphysical point of view. Of course, we do not want to adopt such a view of property entailment at the outset, for this would basically result in the extensional view of properties. However, there can be notions of property entailment 15

For they are constituted by the following sets of properties: {being a Pole, being a Pope}, {being a Cardinal, being born in Wadowice}, {being John Paul II}, respectively. 16 Some authors prefer to talk about the part-whole relation holding between properties instead of talking about entailment between properties.

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independent of what objects exist or may exist. In particular, property entailment is that which mimics propositional entailment in classical logic and which is expressed in the rule R3: a property Į entails a property ȕ if and only if Į Š ȕ is an instance of a tautology expressed by means of property variables. Yet, this definition is purely formal and as such does not capture cases like redness entailing colouredness; and, being based on classical material implication, it inherits its counterintuitive features (in particular, self-contradictory or empty properties entail all other properties). The concept of property entailment should presumably be treated as primitive, in contradistinction to the concept of metaphysical entailment, rendering analytic or synthetic a priori truths. And R3 should be understood only as a sufficient condition for this entailment.17 We may understand property entailment as determined by the ordering d of a particular Boolean algebra of properties in the sense that P entails Q iff I(P) d I(Q). Certainly, in this sense redness may entail colouredness in a particular Boolean algebra of properties. The empty and full properties still, however, remain indistinguishable in this algebra. Yet, given the concept of property entailment Ÿ, whatever it could be, one may adopt another definition of contradictory properties: properties P and Q are contradictory if and only if P entails the negation of Q (P Ÿ ¬Q). In this sense the round square, i.e., an object which is round and is a square turns out to be inconsistent. As we noticed earlier, not all combinatorically possible categories, i.e., categories that result from combinations of some independent conditions P1-P7 and R1-R3, deserve to be studied. However, some categories of objects that lack certain features of individuality nevertheless exemplify a sort of regularity worth considering. In particular, one may distinguish a category of closed objects, objects closed under some sort of entailment (ontological move), or one may also accept a metaphysical thesis that all objects are closed. There is some evidence that every intentional object, i.e., an object we are conscious of, is somehow closed, although the issue of what kind of closure we are dealing with is a matter of logical and phenomenological discussions. Thus, not only the ordinary existing individuals we encounter in the real world, but also objects of imagination and dreams, theoretical and fictional objects, etc. are closed. If we think of the rectangle with equal sides or even of the round square we think of these “objects” as geometrical figures. When we read Arthur C. Doyle’s stories we have no 17

Even this view may, however, be questioned on the basis of the paradoxes of material implication (for example: does roundness entail roundness or brightness?)

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doubts that Sherlock Holmes is a European even if this is not explicitly stated in any story. Let us take into account objects closed under property entailment generated by the ordering of algebra P and consider now objects that fulfil the following condition, which we call the condition of monotonicity: P8

P d Q Š (aP Š aQ)

Assuming P8, let 4 be an object interpreted in P by {1} and let 5be interpreted by P(P)-{0}. Then 4 and 5 can be understood as the universal and particular quantifier objects, respectively. One should here notice that when sets ‡ and {‡} are precluded from representing any objects 18 the following formulas are valid for objects closed in the sense of P8:

(*)

4P Š aP aP Š 5P 4P { ¬5¬P 4(P Š Q) Š (aP Š aQ) R2

This means that closed objects generate a simple version of classical quantifier logic where quantifiers 4 and 5 are treated as categorematic expressions referring to what is sometimes called quantifier objects.19 Notice that P8 is equivalent to (*). The closure of objects can also be expressed by P7. The meaning of this notion of closure is in a sense close to the meaning of the notion of closure under conjunction (see FACT 5 and FACT 6).

5. Extending an Ontology of Properties Should some additional assumptions be imposed on the algebra of properties? In particular, should we assume that P is a complete Boolean algebra (completeness means here that for every subset of P there exists an infimum and a supremum). One can point out to an interesting correlation between individuals and properties in complete P. Let us consider an object a. In complete P the infimum ˆa (the meet of all 18 We do not want to say that something that possesses no properties or something that possesses all properties is an object. 19 A logic of this kind is developed in PaĞniczek (1998).

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elements of a) and supremum ‰a (the join of all elements of a) exist. The following formula will then hold: (*)

if ˆa z 0 and a is an individual then ˆa  a and ˆa is an atom in P.20

An element of Boolean algebra is an atom if there are no smaller elements other than 0. If a is an individual then ˆa can be treated as its haecceity since ˆa directly identifies a in the sense that aP if and only if ˆa d P. Notice that there may be an object constituted by haecceity as its only property. Although such an object possesses the property ˆa which could be read: being identical with a, it does not possesses other properties of a. This may suggest another argument in favour of the closure of objects under the property entailment induced by d: if an object possesses an atom-haecceity then why should it not possess all properties entailed by the haecceities? Clearly, not every object a contains ˆa; in other words, the assumption that a must be an individual is necessary here. Another question is whether the algebra of properties should be atomic. In atomic algebras every element is entirely identified by atoms in the sense that it is a union of atoms (two properties that contain the same atoms are equal). Consequently, the atomicity of the algebra of properties results in a kind of extensional view of properties. But this is not a bona fide extensionalism as long as we do not acknowledge a oneto-one correspondence between atom-haeceities and individuals. That correspondence does not need to be obligatory from an ontological point of view. Maybe there are haeceities that are exemplified by no existent or even possible individual or  and this is perhaps a less conceivable possibility  there are individuals that are correlated with no haecceity at all. We are not going to discuss these ontological issues here.21 Let us here notice, however, that if there were unexemplified haecceities then we would escape classical extensionalism: there could be two properties with the same extension consisting of individuals but with a different “extension” consisting of atoms-haecceities. It is an interesting question whether abandoning the assumption about the atomicity of the algebra of properties would not open another 20

a is an individual if and only if a is an ultrafilter in the algebra of properties generated by ˆa. 21 The first possibility can be understood in such a way that some individuals do not exist or even cannot exist in a given possible “extension” of the real worlds. As we stressed before, we are not committed to all objects that can be constituted by properties that belong to P (which are correlated with the subsets of P).

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possibility of the intensional approach to properties. However, a nonatomic algebra can hardly be interpreted as non-extensional, i.e., as an intensional model of properties. For atomicity seems to be strictly connected with the cardinality of a given Boolean algebra (it is a wellknown point that every finite Boolean algebra is atomic).22 The extensional approach to properties certainly has flaws of its own. Still, it makes many issues connected with structural features of properties, identity, quantification (also generalized quantification), cardinality, etc. much more natural and straightforward than does the intensional approach.23 So let us assume that the algebra of properties P is atomic with D as the set of atoms and that there is a one-to-one correspondence between the set of atoms as haecceities and the respective set of individuals D. So let us assume that D is a non-empty domain of individuals and I is a function assigning to P, Q, etc. subsets of D and assigning to complex properties their Boolean counterparts: I(¬P) = D  I(P), I(P š Q) = I(P) ˆ I(Q), etc. Let I assign subsets of P(D) to a, b, etc. As before, we will then say that the proposition aP is true if and only if I(P)  I(a). What is important, individuals, i.e., members of D, can be embedded in the domain of objects, i.e., can be identified with some subsets of P(D) in the following way: d  D o {X  D: d  X}. Clearly, for any property P, d  I(P) if and only if I(P)  {X  D: d  X}. In what follows, we will indicate another consequence of the assumption that the algebra of properties is atomic. The problem of complex properties can still be considered in a more general ontological framework. Properties are usually treated as a kind of relation, i.e., one-argument relations. Consequently, singular (oneargument) predication can be treated as a special case of plural (manyargument) predication. The structural complexity of relations far exceeds the complexity of properties and a much more sophisticated algebraic description is needed than that based on the Boolean algebra. One may also achieve a more general approach to complex properties by admitting 22 Theoretically, it can happen that two different properties “consist” of the same set of atoms. 23 Keenan and Faltz motivate the adoption of atomic Boolean algebra for the semantics of natural language differently. They need atoms to provide denotations for some proper names and descriptions. Consequently, they treat a property: a vegetarian who is John as an atom. But if this property is an atom of a Boolean algebra then it is the same property as a student who is John, a philosopher who is John or simply John. Thus, their argument is mistaken. Notice, that our approach enables us to distinguish the object a vegetarian who is John from the objects a student who is John, a philosopher who is John or simply John since they are constituted by the following sets of properties: {vegetarian, John}, {student, John}, {philosopher, John}, {John}, respectively.

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other kinds of complex properties. So far we have considered only complex properties built out of simpler ones by means of Boolean operations. But we can also take into account relational properties like “is loved by someone,” “loves Mary” built out of relations by means of several special operations which do not exist in ordinary Boolean algebra (like universalisation, conversion, reflection, plugging, etc. (see in particular: Bealer 1982; Menzel 1986; Zalta 1983, 1989; Swoyer 1997). One can ponder whether possessing a relational property is  or should be  always tantamount to bearing a respective relation to something. Also, in particular, one can ponder whether John possessing the relational property “loves Mary” is tantamount to Mary possessing the property “is loved by John.” Certainly, when a very general ontological setting is adopted then any singular predication does not need to be equivalent to plural predication (and the active form of predication does not need to be equivalent to the passive form of predication).24 In such cases we could further differentiate objects with respect to their category if we found sufficient reasons (evidence) to do so. However, we are not going to explore here this new ontological perspective for it would first require the introduction of a quite complicated formal syntax. We only point to an interesting fact. If an atomic algebra of properties is assumed, i.e., properties are treated extensionally and every relational property is represented in that algebra then P1-P7 turn out to be the only formal characteristics that may be used in distinguishing categories of objects. For instance, if an object a is consistent and closed under conjunction (i.e., if it fulfils P1 and P4) then, if a bears a relation to itself then a possesses the property of bearing the relation to itself. Thus, the ontological categorization of objects based on the Boolean algebra of properties is maximal if that algebra is atomic.

Uniwersytet im. M. Curie-Skáodowskiej Department of Philosophy 20-031 Lublin, Poland email: [email protected]

24 Considering complex properties in the framework of complex relations is not, of course, strictly equivalent to considering complex properties as relational properties. These are two different approaches to the nature of predication (and, as a matter of fact, to the nature of the subject of predication). However, these two approaches are basically equivalent when properties and relations are treated extensionally.

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REFERENCES Armstrong, D.M. (1997). A World of States of Affairs. Cambridge: University Press. Bealer, G. (1982). Quality and Concept. Oxford: Oxford University Press. Grossmann, R. (1972). Russell’s Paradox and Complex Properties. Noûs 6, 153-164. Ingarden, R. (1964). Der Streit um der Existenz der Welt. Tübingen: Max Niemeyer Verlag. Keenan, E.L. and L.M. Faltz (1985). Boolean Semantics For Natural Language. Dordrecht: Reidel Publishing Company. KotarbiĔski, T. (1920). Sprawa istnienia przedmiotów idealnych [The Problem of the Existence of Ideal Objects]. Przegląd Filozoficzny 23, 149-170. LeĞniewski, S. (1911). The Critique of the Logical Principle of Excluded Middle. In: S. LeĞniewski, Collected Papers, edited by S.J. Surma, J.T. Srzednicki, D.I. Barnett and V.F. Rickey, vol. 1, pp. 47-85. Dordrecht: Kluwer Academic Publishers. àukasiewicz, J. (1987). O zasadzie sprzecznoĞci u Arystotelesa [On the Priciple of Contradiction in Aristotle]. Warszawa: PWN. Meixner, U. (1991). On Negative and Disjunctive Properties. In: K. Mulligan (ed.), Language, Truth and Ontology, pp. 28-36. Dordrecht: Kluwer Academic Publishers. Menzel, Ch. (1986). A Complete, Type-Free Second Order Logic of Properties, Relations, and Propositions. Center for the Study of Language and Information, Stanford University. Technical Report #CSLI-86-40. PaĞniczek, J. (1998). The Logic of Intentional Objects. A Meinongian Version of Classical Logic. Dordrecht: Kluwer Academic Publishers. Swoyer, Ch. (1997). Complex Predicates and Conversion Principles. Philosophical Studies 87, 1-32. Swoyer, Ch. (1998). Complex Predicates and Logics for Properties and Relations. Journal of Philosophical Logic 27, 295-325. Zalta, E. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics. Dordrecht: Reidel. Zalta, E. (1988). Intensional Logic and the Metaphysics of Intentionality. Cambridge, MA: The MIT Press.

Andrzej Biáat OBJECTS, PROPERTIES AND RUSSELL’S PARADOX

1. “To be an object” means “to have properties.” Thus, any object has at least one property. A good formalization of this simple conclusion is a thesis of second-order logic: (1)

x P (Px)

This formalization is based on two assumptions: (a) object variables range over the universe of objects; and (b) one-place predicate variables range over the universe of properties. Thesis (1) does not exhaust the ontological content of these assumptions. For example, it does not exclude the existence of properties which themselves have properties  and are therefore objects  but which are not values of object variables. This may give rise to a question how to formalize the following rule in an object language: (2)

Each property that is an object is a value of object variables.

Methods for formalizing this type of statements are given by secondorder type-free systems in which predicates can be arguments of predicates. Especially, for a proper expression of thesis (2) one needs to use a “generalized” identity functor which can take expressions belonging to various syntactical categories as its arguments. (3)

P (Q (Q(P)) o x (P = x))

The traditional definition of an object as the subject of properties gives rise to further questions, the formalization of which requires the use of a “generalized” identity relation, as it is in the case of a hypothesis concerning the existence of objects that are not properties: (4)

x P ¬(P = x)

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 129-142. Amsterdam/New York, NY: Rodopi, 2006.

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Another example of a typical formula of type-free systems is the following proposition: (5)

P x ¬(P = x)

The last formula expresses the main thesis of this paper: there are properties which are not objects. This postulate is also assumed (although only in the metalanguage) in standard second-order logic. In contrast to this logic, however, here we do not exclude the existence of properties which are objects. Our defence of the main thesis of this paper is at the same time an attempt at generalizing the ontology that constitutes the foundation of standard second-order logic.

2. Various type-free systems (first- or second-order), which have recently appeared in great numbers, are in fact developed theories of properties, relations and propositions. In those systems, various theses regarding the quantification over objects of higher “logical types” (such as properties of properties, properties of propositions or relations between objects and properties, etc.) are expressed. However, in contrast to higher-order logic, this type of hierarchy of entities is not fully reflected in the syntax of those theories. A type-free approach gives a more comfortable tool for logicophilosophical analysis in comparison to higher-order logic.1 Moreover, this approach is not as philosophically committed as the latter, which from the start accepts the existence of a hierarchy of properties (one or another, finite or infinite) and excludes the existence of properties belonging to themselves (contrary to the intuition that such properties exist: being cognisable, untouchable, different from Immanuel Kant, being a property, etc.). In spite of the fact that second-order type-free systems are substantially richer than analogical first-order theories (for at least two kinds of variables  objects and predicates  are there distinguished), they are simpler in certain respects. They do not require discerning extra1

The advocates of the type-free approach formulate various arguments for this view. For example, S. Feferman defended the need to use a type-free theory for explaining some crucial aspect of mathematics. N. Cocchiarella and G. Bealer employ their own systems (very different from each other) for reconstructing logicism. G. Chierchia and R. Turner are developing Montague semantics in a type-free manner and argue that it helps in explaining important linguistic phenomena.

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logical constants (such as “exemplify,” “is a property,” etc.) necessary in first-order theories. This, in turn, allows us to treat those systems as rich and convenient conceptions of philosophical logic. 2 The assumption that all properties are objects (i.e., are values of object variables, although not necessary only of them) is very characteristic for the type-free approach, at least in its contemporary forms. By using a “generalized” identity functor, the thesis that the class of properties and relations (the class P) is included in the class of objects (the class O) can be expressed as follows: (a)

Q x (x = Q)

(inclusion of P in O)

Typical examples of other possible relations between extensions of two non-empty classes P and O are described by the following formulas: (b) (c) (d) (e)

Q x (x = Q) š x Q (x = Q) (equivalence) x Q (x = Q) (inclusion of O in P) x Q ¬(x = Q) š x Q (x = Q) š x Q ¬(x = Q) (intersection) x Q ¬(x = Q) (disjointness)

Some of these possibilities were in fact expressed in certain conceptions of logic; some other correspond to notions of object developed in philosophy. For example, in standard logic the disjointness of these two classes (e) is assumed. Type-free systems usually adopt variant (a). Inclusion of O in P (c) corresponds to the idea of Meinongian logic and of the so-called bundle conception of object (where objects are identified with appropriate classes or bundles of properties; O is included in P if and only if those bundles are conceived as complex properties). Variant (b) is adopted in Zalta’s system. It is difficult to indicate a system that would represent the relation of intersection of the classes P and O. This case seems, however, to be the most interesting, for it gives the richest partition of the considered universe: objects that are not properties (class A), objects that are properties (class B) and properties that are not objects (class C).

2

One example of relatively simple systems of second-order type-free logic is presented by T.N. Cocchiarella in “Whither Russell’s Paradox of Predication?” This system results from extending classical second-order logic with atomic formulas in which predicates can occupy the place of arguments (so such formulas as P 1 (R 2 ), R 2 (x, P 1 ), P 1 (P 1 ) are meaningful) and adding a scheme: xD o PD (x/P) (where the formula D (x/P) results from substituting the variable x with a predicate variable P).

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3. The main thesis of this paper says that class C is not empty. Thus, we allow only three of the five presented possibilities: inclusion (c), intersection (d), and disjointness (e). Variant (a), usually adopted in typefree logic, is rejected here. We shall now attempt to show that its acceptance gives rise to certain problems of a logical nature. We shall first define the Russellian property (being a property that does not belong to itself) and then indicate difficulties connected with taking it to be a value of an object and of a predicate variable at the same time. Let us consider the standard second-order language extended to formulas of the form: x = P and P = x. Let us call this language “L*.” Apart from the standard logical laws, we accept the following principle of extentionality for identity: (i)

x P (x = P o (D l D(x // P)))

where D(x // P) is a formula of language L*, which is the result of a legitimate substitution of the object variable x with the predicate variable P in D. Such substitution is legitimate when: (a) it does not lead beyond the set of formulas, i.e., its result is a correctly built formula; and (b) the predicate variable P is free wherever variable x is free in D. Let us also accept the standard comprehension principle for properties: (ii)

P x (Px l D)

where D is a formula in which variable x is free and variable P is not free. On the basis of (i) and (ii), we will show that there exists at least one property that is not an object. In other words, we will prove the following thesis: The set of all formulas of language L* that are instances of schemata (i) and (ii) is inconsistent on ground of standard logical laws and of the postulate: (iii)

P x (P = x)

In the proof, we will use a definition of the Russellian property R, which is an instance of the comprehension principle (ii): 3 3

Strictly speaking, the schema (ii) does not prove that R is one and only one: one needs also a suitable principle of property identification. This fact, however, plays no significant role at this moment since constant R is introduced here only for the purpose of the proof.

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(R) x (Rx l P (x = P š ¬Px)) (R is a property that belongs to all and only those objects which are properties that do not belong to themselves.) Taking the issue in a somewhat simplified way (which assumes the correctness of atomic formulas in which predicates take the place of logical subjects): if the property P were an object (i.e., the value of an object variable), then it would be allowed to substitute variable x with constant R by universal quantifier elimination in (R). This would produce the sentence: R(R) l P (R = P š ¬P(R)), from which, on the basis of the principle of extensionality for identity, follows the counter-tautological equivalence R(R) l ¬R(R). Thus, we obtain a particular form of Russell’s antinomy for properties (the paradox of predication). The detailed proof proceeds in the following way: (a) R=r (iii), assumption (b) Rr l Q (r = Q š ¬Qr) from (R) (c) Q ((r = Q š ¬Qr) o Rr) from (b) (d) (r = R š ¬Rr) o Rr from (c) (e) ¬Rr o Rr from (d), (a) (f) Rr from (e) (g) Q (r = Q š ¬Qr) from (f), (b) (h) R = Q0 š ¬Q0 r from (g), (a), (i) (i) ¬Rr from (h), (i) Contradiction.

4. Premises (i), (ii) and (iii) are essential in this argument. One could, therefore, avoid Russell’s antinomy either by rejecting or modifying one of them or by altering the standard logical principles. In similar forms of Russell’s antinomy, one usually restricts premise (ii) – the general comprehension principle for properties. Premise (i) is sometimes rejected and sometimes logic is radically changed (for example, if àukasiewicz’s or Kleene’s three-value logic is employed). However, as far as the author knows, the view that would reject premise (iii) is absent from the literature. The laws of classical logic and the principle of extensionality for identity (i) build the framework of our considerations. We cannot here evaluate the appropriateness of the framework’s choice. Let us, however, make some comments on typical conceptions that accept certain

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restrictions on the general form of the comprehension principle for properties and which could be then used for modifying schema (ii). Restrictions modelled upon Zermello’s or Neumann’s conceptions of set theory (which consist in substituting formula D in (ii) either with a conjunction of Qx š D, in a Zermellian manner or with a formula Q (Qx) š D, in a Neumannian manner)4 lead to some quite surprising consequences. Either the universal property of being an object does not exist (in the Neumannian sense) or there exist objects that have no properties. Moreover, both versions of the comprehension principle introduce the existence of relevant properties only conditionally and that requires the acceptance of further – as a matter of fact extra-logical – axioms that define which properties exist unconditionally.5 Another way of restricting premise (ii) consists in restricting the range of possible forms of formula D. Cocchiarella’s late suggestions, which restrict that range to formulas stratifiable in Quine’s sense, are among the best known.6 The soundness of this solution has been questioned,7 however, especially since it lacks a satisfactory ontological justification. This answer, moreover, does not give any basis for introducing non-paradoxical properties, for example “being a property exemplified by itself.”8 Those solutions which limit quantification over predicates in formula D are even more restrictive: in Menzel’s (1986) conception, one is not allowed to quantify over predicate variables that 4

G. Bealer in (1982, pp. 96-100) considers Zermellian and Neumannian variants of the comprehension principle for properties. The author of this paper also accepted a certain Neumannian restriction (expressed in a first-order modal theory of classes and properties) “Do Negative States of Affairs Exist?” (in Polish, 1998). 5 The analogy between this conception and set theory is sufficient to allow the former to be considered simply as a version of the latter properly interpreted. This has been done (in a first-order theory) by M. Jubien (1989). 6 Probably the earliest of those propositions was formulated in Cocchiarella’s (1975). His even-earlier attempt at a solution of Russell’s paradox for properties was expressed in system T* (see footnote 3, above). This attempt consists in rejecting the antinomial schema PQ(P(a) l D) (for any formula D of language T* which does not contain a free variable P and contains a free variable Q) with retaining the comprehension principle (ii). Since thesis (iii) is observed in system T*, many essential difficulties occur when the predicate of identity is added to this system. Those difficulties led Cocchiarella to reject this solution and to accept the idea of restricting the comprehension principle with a condition of the stratifiability of formulas. In our considerations we attempt another solution: retaining schema (ii) and rejecting thesis (iii). 7 See, for example, Bealer (1982, p. 14, postulate 14) and H.-N. Castañeda (1976, p. 73). 8 This objection is formulated by Castañeda (1976). Castañeda sees the essence of Russell’s paradox in an over-liberal use of negation. In a schema he proposes, no predicate can be placed within the scope of sentence-negation (although it can be placed within the scope of predicate-negation; see 1976, pp. 79-80).

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stand in the place of the subject9; in Zalta’s (1993) conception, one is not allowed to quantify over predicate variables in D at all. Two general objections against such modifications of schema (ii) can be formulated. First, those modifications eliminate the possibility of defining such properties as R, and this is rather counterintuitive. In those conceptions one assumes that R does not exist. This, however, goes against the intuition that most properties do not belong to themselves (being green, being an even number, etc.). An assumption that there exists such a property which belongs to these properties and only to them, is very natural if we accept any version of the comprehension principle for properties at all. It is then permitted to maintain that the Russellian property exists and is non-empty. Second, those modifications do not sufficiently protect against the possibility of rendering Russell’s antinomy. For in each type-free system one can introduce a predicate of exemplification E that fulfils the following condition: (a)

x (E(P, x) l Px)

Thus, the following postulate would be a special case of the schema of comprehension: (b)

P x (Px l ¬E(x, x))

On the basis of (a) this postulate is equivalent to: (c)

P x (E(P, x) l ¬E(x, x))

If a given system fulfils principle (iii), then on the basis of (c) we obtain: (d)

y x (E(y, x) l ¬E(x, x))

From this, a contradiction follows in a well-known way: (e)

E(y0 , y 0 ) l ¬E(y 0 , y 0 )

To sum up: the conclusion that takes postulate (iii) as the source of Russell’s antinomy for properties seems to dominate over alternative views.

9

This restriction follows from Menzel’s more general assumption that there exist only such complex properties that can be constructed from simpler ones by means of certain specified logical operations. Those operations, however, do not give a basis just for constructing the Russellian property but also for such properties as “being a non-empty property.” An attempt at a modification of Menzel’s system is contained in Orilia’s work (1991).

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5. A similar conclusion may also be arrived at in the case of multi-argument properties. In particular, one can render an antinomy for Russellian relations on the basis of two premises: P2 x (P2 = x) P2 x y (P2 xy l Q2 (x = Q2 š ¬Q2 xy)) Similarly, this can be done for multi-term relations. In each case, it leads to the conclusion that was formulated in the case of properties. Given these conclusions, we assume that Russellian properties and relations are not objects. Any property or relation that is not an object will be called a “quasi-object.” Thus, we divide the totality of entities (i.e. the totality of what exists in the most general sense of that term) into objects and quasi-objects. From the accepted definition of an object as the subject of properties, it follows that each quasi-object is a entity of a special kind: it possesses no properties. This consequence, although at first glance counterintuitive, yields a simple explanation of a certain logical asymmetry between standard and type-free (with predicates in the position of subject in propositions) formulas. This explanation consists in the fact that for the latter the schema of comprehension (analogical to (ii)) is not valid: (*)

P Q (P(Q) l D)

where P is not free in D while Q is free in D.10 The principle that properties belong solely to objects explains why certain instances of schema (*) must lead to contradiction: one attempts to define complex properties that are exemplified by beings which possess no properties. This explanation gives at the same time a general diagnosis concerning Russell’s antinomy for properties. In this solution, the analysis of typical examples of quasi-objects, i.e., properties and relations of the Russellian type, suggests that these are properties (relations), which either (1) possess “overly broad scopes” of application which, in turn, makes it impossible to predicate properties of those properties (in this case it is natural to assume that those broad scopes of applications are proper classes, i.e., classes which are not elements of any class) or (2) are “maximal” in one of two meanings of this term: (a) are properties (relations) of the highest order (in this sense 10 This schemata cannot be accepted under the threat of rendering an ordinary Russell’s paradox for properties: (a) Q (R 0 (Q) l ¬Q(Q)) [from (*)], (b) R 0 (R 0 ) l ¬R 0 (R 0 ) [from (a)], (c) contradiction [from (b)].

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similar to higher-order complete logic) or (b) are not elements of any class (similarly to proper classes).11 One should not assume, however, that the interpretation of quasiobjects as ontologically “naked” beings is the only possible explanation of the mentioned asymmetry. Another possible explanation is given by an interpretation of quasi-objects based on a semantic conception of an object. Within this conception, an object is anything that can be designated by an argument of any predicate in a given language. That is, it is anything of which any property can be predicated in that language. In accordance with the argumentation accepted here, one must then recognize that there are properties, such as the Russellian one, that are not objects in the object language of logic (they are then quasi-objects in the semantic sense). It follows that it is not the case that in a given language one can predicate any property of all properties (although any property can be meaningfully predicated of any object  even if that property does not belong to it). Whereas in the “naive” schema of comprehension (*), it is assumed that all properties generated by its substitution can be meaningfully predicated of all properties  contrary to the fact that some propositional forms (for example ¬Q(Q)) yield properties that do not fulfil this requirement.12

6. On the ontological conception of object assumed in this paper (in which all and only objects possess properties), no extension of the language of logic to formulas in which predicates are arguments of any predicates is needed. For the formula P(Q) can be interpreted here only in the following way: the property Q is an object that possesses the property P. This can in turn be defined by means of identity: (7)

P(Q) = df x (x = Q š Px)

11 A solution of the latter type was presented by the author (1998) within a framework of a certain elementary theory of classes and properties. 12 It seems that a proper understanding of the expression “to predicate meaningfully” is possible on the basis of a semantics in which properties are interpreted as propositional functions (in the sense coming from Frege’s thought, although not necessarily in a strictly Fregean sense, where the values of propositional functions are only zero and one), total or partial in the ontological universe. The expression “a property is meaningfully predicable of a given element of that universe” would then mean that a given propositional function is specified on that element.

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It is now easy to verify that postulate (3) from section 1 follows from (7). This can also be done for relations by, for example, accepting the following definition13: (8)

R(x, P) = df y (y = P 4 Rxy)14

It seems also that there are no special difficulties in using formulas of the type x = P (as well as P = Q). In order to demonstrate this fact, we will define a consistent calculus – a system of logic of objects, properties and relations in accordance with the conclusions formulated above – in which such formulas play an essential role. We will call the language of that system “L.” Its vocabulary accords with the vocabulary of ordinary logic and the predicate of identity. The full list of primitive symbols is as follows: (1) (2) (3) (4) (5) (6)

object variables: x1 , x 2 , x 3 ,  . . (we will also use: x, y, z, . . . ) predicate variables: P1 , Q1 , . . . ,P2 , Q2 , . . . , the identity symbol: = propositional connectives: ¬, š, ›, o, l, quantifiers: , , parentheses: (, ).

The elementary categories of expression in language L are object variables, predicate variables and propositional formulas. The concept of a formula is defined inductively:

13

These definitions have a certain explanatory power: they explain the only logically (in our sense of “logic”) acceptable sense in which a predicate can be used in the position of the subject. In particular, they explain the ambiguity of propositions of the form ¬P(Q): (a) ¬x = Q(Px) (there does not exist an object Q that possesses property P), and (b) x = Q(¬Px) (there exists an object Q that does not possess property P). They also explain the truth of some propositions which are alleged substitutions of negation of the law of excluded middle: ¬P(P(R) › ¬P(R)) (R is here the Russellian property). 14 Within a semantic conception of object (according to which one can meaningfully predicate any properties of all objects in, but only in, the object language) the mentioned extension of language would in turn introduce logically undetermined propositions (for example R(R)) which do not fall under the law of excluded middle. Let us also note that in this conception one does not exclude the possibility of using any predicates as arguments of the identity functor. For from the fact that predicates denoting quasi-objects cannot be arguments of any predicate, it does not follow that predicates which can take any predicates as arguments do not exist. On the contrary, it is reasonable to think that there is at least one such predicate, namely the symbol of identity. This would correspond to a semantic assumption that the propositional function of identity (which takes the value 1 when its arguments are identical and 0 in all other cases) is an element of any model (more precisely, of a class of propositional functions in any model).

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if t 1 , . . . , t n are object variables and P is an n-argument predicate variable for n t 1, then Pt 1 . . . t n is an (atomic) formula; if u, w are (object or predicate) variables then u = w is an (atomic) formula; if D, E are formulas and u is a variable, then ¬D, (D š E), (D › E), (D o E), (D l E), u D, u D are formulas. no other sequence of symbols is a formula.

Two expressions (of elementary categories) belong to the same category if and only if they are either object variables or n-argument predicate symbols, for n t 1, or formulas. As usual, language L is identified with the set of all formulas. We retain the convention that values of metavariables u, w are (object or predicate) variables. The schema D(u / w) signifies an expression obtained from formula D by substituting all free occurrences of variable u with variable w in D. This substitution is legitimate if it does not lead beyond the set of formulas (i.e., its result is a correctly built formula) and w is free in D(u / w) wherever variable u is free in D. The schema D(u // w) signifies any formula obtained from formula D by substituting all or some occurrences of variable u that are not bound variables in D with variable w. This substitution is legitimate if it does not lead beyond the set of formulas and w is free in D(u // w) wherever u is free in D.

7. The axiomatisation of the logic of objects, properties and relations is determined by the following schemata: (PC) D, if D is an instance of a thesis of classical propositional logic. (Q1) u D o D(u / w), if that substitution D(u / w) is legitimate and u, w belong to the same category. (Q2) u (D o E) o (D o uE), if variable u is not free in D (Q3) u D l ¬u ¬D (I1) u = u, where u is either an object variable or a predicate variable (I2) u = w o (D o D(u // w)), where D is any formula, the substitution D(u // w) is legitimate and u, w are variables of the same or different categories. (CP) Pn x 1 . . . x n (Pn x 1 . . . x n l D), for n t 1, where D is any propositional formula in which object variables x1 , . . . , x n are free and the predicate variable Pn is not free.

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The rules of inference are: modus ponens (MP) and the rule of generalization (Gen) (formulated jointly for first- and second-order quantification): (MP) D o E, D / E (Gen) D / u D Two things are characteristic for this axiomatisation: (1) a condition imposed on (Q1), according to which expressions in substitution must not belong to different categories, and (2) a generalization of the principle of extensionality for identity, expressed by (I2), where substitution expressions can belong to different categories. This generalization expresses a more liberal understanding than does a standard theory of identity. The constraint imposed on (Q1) is, in turn, given a more restrictive understanding than that given by type-free second-order systems. The system defined in this way fits in the middle between standard logic and type-free second-order logic. The constrain imposed on (Q1) impedes the possibility of generating antinomial properties and relations by means of schema (CP). Consistency of the formulated axiomatisation can be ascertained by interpreting it in a standard way in any non-empty model (the logical value of any equivalence of the form Pn = x and x = Pn is equal to 0). From (I2) and (Q1), two schemata characteristic of this logic follow: (8) (9) (10)

x (P = x) o (x D o D(x / P)), if the substitution D(x / P) is legitimate.15 P (x = P) o (P D o D(P / x)), if the substitution D(P / x) is legitimate. u (w = u) o (u D o D(u / w)), for all variables u, w (which belong to the same or to different categories), if the substitution D(u / w) is legitimate.

Russell’s antinomial reasoning may be used in an indirect proof of another schema characteristic of this logic: (11)

Pn ¬x (Pn = x)

under which thesis (5) formulated in section 1, among others, fails. Special cases of (CP) are the various theses that state the existence of properties and relations, for example: (12) 15

P x (Px l Q(x = Q š y Qy))

This is a formula analogical x (a = x) o (x D o D(x / a)).

to

a

characteristic

thesis

of

free

logic:

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(there exists a property of being an object which is a non-empty property). (13)

P x (Px l Q (Q = x š Qx))

(there exists a property of being an object exemplified by itself). (14)

R x y (Rxy l P (y = P š Px)

(there exists a relation of exemplifying properties, which are objects). 16 On the basis of the comprehension principle (CP) and of the law of identity (I1), it is possible to infer the condition of object identity: (15)

P (Px l Py) o x = y 17

from which by (I2), we obtain the Russell-Leibniz principle of identity and the “condition of individualization” of objects: (16) (17)

x = y l P (Px l Py) x z y l P (Px š ¬Py)

As usual, by using the law of identity for objects and the schemata (CP) one can prove the existence of non-empty universal properties (which belong to all and only objects) and of empty properties (which belong no objects): (18) (19)

P x (Px) P ¬x (Px)

By transposing the quantifiers in (18) we obtain postulate (1). Translated from Polish by Agnieszka Lekka-Kowalik

Uniwersytet im. M. Curie-Skáodowskiej Zakáad Logiki i Metodologii Nauk pl. Marii Curie-Skáodowskiej 4 20-031 Lublin, Poland email: [email protected] 16 It is worth stressing that these examples of existential theses are not “metaphysical” in the sense that they do not assume the non-emptiness of properties and relations whose existence they postulate. 17 The proof: 1) P (Px l Py) [the premise]; 2) P x (Px l x = y) [from (CP)], 3) P y x l x = y [from 2], 4) P y x l P y y [from 1], 5) P y y l y = y [from 2], 6) P y y [from 5, (I1)], 7) P y x [from 6, 4], 8) x = y [from 7, 3]. Compare with the proof given in Sáupecki and Borkowski (1963, pp. 117-118).

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REFERENCES Bealer, G. (1982). Quality and Concept. Oxford: Clarendon Press. Biáat, A. (1998). Czy istnieją negatywne stany rzeczy [Do Negative States of Affairs Exist]? In: Z. Kowalski, W. Krysztofiak and A. Biáat (eds.), Koncepcje negatywnych stanów rzeczy [Conceptions of Negative States of Affairs], pp. 105-130. Lublin: UMCS. Castañeda, H.-N. (1976) Ontology and Grammar: I. Russell’s Paradox and the General Theory of Properties in Natural Language. Theoria 42, 43-92. Chierchia, G., R. Turner (1988). Semantics and Property Theory. Linguistics and Philosophy 11, 261-302. Cocchiarella, N. (1973). Whither Russell’s Paradox of Predication? In: M. Munitz (ed.), Logic and Ontology, pp. 133-158. New York: New York University Press. Cocchiarella, N. (1975). Second-order Theories of Predication: Old and New Foundations. Noûs 9, 33-53. Feferman, S. (1984). Toward Useful Type-Free Theories: Part I. The Journal of Symbolic Logic 49, No. 1, 75-111. Jubien, M. (1989). On Properties and Property Theory. In: G. Chierchia, B. Partee and R. Turner (eds.), Properties, Types and Meanings, pp. 159-175. Dordrecht: Kluwer Academic Publisher. Menzel, C. (1986). A Complete, Type-Free, “Second Order” Logic and Its Philosophical Foundations. Center for the Study of Language and Information, Stanford University. Technical Report #CSLI-86-40. Orilia, F. (1991). Type-Free Property Theory, Exemplification and Russell’s Paradox. Notre Dame Journal of Formal Logic 32, No. 3, 432-447. Sáupecki, J. and L. Borkowski (1963). Elementy logiki matematycznej i teorii mnogoĞci [Elements of Mathematical Logic and Set Theory]. Warszawa: PWN. Zalta, E. (1993). Twenty-Five Basic Theorems in Situation and World Theory. Journal of Philosophical Logic 22, 385-428.

Joanna OdrowąĪ-Sypniewska

ON THE NOTION OF IDENTITY 1

1. Introduction The notion of identity has been a source of much controversy for a long time. Its origin and ontological status remain unclear. It is difficult to imagine, it has been observed, that the notion of identity comes from the perception of a single object (this perception gives us the notion of unity); but it is also hard to conclude that the perception of a few objects is its source (this perception gives us the notion of multitude). If, in the sentence ‘That object is the same as itself’, the words ‘object’ and ‘itself ’ denoted the same notion then this proposition would be a tautology (Hume 1898, p. 489). “Once one object is supposed to exist, we must either suppose another to exist also; in which case we have the idea of number [i.e. multitude]: or we must suppose it not to exist; in which case the first object remains the same” (Hume 1898, p. 489). Perhaps, then, we should refer to the notion of time. When, at different times, we perceive the same object, which lasts through time and is unchangeable, we acquire first the notion of multitude (because we perceive this object at different moments) and then the notion of unity (because we always perceive the same object). These notions are inconsistent, and from this inconsistency arises the notion of identity, which is “[a] medium betwixt unity and number [i.e. multitude]; or more properly speaking, is either of them, according to the view, in which we take it” (Hume 1898, p. 490). However, according to this interpretation of the origin of identity, we can speak only about the identity of an object that lasts in time. Hence, we can speak, for example, about the identity of a certain insect’s chrysalis and the imago, which has emerged from it; about the identity of 1

This paper is a version of my M.A. dissertation (Warsaw University, 1994).

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 143-167. Amsterdam/New York, NY: Rodopi, 2006.

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a tree which is in bloom in spring but is totally leafless in winter; and about the identity of law student John Brown and the 10-year older attorney whom this student has become. But from where does the identity between several objects or between a certain object and itself at the same time arise? And, above all, doesn’t identity become in a way fictitious in this formulation? It seems that object x at time tƍ and object x at time tƍƍ are simply not identical  after all their properties have changed (a chrysalis becomes an imago, a tree ceases blossoming and becomes leafless, a human is aging). Equally enigmatical and disputable is the ontological status of identity: is identity a property or a relation? Does it hold between objects or between the names of objects? The dilemma of whether an object can be identical only with itself (Descartes, Hume, Kant) or also with other objects is still unresolved. Some philosophers (e.g. Bradley) believed that identity is a property belonging to every object, others (e.g. Frege, Russell) supposed that it is a relation. In the latter case identity can be formulated either as a relation that holds between objects (e.g. Hume, Leibniz) or as a relation that holds between the names of objects (e.g. Leibniz, Frege). There are also philosophers (e.g. Wittgenstein, Savery) who thought that the notion of identity is completely useless in any description of the world. According to them, talking about identity is a hypostasis and “the identity-sign is not an essential constituent of conceptual notation” (Wittgenstein 1963, 5.533). I believe that they are wrong. The importance of the notion of identity consists in the fact that solutions to many philosophical problems depend on us having a way to understand that notion (see Stroll 1967). An explanation of the notions of numerical and qualitative identity is relevant to the solution to questions connected with notions of substance and change as well as the notion of universals. When we say that a certain object is numerically identical with itself; that  in spite of lasting in time and undergoing changes  it preserves its own identity, we touch upon the problem of continuity and admissible changes of objects. Identity as persistence and relative unchangeability, calls for a qualification of what has to remain invariable in an object that undergoes changes to remain the same object. On the other hand, qualitative identity is connected with the problem of universals and the principle of individuation. When we ask whether two red patches that are seen simultaneously and that we are not able to tell apart, represent the same colour, we are asking whether that which appears distinct (here: distinct patches) is in reality one and the same (here: one colour) (see Stroll 1967). On the other hand, the qualitative

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identity of the two objects provokes the question about what convinces us that they are two objects; what is the principle of individuation in this case. A lot of misunderstanding has grown around the principle of the identity of indiscernibles. According to that principle, objects that are not numerically identical cannot be qualitatively identical. The existence of two identical (i.e. having all the properties in common) drops of water would contradict this principle. In my paper I am going to present and analyse different senses of identity. I will try to clarify the terminology relevant to the notion of identity, explicate different types of identity, point out some of their properties and examine the relations between the particular types. Next, I will look briefly at the relationship between identity and other relations such as kinship, similarity, equivalence, sameness and difference. Finally, I will consider the criterion that would make the identification of identity possible. Identity as a relation between expressions, i.e. identity of meaning, has been elaborated on many occasions (e.g. Pelc 1986) and there is no place here to present these analyses.

2. Types of Identity 2.1. Numerical (Concrete, Extensional, Existential) Identity Numerical identity is a relation between every object and itself. It is an equivalence relation (i.e. is reflexive, symmetrical and transitive). It is a class of all pairs consisting of one object taken twice. It concerns not only such pairs as (Socrates; Socrates), (Cicero; Cicero) but also (Cicero; Tully), (the Morning Star; the Evening Star). Numerical identity can be understood in two ways. 2.1.1. Strict Numerical Identity First, numerical identity can be understood as a relation between that which is in fact one object 2 (although it is named, comprehended and perceived in different ways). «Two» objects, x and y, are strictly numerically identical when «they» have the same spatiotemporal characteristics (e.g. Lake Léman and Lake Geneva; the Morning Star and the Evening Star). 2

Here ‘object’ means ‘thing’, i.e., that which exists in time and space and is physically defined, for instance, physically influences something else (KotarbiĔski 1966, p. 426).

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xy [x is numerically identical with y { tm (x is at t at m { y is at t at m)], where variables ‘x’ and ‘y’ range over existing spatiotemporal objects, variable ‘t’ ranges over moments, and variable ‘m’ ranges over places. The expression ‘two objects, x and y, are numerically identical’ should not, however, be understood literally. It should not mean that two distinct objects are at the same place at the same time. It is only an abbreviation of the following situation. We know about the existence of two objects, x and y  e.g. the Morning Star and the Evening Star  having specified properties. Only some of these properties they have in common. For example the Morning Star shines only in the morning and is brightest on the horizon, while the Evening Star is the brightest «star» on the horizon but it shines only in the evening. In this case these two objects x and y (here: the Morning Star and the Evening Star) are in fact one object z (here: planet Venus), because de facto they have the same spatiotemporal characteristics. The properties of objects x and y are mere aspects under which we perceive a single object at different times. And only for epistemological reasons can one talk about two objects being strictly numerically identical. 2.1.2. Genetic Numerical Identity (Genidentity) Secondly, numerical identity can be understood as a relation between a certain object, z, at a specified time, tƍ, and the same object z at a different time, tƍƍ. We say that object z preserves its identity when, after a considerable lapse of time or after a transformation (i.e. after a change of properties), it stays the same object as it was before that period of time or transformation. Were particulars not capable of intrinsic changes, it would be easy to ascertain whether a certain particular still persists after a period of time. In order to check whether the particular a at a time t 1 and the particular b at some later time t 2 are in fact one persisting particular, it would suffice to check whether there is a spatiotemporal continuity between a and b. However, since particulars are subject to change, the continuity of spatiotemporal trajectory between a and b is a necessary but not a sufficient condition for a and b being the same persisting particular. Assume that first a statue is made from clay, and next, the statue is destroyed and a vase is made instead. Clearly there is a spatiotemporal continuity between the statue and the vase, but we do not normally want to say that the statue is the same particular as the vase. We say that a particular z preserves its identity if it preserves its continuity and constancy through changes of its own states and those states of the environment that condition its existence. The fact that z is

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identical with itself does not mean that either its substance or all of its properties remain the same (e.g. during its life a living body changes the majority of its cells and changes most of its properties). The notion of essence of an object is notoriously hard to define. For present purposes, let ‘essence of an object’ denote the set of permanent properties, marking out the totality of a given object’s specific properties. By ‘permanent properties’ we mean not only those properties without which an object would not be what it is, but also those properties that belong to every object which belongs to a certain kind for the whole time of its existence (KotarbiĔski 1966, p. 414); by ‘specific properties’ we mean those properties that are common to all and only objects which belong to a certain kind. Only properties that do not belong to the essence of an object can undergo changes if identity is to be preserved. Let ‘t-time cross-section of an object x’ denote object x at time t. We call the relation between time sections of the same object ‘genetic identity’ (‘genidentity’). Objects x and y, which are different time sections of object z (x is object z at time tƍ, y is object z at time tƍƍ), are genidentical when they have the same essence. Consequently one can say that object z preserves its genidentity through period tƍ-tƍƍ when time section x of object z from time tƍ is genidentical with time section y of object z from time tƍƍ. The realization that time-sections of a persisting object almost always differ in some of their properties has led some philosophers to claim that the relation that occurs between such time-sections is not an identity relation at all. On this view our everyday objects are in fact aggregates of temporal parts. At each time at which, for instance, a vase exists there is a temporal part of it present. And it is such parts that really have properties; particulars have temporary properties only derivatively. Different temporal parts of a given particular are related in many ways, but are never identical (see below remarks concerning the relation of kinship). Thus, according to the present view genetic numerical identity is a notion that no objects satisfy. 3

3

Such a view is called “perdurantism.” See e.g. Lewis ([1976] 1983). Perdurantists argue that all those theorists who think that identity can hold between objects existing at different times have to find a solution to the problem of intrinsic change  i.e. they have to explain how objects having incompatible properties (e.g. a banana that yesterday was green all over and the same banana which is yellow all over today) can nevertheless be identical.

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2.2. Qualitative (Abstract, Intensional, Specific) Identity Qualitative identity is a relation of one or more objects. Objects that are strictly numerically identical are also qualitatively identical. The genetically numerically identical sections of object z are qualitatively identical only if object z is unchangeable. Two kinds of qualitative identity, namely absolute identity and relative identity, are usually distinguished. N.B. Both kinds are equivalence relations. 2.2.1. Absolute Qualitative Identity Two objects x and y are absolutely qualitatively identical if they share all their properties; i.e. if all the properties that belong to x belong also to y, and all the properties that belong to y belong to x as well. So, if x and y are identical in this sense, any property that can be truthfully ascribed to x can also be truthfully ascribed to y and conversely. 4 x y {x is absolutely F [F(x) { F(y)]}

qualitatively

identical

with

y

{

According to this definition of identity, object x (at time t and place m) is above all identical with itself (at time t and place m). Being identical with some other object depends on what is understood as properties. Properties can be understood as containing both intrinsic (generally these are constitutive and consequent features) and extrinsic properties (in most cases not belonging to the essence of an object: relational features and spatiotemporal location). In this sense, according to Leibniz’s principle of the identity of indiscernibles, «two» objects are qualitatively identical only when they are strictly numerically identical. Two objects with different spatiotemporal locations cannot be identical: they differ in certain relative properties as well as in spatiotemporal characteristics. The principle of the identity of indiscernibles can be represented symbolically as follows: 5 xy{F [F(x) { F(y)] o x = y}.

4

Sometimes absolute identity is defined in terms of predicates rather than properties. For instance, Noonan has argued that if there are predicates to which no property corresponds, any definition in terms of properties would result in “a kind of relative identity: a relation which ensures indiscernibility of its terms in some, but not all respects [. . .]” (Noonan 1984, p. 118). 5 There are also interpretations according to which the identity of indiscernibles is a stronger principle in which the conditional is replaced with a bicondotional.

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All the properties connected with identity should be removed (properties like being identical with x; of being different from x (i.e. non-identical with x), or of being identical with x and at the same time being green) from the range of variable ‘F’. If we substitute, for instance, the property of being identical with x for ‘F’, the implication will be a trivial truth (either its antecedent will be false – when y is not x – or it will concern just one object x). It is not clear whether the principle of the identity of indiscernibles was understood by Leibniz as an empirical or a logical principle. The principle of sufficient reason entails  according to Leibniz  that “there are not in nature two real, absolute beings, indiscernible from each other” (Leibniz 1890, Fifth Paper, §21). There cannot exist “in nature” two things differing only in number because there should be a reason for things to be different; and that reason can be deduced only from a difference between those things. The principle of individuation resolves itself into the principle of distinction. “If two individuals were perfectly similar [i.e. absolutely identical] and equal and [. . .] indistinguishable in themselves, there would be no principle of individuation [. . .], no separate individuals” (Leibniz 1981, §230). The fact that Leibniz constantly referred to experience favours the interpretation of the principle of the identity of indiscernibles as an empirical principle. In his opinion, “in sensible things, two, that are indiscernible from each other, can never be found” (Leibniz 1890, Fifth Paper, §23). Leaves or drops of water, which are at first glance identical, will appear completely different on a more thorough examination. If two objects are not numerically identical, it is always possible to find such a property of one of them that does not belong to the other. On the other hand, the fact that Leibniz denied the existence of time and space «outside» of things favours the «stronger» interpretation of the principle of the identity of indiscernibles as a logical principle. Time and space are, according to him, merely the order of, respectively, the succession and the coexistence of things. “I hold space to be something merely relative, as time is; I hold it to be an order of coexistences, as time is an order of succession” (Leibniz 1890, Third Paper, §4). Extrinsic properties are not separable from intrinsic ones. Objects cannot differ solo numero because objects that have different spatiotemporal locations, have different properties. It is, therefore, impossible that those objects are qualitatively identical. Doubts concerning the nature of the principle of identity result from the fact that Leibniz sometimes wrote that existence of two indiscernible objects would be “contrary to divine wisdom” but not “absolutely

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impossible” (Leibniz 1890, Fifth Paper, §25) and sometimes that it would contradict “the greatest principles of reason” (Leibniz 1981, §231). It seems that when something is inconsistent with the principles of reason then it is logically impossible as well, whereas if it is not “absolutely impossible” then it is not logically but only technically or physically impossible. In our world, two different objects that are qualitatively identical in an absolute sense cannot exist. It seems, however, that a universe where those objects would exist is possible. One can imagine a spherical universe composed of nothing but two identical spheres, each sphere being equally distant from the other and from the borders of the universe (Black 1952, p. 156). “We might suppose that each was made of chemically pure iron, had a diameter of one mile, that they had the same temperature, colour and so on” (Black 1952, p. 156). Every feature, every relative property that belongs to one of the spheres, belongs to the other as well. About each of them we can say, for instance, that it “is at a distance of x miles from the other” or that it “is at a distance of y miles from the borders of the universe.” We cannot select one of them and from that chosen one start describing that universe. If one of the spheres were chosen and named, it would get new properties which the other does not have (e.g. property of “being on the left of the observer,” “being called ‘A’,” “being at a distance of x miles from the sphere which is not designated by ‘A’ ” etc.). The spheres would cease to be identical. We could not name one of them because it would require putting an observer into this universe, and this would cause a change in the whole universe, a destruction of the symmetry. It is assumed a priori that the spheres are identical and two. And, in this assumption surely no inconsistencies are hidden. It seems, however, that this universe differs from our world not only in the small number of individuals but in its description of space and time location as well. We are able to place the centre of symmetry in this two-sphere universe but we have no coordinate system there (if there were one, the spheres would not be identical, they would differ in their relative properties). We cannot, therefore, point out any place. The only way of distinguishing places would be to show that these places are occupied by different objects. But that cannot be done. The predicate ‘being at place m’ has to be replaced by a three-place predicate ‘being at a distance of . . . miles from . . . ’. ‘The sphere is located’ means exactly ‘the sphere is . . . miles away from the other sphere’. In our world, the assumption of the existence of two spheres entails the consequence that these spheres have different space locations (one is at a place m, the other at a different

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place, n). In the two-sphere universe, there is no such entailment (we do not know how to distinguish these places from each other). That assumption of the existence of two spheres in that «poor» universe entails only that they are distant from each other. 2.2.2. Relative Identity 2.2.2.1. Indiscernibility in a Language (or Theory) The thesis of the relativity of qualitative identity seems to be connected with a change in the way of talking about identity. Till now we have been talking about identity as a relation between objects. Proponents of the relative-identity thesis talk not about a relation but about a predicate. “Identity” becomes a predicate which can be predicated about objects. And so, it is obvious that it is relativised to a language whose predicate it is. The meaning of the identity predicate is that “two objects are indiscernible by the predicates that form the descriptive resources of the theory” (Geach 1967, p. 5). Predicate I is an identity predicate when, if predicate I is true of x and of y in language L, then every predicate which belongs to L is true of x iff it is true of y. So the two objects, x and y, are identical in this sense when any of language L’s predicates truthfully predicated of x can be truthfully predicated of y, and conversely. We are not talking here about “whatever can be truthfully predicated of x” but about predicates of language L, which can be so predicated. Relative identity in this sense boils down to indiscernibility within a certain language. Identity so formulated cannot be absolute, because it would lead to nonsense. Objects which are indiscernible within theory T ƍ can be discernible in theory T ƍƍ (which contains theory T ƍ). Thus, a predicate which is an identity predicate in theory T ƍ can cease to be such a predicate in theory T ƍƍ. One could argue that although an identity predicate comes from a language and is therefore relative to it, it does in fact express absolute identity. It is not so, however. Theory T can be, for example, a theory of expressions of a given natural language. Concrete word tokens belong to the range of the quantifiers of theory T. Assume that the predicates of theory T do not allow us to distinguish two word tokens of the same word type. So the predicate “being equiform with” (F, for short) is an identity predicate in T. If we now add to theory T even one such predicate that allows us to distinguish between two word tokens of the same shape, predicate F will cease to be an identity predicate. Doing this we did not change predicate F’s meaning. Thus, we can see that what it has

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expressed from the beginning was not an absolute identity but only an indiscernibility relation relative to theory T and its predicates. If we agree with Quine’s interpretation of existence as being a value of a bounded variable, we will be able to see clearly that identifying indiscernibility in a given language with absolute identity leads to absurdity. One can agree that a theory of identity is connected with a theory of quantifiers; one can, for instance, modify a calculus of quantifiers so as to eliminate the identity sign (see Wittgenstein 1963). The range of a limited quantifier should be understood as a class of objects about which the identity predicate of a given theory, T, can be predicated. Let’s assume, for example, that theory T which is ontologically committed to the existence of people has as a part a subtheory T ƍ, in which there are no predicates that would allow a distinction between people with the same first name (see Geach 1973, p. 301). In that subtheory there are no such predicates as ‘is blond’, ‘is tall’, ‘is intelligent’ (each of these predicates would allow us to distinguish one John from another). Predicate ‘is the same surman as’ is an identity predicate in subtheory T ƍ. The range of the limited quantifier ‘every surman’ is a set of all surmen. If we agree that an identity predicate expresses absolute identity, theory T ƍ (and so theory T of which T ƍ is a part) would be committed to the existence of «an absolute surman». Theory T cannot have this ontological commitment without becoming absurd, however. «Absolute surman» would have some of the properties belonging to a man e.g. having a liver, being a mammal; he would also have the property of being called ‘John’, but nobody would know which one of the Johns he is. That relative identity leads to a relativisation of ontology. “We can never so specify what we are quantifying over that we are secure against an expansion of our vocabulary enabling us to discriminate what formerly we could not” (Geach 1973, p. 301), i.e. against a division of the quantifer’s range. 2.2.2.2. Sortal Identity As Frege has pointed out, there is no use in counting objects, if one does not know to which kind they belong (see Geach 1962). If someone said “Count all the red objects which are in that room,” for instance, nobody would know what he should consider as an object. In particular he would not know if he ought to count a table as a whole or its legs, top and drawer separately. Some theorists argue that identity has to be relativised to a kind as well. According to them, it is not enough to say “a and b are the same.” The logical value of that sentence is undetermined, for one

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might ask: “The same what?” That sentence requires a supplementary general name. After the relevant name is added the sentence “a and b are the same river” may turn out to be true and the sentence “a and b are the same water”  to be false.6 The expression ‘x is identical with y’ should be replaced by the expression ‘x is the same A as y’, where ‘A’ is a variable ranging over general names. On that score, identity resembles other relative relations. One cannot say, for instance, that x is better than y, without specifying in what respect it is better. Let x and y be people. One could say, for example, that x is a better football player than y, but y is a better sprinter than x. Ordered pair ¢x; y² can satisfy the relative relation “being better with respect to A” and not satisfy relation “being better with respect to B.” The situation with identity is similar. Object x can be the same A as y but not be the same B as y (even if x and y are Bs). According to this, the relation “being the same water” is not a conjunction of the relations “being the same” and “being water.” There is not one absolute relation of identity but a lot of relative relations; relations of being the same A. The expression ‘is identical with’ denotes different identity relations according to contexts. a is the same word type as b but a and b are not the same word token. The river on which my boat moors today is the same river on which my boat moored yesterday, but not the same water. Dr Jekyll and Mr Hyde were the same man, but not the same personality. John Brown is the same firman as John Smith (i.e. they both have the same first name), but they are not the same man. One can, however, call into question this description of a problem (see Quine 1964). First, from the fact that it is impossible to count red objects situated in a room, it does not follow that it is also impossible to tell that x is the same object as y (without qualifying what x and y are, only pointing them out or giving their descriptions). Besides, every given example raises doubts. If in the whole sentence ‘a is the same word type as b, but a and b are not the same word token’ ‘a’ and ‘b’ refer to word tokens, then the first half of the sentence (i.e. ‘a is the same word type as b’) is not an identity statement but a sentence saying that a and b are similar in some respect: they have the property of being the same word type (i.e. being equiform) (see Perry 1970). This part of the sentence can also be formulated as ‘a and b are word tokens of the same type’. So, for both parts to be identical statements, ‘a’ and ‘b’ should refer to the word type in the first part and to the word token in the second.

6

The example is due to Geach (1962).

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Next, rivers are not only waters. Water constitutes a river but one should not equate them. The river does not cease to exist if its bed has dried out for a while. And the identity of river C is not broken in spite of the fact that at time tƍ different water flows in it than at time tƍƍ. The example of Dr Jekyll and Mr Hyde is similar to that of the Morning Star and the Evening Star. Dr Jekyll and Mr Hyde are the same man. They are numerically identical. ‘Dr Jekyll’ and ‘Mr Hyde’ are names of two personalities (aspects) of one individual. Between John Brown and John Smith there is no identity at all despite both having the same first name. We are asked by the proponents of the relative-identity to believe that there is no such thing as absolute identity. This is a rather counterintuitive view, however. The prevailing opinion has it that there is an absolute identity relation which is different from many relative identity relations that may hold between both absolutely identical and non-absolutely-identical objects. The possibility of x being the same A as y yet not being the same B as y depends on our interpretation of ‘A’ and ‘B’. There is an opinion according to which identity, in spite of being always the identity of an object belonging to a certain kind, entails (absolute) indiscernibility. So understood, identity is an absolute equivalence relation; i.e., if it holds between x and y then there is no equivalence relation that holds between one of x or y and some z and not between x and y. So, if x is the same A as y then it cannot be other B. 2.2.2.3. Identity in Some Respect (Sameness) Sameness is a specific, «weaker» kind of identity – identity in some respect. It allows building equivalence classes. When one identifies objects that belong to a particular class one ignores properties that differentiate these objects. An equivalence class in a given set, A, is fixed by an equivalence relation and a given object x. Let R be an equivalence relation with respect to colour (i.e. a relation of having the same colour) and A  a set of physical objects. So classes of equivalence R in A are: a set of red objects (i.e. identical to x with respect to colour), a set of green objects (identical to y with respect to colour) and so forth. The set of all these classes is a logical division of set A with respect to colour. Equivalence classes (like classes of sets of the same cardinality, classes of similar geometrical figures, classes of objects with the same weight, etc.) divide sets of objects into mutually disjoint subsets of elements that are the same in some respect. Of course, elements that belong to a given

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equivalence class are neither indiscernible nor identical sortaly but only the same with respect to a given property.

3. Identity Statements There is a problem with writing down identity understood as a relation between objects. The sentence ‘a = a’ is an obvious tautology (devoid of informative value). Its subject- and predicate-terms are equiform. The sentences ‘a = aB’ (e.g. ‘That book is the book Paul gave me on my birthday’) and ‘a = b’ (e.g. ‘That rag is the towel I bought last year’) are not tautologies. They are supposed to express an “identity of denotation in difference of intension” (Jones 1908, p. 387); i.e., genetic numerical identity without absolute qualitative identity. ‘b’ in a predicate symbolizes a change of a’s intrinsic properties. The sentence ‘a = b’ allows, however, such substitutions as ‘A pot is identical with the clay [from which it was formed]’ or ‘A tree is identical with paper [which it was processed for]’, although neither for the pot and the clay, nor for the tree and the paper does numerical identity hold in any sense. In the sentence ‘a = aB’ the relation between a and b is not clear – ‘aB’ in the predicate symbolizes a change of a’s extrinsic (not intrinsic) properties. However, if the book Paul gave me now has dog-eared corners, its intrinsic qualities have changed as well. Moreover, in this form that sentence can only be used ostensively: ‘This book is identical with . . . etc’. It should be replaced by the sentence ‘aC = aB’. The difference between sentences of type ‘a = b’, which have a cognitive value (such as Frege’s well-known example ‘the Morning Star is identical with the Evening Star’), and sentences ‘a = a’, which do not have any cognitive value, can be explained  in Frege’s opinion  only when one assumes that identity is a relation between names of objects and not between the objects themselves. If we treat identity as a relation between objects then the sentences ‘a = b’ and ‘a = a’ would not differ with regard to cognitive value, assuming that the sentence ‘a = b’ is true. However, the expression ‘the Morning Star is the Evening Star’ is a synthetic sentence, a valuable enrichment of our knowledge; on the contrary, the sentence ‘the Evening Star is the Evening Star’ is a simple truism. It is the assumption that identity is a relation between names of objects and the introduction of a distinction between the denotation and connotation of a name that makes it possible to show the difference between those statements.

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If we agree that proper names have connotations, we will see that every proper name can be replaced by a composite expression (a definite description). Let us notice that a proper name used as a proper name (i.e. to point directly to what we are talking about) can play only the role of a subject; it is a separate expression (i.e. it has its own meaning) and a logically simple symbol which directly designates an object (see Russell 1952). When we use a proper name as a proper name it is only a symbol through which we express our thought; it does not belong to what we are claiming. However, quite often we use proper names as descriptions, i.e. compound symbols that consist of words which have specific meanings. The name ‘Aristotle’ can be used, e.g., as a short abbreviation for the definite description ‘the teacher of Alexander the Great, born in Stagira’. In identity statements, in which there are two proper names, at least one of them is de facto used predicatively. It is an abbreviation of a description. The sentence ‘Scott is Sir Walter’ in which both ‘Scott’ and ‘Sir Walter’ are used as proper names is a tautology (in Polish, sentences like this cannot even be built). However, if we treat proper names as descriptions we will get a sentence like ‘Scott is called ‘Sir Walter’’ or ‘The man called ‘Scott’ is the same person as the man called ‘Sir Walter’’. In this way proper names merge into descriptions. Thus, every identity statement, if it is to have a cognitive value, should contain at least one description. Expressions like ‘a = b’ or ‘a = a’, where ‘a’ and ‘b’ are proper names, are useless tautologies. Valuable statements that carry some information have the form ‘a = (Lx)(F(x))’ (e.g. ‘Scott is the author of Waverley’) or ‘(Lx)(F(x)) = (Ly)(G(y))’ (e.g. ‘The author of Waverley is identical with the author of Marmion’). If we assume that identity is a relation between language expressions then we get the following substitution of the principle of the indiscernibility of identicals, which is the converse of the principle of the identity of indiscernibles (see Williams 1989, p. 91): (i) (ii) (iii)

for names: a = b o F [F(a) { F(b)], for a description and a name: a = (Lx)(F(x)) o F [F(a) { F(Lx)(F(x)], for two descriptions: (Lx)(F(x)) = (Ly)(G(y)) o h [h(Lx)(F(x)) { h(Ly)(G(y))].

The antecedent of conditional (i) will be true only when it means the same as ‘a = a’, and thus the consequent means F [F(a) { F(a)], which results in the whole implication being a truism. However, substitutions

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(ii) and (iii) are not trivial. They are valid only in extensional contexts. Applying them to an intensional context may lead to an absurdity. From the fact that Warsaw is John Brown’s birthplace and that Peter thinks that Warsaw is the capital of Poland, it would follow immediately that Peter thinks that Warsaw is John Brown’s birthplace. Whereas, obviously, Peter might be unaware even of John Brown’s existence. Concerning intensional contexts, let us notice that one can try to eliminate them from language by a proper interpretation. Expressions like ‘he thinks that’, ‘he believes that’, ‘it is necessary that’ should not be treated as being simply attached to a sentence. On this view, relevant contexts do not have the form ‘x thinks that p’, where ‘p’ is a variable ranging over sentences. Intensional functors break the structure of relative clauses into pieces. The sentence ‘Peter thinks that Warsaw is the capital of Poland’ does not have the form bƍpa (Peter thinks that p) but the form bƍFcda (where bƍ  think that; a  Peter; F  is a capital of ; d  Poland; c  Warsaw). The sentence ‘Warsaw is the capital of Poland’ does not form a syntactical whole just as ‘Cicero was bald’ is not a syntactical part of the sentence ‘The man who was taught by Cicero was bald’ (see Ajdukiewicz 1967 and Williams 1989).

4. Properties of the Identity Relation 4.1. Identity as a Relation There is an opinion, according to which identity is not a relation sensu stricto. Proponents of this view believe that the statement ‘a is identical with b’ does not entail that there is a relation between that which ‘a’ symbolizes and that which ‘b’ symbolizes. ‘a’ and ‘b’ are symbols of the same object, while for a relation there must be at least two objects. So to use the term ‘is identical with’ is to point out that two names are being used to denote one object or to say only that the object is what it is, i.e., to say nothing. However, the assumption that for a relation there should be two objects seems to be groundless. There are after all a lot of relations which are reflexive, i.e. which hold between one object and itself. Moreover, most theorists seems to agree that indiscernibility is a relation but there is no common opinion whether it holds of any distinct objects. It is said sometimes that identity does not hold between two objects, either (see Savery 1942). When stating that two red patches are identical we have in mind only that they are absolutely qualitatively identical and numerically different. If we imagined them apart from a spatiotemporal

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location we would picture one red patch and not two identical ones. So identity is only a handy abbreviation for naming the situation in which nothing except spatiotemporal characteristics make them different. They would be one object, if we abstracted from how they are placed. But this opinion cannot be held either. We would like to have a description of the world as it is, not as it would be, if we could abstract from spatiotemporal location of objects that belong to it. There are qualitatively identical objects that occupy different places and so we should describe them. When we notice that there is nothing that makes two patches different we state that the identity relation holds. It is because we observe that an identity relation holds that we believe that if there were no difference in location then these patches would be one patch. It is said sometimes, however, that identity is not a relation between objects: “to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all” (Wittgenstein 1963, 5.5303). This statement hinges on a few assumptions. First, it applies only to strict numerical identity and absolute identity, where both intrinsic and extrinsic properties are taken into account. With these two interpretations, it really is impossible to say that two things are identical with each other. Two objects about which we predicate strict numerical identity are, in fact, one thing. Also in the case in which it is assumed that only objects that do not differ in spatiotemporal location can be absolutely identical, when we deal with identical objects, we have in fact to do with one object only. However, saying that one thing z, preserves its genetic numerical identity is not “saying nothing at all.” When claiming this identity, we argue that object z preserves its continuity and essence. Similarly, saying that two objects are relatively identical – indiscernible or the same in a certain respect – is not, of course, “nonsense.” 4.2. Identity as an All-or-Nothing Notion It seems that the identity relation  in any of the formulations under consideration – does not come in degrees. There are no degrees of numerical identity. A stove cannot be more or less identical with the cooker, and neither can the Morning Star with the Evening Star, since one object cannot have a more or less identical location with itself. Neither can law student John Brown be more or less identical with attorney John Brown, nor the chrysalis with the imago:

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either they are stages of one object and have the same essence, or they are not. They cannot have «almost the same» essence. 7 Absolute identity also does not come in degrees. The fact that objects are identical when whatever is true of one of them is also true of the other, and conversely, does not mean that the more properties they share, the more identical they are. Likewise, objects cannot be more or less indiscernible in a given language. Either two namesakes are identical (indiscernible) or they can be  at least theoretically  distinguished. The fact that x has more features that differentiate it from y than features that differentiate it from z, does not mean that x and z are «more discernible» than x and y. Similarly, one cannot be more or less the same in some respect. Object x can be the same as y in many respects (e.g. the same in size, weight, volume etc.) and the same as z in only one respect (e.g. being equiform). But this does not mean that identity admits of degrees. Objects x and y are not «more the same» than x and z. There is also a view according to which indiscernibility can have degrees. Objects that stay indiscernible in a language’s successive extensions are «more indiscernible». According to that opinion, two blue mugs are «more indiscernible» than, for example, a blue mug and a little blue flowerpot because the indiscernibility of the former pair survives in an extended language with new predicates (e.g. ‘being equiform’, ‘having the same weight’, etc.). However, indiscernibility is clearly relative to a given language and in this language, in my opinion, it does not admit of degrees. It is of course true that the more similar the objects are, the more difficult it is to tell them apart, but it does not mean that there are degrees of indiscernibility. Either the resources of a given language allow two objects to be discerned, or they do not – either these two objects are discernible, or they are not.

5. Relations between Types of Identity There are a number of relationships between the different types of identity and between identity relations of different types and other relations. Some of them have already been mentioned. Below we will try to present them in a more complete and orderly manner.

7

It may be vague, however, whether they have the same essence or not.

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Every object is strictly numerically identical with itself. Objects numerically identical in that sense are also absolutely qualitatively identical. If x and y have the same spatiotemporal location, then each property of x is also a property of y, and conversely. If we substitute ‘Lake Geneva’ or ‘Tully’ for ‘x’ and ‘Lake Léman’ or ‘Cicero’ for ‘y’, respectively, the fact that they are absolutely identical will seem obvious. But if we substitute ‘the Morning Star’ for ‘x’ and ‘the Evening Star’ for ‘y’, some doubts may arise. It seems that the Morning Star’s property of shining in the morning is not a property of the Evening Star. The same with the property of shining in the evening: it seems that it is a property of the Evening Star but not of the Morning Star. However, the Morning Star and the Evening Star are in fact one object: planet Venus; these properties (among others) are its properties. Objects that are strictly numerically identical (e.g. Cicero and Tully) are genetically numerically identical as well. It can be said that Cicero (at time tƍ) preserves identity with Tully (at time tƍƍ). On the other hand, objects x and y which are genetically numerically identical are not strictly numerically identical unless x and y are both object z considered at the same time. Even if attorney John Brown in year 2000 keeps his genidentity with law student John Brown in year 1990, they will not be strictly numerically identical. Objects that are genetically numerically identical need not be absolutely qualitatively identical. For their genetic numerical identity invariability is not required; the preservation of the same essence is sufficient.8 The truth of the genidentification in the sentence ‘This is the same book Paul gave me on my birthday’ is not influenced by the fact that that book now has dog-eared corners and notes in the margins. The mug from which I drank tea a year ago is genidentical with the flowerpot in which there are now flowers, in spite of the fact that the mug used to be dark blue and had a handle and the flower pot is bluish (because it has faded) and has no handle (because it has been broken). Whether or not they are indiscernible  i.e. relatively identical  depends, of course, on language. They are discernible in common parlance. On the other hand, the mug standing on a shelf today and the same mug standing there tomorrow (provided that «nothing» happens to them and we do not take into consideration the change of temporal location) are  in common parlance  indiscernible. If certain objects are genetically numerically identical then they are also the same in a certain respect. The mug and the flowerpot have, e.g., the same volumes, heights and diameters. 8

It is here that the problem of intrinsic change arises. See footnote 12, below.

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As has been mentioned, the question whether objects that are absolutely qualitatively identical also have to be strictly numerically identical is related to the principle of the identity of indiscernibles. However, this principle does not resolve it. If we suppose that extrinsic properties (at least temporal characteristics) are independent from the others, then objects absolutely qualitatively identical can be genidentical. Objects that are indiscernible in a language can be genidentical. Of course, they are the same in some respect as well. If they have in addition the same spatiotemporal characteristics, then they are absolutely qualitatively identical and they are strictly numerically the same object. Objects that are the same in some respect may be neither relatively identical (either in sense of indiscernibility or in sense of sortal identity) nor absolutely qualitatively identical. If they are only the same in some respect then they are neither strictly numerically identical nor genidentical. Whether or not the principle of the identity of indiscernibles is true is still an open question.9 On the other hand, its converse, i.e. the principle of the indiscernibility of identicals, is generally held to be true.10 According to this principle, two objects which are identical have the same properties. xy {x = y o F [F(x) { F(y)]} However, the principle of the indiscernibility of identicals involves only absolute qualitatively identity. If two objects are absolutely qualitatively identical, something that is true of one of them is also true of the other, and conversely. The principle does not concern, for instance, relative identity. Even if two portions of water are the same river, the property of being clear may pertain to one of them but not to the other.

6. Identity and Similarity, Kindship, Sameness, and Difference Having at least one property, F, in common is the weakest condition for a similarity relation. xy {x is similar to y { F [F(x) š F(y)]} 9

It has been argued that even if spatiotemporal properties are counted among properties that can distinguish objects according to the principle of identity of indiscernibles, quantum mechanics shows that this principle is false. 10 However, some doubts concerning its truth (and, in particular, the truth of its contraposition) have arisen in connection with indeterminate identity.

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If F is such a shared property, it is said then that x is similar to y with respect to F. Similarity is a reflexive, symmetric and nontransitive relation: x can be like y  with respect to F, and y can be like z  with respect to G; it does not follow that x is like z. Unlike identity, similarity admits of degrees: the more properties x and y have in common, the more similar x and y are. It is sometimes supposed that similarity involves difference. Objects that have all their properties in common are not said to be similar  they are said to be identical. In that case, one should agree that the more properties objects have in common, the more alike they are, but when they have all their properties in common, they change from being similar to being identical. The unwanted consequence of that view is the fact that similarity stops being a reflexive relation: no object can be similar to itself. So, it is better to assume that identical objects are similar to themselves  in the highest degree. The term ‘exact similarity’ is sometimes introduced. It denotes indiscernibility. Two objects are exactly similar when they are indiscernible (e.g. two newly minted coins with the same nominal value). As we have already seen, it can be argued that ascribing absolute qualitative identity to objects that last in time is groundless. No object that lasts in time is unchangeable, and that fact that it undergoes changes destroys its absolute qualitative identity. The successive stages of changing objects are not absolutely identical. So, one may claim that instead of talking about identity between stages of a given object, it is better to talk about their kinship (see e.g. Quine 1950). According to that opinion, objects that last in time are really a succession of instantaneous objects (temporal parts) related to themselves. Kinship is reflexive and symmetrical, but its transitivity can be questioned. Between temporal parts various kinship connections hold (see Quine 1950). Let object x be, for example, river C at place m at instant t; let object y also be a momentary stage of river C at the same place m but at time tƍƍ; let object z be a momentary stage at time tƍ at a place mƍ of water molecules which were at time t at place m. Between objects x and z, there is a «water kinship», between x and y – a «river kinship». Such kinship relations consist of relations of spatiotemporal continuity, similarity and causation of a relevant kind. The expressions ‘the Morning Star’ and ‘the Evening Star’ are descriptions denoting related but different objects (temporal stages of the planet Venus).11 Between the consecutive temporal parts of Venus there

11

More precisely, each of the names ‘the Morning star’ and ‘the Evening star’ are names of many temporal parts. For instance, the former is the name of this morning’s temporal

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are a lot of similarities, kinship relations included (see Wilson 1955). When one introduces names such as ‘river’ or ‘planet’ one passes from kinship (between temporal parts of spatiotemporally extended objects) to an identity relation (between those objects themselves). Sameness was mentioned above as a sort of relative identity. Sameness can be complete (in all respects) or incomplete (only in some respects). If objects are the same in a certain respect, they are similar in that respect as well. Difference is the opposite of identity. Numerical identity is contrasted with numerical difference (i.e. disjointness; separateness; having different spatiotemporal characteristics). Numerical identity precludes numerical difference but does not preclude qualitative differences. Objects that last in time and preserve genetic numerical identity, do not preserve absolute qualitative identity. The opposite of qualitative identity is qualitative difference. Sameness allows the existence of qualitative differences. When stating a relation of the same age between certain people or a relation of having the same colour between patches, we stress what is identical and overlook differences that separate those objects.

7. Criterion of Identity To say that a certain object satisfies an identity relation one should first separate it from other objects – isolate it. On the other hand, to isolate one object, one should identify it: You must know what you are isolating. Without introducing the notion of identity our n acts of ostention point to n objects with an undetermined spatiotemporal extention. Only thanks to the assumption that between our pointings the identity of the indicated object is preserved is it possible to specify at which object (with which extention) we are pointing (see Quine 1950). The criterion for strict numerical identity is empirical. The fact that the Morning Star and the Evening Star are planet Venus is an astronomical truth, discovered by the Greeks. The fact that Lake Geneva is identical with Lake Léman can be stated if their geographical position is determined. One can find out that Cicero is the same person as Tully by comparing their personal histories. Genetic numerical identity of a given object can be stated only if one knows which object one is identifying – what its essence is. part of Venus, yesterday’s morning temporal part of Venus, the-day’s-before morning temporal part of Venus, etc.

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According to the view that relativises an object’s essence to language – the essence of given object x is a set of properties co-denoted by name ‘x’ in language L. Looking for an essence resolves itself into discovering the connotation of a general name. The resolution of puzzles about Theseus’ ship, Heraclitus’ river and Ralph, who moved the lawn eight years ago and now is having a dinner will consist in the precise definition of names, in determining what we understand by ‘ship’, ‘river’ and ‘human’ (see Quine 1987). General names can be divided into substantival (sortal) ones, like ‘man’, ‘river’, ‘stone’  and adjectival e.g. ‘object’ (see Geach 1962). The criteria of identity for objects which are denoted by sortal names are connected with the use of those names. Individuals are of a kind. For any given kind of particulars there is a criterion of identity for particulars of that kind. A criterion of identity for a kind F tells us what conditions x has to satisfy in order to be an F. Different kinds often have different criteria of identity. A criterion of identity conveys semantic information, which constitutes a part of the meaning of the relevant sortal term.12 Criteria of identity determine persistence conditions. Roughly speaking, persistence conditions tell us what it takes for a particular to continue to exist; in other words, what it takes for a particular to retain its sort. In order to say that a stone at time tƍ is the same stone it was at time tƍƍ, we have to make sure that it has preserved its substance (material) and (rough) shape. On the other hand, the identity of constituents is unnecessary to say that river x at time tƍ is the same river as it was at time tƍƍ (since a river cannot be identified with the water which flows in it). Similarly, water ceases to exist when its molecules break apart into oxygen and hydrogen, while a river can still exist even when there is no water in it. An object that belongs to sort F can change into an object of sort G, provided that F and G have the same criterion of identity. In this way, the chrysalis of an insect can change itself into the imago of that insect, but Lot’s wife cannot change into a pillar of salt. This semantical criterion refers only to genetic numerical identity. If one assumes that the essence of an object is predicated by its general name then the constant usage of that name testifies to the fact that essential properties are preserved. In this formulation a given object is always an object of a certain sort. Its essential properties are always sortal ones. 12 See Lowe (1989), chapter I. The criterion of identity does not specify the whole meaning of a given sortal though. This is why different sortals can have the same criterion of identity associated with them.

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The criterion of genetic numerical identity mentioned above is also a necessary condition for absolute qualitative identity. In order for x and y to be absolutely qualitatively identical, they have to be, first of all, objects of the same sort; they have to have the same essence (in the sense given above). This is not a sufficient condition, however. It seems that the criterion for absolute qualitative identity can be formulated only in a negative way. Objects x and y are absolutely qualitatively identical if there is no property that belongs to x but not to y. The criterion of indiscernibility of x and y in a language is as follows: x and y are indiscernible when there are no predicates that can be predicated of x and not of y.

8. Conclusion The main results of the preceding analysis can be summarized as follows. There are two main types of identity between objects: numerical identity and qualitative identity. Within the first type there are two subtypes: numerical identity sensu stricto and genetic identity. Objects that have the same spatiotemporal characteristic are strictly numerically identical, objects that exist at different times but are connected by a continuous spatiotemporal trajectory and have the same essence are genidentical. There are two subtypes of the second type of identity, i.e., qualitative identity: absolute identity and relative identity. Objects that have all their properties in common are absolutely identical; it is disputed whether properties connected with a spatiotemporal location should be included among those properties. If they were to be included, a given object at a given time would be absolutely identical with nothing but itself (at that same time). Relative identity can be understood as an indiscernibility in a given language or theory. Two objects are relatively identical whenever every predicate which belongs to one of them also belongs to the other. Alternatively relative identity can be understood as a sortal identity (relativised to a sort). In this interpretation, there is no relation of being (absolutely) identical, but many relations of being the same A, where ‘A’ is a general name. Sameness is «the weakest» relative identity. Two objects are the same with respect to a property in case they both have this property. There are various relations between the types mentioned above. Most controversies are raised by the relation between absolute identity and strict numerical identity. This relationship is expressed by the principle

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of the identity of indiscernibles. According to that principle, only objects that are strictly numerically identical are absolutely identical. Each type of identity has its own criterion of identity. For instance, strict numerical identity has an empirical criterion while genidentity has a semantic criterion.

Uniwersytet Warszawski Department of Philosophy ul. Krakowskie PrzedmieĞcie 3 00-047 Warszawa, Poland e-mail: [email protected]

REFERENCES Ajdukiewicz, K. (1967). Intensional Expressions. Studia Logica 20, 63-86. Black, M. (1952). Identity of Indiscernibles. Mind 61, 153-164. Frege, G. ([1892] 1980). On Sense and Reference. In: M. Black and P.T. Geach (eds.), Translations from the Philosophical Writings of Gottlob Frege, pp. 56-78. Oxford: Basil Blackwell. Geach, P.T. (1962). Reference and Generality. Ithaca, NY: Cornell University Press. Geach, P.T. (1967). Identity. Review of Metaphysics 21, 3-12. Geach, P.T. (1969). A Reply. Review of Metaphysics 22, 556-559. Geach, P.T. (1973). Ontological Relativity and Relative Identity. In: M.K. Munitz (ed.), Logic and Ontology, pp. 287-302. New York: New York University Press. Hume, D. (1898). A Treatise of Human Nature. London: Longmans, Green and Co. Jones, E.E.C. (1908). Precise and Numerical Identity. Mind 17, 384-393. KotarbiĔski, T. (1966). Gnosiology. The Scientific Approach to the Theory of Knowledge. Translated by O. Wojtasiewicz. Oxford: Pergamon Press. Leibniz, G.W. (1890). Letters to Sam. Clarke. In: The Philosophical Works of Leibniz, pp. 238-286. New Haven, CT: Tuttle, Marchouse and Taylor. Leibniz, G.W. (1981). New Essays on Human Understanding. Cambridge: Cambridge University Press. Lewis, D. ([1976] 1983). Survival and Identity. In: Philosophical Papers, vol. 1, pp. 55-72. Oxford: Oxford University Press. Lowe, E.J. (1989). Kinds of Being: A Study in Individuation, Identity and the Logic of Sortal Terms. Oxford: Basil Blackwell. Noonan, H.W. (1984). Indefinite Identity: A Reply to Broome. Analysis 44, 117-121. Pelc, J. (1986). Synonymy. In: T.A. Sebeok (ed.), Encyclopedic Dictionary of Semiotics, pp. 1036-1043. Berlin: Mouton de Gruyter. Perry, J. (1970). The Same F. Philosophical Review 79, 181-201. Quine, W.v.O. (1943). Notes on Existence and Necessity. Journal of Philosophy 40, 113-127. Quine, W.v.O. (1950). Identity, Ostension and Hypostasis. Journal of Philosophy 47, 621-633.

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Quine, W.v.O. (1964). P.T. Geach: Reference and Generality. Philosophical Review 73, 100-104. Quine, W.v.O. (1987). Identity. In: Quiddities. An Intermittently Philosophical Dictionary, pp. 88-92. Cambridge, MA: Harvard University Press. Russell, B. (1952). Descriptions. In: L. Linsky (ed.), Semantics and Philosophy of Language, pp. 277-293. Urbana: The University of Illinois Press. Savery, B. (1942). Identity and Difference. Philosophical Review 51, 205-212. Stroll, A. (1967). Identity. In: P. Edwards (ed.), The Encyclopedia of Philosophy, pp. 121-124. New York & London: The Macmillan Company & The Free Press. Williams, C.J.F. (1989). What is Identity? Oxford: Clarendon Press. Wilson, N.L. (1955). Space, Time and Individuals. Journal of Philosophy 52, 589-598. Wilson, N.L. (1961). Reply to Professor Rescher. Review of Metaphysics 14, 714-720. Wittgenstein, L. (1963). Tractatus Logico-Philosophicus. London: Routledge and Kegan Paul.

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PART III PROGNOSES, NORMS AND QUESTIONS

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Tomasz Placek A PUZZLE ABOUT SEMANTIC DETERMINISM 1 àUKASIEWICZ’S “ON DETERMINISM” YEARS LATER

Do you feel somewhat uneasy with the view that, given that you are reading this paper now, it has always been true that you would have been doing this at this very time? This rather fatalistic claim is a consequence of the so-called eternity of truth, which in turn follows from some intuitive assumptions motivated by classical logic. As it is well known, the idea that, if a proposition is true then it is true at all earlier times, troubled Jan àukasiewicz so much that he decided to revise logic by introducing a third truth value. This is, at least, what one may infer from his “On Determinism,”2 which was first delivered as the chancellor’s address at the opening ceremony for the academic year 1922/23 at Warsaw University and was then was prepared for publication in 1946 and finally published in 1961. For historical reasons, let me add that the first time àukasiewicz mentioned his construction of three-valued logic was in 1918. My principal objective in this paper is to show that the thesis of eternity of truth is philosophically innocuous, as it expresses merely a weak (semantic) determinism and not fully fledged determinism. We start with considering two interpretations of the phrase “it is true at a time t.” Construed one way, the expression stands for an operator; semantic determinism is then a consequence of the incorrect assumption that this operator commutes with negation. Construed the other way, the discussed 1 I read the first draught of this paper at the 6th Convention of the Polish Philosophical Association (PTF). The written version of this talk will appear in Polish in Filozofia logiczna. Since then, my views on the subject have sharpened considerably as expressed in this paper. Because of the significant differences, I have decided to publish the present paper, despite its earlier version being published elsewhere. I am grateful to both the participant of the Convention and my colleagues in Cracow for valuable discussions and suggestions. 2 J. àukasiewicz, “O determinizmie,” in his (1961).

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 171-185. Amsterdam/New York, NY: Rodopi, 2006.

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phrase is assumed to stand for a truth value. In that case, semantic determinism follows from the principle of the excluded middle and the stipulation that truth-at-t distributes over disjuncts of any instance of this principle. However, on this construal truth-at-t is replaceable by classical truth simpliciter and vice versa, which makes semantic determinism rather trivial. Nevertheless, this position turns into fully fledged determinism if one reads into it a seemingly evident but flawed claim concerning truth-makers. Still another way of strengthening semantic determinism is to supplement it with the claim stating that every truth has its cause: If at a time t it is true that p, then at t a fact obtains that causes that p. From our discussion it shall turn out that this thesis is also incorrect. The paper ends with some remarks about how to investigate the issue of determinism in the context of the semantic approach to empirical theories. To begin with, let us investigate in detail àukasiewicz’s reasoning. By p T I understand a sentence about something that happens, obtains or takes place at a time T. According to àukasiewicz, (what we will call semantic) determinism is characterized by the following thesis: DS

If p T , then it is true at any time t prior to T, that p T .

The expression “it is true at t that . . . ” is assumed to mean more or less the same thing as “it is the case at t that . . . ” This formulation suggests that truth is redundant; instead of saying that a sentence is true at a time t, we claim that something is the case at t. For this reason we treat the expression “it is true at t that . . . ” as standing for an operator, instead of introducing a predicate “true at t ” and assigning it to names of sentences. Given sentences are assigned some semantic values, the operator carries pairs: {semantic value of a sentence, time t} into semantic values appropriate for sentences. As an illustration of the DS thesis one may consider the following: If it rains in Cracow on September 18th, 1995, then it is true / the case at any time prior to this very day that it rains in Cracow on September 18th, 1995. For brevity, from now on I assume the convention that a time t is prior to the time T. My second convention concerns enumeration of steps in àukasiewicz’s reasoning. The missing steps that are disclosed are enumerated by asterisks, these which occur explicitly are named by

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letters and my own claims are given Arabic numerals. We start with an instance of the principle of the excluded middle: *

It is true at a time t that p T or it is not true at a time t that p T .

In the second move, it is assumed that negation commutes with the operator “it is true at a time t that . . . ”: **

It is not true at a time t that p T if and only if it is true at a time t that not- p T .

The commutability assumption is far from obvious; indeed this is the most controversial claim of the whole reasoning. We shall consider it later. From (*) and (**) one gets àukasiewicz’s first premise: (a)

It is true at a time t that p T or it is true at a time t that not- p T .

The second premise comes from the intuitive view that whatever is the case at an earlier time, is the case: facts that obtained do not change. In àukasiewicz’s framework this is expressed by the following: If it is true at a time t that \, then \. Taking ‘not-p T ’ for \ one gets the second of àukasiewicz’s premises: (b)

If it is true at a time t that not-p T , then not- p T .

Now let us introduce the following abbreviations: it is true at a time t that not-p T { D it is true at a time t that p T { E. With these abbreviations the premise (a) reads as D›E which is equivalent to (1)

not-D o E.

On the other hand, the premise (b), that is D o not-p T , is equivalent to (2)

p T o not-D

Accordingly, from (1) and (2) it follows that (3)

p T o E,

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which, after introducing the general quantifier to the consequent, yields àukasiewicz’s determinism thesis (DS): 3 If p T , then it is true at any time t that p T . Let us return to the step (**), which is missing in àukasiewicz’s proof. The claim (**) expresses commutability of negation and the operator “it is true/the case at t that . . . ” The commutability encapsulates, so to speak, the determinism thesis, since this thesis can be derived solely from it and the intuitive claim (b). This is of some historical importance, as in “On Determinism” àukasiewicz was hesitant as to what should be blamed for the derivation of the determinism thesis (DS): the principle of the excluded middle or, instead, the principle of bivalence? Our tentative answer is: commutability. To see this assume p T and the premise (4)

It is not true/the case at t that p T .

Given commutability, the above is equivalent to: (5)

It is true/the case at t that not-p T .

This, however, on the assumption of the premise (b) entails (6)

not-p T ,

which contradicts our initial assumption (p T ). Thus, we conclude that (7)

If p T , then it is true/the case at t that p T .

Again, introduction of the general quantifier to the consequent of (7) yields the DS thesis. The problem we face now is how to justify the commutability of negation and the operator “it is true/the case at t that . . . ” To get a grip on this issue, let us see how we can interpret the sentence: It is true/the case at t that p T . One way of reading it is epistemic. On this reading, the above sentence claims that at a time or in an epistemic situation t, p T is verified or conclusively established or is assertible. Even without attempting to disentangle these notions and displaying various theoretical machineries that try to account for them (Brouwer’s theory of the creative subject, 3 More precisely we first obtain: p o for any t, it is true at t that p , from which, given T T “for any t, it is true at t ” means the same as “it is true at any t,” the discussed thesis follows.

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Beth trees, Kripke trees), it can be shown that on the epistemic interpretation commutability is patently false. To make a little alteration, assume that t does not range over times, but rather over states of information, which are numbered in accordance with the relation of comprising. This means that any later state of information comprises all the information from the earlier states. We do not presuppose anything as to how many states of information follow immediately after a given state or whether the sequences of states are finite. For reasons of brevity, instead of saying: “It is conclusively established (verified, is assertible) at t that p T ” we use a neat formula: Ōt pT. Now, does the commutability of ʼn, and Ō t hold? That is, do we have (8)

ʼn(Ō t p T ) iff Ō t (ʼn p T ) ?

The left side of this equivalence is a factual claim, as it states that at t the sentence p T is not established. On the other hand, the right hand side states that the negated sentence has been established at t. What does this latter claim amount to? It usually says that at any state of information following the tth one, p T is not established. So clearly, the right hand side of the equivalence does not follow from its left side. As a curiosity, it is perhaps worth mentioning that on epistemic interpretations the DS thesis cannot be expected to hold, either. In Brouwer’s theory of the creative subject you may find a thesis that was much discussed at one time (see, for instance, Dummett 1978) (9)

\ o  t Ō t \.

If read with the classical interpretation of logical constants, it says that if \ is the case, then \ is established at some state of information t. The intuitionist objects to this, since it appears far too optimistic. However, he agrees with a weaker version of (9), to the effect that if \ holds, then it can never be asserted that \ will never be established. In symbols we have: (10)

\ o ʼnʼn t Ō t \.

The claim (9) resembles our DS thesis, with two differences, however. First, in DS we have the general quantifier in place of the existential. Secondly, in DS quantification is over times prior to the time T to which p T relates. Accordingly, the DS thesis is much stronger than the already controversial implication (9), which means that the DS cannot hold if “true at t ” is understood epistemically.

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Thus, we are driven to an opinion that the expression “it is true at t that \” should be interpreted as presupposing that truth (falsity) at t is an objective matter, of which we may or may not become aware. Thus, the natural move is to assume that the expression should be read as “It is already determined at t that \.” However, on this construal commutability is question-begging, since it simply amounts to fully fledged determinism. Only the most hard-headed determinist would agree that, if it has not already been determined that I will eat the cake in a moment, then it has already been determined that I will not eat it in a moment. Thus, if the discussed expression is construed as “it is already determined at t that . . . ” then commutability amounts to fully fledged determinism. Despite the failure to justify commutability, one may have a persistent feeling that it is a consequence of the principle of bivalence and the classical behaviour of negation, such that it authorizes the principle of excluded middle. The principle of bivalence claims that any sentence has precisely one of two truth values: it is either true or false. Then, the principle of excluded middle permits us to identify the falsity of a sentence with the truth of its negation: \ is false iff ʼn\ is true. These principles yield the thesis that for any sentence, either it or its negation is true. Importantly, for establishing this result both the principles are needed. The case of intuitionism shows that mere adherence to bivalence does not entail the above result.4 On the other hand, in logic of vagueness one accepts the excluded middle, rejects bivalence and the above thesis. At this point, we need to reflect again on the problematic expression “it is true/the case at t.” Since we now talk of the truth values that sentences possess, instead of interpreting “it is true at t that” as standing for an operator, we had better construe it as a predicate assignable to names of sentences, which is intended to stand for a relativised truth value. We form the name of a sentence by putting the sentence in single quotes ‘ ’. To indicate that “true at t” is a predicate, on par with “true” simpliciter, we write it with hyphens: “true-at-t.” Now, one may say that if bivalence holds, so does its temporised relative. The temporal principle of bivalence states that for any sentence, for any given t, the sentence has exactly one of two truth values: it is either true-at-t or false-at-t. The same trick can be applied to the principle of excluded middle. The result amounts to the claim that at any 4

In fact, the intuitionists reject bivalence as it is stated above, since they argue that it is not so that for any sentence, it is either true or false; nevertheless they cannot say that some sentences have a third truth value. This is the reason to claim that they accept bivalence though in a weaker formulation.

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time t any instance of the principle of excluded middle has the value “true-at-t.” Suppose now that ‘p’ does not have the truth value “true-att.” Does it follow that ‘not-p’ has the truth value “true-at-t”? Clearly, ‘p or not-p’ has the value “true-at-t.” What is missing for deriving the desired conclusion is a demonstration that from any instance of the principle of the excluded middle ‘p or not-p’ it follows that ‘p’ has the value “true at t” or ‘not-p’ has the value “true at t.” Clearly, the way classical logic is constructed guarantees this holds for truth simpliciter. However, in introducing a new predicate “true-at-t” that is intended to stand for a truth value one is by no means forced to follow in the footsteps of the classical logician. And again, the intuitionist and the proponent of the logic of vagueness would reject, though for differing reasons, that truth distributes over disjuncts of instances of the principle of excluded middle.5 This shows that the assumption that truth-at-t distributes over disjuncts of instances of principle of the excluded middle is arbitrary. However, the proponent of semantic determinism attempts to make the difference between classical truth and truth-at-t as minimal as possible and at this point does not bother about interpretational problems. Given that the move is made, we have a close relative of àukasiewicz’s thesis (a) (aƍ)

‘p T ’ is true-at-t or ‘not- p T ’ is true-at-t.

àukasiewicz’s second premise reads now: (bƍ)

If ‘not-p T ’ is true-at-t, then not-p T .

The rest of the derivation of the DS thesis easily follows. Hence, we have another culprit for semantic determinism, that is, the principle of excluded middle together with the stipulation that truth-at-t distributes over disjuncts of instances of the principle of excluded middle. Now, it is easy to notice that the rules governing truth-at-t are exactly the same as those governing classical truth simpliciter. Moreover, the DS thesis together with the premise (b) entail the following: (11)

p T iff ‘p T ’ is true-at-t, for any t.

Given Tarski’s T-convention, the above is equivalent to 5

To comment on the intuitionistic case, the intuitionists assert ‘p or not-p’ on the ground of their knowing a decision procedure for the proposition expressed by p. Nevertheless, as long as the results of the application of the decision procedure are not known, they have no right to assert any of these two: p, not-p. A formal representation of this fact is provided by Beth trees. An instance of the principle may be true at the vertex, though none of the disjuncts is true there.

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‘p T ’ is true iff ‘p T ’ is true-at-t, for any t.

Thus, whenever a sentence is true simpliciter, it is also true-at-t and vice versa. Thus, the problem we face is what truth-at-t can mean that is different from the meaning of truth simpliciter. However, given that the two notions are replaceable and truth-at-t obeys the same metalogical principles as classical truth simpliciter, they cannot differ. The celebrated DS thesis amounts merely to a rather trivial claim: (13)

If p T , then ‘p T ’ is true.

That is how the little letter t has made us sail high on the waves of language! It seems, therefore, that àukasiewicz’s reasoning is faulty. On the first interpretation, his main premise (**) does not hold. On the other interpretation, although the proof is formally correct, the DS thesis reduces to a triviality. Nevertheless, despite the faulty arguments, the thesis may be correct. Accordingly, it is our task to consider what the thesis may mean, whether it may express fully fledged determinism and how it can be (mis)understood. To begin with, consider whether the determinism thesis (DS) is philosophically troublesome. Are its consequences so frightening that one needs to renounce the principles of bivalence or excluded middle? From the thesis it cannot be derived that any present fact is determined by some earlier facts, which is the intuition that underlies the deterministic position. This intuition, however, is likely expressed by the statement below: (14)

If p T , then it is true at t that it is not possible that not-p T

This statement by no means follows from the determinism thesis (DS), at least on the basis of the standard possible-worlds semantics with time. Thus, the suspicion arises that there is an inclination to read into the DS thesis an extra-logical claim. Indeed, semantic determinism transforms into a much stronger claim if it is conceived of as a claim about truth-makers. Let us investigate this possibility and its appeal. I am looking through the window and see that today, on September 18th, 1995 it is raining in Cracow. According to the DS thesis, it was true, for example, on September 18th, 1895 that it would rain in Cracow on September 18th, 1995. If this is so, what makes true the sentence

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On September 18th, 1885 it is true that it rains in Cracow on September 18th, 1995? 6

A tempting answer is to assume that what makes this sentence true – its truth-maker – already obtained on September 18th, 1885. In other words, one may be inclined to see in the DS thesis the much stronger claim about truth-makers, to the effect that if p T , then at any earlier time t the truth-maker of the corresponding sentence “It is true at t that p T ” obtained. To check whether this inclination is justified, let us look into truth-makers for a while. For instance, consider what is the truth-maker of the sentence (16)

It rains in ToruĔ at noon on September 18th, 1995

Clearly, it is the rain raining in ToruĔ on this very day. Thus, truthmakers are facts, and if a sentence expresses something that happens at a given time, its truth-maker should obtain exactly at this time. But a sentence like (17)

It is true at t that p T

relates to both the times: T and t. So, when is its truth-maker to take place? I claim that it can only obtain at the time T, so at the time to which the sentence p T relates. The sentence “It is true at t that p T ” does not say about anything that happens, obtains or takes place at the time t. Instead, it refers to a fact that obtains at T. As this argument is not very convincing, I had better show that the appeal of the opposite position is illusory. The opponent to the solution proposed here would be likely to mention the following difficulty. How is it possible that the fact obtaining in the year of 1995 can make true the sentence (18)

It is true on September 18th, 1895 that it rains in Cracow on September 18th, 1995?

The opponent can add that if this were possible, truth-makers would act backwards, so to speak. To this I have to reply that truth-makers do not act; to ascribe to them any action is to confuse them with causes. In the year 1895 the sentence stating that it is true that one hundred years later it rains in Cracow, if it exists at all, then it exists only in a mythical 6 Given “truth at t ” is construed as standing for a truth value, this sentence should be interpreted as stating that ‘p T ’ has the value true-at-t. But the idea that such ascriptions of truth values have truth-makers that are identified with temporal facts does not seem to have any credibility. Thus, in order to be attracted by the discussed (incorrect) reasoning one needs to interpret “true at t ” as referring to an operator.

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Fregean third realm (more precisely, the proposition it expresses might exist there). Accordingly, a fact can hardly act on such an unearthly object. But perhaps, some ingenious meteorologist uttered the sentence (18) in 1885! However, in this case as well, the question of what makes true his assertion, more than 100 years ago, of “It is already true that it rains in Cracow on September 18th, 1995” receives the same answer: it is the rain pouring down in Cracow on September 18th, 1995. The sentence uttered by our Royal-Imperial Meteorologist has a truth-maker that obtains 100 years after its utterance. To draw the moral of this story, a proponent of truth-makers may mistakenly see in the innocuous DS thesis a claim that for any true sentence p T , any corresponding sentence of the form “it is true at t that p T ” has a truth-maker that obtains at the time t. Firstly, this reading of the DS thesis results from mistaking truth-makers for causes. Secondly, the truth-maker of the sentence “It is true at t that p T ” is a fact obtaining at T rather than at t. The proponent of truth-makers assumes that between sentences and their truth-makers a relation holds. This, however, is a relation and not an interaction, though the name suggests that truthmakers make something or act on something. It is instructive to consider still another argument in which the similar confusion is exemplified. This is the well-known argument against the reality of the future that starts with the observation that a sentence ‘p T ’, which relates to a future time T, is either true or false, including at any earlier time. Given this the anti-realist about future asks: How can there be room for choice between different possible courses of action, when it is already the case that one of them will in fact be followed? Why is it, moreover, that we cannot affect the past as we believe we can affect the future? (Dummett 1991, p. 7)

The key word in such argumentation is “to affect.” We commonly believe that we cannot affect the past, but may influence what will happen in the future. This latter belief seems to clash with determinate values of sentences about the future. But this apparent clash stems from a confusion similar to the one already encountered. In the first move, the envisioned anti-realist translates the thesis that p T is either true or false into: (19)

Either it is true at t (now) that p T or it is true at t (now) that not-p T .

In the second, controversial step, he assumes that (19) is a claim about truth-makers, namely, that there is at t (now) some fact that makes the above sentence true. As it is a disjunctive sentence, the fact in question should be a truth-maker of exactly one of the two sentences below:

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It is true at t (now) that p T It is true at t (now) that not-p T . Finally, he concludes that as one of these truth-makers obtains, we do not have any chance to influence whether or not p T holds in the future time T. However, the mistake in this argument lies ready at hand: the truth-maker of (19), if there is such a thing, obtains in the future, at the time T, and neither earlier nor later. There is, however, a certain oddity, loosely linked to the issue of determinism, and that originates from the already mentioned doctrine of the Third Realm. Let us suppose with Frege, that this realm is inhabited by all thoughts, no matter whether they have been expressed or not, and they exist there eternally, each with a determinate and unchangeable truth value. The proponent of eternal and logically determinate thoughts may read into semantic determinism the following reasoning. If p T holds, then the thought expressed by p T is true, and moreover it has always been in the Realm and always been true. Accordingly, the world could not be different, it must have happened that p T ! As it is easy to notice, this reasoning does not appeal to the DS thesis, it is logically independent from it. I will not investigate it, since I believe there are no reasons to accept its premises, that is, to assume eternality of all thoughts, including those never expressed. Both Frege’s argument for the eternality of thoughts and Meinong’s reasoning that introduces the eternal Obiektive have been analysed many times and shown invalid. This does not mean, obviously, that a new and better argument for the above thesis will never crop up. But, if we invent such convincing arguments, we will be in for trouble: if there are eternal thoughts, determinately either true or false, then undoubtedly everything is determined! The confusion between truth-makers and causes is instructive, as it points out the rationale that made àukasiewicz complement the DS thesis with causal determinism (CA). After this supplementation determinism is characterized by the following two theses: DS CA

If p T , then ‘p T ’ is true-at-t, at any time t prior to T; If ‘p T ’ is true-at-t, then at t a fact obtains that causes that p T .7

Let us illustrate it with the following example. On July 5th, 1995 an eclipse of the Sun was visible from Singapore, as the moon found itself between the Sun and the Earth. Assume that this fact is expressed by p T . According to the DS thesis, ‘p T ’ has the truth value true-at-t, for any time 7

As the more appropriate name for DS is “fatalism,” the above may be summed up in an equation: fatalism + causation = determinism. I owe this point to Kevin Mulligan.

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t prior to T. For instance, we may say: ‘p T ’ is true at noon GMT, on July 5th, 1002. Now, the CA thesis requires that at this very time a fact obtained that caused the Sun’s eclipse on July 5th, 1995. Indeed, from the point of view of physics, the states of the motion of the planets of our planetary system at noon GMT, on July 5th, 1002, taken together with the laws of gravitation and the laws of dynamics “determined” the locations of the Earth, the Sun and the Moon on July 5th, 1995. We can say this because the mentioned laws yield a functional dependence that holds between the state of motion at a given time and a state of motion at any earlier time. On the other hand, the above claims are not satisfied by quantum events, or at least, there are serious reasons to think so. Suppose that just now, in the Cracow Institute for Nuclear Research, an experiment is carried out, in which electrons moving in the magnetic field are deflected, either up or down, in accordance with the projections of the spin they take. The mathematical algorithm of quantum mechanics permits us only to calculate the probability that the electron will go “up” (or “down”). Nevertheless, imagine that just at this very moment the investigated electron was registered by the “up” counter. By DS we get ‘The electron investigated at T goes up’ is true-at-t, for any t < T. I take this claim is not controversial since it does not entail that at some time prior to T it is impossible that the discussed electron is not registered by the “up” counter at T. That controversial claim, however, is suggested by CA. Indeed, CA requires that at any time t prior to T something obtains that causes that the electron goes up at T. This requirement boils down to the postulate of constructing a viable local hidden variable theory or, equivalently, a causal explanation of quantum processes such that the assumed causes respect Reichenbach’s common-cause condition. These possibilities, however, are excluded within limits of experiment error by the results obtained by Aspect and collaborators. Let us go back to àukasiewicz’s characterization of determinism. Its main virtue consists of splitting the position of determinism into two ingredients. The first is trivial as it only amounts to the claim: if p T , then ‘p T ’ is true. This ingredient may be called “logical.” The second part is metaphysical, although, as recent results in philosophy of physics show, it is empirically testable as well. Moreover, experiments testify against its tenability. Secondly, this analysis of determinism sheds light on the cogency of àukasiewicz’s rationale for revising classical logic. Facts that speak against determinism, testify only against the metaphysical and empirical thesis CA, and have no bearing whatsoever on the rather empty claim DS. Thus, the introduction of a third truth value as a defensive

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weapon against determinism was superfluous and much too prudent a move. Yet another advantage of the above characterization is that it corresponds well with the received views on determinism in physical phenomena. For instance, as we showed, gravitational motions are classified as deterministic while some quantum events are not. The most serious of the drawbacks of this position is its appeal to causation. Not only is this concept not scientific, as the author of “On Determinism” complains, but also its acceptance commits one to a rich ontology. Many would prefer ontological theories that do not invoke causality. Instead of causes one may, as we did above, talk of laws of nature which, given initial conditions, determine the state of a system at any time. But again, this type of discourse is meaningful only to those who believe that apart from things, their properties, relations and interactions there are also some platonic laws. One may convincingly argue that the so-called laws of nature are either formulas written down in books and journals, or thoughts in the minds of people, whereas on the side of physical reality there are only regularities, no laws. Another troublesome aspect is the concept that laws determine. If we are not to contend ourselves with explaining determinism in terms of determining, we had better not appeal to this latter notion. Whenever it is said that a law, given initial conditions, determines the states of a system, the expression concerns the mathematical form of the law, namely, that it operates like a function from a set of states of a system to this set. When hearing this, the proponent of an economic ontology will most probably rise his eyebrows: not only a platonic law has been assumed, but it is supposed to have a mathematical form, and operate like a function whose arguments and values are identified with rather mysterious states of systems! The talk of mathematical forms of laws, however, gives a hint of how to pose the issue of determinism more precisely, in such a way such that the abovementioned difficulties do not arise. One may do this in the framework of the so-called semantic approach to empirical theories. This approach relies on a significant rewording of questions. Instead of asking whether the world is deterministic, one queries if a given empirical theory is deterministic. By “theory” the proponent of the semantic approach does not understand a reconstructed and then formalized language, but rather a set of mathematical structures (models) that satisfy a given predicate. To decide whether or not a theory is deterministic, one considers the following questions: (i)

By what mathematical structures are states of systems represented?

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What sort of mathematical relations hold between structures representing states at various times? What dependence does a given theory assume to hold between structures representing states of a system and measurable quantities like location, momentum, energy, spin, etc.?

At first sight it looks as if answers to the first two questions are sufficient to decide whether or not a theory is deterministic. Accordingly, one may say that a theory is deterministic if only it assumes a functional dependence between the states of a system such that for a given state \(T ) there is at any earlier time t a state \(t) that, if taken as an argument of the function, yields the state \(T ). In this formulation one may easily recognize a similarity to the thesis CA that occurs in àukasiewicz’s characterization of determinism. However, the case of quantum mechanics commands carefulness. Quantum states indeed evolve continuously (Schrödinger equation), so if one were only concerned with questions (i) and (ii), they would have to classify quantum mechanics as deterministic. On the other hand, however, the relation between states and results of measurements is only probabilistic, and this is the crucial difference between quantum and classical theories. This means that for a system in a given state the algorithm of quantum mechanics yields only a set of possible results of measuring an observable and their probabilities. Accordingly, if it is assumed that results of measurements are real whereas states are merely mathematical fictions, then quantum theory turns out to be indeterministic. Thus, the answer as to whether or not a theory is deterministic may as well depend on the standpoint of what is real: states or results of measurements. The semantic approach to the issue of determinism makes clear the challenges that àukasiewicz posed by splitting determinism into two claims. Advances in physics point out that the issue is inevitably linked to the query of what is real. The changing mathematical language of physics requires us to translate intuitions underlying causal determinism (CA) into mathematical conditions appropriate for new theories. Looking from a mathematical perspective, an ambitious task emerges: to find the conditions whose satisfaction guarantees that a theory is deterministic.

Uniwersytet JagielloĔski Department of Philosophy ul. Grodzka 52 31-044 Kraków, Poland e-mail: [email protected]

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REFERENCES Dummett M. (1978). Philosophical Basis of Intuitionistic Logic. In: Truth and Other Enigmas, pp. 215-247. London: Duckworth. Dummett M. (1991). The Logical Basis of Metaphysics. Cambridge, MA: Harvard University Press. àukasiewicz J. (1961). O determinizmie [On Determinism]. In: Z zagadnieĔ logiki i filozofii: Pisma wybrane [Some Problems of Logic and Philosophy: Selected Papers], pp. 114-126. Warszawa: PWN.

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Max Urchs CAUSALITY IN CHAOTIC ENVIRONMENT: DOES STRONG CAUSALITY BREAK DOWN IN DETERMINISTIC CHAOS?

1. Introduction “Cause” is an old concept, and “chaos” is a new one, yet they have much in common. Both of these notions are alleged to function universally in physical reality. On the one hand, causal analysis is claimed to be universally applicable as a description of open dynamical systems and on the other chaos was discovered in almost every part of the world within the last years, for instance, in all possible kinds of chemical and biological evolution, in meteorological, cosmological and even in stock market processes. On that background the thesis “there is no causality in chaotic dynamic systems” seems to be an open rebellion against the established paradigm, what may even force us to change our view of the physical world. Claiming the breakdown of causality might be just another instance of skilful publicity making of chaos theory. But anyway, it seems worth while to ask for the compatibility of the universality of chaos on the one hand, and the universal efficacy of causal nexus on the other hand.

2. Invariance of Similarity Following Hume’s famous passage from his Inquiry one sometimes takes the cause as an object followed by another and where all the objects similar to the first are followed by objects similar to the second. For Hume this principle (let us call it the invariance of similarity) plays the role of an (alternative) explication of causal dependency, it is not only its mere description. This similarity invariance of the causal nexus seems to

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 187-194. Amsterdam/New York, NY: Rodopi, 2006.

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be one of our basic assumptions concerning physical reality. It lies on the very core of all human strategies of planning and rational behaviour. Chaos theory, however, denies the universal validity of this apparently most plausible principle. What is more, it denies its validity in central areas of previous application: the principle of invariance of similarity is said to fail to describe the typical long-time behaviour of natural processes. That needs some more detailed comment.

3. Chaos It is by no means easy to say what exactly is meant by “chaotic behaviour of a system.” 1 On the one hand, there are a few oversimplified explications which in fact do explain nothing, e.g. “Chaos is the deterministic production of behaviour that is unpredictable over long time.” On the other hand, there is a still ongoing fundamental mathematical discussion which I am not able even to outline. So the only thing one can do is to give a description confining this concept (I make use of some points from a similar characterisation proposed by Hans Rott in 1994). (1)

Even very simple deterministic systems with only few elements may exhibit a truly contingent conduct. The large majority of deterministic systems exposes that kind of behaviour, instead of a neglectedly small number of them, as it was hitherto always believed. What is more, this contingency is essential in two respects: (a) (b)

1

It holds not only for some exotic initial conditions of these systems, but over a wide range of possible states. Gathering more information about the system and its initial conditions does not make it go away  it does not depend on our subjective ignorance, on the lack of knowledge about the system. As Prigogine and Stengers (1984, p. 263) put it “Absolute precision  the fixation of absolutely correct values of the relevant parameters  would help, but it is unattainable.” A perfectly exact statement of the initial

Of course, we talk about chaos in the modern sense of the word, first used perhaps by Maxwell when writing about the state of “molecular chaos.” In the literature of the discipline called Chaos Theory later on, the concept appeared for the first time in the title of a paper by Li and Yorke “Period Three Implies Chaos” in (1975).

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conditions cannot be obtained  unavoidable measurement errors, approximate representation of data by the information processing machine, and even the mathematical form of otherwise equivalent equations of motion lead to deviations from the actual state of the system. (2)

(3)

(4)

(5)

These small deviations do not tend to fade out in the further course of the process. On the contrary, they will quite often build up to clearly perceptible effects. This leads to disastrous consequences for the calculation of the further development of the system: the initial errors are amplified to such an extend that the behaviour of the system cannot be calculated with any reasonable certainty after passing some time horizon. Especially iterative processes are characterised by errors exploding in time. All information get lost when, as Edward Lorenz had it, “error overrides signal.” The weather forecast: “Temperature on Saturday will be 18°C/24°C” is quite worthless. Let us mention however that in case of chaotic systems only the long time behaviour is unpredictable  it might well be possible to anticipate close future states of the development of the system. That distinguishes chaotic systems from truly contingent ones. Thus determinism and predictability, which had so often been taken to mean the same thing, get separated. It is only the latter which should be associated with the concepts randomness and chance. As a consequence, there is no incompatibility between determinism and chance. Chaotic systems may react sensitive to minimally changing initial conditions. Alike causes do not always have alike effects: the similarity invariance of causation is violated. What is more, there may appear completely “unforeseen,” i.e., contingent events in the process, which are “caused” by modifications of the initial conditions that were neglected (sometimes called “variable causes”). Chaos theory offers the hypothesis that in case of chaotic systems the so called “basins of attraction,” i.e. the realms of initial conditions leading to the same future state of the system, have extremely complex and subtle, self-similar geometric structure: they form fractals. The present situation of such a system does not reveal what will happen in near future.

Whoever experienced a zooming into the Mandelbrot set will feel the analogy: “To measure more precisely” means to enlarge the scale of

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magnification. However, by each enlarging of the image we obtain the same filigree of this self-similar figure. No measurement, as precise as ever you like, will lead us to “safe” areas: always there will be a border line close by at the back of which the behaviour of the system changes significantly. Of course it is (4) our special interest is devoted to. If the interpretation putted forward by chaos theory is right and if in fact the principle of invariance of similarity is characteristic for causal nexus, then we have a problem when putting things together.

4. Similarity Imagine a game of darts. To hit the centre of the target and to punch the innkeeper are very dissimilar events, though the throwing angle of the dart and the punch may be almost the same. Are the causes of these dissimilar events similar? Does that mean a failure of the Humean principle of invariance? These kinds of example were lively discussed long before chaos theory became an autonomous discipline. Maxwell investigated the case of a switch-man who brought about either a railway accident or its avoidance by minimal alteration to the switches on his desk. It was quite obvious that causal analysis has to deal with such questions. If seemingly alike situations give rise to completely different effects, then it is a natural idea to assume that something essential was disregarded. Yet, taking into account what was previously left out brings it out that actually both situations where not similar at all. That is Anscombe’s assumption of relevant differences: “If an effect occurs in one case and a similar effect does not occur in an apparently similar case, there must be a relevant further difference” (see 1971, p. 63). So far so good. But what, if there is no reasonable difference? Examples abound. Just recall the notorious butterfly at the Amazon river that causes a tornado in Boston, MA. How to argue for a convincing dissimilarity of an alternative situation in which anything in the Americas remains the same except that one butterfly will be caught by a Brazilian spider just in time? One idea is to agree that we are in trouble here and therefore one should try to save whatever is left from the traditional concept of causality. To that purpose, let us call a causal connection which fulfils the invariance principle (Similar causes always bring about similar effects) a “strong causal relation.” It is this kind of strong causality which allegedly breaks down in chaotic environments. Yet one has still the

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canon The same cause always brings about the same effect. That is the kind of causality, sometimes called “weak causality,” which suffices for the Laplacean demon. Needless to say, that (although not pointless as a methodological principle) it is entirely futile from an epistemic point of view: Since the antecedence conditions of the sentence describing the demon are never realised, this sentence is true for purely logical reason (because its antecedent is false). It is a tautology and thus, like all tautologies, “epistemologically empty.” No predictions or explanations of real facts can be based on it. It is thus not a good idea to confine ourselves to weak causality. At least as long as one is interested in doing theory of science instead of, say, formal metaphysics. So, perhaps, one could try to work out an accordingly modified variant of the notion of similarity? A first idea points to metrical or geometrical correspondence of properties. Yet, it is not perfect agreement, not indiscernibility of any differences, that we are interested in: from such a point of view only identical things would turn out to be similar. What we are interested in are “similar things” which may multiply occur, which may recur regularly. Otherwise we would be left with the tedious demon again. Therefore we cannot simply adopt the mathematical notion of similarity. What we need is rather a kind of “fuzzy agreement” of relevant properties, a limited variation thereof. If you try to make explicit what is meant by such a threshold value, you will face a threatening circular argument. Even if you manage to avoid circularity, you will run into trouble anyway: think about a human face which is not recognised because of varying make up and illumination. Yet at least identical things should turn out to be alike. The strategy to concentrate on selected, relevant properties only is no remedy at all. How to understand “relevant” differently from “having influence,” “being efficient”? It would be preferable if a definition of causality would not involve such ambiguities. Maybe the story is much more simple. Maybe there is no “good” definition of causality based on similarity. Events are similar if and only if their effects belong to the same class. Similarity of events means nothing but leading to the same effects. Then the principle of invariance of similarity is a plain tautology and the so called breakdown of strong causality is just a misunderstanding. We return to Hume’s claim that there is no such thing like causality. If somebody were to keep asking him what a cause is, he’d be given just that circular definition of causality by means of similarity. Hume’s joke would consist in prompting the ongoing exertions of philosophers who try to make sense of it – without any chance to succeed. Often enough articles come up

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with the reproach of circularity, since causal motifs are employed in order to explicate similarity and the notion of similarity is intended to define causality subsequently. So, why not to give up these futile efforts and to admit that the proposal just does not work?

5. Events for Causes If one does not agree with this radical suggestion, then one should think about the tornado – causing butterfly once again. Suppose some specific causal analysis reveals the butterfly as the true cause of some specific tornado taking place in the other hemisphere. Then, obviously, we have something like a “physical categorical failure”: The behaviour of a single butterfly is never the cause of a tornado, both events have nothing to do with each other. Let us try to make more explicit the feeling of unsuitability between the events “the tornado destroys Boston” and “the butterfly flutters by” as its alleged cause. Neither common sense nor any of the empirical sciences deal with relations of that kind. Meteorology, on the one hand, does not care about individual butterflies, and zoology, on the other, has nothing to do with weather processes on other continents. An image of the world, which reveals global weather-conditions is much too coarse grained to show individual butterflies. In order to be more specific, we first of all need an appropriate notion of event. In what follows, I shall draw on the concept of event developed in Urchs (1993). The most important point of that definition was its relativisation to what I called an epistemic system  some general information  gathering and utilizing system, characterized by its knowledge basis, its inference apparatus and its specific perceptual abilities. Thereby the concept of event (as well as these of a state of affair, fact, object etc.) splits up into a manifold of concepts, indexed by the underlying system: “to be an event” means “to be an event with respect to the considered system” or more concisely “to be an event for a system.” Some of the events (for a system) are connected and some of those connections may be causal (for the considered system). A claim like “causality is universal” is surely not a true statement: it occurs for some systems only – according to the kind of filtrations constituting the system. If these filtrations are sufficiently weak then no causal relationships can be isolated among the chaotic interactions figuring on the screen. If they are too strong then the world-view may not contain

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any events at all. All that remains in such a image of the world is functional dependence between its abstract objects. An example of the latter is any mathematical theory or some parts of theoretical physics (there is no causality in it any longer), the former is illustrated e.g. by some primitive forms of purely descriptive accounts in social sciences (causality does not yet appear therein). Only if the filtrations are in between, i.e. if the abstraction level is neither too low nor too high, is the causal nexus screened. It goes without saying that there is nothing on the screen without an origin in the world. In that sense, causality is a projection of something real on the screen. What is essential is that it depends on the screen (i.e. on the epistemic system) whether causality appears in the world view.

6. Either Causes or Chaos It is the same with chaos. Chaotic behaviour of dynamical systems appears on the screen but it has its origin in physical reality. We already mentioned the trouble with a precise explanation of deterministic chaos. Sometimes chaos is exemplified by kneading dough, sometimes it is helpful to make use of aesthetically appealing Julia sets. In any case, in a final definition, one has to employ solid (and hard) mathematical theories: ergodic theory and the theory of dynamical systems. To that purpose a dynamical system is characterised by appropriate mathematical modelling in the phase space of its strange attractor, which itself is determined by differential equations. Chaotic behaviour of the dynamical system is then depicted by specific properties of the describing system of non-linear differential equations. Whatever are the mathematical details of this story, one thing should be pretty obvious: we can talk about chaotic behaviour of dynamical systems only after appropriate mathematical modelling. The claim that some real system reveals chaotic behaviour thus presupposes a description of the system given on relatively high level of abstraction. Only on this high level of abstraction “chaos is screened” – whereas causality has already become extinct on such a level. You may have either causality or chaos on the screen or, if you like, in the world view – yet not both of them at the same time. So the problem gracefully disappears: there is no such thing as chaotic causality nor causal chaos.

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Universität Konstanz Department of Philosophy PF 5560 D22 D-78-434 Konstanz, Germany e-mail: [email protected]

REFERENCES Anscombe, G.E.M. (1971). Causality and Determinism. Cambridge: Cambridge University Press. Reprinted in: E. Sosa (ed.), Causation and Conditionals, pp. 63-81. Oxford: Oxford University Press. Gell-Mann, L. (1994). Das Quark und der Jaguar: Vom Einfachen zum Komplexen – die Suche nach einer neuen Erklärung der Welt. München: Piper. Prigogine, I., I. Stengers (1984). Order Out of Chaos: Man’s New Dialogue With Nature. Foreword by A. Toffler. Toronto / New York: Bantam Books. Rott, H. (1994). Chaos: The Reason for Structural Causation. In: J. Faye, U. Scheffler and M. Urchs (eds.), Logic and Causal Reasoning, pp. 191-215. Berlin: Akademie Verlag. Stelzner, W. (1984). Epistemische Logik – Zur logischen Analyse von Akzeptationsformen. Berlin: Akademie-Verlag. Urchs, M.P. (1994). Causal Priority: Towards a Logic of Event Causation. In: G. Meggle and U. Wessels (eds.), Analyomen 1: Proceedings of the 1st Conference “Perspectives in Analytical Philosophy,” pp. 386-396. Berlin / New York: de Gruyter.

Jan WoleĔski THREE CONTRIBUTIONS TO LOGICAL PHILOSOPHY 1

1. Permissions, Prohibitions and Two Legalisms Any normative system, specifically a legal one, tacitly assumes certain general closure principles that establish relations between the spheres of actions subjected to particular modes of normative regulation. Above all, it is important to have a clear account of the relations between actions that are permitted and those that are prohibited. The closure principles related to this question are modelled on two statements, namely: (I)

if something is not prohibited then it is permitted

and (II)

if something is not permitted then it is prohibited.

Principle (I) is usually attributed to British-style legal systems, which are said to be oriented towards permissions as something primary. On the other hand, (II) is often considered to be an expression of German legal consciousness, which is based on the idea of the primacy of obligations. For brevity, I will refer to these views as British legalism and German legalism, respectively. In this section, I offer a logical analysis of (I) and (II). I take the concept of legal requirement (Q) as primitive. Now legal requirements are divided into obligations (O) and prohibitions (F); we could also speak about positive obligations (obligations) and negative obligations (prohibitions). These normative modes are mutually related in the following ways (the letter A is a schematic propositional variable): 1

The term ‘logical philosophy’ was recently proposed by Jerzy Perzanowski as referring to the results of philosophy obtained by normal logical analysis. According to Perzanowski, logical philosophy goes back to the metaphilosophical ideas of Bertrand Russell and Jan àukasiewcz.

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 195-213. Amsterdam/New York, NY: Rodopi, 2006.

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(a) (b)

FA œ O™A OA œ F™A.

Assuming the standard deontic logic, there are three possibilities of defining the concept of permission (P): (2)

(a) (b) (c)

PA œ ™FA PA œ ™OA PA œ ™FA š ™OA.

Formula (2a) defines permission as non-prohibition, (2b) as nonobligation. Now the concept stemming from (2c) is commonly regarded as not particularly important. Thus, (2a) and (2b) remain here and we need to distinguish both. Let us replace ‘P’ with ‘I’ in (2c), which gives: (3)

IA œ ™FA š ™OA

where ‘IA’ means ‘it is indifferent that A’. The formal relations which hold between the introduced concepts are represented by the following diagram (D l):

H

D

E

J

G

M Diagram (D 1 )

Let the Greek letters serve as abbreviations in the following way: D = OA, E = FA, J = PA, G = ™OA, H = QA (note that QA œ OA › FA),

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and I = IA. We have several dependencies as theorems of (standard) deontic logic: (4)

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)

DŸJ ™(D œ G) ™(E œ J) ™(D š E) DŸG J›G DŸH EŸH IŸJ IŸG ™(H œ I) D › E › I.

From (4c) we immediately derive (5)

(a) (b)

PA œ ™FA ™PA œ FA.

Now (5a) says that both sentences ‘A is permitted’ and ‘A is not prohibited’ are exactly equivalent. According to (5b), the same holds for ‘A is not permitted’ and ‘A is prohibited’. 2 The extension of the formula ‘PA’ is not homogeneous in a sense, because (6)

PA œ OA › IA.

Thus, when we say that something is permitted, we do not determine whether it is obligatory or indifferent. This defect (pragmatic, but not logical) is not inherent in the formula (7)

IA œ ™QA,

which expresses the thought that “something is not required if and only if it is indifferent.” (7) entails (8)

™IA œ QA,

i.e. “something is not indifferent if and only if it is required.” Assume that (7) is a formal wording of British legalism, and that (8) captures the 2 I regard deontic sentences de dicto (for example, ‘it is permitted that A’) and de re (for example, ‘A is permitted’) as equivalent. Nothing in this paper depends on the de dicto / de re distinction.

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German way of thinking. Are both legalisms logically indistinguishable? A purely formal answer is: yes. Yet something can be said about (7) and (8) from a pragmatic point of view, when we consider the concept of a normative regulation. Let U be a universum of possible human actions. By a normative regulation of U, I mean its division into three, for consistency, mutually exclusive domains (see (41)): of obligations, of prohibitions, and of indifferences (or simpler, into two domains: of requirements, and of indifferences).3 How can it be done? The only effective way consists in giving a finite set of primitive requirements S0 . The full system of requirements S can be defined by (9)

S = CnS0 ,

that is, as the set of logical consequences of S0 . The domain of indifferences is defined as the complement of S. Formally speaking, the domain of requirements is the complement of the domain of indifferences. However, normative regulation by establishing a set of indifferences is not effective. In order to see how it works, let us consider a completely free society, that is, such that it was not already subjected to any normative regulation in which requirements were involved. This means that any possible action is free in this society in the sense that one can do it and also abstain from doing it. If we say that an action A is indifferent in this society, we do not assert anything new. 4 Therefore, the domain of indifferences is secondary with respect to that of requirements, not conversely. In general, saying that something is indifferent in isolation from any requirement cannot be regarded as a normative regulation; its sphere of indifferences is rather a by-product of an already given regulation. The primary role of obligations or prohibitions is derivable from logic itself because I cannot serve as the sole deontic primitive. This fact should not be confused with the primacy of obligations, according to German legalism. The above analysis shows that although the closure principles associated with both legalisms are expressed by the tautologies of deontic logic, they are not equivalent. Then we look at them as pointers for normative regulations. In particular, (II) is sterile in this regard. Hence, (I) closes normative systems in a rational way, but (II) does not, because the catalogue of indifferent 3

More precisely: of obligatory, prohibited and indifferent actions. The context also determines whether the matter concerns arguments of deontic operators, that is, sentences or actions described by those sentences. See also footnote 2, above. 4 Probably it would be very difficult to teach the members of our “free” society what the adjective ‘indifferent’ actually means.

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actions is a result of the stock of requirements. This means that it is impossible to give an effective list of indifferences without looking at obligations and prohibitions. Briefly: pragmatic aspects of normative regulations are not reducible to logical dependencies between deontic operators. However, my analysis of German legalism may be subjected to criticism because one can observe that it is based on an improper concept of permission. According to this view, the correct formulation of German legalism is: (10)

if A is not explicitly permitted then it is prohibited.

We need to explain the concept of explicit (or strong) permission.5 The main claim is that that strong permission is irreducible to negation and the concepts involved in diagram (Dl). This means that we have a new concept of permission, which must be taken as a new primitive. It is intuitively captured in the following way: (11)

A is strongly or explicitly permitted if and only if there is a rule that says “you can do A.”

Unfortunately, (11) generates no explicit logical properties of strong permission. Assuming consistency as a desirable property of any normative system, we have only (‘Ps’ stands for ‘it is strongly permitted that’) (12)

PsA Ÿ ™FA.

The converse of (12), that is, the formula (13)

™PsA Ÿ FA,

is not admissible for strong permission because it would result in the collapse of Ps into P. However, (13) is fairly indispensable as a logical justification of German legalism in version (9), unless we admit that this form of legalism is based only on an informal closure principle. One can say: why not? Well, although a priori there is nothing wrong here, but perhaps it is more rational to rely on principles that have clear logical justifications. Do strong permissions produce normative regulations? It seems that they do not because (12) leads only to a conclusion that if something is strongly permitted then it is not prohibited (we can put forward an even stronger thesis: if something is strongly permitted then it is not required). 5

My further analysis of strong permission repeats main points of Opaáek and WoleĔski (1973, 1986, 1991).

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However, neither (12) nor its extension to requirements suffices to give a normative regulation of U. On the other hand, the view that strong permissions are needed for an adequate description of normative systems is quite common. I think that this view has two sources. Firstly, in normative discourse, we encounter phrases which suggest that ‘can’ means something else than ‘it is permitted’ or ‘it is indifferent’. Consider the message “you can smoke here.” Apparently, it seems to give the right to smoke in this place. But it means only: outside the area of a certain compact public space (for example, a railway-station), smoking is prohibited. Similarly, if a commander says to his soldiers “well, you can smoke now,” he cancels an explicit or tacit earlier order of non-smoking. Thus, phrases with ‘can’ can be interpreted as modifications of requirements, in particular, as exceptions from general obligations or prohibitions. These exceptions lead to indifferences because if there is a cancellation of an order, for example, of non-smoking, it refers to an action which is fairly indifferent like smoking, in our example. The only serious problem concerns ‘can’ in procedural regulations (see ZiembiĔski 1970, for a more complete analysis). Assume that a committee can dissolve itself before the end of its tenure. This is an example of a competence rule, which is usually analyzed as a strong permission, irreducible to requirements. However, we can treat this rule as a conditional obligation: if a committee dissolves itself etc. then this must be respected by everybody, that is, by every individual and every institution. Even if we agree that strong permissions are actually conditional obligations, we must face the question of a closure principle in this case. The problem is that actions governed by conditional procedural obligations (or strong procedural permissions, if one prefers) do not seem to be indifferences. Hence, (I) is not applicable to such cases. On the other hand, we have the following good intuition: (14)

an institution has only those competences that are explicitly stated in a related normative system.

Now, we see that (14) is something very similar to (9) because it can be transformed into (15)

if an action of an institution is not strongly permitted then it is prohibited.

Since (9), as I already noted, is not a logical rule, (15) seems to have no logical justification, unless we extend our standard deontic logic in order to cover conditional obligations. Since I do not know any unquestionable

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logic of such obligations, my reasoning will be only semiformal. Let the formula ‘A Ÿ OxA’ mean ‘if A is done, then obligations x come into being’.6 Since I regard strong permissions as conditional obligations, we can adopt the definition: (16)

PsA œ (A Ÿ OxA).

Also I assume that (17)

(a) (b)

if O xA then ™IA ™(A Ÿ O xA) œ A š ™OxA.

Now we are in a position to derive (13). First, we observe that: (18)

™PsA œ A š OxA.

Since the operator Ox satisfies the usual deontic laws, we have: (19)

™OxA œ FA › IA.

By (17a), we obtain that A must be prohibited because it cannot be indifferent. And this was to be proved. Perhaps a more proper reading of this result is: it is prohibited to regard obligations x as binding but I will not discuss other possible interpretations. If one wants to accept German legalism as something different from English legalism, it is reasonable only with respect to institutional competences. It is interesting that English legalism is of no use in this case because the lack of prohibitions does not create any competence. It is sometimes said that German legalism restricts personal freedom. In fact, making personal freedom dependent on explicitly stated permissions automatically leads to its limitations since, in the sphere of actions of individuals, (9) is not only logically unjustified but also inconsistent with the principle of the rule of law. On the other hand, this principle seems to invalidate English legalism in the region of institutional competences. The above analysis leads to a conclusion that is perhaps unexpected: the principle of the rule of law has its explicit logical foundation. English legalism has been recently questioned by Catholic authorities due to its connection with the so-called permissivism. More specifically, (I) is seen in this context as a justification for violating the natural law. This criticism suggests the following closure principle: 6

Note that this formalization admits so-called mixed formulas, in which sentential variables occur as self-contained parts of deontic formulas. For example, the expressions ‘A › OA’ and ‘ A › PA’, are examples of mixed formulas. Such inscriptions are not wellformed in standard deontic logic.

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A is legally permissible if and only if it is permitted by the natural law system, otherwise it is prohibited.

However, (20) is not a closure principle. It is rather a claim that positive legal systems should agree with the natural law, that is, a demand that the two normative systems should be mutually consistent. If (20) were to be interpreted as a closure principle, no other way would remain than to interpret it as (9), that is, in terms of German legalism. The point is that the phrase ‘otherwise it is prohibited’ is ambiguous. If it refers to a concrete positive law, it introduces external duties that can be interpreted only as derivatives of the natural-law permissions, which serve as strong permissions for legal systems. On the other hand, the natural law itself is also subjected to closure principles, in particular to (I). Thus, it is incorrect to say that (I) has its applications only with respect to the positive law, but (20) is valid for the natural law. My remarks do not suggest that the content of legal systems is to be arbitrary. However, we must sharply distinguish between two things: closure principles and claims concerning the content of the law. Their confusion can produce similar dangers to the rule of law as German legalism applied to personal freedom.

2. God, Foreknowledge and Freedom Probably the question of human freedom and responsibility in the context of God’s attributes is the most dramatic horror theologicus. Of course, I am not the first who doubts that God’s omnipotence, omniscience, foreknowledge together with the free creatio ex nihilo leave any space for human freedom (or free will, if one wish) and responsibility. Let me abbreviate this view as (V). This section is devoted to a refutation of Plantinga’s criticism of (V) (see Plantinga’s 1974, pp. 66-73; the quotations are from p. 67). According Plantinga, this view is based on two assumptions: (21)

if God knows in advance that a person X will do A then it must be the case that X will do A

and (22)

if it must be the case that X will do A then X is not free to refrain from doing A.

Now, both (21) and (22) entail:

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if God knows in advance that someone will take a certain action, then that person isn’t free with respect to that action. But [(21)] bears further inspection. Why should we think it’s true? Because, we shall be told, if God knows that X will do A, it logically follows that X will do A: it’s necessary that if God knows that p, then p is true. 7 (Plantinga 1974, p. 67)

Plantinga then says that “this defense” of (21) actually suggests that (21) is ambiguous. It may be understood either as: (23)

Necessarily, if God knows in advance that X will do A, then indeed X will do A

or as: (24)

If God knows in advance that X will do A, then X will do A.

Plantinga refutes (V) by pointing out that it requires (24), but the quoted reasoning justifies only (23). I will argue that (23) is sufficient for (V). In particular, I will prove that (25)

it is necessary that X will do A

is true, provided that God is omniscient and that He has universal foreknowledge. The demonstration is this (for simplicity, I will write ‘KG(B)’ instead of ‘God knows in advance that B’, ‘w(X, A)’ instead of ‘X will do A’, ‘LB’ instead of ‘it is necessary that B’, and ‘LCKG(w(X, A))w(X, A)’ instead ‘it is necessary, that if KG(w(X, A)), then w(X, A)’. (26)

if G is omniscient, KG on X ’s future actions

(27)

G is omniscient

(28)

KG on X ’s future actions

(29)

assume that w(X, A)

(30)

KG(w(X, A))

(31)

if KG(B) then LKG(B)

(32)

if KG(w(X, A)) then LKG(w(X, A))

(33)

LKG(w(X, A))

(34)

LCKG(w(X, A))w(X, A)

(= (23))

7 This line means: if it is necessary that if God knows that X will do A, then X will do A, then if God knows that it is necessary that X will do A, then it is necessary that X will do A.

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(35)

if LCKG(w(X, A))w(X, A) then CLKG(w(X, A))Lw(X, A)8

(36)

if LKG(w(X, A)) then Lw(X, A)

(37)

Lw(X, A)

(= (25))

Step (26) follows from the definition of omniscience; (27) states an attribute of God; (28) follows from (26) and (27) by Modus Ponens; (29) is an assumption; (30) is an instantiation of (28) with respect to A; (31) follows from the definition of God’s knowledge: if God knows something, then necessarily God knows it; (32) is the result of substituting ‘w(X, A)’ for B in (31); (33) follows from (30) and (32) by Modus Ponens; (34) is Plantinga’s explicit assumption; (35) is an application of the principle of the distribution of necessity over implication: L(B Ÿ C) Ÿ (LB Ÿ LC); (36) follows from (34) and (35) by Modus Ponens; (37) follows from (33) and (36) by Modus Ponens. I see no objection against (26), (27) or (31). Assumption (29) must be made because the entire discussion would be pointless without it. Moreover, (29) is accepted by Plantinga himself. The rest is just logic. If my refutation of Plantinga is right. (V) has firm grounds. Hence, it is very difficult to maintain that the Christian conception of God is compatible with human freedom. Now, if we claim that responsibility requires freedom, the idea that people are responsible for their actions before God is actually absurd. This situation can be perhaps overcome by an act of faith, according to the maxim “credo, quam absurdum,” not by means of rational argumentation.

3. Determinism and Sentences about the Future The problem considered in this section has some affinities with the discussion in §2, but it is free from any theological connotations. 9 Consider the following claim (in this section, variable A ranges over states of affairs): (38) 8

It will happen that A.

The question of “determinism and the logical value of sentences about the future,” has a very long history. I have no possibility here to touch on previous discussions even in a modest way. Perhaps I will mention only that my considerations refer to Twardowski’s defence of the absolute character of truth, LeĞniewski’s argumentation against indexing truth by time (although we can say that ‘a is b at moment t’ is true, we should not say that ‘a is b’ is true at t), and Jordan’s (1963) claim that ontological and logical issues must be sharply distinguished. Some related problems are discussed in Placek (2004). 9 Thus, indexing the truth predicate by time is not essential.

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Now, the question arises whether (39) is true or false at the present time, i.e. at the moment of issuing this sentence. There is a well-known view, called “logical determinism,” that if every sentence is either true or false at every moment t, then the future is completely determined by the past. This determinism is called “logical” because it links the (semantic or metalogical) principle of the excluded middle or the principle of bivalence with determination of the future by the past. Let (39) be specified as: (39)

I will be in Warsaw next week.

Logical determinism maintains that if (40) is either true or false at every time, in particular before “next week,” my visit (or its lack) in Warsaw is sempiternally decided, and this justifies radical determinism (fatalism). It is a very serious problem for the advocates of the absolute character of truth (ACT) who would simultaneously like to reject fatalism. (ACT) is the view that (40)

if a sentence is true at a moment t, it is also true at any other moment t 1 such that t z t 1 .10

(ACT) can be split into: (41)

(a) (b)

if a sentence is true at t then it is also true at t 1 , where t 1 > t. if a sentence is true at t then it is also true at t 1 , where t 1 < t.

Statement (42a) asserts that truth is eternal (if something is true now, it never becomes false), but (42b) says that truth is sempiternal (if something is true now it has always been true). The eternity of truth does not cause any trouble for the opponents of determinism. Thus, the sempiternality of truth (and falsity, of course) is the main problem here. My further remarks are intended to show that (ACT) is completely independent of the determinism/indeterminism controversy. As far as the matter concerns the principle of the excluded middle (EM) and the principle of bivalence (BI), I will argue that none of them implies radical determinism. This means that in order to defend human freedom (or free will) we do not need to reform semantics or metalogic. I explicitly indicate that my task consists in such a defence of (ACT) that would be consistent with determinism, on the one hand, and with indeterminism, on the other.

10

I make no specific assumptions on determinism, necessity and related concepts. My reasoning works for any mechanism invented for showing that the future is completely dependent on the past as well as for absolute or relative understanding of necessity.

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Let us consider (40) once more. When I utter this sentence, I can have various things in mind, for example, that I will probably visit Warsaw next week, or that I intend or hope to do so, or that I should be there on business or because of my friend’s wedding ceremony, etc. All these circumstances are of no importance to our issue as a philosophical problem. On the other hand, I definitely mean that this event (that is, my visit in Warsaw) is possible. I do not say that (40) has the same meaning as: (42)

It is possible that I will be in Warsaw next week.

However, I think that whatever relation holds between (40) and (43), that is, whether the latter is regarded as a part of the meaning assumption or consequence of the former, (43) is a sentence that has the basic import for the truth (or falsity) of (40). In general, I guess that the problem of the truth of sentences about the future is essentially dependent on our treatment of: (43)

It is possible that A will happen.

Therefore, if we intend to defend (ACT), we need to offer an interpretation of (44). It is sufficient to read it as: (44)

It is possible at t d t 1 that A will happen at t 1 .

In particular, if we apply (45) to (40) we obtain: (45)

It is possible today that I will be in Warsaw next week.

It is rather unquestionable that (46) expresses a part of the intentions of a person who utters (40) in a precise way. Now we have no reason to reject: (46)

If the sentence ‘it is possible at t d t 1 that A will happen at t 1 ’ is true at t, then it is also true at any t 2 such that t 2 d t 1 .

The last formula expresses (ACT) for sentences such as (45), and a fortiori for sentences such as (46) and (39). Disregarding some simplifications, (47) means that if something is possible then it has always been possible. What remains is to show that (47) is equally consistent with determinism as well as indeterminism. Let us return to diagram (Dl) from §1. We can replace the symbols O, F, P, Q, I by L, S, M, D, C, respectively. We can interpret ‘LA’ as ‘A is necessary’, ‘SA’ as ‘A is impossible’, ‘MA’ as ‘A is possible’, ‘DA’ as ‘A is determined’, and ‘CA’ as ‘A is contingent (accidental)’. Moreover, all logical dependencies

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derived from (Dl) are preserved on this interpretation. We can propose the following definitions (the symbol f in (48g-h) denotes a fixed state of affairs): (47)

(a) (b) (c) (d) (e) (f) (g) (h)

radical determinism: ADA (¾ A (LA › SA)) radical indeterminism: ACA minimal determinism: ADA minimal indeterminism: ACA moderate determinism: ADA š ACA moderate indeterminism: ACA š ADA determinism with respect to f: Df indeterminism with respect to f: Cf.

Thus, radical determinism claims that everything is determined (every state of affairs is necessary or impossible); radical indeterminism – that everything is contingent (every state of affairs is contingent); minimal determinism – that at least something is determined (there are determined states of affairs); minimal indeterminism – that at least something is contingent (there are contingent states of affairs); moderate determinism – that at least something is determined and at least something is contingent (there are determined as well contingent states of affairs); moderate indeterminism – that at least something is contingent and at least something is determined (there are contingent as well as determined states of affairs); determinism with respect to a given state of affairs f – that f is determined; indeterminism with respect to a given state of affairs f – that f is contingent. Moderate determinism and indeterminism can differ only in the distribution of determinacies and contingencies.11 It is sufficient to consider our problem in the context of (48g) and (48h). Let f be a fixed state of affairs. Like operator P, operator M is not homogeneous, i.e.: (48)

Mf œ (Lf › Cf ).

If we consider Mf and Lf to be equivalent then, in effect, we decide that determinism with respect to f is right, but if we consider Mf and Cf to be

11 This means that I reject the principle “unumquodque, quando est, oportet est.” The meaning of ‘if A is the case, then ™ A is impossible’ is not clear for me. Using the operator R, we have ‘if A is real, then ™ A is not real’. The phrase ‘unumquodque quando est, est’ is sufficient for expressing the ontological principle of non-contradiction. I do not deny that ‘unumquodque, quando est, oportet est’ holds for tensed necessity, but even if the sentence ‘if A is real now, it is necessary now’ is true, it does not exclude that ™ A was possible in the past.

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equivalent, we choose indeterminism with respect to f. Now, if we want to be neutral, we must stay with Mf. Since: (49)

(a) (b)

Lf Ÿ Mf Cf Ÿ Mf,

admitting that f is possible is consistent with determinism as well as indeterminism. Further, if (47) states the truth – condition for sentences about future events, determinism and indeterminism are also consistent with (ACT). Quod erat demonstrandum. The alternative view, namely that future contingencies require the rejection of sempiternality of truth, results from taking into account only A (™A, respectively) instead of ‘A is possible’ (‘™A is possible’, respectively). Now I pass to (EM) and (BI). The formal definition of radical determinism can be written as (50)

A (LA › L™A).

It is easy to prove that the usual (EM), that is, the formula (for a moment, let A be a sentential variable) (51)

A › ™A

does not entail (51). If we apply the rule of necessitation (52)

A o LA

to (52), we obtain (53)

L(A › ™A).

However, (54) does not entail (51). Due to: (54)

(LA › LB) Ÿ L(A › B)

and the rule of eliminating the necessity sign, we can derive (52) from (51). This shows that if there is a logical link between radical determinism and (EM), it is just the reverse of what one usually maintains because it is determinism that implies (EM) rather (EM) that implies determinism. This result is philosophically interesting in itself but it is not satisfactory for four reasons. First, I claim (see footnote 10, above) that logic and ontology should be separated at least logically, that is, logical theorems are not premises for ontological theses, and vice versa. Second, (52) has no clear ontological meaning. Third, I would like to prove that also indeterminism entails (EM). This means that the rejection of (EM) should not entail indeterminism without further ado.

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Fourth, as will be soon clear, diagram (D2 ) introduced below gives no way to explain the role of (BI) for the issue under consideration. In order to go deeper, let me extend diagram (Dl) to (D2 ). The letters F and O refer to sentences ‘RA’ and ‘R™A’ which means respectively ‘A is real’ and ‘™A is real’ (œ ‘A is not real’). We have: (55)

(a) LA Ÿ RA (b) L™A Ÿ R™A (c) RA Ÿ ™R™A (d) RA Ÿ MA (e) R™A Ÿ M™A

H D

E

F

O

J

G M Diagram (D 2 )

However, diagram (D2 ) does not exhaust all relations between its modalities. In particular, it suggests nothing about the relation between the real and the contingent. In fact, (D2 ) describes the contingent mainly as the future contingent, possible but unrealized as yet. Let us look once more at (40). My visit to Warsaw is something future and contingent at every moment before its planned date. But what about the same fact after

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the date when it happened? It is something real and still contingent.12 In general: (56)

A is real and contingent if and only if A is real, ™A is not real, A was possible and ™A was possible.

Since LA implies RA and RA implies MA, the real comprises the determinate real and the contingent real. However, this observation does not entail that there is either something determinately real or contingently real. (D2 ) exhibits only the necessary relations between modalities, but the distribution of determinacies and contingencies is a factual matter. Logically speaking, we have only that if something is necessary, it is automatically real. As I already noted (52), that is the usual (EM) has no clear ontological import. We rather need the ontological excluded middle. Let the symbol ‘K’ denote any modality involved in diagram (D2 ). Thus the ontological (EM) is represented by the scheme (57)

A(KA › ™KA)

and its particular instances are given by (58)

(a) (b) (c) (d) (e)

A A A A A

(LA › ™LA) (SA › ™SA) (RA › ™RA) (MA › ™MA) (CA › ™CA)

None of (59a)-(59e) implies (51). This means that the ontological excluded middle does not entail radical determinism. What about the reverse connection? Of course, (59a) leads to (54), universally or existentially quantified, that is, AL(A › ™A) or AL(A › ™A). However, none of them expresses (EM). The former is a consequence of radical determinism, but the latter states minimal determinism. If one rejects the universal closure of (54), minimal indeterminism is admitted, but if the existential closure of (54) is abandoned, radical indeterminism is admitted. In fact, (59a), (59b), (59e) and (59e) have never been questioned by the advocates of indeterminism. (59c) is the statement that raises problems, perhaps because it is connected with (BI). This principle is expressed in our language by 12 These assumptions are motivated by an analogy between ‘is real’ and ‘is true’. Note, however, that the analogy is only partial because I do not accept a counterpart of (ACT) for R; see the closing remarks of this paper.

Three Contributions to Logical Philosophy

(59)

211

A ((RA › R™A) š ™(RA š R™A)),

which is equivalent to (60)

A (R™A œ ™RA),

Haying (61) we can strengthen (56c) to: (61)

A (RA œ ™R™A).

Thus, (EM) has its other version in: (62)

A (RA › ™RA).

Of course, neither (EM) nor (BI) imply radical determinism. Let me investigate what will happen if we reject the ontological version of (EM) for operator R, i.e. (63). First, I shall introduce two assumptions (the symbol A denotes the contradictory state of affairs): (63)

(a) (b)

SA (RA š RB) œ R(A š B).13

Now, let us reject (59c). We obtain: (64)

A (RA š R™A).

Apparently (65) opens the door for contingencies because it entails (65)

A (MA š M™A).

However, (65) leads to a contradiction. At this point, let us observe that it gives (see (64b)): (66)

R( f š ™ f ).

Then, using (56d), we obtain (67)

M( f š ™ f ).

which contradicts (64a). Thus, the rejection of (EM) does not lead to indeterminism, but to inconsistency if (64a) is adopted. If we drop (BI), (64a) is also to be discarded. In this case (68)

A (RA š R™A).

does not lead to any trouble. Of course, it allows contingencies as something between reality and unreality. However, our basic logic is 13 In fact, àukasiewicz (1953, pp. 377-378) rejects (64a). In order to show that the rejection of (EM) leads to troubles, we can also weaken (64a) to RA. In this case, (67) ends the reasoning.

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changed. Let me observe that the controversy over the relation of logic to determinism also concerns (64a), that is the concept of possibility.14 Changing logic is one possibility of defending indeterminism. However, we can stay with either determinism or indeterminism on the basis of the modal extension of classical logic. What I have tried to show is that we can admit contingencies without abandoning classical logic, because (ACT), (EM) and (BI) are consistent with determinism as well as with indeterminism. This means that all instances of (58) are equally valid in our formal ontological schemes. The scheme displayed in diagram (D2 ), generates a certain formal ontology, which has various concrete solutions with respect to the determinism/indeterminism issue (see (48)). None of them is forced by logic. How to interpret (40), if (59c) holds? It is very simple, because until my visit in Warsaw is realized it is unreal without being determined. If it happens to be realized, it becomes real and still contingent, if it was such earlier. Necessity and contingency are timeless, reality is not. However, the truth that something is necessary (contingent, real, etc.) is always timeless. I do not decide here whether reality is eternal (the reality of the past is preserved over the future), but I reject the claim that reality is sempiternal. To sum up, the possibility of contingencies is saved when (ACT) is combined with the view that the real is at most eternal. This position is fully consistent with my earlier account of (ACT). I think that we can find a source of the view that a proper treatment of future contingencies requires the rejection of (EM) and (BI). It consists in reading J and G as statements which assert that A (™A, respectively) is possibly true, that it has a logical value which is different from truth and falsity. This way of thinking suggests that A may be outside of the kingdom of truth and falsity. However, possibility is not a logical value but a mode of being. If we read modalities ontologically, we would apply the predicate ‘is true’ to statements about various modes of being.15 Thus, if we consider the truth (or falsity) of the sentences ‘it is possible that A’ and ‘it is possible that ™A’, both these sentences can be accepted or rejected without violating (EM) or (BI) because they are not mutual negations. The trouble (for (EM) and (BI)) appears when A and ™A are taken as sentences about future contingencies. It forces a restriction of (EM) or (BI) to sentences about the past and the present. My way out consists in regarding A and ™A as nominalized sentences which eventually refer to future 14

‘Necessarily (possibly) true’ has a clear meaning for logical modalities. àukasiewicz observed (1971, p. 124) this possibility for the operator ‘it is true at t’ but he favored many-valued logic. 15

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contingencies but sentences with M as statements about future contingencies.

Uniwersytet JagielloĔski Department of Philosophy ul. Grodzka 52 31-044 Kraków, Poland e-mail: [email protected]

REFERENCES Jordan, Z. (1963). Logical Determinism. Notre Dame Journal of Formal Logic 4, 1-38. àukasiewicz, J. (1971). On Determinism. In: Selected Works, pp. 110-128. Amsterdam: North Holland. àukasiewicz, J. (1953). A System of Modal Logic. The Journal of Computing System 1, No. 3, 111-149. Reprinted in: Selected Works, pp. 352-390. Amsterdam: NorthHolland, 1970. Opaáek, K. and J. WoleĔski (1973). On Strong and Weak Permissions. Rechtstheorie 4, 369-384. Opaáek, K. and J. WoleĔski (1986). On Weak and Strong Permissions Once More. Rechtstheorie 17, 83-88. Opaáek, K. and J. WoleĔski (1991). Normative Systems, Permissions and Deontic Logic. Ratio Juris 4, 334-348. Placek, T. (2004). The Puzzle of Semantic Determinism. This volume, pp. 171-185. Plantinga, A. (1974). God, Freedom and Evil. Grand Rapis: Wiliam B. Eerdmans Publishing Company. ZiembiĔski, Z. (1970). Norms of Competence as Norms of Conduct. Archivum Iuridicum Cracoviense 3, 21-30.

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Andrzej WiĞniewski REDUCIBILITY OF SAFE QUESTIONS TO SETS OF ATOMIC YES-NO QUESTIONS

1. Yes-No Questions By a yes-no question we mean a question which can be satisfactorily answered by saying either “yes” or “no.” Usually the answer “yes” is regarded as an abbreviation of the sentence which can be obtained from the question by the conversion from the interrogative to the declarative mood, whereas the answer “no” is interpreted as an abbreviation of the negation of this sentence. According to the analysis which can be found in most textbooks, negation should be understood here in the sense of classical logic. For example, the interrogative sentence: (1)

Did John marry Joan?

is interpreted along the lines sketched above as expressing the question which has the following possible and just-sufficient (i.e. direct) answers: (2) (3)

John married Joan. It is not the case that John married Joan.

These answers contradict each other; the question must have a true direct answer and thus is a safe question. Let us call a question whose set of possible and just-sufficient answers consists of a sentence and its (classical) negation a simple yes-no question. It can hardly be said, however, that each natural-language yes-no question can be adequately analyzed as a simple yes-no question. Even the interrogative sentence (1) can be interpreted differently. Let us observe that (1) can be pronounced with the following intonations (among others): (4) (5)

Did John marry Joan? Did John marry Joan?

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 215-236. Amsterdam/New York, NY: Rodopi, 2006.

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Did John marry Joan?

In all of the above cases the answer “yes” is an abbreviation of (2). The meaning of the reply “no,” however, changes from case to case. As far as (4) is concerned, the meaning of “no” is expressed by the sentence (3). Yet, in the case of (5) “no” means: (7)

Someone else (not John) married Joan.

The sentences (2) and (7) share the same background assumption, namely that someone married Joan; what is questioned is whether it was John. Similarly, in the case of (6) the reply “no” means: (8)

John married someone else (not Joan).

Both (2) and (8) presuppose that John married someone; what is inquired via (6) is whether it was Joan. So the questions (5) and (6) interpreted along the lines sketched above are no longer safe questions: it may happen that no direct answer to them is true (John may be a monk, and Joan a declared feminist). The questions (5) and (6) construed in the above manner are examples of focussed yes-no questions. Let’s now consider the following interrogative sentence: (9)

Did John marry Joan and love her?

At first sight it may look like (9) expresses a yes-no question. Clearly the reply “yes” means: (10)

John married Joan and loved her.

But what is the meaning of the reply “no”? The negation of (10), viz.: (11)

It is not the case that John married Joan and loved her.

is a possible answer, but usually not a just-sufficient one. The justsufficiency condition is fulfilled by any of the following: (12) (13) (14)

John married Joan, but he didn’t love her. John didn’t marry Joan, but he loved her. John neither married Joan nor loved her.

Taking this interpretation for granted (of course, it is not the only possible one), (9) expresses a conjunctive question and not a yes-no question: it cannot be satisfactorily answered by saying either “yes” or “no.” Yet, its grammatical form is similar to that of yes-no questions. And it is still a safe question: it must have a true direct answer. Let’s now consider the famous spouse-beating question:

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(15)

217

Has Joan stopped beating her husband?

It is well-known that the reply “yes” should be construed as something like: (16)

Joan has beaten her husband but has stopped.

whereas the reply “no” means: (17)

Joan has beaten her husband and has not stopped.

If Joan never beat her husband, both (16) and (17) are false. So (15) is not a safe question. It is also clear that (15) is not a simple yes-no question: it is a conditional yes-no question. It is worth emphasizing that not only questions about forbearance can be regarded as conditional yesno questions. Let’s consider: (18)

Given that Russia will constantly oppose NATO’s enlargement, will Ukraine join NATO?

The reply “yes” to (18) means: (19)

Russia will constantly oppose NATO’s enlargement but Ukraine will join NATO.

The reply “no” to (18) means, in turn: (20)

Russia will constantly oppose NATO’s enlargement and Ukraine will not join NATO.

The above considerations show that it is not the case that each yes-no question of a natural language should be construed as a simple yes-no question and thus a safe question. Of course, this fact is known to many logicians and linguists.1 It is also completely obvious that most questions cannot be construed as yes-no questions of any type. But the simple yesno questions are customarily regarded as the epistemologically prior questions. Among simple yes-no questions, in turn, questions whose sets of direct answers consist of an atomic sentence and its (classical) negation seem to have the logical priority; we shall call them atomic yesno questions. So the following questions arise: is it possible to reduce any question to simple yes-no questions? If not, what questions can be reduced in this way? And is it the case that each question that can be 1

See, e.g., Koj (1972), Belnap (1969), Hajièová (1983), Kiefer (1980, 1988). This is not to say, however, that all of them would accept the analysis of the above questions presented here. For details of this analysis, see WiĞniewski (1995), Chapter 3.

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reduced to simple yes-no questions can be also reduced to atomic yes-no questions?

2. Reducibility As usual, the answer depends on the meaning of the crucial term. Let us observe that one can speak of: (a) reducibility of a (single) question of some kind to a (single) question of another kind, or (b) reducibility of a (single) question of some kind to a set of questions of some kind or kinds. By and large, the former concept has been clarified in two ways: as an equivalence within a given calculus (see e.g. Åqvist 1965) or as some equivalence relation between questions which is defined in terms of (set-theoretic or semantic) relations between sets of their direct answers or in terms of relations between sets of presuppositions (cf. mainly KubiĔski 1980, but also Belnap and Steel 1976). It is not surprising that no general results have been obtained in this perspective: it would have been rather strange if one had proved that, for example, a which-question can be reduced, in any reasonable meaning of the word “reduction,” to a simple yes-no question. The situation is different, however, in the case of the latter concept of reducibility. Now reducibility of a (initial) question to a set of (auxiliary) questions is under consideration. In order to find the correct answer to an initial question we usually pass to a number of auxiliary questions and try to answer them. This can be done in many ways; yet, there are some underlying logical relations which enable us to do this in a (relatively) safe and efficient way. In WiĞniewski (1994), the concept “a question is reducible to a non-empty set of questions” is defined in semantic terms. The proposed definition pertains to a formalized language: it is a first-order language enriched with questions and supplemented with a model-theoretic semantics. Some conditions are imposed on the language under consideration. In particular, it is assumed that to each question of the language there is assigned a set of declarative sentences of the language which contains at least two sentences; elements of this set are called direct answers to the question. Direct answers are defined syntactically; on the other hand they are regarded as the possible and just-sufficient (providing neither less nor more information than is called for) answers. They may be true or false. The intuitions which underlie the proposed definition of reducibility can be briefly described as follows. First, an initial question and the questions to which the initial question is reducible must be mutually sound: it is required that if a question Q is reducible to a set of questions ), then Q has a true direct

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answer if and only if each question in ) has a true direct answer. The second requirement is the efficacy condition: it is required that an initial question can always be answered by answering the questions to which it is reducible. To be more precise, it is required that if a question Q is reducible to a set of questions ), then each set made up of direct answers to the questions of ) which contains exactly one direct answer to each question of ) must entail some direct answer(s) to Q. The last requirement is the relative simplicity condition: all questions to which a given question is reducible are supposed to be no more complex than the initial question in the sense that no one of those questions has more direct answers than the initial question. The relevant concept of reducibility is then defined in semantic terms: we will introduce the definition below. Yet, in order to give a simple example let us observe that the question: (9)

Did John marry Joan and love her?

interpreted as a conjunctive question is reducible int.al. to the set of questions whose elements are: (21) (22)

Did John marry Joan? Did John love Joan?

provided that these are construed as simple yes-no questions. When the concept of reducibility is clarified in the above manner, some solutions to our main problem emerge. First, it may be proved that each safe question (i.e., roughly, a question which must have a true direct answer; see below) is reducible to some set of questions exclusively made up of simple yes-no questions. It can also be proved that the relevant set of simple yes-no questions is finite if the initial question has a finite number of direct answers or entailment in the language is compact. We may even go further in this direction: one may prove that safety amounts to reducibility to sets of simple yes-no questions. Yet, in the case of risky questions, that is, questions which are not safe, the situation is far more complicated. It can be proved that each risky but proper and regular question 2 is reducible to some set of questions made up of simple yes-no question(s) and exactly one question which is not a simple yes-no question, but nevertheless has exactly two direct answers. 2

Roughly, a question Q is proper and regular if no direct answer to it is entailed by the set of presuppositions of Q, but nevertheless the question Q has a presupposition whose truth guarantees the existence of a true direct answer to it (that is, which multipleconlusion entails the set of direct answers to Q). By a presupposition of a question Q we mean here a d-wff which is entailed by each direct answer to Q.

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The negative result, however, is that no risky question is reducible to a set of questions whose elements are only simple yes-no questions. (For proofs and further results, see WiĞniewski 1994; see also WiĞniewski 1995, Chapter 7.) Risky questions are thus not reducible to homogenous sets of simple yes-no questions. But safe questions are reducible that way. So we may ask: is it the case that each safe question is reducible to some set of logically prior simple yes-no questions, that is, atomic yes-no questions? As we will see, the answer is rather complicated: there are conditions under which it is the case and conditions in which it is not. In order to go on we have to introduce some logical apparatus.

3. The Logical Basis First, we need some formalized language whose meaningful expressions are either declaratives or questions. There are many methods of constructing such languages (for an overview see, for example, Harrah 2002, or WiĞniewski 1995). Most of the details of the construction are irrelevant for the purposes of this analysis, however. Thus we only assume that we have at out disposal some formalized language L which consists of a declarative part as well as of an erotetic part and fulfills certain conditions. The declarative part of L is a first-order language with identity whose vocabulary contains all the connectives ¬, ›, &, Š, {, the quantifiers , , and at least one closed term (i.e., a term with no individual variable). As far as the declarative part of L is concerned, the concepts of term, atomic well-formed formula, (declarative) well-formed formula (d-wff for short), freedom and bondage of variables, etc., are defined as usual; by a sentence of L we mean a d-wff of L without free variables and by a sentential function we mean a d-wff which is not a sentence. The vocabulary of the erotetic part of L contains some expressions which enable us to form questions of this language. Questions of L are not d-wffs but they are the meaningful expressions of the erotetic part of L . We do not decide, however, what is the particular form of questions of L : they may be constructed in some way or another (see, inter alia, Belnap and Steel 1976; KubiĔski 1980; Harrah 2002; or WiĞniewski 1995). Yet, we assume that the following conditions are met: (1) the syntax of L assigns to each question of L an at least two-element set of sentences of L ; these sentences are called direct answers to the question and, looking from the pragmatic point of view, are regarded as the possible and just-sufficient answers; (2) for each sentence A of L

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there exists the corresponding simple yes-no question of L whose set of direct answers consists of the sentence A (called the affirmative direct answer) and the sentence ¬A (called the negative direct answer); (3) there are questions of L which are not simple yes-no questions; (4) the set of questions of L is denumerable. 3 Some comments on the condition (1) are in order here. We require the question-answer relationship in L to be purely syntactical. Moreover, we claim that it is the logical form of a question of L which determines what counts as not only possible but also just-sufficient answer to the question. It is obvious that things look differently in natural languages. Although the syntactical form of a question is usually an important factor, it is not always the decisive one. There are cases in which some meaning components of a natural-language question play an important role. And in general, as many linguists and philosophers pointed out, there are cases in which it is strongly context-dependent what sentences may be counted as the possible and just-sufficient answers to some question: such pragmatic factors as, for example, intonation or other focus indicators, or the position of a question in a text or in a utterance, or the state of knowledge of the questioner, or his/her intentions, or the speech situation are relevant. Yet, questions of L are questions of a formalized language and they only represent natural-language questions. The relation of representation we have in mind can be briefly described as follows: a question Q of L represents a question Q* of a natural language construed in such a way that the possible and just-sufficient answers to Q* have the logical form of direct answers to Q. Thus we do not say that there is one-to-one correspondence between natural-language interrogative sentences and questions of L : if a natural-language question admits many readings, it has many representations. The richer the interrogative part of L is, the more natural-language questions and/or their admissible readings can be represented. However, beyond the scope of such a representation system (and thus also beyond the scope of our analysis) are the so-called open natural-language questions, that is, questions whose possible and just-sufficient answers cannot be defined even if all the relevant syntactical/semantical/ pragmatical factors are known. The condition (1), however, not only requires that the questionanswer relationship in L is purely syntactical, but also claims, first, that each question of L has at least two direct answers and, second, that each In WiĞniewski (1994), some additional condition are imposed on L as well. Yet, we will not make use of them in this paper.

3

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direct answer is a sentence. The motivation for the first claim is philosophical: we think that a necessary condition of being a question is to present at least two “alternatives” or conceptual possibilities among which some selection can be made. The “Hobson choice” questions are thus excluded, but rhetorical questions are allowed – the selection need not be rational.4 Concerning the second clause: we want the direct answers to be the just-sufficient answers and those can be expressed only by sentences. We shall use the letters A, B, C, . . . (with subscripts if needed) as metalinguistic variables for d-wffs of L , and the letters X, Y, Z as metalinguistic variables for sets of d-wffs of L . The symbols Q, Q1 , . . . will be used as metalinguistic variables for questions and the capital Greek letters (with or without subscripts) as metalinguistic variables for sets of questions. The set of direct answers to a question Q will be referred to as dQ. In the metalanguage of L we assume the Bernays-von Neumann-Gödel version of set theory; we adopt here the standard settheoretical terminology and notation. Sometimes we shall write “iff ” instead of “if and only if.” The semantics of L is basically the model-theoretic one. By an interpretation of L we mean an ordered pair , where U is a nonempty set (the universe) and f is the interpretation function defined on the set of non-logical and non-erotetic constants of L (that is, predicates, individual constants and  if there are any  function symbols) in the standard way. Of course, there are many interpretations of L . If I is an interpretation, then by a I-valuation we mean an infinite sequence of the elements of the universe of I. The concepts of value of a term under a I-valuation and of satisfaction of a d-wff in an interpretation I by a I-valuation are defined in the standard manner. A d-wff A is said to be true in an interpretation I if and only if A is satisfied in I by all I-valuations; by a model of a set of d-wffs we mean an interpretation in which all the d-wffs of this set are true. Note that the concept of truth does not apply to questions of L . In the case of questions, however, we use the concept of soundness. A question Q is said to be sound in an interpretation I if and only if at least one direct answer to Q is true in I. The further semantical concepts pertaining to L are defined by means of the concept of normal interpretation of L . Yet, the language L was characterized only in a schematic manner and in fact there are many languages which fulfil the conditions specified so far. For that reason we 4

Some logical theories of questions (for example, Belnap’s theory or KubiĔski’s theory) allow questions of formalized languages which have only one direct answer, but it seems that this step is motivated rather by the pursuit of generality than other reasons.

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only assume that the class of interpretations of L includes a non-empty subclass (not necessarily a proper subclass) of normal interpretations, but we do not decide what the normal interpretations are in each particular case. If the declarative part of L is an applied first-order language, normal interpretations can be defined as those in which some meaning postulates and/or axioms are true. Normal interpretations can also be defined for purely erotetic reasons. If L contains questions about objects satisfying some conditions, it would be natural to define normal interpretations as those in which all the objects called for have names (are values of some closed term(s)): by doing so we would avoid the paradoxical consequence that there are objects which satisfy the appropriate conditions, but nevertheless the corresponding questions have no true answers. There are also other possibilities of defining normal interpretations (for more information, see WiĞniewski 1995, pp. 104-105). We do not even exclude that the class of normal interpretations of some language of the considered kind is equal to the class of all interpretations of this language. Yet, for the purposes of this analysis the assumption about the existence of a non-empty class of normal interpretations is sufficient. By means of normal interpretations we shall define the relevant concepts of entailment in L . We will introduce two concepts of entailment: multiple-conclusion entailment being a relation between sets of d-wffs and (single-conclusion) entailment understood as a relation between sets of d-wffs and single d-wffs. DEFINITION 1. A set of d-wffs X of L multiple-conclusion entails (mc-entails for short) in L a set of d-wffs Y of L iff the following condition holds: (#) for each normal interpretation I of L : if all the d-wffs in X are true in I, then at least one d-wff in Y is true in I. DEFINITION 2. A set of d-wffs X of L entails in L a d-wff A of L iff A is true in each normal interpretation of L in which all the d-wffs in X are true. Note that the above concepts are defined in terms of truth and not of satisfaction. Note also that in the general case mc-entailment cannot be defined in terms of (single-conclusion) entailment. For instance, assume that the declarative part of L is the language of Classical Predicate Calculus and that each interpretation of L is a normal one. Then the singleton set {A › B}, where A, B are atomic sentences, mc-entails the set {A, B}, but neither A nor B is entailed by the set {A › B}. On the other

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hand, entailment can be defined in terms of mc-entailment as multipleconclusion entailment of a singleton set. The concept of multiple-conclusion entailment proved it usefulness in the logic of questions in many ways. (For the properties of mc-entailment, see Shoesmith & Smiley 1978; see also WiĞniewski 1995, pp. 107-113.) Since the concept of normal interpretation was left unspecified, the same pertains to the above concepts of entailment. But since the class of normal interpretations was assumed to be a subclass of the class of all interpretations, logical entailment (defined in the manner similar to that of Definition 2, but with respect to any interpretation) is a special case of entailment in L . We may also say that, in particular, any disjunction of sentences (or, to be more precise, a singleton set containing this disjunction) mc-entails in L the set made up of the appropriate disjuncts. In what follows the specification “in L ” will normally be omitted. We shall use the symbol š for mc-entailment in L and the symbol ş for entailment in L . We shall write Aš Y instead of {A}š Y. The relation š is said to be compact if and only if for any sets of d wffs X, Y such that Xš Y there exist a finite subset X1 of X and a finite subset Y1 of Y such that X1 š Y1 . In the case of ş the concept of compactness is understood in the standard way. It may be proved that mc-entailment in a language is compact if and only if entailment in this language is compact. However, we neither claim nor deny that entailment in L and mc-entailment in L are compact. Compactness of entailment in a language depends on the conditions imposed on the class of normal interpretations of the language and there are languages of the considered kind in which entailment is compact and languages in which it is not. We are now ready to define the concept of reducibility of questions. DEFINITION 3. A question Q is reducible to a non-empty set of questions ) iff for each direct answer A to Q, for each question Qi of ): A mc-entails the set of direct answers to Qi, and (ii) each set made up of direct answers to the questions of ) which contains exactly one direct answer to each question of ) entails some direct answer to Q, and (iii) no question in ) has more direct answers than Q. (i)

For conciseness, the non-emptiness clause will be omitted in the sequel. Also for the sake of brevity we shall introduce the notion of a P())-set. Let ) be a non-empty set of questions. By a P())-set we mean a set made up of direct answers to the questions of ) which contains

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exactly one direct answer to each question of ). By means of this concept the second clause of Definition 3 can be expressed as follows: each P())-set entails some direct answer to Q. When saying that no question in ) has more direct answer than Q we mean that the cardinality of the set of direct answers to any question of ) is not greater than the cardinality of the set of direct answers to Q. Let us finally clarify the erotetic concepts of safety and riskiness. A question Q of L is said to be safe if and only if Q is sound (has a true direct answer) in each normal interpretation of L ; otherwise Q is said to be risky. (It is easy to observe that safety can be also defined in terms of mc-entailment: a question Q is safe iff the set of direct answers to Q is mc-entailed by the empty set.) Note that a question can be safe although no direct answer to it is valid (i.e. is true in each normal interpretation of the language)! Of course, each simple yes-no question is safe, but there are also safe questions which are not simple yes-no questions. In what follows we will be frequently speaking of simple yes-no questions, so, to simplify matters, we need a temporary notation for them. We shall write them down as ? {A, ¬A}. Under this notational convention the signs ?, {, } belong to the (erotetic part of the) objectlanguage5. Yet, we might have adopted some other notational convention for simple yes-no questions as well. The advantage of this one is that it makes explicit what the direct answers to a simple yes-no question are: these are the sentences enclosed in { }. Let us finally recall that an atomic yes-no question is a simple yes-no question whose affirmative direct answer is an atomic sentence (i.e., a sentence built up of a predicate and closed term(s)) and whose negative direct answer is the negation of this atomic sentence. In other words, an atomic yes-no question has the form ? {B, ¬B}, where B is an atomic sentence.

4. The Quantifier-Free Case It can be shown that in the case of quantifier-free safe questions the reduction to homogenous sets of atomic yes-no questions is always possible. By a quantifier-free question we mean a question whose direct answers contain no occurrence of a quantifier. Let us prove 5

Of course, the brackets { } also occur in the metalanguage in their normal roles.

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LEMMA 1: Let A be a quantifier-free sentence. Then the simple yes-no question ? {A, ¬A} is reducible to some finite set of atomic yes-no questions. PROOF: Let us observe that the clauses (i) and (iii) of the definition of reducibility are fulfilled by each set made up of atomic yes-no questions with respect to any “initial” question. The clause (i) is fulfilled because the set of direct answers to an atomic yes-no question is mc-entailed by any d-wff; the clause (iii) is satisfied since each question has at least two direct answers and an atomic yes-no question has exactly two direct answers. So it remains to be shown that for each quantifierfree sentence A there exists a finite set of atomic yes-no questions ) such that for each P())-set Y, Y entails the sentence A or the sentence ¬A. The proof goes on by induction on the structure of A. (1)

(2)

(3)

Assume that A is an atomic sentence. Let Q = ? {A, ¬A}. So {Q} is a finite set made up of atomic yes no-questions. On the other hand, it is obvious that {A} ş A and {¬A} ş ¬ A. Assume that A is of the form ¬B. By induction hypothesis there exists a finite set ) of atomic yes-no questions such that for each P())-set Y we have Y ş B or Y ş ¬B. But B ş ¬A and ¬B ş A. So each P())-set entails A or entails ¬A. Assume that A is of the form B & C. By induction hypothesis there are a finite set ) 1 of atomic yes-no questions and a finite set )2 of atomic yes-no questions such that for each P()1 )-set Y we have Y ş B or Y ş ¬B, and for each P()2 )-set Z we have Z ş C or Z ş ¬C. On the other hand, the following hold: (a) (b) (c) (d)

{B, C} ş A, {B, ¬C} ş ¬A, {¬B, C} ş ¬A, {¬B, ¬C} ş ¬A.

Let ) = )1 ‰ )2 . Each P())-set equals to a union of a P()1 )-set and a P()2 )-set. But each P()1 )-set entails B or entails ¬B, and each P() 2 )-set entails C or entails ¬C. So by the conditions (a) – (d) each P())-set entails A or entails ¬A. It is obvious that ) is a finite set made up of atomic yes-no questions. (4)

Assume that A is of the form A › B. We proceed as above by means of the following facts: (e) (f)

{B, C} ş A, {B, ¬C} ş A,

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(g) (h) (5)

{¬B, C} ş A, {¬B, ¬C} ş ¬A.

Assume that A is of the form B Š C. We proceed analogously as above by using the following: (i) (j) (k) (l)

(6)

227

{B, C} ş A, {B, ¬C} ş ¬A, {¬B, C} ş A, {¬B, ¬C} ş A.

Assume finally that A is of the form B { C. We use the following: (m) (n) (o) (p)

{B, C} ş A, {B, ¬C} ş ¬A, {¬B, C} ş ¬A, {¬B, ¬C} ş A. Ŷ

Lemma 1 yields THEOREM 1. Each quantifier-free simple yes-no question is reducible to some finite set of atomic yes-no questions. Thus quantifier-free simple yes-no questions are reducible to sets of logically prior questions, that is, atomic yes-no questions. Let us now consider the possibility of reduction of any quantifier-free safe question to a homogenous set of atomic yes-no questions. We shall first prove THEOREM 2. If Q is a quantifier-free safe question, then Q is reducible to some set of quantifier-free simple yes-no questions; if moreover Q has a finite number of direct answers or entailment in the language is compact, then Q is reducible to some finite set of quantifier-free simple yes-no questions. PROOF: Let Q be a quantifier-free safe question. Direct answers are sentences and the set of direct answers to each question is at most countable. Let s = A1 , A2 , . . . be a fixed sequence without repetitions of direct answers to Q such that each direct answer to Q is an element of s. Let us then define the following set of simple yes-no questions: ) = {Q*: Q* is of the form ? {Ai, ¬Ai}, where i > 1} In other words, ) consists of the simple yes-no questions based on the elements of the sequence s with the exception of the simple yes-no

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question based on the first element of s. It is clear that the clauses (i) and (iii) of Definition 3 are fulfilled by ) with respect to Q. Let Y be a P())set. There are two possibilities: (a) the set Y contains some affirmative direct answer(s) to the questions of ), (b) the set Y is made up of the negative direct answers to the questions of ). If the possibility (a) holds, then – since the affirmative direct answers to the questions of ) are also direct answers to Q – the P())-set Y entails some direct answer(s) to Q. Suppose that the possibility (b) takes place. Since Q is a safe question, then ‡ š dQ. It follows that Y ş A1 . So there is a direct answer to Q which is entailed by Y. But Y was an arbitrary P())-set. Therefore Q is reducible to ). It is obvious that the set ) constructed in the above manner consists of quantifier-free simple yes-no questions. Moreover, it is also clearly visible that if Q has a finite number of direct answers, then – since each question has at least two direct answers – the set ) constructed according to the above pattern is finite and nonempty. Let us now assume that entailment in L is compact and that Q is an arbitrary but fixed quantifier-free safe question. If entailment is compact, so is mc-entailment. So there is an at least two-element subset Z of the set of direct answers to Q such that ‡ š Z (if ‡ entails some direct answer to Q, it also mc-entails each at least two-element subset of dQ which contains this answer; if ‡ does not entail any single direct answer to Q, then by compactness there is an at least two-element finite subset of dQ which is mc-entailed by ‡). We fix some at least two-element finite subset of the set of direct answers to Q which is mc-entailed by the empty set and then proceed as above; as the outcome we obtain a finite set of quantifier-free simple yes-no questions such that Q is reducible to this set. Ŷ We can now prove THEOREM 3. Each quantifier-free safe question Q is reducible to some set of atomic yes-no questions; if moreover Q has a finite number of direct answers or entailment in the language is compact, then Q is reducible to a finite set of atomic yes-no questions. PROOF: Assume that Q is a quantifier-free safe question. According to Theorem 2, Q is reducible to some set of quantifier-free simple yes-no questions (a finite set if Q has a finite number of direct answers or entailment in L is compact). Let ) be a fixed set of quantifier-free simple yes-no questions such that Q is reducible to ); if Q has a finite

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number of direct answers or entailment in the language is compact, ) is supposed to be a finite set. By Theorem 1 each question in ) is reducible to some finite set of atomic yes-no questions. We associate each question in ) with exactly one finite set of atomic yes-no questions such that the considered question of ) is reducible to this set. Let < be the union of sets of atomic yes-no questions associated with the questions of ) in the above manner. The set < is a homogenous set of atomic yes-no questions. It is clear that the clauses (i) and (iii) of Definition 3 are met by < with respect to Q. Let Y be a P(<)-set. It is easily seen that Y entails some direct answer to each question of ). So there exists a P())-set, say, X, such that each normal interpretation which is a model of Y is also a model of X. But since Q is reducible to ), the P())-set X entails some direct answer to Q. Therefore the P(<)-set Y entails some direct answer to Q. But since Y was an arbitrary P(<)-set, it follows that the clause (ii) of Definition 3 is also fulfilled by < with respect to Q. Therefore Q is reducible to <, where < is a set of atomic yes-no questions. It is clear that if Q has a finite number of direct answers or entailment in the language is compact, then < is a finite set. Ŷ Theorem 3 shows that in the case of quantifier-free safe questions the reduction to sets of atomic yes-no questions is always possible; moreover, it shows that in some cases the reduction to finite sets of atomic yes-no questions is possible as well. Note that no assumptions concerning the particular form of the semantics of L have been used in the proofs of the above theorems; it follows that the reducibility of quantifier-free safe questions to sets of atomic yes-no questions takes place in every language of the considered kind.

5. The General Case So far we have restricted ourselves to quantifier-free safe questions. But what happens if the initial question is not quantifier-free? Sometimes the initial safe question Q is not quantifier-free, but nevertheless it is reducible to some set of quantifier-free simple yes-no questions. One can easily prove THEOREM 4. Let Q be a safe question. If Q is reducible to some set of quantifier-free simple yes-no questions, then Q is reducible to some set of atomic yes-no questions; if moreover Q is reducible

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to some finite set of quantifier-free simple yes-no questions, then Q is reducible to some finite set of atomic yes-no questions. PROOF: Similar to that of Theorem 3. It is obvious that if Q is reducible to a finite set of quantifier-free simple yes-no questions, then the resultant set < is finite. Ŷ Yet, there is no guarantee that each safe question is reducible to a set of quantifier-free simple yes-no questions (although, as we shall see, there is a guarantee that each safe question is reducible to some set of simple yes-no questions). Moreover, there are examples which show that Theorem 3 cannot be generalized to any safe question with respect to any language of the considered kind. Here is a very simple example: assume that the declarative part of L is the language of Monadic Classical Predicate Calculus and that each interpretation of L is a normal one. Let us then consider a simple yes-no question of the form ? {P(x), ¬P(x)}, where P is a predicate. At first sight it may look as if the above question is reducible to the set made up of atomic yes-no questions of the form ? {P(t), ¬P(t)}, where t is a closed term. Yet, the set which contains only the affirmative direct answers to the above questions does not entail any direct answer to the initial question. The reason is that there may exist some “unnamed” elements of the domain which do not satisfy the sentential function P(x). The above example not only shows that the reducibility to sets of atomic yes-no questions does not always hold, but also suggests a certain sufficient condition whose satisfaction enables reducibility of any safe question to a set of atomic yes-no questions. Let Ax be a sentential function with exactly one free variable. Let us designate by S(Ax) the set of sentences which result from the sentential function Ax by proper substitution of a closed term for the variable which occurs free in Ax (i.e. the set of sentences which have the form A(x/t), where t is a closed term). Let us now consider the following condition: (Z) for each sentential function Ax with exactly one free variable,  x Ax š S(Ax). The condition (Z) says that for each sentential function with exactly one free variable, the existential generalization of this sentential function multiple-conclusion entails the set of sentences which are instantiations

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of this sentential function.6 In other words, the condition (Z) requires that for each normal interpretation of the language the truth of the existential generalization of a sentential function with exactly one free variable guarantees that at least one sentence which results from this sentential function by proper substitution of a closed term for the free variable is true. It follows that the set of normal interpretations of the considered language must be a proper subclass of the class of all interpretations of it: as a matter of fact the normal interpretations are those, in which all the elements of the universe have names (to be more precise, the condition (Z) is fulfilled if for each element y of the universe there exists a closed term t such that for any valuation s, y is the value of t under s). It is clear that the condition (Z) is met only by some languages of the considered kind. But we may prove that if the condition (Z) does hold in the case of some language, each simple yes-no question of this language is reducible to a set of atomic yes-no questions. In order to continue we need the concept of prenex normal form of a d-wff: this concept is understood here in the standard sense. It is a well-known fact that for each d-wff there exists a logically equivalent d-wff in prenex normal form which contains the same free variables as the initial d-wff. Since logical entailment yields entailment in a language, then for each sentence A there exists a sentence B in prenex normal form such that A ş B and B ş A. One can easily prove LEMMA 2. If A is a sentence and B is a sentence in prenex normal form such that A ş B and B ş A, then the question ? {A, ¬A} is reducible to a set of questions ) iff the question ? {B, ¬B} is reducible to the set ). Now we shall prove LEMMA 3. If the following condition holds: (Z) for each sentential function Ax with exactly one free variable, x Axš S(Ax) then for each sentence B in prenex normal form, the simple yes-no question ? {B, ¬B} is reducible to some set of atomic yes-no questions. PROOF: Let B be a sentence in prenex normal form. As above, let us observe that the clauses (i) and (iii) of the definition of reducibility are Let us recall here, however, that the set of closed terms of L need not be (but of course can be) infinite – we only imposed the non-emptiness condition on it. 6

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fulfilled by any set of atomic yes-no questions with respect to the question ? {B, ¬B}. So it remains to be proved that for each sentence B in prenex normal form there exists a set of atomic yes-no questions ) such that for each P())-set Y, Y entails the sentence B or the sentence ¬B. The proof will go on by induction on the number of layers of quantifiers in the prefix of B. Assume that the sentence B contains no layers of quantifiers. Since B is in prenex normal form, it follows that B is a quantifier-free sentence. So by Lemma 1 the question ? {B, ¬B} is reducible to some set ) of atomic yes-no questions and thus for each P())-set Y, Y entails the sentence B or the sentence ¬B. Assume now that B contains n layers of quantifiers in its prefix, where n > 0. By induction hypothesis for each sentence C in prenex normal form that contains n-1 layers of quantifiers in its prefix there exists a set of atomic yes-no questions 6 such that for each P(6)-set X, X entails C or entails ¬C. We have four possibilities: (a) (b) (c)

(d)

B is of the form xD, where D is in prenex normal form, contains n-1 layers of quantifiers in its prefix and x is not free in D, (b) B is of the form xD, where D is in prenex normal form, contains n-1 layers of quantifiers in its prefix and x is free in D, (c) B is of the form xD, where D is in prenex normal form, contains n-1 layers of quantifiers in its prefix and x is not free in D, (d) B is of the form xD, where D is in prenex normal form, contains n-1 layers of quantifiers in its prefix and x is free in D.

If the possibility (a) holds, then – since B is a sentence – D is also a sentence; moreover, we have D ş B as well as B ş D and thus ¬D ş ¬B. The sentence D contains exactly n-1 layers of quantifiers and is in prenex normal form. Thus, by induction hypothesis there exists a set of of atomic yes-no questions, say, ), such that for each P())-set Y, Y entails the sentence D or the sentence ¬D. Therefore for each P())-set Y, Y entails the sentence B or the sentence ¬B Suppose that the possibility (b) takes place. Now D is a sentential function with x as the only free variable; let us designate it by Dx. Let us now introduce the set S(Dx), i.e., the set of sentences of the form D(x/t), where t is a closed term. Since the set of closed terms is nonempty, so is the set S(Dx); moreover, this set is made up of sentences in prenex normal form which contain exactly n-1 layers of quantifiers in their prefixes. So by induction hypothesis for each sentence C in the set S(Dx)

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there exists a set < of atomic yes-no questions such that each P(<)-set entails the sentence C or the sentence ¬C. Let us now consider a union of such sets of questions; to be more precise, for each C  S(Dx) choose exactly one set of atomic yes-no questions which fulfill the inductive hypothesis and consider the set, say, ), which is the union of these sets. Let Y be a P())-set. There are two possibilities: (1) the set Y entails each sentence from the set S(Dx), i.e., each sentence of the form D(x/t), where t is a closed term; (2) the set Y entails some sentence of the form ¬D(x/t). In the case of (2) Y entails the sentence ¬B. Let us now consider the case (1). Suppose that Y does not entail the sentence B, i.e., the sentence x Dx. So there is a normal interpretation I such that I is a model of Y and the sentence x ¬Dx is true in I. By assumption we have x ¬Dx š S(¬Dx). So at least one sentence of the form ¬D(x/t) is true in I. But Y entails each sentence of the form D(x/t). Therefore each such sentence is true in I. We arrive at a contradiction. So Y entails the sentence B. Thus we may say that each P())-set entails the sentence B or the sentence ¬B. If the possibility (c) holds, we proceed as in the case of (a). Suppose finally that the possibility (d) takes place. Again, D is now a sentential function with x as the only free variable. We designate it by Dx. Then we construct the set ) as in the case of (b). Let Y be a P())-set. There are two possibilities: (1) the set Y entails some sentence from the set S(Dx), i.e., some sentence of the form D(x/t), where t is a closed term; (2) the set Y entails each sentence of the form ¬D(x/t). In the case of (1) Y entails the sentence B. Let us now consider the case (2). Suppose that Y does not entail the sentence ¬x Dx. So there is a normal interpretation I such that I is a model of Y and the sentence x Dx is true in I. By assumption we have xDx š S(Dx). So at least one sentence of the form D(x/t) is true in I. But Y entails each sentence of the form ¬D(x/t). Therefore each such sentence is true in I. We arrive at a contradiction. So Y entails the sentence ¬x Dx. This sentence, however, is equal to ¬B. Thus we may say that each P())-set entails the sentence B or the sentence ¬B. Ŷ Next we shall prove THEOREM 5. If the following condition holds: () for each sentential function Ax with exactly one free variable, x Ax š S(Ax)

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then each simple yes-no question is reducible to some set of atomic yes-no questions. PROOF: Let ? {A, ¬A} be a simple yes-no question. It is obvious that there exists a simple yes-no question ? {B, ¬B} such that B is in prenex normal form and A ş B as well as B ş A. By Lemma 3 the question ? {B, ¬B} is reducible to some set of atomic yes-no questions; so by Lemma 2 the question ? {A, ¬A} is reducible to some set of atomic yes-no questions. Ŷ Now we need the following THEOREM 6. Each safe question is reducible to some set of questions made up of simple yes-no questions. The proof of Theorem 6 is similar to that of Theorem 2. For details, see WiĞniewski (1994). By means of Theorem 5 and Theorem 6 we can prove THEOREM 7. If the following condition holds: (Z) for each sentential function Ax with exactly one free variable, x Axš S(Ax) then each safe question is reducible to some set of atomic yes-no questions. PROOF: Let Q be a safe question. By Theorem 6 Q is reducible to some set of simple yes-no questions. Let ) be an arbitrary but fixed set of simple yes-no questions such that Q is reducible to ). By Theorem 5 each question in ) is reducible to some set of atomic yes-no questions. Let us then pair each question in ) with exactly one (arbitrary but fixed) set of atomic yes-no questions to which the considered question in ) is reducible. Let < be the union of sets of atomic yes-no questions associated with the questions of ) in the above manner. The set < is a homogenous set of atomic yes-no questions. Then we proceed as in the proof of Theorem 3. Ŷ Thus, if the condition (Z) holds, then each safe question is reducible to some set of atomic yes-no questions. But it is not the case that the condition (Z) holds for any language of the considered kind. So designing the semantics in such a way that the condition (Z) would be met is the price which, if paid, gives us the unrestricted reducibility of

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safe questions to homogenous sets of atomic yes-no questions. And it may be a high price: as a by-product we may obtain the lack of compactness of entailment as well as of mc-entailment.7

Adam Mickiewicz University Section of Logic and Cognitive Science Department of Psychology ul. Szamarzewskiego 89a 60-568 PoznaĔ, Poland e-mail: [email protected]

REFERENCES Åqvist, L. (1965). A New Approach to the Logical Theory of Interrogatives. Uppsala: Almqvist & Wiksell. Belnap, N.D. (1969). Questions, Their Presuppositions and How They Can Fail to Arise. In: K. Lambert (ed.), The Logical Way of Doing Things, pp. 23-37. New Haven, CT: Yale University Press. Belnap, N.D. and T.B. Steel (1976). The Logic of Questions and Answers. New Haven, CT: Yale University Press. Hajièová, E. (1983). On Some Aspects of Presuppositions of Questions. In: F. Kiefer (ed.), Questions and Answers, pp. 85-96. Dordrecht: D. Reidel. Harrah, D. (2002). The Logic of Questions. In: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. 8, 2nd ed., pp. 1-60. Dordrecht: Kluwer. Hintikka, J. (1985). A Spectrum of Logics of Questioning. Philosophica 35, 135-150. Kiefer, F. (1980). Yes-No Questions as Why-Questions. In: J.R. Searle, F. Kiefer and M. Bierwisch (eds.), Speech Act Theory and Pragmatics, pp. 97-119. Dordrecht: Reidel. Kiefer, F. (1988). On the Pragmatics of Answers. In: M. Meyer, F. Kiefer and M. Bierwisch (eds.), Questions and Questioning, pp. 255-279. Berlin: Walter de Gruyter. Koj, L. (1972). Analiza pytaĔ II: RozwaĪania nad strukturą pytaĔ. Studia Semiotyczne 4, 23-39. English translation: Inquiry into the Structure of Questions, in: L. Koj and A.WiĞniewski (eds.), Inquiries into the Generating and Proper Use of Questions (Lublin: Wydawnictwo Naukowe UMCS, 1989), pp. 33-60. KubiĔski, T. (1973). Twierdzenia o relacjach sprowadzalnoĞci operatorów pytajnych [Theorems about the reducibility of interrogative operators]. Ruch Filozoficzny 31, 313-320. KubiĔski, T. (1980). An Outline of the Logical Theory of Questions. Berlin: Akademie Verlag. 7

I am grateful to the Netherlands Institute for Advanced Study in the Humanities and Social Sciences, and to the Foundation for Polish Science for their support.

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LeĞniewski, P. (1997). Zagadnienie sprowadzalnoĞci w antyredukcjonistycznych teoriach pytaĔ [The problem of reducibility in anti-reductionistic theories of questions]. PoznaĔ: Wydawnictwo Instytutu Filozofii UAM. Shoesmith, D.J. and T.J. Smiley (1978). Multiple-Conclusion Logic. Cambridge: Cambridge University Press. WiĞniewski, A. (1994). On the Reducibility of Questions. Erkenntnis 40, 265-284. WiĞniewski, A. (1995). The Posing of Questions: Logical Foundations of Erotetic Inferences. Dordrecht: Kluwer.

PART IV CATEGORIAL GRAMMAR

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Peter Simons LANGUAGES WITH VARIABLE-BINDING OPERATORS: CATEGORIAL SYNTAX AND COMBINATORIAL SEMANTICS The analysis of an expression that contains an operator, e.g. the general proposition ‘(3x).fx’, into functors and arguments with appropriate semantic categories, seems to meet with insuperable difficulties. Kazimierz Ajdukiewicz, “Syntactic Connexion” (1967), pp. 220-221.

Artur Rojszczak in memoriam

1. Introduction Many who have come across categorial grammar (CG), whether in the original paper by Ajdukiewicz (see Ajdukiewicz 1935) or elsewhere, were probably, like myself, initially captivated by its elegance, only later to grow disillusioned in the face of the complications and shortcomings of the available theories. Although CG has revived healthily in recent years,1 one problem has resisted solution: CG as it stands cannot cope adequately with variable-binding operators. To the extent that natural languages manage without variable binding they are not affected by this problem, and CGs, especially the more flexible versions developed since Ajdukiewicz’s time, have many virtues in the description of natural languages. As the opening quotation shows, Ajdukiewicz was himself acutely aware of the difficulty posed by variable-binding operators, devoting a good part of his famous paper to it, but failing to solve the problem. One 1

Signs of this revival are Buszkowski et al. (1986) and Oehrle et. al., eds. (1988).

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 239-268. Amsterdam/New York, NY: Rodopi, 2006.

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reason for the problem’s intransigency lies in the mixed ancestry of CG. Ajdukiewicz was building on the work of more than one thinker, and there is a tension between two ways of regarding functors. On the one hand, they may be considered as quotable expressions, detachable unitary pieces of a syntactical whole. This line of thought informs the work of LeĞniewski, who was the first to consciously formulate a symbolic language along categorial lines. On the other hand, functors may be considered as non-detachable moments or aspects of a whole. The latter view stems from Frege, and tends to push one in the direction of seeing variables as a merely superficial device, rather like parentheses, not present in the “deep structure” of a language, so that functors and operators are only superficially different in kind. Although attractive, this view avoids the problem, which is that of providing a syntactic and semantic analysis of languages which actually contain variables, rather than “explaining them away.” The structure of the rest of this paper is as follows. §2 gives notation and principles for strict or inflexible CGs, §3 for more flexible CGs. In neither case are operators included. §4 reviews the problems involved in describing operator/variable languages using plain CG. §5 shows how to give a syntax for such languages. It includes a new analysis of the nature of variable binding, which distinguishes the roles of place marking and place filling. I call the resulting kind of theory “extended categorial grammar” (ECG), since while it goes beyond the functor/argument scheme, the notation and principles governing operators are recognizably similar to those of CG. One of the principal attractions of CG is the close connection it establishes between syntax and semantics. As an alternative to more usual function- or set-theoretic semantics, in §6 I present the outlines of combinatorial semantics for languages without operators, and in §7 the semantics is extended to languages with operators.

2. Syntax for Strict Categorial Languages Ajdukiewicz’s first and major success was the application of his notation for CG to the simple case of propositional logic notated in the style of àukasiewicz. I shall use a slightly different notation for the categorial indices from Ajdukiewicz, and generalize the analysis to a wider but essentially similar class of possible languages. I shall be brief, as much of this is very familiar (for a more detailed account, see Simons 1989, §2).

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2.1. Categorial Notation There are a finite number of basic category symbols, each standing for a basic category. Each basic category symbol is a category symbol. If a, b 1 , . . . , b n are category symbols (n • 1), then the symbol a is a functor category symbol, standing for the functor category with input (argument) categories b 1 . . . b n in order and output (value) category a. Nothing is a category symbol which is not a basic or functor category symbol so formed. Let ‘s’ be the category symbol for the basic category SENTENCE. Then the category symbols for the categories of one-place and two-place connectives are respectively ‘s<s>’ and ‘s<ss>’. If ‘n’ is the category symbol for the basic category NAME , then s is the category of three-placed verbs,2 n is the category of two-placed function symbols, while s<s> is a category of functors taking one-place verbs to sentences, a category often assigned to determiners in CGs. We assume that if category symbols are different, they denote distinct categories. This distinguishes multi-place functors from multi-link functors – as in LeĞniewski.3 2.2. Categorial Languages Suppose we have a system of categories based on a finite set of basic categories as above. A categorial language (CL) describable by this system of categories is one which fulfils the following conditions: (CL1) There are a number of indecomposable or simple expressions of the language, which we follow LeĞniewski in calling “words” (they need not coincide with words in the everyday sense). Each word belongs to exactly one category of the system generated by the basic categories, and no word can be construed as a complex expression compounded of other simple signs, where a simple sign is either a word or some other indecomposable syncategorematic symbol (one which is not a word, such as a bracket) which the language might possess. 2 The distinction between verbs and predicates is a special case of that between functors as pieces of an expression and hence themselves expressions (here: verbs) and functors as undetachable moments of expressions (here: predicates). See Simons (1981, 1983), for further details. 3 A multi-link functor is one whose output is a functor. For example, s is a category of multi-link functor distinct from multi-place functors of category s. For a clear account of multi-link functors, see Lejewski (1967), pp. 73-75.

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(CL2) If X, Y1 , . . . , Yn , are any expressions belonging to the categories a, b 1 , . . . , b n respectively, there is a determinate way of constructing an expression of category a in which X, Y1 , . . . , Yn are parts, in which the positions of these parts are so determined that the only way other than this to obtain an expression of category a out of these expressions alone is if two or more of the Yi are of the same category and can be permuted. Conversely, given a complex expression of category a formed in this way, there is no way to break it down into constituents without first obtaining constituents having the positions of X, Y1 , . . . , Yn in the compound. (CL3) Every expression of the language is either simple, in accordance with CL1, or compound, and constructed according to CL2. The actual way in which expressions are compounded has been left unspecified, provided they fulfil the uniqueness conditions mentioned. We are abstracting from what Suszko has called the “calligraphy” of the language (see Suszko 1958, p. 222). We use in the metalanguage the symbol ‘X(Y1 , . . . , Yn )’ to name the unique compound expression constructed from X, Y1 , . . . , Yn in order, in whatever way this is actually (calligraphically) done. This enables us to offer a schematic metalinguistic account of syntactic well-formedness in the language in question which could hardly be simpler: (WF1) Each word in a category is a well-formed expression (wfe) of that category. (WF2) If X, Y1 , . . . , Yn are wfes of categories a, b 1 , . . . , b n respectively, then X(Y1 , . . . , Yn ) is a wfe of category a. (WF3) Nothing is a wfe except in virtue of WF1 and WF2. This covers a multitude of possibilities of realization in actual concrete expressions. For example, the various ways in which classical propositional calculus may be notated should all conform to it. Nevertheless the bracketless àukasiewicz notation is particularly simple, because the complex expression X(Y1 , . . . , Yn ) is written simply as the string ‘XY1 , . . . , Yn ’. The mere concatenation of a functor with its arguments in linear order takes care of both the order of arguments and the application of functor to arguments. Any string of simple expressions can be automatically checked for well-formedness, and if it is well-formed, it can be so in only one way.

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3. Greater Flexibility Natural languages do not display the tidy categorial behaviour outlined in the previous section. For one thing, different tokens of one type expression may belong to different categories. A clear example is provided by the conjunction ‘and’, which can conjoin any pair of expressions of like category. For example, in the logicians’ complex connective ‘if and only if ’, the ‘and’ has effective category s<ss> <s<ss> s<ss>>. In languages with linear structure where binary functors have mediate position between their arguments, it is natural to view not only the whole string Arg + Funct + Arg as having a ternary structure, but also to regard the two partial strings Arg + Funct and (more usually) Funct + Arg as being well-formed syntactic units, so we can analyse such strings as having an extra level: (Arg + Funct) + Arg, or Arg + (Funct + Arg). The view that an expression may be analysed in more than one way for logical purposes goes back (at least) to Frege. What it means for CG is that for certain purposes we do not or cannot distinguish between categories a, a , and a . For Frege at any rate there can be no distinction between multi-link and multi-place functors, or, to put the same point procedurally, simultaneous complete saturation and successive partial saturations yield the same result. For LeĞniewski, by contrast, there is a syntactic difference. There is system to the categories a single expression type may adopt in different contexts, and this system is recognized in CGs which admit what are called “type-change” rules. Of these, the best-known and most widely accepted is Geach’s Rule. 4 A special instance is If an expression can have category a, it can have category a >. Expressed in the quotient notation of Ajdukiewicz, this corresponds to “cancelling out”: a / b = (a/c) / (b / c). More generally we might have GR If an expression can have category a, it can have category a . . . c mnm >> 4

The source of Geach’s Rule is Geach (1971).

. . . b m
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Another rule which has been found occasionally useful is one which allows the roles of (single) argument and functor to be switched: it is known as Montague’s Rule: 5 MR An expression of category a may also be of category b>. Given this more flexible kind of category assignment, we may begin to offer more realistic analyses of fragments of natural languages. Many CGs distinguish between left-seeking and right-seeking functors to account for certain word-order phenomena. This distinction clearly makes sense only for linear realization, and at the level of generality we are considering we prefer to remain with the linearly unordered functor/argument structure, which reflects structural hierarchy alone. Our familiarity with natural languages may lead us into thinking that any realization of a syntactic structure must be linear, but this is mistaken. An example is given by Frege’s notation for propositional connectives. 6 Hierarchically organized structures may be partly or wholly alinear.

4. Problems One of the prices of linearization is that no element can be contiguous with more than two elements, so any functor with more than two arguments or any two-argument prefixed functor will be separated from one or other of its arguments, and iteration of infixed binary functors will lead to structural ambiguity unless bracketing is introduced. Both separation and bracketing involve the idea of separated components of the linear realization being structurally connected across others, or syntactic connection at a distance. Brackets may of course be arbitrarily widely separated from their partners in symbolic languages with infix notation, but the same phenomenon is observable in ordinary language, 5

The source is Montague (1973), which “lifts” proper names from the category n to the category s<s>. A more systematic account of admissible type-changes based on the analogy between type-derivation and inference goes back earlier to Lambek (1958). For a survey, see van Benthem (1988). 6 Frege’s two-dimensional notation for implication and negation is effectively constituency trees turned on their side. Since implication is the only binary connective he uses, there is no need to notate it separately from the structural lines themselves, and because of the configuration of these lines, it is not necessary to retain the standard order of writing (antecedent below consequent). Pivoting the constituents freely about their joins does not alter the constituent structure; the propositions could be realized physically as three-dimensional mobiles where no linearity, only hierarchy remains.

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for instance in the word pairs both/and and either/or which are functional units sometimes separated by long stretches of text. Also well known are German separable verbs, as in Wir stellten die Suche bei Dämmerung ein where ein and stellten (‘discontinued’) belong together despite the two intervening phrases. Any attempt to take account of these facts using CG must either be reconciled to the idea of structure-crossing or disconnected constituents, which are not well-loved in syntax, or else may consider the idea of movement transformations which shift arguments originally found next to their functors to other final positions. In neither case do we remain within the bounds of CG alone in giving an account of the resulting linear structure. If on the other hand the linear string itself is taken as the object on which our syntactic theories are to work directly, then to what category are brackets to be assigned? Although a piece in a string, a bracket without its mate or even a pair of mated brackets is not of any category, and is not (usually) regarded as a functional part at all. But clearly it is (again, usually) doing the job of delimiting or scope marking, or it would not be there. I do not see how this job can be explained in purely CG terms.7 So those versions of CG which simply ignore brackets are not giving a complete description of the syntactic phenomena. There is another aspect of the syntax of many artificial languages which is like brackets and other separable units in that it also involves “action at a distance,” but which throws up more fundamental problems for CG. This is the use of bound variables and the operators which bind them. Consider for example the universal quantifier. Frege interpreted this quite naturally as a function of higher order. In Everything flows the quantifier everything of category s<s> is saturated by the verb flows of category s to yield a sentence. Frege of course used not the words of an ordinary language but a symbolic notation with variables to say this: we have simply x(x flows). Superficially, we do not seem to have changed anything essential. A closer look however shows that there is a notable difference between the natural language expression ‘everything’ and the quantifier-variable notation. The latter allows quantifiers to bind variables in expressions of arbitrary complexity, and quantifiers may also 7 In fact brackets may do more than one job. The usual one is structural grouping, but they can contribute to determining the category of an expression, as in LeĞniewski’s formal languages. See Luschei (1962), p. 97.

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be embedded to any depth. For instance, the classic definition of continuity of a function F at number a is 8 H{H > 0 o G{G > 0 & x{|x – a| < G o |F(x) – F(a)| < H}}} and here each of the variables H, G, x occurs more than once within the scope of its quantifier. Because of the embedding of the quantifiers, each quantifier must be assigned a different effective category: the innermost scope contains three variables (assuming ‘F’ and ‘a’ are constants) so the innermost quantifier has effective category s <s>, the middle quantifier has effective category s <s> and only the outermost one has category s<s>. Of course, given Geach’s Rule, we can accept this categorial flexibility, just as we have to accept it in natural languages. But the difference is that whereas in natural language we have a connected expression, e.g. everything, some dog, most heads of state as the quantifier, here we cannot take the mere symbol ‘’ as the quantifier, because if we did, an expression like ‘(x = x)’ would be well-formed, which it is not. So we must add the variable to the symbol ‘’. But the expressions ‘H’, ‘G’ and ‘x’ are not the quantifiers either, because in an expression like ‘H(a = b)’ the quantifier does not close any gaps, as it should do if it is a functor. If a and b are constants, a = b is a sentence, and the combination s<s> + s is not syntactically connected. Ajdukiewicz gave this account of variable binders (see Ajdukiewicz 1935, § II), but it fails to account for the difference between vacuous and non-vacuous quantification. Many logical languages allow vacuous quantification, treating it as a harmless by-product of the symbolism, but logicians with a more finely developed sense of propriety, such as Frege and LeĞniewski, did not allow vacuous quantification. Indeed for Frege the scope of the quantifier cannot be a chunk of language, that is, a separable, quotable expression, but must be a pattern. If we choose to see variable-binding operators together with their variables as functors, we shall have to assign functor and argument status to something other than quotable expressions as has generally happened hitherto. The variables of such languages then turn out to be much more like brackets than they appeared to be at first sight: they come in groups which function only together, they have no meaning on their own, and they may be widely separated from their associates. But they differ from brackets in being associated with a category and being able to stand where and only where constant expressions of this category stand, excepting only their tag position alongside the operator symbol. 8

The braces are operator scope markers. See Ajdukiewicz (1935), § II.

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It might be objected that, just as brackets are dispensable, provided a suitable àukasiewicz-type notation is chosen, so we can manage without variables, provided we choose a suitable notation. This would show that at a deeper level than that found in the superficial syntax of such languages, the principles of CG indeed apply, the difference being then one of realization, as in the case of propositional calculi. And indeed variables are dispensable. Given a suitable supply of combinators, variables of a categorially-based language can be eliminated, and the resulting language is adequately describable by CG (see Simons 1989). However, there are problems arising from this. Do the combinators merely constitute a different kind of notation from the operator/variable notation, there being some “deep structure” they share, or do we not in fact have languages with completely different syntactic structure? On the face of it, the second view is more plausible. In any case, no matter how “deep” the supposed structures are which may be postulated to lie beneath all the variants, their postulation does not help us to give a syntactic description of the sign systems we see in front of us when we read them, brackets, dots, bound variables, left-right ordering, suffixes, primes, commas, and all. Here I think we must concede that CG rarely, if ever, tells more than a part of the story. The most extensive use of CG is found among those logicians who have developed model-theoretic semantics in the wake of Richard Montague. The syntax and semantics in such languages are of course designed to mesh as closely as possible, but to take account of variable binding again it is usual to go beyond pure CG and introduce lambda abstraction (see, for example, Cresswell 1973, pp. 80 ff).

5. Extended Categorial Grammar: Accommodating Operators If we are to find an account of something like CG which can cope with variable-binding operators as they stand, rather than postulating a deep structure where they are no longer needed, we shall need to take account of the fact that operators can bind variables which are embedded to any finite depth in a (finitely) complex structure. The operators can penetrate as many levels of structure as we like and still operate in the same way. This alone makes them different from functors, which simply add a further layer of structure but do not reach inside their arguments. But operators are like functors in having a scope and determinate categories of input and output.

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To provide a notation for operators adequate to cope with all of these facts, we must signalise more than just scope, output category, and input categories, as we do with functors. We need to know which categories of variables the operator binds. For complete generality we cannot confine ourselves to operators binding a single variable, nor need all variables bound be of the same category. Let us call the categories of variables bound marker categories, for reasons which will become apparent later. So we need to signify in addition (1) (2) (3)

Marker categories Full syntactic structure (not just category!) of the expression to which the operator is applied Indication of the places within this structure where variables of the relevant marker categories are bound.

To achieve all this at one go we need to substantially extend the notation of CG. We can represent the full syntactic structure of a complex expression even for CG without considering variable binding. The categories are written as in §2, and the schematic notation a(b 1 . . . b n ) is now used to signify the structure (however concretely realized) of a complex expression where a functor of category a is applied to arguments of category b 1 , . . . , b n . The resulting complex is then enclosed in square brackets and its category written to the left. Thus the structure of a sentence consisting of the application of a binary predicate to two names is signified s[s(nn)] while that of the predicate logical expression ~F(a) › G(bc) is s[s<ss>(s[s<s>(s[s(n)])]s{s(nn)])] The notation to date is redundant, in that we do not need to have the square brackets or the symbol for the output category to the left of them. We can leave these off in the context of a rigid CG of the type considered in §3. The structure of the last example would then be represented compactly by s<ss>(s<s>(s(n))s(nn)). Bracketed linear notations of this sort are compact but difficult to survey, and for many purposes we may resort to the familiar sort of constituency trees, which however we omit in this paper for economy and simplicity. We stipulate that functors in principle always appear to the left of their

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argument(s) though in giving concrete examples we often resort to the most usual infix notation. If we wish to allow for categorial flexibility as illustrated by Geach’s Rule, and the partial filling of slots, then this part of the notation can be turned to good use and is no longer redundant. We do not need to alter the category of a functor according to its context, but can let the context partially determine the category of the outcome. For example, a functor of the category a (or a, this amounting to the same thing for our purposes) can be partially filled by an expression of, say, category b<de>. We may notate the complex arising as a <de>[a (b<de>)]. the unfilled slots being carried across and retained in the “denominator” of the result. This context-dependence of outcome is much more akin to the way we actually determine effective category of expressions than one which assigns them a potential infinity of categories and bids us pick the right one for the context. Returning to the case of a strict CG without type-changes, we now have a way to notate the full syntactic structure of the expression or expressions to which the operator is applied. To obtain the category symbol for the operator we add to its output category and input categories written in the usual way a list of its marker categories. We place these in a new kind of bracket, here written with left- and rightslanted slashes \ and / and place this between the input and output symbols. The marker category of a quantifier of first-order predicate logic being n, the notation for such a quantifier’s category is s\n/<s>. Now to mark the effect of the operator on its operand expression we enclose the full syntactic description of the operand again in new brackets, this time braces { and }, which thus show the operator’s scope.9 All we need to do now is to indicate which of the marker slots inside the operand are filled by variables belonging to which place in the operator’s list of marker categories. There are various ways to do this. We could connect the appropriate slots with operator places by linking lines, for instance (see Quine 1951, p. 70). We choose instead to show the links using subscripts, for which we use lower case Greek letters. A Greek letter subscript next to a place in the structure pairs up with one next to a letter in the operator’s marker category list and shows that the operator binds with this category into this place in the structure. Of course the marker category and the category of the place(s) bound must match, and 9

LeĞniewski used lower corners around variables to signify the universal quantifier: we have used inward slanting brackets. LeĞniewski used upper corners as we use braces to mark operator scope. See Luschei (1962), pp. 194 ff.

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the letters must be chosen with the usual regard to avoiding ambiguities due to overlapping scope. So for example the following simple sentence of predicate logic x{F(x) & G(x)} has the structural analysis s[s\n D /<s>{ss<ss>(s[s(n D )]s[s(n D )])}] or in the more economical notation s\n D /<s>{ss<ss>((s(n D ))s(n D ))}. Note that the more economical notation allows us to replace each symbol of the syntactic description by an expression of the language according to obvious replacement rules. Here is another example: a definite description operator in a simple sentence: G(the x{F(x)}) This has structure (in the compact notation) s(n\n D /<s>{s(n D )}) A more complex example is provided by the Cauchy definition of continuity mentioned above. The resulting sentence has structure s\n H/<s>{s<ss>(s(n Hn)s\n G/<s>{s<ss>(s(n Gn)s\n D /<s> {s<ss>(s(n(n(nn D ))n G)s(n(n (n(n)n(n D )))n H))})})} The final example is a binary operator. What in English is said with There are more Fs than Gs can be rendered in predicate logic with a binary quantifier as more x {F(x), G(x)} which has structure s\n D /<ss>{s(n D ) s(n D )}. The kind of categorial grammar obtained by adding notation and principles of syntactic connection for operators I call extended categorial grammar (ECG). This designation is warranted by the facts that CG provides the basis for the grammar and the extension is made on recognizably similar lines. I shall consider below the extension of CGs with type-changes to more flexible ECGs. However in the more restricted

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and therefore more perspicuous context of a strict ECG without typechanges the main features of operators and variable binding can be brought out more clearly. We must consider a possible objection which threatens to undermine the whole enterprise of ECG. We are trying to explain how variable-binding operators work and yet in our notation we use precisely such a device in the form of Greek suffixes, or, if we did not have them, we should have to resort to some obvious equivalent, such as Quine’s connecting lines. How can variable binding be satisfactorily explained by variable binding? There are two things to be said in answer. Firstly, since variable binding has proved resistant to incorporation within the functor/argument pattern of ordinary CG, it is not very surprising that its like shows up in our metalanguage. This is just evidence that variable binding is a syntactic phenomenon sui generis. Secondly, there is an importance difference between the variables of object languages such as predicate logic and the Greek suffix letters we use in the metalanguage when describing the syntactic structure of complex expressions containing operators. The Greek suffix letters are not variables bound by an operator: they are place-markers showing which places in a structure are linked with which operator. They cannot be variables of the usual kind because they neither have, nor are invariably associated with, a particular syntactic category. The very same Greek letters serve the very same purpose no matter which category symbol they suffix. Bound variables on the other hand are divided into categories in just the same way as simple constants. Our Greek letters mark places but do not fill them. Places in object language expressions are filled by variables, whereas places in the metalinguistic descriptions are filled by category designations. In the object language, a bound variable both marks and fills a place. It is crucial to distinguish these two functions. That they can be not only distinguished but also separated may be seen by considering a wider range of formal devices than those usually employed. Place marker suffixes occur alongside normal bound variables in the object language of one (and, to my knowledge, only one) system of modern logic, but one whose historical importance is unsurpassed: it is the system of Frege’s Begriffsschrift (1879). Alongside the universal quantifier and other operators Frege introduces operators yielding the proper and improper ancestrals of a relation. To signify the holding of the ancestral of a relation F between two objects, Frege places the symbol for the ancestral before a sentence in which the sign for the relation F occurs, with the

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names of the two objects in the appropriate places. A relation is signified in Frege’s logic not by an expression as such but by an expression function, a sentence in which two classes of places are marked as variable. The places may themselves be occupied by constants, variables or functional expressions, provided these are categorially suitable, but in a sentence in which the places are filled we do not know which places are relevant unless they are marked in some way, and for this Frege uses Greek suffixes, putting copies of the relevant Greek letters next to the sign for the ancestral operator. An example shows how the idea works. To express that the number a is a multiple of the number d Frege uses the (improper) ancestral: a is obtained by adding d to 0 some finite number of times (possibly zero). Using ‘Anc’ for this operator and letting its Greek suffix place-markers follow it, instead of using Frege’s awkward sign, we can write this then as 10 Anc\EJ/{0 J + d = a E} Frege’s invention did not survive into his later logical system (Grundgesetze, 1893–1903) and it appears that even he was not fully aware of its nature. At all events it has found no imitators. ECG copes with Frege’s operators as effortlessly as it copes with normal variable-binders. The above sentence has the syntactic structure s\n En J/<s>{s(n(n Jn)n E)} How then are we to interpret the difference between operators with mere place-markers, like these ancestral operators of Frege, and the more familiar operators whose variables both mark and fill their places? The difference is, I suggest, merely one of notational convenience. We could easily manage without bound variables in a formal language, provided we have place-marker suffixes instead. For instance, the predicate-logical sentence F{x{y{F(x) & F(y) & x z y}}} could be replaced by a notation such as  s\s I/{ n \n D /{ n \n E/{PI(ND ) & PI(NE) & Na z NE}}} where ‘N ’ is a constant name and ‘P’ a dummy constant one-place predicate and the category of variable the quantifiers bind is shown by appending the category symbols to the quantifier symbols. Here the roles of place-marking and place-filling are separated. It is in this case quite 10

See Frege’s Begriffsschrift, §26, §29. For a commentary, see Simons (1988).

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irrelevant to the meaning of the result which expressions we use to fill the appropriate places in the structure. As the example shows, any meaning of ‘N ’ as a proper name plays no role in determining the meaning and truth-value of the result. We could replace ‘N ’ either uniformly or non-uniformly by any other simple or complex name or names and the meaning of the whole would be unaffected, and the same goes for the predicate place occupied by ‘P’. This is what is peculiar about the places marked by such operators as distinct from those marked by the ancestral operator. So we could instead adopt meaningless dummy expressions (name, predicate, etc.) to fill the relevant places, and the system would not suffer at all if we had only one dummy expression per category. The pattern of binding would still be shown by the Greek suffixes. Normal bound variables simply fulfil both of these roles. Rather than having a single meaningless dummy expression of each category we have arbitrarily many, and we leave it to their typographical equiformity and difference to connote the binding pattern. The Greek suffixes are then redundant in normal variable-binding languages, which is why neither they nor anything like them need occur. The effect of Frege’s place-marking operators may alternatively be achieved using lambda abstraction, and in that case we can use ordinary bound variables. It is instructive to see why. Lambda operators, rather like ‘and’, can appear in many different categorial guises. We could let the single categorially ambiguous lambda be replaced by a sheaf of fixed-category lambdas to conform with the strict ECG we are considering. Any such lambda operator has a category of the form a {b 1 . . . b n } Consider the special case s <s> { n } as found in the lambda abstract Ox{F(x) & G(x)} with syntactic structure s\n D /<s>{s<ss>(s(n D ) s(n D ))} We can then, as Ajdukiewicz realised, leave all the variable binding to be done by abstractors (see Ajdukiewicz 1935, § III), since they replace marker categories by argument gaps in functor categories and leave a unified functor expression which may be applied as it stands to arguments and to which further functors may be applied. Any other variable-binder can be expressed as the product of a functor and a lambda abstractor. In this case we may add a universal functor of category s<s> and obtain the same result as the universally quantified sentence

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given earlier. This works for Frege’s non-binding operators too, which can be seen as compressing into one notation the three steps of lambda abstraction, application of a functor to the abstract so formed, and then application of the modified functor to suitable arguments. This flexibility of lambda abstraction is another reason for the neglect of Frege’s device, since it shows we can manage well enough without it. For instance, the ancestral can be represented by a functor ANC of category s <s>, and we can express the sentence for which Frege used his ancestral operator by ANC(Oxy{x + d = y})(0, a). We can extend our analysis of operators to the case of a more flexible CG involving type-changes. One thing we can do with this more flexible ECG is to explain something which everyone does without being able to justify it syntactically, namely defining one quantifier in terms of another.11 It is common to define  as ~  ~, but this is not covered by the usual principles of definition for functors, since it is not a functor. If however we extend Geach’s Rule in the obvious way to operators, we have s\n D /<s> + s<s> = s\n D /<s> and s<s> + s\n D /<s> = s\n D /<s> so s<s> + s\n D /<s> + s<s> = s\n D /<s> and hence the sequence ~ +  + ~ is a syntactic unit of the right operator category. This is just a case of a more general phenomenon, namely the extension of Boolean functors like conjunction and negation to apply to operators. We are familiar in natural language with such complex quantifiers as some but not all so we can not only define a complex quantifier 11

See Quine (1951), p. 102: “The parts of ‘~(x)~’ do not, of course, hang together as a unit [. . .]. But the configuration of prefixes ‘~(x) ~’ figures so prominently in subsequent developments that it is convenient to adopt a condensed notation for it.” This is a fudge by Quine’s own high standards. Others were more scrupulous: Frege never defined the dual to the universal quantifier, LeĞniewski never used it in his “official” logical language. Perhaps just such reservations hindered them as are expressed by Quine. The price of scruple is inconvenience, so our demonstration of the acceptability by the highest standards of the combination justifies usual practice.

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Def.

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in context like this, but we can do so with a clear syntactic conscience and out of context, assuming we know what & and ~ mean, because & has effective category s<s>\n D /<s<s>\n D / s<s>\n D /> and the whole complex ( & ~) then has the structure s \n D /<s> [s \n D /<s> < s \n D /<s> s \n D /<s> > (s \n D /<s> s \n D /<s> [s<s>(s \n D /<s>)])]. We might even go so far as to allow variables to take the place of operator symbols and introduce operators which bind operator variables. But how this would go in practice and what benefits it might bring are something I have not gone into. We can now see why quantifiers appear to hover uncertainly between the categories s<s> and s<s>. In their effect they are closely akin to functors of category s<s>, and in simple cases where the variable occurs only once in the matrix they can be replaced by such functors. On the other hand, disregarding their marker categories, they look like ordinary s<s> functors, and indeed this is the way they are effectively treated by those logicians who allow vacuous quantification as well-formed. We can now see that there is nothing syntactically objectionable in this. On the other hand the meaning of quantifiers makes it understandable why Frege and LeĞniewski should have disallowed vacuous quantification, insisting that some marker categories be connected to places within any quantifier’s scope. Both alternatives can be accommodated within ECG, though of course the more usual grammar allowing vacuous quantification is simpler. The Frege/LeĞniewski syntax places restrictions not only on the overall category of the operand but also on its admissible structure: it must contain at least one place of relevant marker category and this place must be linked to that in the operator’s marker category list. Now that we see that variable-binding quantifiers are not in fact of category s<s>, we do not need to explain why they also appear to be of category s<s>, s<s> etc. So although with Geach’s Rule such flexibility is already explicable within ordinary CG, that is here irrelevant, and our account works also for strict ECGs where Geach’s Rule does not hold.

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6. Combinatorial Semantics ECG is a kind of syntax, and tells us nothing about the semantics of formal languages. However, the close parallels between syntax and model-theoretic semantics have been one factor encouraging the widespread use of CGs, so it is important to consider how such methods may be extended to accommodate operators as conceived in ECG. The next two sections present a new kind of semantics for a range of formal languages. The approach, which I call combinatorial semantics, was initially developed to provide a theory of extensional meaning for LeĞniewski’s Ontology (see Simons 1985). The motivation there was to avoid obvious ontological commitment to abstract entities, which as a nominalist LeĞniewski could not accept. Combinatorial semantics appears on the face of things also to be Platonistic. The question whether this appearance is deceptive will be here left aside: independently of the ontological issue the approach has a naturalness which recommends it for itself. In my previous paper I considered a language, that of LeĞniewski’s Ontology, containing only one variable-binding operator, a categorially flexible universal quantifier (so flexible that LeĞniewski regarded it as syncategorematic). So although the general methods of combinatorial semantics there developed do not accommodate such operators, the single exception could be coped with ad hoc. Semantics for formal languages is usually done in pursuit of an account of logical truth and validity, soundness and completeness for a particular logic. This is not what is at stake here, where, except for some remarks on logical constants, we are concerned not with logic as such but with investigating the connections between syntactic and semantic combination. The goal is to present a pattern of what Kaplan calls a “logically perfect language” (see Kaplan 1970, p. 283), that is, one in which syntactic and semantic combination and evaluation are in perfect harmony. Such a language may, but need not, be suitable as a basis for a logical system. 6.1. Ways of Meaning Basic to the theory is the unanalysed concept of an (extensional) way of meaning. This choice of primitive embodies rejection of the idea that being meaningful always or usually involves being assigned some kind of object as meaning (extension). This kind of denotational semantics, which we find in embryo in Frege, and which is continued in Carnap and the model-theoretic tradition, treats all expressions as names, the

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differences amongst them being only in what they name. The most common kind of semantics assigns each categorematic expression some object from a set-theoretic hierarchy. This not only couples us from the start with a Platonistic ontology, it also underrates the semantic differences between categories of expressions. Thus I say not that an expression has this or that object as its meaning, but rather that it means in this way or that, thus or so. Nominalizing these “thuses,” one may then speak of ways of meaning. However, having made this point, I shall also now allow myself to speak simply of meanings. Ways of meaning come in kinds, called modes of meaning. By design, there is one mode per syntactic category. So there is one mode for sentences, another one for names, another for intransitive verbs, and so on. We might alternatively call modes of meaning semantic categories. The mode of meaning of a functor expression (and hence of its category) is determined uniquely by the modes of its inputs in order and of its output. So the modes of functor categories are constructed recursively in parallel to the categories. For most formal languages we need only consider the two basic categories of sentence and name. Since we are concerned only with extensional semantics, the mode of sentences is to have a truth-value, or, as one might say, be truth-valued. The mode of meaning of names is to designate. Coextensional expressions mean in the same way (this is an extensional semantics), so for sentences to be coextensional is for them to be materially equivalent, while for names to be coextensional is for them to designate the same object(s), and coextensionality for functor categories is defined in terms of coextensionality for their simpler input and output categories. If we assume given a fixed supply of meanings for each basic category, then the meanings deriving from these for functor categories are given as follows. Assuming given meanings for the categories a, b 1 , . . . , b n , we obtain a meaning for category a by specifying for each possible combination of input meanings of categories b 1 , . . . , b n in order, which meaning of category a is obtained as output. For example, the meanings for category s<ss> may be represented by the usual truth-value tables, since these precisely tell us which truth-value we obtain as output for each combination of truth-values as inputs. Combinatorial semantics simply generalizes this idea. Where meanings for functor categories may be tabulated, we may speak in general of semantic tables. If there are M(a) meanings of category a and M(b i) meanings for category b i, 1 d i d n, then there are altogether M(a) M(b 1 )x . . . xM(b n )

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meanings of category a. To take a familiar example: if we have just two truth-values, then there are 22x2 = 16 ways of meaning for binary sentential connectives. 6.2. Categorial Languages Interpreted A language is (fully) interpreted iff each of its categorematic words means in some way appropriate to its category. We require an interpretation to be unambiguous, that is, no expression may mean in more than one way at once on this interpretation. We do not consider partial interpretations, where some words are left meaningless. However, by allowing as a way of meaning for names that of designating no individuals, and as a way of meaning for sentences that of having no truth-value, we are able to accommodate reference gaps and truth-value gaps. This is an advantage of considering ways of meaning rather than entities meant. The ways of meaning actually available for expressions depend, as we have seen, on those available for the basic categories. In most cases we have to consider, we need worry only about the truth-values and the domain of individuals. A bivalent interpretation, on which sentences may only be true or false, offers fewer possibilities for sentences and functors derived using s than multi-valued interpretations and/or those admitting truth-value gaps. Likewise if we consider two domains of individuals, one properly containing the other, then the larger offers a wider range of ways of meaning for names and functors derived using n than the smaller. We may also restrict interpretations to a subset of the ways of meaning which are available. For instance, we might deliberately restrict ways of meaning for sentences to being true and being false, even though others are available. Or we might restrict names to designating exactly one individual, as in standard predicate logic, or to designating not more than one individual, as in certain free logics, or to designating at least one but not all individuals, as in Aristotle’s syllogistic. Given a larger number of truth-values or a larger domain, we can always simulate the effect of having available only a smaller number of truth-values or a smaller domain by restricting interpretations in this manner. The ways of meaning interpretations allow among those available are the admissible ways of meaning. When no restrictions are made, the admissible ways are all those available. On a given interpretation the meaning of a complex expressions is jointly determined by the meanings of its words and its syntactic structure. Thus if expression X, Y1 , . . . , Yn of categories a, b 1 , . . . , b n have meanings, then the meaning of X(Y1 , . . . , Yn ) is the

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output meaning of X for the input meanings of Y1 , . . . , Yn in that order. By induction, every expression we can form in the language has a unique meaning on the interpretation, and the meaning of the whole is evaluated in terms of the meanings of its parts in the same order in which the whole expression is syntactically constructed of its parts. In these respects combinatorial semantics resembles the more familiar semantics of functions. We must distinguish the two kinds of semantic simplicity: the simplicity of meanings of expressions in basic categories and the simplicity of words. A word simply has its meaning (means in its peculiar way) without further ado, but a word may be of any category and there is no necessity that a CL have words in all of its basic categories. The meanings in the basic modes are simple in that they do not involve other meanings, whereas the meanings of a functor mode are defined in terms of those of its inputs and outputs. It will be noted that no expression is variable in meaning within a given interpretation, and any expression can be given a different meaning on another interpretation. This may seem strange if we are accustomed to distinguishing as special those expressions whose meaning is the same on all interpretations, such as the logical constants. That certain symbols are given the same meaning on all interpretations is however dictated in the semantics of logical systems by the need to distinguish between logical and non-logical expressions for the purposes of defining logical truth and validity. It would not be to the purpose if for example the sign ‘&’ sometimes meant conjunction, sometimes disjunction, sometimes material implication etc. But of course there is nothing inherently forbidden about assigning signs for connectives different meanings. 6.3. Logical Constants Within the present framework we can nevertheless make a natural distinction between logical and non-logical meanings, which can then be used to define the concept of a logical constant for expressions of extensional categorial languages based on the categories s and n. Given that logic is about truth-preservation of arguments, the truth-values are logical meanings, so any sentential word is a simple logical constant, as is any word of a purely propositive category, that is, a category whose index contains no letters other than s’s. Suppose we have a fixed domain of individuals for interpreting names. A permutation of the domain is a one-one correspondence of the domain with itself. Permutations of the domain induce in a well-defined way permutations on the meanings of those categories which are not purely propositive, that is, those whose

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index contains an n. For instance, if i is an individual in the domain, then designating i is a way of meaning for names, and if a permutation P carries i to j = P(i) then it carries the way of meaning of designating i to that of designating j. Similarly if c is a class of individuals then one way of meaning for one-place predicates is to yield truth as output for designating a member of c as input and falsehood otherwise (i.e. having c as its extension). The permutation P sends c to another class d, so it sends the predicate meaning to that for which d is the extension. And so on. Those meanings which are invariant under all permutations of the domain (such as designating nothing, or all individuals, or the meaning of the identity predicate) are logically constant with respect to the domain in question, and any word which has one of them as its meaning is a simple logical constant with respect to the domain. The most obvious example is the identity sign ‘=’. We may then define further complex logical constants in terms of those obtained hitherto (e.g. difference as non-identity). This idea is due to Tarski (see Tarski 1986), and it can be used in conjunction with Bolzano’s procedure of variation to explicate the concepts of logical constant, logical truth and validity for a range of bivalent extensional languages (see Simons 1987). However this characterization of logical constants will not suffice as it stands to exhibit as logical such operators as the usual quantifiers or Russell’s description operator, because we have yet to cover their semantics. To this we now turn.

7. The Same, Extended An integrated linguistic theory of operator/variable languages has to provide both a syntax and a semantics adequate to account for them. The formal semantics of logical languages have usually managed to cope well enough with variable-binders because these have been kept to a small number (typically a few quantifiers and abstraction operators) which are reckoned among the logical constants, so their semantic effect can be given in each case ad hoc by special clauses. On the other hand a general semantics for variable-binders has not been forthcoming. Variable-binders do not have to be logical constants. We now extend the combinatorial semantics of the last section to accommodate operators of both kinds mentioned earlier: the variable-binding and the place-marking, in such a way that the semantics as far as possible respects and mirrors the syntax, as for unextended CGs.

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7.1. Meanings for Operators Despite the differences between functors and operators, it is worth always bearing in mind the close connections between them. Variable-binding operators would be unnecessary if we only ever bound one occurrence of each variable in the outermost level of structure: instead of the quantifiers of category s \n/ <s> we could use the associated functors of category s<s>, for instance. Reflecting on this similarity helps us to see how operators work. Taking the universal functor everything as our example, we see that it yields truth as output for the universal s functor as input, and falsehood as output for all other inputs. The universal s functor is the one which yields truth for every singular nominal input. We know that a sentence of standard predicate logic of the form x{A(x)} is evaluated as true iff no matter what singular designating meaning we give to ‘x’, the truth-value of the result of evaluating the open sentence ‘A(x)’ with this meaning of ‘x’ is truth. So the effect of the universal quantifier can be pictured in the following procedural terms: the quantifier as it were successively inputs “messenger” values of ‘x’ to all the relevant positions in the formula within its scope, evaluates the resultant truth-value and stores it next to the value of ‘x’ input. Call this procedure tabulation. When tabulation is complete for all admissible values of ‘x’, the quantifier then delivers its own output: truth if all the resultant values are truth, falsehood otherwise. We can then predict that if the open sentence has the same outputs for inputs as the universal s functor, the quantifier will output truth, otherwise falsehood. In contrast to the functor however, the operator carries with it in the form of its marker variable the means to penetrate to arbitrary depths of structure within its scope. The procedure is the same no matter how complex the structure of the scope is, only the evaluation is more or less involved. The nesting of operators is also no problem: the results issuing from inner operators are passed on in the tabulations carried out for those lying further out, the values for bound variables issuing from outside the scope of the operator currently being evaluated being temporarily held fixed in the course of the inner variations. We have variations within variations. Consider a second example: a definite description operator, taken as a genuine nominal operator of category n \n/ <s>. The complex term (name) is Lx{A(x)} for some suitable open sentence A(x) and we assume L inputs as “messengers” to its matrix singular designating nominal meanings. If on

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tabulation exactly one such meaning input yields the resultant truth, the operator outputs this nominal meaning; otherwise it outputs a suitable “default” value, which may vary according to the description theory. The alternatives will be considered more fully below. For the last example we take a Fregean-style place-marking operator, one which could be used in mathematics. Consider how we may write the value of the derivative of a function at a point, say the function sin(x) + cos(S/3) for the argument S/3 (the value is 0.5). It would not do to write d/dx(sin(S/3) + cos(S/3)) because it is not clear which function it is which is being differentiated. A better notation, which says all we need and is widely used, is d/dx[(sin(x) + cos(S/3))] x = S/3 However, if we indicate the arguments of the function with respect to which we are differentiating by means of a Greek suffix to a differential operator D, and mark their places within the scope using the same Greek letters as suffixes, then we can insert the argument expression ‘S/3’ at this very place, thus: D\Į/ {sin((S/3)Į ) + cos(S/3)} and we have achieved the same result in a more compact way. The subscript enables us to distinguish the two places where ‘S/3’ occurs, the first as being bound into, the second as not. The meaning of ‘D’ then works as follows. For each possible meaning of any expression which can occupy the places marked with its Greek letter, it evaluates the result of combining this meaning at the places marked by the Greek letter with the others meant by the expressions within the operator’s scope, as directed by the structure of the matrix. It tabulates the resulting output meanings against the input meanings and so arrives at the resultant meaning of its matrix (which will be of category n, a monadic function). In our case it will “recognize” by its table the function sin(x) + cos(S/3). Then the differential operator will operate on this function, yielding another function, in this case the derivative cos(x). To complete the process the meaning of the expression in the place(s) marked by the Greek letter is fed into the derivative and its value obtained. We get cos(S/3) = 0.5. This is then finally output as the result of the operation.

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All of this sounds very long-winded and time-consuming. We have expressed the action of the operator in procedural terms, but of course the stages are not run through temporally. However, the order of the three stages is essential and cannot be changed: (1) tabulate the operand, (2) operate on it, (3) apply the result to the value(s). This is exactly the sequence carried out if we use lambda-abstraction to extract a function, modify it with a functor and then apply it. It is an indication that our intuitive view of Fregean operators as telescoping three distinguishable moves is correct. 7.2. Application: Some Logical Operators We noted above that Tarski’s idea of the permutation of the domain of individuals can be used to provide necessary and sufficient conditions for an expression to mean a logical constant. This may be extended to variable-binding operators. We shall show that the meanings of the universal quantifier binding individual variables (category s \n/ <s>) and some (but not all) definite description operators (category n \n/ <s>) are logical constants. Recall that meanings in a simple CG are logical iff they are invariant under permutations of the domain of individuals. That is, if M is a meaning and P a variable ranging over permutations extended to all meanings, M is logical iff for all P, P(M) is coextensional with M. Coextensionality for sentences is material equivalence, for names is denoting the same individual(s), and for a functor category two meanings are coextensional (coex) iff they always give coextensional output for coextensional inputs. The effect of a permutation P on a meaning M of functor category a is given by for all B1 in b 1 , . . . , Bn in b n : P(M)(B1 . . . Bn ) coex P(M(P–1 (B1 ) . . . P–1 (Bn )). Now we need to say how permutations affect operators. Again it is instructive to recall how operators of category a \de . . . / are related to functors of category a c<de . . . > . . . >. The effect of P on a functor F of category a c<de . . . > . . . > is given by for all B, C, . . . : P(F)(BC . . . ) coex P(F(P–1 (B) P–1 (C) . . . )). So suppose we have an operator Q of category a\de . . . /. The effect of P on it in context, where we have an expression of the form Q \DE . . . / {B(D, E, . . . ), C(D, E, . . . ) . . . }

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where ‘B(D, E, . . . )’ and ‘C(D, E, . . . )’ stand for possibly complex contexts of categories b and c, in which variables of categories d, e, . . . may occur, is given by P(Q)\DE . . . / {B(D, E, . . . ), C(D, E, . . . ) . . . } coex P(Q\DE . . . / {P–1 (B(D, E, . . . ), P–1 (C(D, E, . . . ), . . . }) Since the output of Q is of category a it is clear how the first P works, but we must say what is meant by P–1 (B(D, E, . . . ) etc. It cannot mean just the result of applying P–1 to the complete expressions of categories b, c, etc. For if this were so, all operators of the category s\n/<s> for instance would be logical because the truth-values are logical constants. But not all quantifiers of this category are logical. Consider the quantifier ‘Eng’ defined by Eng \ x/ {A(x)} = Def. for all x, if x lives in England, then A(x). Clearly this is not logical, since one permutation switches England and France and nothing else, and it is certain that the attributes shared by all inhabitants of England are not the same as those shared by all inhabitants of France (‘x lives in England’ for instance) . So the effect of the internal P–1 s must take account of the fact that the matrix expressions contain bound variables. The effect is this: the outputs for inputs are tabulated for each matrix as usual. The categories of the resulting meanings are indeed b<de . . . >, c<de . . . > etc. We then apply P–1 to these meanings in the usual fashion of functor expressions. The resulting meanings are likewise of the categories b<de . . . >, c<de . . . > etc. It is on them that the meaning of Q acts, and then the result is acted on by P. So let us consider the universal quantifier , of category s \n/ <s>, and write the variable now in inward slanting brackets because we shall be operating on the ‘’ part of the symbol. We need only consider the case where the matrix A(x) contains no other bound variables than x, because in any other cases, where other operators whose scope includes the quantifier expression  \ x/ {A(x)} and other bound variables occur in the matrix, whenever the meaning of this quantifier takes effect, all other variables are being held constant in the course of compiling meaning tables for the outer operators. We assume bivalence. The meaning of  is that it yields truth if the matrix yields truth for all nominal meaning inputs confined in standard fashion to the designating of a single existing individual. Otherwise, we get falsehood. We suppose given an arbitrary domain D and a permutation P of D. The case of the empty domain is trivial so suppose D non-empty. The effect of P on  is

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P()\ x/ {A(x)} l P(\ x/ {P–1 (A(x))}) (coextensionality for category s is material equivalence). The effect of P on truth-values is nil, so we have P()\ x/ {A(x)} l \ x/ {P–1 (A(x))} What then is the effect of P–1 on the matrix A(x)? The effective category of this matrix is s and the effect of P–1 on functor meanings F of category s is given by for all x: P–1 (F)(x) l P–1 (F(P(x)) since (P–1 ) –1 = P. But P–1 for category s has again no effect, so for all x: P–1 (F)(x) l F(P(x)) The extension of P–1 (F) is the class of all individuals which P sends into the extension of F, i.e. it is the image of the extension of F under the inverse of P, as we should expect. So ‘\ x/{P–1 (A(x))}’ is true if and only if matrix P–1 (A(x)) yields truth for all singular nominal meaning inputs. But this is so if and only if A(x) yields truth for all such inputs, since permuting the individuals then makes no difference. Hence \ x/ {P–1 (A(x))} l \ x/ {A(x)} whence P()\ x/ {A(x)} l \ x/ {A(x)} so the meaning of ‘’ is invariant, which was to be shown. Hence ‘’, being a simple expression, is a simple logical constant. Consider now a definite description operator L of category n\n/<s> whose effect on any sentential matrix A(x) is as follows: if the matrix yields the output truth for exactly one singular nominal input, this nominal input is the output of the operator. Otherwise, the output is the null nominal meaning, that of designating no individuals. We have then P(L)\ x/ {A(x)} = P(L\ x/ {P–1 (A(x))}) where ‘=’ here expresses coextensionality for (non-plural) names: designating the same individual or both designating no individuals. Take the two possible kinds of case. Suppose the matrix A(x) is not uniquely satisfied. Then neither is P–1 (A(x)), since permutations preserve cardinality. So the effect of L on P–1 (A(x)) is to output the null nominal

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meaning, and P, which permutes individuals, leaves this alone. Again, since permutations preserve cardinality, P–1 (A(x)) is uniquely satisfied iff A(x) is, and the output is the image of the output of A(x) under P–1 . If the individual a is the unique satisfier of A(x) and N(a) is the nominal meaning of designating precisely a, then the nominal meaning designating the unique satisfier of P–1 (A(x)) is N(P–1 (a)). This is then output by L and the effect of P on it is of course just to take us back to N(a) again. Hence L is invariant under P, which was to be shown. There are other definite description operators which differ from this on their default values, i.e., in their action for improper descriptions. 12 One sort assigns improper descriptions as output in effect the class of individuals satisfying the matrix. Since we are not guaranteed to have classes here we must consider plural nominal meanings, those of designating more than one individual. It can easily be seen that this sort of description operator is a logical constant too. Another sort makes the default value that of designating the whole domain. Since permutations trivially leave the domain unaltered this too is a logical constant. Another kind is the selected object theory, in which all improper descriptions are assigned the same nominal meaning of designating a selected individual. This sort of description operator is, interestingly, not a logical constant, for having assigned the improper description P-1 (A(x)) the selected object *, P acts on *, and for many permutations P, P(*) z * (* is not a fixed point of the permutation). Some description theories insist that * be outside the domain. As a piece of trickery, this solves the problem, but it is arbitrary: it amounts to saying that some individual is not “really” an individual. So logicians should disapprove of selected object description operators: they are not logical constants. In similar fashion we can show that other familiar variable-binders such as numerical quantifiers, the binary quantifier more from §5, and the lambda abstractors, are all logical constants. These are not surprising results, but it is reassuring to have support for one’s intuitions and prove what one previously assumed. On the other hand the set abstraction operators are not logical constants if sets are construed as (abstract) individuals subject to the permutations of the domain.13 It is not clear that this is how sets are to be construed, but that is an issue best left for another occasion.14 12 Both classical and free logics have offered several theories of descriptions. For a survey of the possibilities in free logics, see Bencivenga (1986), pp. 415-421. 13 As in Scott (1967). 14 Tarski (1986, pp. 151-153) notes that if sets are construed as individuals and set-membership as an undefined primitive, it is not a logical relation, whereas if

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University of Leeds School of Philosophy Leeds, LS2 9JT, UK e-mail: [email protected]

REFERENCES Ajdukiewicz, K. (1935). Die syntaktische Konnexität. Studia Philosophica 1, 1-27. Reprinted in: D. Pearce and J. WoleĔski (eds.), Logischer Rationalismus (Frankfurt am Main: Athenäum, 1988), pp. 207-226. English translation: Syntactic Connexion, in: S. McCall (ed.), Polish Logic 1920-1939 (Oxford: Clarendon Press, 1967), pp. 207-231. Bencivenga, E. (1986). Free Logics. In: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. III, pp. 373-426. Dordrecht: Reidel. Benthem, J. van (1988). The Lambek Calculus. In: Oehrle et. al., eds. (1988), pp. 35-68. Buszkowski, W., W. Marciszewski and J. van Benthem, eds. (1986). Categorial Grammars. Amsterdam: Benjamin. Cresswell, M.C. (1973). Logics and Languages. London: Methuen. Geach, P.T. (1971). A Program for Syntax. Synthese 22, 3-17. Reprinted in: D. Davidson and G. Harman (eds.), Semantics of Natural Language (Dordrecht: Reidel, 1972), pp. 483-497. Kaplan, D. (1970). What is Russell’s Theory of Descriptions?. In: W. Yourgrau and A.D. Breck (eds.), Physics, Logic and History, pp. 277-285. New York: Plenum. Lambek, J. (1958). The Mathematics of Sentence Structure. American Mathematical Monthly 65, 154-169. Lejewski, C. (1967). A Theory of Non-Reflexive Identity and Its Ontological Ramifications. In: P. Weingartner (ed.), Grundfragen der Wissenschaften und ihre Wurzeln in der Metaphysik, pp. 65-102. Salzburg: Pustet. Luschei, E.C. (1962). The Logical Systems of LeĞniewski. Amsteram: North-Holland. Montague, R. (1973). The Proper Treatment of Quantification in Ordinary English. In: J. Hintikka, J. Moravcsik and P. Suppes (eds.), Approaches to Natural Language, pp. 221-242. Dordrecht: Reidel. Reprinted in: R. Montague, Formal Philosophy (New Haven, CT: Yale University Press, 1974), pp. 247-270. Oehrle, R.T., E. Bach and D. Wheeler, eds. (1988). Categorial Grammars and Natural Language Structures. Dordrecht: Reidel. Quine, W.V.O. ([1940] 1951). Mathematical Logic. Revised edition. Cambridge, MA: Harvard University Press. Scott, D. (1967). Existence and Description in Formal Logic. In: R. Schoenman (ed.), Bertrand Russell, Philosopher of the Century, pp. 181-200. London: Allen & Unwin. Simons, P.M. (1981). Unsaturatedness. Grazer Philosophische Studien 14, 73-96. Simons, P.M. (1983). Function and Predicate. Conceptus 17, 75-90. Simons, P.M. (1985). A Semantics for Ontology. Dialectica 39, 193-216. Simons, P.M. (1987). Bolzano, Tarski, and the Limits of Logic. Philosophia Naturalis 24, 378-405. set-membership is construed as a kind of relation crossing adjacent logical types in the sense of Russell, then all such relations (there are many) are logical.

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Simons, P.M. (1988). Functional Operations in Frege’s Begriffsschrift. History and Philosophy of Logic 9, 35-42. Simons, P.M. (1989). Combinators and Categorial Grammar. Notre Dame Journal of Formal Logic 30, 241-261. Suszko, R. (1958). Syntactic Structure and Semantical Reference I. Studia Logica 5, 213-244. Tarski, A. (1986). What Are Logical Notions? History and Philosophy of Logic 7, 143-154.

Urszula Wybraniec-Skardowska ON THE FORMALIZATION OF CLASSICAL CATEGORIAL GRAMMAR

1. Ajdukiewicz’s Assumptions The languages of formal logic and mathematics are most often built according to the basic principles of the theory of syntactic categories. The theory, which was constructed by LeĞniewski (1929, 1930) for symbolic languages of systems of protethetics and ontology, was improved upon by Ajdukiewicz (1935) and made use of in the general description of language, including fragments of natural language. When considering Ajdukiewicz, and to some extent also Bar-Hillel (1953), in characterizing language and the categorial grammar that generates it, several interrelated syntactic factors must be taken into consideration. We shall discuss them below, explicating them for the needs of formalization. 1.1. Categorisation Every word in the vocabulary of a given language is taken as belonging to a definite syntactic category. Everyone of the compound expressions formed from the words in the vocabulary – that is the simple words of the language – is assigned exactly one syntactic category. Syntactic categories of expressions are distinguished with respect to the syntactic role which the expressions perform. They are divided into the basic and other functorial ones. Functorial categories create a branched hierarchy similar to that of Russell’s types. They are used for showing the general principles of the concatenation of words in syntactic entities.

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 269-288. Amsterdam/New York, NY: Rodopi, 2006.

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1.2. Functorialibility Every well-formed compound expression of language has a functorargument structure: one may distinguish in it one constituent called the main functor and other constituents, called arguments of the functor, with which it forms the expression as a linguistic entity. Also, every constituent of a given expression which is not a simple word has a functor-argument construction. Well-formed compound expressions of language are functorial in this sense. In the “functorial analysis” of a given expression each of its functors is an operation, a function which together with other expressions of the language yields the new, more compounded expression. The value of every such function-functor of a given expression for arguments which are its constituents, is always a constituent of the given expression (the expression itself may be a constituent of the given expression). The “functorial analysis” of every concatenation of the words in the vocabulary leads, in Ajdukiewicz’s framework, to examining whether a function-argument recording corresponds to the concatenation: all the functors precede their arguments as appropriate (here in parentheses). Let us consider, for instance, compound expressions of the language of arithmetic: a. 4 > 2  1,

b. 2  1 > 1,

c. (5 > 2) = 1.

The parenthetical recordings aƍ, bƍ, cƍ and diagrams of trees meant to explicate them, Ta, Tb, Tc, show a natural, phrasal “functorial analysis” of these expressions. The dotted lines show functors. Ta. 4 > 2  1

4

>

21

2



aƍ. (4) > ((2)  (1))

Tc. (5 > 2) = 1

Tb. 2  1 > 1

1

2

21

>

1



1



1

bƍ. ((2)  (1)) > ( (1))

5

5>2

=

>

2

1

cƍ. ((5) > (2)) = (1)

Appropriate function-argument recordings a f, b f, c f and diagrams of trees: Ta f, Tb f, Tc f show “functional analysis” of expressions a, b, c in Ajdukiewicz’s prefix notation.

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Ta f. 4 > 2  1

>

4

21



2

a f. > (4,  (2, 1))

Tc f. (5 > 2) = 1

Tb f. 2  1 > 1

1

>

21

1



2

1 

b f. > ( (2, 1),  (1))

1

=

5>2

1

>

5

2

c f. = (> (5, 2), 1)

The “functorial analysis” of expressions a, b, c leads to the statement that they are functorial expressions of the language of arithmetic. The functorial expressions of a language may (though they need not) be its well-formed expressions, in short wfes. a and b are such expressions, while c is not. Let us note here that the structurally ambiguous expression 5 > 2 = 1 is not a functorial expression as it is impossible to distinguish the main functor in it. It is also easy to check that, e.g. the expression 5 > (= 2) is not a functorial expression. Let us note that the “functorial analysis” of a, b, c given here is unambiguously determined due to the semantic functions of the signs ‘>’, ‘=’ and ‘’: the first two are signs of two-argument relations between numbers, the third one in a denotes a two-argument number operation, while in b it also denotes a oneargument operation. The mentioned signs, as functors, and thus as functions on signs of numbers, have as many arguments as their semantic correlates have. Unambiguous “functorial analysis” is a feature of the languages of formal sciences. In relation to natural languages the analysis depends on linguistic intuition and often allows for a variety of possibilities (see e.g. Marciszewski 1988b). Traditional phrasal linguistic analysis, formalized by Chomsky (1957) in his grammars of phrasal structures, takes into consideration grammatical phrasal analysis. It imposes in particular the “functorial analysis” of sentence a which is shown in parenthetical recordings Ɨƍ and a f and suitable trees TƗƍ and TƗ f. Expressions Ɨƍ and Ɨf indicate that a is treated here as the functorial expression in which the sign ‘>’ is not the main functor. It is here a functor-forming functor whose only argument is the term ‘2  1’.

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TƗƍ. 4 > 2  1

4

TƗ f. 4 > 2  1

>21

>

>

21

2



4

>21

1

21



Ɨƍ. (4) ( > ((2)  (1)))

2

1

Ɨ f. ( > ( (2, 1))) (4)

The two different “functorial analyses” of sentence a given above are equipollent from the point of view of the syntax of logic. In first-order predicate calculus two logical forms, R(x, y) and P(x), correspond to sentence a. In Ajdukiewicz’s conception categorization does not lead, however, to maintaining the view that “functorial analysis” of a linguistic expression must be determined unambiguously but to the statement that the categorization refers to expressions of a determined functor-argument structure. 1.3. Assigning Indices The method of categorizing functorial expressions by assigning them categorial indices that determine whether they belong to a given syntactic category was introduced by Ajdukiewicz (1935) and is a fundamental idea of categorial grammar. Every categorial index is here a kind of metalanguage variable symbol whose scope is an established syntactic category. The index of an expression of a basic category is a single letter, e.g. ‘s’ stands for expressions of sentential category, ‘n’ for the category of nominal expressions. In Ajdukiewicz’s conception indices of functors of functorial categories have a fractional notation, which is quasi-arithmetical: the index of the expression which is formed by a functor and its argument is placed above the line, indices of its subsequent arguments are placed below the line. Categorial indices permit description of the categorial structure of every functorial expression. This is so because to every simple word which is part of the expression there corresponds exactly one index appropriate to its form and functorial indices determine indices of its compound constituents. Indices of simple words which are parts of the expression determine its

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indexing term, which originates from its function-argument notation (possibly from a parenthetic notation complying with the natural order of simple words) by substituting in it corresponding indices for all its simple words. The indexing term thus has not only a structure wholly corresponding to its functorial structure but also determines its categorial structure (both internal and external). This makes possible determining the syntactic categories (indices) of all the compound constituents of the expression (including the expression itself) by means of the syntactic categories (indices) of their main functors, and, more precisely, of the syntactic categories (indices) of the expressions formed by them, according to the following principle of reduction of indexing terms to simple indices: (p)

If f is an index of the main functor, whose arguments have indices y x 1 , . . . , x n and if f takes the shape of , then z1 , . . . , z n

f (x1 , . . . , xn ) = y (or (x1 , . . . , xk) f (xk+1 , . . . , x n ) = y). Let us consider, for example, indexing terms i(b f), i(bƍ), i(cf), i(Ɨƍ) of expressions b, c, a in notations b f, bƍ, cf and Ɨƍ, respectively. The “categorial analyses” of expressions b, c, a in this notation are explicated by diagrams of trees of categorial indices Ti(b f), Ti(b), Ti(cf), Ti(Ɨƍ) formed from diagrams of trees Tb f, Tb, Tcf, TƗƍ by substituting for each expression its categorial index. For fractional indices we use here slash signs instead of horizontal lines. s

Ti(b f)

s/nn

n

n/nn

n

s

Ti(b)

n

n n/n

i(b f) s/nn (n/nn (n, n), n/n (n))

n

n

n

n/nn

s/nn

n

n

n/n

n

i(bƍ) ((n) n/nn (n)) s/nn (n/n (n))

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s

Ti(cf)

s

Ti(Ɨƍ)

s/nn

s

n

s/nn

n

n

i(cf) s/nn (s/nn (n, n), n)

n

s/n

s/n//n

n

n

n/nn

n

(Ɨƍ) (n)(s/n//n ((n) n/nn (n)))

It is clear that “moving” in the diagrams of trees of categorial indices from the bottom to the top in order to determine the indices of compound constituents of defined expressions makes use of principle (p). Ordinary transformations of functional formulas are made use of here according to the principle. For i(b f) and i(cf) we obtain: (1) (2)

s/nn (n/nn (n, n) n/n (n)) = s/nn (n, n/n (n)) = s/nn (n, n) = s, s/nn (s/nn (n, n), n) = s/nn (s, n) = s.

For i(bƍ) and i(Ɨƍ) we have: (3) (4)

((n) n/nn (n)) s/nn (n/n (n)) = (n) s/nn (n/n (n)) = (n) s/nn (n) = s, (n (s/n//n ((n) n/nn (n))) = (n) (s/n//n (n)) = (n) s/n = s.

Let us note that the symbol ‘ ’ in the “functorial-categorial analyses” of expressions b f and bƍ given here is assigned two different syntactic categories; likewise, the symbol ‘>’ in the analyses of expressions c f and Ɨƍ. Each of these symbols performs here not only two different syntactic functions but also two semantic functions, which allows each one to be treated as two distinct simple words of the language of arithmetic. Similarly we distinguish between homonyms as structurally ambiguous expressions in natural languages. Many equiform or indiscernible expressions may be assigned more than one categorial index. We differentiate them, e.g. by investing them with some additional signs and assign exactly one categorial index to them according to their form. Let us note that the analysed expressions a, b, c belong to the category of sentences. This does not, however, mean that they are all sentences of the language of arithmetic. The functorial expression c is not a sentence.

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1.4. Syntactic Connection Whether a given functorial expression with a set index is a well-formed expression of the language (wfe), i.e. whether it is syntactically connected, and whether it is, in particular, a sentence of the language, depends on whether each of its constituents is well-formed, i.e. it is a simple word or functorial expression which satisfies the following grammatical rule establishing the relation between the index of the main functor of the given expression and the indices of this expression and of the subsequent arguments of the functor: (r)

If f (x1, . . . , x2 ) = y (or (x1 , . . . , xk) f (xk+1 , . . . , xn ) = y), then y f= x1 , . . . , x n

Rule (r) may be called the rule of syntactic connection. It states that the index of the main functor of any functorial expression is the functorial index whose numerator is the index of expression that the functor forms and the indices of the subsequent argument of the functor are the denominator. It may be seen that expression c is not a wfe: one constituents of c does not satisfy rule (r) (formula s/nn (s, n) = s, holds here, see Tcf and (2), and s/nn  s/sn). On the other hand, all the constituents of expressions bf, bƍ and Ɨƍ satisfy rule (r) (cf. (1), (3), (4) and Tb f, Tb, TƗƍ) and accordingly, expression b as well as a, according to the “functorial analysis” Ɨƍ, are wfes. The algorithm for the examination of the syntactic connection of expressions, similar to the procedure used by Ajdukiewicz, amounts to checking whether functional formulas whose starting point is the indexing term of the expression are transformed successively according to principle (p) and yield a single index according to rule (r). We do not obtain, in particular, a single index by applying rule (r) for the indexing term of expression c (cf. (2)). We have only: s/nn (s/nn (n, n), n) = s/nn (s, n). Let us note that the converse implication to (r) holds for any constituent of functorial expression in virtue of (p) (r 1 )

If f =

y , then f (x1 , . . . , xn ) = y x1 , . . . , x n (or (x1 , . . . , x k) f (xk+1 , . . . , xn ) = y).

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When a functorial expression of the language is syntactically connected, rule (r1 ) may be strengthened to an equivalence. That rule (r1 ), which we will call the rule of syntactic connection in Ajdukiewicz’s sense, should hold for any constituent of a functorial expression is a necessary but not sufficient condition for its syntactic connection: rule (r) does not follow from rule (r1 ) ((r1 ) is true when its antecedent is false). Rule (r1 ) corresponds to the following formulas of index transforming: y (x1 , . . . , xn ) = y, x1 , . . . , x n y (x1 , . . . , xk) (xk+1 , . . . , xn ) = y, x1 , . . . , x k x k 1 , . . . , x n which in non-parenthetic recording of indexing terms corresponds to the laws of index reduction: y x1 , . . . , xn o y, x1 , . . . , x n y x1, . . . , xk x k+1 , . . . , xn o y. x1 , . . . , x k x k 1 , . . . , x n The first was applied by Ajdukiewicz in his procedure for checking syntactic connection, the other derives from Bar-Hillel (1953). 1.5. Substitutability The categorization of the expressions of language in LeĞniewski’s and Ajdukiewicz’s conceptions is based on the views of pure grammar furnished by Husserl (1900-1901), and, especially, on the notion of semantic (syntactic) category as a class of mutually substitutable expressions in contexts possessing uniform meaning. These contexts in a LeĞniewski-Ajdukiewicz’s framework are sentential contexts or, more generally, well-formed ones. The definition of a syntactic category given by Ajdukiewicz may be comprised under the scheme: D Ÿ ( E œ J ),

where D, E , J denote the following expressions: D. E. J.

Expression S(B) is formed from expression S(A) by substituting its for its constituent A the constituent B, A, B are expressions of the syntactic category of sentences. S(A), S(B) are expressions of the syntactic category of sentences.

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Introducing syntactic categories in this way causes some difficulties. The definition given above refers to the concept of substitutability and that of a sentence or, more generally, to the notion of a well-formed expression, wfe. These concepts, however, hadn’t been defined earlier. Distinguishing the category of sentences and, generally, the class of wfes determines the language. The concept of a sentence or, more generally, the notion of a wfe of a given language must, therefore, be defined and in such a way as to make it possible to use the algorithm for checking syntactic connection that was given by Ajdukiewicz and discussed in Section 1.4.

2. The Aims of Formalization and the Principles of Their Realization

The concept of categorial grammar introduced by Bar-Hillel, Gaifman and Shamir (1960) for describing and revealing the syntactic structures of language by means of indices (types) has already had its own history (see Marciszewski 1988a, van Benthem 1988). It has also seen the development and elaboration of its formal bases (see Buszkowski 1988, 1989; Wybraniec-Skardowska 1989, 1991). Concentrating here on the formalization of so-called classical categorial grammars we focus upon the axiomatic theory whose assumptions have been described in Section 1 and whose construction is a realization of the aims given in the points (1)-(5) below. The overriding aim of such a formalization, as was stressed in the end of Section 1.5, is: (1)

to provide an exact but, at the same time, general definition of wfe which would support the “functorial-categorial analysis” of its syntactic connection.

Such an analysis, in Ajdukiewicz’s framework, is a set of psychophysical activities, has a functional character, depends on pragmatic conditioning and on treating language expressions as well as indices (which expose their functorial-categorial construction in syntactic analysis) as linguistic concreta, i.e., physical, temporal objects, the socalled object-tokens. This approach pays regard to: (2)

elaborating a concretistic conception of language, as accepted by LeĞniewski, which is based on a primitive metalinguistic characterization of language solely by means of object-tokens and the relations between them.

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At the basis of a formalized theory of language (or grammar) lies: (3)

the striving for a general functorial-categorial characterization of language.

The universal set T of all tokens and the concatenation relation c, which in practice entails connecting tokens either linearly (from left to right, e.g. in European languages, right to left, e.g. in Hebrew or Semitic languages) or non-linearly (e.g. in hieroglyphs and various mathematical formulas), are then the starting point of the formalization. The following two initial vocabularies: the vocabulary V0 1 of a language and its auxiliary vocabulary V0 2 are singled out from set T. Both may include structural symbols, e.g. parentheses, punctuation marks. The first includes the vocabulary V1 of a language, consisting of its simple words, the second its auxiliary vocabulary V2 , consisting of the basic indices that belong to the metalanguage. By means of the concatenation relation c it is possible to generate from the initial vocabularies the set W1 of all the words of the language and the set W2 of all its auxiliary words, respectively. Then, we single out, from W1 , the set E1 of all its expressions, which is the sum of V1 and the set Ec1 of all its compound expressions. On the other hand, the set E2 of all categorial indices, i.e. the sum of V2 and Ec2 of all functorial indices is distinguished from W2 . In a general syntactic characterization of a language, determining the set S of all wfes can not depend on the number or on the shape of simple words, it also can not depend on symbolism, parenthetic or nonparenthetic notation, or the chosen method of connecting words through concatenation. In a general functorial-categorial characterization of the type Ajdukiewicz would approve of, the generation of compound wfes of the set S \ V1 can depend neither on their specific construction nor on their specific notation and the internal structure of categorial indices which allows one to determine their categorial structure (apart from the notation of functorial indices deriving from Ajdukiewicz, various other ones, e.g. quasi-fractional or parenthetic, are also known; see Marciszewski 1981). We assume with respect to the compound expressions of Ec1 and the functorial indices of Ec2 that they belong to the counterdomains of the relation r1 forming compound expressions and, respectively, the relation r2 forming functorial indices. Relation r1 replaces any rule for obtaining expressions of functor-argument structure as defined by concatenations built up from the main functor and its arguments. The relation r 2 replaces, on the other hand, any rules for forming functorial indices from basic indices.

On the Formalization of Classical Categorial Grammar

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The relation r 1 is defined on all finite sequences of words possessing categorial indices. Assigning indices (typisation) of these words is carried out by means of the relation i which indicates the indices of words and which consists in assigning only one index to defined words, in particular to all the simple words of vocabulary V1 according to their form, or even better, to their indiscernibility. Assigning indices also concerns compound expressions which belong to the counterdomain of relation r1 . It allows every expression of the language which has the same index (with respect to indiscernibility) to be placed in a determinate syntactic category and examines if it is a wfe. The set S is distinguished from the set E of all functorial expressions of set E1 . The set E is the sum of the vocabulary V1 and the set of compound expressions each of whose constituents belongs to E 1 . Syntactic connection of functorial expressions of E is obtained in virtue of the rule of syntactic connection (r) holding for all of their constituents (see Sec. 1.4): (r)

the index of the main functor of a given functorial expression is formed from the index of the expression and the indices of the consecutive arguments of this functor. The set S of all wfes may then be generated by the following system: G =

which may be treated as a reconstruction of Ajdukiewicz’s categorial grammar, or of any other classical categorial grammar. When a functorial expression is a wfe, an analogue of rule (r1) of syntactic connection in Ajdukiewicz’s sense (see Sec. 1.4) must hold for each of its constituents (if the rule does not hold, the functorial expression can not be included in S). The following is an important element of the formal characterization of categorial grammar: (4)

the notion of substitutability of expressions must be connected with the notion of syntactic category, in accordance with Husserl’s ideas.

It may be achieved by an exact definition of the concept of substitutability and it leads to the basic theorem of the theory of syntactic category which has the following scheme: (s)

D Ÿ ( E œ J ƍ),

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Urszula Wybraniec-Skardowska

where D and E denote expressions as in Section 1.5, while J ƍ is the expression: (J ƍ)

S(A) and S(B) are expressions of the same syntactic category.

The aims of formalizing classical categorial grammars discussed so far comprise characterization on the “level of tokens.” In semiotic practice such a characterization is not sufficient. The linguistic objects analysed here have a double nature: they may be either tokens or types – abstract, ideal objects, whose concrete realizations are object-tokens. In pragmatics, types are made use of in explaining the communication process between people, in semantics for explaining such notions as meaning or denotation, in syntax for describing grammatical rules. The token-type distinction derives from Peirce who treated types as classes of equiform or equisounding tokens. Let us note that the notion of equiformity may, here, lead to some misunderstanding. In syntactic analysis tokens which are of different fonts may be considered as equiform, while signs that do not differ in their shapes may none-the-less be regarded as variform (see the final remarks in Sec. 1.3). We shall replace the concept of equiformity by the indiscernibility relation ~ and show: (5)

the possibility of a bi-level characterization of language, not only on the “level of tokens” but also on the “level of types,” where language is characterized as language of expression-types of the set S generated by the categorial grammar G

 T , c, V 1 , V 2 , i, r1 , r2 , (r ) !,

whose notions correspond to the ones of grammar G and are defined as follows: sets T , V 1 , V 2 as respectively the sets of types, i.e., families of classes of indiscernible (identifiable) tokens, and relations c, i, r1 , r2 – as relations defined on types by means of the appropriate relations defined on tokens; rule (r) corresponds here to rule (r) and refers to the syntactic connection of functorial expression-types.

The aim of formalizing a language on the “level of types” is to provide a characterization that will allow the language to be described in an analogous way to the way it is described on the “level of tokens.”

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3. Foundation of Formalization

We shall present below the axioms and basic definitions of the theory of classical categorial grammars. They do not differ essentially from the ones given in Wybraniec-Skardowska (1989, 1991). We have used the commonly accepted set-theoretical formalism. 3.1. Level of Tokens The set T of all tokens, the binary relation of indiscernibility ~ in T, the ternary concatenation relation c in this set, the initial vocabularies V01 and V02 , the relation i indicating indices, and the binary relations r 1 and r 2 forming compound expressions and forming functorial indices, respectively, defined upon some finite sequences of T and assigning them defined elements of the set are primitive concepts of the theory under discussion, on the ‘level of tokens’. Let A, B, C, D, A0 , A1 , A2 , . . . , B0 , B1 , B2 , . . . range over tokens of T, and x, y, z, v, x0 , x 1 , x2 , . . . , y0 , y1 , y2 , . . . index-tokens of T, in particular. We postulate for the relation of indiscernibility that A1.

a. b.

A~BŸB~A A~BšB~CŸA~C›A=C

 it is symmetric,  it is quasi-transitive

and we use it to define the relation of identifiability | : D1.

A | B œ A ~ B › A = B.

It is easy to check that the relation | is an equivalence in T. The concatenation relation c is characterized by the axioms: A2. A3. A4.

 Cc(A, B; C), c(A1 , B 1 ; C1 ) š c(A2 , B2 ; C2 ) š A1 | A2 š B1 | B2 Ÿ C1 | C2 , c(A, B; C) š D | C Ÿ c(A, B; D).

We read the expression ‘c(A, B; C)’ as follows: C is concatenation of A and B. It follows from A3 that the concatenation relation on tokens is not a function: however, only indiscernible or identical tokens may be a concatenation of the same pair of tokens. For the initial vocabularies we postulate, for k = 1,2, that A5.

V0 k Ž T,

282 A6. A7.

Urszula Wybraniec-Skardowska

A  V0k š B | A Ÿ B  V0k, c(A, B; C) Ÿ C  V0 k.

The set W1 of all the words of the language and the set W2 of all its auxiliary words are defined as the smallest sets which include vocabulary V0 1 and vocabulary V0 2 respectively and which are closed under the concatenation relation, i.e. for k = 1,2, D2.

W k = ­½ {X ° V0 k Ž X š  A, B  X C (c(A, B; C) Ÿ C  X)}.

We accept, for k = 1,2, that A8.

C  W k \ V0 k Ÿ  A, B  W1 c(A, B; C).

The relation i of determining indices satisfies the axioms: A9. A10. A11. A12.

i  W1 u W2, D1 (i) ˆ D2 (i) = ‡, i(A, x) š i(B, y) š A | B Ÿ x | y, i(A, x) š B | A š y | x Ÿ i(B, y).

We read the expression i(A, x) as follows: x is an index of the word A. Relation i is therefore not a function, but any word may possess only one index, in accordance with identifiability. Obviously, D1 (i), D2 (i) denote here the domain and counterdomain of the relation i, respectively. The vocabulary V1 of language and its auxiliary vocabulary V2 are defined as follows (k = 1,2): D3.

V1 = V 0 1 ˆ D1 (i).

The set Ec1 of all compound expressions of language and the set Ec2 of all functorial indices are defined by means of relations r k (k = 1,2): D4.

Eck = D2 (r k).

The following axioms characterize these relations (k = 1,2): f

A13. D1 (r k) Ž  Dk(i)n š D2 (r k) Ž Dk(i) \ V k, n 1

A14. r k(A0 , A1 , . . . , Am ; A) š r k (B0 , B1 , . . . , Bn ; B) Ÿ (A | B œ m = n š  0djdn (Aj | B j)); A15. r k(A0 , A1 , . . . , An ; A) š 0djdn (Aj | Bj) š B | A Ÿ r k(B0 , B1 , . . . , Bn ; B) The expression (eA )

r 1 (A0 , A1 , . . . , An ; A)

On the Formalization of Classical Categorial Grammar

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reads: A is a compound expression formed from the main functor A0 and its consecutive arguments A1 , . . . , An . The expression (i)

r 2 (y, x1, . . . , xn ; x0 )

reads, on the other hand: x0 is a functorial index formed from the index y and consecutive indices x1 , . . . , xn . The relations r 1 and r 2 are binary relations defined, respectively, on finite sequences of words which possess indices or on indices of such words. They are not functions but they assign to the same sequences identifiable compound expressions or, respectively, functorial indices. We should remind (see Sec. 2) that expression A in (eA ) may be treated as a substitute for any compound expression formed from the main functor A0 and its consecutive arguments A1 , . . . , An , irrespective of the rules according to which it was possible to form it from them, the syntactic position they occupy or the applied notation. It must only have this main functor and these n (n > 1) arguments. Expression A which is present in (eA ) may then replace, for instance, both expression a and expressions aƍ and a f (see Sec. 1), as well as fragments of them: 2  1, (2)  (1),  (2,1). Expression A in (e A ) may also replace the synonymous expressions of various languages of the sentential calculus, e.g. p Ÿ (q š r); CpKqr; p implies (q & r); (P o (Q š R)) recorded here in the following notations: the one usually applied, àukasiewicz’s bracket-free notation, and computer programming languages for teaching logic, Mizar and Tableau, respectively. Similar observations pertain to expression (i). Index x 0 which appears in it may replace any functorial index formed from index y and consecutive indices x1 , . . . , xn . Thus, the various indices in Ajdukiewicz’s notation, e.g. s/nn, n/nn, s/n // s/n s/n as well as indices recorded in other notations applied by other categorial grammarians (see Marciszewski 1981) and corresponding to e.g. the ordinary sign of equality ‘=’: s/nn, s:nn, n\s/n, (s; n . . . n), (s,(n, n)), (nsn)2 , F2 NNS may correspond to x0 in (i). The fact that they are indices of functors which form sentential expression with two nominal arguments is the common feature of these indices.

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The set E1 of all expressions of a language and the set E2 of all its indices are determined by the following definition (k = 1,2): D5.

Ek = V k ‰ Eck.

We introduce the set C (A) of all constituents of the expression A and of the set C n (A) of all constituents of the n-th order of expression A (n t 0), which are applied in the procedure of checking syntactic connection, for the expressions of set E1 and, in particular, for its compound expressions. D6.

Let A  E1 . Then a. b. c. d.

B  C o (A) œ B | A, B  C 1 (A) œ  nt1 A0 , A 1 , . . . , An  D1 (i) (r 1 (A0 , A1 , . . . , An ; A) š  ndjdn (B | Aj)), B  C k+1 (A) œ  D  C k(A) (B  C 1 (D)), C (A) = ¯¿{B°B  C n(A), for some n}.

The set E of all functorial expressions is defined as follows: D7.

E = V1 ‰ {A  Ec1 °B  C (A) (B  E1 )}.

We accept that & A16. i (E) Ž E2 . & i (E) denotes here the image of the set E with respect to the relation i. From the axioms given it follows that (*)

A  E Ÿ i( C n (A)) Ž i( C (A)) Ž E2 .

Thus, every constituent of a functorial expression is assigned a certain categorial index (see Sec. l, diagrams of trees of categorial indices Ti(b f), Ti(b),Ti(cf ), Ti(Ɨƍ)). In the theory given the following expression (rA )

0djdn i(Aj, x j) š i(A, y) Ÿ (i)

is a formalized recording of rule (r) of syntactic connection (see Secs. 1.4 and 2) for expression A satisfying (e A ). For expression A satisfying (eA ), from the axioms A13, A14 and A12, on the basis of (r A ), we obtain the following expression which is an equivalent of rule (r 1 ) of syntactic connection in Ajdukiewicz’s sense (see Sec. 1): (r1A ) 0djdn i(Aj, x j) š (i) Ÿ i(A, y).

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The set S of all the well-formed expressions of language (wfes) may be defined as follows: D8.

S = ­½ {X Ž E° V1 Ž X š A  A 0 , A1 , . . . , An  X (e A ) š (r A ) Ÿ A  X}.

The set S is therefore the smallest set which includes the set of all simple words of the language and to which belong all its functorial expressions A satisfying, for (e A), condition (rA ). Definition D8 makes it possible to use in practice the algorithm for checking the syntactic connection of expressions which would come near to the one given by Ajdukiewicz (see Sec. 1). On the basis of the assumptions accepted we obtain from the definition the following formula: (**)

A  E Ÿ (A  S œ n C n (A) Ž S).

Thus, speaking loosely, if every constituent B of functorial expression A satisfies the appropriate condition (rB), then A is wfe; if this does not hold, that is if conditions (rB ) or (r1B ) are not satisfied for any constituent B of expression A, A cannot be counted as a wfe. A precise description of the algorithm for checking the syntactic connection of expression A involves introducing a linear ordering in set C (A) of all its constituents. It is obvious that we do concrete checking by beginning with the constituents of the highest order and that constituents of the same order may be ordered in various ways (see examples in Sec. 1). We omit here a formal definition of the ordering in set C (A) (see Wybraniec-Skardowska 1991). For the set S \ V1 of compound wfes we axiomatically assume that some of them have a basic index: & A17. i (S \ V1 ) ˆ V2  ‡. The axiom assures the non-emptiness of all the subsets of tokens of the set T discussed here. The basic expressions of the language are introduced as wfes which possess a basic index, its functors as wfes which possess a functorial index. Syntactic categories of expressions (basic expressions or functors) are defined as sets of functorial expressions of the same categorial index (basic index or functorial index) with respect to indiscernibility. Formal definitions of these concepts are easy to give. Some more important theorems of the theory sketched here are to be found in WybraniecSkardowska (1989, 1991), among them the basic theorem of the theory of syntactic categories of schema (s) (see Sec. 2). The definition of the

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substitutability of expressions is introduced here with the help of the concept of the substitutability of constituents of the n-th order (n>0) of a given expression. This concept is defined inductively. 3.2. Level of Types Each of the sets of tokens X introduced by an axiom or definition in the previous part has its dual counterpart X , which is a quotient family of equivalence classes of identifiability relations | on set X: DX.

X

X / |, i.e. A  X œ A  X ( A [ A]| ).

Syntactic categories of expression-types (basic expression-types and functor-types) are defined as families of equivalence classes of identifiable tokens belonging to the syntactic category of expressions of identifiable index-tokens (basic index-tokens and functorial index-tokens, respectively). Every relation R, introduced in the previous part, has its dual counterpart in the form of relation R defined on types as follows: D R.

R( A o , A1 , . . . , A n ) œ Ao , A1 , . . . , An ( A o . . . š An

[ Ao ] | š A1

[ A1 ] š

[ An ] | š R ( Ao , A1 , . . . , An )), for n t 1.

Definitions DX, DR and definitions of syntactic categories allow one to state, on the basis of the axioms and definitions given in Section 1 on the “level of tokens,” that all their counterparts formulated for dual concepts are theorems of the theory presented on the “level of types” (cf. Wybraniec-Skardowska 1989, 1991). Hence, it follows immediately that all axioms, definitions and theorems which are valid on the “level of tokens” and which characterize categorial grammar G, as well as the language of expression-tokens generated by it, have on the “level of types” their counterparts which analogously describe a categorial grammar G and thus also the language of expression-types generated by it.

4. Final Remarks

The presentation of the foundations of the theory of classical categorial grammars provided in the present paper has not been meant to be a lecture on it. It does, however, take into account a certain pattern of bi-level formalization of language syntax in the spirit of LeĞniewski and Ajdukiewicz, and partially also Bar-Hillel. Such formalization may be

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developed in a similar spirit for languages of expressions including variable-binding operators (Wybraniec-Skardowska 1991). It can also be developed by equipping it with a semantic characterization. Further, it can be modified for use in describing other formal models of language. Lastly, it is worth noting that the concretistic approach to language presented here was first taken up by the author of the present paper in 1985 (see 1991) on Jerzy Sáupecki’s suggestion and is a rarely used ontological approach in the formal characterization of language. It was shown by the author of the present paper in Wybraniec-Skardowska (1989) that the opposite, platonizing approach, consisting in exchanging the levels of formalization, i.e. from token-type into type-token, is an equally good philosophical approach from the theoretical point of view.

PoznaĔ School of Banking Branch in Chorzów ul. Wandy 66 41-500 Chorzów e-mail: [email protected]

REFERENCES Ajdukiewicz, K. (1935). Die syntaktische Konnexität. Studia Philosophica 1, 1-27. Bar Hillel, Y. (1953). A Quasi-arithmetical Notation for Syntactic Description. Language 29, 47-58. Bar Hillel, Y., C. Gaifman and E. Shamir (1960). On Categorial and Phrase Structure Grammars. Bulletin Research Council Israel 9F, 1-16. Buszkowski, W., W. Marciszewski and J. van Benthem, eds. (1988). Categorial Grammars. Amsterdam & Philadelphia: John Benjamins Publishing Company. Buszkowski, W. (1988). Three Theories of Categorial Grammar. In: Buszkowski et al. (1988), pp. 57-84. Buszkowski W. (1989). Logiczne podstawy gramatyk kategorialnych AjdukiewiczaLambeka [Logical Foundations of Ajdukiewicz-Lambek Categorial Grammars]. Warszawa: PWN. Chomsky, N. (1957). Syntactic Structures. The Hague: Mouton & Co. Husserl, E. (1900-1901). Logische Untersuchungen. Halle: Niemeyer. LeĞniewski, S. (1929). Grundzüge eines neuen Systems der Grundlagen der Mathematik. Fundamenta Mathematicae 14, 1-81. LeĞniewski, S. (1930). Über die Grundlagen der Ontologie. Comptes Rendus des Sciences de la Société des Sciences et des Lettres de Varsovie 23, No. 3, 111-132. Marciszewski W. (1988a), A Chronicle of Categorial Grammar. In: Buszkowski et al. (1988), pp. 7-21. Marciszewski W. (1988b). How Freely Can Categories Be Assigned to Expressions of Natural Language? In: Buszkowski et al. (1988), pp. 197-220.

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Marciszewski, W., ed. (1981). Dictionary of Logic as Applied to Study of Language: Concepts, Methods, Theories. The Hague: Martinus Nijhoff. Van Benthem, J. (1988). New Trends in Categorial Grammar. In: Buszkowski et al. (1988), pp. 23-33. Wybraniec-Skardowska, U. (1989). On the Eliminability of Ideal Linguistic Entities. Studia Logica 48, No. 4, 587-615. Wybraniec-Skardowska, U. (1991). Theory of Language Syntax, Categorial Approach. Dordrecht/Boston/London: Kluwer Academic Publishers.

PART V INTENSIONALITY, SENSE AND CONSEQUENCE

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Liliana Albertazzi RETRIEVING INTENTIONALITY: A LEGACY FROM THE BRENTANO SCHOOL 1

1. An Ambiguous Noun At first sight, the term ‘intentionality’ bears all the features of an ambiguous noun, that is, of those nouns that lend themselves to numerous interpretations. In modern philosophy the term is inextricably bound up with phenomenology and today a philosophy of intentionality comprises an area of analytic philosophy close to philosophy of mind. Analytical philosophy examines intentionality by starting from linguistic utterances, on the assumption that language represents the various ways in which we are intentionally directed towards the objects of our experience. A philosophy of intentionality is therefore widely viewed as analysis of the propositions and/or sentences that remand to and report on psychic acts. In particular, analytic philosophy considers linguistic analysis to be an essential point of departure for the identification of those general psychic acts and cognitive systems of which language is a particular instance. Put in these terms, the philosophy of intentionality brackets off a constitutive part of its original theory, if not denying it nevertheless delegating some parts of it to other scientific fields, in particular psychology or the broader area of the cognitive sciences. I refer to the nature of psychic acts, their spatial, temporal and qualitative structure, as well as to their modes of representation of objects. Nowadays, in fact, these aspects are usually addressed not by philosophers but by scientists, for example in the analysis of categorial perception, distribution of attention or of imagery processes.

1

This paper is an expanded and revised translation of Albertazzi (1999).

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 291-314. Amsterdam/New York, NY: Rodopi, 2006.

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The term ‘intentionality’, by contrast, in its original sense (and, it seems, in its modern revival in science), embraces a theory much broader than the one now current in philosophy. The Latin etymon denoted both the directing of a psychic act towards an object and the object towards which the act is directed. One of the controversial aspects of this definition therefore lies in defining what constitutes an objective intention, whether this is a seeing, hearing, imagining, etc. of an object of some kind. A second difficulty arises from the fact that the Latin term actus translates both the Aristotelian terms energheia and entelecheia, which denote respectively coming-into-being and the enduring being that results, i.e. accomplished being as such or one of its effective, actual or present properties. The expression ‘actus mentis’ then has a metaphysical connotation and as such constituted the basis of Franz Brentano’s theory of intentional reference, which became the standard source for subsequent and even contemporary citations on intentionality (Brentano 1874). Brentano, moreover, despite the widely held belief to the contrary, did not develop a thoroughgoing theory of intentionality, but rather one in only embryonic form, and especially in his unpublished writings. A thoroughgoing theory of intentionality, in fact, must fulfil a number of conditions, namely: (1)

(2) (3)

The moment-now of the intentional presentation must extend through a continuing set of durations which comprise fringes of the past and of the future contents. Distinctions must be made among the various ways in which the psychic act is directed towards an object. The relation between psychic act, object and content must be reconstructed, with a precise distinction being drawn between their distinctive parts.

A modern version of a theory of intentionality of this type, which focuses on the relationship between act, object and content, has been developed in Poland by Twardowski, a pupil of Brentano. Twardowski’s theory was then resumed by Husserl with some modifications which accentuated the feature of temporal dynamicity.2 Bearing these

2

The first outline of a theory of intentionality, in fact, is to be found in an essay written by Husserl in 1894, Intentionale Gegenstände, in reply to a question raised by Twardowski in §§ 13 and 14 of his (1894). On this, see Schuhmann (1993), Albertazzi (1993).

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developments of Brentano’s theory in mind, the argument that I wish to develop below is the following: (1) (2)

Reference to a theory of intentionality is much more complex than the currently canonical versions employed by analytic philosophy. It can serve the purposes of cognitive science and in particular the development of an empirical-experimental theory of cognitive space (see Albertazzi 2002).

When connected with experimental data, in fact, Twardowski’s and Husserl’s theories of intentionality3 are able to explain certain aspects of categorial perception, and a number of problems to do with the nature of imagery and the theory of pictorial representation.4 Specifically, they are able to account for the substantially temporal nature of spatial perception. The aim of this paper is therefore to show the relevance of the core of the theory firstly developed by Twardowski to contemporary cognitive analysis by applying it to various examples of visual perception.

2. Murals Those travellers waiting for a train connection at the station of Civitanova Marche, Italy, cannot fail to notice the two large murals next to the control cabin. At first sight they are not noteworthy, apart from some signs of originality, because many similar murals can be seen up and down the Adriatic coast. These are naïve frescoes in pastel tints, mainly blue, with a few stylised figures set against a broad background: horses in the foreground with a high-speed train on the horizon. In one of the murals, the horse in the foreground stands with its mane and tail held vertically: the mane falling onto its back, the tail onto its flanks, three of its feet on the ground, one raised in a semi-circle. The horse is depicted at repose in a field. The mane and the tail of the foal in the background are instead horizontal, and its legs are stretched. The foal is depicted as running against the wind in the same direction as the highspeed train on the horizon

3 4

On the ‘philosophy of mind’ of the Lvov-Warsaw School see ĩegleĔ (1995). On the imagery debate see, for example, Finke (1989).

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The two horses are therefore inserted in a homogeneous visual field endowed with spatial and temporal indices of ordering. Besides the visual field there is a field of positions super-ordinate to it. In fact, from the point of view of the observer, one horse is behind the other, more to the left than the other, and both horses are in front of the high-speed train. Moreover, the depiction of the horse at rest resembles a static snapshot of a punctiform event of reality, whereas the depiction of the running horse refers to a duration which enables identification of movement in a direction. In other words, the first horse is depicted in an a-temporal frozen present while the second is depicted in a temporal flow which contains non-independent moments of the past and of the future states. These different features are conveyed by the figures, independently on the fact that they are part of the same naïve pictorial depiction and of the same visual perception.5 In the case of the mural, the contrast is given by indices of directionality which apparently refer to two snapshots of reality in so far as they are endowed with all the features of staticity and depictive permanence: however many times I may look at them, in fact, I always obtain the same perceptual return. My gaze only yields two extremely similar images of the same scene, ones that are wholly static and identical in terms of distance, position, perceiver’s point of view, perspective, and so on. Moreover, the simplicity of the technique prevents textures, perceptual gradients and the interplay of light and shade from functioning as expressive indices of staticity or movement. 6 Leaving aside a number of problems concerning the relationship between a pictorial representation and a instantaneous snapshot of 5 Husserl would say that there is a similarity by contrast between the mural figures (mane, foal) akin to that between two triangles of the same colour: for example, between two green triangles of different sizes placed against a red background, or between two congruent triangles of different colours, one yellow and one blue. See Husserl (1939). 6 On textures as source of information, see Gibson (1979).

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reality, in which case the painted foal would show all its depictive “falsity” (see Arnheim 1954), what I wish to stress here is that both scenes in the mural bear the traces of time, but in different ways. The saliencies of the features of the foal, in fact, convey the perception of movement and of elapsing time. In other words, the picture of the horse (at least in the first instance) stands for itself, while that of the foal does not, because it implicitly and intuitively presupposes the former as a scene or state temporally previous or subsequent to it. Overall, different perceptual temporal schemes, which nevertheless remand to each other, operate in the depiction of the two horses. These are primitive pictures of the phenomenal level of reality, although what is depicted in the case of the foal is the arrow of time, the directionality of which is tied to the duration of the actual perception or to its reconstruction in a visual image. In other words, it is not so much the arrow of physical time that is apparent in this mural as the arrow of phenomenal time, in so far as the two are connected.7 Moreover, the field comprises numerous direction lines or escape points corresponding to the locus positions filled by the various types of phenomenal content (colour, shape, texture, etc.). The visual field as a whole, therefore, takes the form of a system of positions with a continuous ordering.8 In this case, besides similarity by contrast, various other phenomenal principles are in operation, viz.: (1) (2) (3)

The principle of cohesion, according to which a surface lies on a single object if and only if they are connected. The principle of contact, according to which surfaces move together if and only if they are connected. The principle of continuity, according to which an object exactly follows a spatially-temporally connected trajectory.9

The mural of Civitanova therefore poses a problem of a naïve or intuitive physics: that is, a problem of the kinetic or dynamic representation of perceptual events.10 In every representation of a visual 7

On this difference see Husserl (1966); Albertazzi (1999b). For this concept, see Brentano (1979, 1982). 9 See Eilan et al., eds. (1993), p. 103. As regards perception, rather than spatial extension in the strict sense, involved here is a sort of perceptual extendedness (Extensität) with characteristics similar to those of pictorial space or of a drawing. See Brentano (1979); Albertazzi (2002). 10 The term event denotes various forms of occurrences of the phenomenal world: (i) instantaneous events, (ii) stationary events (objects) or (iii) quasi-stationary events (perceptions of movement). Given this specification, here I shall use the terms ‘phenomenal event’ and ‘phenomenal object’ interchangeably. 8

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scene, whether it psychically presents it or whether it depicts it in naïve forms like children’s drawings or murals, the fact remains that certain spatial properties like distance and position implicitly remand to phenomenal properties of the visual field like movement, velocity or direction.11 Also the indices of direction, in fact, as well as those of clarity, texture, shape and colour, are elementary phenomena – as they are defined by painting textbooks – and as such they can be considered to be primitives of the phenomenal world. 12 In the course of time, the depiction of pictorial space has employed diverse visual cues to represent, for example, a process, a story, or a sequence of interconnected events (see Massironi 2002, Ch. 7). Various devices have been contrived to portray different aspects of the unfolding of events, as well as devices to render the perception of an action ongoing in very brief intervals, or devices to connect two sequential images together. These devices convey the persistence of a scene in time despite partial changes in the context. In these cases attempts have been made to explain intentional comprehension of the temporal relation among images in terms of cognitive drag, i.e. as due to the working memory (see Baddeley 1986). The most interesting situation from the intentional and cognitive point of view, however, concerns the use and comprehension of the visual cues that enable us to perceive previous temporality in a single scene – as exemplified by the mural of Civitanova. In the case of that mural, in the first instance it would seem that principally at work in its depiction of events are the kinetic principles that connect position, velocity and acceleration without taking account of force and mass. But the foal’s mane, horizontally directed, or its legs stretched to their fullest extent in the effort of running, restore the presence of a rapid shift in the action which produces a force – the wind raised by the foal as it runs. At the same time, the above-mentioned principles of solidity, continuity and similarity instead suggest that this is the same horse moving along a spatiotemporal trajectory. In this sense, also the subsequent depiction of the same scene characterized by different indices of directionality involves not only kinetic principles but force-dynamic principles as well. In other words, if a representative entity of the phenomenal world is to hold as the representation of an object of the physical world, also required is the intuition of the relevance of a force to the behaviour of the entity represented (see Neu 1978, p. 107). Hence the principles of the physical 11 12

On these concepts in perceptual continua, see Albertazzi (2002). On the primitives of the phenomenal world, see Albertazzi (1998, 2001, 2002).

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world are woven and internalised into phenomenal perception at the intuitive level, however modified by the structures of the intentional reference.13 Key questions still remain, however. How do we know that different temporal schemes operate in the same mural? How do we know that it is the same scene seen referring to two different moments, one of which is temporally previous or subsequent to the other? The features of the picture analysed so far are unable to explain this particular type of connection; they can only show it in the form of signs. These in fact are not structurally temporal figures or forms, like, for example, a rhythm, an escape, a clash, a fall, a push, a sudden change of colour on a surface or light in a space; they are merely static visual images which manifest embedded temporal traces connected to indexes of direction. To shed light on the temporal structure of spatial representation, let us look at some further common examples.

3. Warning Signs Public buildings are usually fitted with warning signs, for instance arrows indicating “emergency exit,” “no way up,” “no way down” and the like. Let us think of the sign denoting “emergency exit.” This sign usually consists of a stick figure represented in a static act of running indicated by the directionality of the legs, which evoke the perception of movement. The same applies to the sign “children,” although in this case the movement is less evident:

13 For this concept see Shepard (1984). To use an expression from Husserlian phenomenology, material and formal a priori elements are deposited in phenomenal structure. See Husserl (1929).

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Unlike the murals described in the previous section, what we have in this case is not a pictorial depiction, however naïve, as in a mural, but a graphic symbol. As in the case of the mural, however, this is not a snapshot. A similar effect is achieved by the Italian road sign for “falling rocks,” however, where the scheme of the falling rocks is symbolized by the descending direction of the boulders from left to right, or vice-versa:

Or, in the warning sign “beware hump,” where the danger is suggested by a stylised graphic symbol of a “camel hump.” Unlike the previous sign, however, there is no indicator of downwards directionality from left to right to suggest a continuity of direction: the symbol is perfectly symmetrical and can be read from right to left or from left to right. Why, then, do we still perceive an upward or downward movement or process? Also, the warning sign for “slippery surface” gives some indication of direction, but the sign “beware hump” does not even do this. Described exactly, the sign “beware hump” simply consists of an undulating line on the two-dimensional plane of a metal plaque. And yet, when grasping that the line signals danger, we see the points of the line as if they had different depths, as if they were at different distances from us. The perceptual schemes acting at the level of road signs operate equally well in a photograph of a discotheque dance movement, for example, or in a photograph of a military parade which provides a static snapshot of the moment of the goose-step. However, the example of the warning signs, in its simplicity, evidences that woven into the perception of the events under observation are dynamic structures (or parts of structures) whose traces are deposited in the graphic symbolizations of reality. Consider the following picture (see Massironi 2002, p. 207; Massironi and Bonaiuto 1966):

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This figure immediately conveys the perception of temporality and, specifically, a series of events connected in a before/after relation. Even the perception of causality is therefore founded on cognitive processes able to reconstruct the scene by inserting the present content into a duration of pre-existence and succession which is intrinsically intentional, thereby confirming the theory put forward by the Brentano school.14 In this connection consider the photograph of the dancer Gret Palucca reproduced by Kandinsky in his Point Line to Surface (see Kandinsky 1926, Ch. 1):

14

On the perception of causality, see Michotte (1946); Albertazzi (2002b).

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On the basis of this photograph Kandinsky drew the following graphic scheme of the leap:

The photograph and the scheme were part of Kandinsky’s graphicconceptual theory of the point, whose nature and dynamic tension towards an implicit spatial development (in line and surface) he described. This type of spatial development has an intrinsically temporal progression. In fact, a point has the implicit and potential function of determining place and direction of the phenomenal space. Independently

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of the numerous plastic, phenomenal and metaphysical implications of Kandinsky’s argument, what I wish to stress here is that any phenomenal event, even one reproduced in a snapshot, further relates both to elementary forms of the tension of the underlying materials – materials which display an intrinsic dynamicity – and an essentially temporal field structure. A point, in fact, is potentially and intrinsically a punctiform surface, just as a line is a linear form surface. We are told that Palucca actually is leaping not by the snapshot but by a series of contents and psychic images in succession, of which the snapshot is a non-independent part and to which it implicitly remands.15 Every perceptual content, whether visual, auditory or tactile-kinetic, then, displays a synthetic internal structure which is manifest in its persistence. Once again, key questions still remain. How do I recognize the implicit evolution of events in a warning sign? How do I recognize that Gret Palucca is leaping, or was leaping at the moment when the photograph was taken? At issue in this case, too, is the role of the elementary visual phenomena as schemes of experience which function as props for the development of thought. In favour of the hypothesis of an essentially temporal scheme operating in visual perception, however, there are other examples.

4. Leaning Towers Why is the Tower of Pisa so impressive? Because it seems to be falling on top of us as we walk beneath its leaning side, and also because at first sight it is decidedly crooked and – apparently at least – in breach of the laws of physics. However, even though we are reassured that it will not fall (and so far it has not), because there are other physical laws that ensure its stability, a number of problematic features remain. There is verticality, for instance, which is a typical directional index of many objects in the phenomenal world, like towers. In other words, we expect towers to be vertical, possibly solid, and not oblique. For this reason the overestimation of verticality is a general phenomenon of experience. In fact, the height of fifteen- or twenty-metre-high buildings is usually overestimated, compared with the same lengths in the horizontal, and rectangles are often perceived as slightly over-tall 15

I use the concept of a non-independent part in the same sense with which it is employed in Husserl’s Third Logical Investigation. See Husserl (1900-1901).

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squares. An extremely simple example of this phenomenon, first introduced by Klemm, is provided by the following figure:

In this case, the subject of the experiment perceives the vertical segment as being the same length as the horizontal segment, although it is in fact shorter by one-fifth (see Klemm, cit. in Brunswik 1935, p. 86). Hence there is an asymmetry and a heterogeneity in the estimation of directions which suggests a mix between visual space and kinaesthetic functional space – as already Allesch pointed out in his time (see Allesch 1931). More in general, these findings display interesting features and raise a number of problems, including: (1) (2) (3)

The phenomenal problem of the connection between verticality, solidity and incumbency. The problem of the verticality of solids as their canonical state but virtually prone to alteration over time. The problem of justifying the sensate connection – to use Koffka’s expression – we experience between the spatial geometry of vision and the dynamic temporality of kinaesthesis.

Marigonda, for example, has provided an extremely significant example with regard to the first point (see Marigonda 1986, pp. 297-327). In fact, considering two triangles of different heights, shapes, positions in space, and sizes, descriptions by observers confirm that static objects convey social meanings like command, incumbency, obedience, etc. The second point concerns the presence of an implicit temporal scheme even in apparently static visual perceptions. The problem of the leaning tower, indeed, can be summed up in the fact that at every single actual moment of visual perception we are dealing with a single state of an imagery process. In this case, the leaning tower is a “parallelogram”

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in a certain state of rotation, in one single static presentation of the whole virtual process. In other words, the leaning tower remands to a straight vertical tower as the initial state of the process of rotation and inclination. The actual static visual presentation of the Tower of Pisa obtained whenever we look at it, therefore, virtually contains within itself, as traces, potential multiple states prior and subsequent to the actual state. Hence visual spatial perception is intrinsically temporal. A second consequence of assuming a potential and temporal viewpoint in visual perception is that every distinguishable event endowed with certain characteristics of symmetry is virtually derivable from a principle of asymmetry, and vice versa (see Leyton 1993, Ch. 2, p. 40; Massironi 2002, Ch. 7, esp. p. 202). For example, the leaning towers are interpreted as the same parallelogram seen in successive temporal moments of its rotation. This fact can be explained from the subjective point of view as the necessary integration and completion of the structural incompleteness of every actual perception, and from the perceptual point of view as the presence of dynamic elements intrinsic to the texture or phenomenal material of experience. Put simply: I implicitly judge the Tower of Pisa to be leaning because I recognize it as a non-independent moment of a process of rotation, or of deviation from a canonical state, as in the case of the rotation of a parallelogram, of which the tower is a paradigmatic example. Secondly, I judge the tower to be leaning because the materials of which it is made (stone, brick, etc.) intrinsically convey, from a phenomenal point of view, a sense of staticity and solidity. These are non-volatile materials, ones that are “heavy” and which therefore tend to assume a position of verticality. From this point of view, the “Tower of Pisa,” as content of an act of seeing, is part of a multiple process of intentional states: it is a state internal to the present which presupposes a duration (see Brentano 1928, Part I, Ch. V, §4).

5. Acts, Contents and Phenomenal Objects The murals of Civitanova, warning signs and leaning towers, however diverse they may be, have structurally something in common: an embedded dynamic temporal scheme to which they remand despite the apparent staticity of the actual visual image. This conclusion gives evidence of the presence of the modes of temporal presentation implicit in the perception of phenomenal events,

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be these spatial or temporal. The fact that the examples presented display a temporality of traces, i.e. a history of images, does not explain what it is that engenders these traces: what a general theory of intentionality of the Twardowski-Husserl type is instead able to do. Our understanding of the objects of experience, in fact, is conditioned by subjective integrations tied either to psychic content or to phenomena of completion due to the acts of perception: 16 in other words, a key contribution to our perception of phenomenal events is made by psychic phenomena defined as intentional acts.17 Let us once again take the example of the Leaning Tower, on the basis of parallelograms in successive rotation, and simplify the example further by considering only the presentation of two different rectangles, one “squat” and “seated,” the other “slim” and “standing.” If we ask the experimental subjects to draw a third median rectangle between the two presented to them, according to the type of characteristics considered – i.e. expressive or geometric ones – different types of rectangles are drawn. Taken individually, these rectangles variously emphasise one or other set of characteristics mentioned above. This experiment is important firstly because it distinguishes among the different types of quality of phenomenal objects, among which the tertiary or expressive qualities play a crucial role in the decodification of experience in general.18 Secondly, it is important because it highlights the psychic processes that underlie the performance of a task or the solution of a problem. In short, the third median triangle is constructed, imagined and drawn according to the psychic contents that the subject considers to be either the most important or the canonical one. Hence it follows that the representation is implicitly derived from, and dependent on, the psychic contents employed by the psychic acts of seeing, hearing, thinking, imagining, and so on. To use an expression from Benussi – a pupil of Meinong’s – the reality of the objects of experience is given by an invariance among: (1) (2)

16

The objects of experience. The psychic acts directed towards them.

The issue of cognitive integrations in perception theory is a matter of heated debate. For a brief summary, see Palmer (1997). 17 This concept is at the basis of Brentano’s theory of intentional reference. See Brentano 1874. 18 For this distinction see Metzger (1941), Ch. 2, §8.

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The psychic contents that enable the perception itself of objects, which are an integral part of it (see Benussi 1925, §§2-4).

In this sense also the perception of apparently static ‘objects’ like rectangles, parallelograms or towers (straight or leaning) belongs to a more general theory of intentionality and of the modes of presentation of objects.

6. Moving Spirals The foregoing conclusions comprise a more general point of view which concerns spatial shape as an intrinsically dynamic form and warrants analysis. The nature of this dynamicity of shapes has been well evidenced by pictorial studies, in particular those by Kandinsky mentioned already, which highlight that spatial shape, due to its intrinsic dynamicity, is a structurally temporal form. 19 The fact that events or processes in movement are intrinsically endowed with a temporal scheme has been invariably admitted, except by committed supporters of Parmenides. Apples falling from trees, weights being thrown, the waves of a stormy sea, bicycles crossing the street, but also balls colliding on a billiard table, are naturally considered to be perceptions of movement with various levels of complexity. More controversial is the intrinsic dynamicity or the intrinsic potential for movement of spatial figures endowed with staticity. I have discussed some aspects of this problem in the case of leaning towers, as perceptions which implicitly relate to a virtual dynamic process of deviation from the canonical state. We have also seen that things have points of view which in turn are susceptible to further analysis and development, especially in consideration of the role played by implicit psychic contents in the apprehension of the phenomena of reality under examination. One of these developments nowadays concerns cognitive linguistics, and specifically the notion of a conceptually moveable object, in the sense that movement can be conceptually attributed to objects that appear to be intrinsically stable. Examples of this are extremely frequent in language, for example when we say that “the road runs along the coast” or “the road moves through the trees” (see Talmy 1991). However, the most interesting problem, I believe, is finding realist grounds to justify this substantially conceptualist position. For this 19

The distinction between form (temporal) and shape (spatial) is in Albertazzi (1999a), p. 261.

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purpose I shall examine a specific case, that of static phenomenal figures which immediately arouse a perception of movement. These are stereokinetic movements, and once again the theory of intentionality plays a role in their explanation. Firstly consider the following figure:

If we rotate the disk slowly and evenly first clockwise and then anticlockwise, we see, in expansion, in the first case an evolutive spiral movement and in the second an involutive one. These are a specific type of the so-called apparent movements (see Benussi 1925, p. 11; Musatti 1924, pp. 105-120). Also belonging to this specific domain is the so-called negative reproduction of movement: the phenomenon, that is, whereby after prolonged observation of an object moving in a certain direction, an apparent movement in the opposite direction is observed when it comes to rest. This specific type of visual perception has at least two important aspects: the first is the absolute evidence of the perception of movement, where the movement is as clear and indisputable as in the case of a cyclist riding along a country road or a hare running along its grass verge. In both cases we have de facto perceptions and not judgements about perceptions made subsequently to them. These kinds of phenomena, therefore, give rise to a distinction in perception between perceptually present phenomena and perceptually absent ones, independently of the ‘apparent’ status of certain phenomena of the first type. 20

20

For this distinction, see Benussi (1925), p. 12; Kanizsa (1991), Ch.2, §2.

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Secondly, this type of phenomenon enables us to recognize cases of transformation of certain ‘perceptually absent’ phenomena into ones of ‘perceptual presence’, a case in point being stereokinetic phenomena. Consider, in fact, the following photograph:

Used for the experiment is an iconoscope with black discs revolving once a second. The figures are: circle + dot (or ellipse + dot).21 A slight corporeity is apparent if the figure is still. But if it rotates, we see a solid, a cone, which moves freely in a three-dimensional space, and appears of a well defined height. In particular, there is a physical rotation and a perceptual translation: what we perceive is solely the translation, however. When the disc rotates, various types of movement can be seen. First of all, an apparent relative movement (the dot moves along the circumference of the circle, and the distance between the dot and the circumference is a constant).

21

The image has been realized by ATI-AM/S.A.M. of Trento University.

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In this case, a solid object with rigid transparent walls is seen moving in space. Moreover, the solid object oscillates in space in such a manner that its principal axis coincides with the height of a cone, the base of which is turned towards the observer. 22 While it is being perceived, the solid may assume partially different shapes, ranging from a cone to a funnel. The important fact, however, is that without a shadow of doubt we perceive a volumetric, however transparent, object. Experiments in this area are of especial importance for the light that they shed on the problem under consideration here. They demonstrate, in fact, that in this case, too, the construction of perceptual reality – specifically, visual reality – involves a complex series of subjective operations of scene completion, and as such belongs to a theory of intentional acts. Moreover, the fact that during perception a particular shape may vary (obviously to a limited extent) and assume specific features at a certain moment of the time of its development (internally to the so-called time of presentness) provides further evidence of a temporal scheme structurally implicit to visual perception (see Albertazzi 1995, 1999b). The structural importance of the temporal modes of acts of presentation is borne out by a further phenomenon which exemplifies the temporal nature of every intentional reference. Imagine a glowing coal: if we hold it still we see a glowing point; if we slowly rotate it, we see a glowing point in movement. If, however, we rotate the coal rapidly and then quickly change position, what we see is no longer a glowing point in movement but a glowing circle in stasis. All these perceptions – of the still point, of the point in movement, and of the circle in stasis – are due to the continuity and the succession of the temporal modes with which the object is presented. Without continuity in the modes of presentation we are unable to see either the point in movement or the circle in stasis. Both cases involve temporal differences because if an object is at rest something appears the same both before and after; conversely, in the case of an object in movement, something appears to be differently localized before and after (see Brentano 1928, Part I, Ch. V, §4; Benussi 1913, esp. Ch. 9). Consequently, in this case too the actual perception of an object of some kind perceived at rest or in motion involves both a succession of temporal modes and the perception of this succession. In other words, both phenomenal objects at rest and phenomenal objects in motion are 22

Benussi (1925), p. 13. For a summary and a development of these experiments, see Zanforlin (2003), Albertazzi (2004).

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perceived as such by virtue of the temporal structure of the presentation that is directly addressed to the phenomenal object actually given in the actual presentation, and which indirectly maintains the individual phases of the presentation of the object in the overall duration of the time of presentness (see Brentano 1928, Part I, Ch. V, §4; Husserl 1985).

7. Theory of Intentionality All these examples taken from the perception of visual events, or from their essentially spatial schematic depiction, have features in common. All of them, even those most closely connoted by staticity and schematicity, display kinetic and sometimes dynamic properties: these are tied both to their virtual capacity for movement in space and time and to the implicit change due to the more general field structure that comprises them, as in the case of apparent movements. Indeed, with the sole exception, perhaps, of events so temporally restricted that they are instantaneous (fleeting objects, as Meinong called them), even the spatial shapes of visual perception have a component of process-based form, although this may go entirely unobserved at first sight. The fact that a spatial shape remains such, at rest, is nevertheless a temporal process, from the point of view of vision, although it does not involve any apparent change. Some sort of reciprocal relation holds whereby essentially temporal forms like a melody or the undulating motion of a sphere can be statically frozen in musical notation or in a sinusoidal line and thus be schematised into a static spatial form. The shape of a vault, with its recesses and protuberances, conveys this cross-reference. We all have had the experience of playing the comic-book game of joining up numbered dots to create a figure. Although extremely elementary, this operation demonstrates the existence of highly sophisticated temporal relationships within the spatial shape.23 In other words, the entire virtual sequence of the possible movements of a figure, or of its depiction, acts as a background or substrate for every possible place that the movement may reach or is currently passing through. We have a similar experience when watching a long pass during a football game, the movement of which we complete by means of a perception which includes the imminent outcome of the ongoing 23

In mathematical terms, this is the phenomenon of extreme curvature. See Leyton (1993), Ch. 1, p. 17.

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movement. Note that this is only possible on the basis of a presentation of the simultaneous shape of the trajectory of the movement about to be accomplished, although in confused and still partially indeterminate manner (see Volkelt 1962, p. 148). This dual presence of the phenomenological characteristics of the stability and processuality, of the staticity and temporality of perceptual objects – whether they are objects at rest or in motion – raises the problem of whether the visual perception of objects is really independent of the presence of the observer in the field. Or, in other words, whether it is independent of a theory of intentionality; that is, independent of the fact that the overall structure of the perceptual field of phenomenal events is based and dependent on structural forms of subjective proprioception (see Albertazzi 2002). To return to the case of the mural, how do we know that the same scene is depicted in two different temporal moments? What are the indicators, the discontinuities, that enable us to represent the temporal continuity of the scene to ourselves? Or how can an essentially static representation like “falling rocks” or “emergency exit” act immediately as a signal of danger and immediately communicate the idea of movement? How are movement and velocity woven together in those graphic symbols so that they immediately constitute evidence of “danger” and of “escape”? If the representation of the events of the phenomenal world were a substantially static representation enacted by an eye constructed to passively record events in a sequence of unconnected snapshots, we would never recognize these events as belonging to a continuing duration, and therefore as indexes of danger. However, it is not sufficient to acknowledge that a duration is necessary to elaborate sensory outcomes into meaningful events of experience. Other problems still remain: for instance, is the temporality of events due to our judgement of them or is it a characteristic of the modes of presentation of events themselves? In other words, does the object that I see have an immediate temporal connotation, or is the intervention required of some form of recognition ontologically subsequent to its presentation? 24 One of the principal problems to solve for a theory of intentionality, therefore, is the relationship between the continuity of a sequence of different perceptions and the unitary nature of these perceptions taken individually. A second problem is explaining the relationship between 24

This question was subjected to detailed discussion and experimentation by Meinong’s school.

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the actuality of the perceptions that we experience as directly present and the presence in the same situations of indices of directionality addressed to the past or the future, i.e. to absent states. The temporal structure of the presentation, understood as a dynamic unfolding of consciousness, therefore constitutes at least a constitutive component of the field structure of perceptual phenomena, since it is in the extension of the time of presentness that we apprehend the beingbefore or the being-after of phenomena, or simply their permanence (see Husserl 1936). Every phenomenal spatial ordering into above-below, to the right-to the left, in front-behind, before-after, and so on, presupposes a simultaneous flow of discretely ordered impressions, the temporal traces of which are also retained by the visual image and frozen in indices of direction which relate to the dynamics of the actual presentation of the event. In other words, the reference system usually exemplified in terms of spatial ordering can only be understood within a temporal subjective completion of the scene. Every temporal form experienced (with the already mentioned exception of those presented for extremely brief instants of time), for example, every spatial shape of vision is experienced with a particular duration and therefore with a processual form, although this aspect is usually unobserved. Experience of its permanence, of its duration, is a process in time, even if it does not involve a change of shape, or more exactly, essential changes which we perceive. Even the most ingenuous perception of spatial shapes therefore implicitly remands to the structure of a temporal scheme, which can be explained only in terms of a theory of intentionality of presentations.

Università degli Studi di Trento Rovereto Branch Department of Cognitive Science and Education Matteo del Ben Street, 5 I-38048 Rovereto, Italy e-mail: [email protected]

REFERENCES Albertazzi, L. (1993). Brentano, Twardowski and Polish Scientific Philosophy. In: Coniglione et al., eds. (1993), pp. 11-40. Albertazzi, L. (1995). Forms of Completion. Grazer Philosophische Studien 50, 321-340.

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Albertazzi, L. (1997). Murali, indici di pericolo e torri pendenti. In: La natura della visione, pp. 85-99. Rome: Montag. Albertazzi, L. (1998). The Aesthetics of Particular: A Case of Intuitive Mechanics. Axiomathes 9, 169-196. Albertazzi, L. (1999a). Form Metaphysics. In: L. Albertazzi (ed.), Shapes of Form: From Gestalt Psychology and Phenomenology to Ontology and Mathematics, pp. 257-305. Dordrecht: Kluwer. Albertazzi, L. (1999b). Time of Presentness: A Case of Positivistic and Descriptive Psychology. In: S. Cataruzza (ed.), Vittorio Benussi. Axiomathes 10 (special issue), 49-74. Albertazzi, L. (2001). The Primitives of Presentation: Wholes, Parts, and Psychophysics. In: L. Albertazzi (ed.), The Dawn of Cognitive Science: Early European Contributors, pp. 29-60. Dordrecht: Kluwer. Albertazzi, L. (2002a). Towards a Neo-Aristotelian Theory of Continua: Elements of an Empirical Geometry. In: L. Albertazzi (ed.), Unfolding Perceptual Continua, pp. 29-79. Amsterdam: Benjamins. Albertazzi, L. (2002b). Kinetic Structures and Causatives. Axiomathes 13, 17-37. Albertazzi, L. (2004). Stereokinetic shapes and their shadows. Perception 33, 1437-1452. Allesch, G.J. von (1931). Zur nichteuklidischen Struktur des phänomenalen Raumes. Jena: Fischer Verlag. Arnheim, R. (1954). Art and Visual Perception: A Psychology of the Creative Eye. Berkeley, CA: University of California. Baddeley, A. (1986). Working Memory. Oxford: Oxford University Press. Benussi, V. (1913). Die Psychologie der Zeitauffassung. Heidelberg: Winter. Benussi, V. (1925). La suggestione e l’ipnosi come mezzi di analisi psichica reale. Bologna: Zanichelli. Brentano, F. (1874). Psychologie vom empirischen Standpunkte. Leipzig: Duncker & Humbolt. English translation: Psychology from an Empirical Standpoint (London: Routledge & Kegan Paul, 1973). Brentano, F. (1928). Vom sinnlichen und noetischen Bewusstseins. Edited by K. Kraus. Hamburg: Meiner. Brentano, F. (1976). Philosophische Untersuchungen zu Raum, Zeit und Kontinuum. Edited by S. Körner and R.M. Chisholm. Hamburg: Meiner. English translation: Philosophical Investigations on Space, Time and the Continuum (London: Croom Helm, 1988). Brentano, F. (1982). Deskriptive Psychologie. Edited by R.M. Chisholm and W. Baumgartner. Hamburg: Meiner. English translation: Descriptive Psychology (London: Routledge, 1982). Brunswick, E. (1935). Experimentelle Psychologie. Wien: Springer. Coniglione, F., R. Poli and J. WoleĔski, eds. (1993). The Scientific Philosophy of the Lvov-Warsaw School. PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 28. Amsterdam: Rodopi. Eilan, N., R. McCarthy and B. Brewer, eds. (1993). Spatial Representation: Problems in Philosophy and Psychology. Oxford / Cambridge: Blackwell. Finke, R.A. (1989). Principles of Mental Imagery. Cambridge, MA: The MIT Press. Gibson, J.J. (1979). The Ecological Approach to Visual Perception. Boston: HoughtonMifflin. Husserl, E. (1900-1901). Logische Untersuchungen. Halle: Niemeyer. English translation: Logical Investigations (London: Routledge & Kegan Paul, 1970).

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Husserl, E. (1990/1). Intentionale Gegenstände. Edited by K. Schuhmann. Brentano Studien 3, 137-176. Husserl, E. (1929). Formale und transzendentale Logik: Versuch einer Kritik der logischen Vernunft. Halle: Niemeyer. English translation: Formal and Transcendental Logic (The Hague: Nijhoff, 1969). Husserl, E. (1936). Erfahrung und Urteil: Texte zur Genealogie der Logik. Edited by L. Landgrebe. Prag: Akademie. English translation: Experience and Judgment (Evanston: Northwestern University Press, 1973). Husserl, E. (1966). Analysen der passiven Synthesis. Den Haag: Njihoff. English translation: Analyses of Passive Synthesis (Dordrecht: Kluwer, 2001). Husserl, E. (1985). Texte zur Phänomenologie des inneren Zeitbewußtseins. Edited by R. Bernet. Hamburg: Meiner. English translation: Lectures on Internal Time Consciousness (Dordrecht: Kluwer, 1991). Kandinsky, W. (1926). Punkt und Linie zu Fläche. München: Bauhaus Bücher, Langen. Kanizsa, G. (1991). Vedere e pensare. Bologna: Il Mulino. Kanizsa G. and G.B. Vicario, eds. (1968).Ricerche sperimentali sulla percezione. Trieste: Università degli Studi di Trieste. Kleint, H. (1940). Versuche über die Wahrnehmung. Zeitschrift für Psychologie 149, 31-82. Kraus, O. (1930). Zur Phänomenognosie des Zeitbewusstseins. Aus dem Briefwechsel Franz Brentanos mit Anton Marty nebst einem Vorlesungsbruchstück über Brentanos Zeitlehre aus dem Jahre 1895. Archiv für die gesamte Psychologie 75, 1-22. Langacker, R. (1987). Foundations of Cognitive Grammar, vol. 1: Theoretical Prerequisites. Stanford: Stanford University Press. Langacker, R. (2000). Grammar and Conceptualization. Berlin: DeGruyter. Leyton, M. (1993). Symmetry Causality Mind. Cambridge, MA.: The MIT Press. Marigonda, E. (1968). Contributo allo studio della percezione dei rapporti interpersonali. In: Kanizsa G. and G.B. Vicario, eds. (1968), pp. 145-159. Massironi, M. (2002). The Psychology of Graphic Images: Seeing, Drawing, Communicating. London: Laurence Erlbaum Ass. Massironi, M. and P. Bonaiuto (1966). Ricerche sull’espressività: Qualità funzionali, intervalli e relazione di causalità in assenza di movimento reale. Rassegna di Psicologia Sperimentale Clinica 8, 3-42. Metzger, W. (1941). Psychologie: Die Entwicklung ihrer Grundannahmen seit der Einführung des Experiments. Darmstadt: Steinkoppf. Michotte, A. (1946). La perception de la causalité. Etudes de Psychologie, vol. 8. Louvain. English translation: The Perception of Causality (London: Methuen, 1963). Musatti, C. (1924). Sui fenomeni stereocinetici. Archivio italiano di psicologia 3, 105-120. Neu, T. (1978). Von der Gestaltungslehre zu den Grundlagen der Gestaltung: Von Ittens Vorkurs am Bauhaus zu wissenschaftenorientierten Grundlagenstudien; eine Lehr und Wahrnehmungstheoretische Analyse. Ravensburg: Mayer. Palmer, S. (1997). Foreword. To: I. Rock, Indirect Perception, pp. xi-xxviii. Cambridge, MA.: The MIT Press. Sander, F. and H. Volkelt (1962). Ganzheitspsychologie. München: Beck. Shepard, R.N. (1984). Ecological Constraints on Internal Representation: Resonant Kinematics of Perceiving, Imagining, Thinking, and Dreaming. Psychological Review 91, 417-447. Schuhmann, K. (1993). Husserl and Twardowski. In: Coniglione et al., eds. (1993), pp. 41-58. Amsterdam: Rodopi.

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Talmy, L. (1975). Semantics and Syntax of Motion. In: Kimball (ed.), Syntax and Semantics, vol. 4, pp. 181-238. New York: Academic Press. Talmy, L. (1991). Path to Realization: A Typology of Event Conflation. Berkeley Linguistics Society 17, 480-520. Twardowski, K. (1894). Zur Lehre vom Inhalt und Gegenstand der Vorstellungen. Edited by R. Haller. München-Wien: Philosophia Verlag, 1982. English translation: On the Content and Object of Presentations (The Hague: Nijhoff, 1977). Zanforlin, M. (2003). Stereokinetic Anomalous Contours: Demonstrations. Axiomathes 1-2 (The Legacy of Gaetano Kanizsa in Cognitive Science, ed. L. Albertazzi), 389-398. ĩegleĔ, U., ed. (1995). The Philosophy of Mind of the Lvov-Warsaw School. Axiomathes 1-2 (special issues). Trento: Centro Studi per la Filosofia Mitteleuropea.

Kazimierz TrzĊsicki LOGICAL AND METHODOLOGICAL ASSUMPTIONS OF AJDUKIEWICZ’S AND KRIPKE-PUTNAM’S VIEWS OF MEANING

Of all the philosophers of the Lvov-Warsaw School it is K. Ajdukiewicz who gives the question of meaning the most attention; the concept of meaning is essential to his philosophical investigations. During the whole period of his scientific activity Ajdukiewicz ascribed to language “a definite and very important role in the process of cognition” (1931, p. 105). The importance of the concept of meaning [. . .] of expressions for methodology and epistemology results also from that, that theorems of science are nothing other than meanings of certain sentences which these sentences have in a definite language and the process of cognition (as opposed to knowledge as a result of this process), at least in its most developed form, is just that meaning of certain sentences and of other possible expressions. (1934a, p. 145)

In the paper on the meaning of expressions (1931), Ajdukiewicz writes that there are two approaches to concept of meaning and looks for a third one. The way chosen by Mill is the one which properly points out the direction for further investigation, as Ajdukiewicz writes, “in order to grasp that which is usually understood by the expression ‘meaning’ in the sense which we are interested in” (1931, p. 123). In the paper on the problem of empiricism (1964), which to some extent is his philosophical testament, Ajdukiewicz returns to Mill’s approach, regarding it as the starting point for the proper solution to the problem of meaning: “My new approach is near to Mill’s own, it can even be regarded as its development and precise restatement” (1964, p. 399). But this new “third way” has not been elaborated even in the phase of basic ideas.

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 315-323. Amsterdam/New York, NY: Rodopi, 2006.

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Seeing Mill’s approach as the starting point for the creation of their own approach is common to Ajdukiewicz and Kripke, one of the creators of a currently much discussed solution to the problem of meaning which I will call the Kripke-Putnam’s view. Kripke writes: The classical tradition of modern logic has gone very strongly against Mill’s view. Frege and Russell both thought, and seemed to arrive at these conclusions independently of each other, that Mill was wrong in a very strong sense: really a proper name, properly used, simply was a definite description abbreviated or disguised. [. . .] Nevertheless, I think it’s pretty certain that the view of Frege and Russell is false. (1972, pp. 255-257)

In the last part of his paper, Kripke compares the view he advocates to that of Mill. The modern logical tradition, as represented by Frege and Russell, seems to hold that Mill was wrong about singular names, but right about general names. [. . .] My own view, on the other hand, regards Mill as more-orless right about “singular” names, but wrong about “general” names. (1972, p. 322)

However, that is not the most important thing. Far more significant is the fact that Ajdukiewicz was strongly influenced by the formal logicians in the Lvov-Warsaw School. His view of language was designed on the pattern of languages of formal systems and strongly dependent of them. This resulted in e.g. a rejection of this view of meaning in connection with Tarski’s argument (Ajdukiewicz 1964, p. 397). Kripke’s view of meaning is connected with his semantics of possible worlds for modal logics. Both Ajdukiewicz and Putnam are interested in the methodology of science. We can also question how logic and methodology of science have affected the Ajdukiewicz’s and Kripke-Putnam’s views of meaning.

1. Natural Language from the Point of View of Logic and of Methodology of Science Ajdukiewicz’s view of language is created on the pattern of the languages of deductive systems, i.e. discursive languages. All the concepts used to describe a certain language are permanently related to just that language. Thus, Ajdukiewicz writes not simply about the meaning of the expression E but about the meaning of the expression E in language L; not about the rules of meaning but about the rules of meaning of language L. This is the custom of formal logicians already rigorously practised in the Lvov-

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Warsaw School. In the structure and properties of discursive languages there is the source of the fundamental distinction between open and closed languages. This division was abandoned in connection with the rejection of the primary conception of meaning. Independently of this and not known at that time, Gödel’s theorem makes the existence of rich, coherent (and especially) closed languages doubtful. Gödel’s incompleteness theorem plays an important role in Putnam’s considerations and as it seems could be used to explain Kripke’s stipulation that the main aim of his investigation is not a theory of meaning but only that he is trying to give a better picture of the problem. For Ajdukiewicz the properties of natural language and the languages of natural sciences, i.e. empirical languages, are to be regarded as extrapolations of the properties of the languages of formal systems. Language is an ideal object and all that can be said about it applies to an empirical language only in as much as the empirical language approximates the ideal language. No doubt this approach is characteristic for the logicians of the Lvov-Warsaw School. Consider, for example, comments in Tarski’s work (1933). Incidentally speaking, some logicians in opposition to Tarski employ his semantics to natural language. D. Davidson (1970) is the chief advocate of this approach. Ajdukiewicz justifies the view that only the languages of formal systems are languages in the proper sense of the word as used in the investigation of meaning and rejects the possible objection that such languages cannot be used in methodology and epistemology. In almost every science there is such an “idealization tendency.” Physics states its thesis for ideal gases though it is well-known that no gas is ideal; mechanics occupies itself with motions which have to take place in conditions that in reality are never fulfilled. Physics conducts itself in this way maybe because only in this manner knowledge can be brought closer to reality. At first theorems are stated which are strictly valid only for ideal gases whereas for real gases there is a rather considerable margin of error. Only later these laws are changed in order to make this margin of error smaller. If one started with the requirement of absolute matching with reality this would be a difficult task. We point this out to defend our dissertation. In it we begin by considering an ideal case which only approximately agrees with the cognisable reality. Maybe it is the first step, after that further ones follow which decrease the error of approximation. (1934a, p. 195, my translation)

In his uncompleted last book Logika pragmatyczna (1965), Ajdukiewicz repeats that thought in a less radical way.

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Paying attention to differences between the logical conception of language and those with which linguists usually operate, we would stress that the logical conception is much simpler and the analyses connected with it prepare the conceptual apparatus which is indispensable to give clarity to the works of linguists. (1965, p. 24)

It is worth mentioning here that Putnam changed his mind many times in philosophy. However, as he writes “Philosophy is not a subject that eventuates in final solutions” (1988, p. XII). Carnap – a great philosopher who had an aura of integrity and seriousness which was almost overwhelming – would stress that he had changed his mind in philosophical issues and changed it more than once. ‘I used to think [. . .] I now think’ was a sentence construction that was ever on Carnap’s lips. (1988, pp. XI-XII)

Kripke and Putnam do not construct an ideal language but they consider natural and empirical languages a proper subject of investigations. This does not mean that they give a definition of meaning. Putnam even proves that such a definition is impossible. This does not mean that they have adopted a naive picture of language. They are aware that natural languages have their specific features. Different statements – in some cases, even statements that are “incompatible” from the standpoint of classical logic and classical semantics – can be true in the same situation because the words – in some cases, the logical words themselves – are used differently. (1988, pp. 115116)

Ajdukiewicz’s view of language was influenced by the “aversion” of the Lvov-Warsaw School to intensional contexts motivated by the wellknown troubles with such contexts. Usually, the school was open to new ideas but in this case it was conservative. It did not perceive the changes in philosophy of language and missed a great deal of interesting semantic problems, which were created by the problems of intensionality. The Lvov-Warsaw School’s approach to natural languages was characterized by typical limited reconstructivism – natural language is a greatly imperfect composition which by logical analysis should be improved. In the case of Kripke and Putnam the question is opposite. They aim at the elaboration of a conception of meaning for natural language. Of course, they do not have any problems with intensionality which have been solved by Kripke’s semantics of possible worlds. Nevertheless, they do not consider a natural language as a language to which “negative” theorems about formal languages are not applicable. The fact that

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[. . .] no formal system of mathematics can define what it is to be a mathematical proof, and no formal system of inductive logic can define what it is to be ‘confirmed’, so no program for interpreting utterances in a natural language can define what it is for utterances to be synonymous or even coreferential. A complete computational characterization of ‘proof ’, ‘confirmation’, ‘synonymy’, and so on, will always be an impossibility. (Putnam 1988, p. 119)

Ajdukiewicz was not interested in the origin of a language but only in the language as a product. A language is for him an autonomous product. He inherited the distinction between acts and products from his teacher, Twardowski. The objective assignment of meanings to expressions, in point of fact, even speaking about such an assignment, is possible only if the language is regarded as a product. Kripke maintains that the reference of expressions is determined by an initial baptism: it fixes the reference by some contingent marks of the object. The name denoting that object is then used to refer to that object and a chain of communication has been established which passed the name from link to link. Most important is the fact that a speaker is a member of a community of speakers: the name has been passed to him by tradition. Putnam also stresses the social aspects of language. “Language is a form of cooperative activity, not an essentially individualistic activity. [. . .] Reference is socially fixed and not determined by conditions or objects in individual brains/minds” (1988, p. 25).

2. Meaning of Expressions in Logic and Methodology In “O znaczeniu wyraĪeĔ” (1931), Ajdukiewicz distinguishes two ways in which the notion of meaning is defined. Associationists treated meaning as a mental item. For Ajdukiewicz, the decisive argument against this conception is its consequence that in order to use a phrase as an expression of a language, it is sufficient and necessary that, in the mind of the user, the phrase be accompanied by a thought of the type assigned to this type of phrase in this language (1931, pp. 113-120). This argument is based, as Ajdukiewicz points out, on Husserl’s considerations in Logische Untersuchungen (vol. II, part I, chapter “Ausdruck und Bedeutung”). The meaning of expressions can be sought in the objects to which these expressions refer. This is Mill’s view of meaning. For Ajdukiewicz, there are three reasons to reject it: 1. the expressions which are not names would not have a meaning; 2. we would have to accept that some names

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would be without meaning; and 3. and, most importantly, different meanings would be assigned to names such as ‘rectangle’ by different persons. The third way chosen by Ajdukiewicz, the way fully developed in “Sprache und Sinn” (1934a), consists in searching for meaning in language. What is the origin of the view? Without doubt it is inspired by the description of language in formal systems. A language is not only characterized by the definition of a well-formed expression but by the definition together with the rules of the system. From this point of view, the full characteristic of a language consists of a vocabulary, grammatical rules and rules of meaning. The difference between artificial languages, such as the language of logical calculus, and natural languages, such as Polish, consists exclusively in the fact that the rules of an artificial language are uniquely and strictly determined whereas the rules of natural languages are not precise and thus these languages are not languages in the proper sense of the word. The languages of logic and mathematics are characterized only by discursive rules, in the case of empirical languages there are also empirical rules. Discursive rules govern the relation between a sentence and a sentence. Empirical rules govern the relation between experiential data and sentences. Ajdukiewicz’s view has been expounded exclusively by means of pragmatic concepts (acceptance of sentences) and syntactic ones – the latter are first of all used in the description of the matrix of a language. Ajdukiewicz does not use semantic concepts at all. He was doing it consciously because he feared involvement in semantic antinomies (Tarski’s works were not known then). The main aim of Logika pragmatyczna (1965), a textbook which was intended to fill the gap in Polish scientific literature, and provide an exposition of the fundamental concepts of semantics and the methodology of science in a manner satisfying current requirements. In this uncompleted work, Ajdukiewicz did not give the definition of meaning, but, as he usually did, he defined the sameness of meaning of an expression: whether two people are using (in a given case) some expression in this or another sense is decided by those aspects of the thoughts by means of which they understand, which are the same (1964, p. 19). The act of understanding an expression takes place when somebody hearing it directs his thought at an object different from it. It has been mentioned that Ajdukiewicz rejected his first conception of meaning as fiction. The decisive moment for this opinion was Tarski’s argument, which soon after publishing “Sprache und Sinn” (1934a), was

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presented to Ajdukiewicz. In the case of an axiomatic system with two axioms: AzB BzA all the theorems of this system are invariant with respect to A and B. On the ground of the conception of meaning under discussion, both the names should have the same meaning. Of course, this is a contradiction. Ajdukiewicz concluded that the concept of meaning of an expression is not definable by syntactic means and that semantic concepts are indispensable. This argument, without reference to Tarski and formulated differently, is used by Kripke and Putnam. Kripke considers as an example a situation in which we find a yellow metal with all the properties by which we originally identified gold. Would we say, in this counterfactual situation – let the substance be iron pyrites – that in that situation gold would not even be an element because pyrites is not an element? It seems that we would not (Kripke 1972, p. 320). In Putnam’s case, his fabulous version of Tarski’s argument is used against a lot of “isms” especially against the thesis advanced by himself that the computer is the right model for the mind. Under the name “functionalism,” it has become the dominant view, some say the orthodoxy, in contemporary philosophy of mind. In his thought experiment involving Twin Earth, Putnam supposes that the people on Twin Earth have brains identical with ours, a society virtually identical with ours, and so on. The only relevant difference between Earth and Twin Earth is that the liquid that plays the role of water on Twin Earth is supposed not to be H2 O but a different compound, call it XYZ. Putnam claims that the term ‘water’ does not have the same reference in Earth English and Twin-Earth English. Ajdukiewicz, in the period of “Sprache und Sinn” (1934a), elaborated the idea of a conceptual apparatus (that can be conventionally chosen) as a component of our picture of the world (Ajdukiewicz 1934b). A priori elements in science are due to its language. The same is true of logic. The logic is embedded in the axioms of language. He did not agree with àukasiewicz who maintained that experience would decide if two- or one of the many-valued logics invented by him would be true. Since 1947 Ajdukiewicz evolved from methodological apriorism to empiricism. He was aware that to reject logic as an a priori element of a language, the concept of meaning has to be changed. The new idea of the concept of meaning, meaning as codenotation, as it was cited, could be regarded as a development and a precise statement

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of Mill’s view. It has not been elaborated even to its initial phase. The codenotation of compound expressions is described as the set of designators and their syntactic structure. The codenotation of a simple expression should be based on its denotation and the subjective sense that that expression has for a particular person. For Kripke, a description is used by a person to refer to some object, to fix a reference. In some cases of initial baptism, the referent is determined by a description, by some uniquely identifying property. The description is not giving a synonym or something for which the name is an abbreviation. Contingent marks are used to refer to the object even if that object does not have the properties in question in some counterfactual situations (Kripke 1972, p. 309). Thus, it seems that there is a sufficient resemblance between Ajdukiewicz’s subjective sense of an expression and Kripke’s description of an object to fix a reference to see both solutions as realizing the same idea. It seems that the difference between Ajdukiewicz’s view of language (and meaning) from the period of “Sprache und Sinn” and Putnam’s idea of language in the “functionalism” period is not substantial. For Ajdukiewicz the idea of a language with the rule of meaning was fiction and for Putnam a complete computational characterization of ‘proof’, ‘confirmation’, ‘synonymy’, and so on, will always be an impossibility (Putnam 1988, p. 119). In spite of all the differences of time, tradition and school paradigm thorough consideration of the concept of meaning has led to the conclusion that the concept of meaning can be understood only if one understands the decisive role played by the empirical component in our knowledge.

Uniwersytet w Biaáymstoku Department of Logic, Computer Science and Philosophy of Science pl. Uniwersytecki 1 15-420 Biaáystok, Poland e-mail: [email protected]

REFERENCES Ajdukiewicz, K. (1931). O znaczeniu wyraĪeĔ [On the Meaning of Expressions]. In: KsiĊga Pamiątkowa Polskiego Towarzystwa Filozoficznego we Lwowie [Commemorative Book of the Polish Philosophical Society in Lvov] 12.II.1904-12.II.1929, pp. 31-77. Lwów: PTF.

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Ajdukiewicz, K. (1934a). Sprache und Sinn. Erkenntnis 4, 100-138. Ajdukiewicz, K. (1934b). Das Weltbild und die Begriffsapparatur. Erkentnis 4, 257-287. Ajdukiewicz, K. (1964). Zagadnienie empiryzmu a koncepcja znaczenia [The Problem of Empiricism and the Conception of Meaning]. Studia Filozoficzne 1, No. 36, 3-14. Ajdukiewicz, K. (1965). Logika pragmatyczna [Pragmatic Logic]. Warszawa: PWN. Ajdukiewicz, K. (1985). JĊzyk i poznanie [Language and Knowledge]. 2 vols. Warszawa: PWN. Davidson, D. (1970). Semantics for Natural Language. In: Linguaggi nella societa e nella technica, pp. 177-188. Milano: Edizioni di Comunità. Kripke, S.A. (1972). Naming and Necessity. In: G. Harman and D. Davidson (eds.), Semantics of Natural Language, pp. 23-355, 763-769. Dordrecht: Reidel. Pelc, J. ([1981] 1984). WstĊp do semiotyki [Introduction to Semiotics]. Warszawa: PWN. Putnam, H. (1988). Representation and Reality. Cambridge, MA: The MIT Press. Tarski, A. (1933). PojĊcie prawdy w jĊzykach nauk dedukcyjnych [The Notion of Truth in the Languages of the Deductive Sciences]. Warszawa: TNW. Tarski, A. (1935). Das Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica 1, 261-405. WoleĔski, J. (1985). Filozoficzna Szkoáa Lwowsko-Warszawska [The Philosophical LvovWarsaw School]. Warszawa: PWN.

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Anna Jedynak ON LINGUISTIC RELATIVISM

There are some misunderstandings and misinterpretations concerning linguistic relativism (abbreviated here as LR), a view originally put forward during the twenties by Sapir and Whorf in their research on various ethnic languages. LR states that some languages are radically different and, consequently, so are worldviews commonly shared by the users of these languages, and that these visions cannot be correlated. Hence, the following questions arise: Which of these visions  if any  is the “true” one? Do the different cognitive viewpoints corresponding to different languages mutually exclude one another and is a universal cognitive perspective impossible? Or, does LR find some languages untranslatable only because they are too simple and only suitable for speaking about disparate domains of life? If so, would it be possible to make them translatable by enriching them, thus approximating them to the universal object language? The aim of this paper is to consider these problems.

1. What does Linguistic Relativism Claim The very word ‘relativism’ can be used in a stronger or weaker sense. Let us consider moral relativism as an example: (MR1 ) In the weakest sense, relativism sometimes means that people coming from different societies adhere to different moral norms, which shows the natural and cultural conditioning of these norms. This descriptive statement is empirically verifiable (though evidence is stronger for its first part than for the second one). (MR 2 ) In a stronger sense, one may add that moral norms adhered to by people from different societies are sometimes mutually inconsistent. This analytical statement is linguistically verifiable.

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 325-344. Amsterdam/New York, NY: Rodopi, 2006.

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Thus both statements (MR1 ) and (MR2 ) are decidable. None of them expresses any controversial philosophical view. One can question if they really represent relativism as they only contain a pure description of human beliefs, their conditioning and mutual analytical relations. (MR3 ) In a still stronger sense, one may add that people in different societies  adhering to different moral norms  are simultaneously right. It is this statement that represents relativism in its fullest sense and sometimes gives rise to strong objections. However, it is only controversial when thesis (MR1 ) is strengthened to thesis (MR2 ). Then, it says that people adhering to mutually inconsistent norms are simultaneously right and, consequently, that these norms are equally good. Thesis (MR3 ) is neither empirically nor analytically decidable, for it evaluates people’s moral beliefs. In the same way one can distinguish three versions of cognitive relativism, of which only the third one fully deserves to be called relativism. And, analogously, the following three versions of LR can be distinguished: (LR1 ) Some languages spoken by various ethnic groups differ considerably, not only in terms of the sound of words, but, most importantly, their meaning and the worldviews commonly accepted in those societies differ as well. This is because language influences the way people perceive and interpret the world. (LR2 ) Moreover, it is impossible to harmonize or correlate the worldviews in question. Acceptance of one of them excludes the possibility of accepting another one. (LR3 ) Among languages influencing various views, there are no better or worse ones  all of them are equally good. Contrary to other types of relativism, LR is usually presented in its weakest version. Why is it so? It is because weak versions of moral or cognitive relativism are well documented and only the strongest one is controversial. However, in the case of LR the controversy already appears in the weakest version, expressed in thesis (LR 1 ), which can be presented as a conjunction of the following two statements: (LR1a) Some languages spoken by various ethnic groups differ considerably, not only in terms of the sound of words, but, most importantly, their meaning; the views commonly accepted in those societies differ as well.

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(LR1b ) Language influences the way people perceive and interpret the world and that is why people speaking different languages have different views on world and life. Thesis (LR1a) is an empirical one. It says that the fundamental difference between ethnic languages goes together with the difference in the worldviews of the language users. Speaking about differences between languages, Sapir meant differences in sets of concepts, while Whorf rather meant differences in grammars. In the latter case, grammar was presented as including various ways of fragmentation of direct empirical data, enabling the distinction of particular objects which were to be named. Such a grammar is thus not limited to syntax, but also includes the differentiation of sets of concepts. This conceptual differentiation of languages results from different conditions of life in various societies. People tend to make exacting and subtle conceptual distinctions in those domains of life which are especially important for them in terms of fulfilling their needs, especially the utilitarian ones. Thus, Native American tribes for whom potatoes are the main source of nutrition have two hundred names for potato varieties, and one of the Siberian ethnic groups has a similarly elaborate vocabulary covering reindeer fur colours. However, the differences between languages go deeper and concern e.g. the segmentation of impressions about the flow of time or isolating essential ontological elements of the world from the stream of sensations. Thus, the Hopi Indians do not categorize time into past, present and future, but into actual and potential, and perceive reality not as static and composed of things, but as dynamic and composed of events. Whorf ’s ethno-linguistic research, which aims at proving Sapir’s views empirically, primarily confirms thesis (LR1a). To some extent it also confirms thesis (LR 1b ), but in a much weaker sense. It was not sufficiently tested and was neither sufficiently confirmed nor refuted (except for a partial confirmation resulting from a 1954 experiment which proved that perception and recollection of various colour tones depend on which tones can be named in the languages used by tested people). As is the case with other unresolved problems, already the weakest LR thesis gives rise to philosophical considerations. An analogous component of thesis (LR1 ) in the case of moral or cognitive relativism is less controversial: for according to those views, people’s beliefs are influenced not by language alone, but also by natural, social and cultural conditioning. Thesis (LR 2 ) seems to be quite enigmatic, contrary to analogous theses of moral or cognitive relativism. In those cases one could compare

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and find as inconsistent views adhered to in different societies, for those views could be expressed in one and the same language; actually, the differences between them only concerned the meaning and not the linguistic form of the expressions. On the other hand, in case of LR the differences concern the very language and its set of concepts  how then can one compare the views in question? Careful consideration of thesis (LR2 ) is especially important, for thesis (LR3 ) is only controversial if thesis (LR2 ) is accepted. Languages that do not lead to contradictory worldviews can turn out to be equally good; but how could languages that do result in opposite views be equally good? According to the principle of contradiction, one of such views must be false, which makes the value of its underlying language problematic.

2. Relativism in the Philosophy of Science The problem mentioned above concerns not only the LR of Sapir and Whorf, but also some similar epistemological views in the philosophy of science which, though they have not been corroborated by empirical research, have nevertheless been much more precisely expressed. They can be helpful in the proper evaluation of thesis (LR 2 ). The idea of languages based on different concepts appeared not only in LR, which was proposed in the 1920s and developed in the 1940s, but also in the philosophy of science. Radical conventionalism, formulated by K. Ajdukiewicz in the 1930s, is based on the notion of closed and connected languages. The sociological interpretation of the development of science, formulated by T. Kuhn in the 1960s, involves the notion of incommensurable languages corresponding to different paradigms in science. Both these conceptions and LR develop similar philosophical intuitions; we can acquire knowledge using different languages and consequently interpret the world in different ways. And all of these theories were sometimes misunderstood from the perspective of thesis (LR2 ), i.e. as follows: views formed in different conceptual frameworks are mutually inconsistent for they present the same reality in different ways. One cannot accept all such frameworks or paradigms simultaneously. Thus it is impossible to correlate alternative conceptual frameworks for they lead to contradictory consequences. Each conceptual framework represents a specific conceptual viewpoint in terms of which

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one can acquire knowledge about some aspects of reality. However, one cannot maintain both such viewpoints simultaneously. Yet such an interpretation is based on a misunderstanding. If two languages use completely different sets of concepts, sentences derived from them cannot contradict each other. Actually, these sentences can in no way be logically related. Thus, Kuhn and his followers call such languages “incommensurable.” Indeed, users of such different languages cannot correlate their views. However, we must distinguish the impossibility of such correlation, caused by the lack of common concepts, from the mutual inconsistency of such views, where each of them leads to the negation of the other. It is quite easy to overlook this difference and replace the former by the latter. Ajdukiewicz’s papers show this misunderstanding quite clearly. He often had to refute the objection that, according to his views, truth depends on the choice of language. He used to repeat that it is not the case, that there are statements which are true in some languages and false in others, since completely different languages cannot contain the same statements. Inconsistency of statements means much more than their dissimilarity: using different and disconnected concepts, we can express statements that are different (since they refer to different matters) but not ones that are inconsistent. Only sentences belonging to languages based on the same concepts can be inconsistent. However, neither LR nor relativistic philosophy of science deals with such languages.

3. Conceptual Relativity in Logical Calculi However, the belief that the same statements are true in some languages and false in others still lingers in philosophical reflection on cognition. Different logical calculi are mentioned as an example of languages where the same statement is claimed to be either true or false depending on the language. (Whorf himself mentioned various logics as an example of languages which represent different ways of thinking.) For example, the law of excluded middle (p › ~p) is said to hold in classical logic (further: CL) and not to hold in the three-valued logic formulated by Jan àukasiewicz (further: àL) in the second decade of the 20th century as the first non-classical logic. This suggests that the truth-value of a statement is determined by the choice of a language. Let us investigate this more closely using the example of the formula p › ~p in CL and àL. While CL is based on two truth-values: truth (denoted by 1) and falsity (denoted by 0), àL is based on three truth-values: truth, falsity,

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and possibility (denoted by ½). A law of logic is a statement form that is true for any truth-values of the variables in it. Since in CL there are two truth-values that can be substituted for variables, while in àL there are three of them, a statement form must fulfil stronger conditions to be a law of àL than to be a law of CL. There are statement forms that fulfil such stronger conditions, e.g. the principle of identity: p o p, and these are laws of CL as well laws of àL. There are also statement forms that fulfil only the weaker conditions, e.g. the law of contradiction or the law of excluded middle. They are true when truth or falsity are substituted for the variables, but not when possibility is substituted, and that is why they are said to hold in CL, but not in àL. The set of CL laws seems to include the set of àL laws. If so, some questions arise: How to explain the differences in status of the same formula in the two calculi? Which set of laws should we accept and use? How can we know which logic is the proper one? (àukasiewicz himself claimed that the choice of logic should depend on the nature of reality; in the post-war period he changed his mind and assumed that the choice of logic is based on convention.) Indeed, can we rely on any logic at all, if various calculi contradict one another at certain points? Finally, why should we replace CL by àL, if we “lose” some useful laws as a result? However, the claim that the set of CL laws includes the set of àL laws is true only if the respective statement forms in both logical calculi are deemed identical on the basis of their logical form and not their meaning. Indeed, the statement form representing the law of excluded middle in CL can be written in the same way in àL, i.e. as p › ~p, but it is not a law in àL. In each of the two calculi, it means something else. The meaning of a statement form depends on the meaning of logical constants, and the meaning of logical constants differs in the two logics. The meaning of a logical constant is defined by its truth table, which shows how the truthvalue of a compound sentence that contains the constant in question, depends on the truth-values of its arguments. The following truth tables define, in CL, the meaning of those functors that are involved in the law of excluded middle and the principle of identity (i.e. the functor of negation, alternation and implication): p 1 0

~p 0 1

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p 1 1 0 0

q 1 0 1 0

p›q 1 1 1 0

poq 1 0 1 1

The formulas p › ~p and p o p are CL laws, which can be demonstrated as follows: › 1 1

p 1 0

o 1 1

p 1 0

p 0 1

~ 1 0 p 1 0

The following truth tables define analogous functors in àL:

p 1 1 0 0 ½ ½ ½ 1 0

p 1 0 ½

~p 0 1 ½

q 1 0 1 0 1 ½ 0 ½ ½

p›q 1 1 1 0 1 ½ ½ 1 ½

poq 1 0 1 1 1 1 ½ ½ 1

We can see that the truth tables in àL are bigger than the ones in CL, since àL must define the truth-value of a sentence for an additional argument value: possibility. The differences between the matrices in CL and in àL determine the differences in the meaning of logical constants in both calculi. In àL, for example, some sentences (namely, the possible ones) are equivalent to their negations. This is clearly inconsistent with both the classical and the common meaning of negation. Thus, formally

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identical statement forms mean something else in those logics, hence their different status. There are statement forms in àL which represent true sentences, regardless of whether their arguments are true, false or possible (e.g. p o p), and there are other statement forms which represent sentences that are true only for true or false arguments but not for possible ones (e.g. p › ~p). o 1 1 1

p 1 0 ½ p 1 0 ½

› 1 1 ½

p 1 0 ½ ~ 1 0 ½

p 0 1 ½

However, the claim that the law of excluded middle does not hold in àL is not complete. It does not specify what is meant by ‘the law of excluded middle’, whether the phrase refers to the syntactical shape or to the meaning of p › ~p. This claim is only true when what is means is the logical form, not the meaning. If the equivalence of the statement forms in the two logics is to be determined not by their logical form but by their meaning, then the respective sets of laws turn out to be disjoint. Statement forms of various calculi that have the same logical form, differ in meaning because the logical constants involved have different meanings in each of the calculi. Thus, the calculi represent untranslatable languages. We cannot evaluate one of them using the language or meta-language of another one and, specifically, we cannot revoke its laws. (We can only try to interpret one logic using the terms of another one but that is quite a different issue.) The calculi do not contradict each other. They differ in conceptual tools and, consequently, in the very topics they aim to discuss, but not in the opinions about the same topics. Thus, the decision to qualify a statement form as a law of logic is not arbitrary. What is arbitrary is the choice of the logical calculus, i.e. the choice of the language of a logic. But having established the language, we cannot freely decide which statement form is, and which is not, a law of logic for this depends on the meaning of logical constants.

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4. The Possibility of Compiling Different Languages As was evident in the example of logical calculi, languages based on different concepts cannot assign different truth-values to the same statements. Examples given by Whorf can be used to reach the same conclusion. He analysed and compared ethnic languages based on different sets of concepts. If a statement can be expressed in one language L1 and cannot be expressed in another language L2 because L2 lacks suitable concepts, then it is not possible to deny it in L2 . Thus, thesis (LR2 ) proves to be false: worldviews resulting from different languages are different but not mutually inconsistent. Consequently, thesis (LR3 ) loses its controversial character: the problem of equally good languages leading to mutually inconsistent worldviews just disappears. However, we can still evaluate languages as equally good, as Whorf does, if each of them serves equally well as a means of communication, thinking and fulfilling practical needs, and if they are comparable in their degree of complexity, subtlety and precision. Since thesis (LR2 ) is false, we can ask whether it would be possible to create a language based on the sum of the conceptual tools from the different languages that are actually used. Sometimes, these languages are not completely translatable but maybe their conceptual tools could be combined, approximating the universal object language? Conceptual distinctions specific to particular languages should be preserved in the wider language. Could we not divide time both into past, present and future (as European nations do) and into actual and potential (as the Hopis do), and then combine these divisions? It would mean that we would not have to choose between languages but could simultaneously take advantage of the various possibilities they offer. Of course, such a compilation of languages can only be considered on the object level, for if the universal language were to contain meta-levels it would generate antinomies. Again, the views of Ajdukiewicz can be helpful. Some years after having formulated radical conventionalism he gave it up, for he stopped using the concepts necessary to express it, namely the concept of closed and connected languages (a closed language is a language that cannot be enriched by any new meanings connected with the old ones; a connected language is a language that does not contain any isolated parts, i.e. every two of its expressions are mutually meaning-related either directly or indirectly). He found that such languages do not exist. However, this self-critical revision was followed neither by a re-evaluation of the sociological interpretation of scientific development nor by a re-

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evaluation of the validity of LR strengthened by thesis (LR2 ). Nevertheless, relativism in the philosophy of science seems to be losing popularity and the search for new solutions tends towards a compromise between relativism and realism. Thus, we face the problem of compiling languages that are considered untranslatable (at least in some parts). Since every language can be enriched, it can be specifically enriched with concepts taken from other languages. Ajdukiewicz did accept such a possibility in his radical conventionalism period, but he considered such a compilation a disconnected language, i.e. an artificial conglomerate of languages that have no mutual conceptual connections and function separately. He did not find such a compilation to be useful at all. The user of such a disconnected language would have to divide his attention between various isolated parts of it and would not be able to make simultaneous use of them. But since every language can be enriched by new concepts connected with the old ones (a view later accepted by Ajdukiewicz himself), we can assume that the compilation of languages should lead to a more fruitful cognitive perspective and previously unavailable cognitive possibilities, which validate in particular those statements that could not be validated on the grounds of any of the individual languages. Whorf did not deal with the problem of compilation of languages based on different concepts, perhaps because he was interested in the actual usage of languages and not in their inherent possibilities. Yet, we can presume that he would be rather critical of it. He admitted that the study of the languages and the viewpoints of different cultures extended our cognitive perspective. He hoped that a language suitable for acquiring knowledge about the most essential aspects of reality would come into use in the future. However, he did not suggest a compilation of various existing languages. In any case, an imaginary inconsistency of viewpoints resulting from the usage of different languages does not preclude the possibility of their compilation. It would not be the results of using different languages that could be an obstacle here, but rather their conditioning. Whorf mentioned the natural, social and cultural conditioning of languages, but he did not develop the subject. He was convinced that we would never actually be able to reach the sources of the languages. Probably, the conditioning of languages would preclude their compilation only if: (1)

the conditioning of languages that are to be compiled could not occur conjointly,

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and (2)

the respective ways of conditioning were necessary for the origination of these languages.

5. Compiling the Languages of Logical Calculi Let us go back to the example of different logical calculi to check the problem of compiling their languages. It is commonly assumed (as Whorf assumed referring to different ethnic languages) that we can only stand on the ground of either one or the other logic, but not on the ground of both of them simultaneously. Indeed, àL and CL define the same logical constants in different ways. However, we can assume that they provide different explications of the vague expressions of the natural language. And different explications of the same concepts, admitted for different purposes, can still coexist in one and the same language. Of course, there is a problem with logical constants which have the same form but different meanings. But we can avoid it applying different symbols, e.g. we can use the traditional notation to denote constants used in CL, and the Polish notation (àukasiewicz’s parenthesis-free notation) to denote constants used in àL. But still, could logical constants with such different meanings belong to one connected language? Their meaning would be defined heterogeneously. The functors from CL would be defined using two truth-values, and those from àL  using three values. This heterogeneity is not accidental. It results from different assumptions on which the calculi are based. CL: there are exactly two truth-values: truth and falsity. àL: there are exactly three truth-values: truth, falsity and possibility. These assumptions seem to be mutually inconsistent as well as necessary to define the functors in both logics. Thus we might readily suspect that these logics are radically different and their connected compilation is impossible. However, such a conclusion would be too hasty for, as we will see, the assumptions are based on different concepts of truth and falsity – on different concepts of truth-value, in general. According to the classical conception, truth is the correspondence of a sentence or thought to reality. This definition is not complete and it leaves some details open. One of them concerns the scope of the concept of reality which the sentence should correspond to: should this scope also

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include expressions of the language on which the sentence is based on or not? The tacit positive answer of this question gave rise to antinomies and Tarski’s negative answer made it possible to eliminate them. Another unresolved point, widely discussed nowadays, refers to the explication of the concept of the correspondence of sentences to reality. But there is still another important unresolved point in the classical conception of truth. It is connected with the philosophical motivation that brought àukasiewicz to non-classical logic. He was a mild determinist: he assumed that every event had its cause but in contrast to strict determinists he denied the idea that such a chain of causes reaches into an unlimited past. The chain can only form an unlimited sequence convergent on a certain moment in time. Before that moment, the causes of a given event had not yet started to act. àukasiewicz assumed that some future events were determined at the present moment because their causes had already started to act (e.g. that, in a year from now, the position of the Earth with respect to the Sun will be the same as it is now). Some other events are excluded at this very moment if the causes that exclude their occurrence have started to act. And the third group of events includes those whose occurrence or whose impossibility is not yet determined. They constitute an indeterminate future. And it is those sentences about the indeterminate future that àukasiewicz found neither true nor false, but possible. Let us return to the unresolved point in the classical conception of truth. A true sentence is one that corresponds to reality. However, since reality is changeable in time, which moment in time should be taken into account in the evaluation of the correspondence? If the evaluation of a sentence about a certain event is made at a different time than the occurrence of the event itself, which of the two moments should be taken as the point of reference for reality? Traditionally, it was tacitly assumed that it was the moment at which the event, described by the sentence, occurred. As a result, two truth-values were adopted. At the moment at which the event described by the sentence occurs, reality can either be such as the sentence describes it, or not. However, àukasiewicz tacitly took the moment of the logical evaluation of the sentence as the point of reference. If an event described by the sentence is determined or excluded now, at the moment of evaluation, the sentence is either true or false (this is the case with sentences concerning the past, the present and the determinate future). If the event is now neither determined nor excluded, the sentence is neither true nor false, but can only be deemed to be possible (this is the case with the indeterminate future). In this way, the third truth-value is brought into use. Sentences about the

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indeterminate future are neither in correspondence nor not in correspondence with present reality. In the course of time, they change their truth-value: what is now possible, will later become true or false. The assumption of two or three truth-values is based on different explications of the concepts of truth, falsity and truth-value. Assumptions underlying the two logics are not inconsistent, for they assign different meanings to expressions that have the same form. Hence, the impression that they are inconsistent is based on an equivocation. They can be, and actually are, conjointly true. If by ‘truth-value’ we mean the correspondence of a sentence to reality at the point at which the event described by the sentence occurs, then we can evaluate sentences dichotomously. And if by ‘truth-value’ we mean the correspondence of a sentence to reality at the point at which the sentence is evaluated, then we can evaluate sentences trichotomously. CL and àL are neither assumed completely arbitrarily nor are they adequate or inadequate to reality. Actually, they result from the above explications. It is not the case that àukasiewicz added a new truth-value to the two familiar ones since his concepts of truth and falsity are not the same as the respective concepts on which CL is founded. The picture below presents the scope relations between truth-values taken from both calculi. Although the concepts of truth and falsity, when limited to determined sentences, are equivalent, they are still not synonymous. CL: àL:

truth past truth

present Ļ determined sentences

future possibility Ļ undetermined sentences

present

falsity past falsity

Ļ determined sentences

The distinction based on CL qualifies every sentence as true or false, regardless of whether it describes the past, the present or the future. The division based on àL qualifies as possible some of the sentences about the future (namely those sentences that are undetermined at the moment of evaluation, among them those that will become true as well as those that will become false). Although among the determined sentences, the concepts of truth, falsity and truth-value are equivalent, they are still not synonymous. It is these new concepts of truth and falsity, not the old concepts, that need to be supplemented by the third truth-value. àukasiewicz’s understanding of truth and falsity is close to modal concepts. In the context of his views on causality, his conception of truth refers not so

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much to what happened, is happening or will happen, but is stronger and refers to what is now determined and thus has to be. His concept of falsity refers not so much to what did not happen, is not happening or will not happen, but is stronger and refers to what cannot happen because the opposite state is determined. For àukasiewicz, truth is close to necessity and falsity to impossibility, where both modal concepts are interpreted physically. Facing the incompleteness of the classical conception of truth, àukasiewicz presented an explication of truth and falsity different from the commonly accepted one. However, neither was the common custom explicitly expressed nor did àukasiewicz explicitly express his own view. His novel approach seemed to contradict common views only at the stage of the consequences of both approaches. Thus, the assumptions of CL and àL are not mutually inconsistent and there is no obstacle to the compilation of their languages. The second factor that could preclude the compilation (namely, that the given assumptions are necessary to found the language) seems to lose its importance since only the two factors taken together could preclude it.

6. The Usefulness of the Compilation of Different Languages It is quite another issue whether having these two explications coexisting in one and the same language is useful, and, more generally, whether it is useful to create one conceptual apparatus from two different ones that correspond to cognitive viewpoints regarded as inconsistent. A. Grobler (2000) claims, with reference to cognitive relativism, that what is actually used are not isolated conceptual systems but the whole network of them including their mutual connections and interdependencies. On his view, changes in one system lead to changes in the others so that the choice of a system depends on the context. Conceptual systems do not have a global range but only a local one, established by particular, usually short-term, cognitive needs. A system changes according to the changing needs (see Grobler 2000, Ch. 3). M. Tempczyk’s (1998) view is similar. He claims that two different paradigms in science, correlated to languages regarded by relativists as untranslatable, may be conceptually and genetically similar and complementary to one another. It is possible to compare them and to choose rationally between them or to decide to use them both (see Tempczyk 1998, Ch. 13). Simultaneous use of conceptual tools from different languages is in accordance with the spirit of Grobler’s and Tempczyk’s considerations.

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The language containing logical constants from both CL and àL would represent pluralistic logic (further: PL), i.e. a logic which defines functors heterogeneously. Since truth and falsity in àL are different from truth and falsity in CL, PL would be based on two types of truth-values and five ways of evaluating sentences with respect to their adequacy to reality. Some functors would be defined using two truth-values, and some using the other three. Yet the language of PL would not be disconnected, i.e. composed of isolated parts (we will deal with this issue in more detail later). Actually its parts would have mutual conceptual interconnections, just as in the network of conceptual systems considered by Grobler. It seems that everyday conversations are governed by PL – does natural language not aspire to universality? Some questions anticipate one of the two answers: “yes” or “no,” and these are parts of the language of CL. Others are parts of the language of àL for they anticipate one of the three answers: “yes,” “no,” or the third one which is either “maybe,” if applied to indeterminate future, or “to some extent,” if applied to vagueness. If a third answer becomes acceptable, “yes” and “no” become stronger answers than they would be otherwise. If people were divided into tall, medium and short and if the question whether John was tall was answered in the affirmative, the answer would bring more information than in the case of people being divided only into tall and short. And if John is tall according to the trichotomous division, he is also tall according to the dichotomous division, though the reverse does not hold. Such relations can be expressed. It is not the case that having assumed one of the two divisions, we cannot refer to the other one. Analogously, it is not the case that having assumed CL or àL we cannot refer to the other. Every formula that is true according to the àL meanings of the logical constants is also true according to the CL meanings of the logical constants, but not conversely (and the same holds for false formulas). This probably expresses adequately the intuition which is sometimes not quite adequately expressed as follows: every law of àL is a law of CL, but not conversely. In everyday conversations, we unconsciously change the meaning of the functors we use and the number of truth-values we accept. PL could throw some light on the matter. We should distinguish domains of applicability of the different logical concepts (not only those from CL and àL), define ways of recognizing the changes in meaning of functors in the course of conversation, and analyse the relations between the laws of different logics. Then we would be able to recognize whether the conversation is or is not interconnected.

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Thus actually in common ways of reasoning and conversations we conjointly use concepts that correspond to different logical calculi. Chronologically speaking, the logical pluralism of everyday speech, which arose as a result of the plurality of cognitive situations, was prior. It was followed by the various precise explications of commonly used connectives. It is from those explications that various calculi originated. So it seems obvious that one should not question the usefulness of compiling languages of different calculi. Rather, there is a need to examine formal aspects of the functioning of such a compilation. But not all languages regarded as based on different conceptual tools originated from a situation of a connected coexistence. Ethnic languages from different cultures did not, for example. In these cases, it is quite legitimate to question the usefulness of their compilation. Whorf encouraged people to learn foreign languages and to acquire knowledge about foreign cultures, viewpoints and manners of reasoning, i.e. to become more broad-minded. However, a person following his advice would still be just a polyglot using two or more disconnected languages and not a connected enrichment of any language. Still, it is difficult to imagine someone who uses various languages in such a way that they are completely isolated, with no mutual connections or influences between them. Such connections arise at least because all these languages serve to describe the same or similar facts. (In a similar way, M. Tempczyk feels connections between paradigms in science exist because they explain the same phenomena and the same people are able to know and use more than one of them.) Indeed, compiling conceptual tools of different languages enables us to enlarge the cognitive perspective. The key point is to recognize connections between concepts from different languages. On the object level, the compilation enables us to acquire and express new knowledge that would not be available while using any of those languages taken in isolation. On the meta-level, it provides simultaneous access to conceptual tools of different languages, so that we can compare and evaluate their usefulness in various situations and make rational choices between them.

7. Beliefs and Attitudes In spite of the language-incommensurability thesis formulated in the philosophy of science, we can compare our language with those of other languages. Whorf himself did so, otherwise he would not have formulated the LR thesis. Simultaneous accessibility to the conceptual

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tools of various languages seems possible and useful, if considered only in respect of people’s beliefs. There are no logical obstacles to compiling languages, for the use of different languages does not entail inconsistent beliefs. There are, however, pragmatic obstacles. Even if we do not need to change our beliefs when enriching our conceptual apparatus, sometimes we do need to change our values or attitudes, or at least we need to weaken them. It seems that different languages are based not on mutually inconsistent beliefs but, actually, on mutually inconsistent attitudes toward reality. (Emotivism has developed a similar thought in metaethics: inconsistent moral evaluations are due not to inconsistent beliefs, including beliefs about attitudes, but due to inconsistent attitudes themselves.) A conceptual apparatus usually reflects the hierarchy of values or at least the hierarchy of significance assigned to various matters that is accepted by its users. What is important needs to be named. The most subtle conceptual distinctions are made in the areas most important for a language group. The enrichment of a conceptual apparatus so that distinctions in other areas of reality become equally subtle, spreads thin the hierarchy of significance which is accepted by the group and which lies at the foundations of the original unenriched language. Attitudes towards reality are conditioned by the cognitive context. It is not the case that if the components of two such contexts occur conjointly then they generate attitudes specific for both these contexts. The whole is not just a sum of its parts: adding to a given cognitive context some components specific to another context can considerably change the function of the elements of the former and, in particular, weaken or eliminate its influence on people’s attitudes. This concerns not only the frequently described conceptual distinctions made by aboriginal communities living in various conditions. It also concerns the distinctions made by civilized societies in various abstract domains. For example, the choice of a specific explication of the classical conception of truth shows what manner of evaluating sentences seems essential to the chooser. Enriching the conceptual apparatus by accepting another competitive explication (simultaneously enlarging the vocabulary to make the terms univocal) interferes to some degree with the originally established value hierarchy. There is then not just one way of evaluating sentences but also another slightly different way. Adding the latter obviously weakens the function of the former. This is probably the reason why people engage in polemics on various explications of important philosophical notions. Quite often the parties cannot agree on a

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harmonious coexistence of two or more explications because each party believes that accepting a competitive explication will weaken the importance of the “correct” one and diminish the significance of what the “correctly” explicated word designates. While examining the possibility of compiling the languages of two logical calculi, we have been investigating whether certain verbal assumptions underlying these calculi are inconsistent or not. However, we cannot mechanically apply this to the problems considered by Whorf because ethnic languages are not based on verbal assumptions. Before the origin of natural language, no assumptions could be expressed verbally. The matter changes if we take into account preverbal assumptions: people’s attitudes, values, needs, aims and aspirations, their sense that various matters are significant and their consequent tendency to differentiate the particular objects that are worthy to be named. Suppose that we enrich our conceptual apparatus with another one that originates from an attitude different than ours. This means doing something artificial since a language is a response to a specific attitude that results from our cognitive needs. If we do not share the attitude, we do not really need a language geared to that attitude. Moreover, if the attitude is inconsistent with ours then a serious compilation of languages can lead to significant confusion in the usage of our own language as it weakens the attitude from which our language arose. It is true that the new language, even if originating from a foreign attitude and hence “unnecessary,” makes us broader-minded, helps in understanding the mentality of its users and enables us to compare their beliefs with ours. But it does not enable us to be “in their shoes” and to perceive the world in exactly the same way, so in a psychological sense it will always remain a foreign language. We feel best “at home” in our own language, among concepts that arise from our own values and cognitive attitudes. The price we have to pay for enriching our language is that we distance ourselves from the attitude the language originated from. The enriched language seems to be devoid of a subject, in the sense of not originating from anyone’s actual attitude. However, maybe such an enriched language, if actually used, gives rise to new attitudes later. In such a way we assimilate new languages and cultures, elements of which find their way into our own language and culture. And so we create our global village. It thus seems useful to develop studies on linguistic pluralism (including PL), which involves the coexistence of cognitive tools of various languages originally derived from inconsistent attitudes toward the world.

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Indeed, maybe it is the dependence of various languages on radically different attitudes towards reality that led to the formulation of LR in the form of thesis (LR2 ), a form which was false, as explained above. This thesis, however, can be reformulated to make it valid. The changes should go in two directions: (1)

(2)

It cannot refer to worldviews (i.e. general, verbalized beliefs about the world), but to people’s attitudes toward the world: to needs, wishes, aims, focusing on certain areas etc. While logical inconsistency of beliefs is a clear-cut notion and definitely precludes their simultaneously true, psychological inconsistency of attitudes is a vague and, it seems, weaker one. The latter does not assume logical inconsistency; attitudes giving rise to different languages do not mutually preclude one another but adopting one of them is not conducive to adopting other ones.

Thus instead of thesis (LR2 ), which was too strong and dealt with beliefs, let us formulate thesis (LR2 ƍ), which is weaker and deals with attitudes, and which seems to be true: (LR2 ƍ) The formulation of a language involves a certain hierarchy of importance of matters, a certain cognitive attitude towards reality and such attitudes are sometimes inconsistent in the sense of their being quite difficult to adopt conjointly in a serious manner. One can similarly modify thesis (LR3 ) which also deals with attitudes: (LR3 ƍ) Languages originating from different attitudes serve the users adopting these attitudes equally well.

Uniwersytet Warszawski Department of Philosophy ul. Krakowskie PrzedmieĞcie 3 00-470 Warszawa, Poland e-mail: [email protected]

REFERENCES Ajdukiewicz, K. (1978). “The Scientific World-Perspective” and Other Essays. 1931-1963. Edited by J. Giedymin. Dordrecht: Reidel. Grobler, A. (2000). Prawda i wzglĊdnoĞü. Kraków: Wydawnictwo Aureus. Kuhn, T. (1996). The Structure of Scientific Revolutions. Chicago & London: The University of Chicago Press.

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àukasiewicz, J. (1919-1920). On Three-Valued Logic. In: àukasiewicz (1970), pp. 87-88. àukasiewicz, J. (1922-1923). On Determinism. In: àukasiewicz (1970), pp. 110-128. àukasiewicz, J. (1930). Philosophical Remarks on Many-Valued Systems of Propositional Logic. In: àukasiewicz (1970), pp. 153-178. àukasiewicz, J. (1970). Selected Works. Edited by L. Borkowski. Amsterdam & Warszawa: North-Holland Publishing Company & Polish Scientific Publishers. Stevenson, C.L. (1944). Ethics and Language. New Haven: Yale University Press. Tempczyk, M. (1998). Teoria chaosu a filozofia. Warszawa: Wydawnictwo CiS.

Dale Jacquette TARSKI’S ANALYSIS OF LOGICAL CONSEQUENCE AND ETCHEMENDY’S CRITICISM OF TARSKI’S MODAL FALLACY

1. Tarskian Consequence In his philosophical critique of Tarskian model set truth and validity semantics, The Concept of Logical Consequence, John Etchemendy attributes a modal fallacy to Tarski’s argument that logical consequences in the technical sense of his formal criterion are also consequences in the ordinary sense.1 Tarskian consequences in Etchemendy’s exposition are conclusions in ordered sentence sets that are satisfaction-preserving on all sequences. Sequences are functions that assign objects from appropriate satisfaction domains to variable terms in an argument’s assumptions and conclusions. An argument is Tarski-valid if and only if all such substitutions preserve the truth of the conclusion or falsify at least one assumption. If ‹K, S› is an argument in which K is the (set of) assumption(s), and S the conclusion, then S is a Tarskian consequence of class K if and only if ‹K*, S*› is satisfaction-preserving on all limited sequences f *, where: (i)

K* and S* are produced from K and S respectively by uniformly replacing all variable terms in K and S (those not exempted from substitution by inclusion in a designated fixed term set F ) by corresponding object, predicate, connective, and operator variables

1

It is customary to speak loosely of model set theoretical semantics as ‘Tarskian’. But Etchemendy (1988), pp. 51-68, 72, argues persuasively against the historical accuracy of referring to such theories as Tarski’s or directly related to Tarski’s theory of models. Etchemendy (1990), pp. 51-55, explains how Tarski’s analysis must be supplemented to produce what is recognized today as a model set theoretical account of logical truth and consequence. See note 2 on next page.

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 345-368. Amsterdam/New York, NY: Rodopi, 2006.

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from within the same term-type categories of some language L, to yield from each sentence S i in ‹K, S› the sentential function, & && & & && && ‘ S i* (n /x, p/g, b /c , u /o ) ’ (I I1 , . . . , I k ) ; ‹ K*, S *› is satisfaction-preserving on all limited sequences f * if and only if every f * either satisfies S * or does not satisfy some member of K*; && & & && && f * satisfies sentential function ‘ S * (n /x, p/g, b /c , u /o ) ’ in ‹ K*, S *› if && && & & & & i and only if ‘ S i ( x/n , g/p, c /b , o /u ) ’ is true in language L.2

Here is an application. We begin with an argument whose Tarskian logical consequence relation is to be determined: ‹{Grass is green, If snow is white then grass is green}, Snow is white›. The premises and conclusion are true, but the inference pretheoretically is invalid. How does Tarski’s criterion establish this? Modulo some chosen set of fixed terms F , the (remaining) variable terms are uniformly replaced by variables from cross-term-restricted categories for object, predicate, connective, and operator terms. Let F = {‘If . . . then’}; then this connective term is fixed and not allowed to vary for purposes of testing the Tarskian sequence satisfaction-preservation of the argument. The first-stage substitution of variables for variable terms produces ‹{x1 g 1 , If x2 g 2 then x1 g 1 }, x2 g 2 ›. The test is complete when appropriate resubstitutions of variable terms for variables are made for all sequence functions assigning objects in respective categories to x, g , c, and o in the sentence function, and the resulting sentences checked for truth in language L. The method provides counterexamples to prove that the sentence function form ‹K*,S*› is not satisfaction-preserving on all limited

2

This somewhat simplifies Etchemendy’s account. Etchemendy introduces further complications concerning direct or d-sequences and cross-term restrictions on the set of models. The d-sequences are supposed to effect a more direct (‘though potentially more confusing’ Etchemendy 1990, p. 53) one-one correspondence between variable terms and variables, and sentences and sentential functions when testing sentences for Tarskian consequencehood. These elements of Etchemendy’s exposition are omitted here, since nothing important in the evaluation of his criticism of Tarski depends on them. Etchemendy 1990, pp. 53-54: “Let us say that a direct or d-sequence is any function that assigns to each variable term an object from the appropriate satisfaction domain [. . .]. For any d-sequence f, let f * be the corresponding limited sequence – that is, the function that assigns the same object to a chosen variable (for example, ‘x 1 ’, ‘g 1 ’) as f assigns to the corresponding variable term...We can now introduce a relation, parallel to satisfaction, which holds between d-sequences and sentences. Specifically, say that a d-sequence f d-satisfies sentence S if and only if the corresponding limited sequence f * satisfies the distinguished sentential function S*.” Etchemendy’s cross-term restrictions are explained on pp. 68-69.

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sequences f * modulo F , including the resubstitution instance ‹{Water is wet, If snow is red then water is wet}, Snow is red›. This alone is sufficient to prove that the conclusion of the argument modulo F is not a Tarski-consequence of the assumptions. Where ‹K, S › = ‹{Grass is green, If grass is green then snow is white}, Snow is white›, on the contrary, there is, modulo F as before, evidently no such counterexample.

2. Etchemendy’s Critique of Tarski’s Fallacy

Etchemendy is dissatisfied with Tarski’s definition of consequence, among other reasons, because it does not account for the logical necessity of valid deduction; or, more generally, for the specific modality linking assumptions and conclusions in valid inferences that characterizes the intuitive concept of logical consequence. This requirement is reflected in the standard definition by which an argument is said to be valid if and only if, if the assumptions are true, then the conclusion must be true; or, if and only if it is logically impossible for the assumptions to be true and the conclusion false. Tarski’s concept of consequence, according to Etchemendy, does not explain the fact that premises of deductively valid arguments guarantee the truth of their deductive consequences. 3 The problem as Etchemendy sees it is that Tarski’s analysis does not give us all that we have a right to expect from the intuitive concept of consequence. The desired equivalence between Tarskian and intuitive logical consequence states: S is a consequence (in the ordinary sense) of K if and only if S is a Tarskian consequence of K on some selection of F. (1990, p. 86)

Etchemendy believes that Tarski’s definition at most supports the unproblematic conditional that any intuitively valid argument ‹ K, S › is also Tarski-valid on some arbitrary choice of fixed terms F . If S is a consequence (in the ordinary sense) of K, then S is a Tarskian consequence of K on some selection of F. (p. 86)

For the equivalence to hold, the converse of the conditional, that if S is a Tarskian consequence of K on some selection of F , then it is also a 3

Etchemendy 1980, pp. 81-82. Etchemendy documents the intuitive concept of logical consequence with the standard definition of validity appearing in logic texts of three grades of difficulty. A useful technically accurate historical exposition of Tarski’s theory of logical consequence is given by Gomez-Torrente (1996).

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consequence in the ordinary sense, must also be true, which Etchemendy denies. Yet he admits that if the equivalence were demonstrated, Tarski’s analysis would be acceptable: Needless to say, if [the] equivalence [. . .] could somehow be shown, then Tarski’s definition of consequence could hardly be faulted. But in order to show that the equivalence holds, we must show that if S is a Tarskian consequence of K, then it is a consequence “ in the ordinary sense.” That is, we must show that if all the members of K are true, S must be true as well. (p. 86; emphasis added)

It is here that the modal linkage of assumptions and consequences is supposed to be lacking in Tarski’s analysis. And it is here that Etchemendy maintains Tarski regards the analysis as adequate only by dint of a modal fallacy. Etchemendy quotes Tarski’s reasoning in this passage: It seems to me that everyone who understands the content of [my] definition must admit that it agrees quite well with ordinary usage. This becomes still clearer from its various consequences. In particular, it can be proved, on the basis of this definition, that every consequence of true sentences must be true, and also that the consequence relation [. . .] is completely independent of the sense of the extralogical constants which occur in these sentences. (p. 91; Etchemendy’s translation; see p. 167, n. 8; Tarski 1956, p. 417)

This appears to provide the needed second half of the equivalence. Etchemendy, however, finds it unsatisfactory, and regards the equivalence as “obviously false” (p. 87). The missing component states that: If S is a Tarskian consequence of K on some selection of F, then S is a consequence (in the ordinary sense) of K.

There are certainly some selections of F for which S is a Tarskian consequence of K, but not a logical consequence in the ordinary sense. Etchemendy gives the counterexample ‘Lincoln had a beard’ as a Tarskian logical consequence of ‘Washington was president’ when all the component expressions are fixed in F , in effect freezing up cross-termrestricted substitution in the limiting case. He reports: “We would hardly say that, provided ‘Washington was president’ is true, ‘Lincoln had a beard’ must be true as well ” (p. 87). Etchemendy does not restrict himself to the limiting case where F is maximal, but indicates that counterexamples obtain for other sets of fixed terms (Etchemendy 1990, p. 88). Take any intuitively invalid inference ‹ K, S › where S happens to be true in L, where there are terms in K not in

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S , and where the sentences in K are not logically false, and fix all terms of S in F ; or, where some sentence in K happens to be false in L, where there are terms in S not in K, and where S is not logically true, fix all the terms of K in F . By the disjunctive form of Tarskian satisfactionpreservation, it follows in either case that S is a Tarskian consequence of K, but not an intuitive consequence in the ordinary sense. Here are two such counterexamples: ‹{Grass is green}, Snow is white›, where F = {‘Snow’, ‘is white’} ‹{Snow is red}, Snow is blue›, where F = {‘Snow’, ‘is red’} Etchemendy is right to observe that a Tarskian consequence modulo F does not carry over to the intuitive sense of logical consequence. Consequence in the ordinary sense is just consequence, modulo nothing . Etchemendy’s diagnosis of the difficulty in the equivalence between the Tarskian and the intuitive concept of logical consequence is that Tarski’s analysis misplaces the modality governing the logical connection between assumptions and conclusions in valid inferences, symptomatic of which is the fact that Tarski’s argument for the second conditional in the equivalence appears to contain a modal fallacy. It is perfectly clear that with many selections of F, there are Tarskian consequences that are not genuine consequences, and hence that [the equivalence] is simply false. Yet our proof that every Tarskian consequence of true sentences must be true is perfectly correct. The problem is not with our proof, but with thinking that this proof shows that any modal relation holds between the premises and conclusion of the argument ‹K, S›. (p. 87)

This is the missing link required to guarantee the truth of logical consequences from assumptions in deductively valid arguments. It is the modal relation implied by the ordinary intuitive concept of logical consequence. Etchemendy claims that Tarski’s argument needs but fails to establish this “embedded modality,” and that it gives the impression of doing so only because of an invalid modal inference. To show that all Tarskian consequences are consequences in the ordinary sense, we would need to prove a theorem with an embedded modality. Specifically, we would have to show that, for any K and S, if (1) S is a Tarskian consequence of K (for some F ) then the following are jointly incompatible: (2) All the members of K are true (3) S is false.

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But of course all we can show is that for any K and S, the [above] three conditions are jointly incompatible. (p. 87)

Tarski’s fallacy is then said to be the confusion of thinking that (2) and (3) are logically incompatible if (1) is true, whereas at most if (1) is true, (2) and (3) cannot both be true (but might both be false), in which case the desired modality linking S to K as in the ordinary concept of logical consequence is not reflected in Tarski’s consequence definition. Etchemendy remarks: Now, it should be clear from a purely abstract point of view that the joint incompatibility of (1), (2), and (3), plus the truth of (1), does not entail the joint incompatibility of (2) and (3). Here we need only note the fallaciousness of any inference from Necessarily (if P and Q then not R) to If P then necessarily (if Q then not R). (p. 87)

The translation to Etchemendy’s discussion of Tarski’s argument equates P with proposition (1), that S is a Tarskian consequence of K (for some F ); Q with proposition (2), that all members of K are true; R with (3), that S is false. Unquestionably, there is a modal fallacy in the inference. Etchemendy grants the soundness of Tarski’s demonstration that every consequence of true sentences must be true. He regards this as misplacing the modality needed to capture the intuitive sense of logical consequence. Etchemendy in effect accuses Tarski’s argument of embodying an error in the scope of its modal operator. Tarski’s proof places necessity outside the conditional needed to prove the equivalence, whereas it should occur inside the main conditional, governing the subordinate conditional relation holding between the truth of assumptions and conclusion in a deductively valid inference. Etchemendy continues: Obviously, the proof in question does not show that every Tarskian consequence is a consequence “in the ordinary sense.” It is only through an illicit shift in the position of the modality that we can imagine ourselves demonstrating of any Tarskian consequence that it is entailed by the corresponding set of sentences. (p. 88)

Etchemendy later conjectures about the meaning of the modal fallacy he has uncovered, and about Tarski’s intentions in formulating his argument for the definition of consequence in its relation to the ordinary or intuitive concept:

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Was Tarski guilty of the modal fallacy I have described? Did he really believe that his proof that “every consequence of true sentences must be true” assures us that the right sort of relation will hold between the premises and conclusion of arguments satisfying the definition? Or was he simply making a very weak claim for his definition  namely, that it will not designate as “logically valid” any argument with true premises and a false conclusion? This latter claim follows trivially from the definition, but hardly seems much evidence that it “agrees quite well with ordinary usage.” (p. 90)

He dismisses the possibility that Tarski did not intend an equivalence strong or interesting enough to engender the fallacy on the basis of internal textual evidence (pp. 90-92), and concludes that: “Set next to [Tarski’s] earlier remarks, it is hard not to see the fallacy at work in this justification” (p. 91). Deciding that the fallacy is not excusable as a misunderstanding of Tarski’s intentions, Etchemendy further speculates about the real source of the modal fallacy and misplaced modality in Tarski’s argument for the equivalence of ordinary or intuitive logical consequence and consequence according to the model set or satisfaction-preservation definition, and concludes that Tarski was satisfied to inject necessity anywhere at all into the account, given his vague grasp of the modality. Did Tarski really think that the right modality followed from his definition? Or did he see that it did not, but still try to convince his readers that it did? I think the most likely explanation is much less dramatic than either of these. Although Tarski recognized the importance of some intuitive modality to the relation of logical consequence, he also recognized that this modality is obscure and poorly understood. Given this fact, he may well have thought that the modality appearing in his justification, though perhaps not quite right, was close enough to count as capturing this essential but ill-understood feature of the consequence relation. Better a misplaced modality than no modality at all. (p. 92)

If the dilemma is to discredit Tarski’s intellectual honesty or modal competency, I too would prefer to regard him as failing to understand the subtleties of modal inference. Many of these modal relations have been clarified only since the time of Tarski’s formulation of his definition with the development of model set theoretical semantics for modal systems (directly inspired, it must be recalled, by Tarski’s own work).4

4

See Suppes (1988), p. 86: “Tarski’s study of the relation between modal logic and the algebra of topology with McKinsey (1944) marks an early anticipation of the application of the theory of models to modal logic. In any case, the characteristically philosophical

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Fortunately, there is an alternative, a way of interpreting Tarski’s definition of consequence so that it avoids Etchemendy’s counterexamples and the modal fallacy that seems to plague Tarski’s proof for the equivalence of his analysis with the intuitive concept.

3. The Necessity of Tarski-Validity

That the inference Etchemendy attributes to Tarski is fallacious is indisputable. By its means Etchemendy neatly pinpoints a difficulty in understanding Tarski’s argument. The question remains of the real depth and significance of the problem, the extent to which it applies to Tarski’s defence of the equivalence, and whether there are ways of avoiding the modal fallacy in the general framework of Tarski’s concept of logical consequence. There is as a matter of fact a relatively straightforward way of revising or, if necessary, reinterpreting Tarski’s argument, so that the modal fallacy Etchemendy ascribes to it does not arise. It is of course modally fallacious to deduce: (( P & Q) Š ~R)

P Š (Q Š ~R) A similar perfectly valid derivation uses the same assumption, which Etchemendy accepts: (( P & Q) Š ~R) P Š (Q Š ~R)

The inference is obvious, but it may be worthwhile to exhibit an easy natural deduction proof in the weakest alethic modal logics. 5

subject of modal logic is now dominated by semantical methods that in a general form owe much to Tarski.” 5 The natural deduction application for modal logic is given in Hughes and Cresswell (1972), Appendix One, pp. 331-334.

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(( P & Q) Š ~R)

2.

P

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4.

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(1)

P

(2)

5.

Q

6.

~R

Q Š ~R

7. 8. 9.

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(Q Š ~R) P Š (Q Š ~R)

(3-5) (5-6) (7) (2-9)

To restore validity to Tarski’s argument in light of Etchemendy’s criticism, it is sufficient to regard proposition (1) as a necessary truth, or to allow that from the fact that S is a Tarskian consequence of K (for some F ), it follows that necessarily S is a Tarskian consequence of K (for some F ); that is, not for every proposition, but for P specifically as interpreted above, P Š P. Intuitively, it may at first appear that the proposal is or ought to be well-justified. Tarskian consequencehood via satisfaction-preservation on limited sequences is at least partially an abstract set theoretical matter. Its main advantage over Bolzano’s substitution theory of logical truth and consequence, even as Etchemendy explains it, is that it is independent of the contingencies of linguistic expansion or contraction in particular logically possible worlds (Etchemendy 1990, pp 27-32). This suggests that Tarski might have understood logical consequence under his definition as implying the logical necessity of logical consequence, the modality for which any adequate account of consequence must strive. Later, I shall argue in detail for the Tarskian necessary consequencehood principle. For now, we may consider the matter hypothetically. If the necessity of Tarskian logical consequence is admitted, then the modal fallacy Etchemendy attributes to Tarski’s proof for the equivalence of his

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analysis and the intuitive concept of logical consequence is avoided. But it is because of the modal fallacy (in various guises) that Etchemendy repudiates Tarski’s argument for the equivalence. To eliminate the fallacy is therefore to eliminate Etchemendy’s only ostensible reason for rejecting Tarski’s analysis.

4. Resurgence of Etchemendy’s Counterexamples

The trouble with this line of defence is that the logical necessity of Tarskian consequencehood is jeopardized by roughly the same category of counterexamples with which Etchemendy’s criticism begins. Consider one of the problems raised as an illustration in support of Etchemendy’s claim that Tarski’s definition is inadequate even for submaximal assignments of fixed terms to F : ‹{Grass is green}, Snow is white›, where F = {Snow, is white} This argument counterintuitively is Tarski-valid. The argument’s conclusion ‘Snow is white’ follows as a Tarskian consequence of the assumption ‘Grass is green’, modulo F , in just those logically possible worlds where snow is white (where ‘Snow is white’ is true in L). But in other logically possible worlds where snow is not white (where ‘Snow is white’ is not true in L), the argument is not logically valid by Tarski’s definition, and, modulo F , ‘Snow is white’ does not follow as a Tarskian consequence of the premise ‘Grass is green’. This is an unwelcome result, which Etchemendy calls attention to only indirectly. Intuitively, it is part of the concept of consequence in the ordinary sense that valid arguments are necessarily valid, or logically valid in every logically possible world. This is particularly true where logical relations are standardly understood to be logically necessary, holding in every logically possible world, and where propositions are standardly understood to be abstract eternal Platonic vehicles of meaning and truth, belonging to every logically possible world, in the manner of Bolzano’s Sätze an sich (1972, Part I, §§19-33, pp. 20-42) or Frege’s Gedanken (1977, pp. 5-8, 24-30). As Etchemendy characterizes the situation, the misplaced modality resulting from the alleged modal fallacy in Tarski’s argument not only prevents Tarskian consequence from holding across all logically possible worlds, but even validates or invalidates the same argument at different times within the same logically possible world.

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What is important for our purposes is that we recognize that no real modality, obscure or otherwise, follows from Tarski’s definition. Thus, suppose that some argument ‹K, S› satisfies the definition. Can we show, for example, that ‘it can never happen’ that the members of K will all be true while S is false-that is, that truth preservation is in any way an enduring feature of this argument? The answer is no: for all we know, the same argument may have true premises and a false conclusion tomorrow. Of course, should this come to pass, Tarski’s definition guarantees that the argument will no longer qualify as logically valid. But this is a guarantee of entirely the wrong sort [. . .]. (p. 92)

This is true for counterexamples where valid Tarskian consequences hold because they happen to be contingently true, the assumptions are not logically false, and all terms in the argument or those in the conclusion are fixed in F . Thus, if snow is white is true in L for the actual world, it makes that proposition a valid Tarskian consequence of any such argument as long as snow continues to be white. But if snow suddenly becomes red or blue, then an argument that had once been Tarski-valid in the past will not continue to remain valid in the future. Similarly for arguments where the assumptions happen to be false, the conclusion is not logically true, and all terms or at least those in the contingently false assumption, are fixed in F . There is a convergence of two intimately related problems in Tarski’s theory that Etchemendy’s criticism brings forward. Arbitrary stipulation of fixed terms in F permits counterexamples to indicate what Etchemendy has identified as a modal fallacy in Tarski’s justification for the equivalence between his definition of consequence by satisfactionpreservation on all sequences and the intuitive concept of consequence. The modal fallacy in turn is avoidable only if Tarskian consequence is logically necessary, which is thwarted by the same counterexamples obtained from unrestricted stipulation of fixed terms in F . This suggests the possibility of amending Tarski’s account (if necessary) to provide a Tarsk ian analysis of consequence in which the specification of fixed terms in F is restricted in some plausible principled way to avoid the counterexamples. It also makes it possible for Tarski’s justification of the equivalence of his definition of consequence with the intuitive notion to avoid modal fallacy by the proposed method of treating Tarski-validity and Tarski-consequencehood as logically necessary. Yet the revision of Tarski’s definition into a cousin Tarsk ian analysis may not be needed, as Etchemendy’s criticisms invite a new look at Tarski’s original statement of the definition to see whether or not he countenanced unrestricted fixed term specifications.

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5. Logical Defence of Tarski’s Analysis

Etchemendy is right in one way to insist that the modal fallacy he attributes to Tarski is independent of the particular choice of fixed terms by which the definition is implemented. He maintains: This fallacy becomes quite apparent when we consider the arguments that come out valid when we include all expressions in F. But it is crucial to recognize that the inference remains fallacious, and for exactly the same reasons, regardless of our choice of fixed terms. The fallacy may be easier to spot when we include names and predicates in F, but the inference is no less fallacious when we only hold fixed (say) the truth functional connectives. The argument does not depend on F, and it does not get better or worse according to what we suppose the members of F to be. (p. 88)

The disclaimer is also somewhat misleading in suggesting that there is no real connection between choice of F and the modal fallacy in Tarski’s argument. What is true is that the modal fallacy of trying to derive the conditional P Š ( Q Š ~R) from (( P & Q) Š ~R) stands alone, regardless of how set F of fixed terms is specified in applying Tarski’s definition of logical consequence, and that changing or restricting F does not change the fact that the inference is fallacious. There is more to the problem than this. If set F is properly restricted to assure that the Tarskian consequence relation itself is logically necessary, or in such a way that if the Tarskian definition holds in any, then it is sure to hold in every, logically possible world, then the modal fallacy does not arise, and Etchemendy’s only objection to Tarski’s theory of logical consequence is eliminated. Thus, while it is true that the fallacy abstractly considered does not go away even if F is restricted, the question of whether Tarski’s argument should be regarded as committing the fallacy very much depends on whether and how set F is restricted, or whether, as in Etchemendy’s exposition of Tarski’s theory, it is arbitrary. We can also consider a way of limiting F so that the modal fallacy alleged to invalidate Tarski’s argument for the equivalence of his definition and the intuitive concept of logical consequence is avoided. It further serves to explain the sense in which by this method Tarski’s definition does after all correctly place the right modality in the right location, exactly where it belongs for the derivation of logical consequences from assumptions in deductively valid inferences, and plausibly accounts for the source of this necessity in Tarski’s concept of satisfaction-preservation on all possible (appropriately limited) sequences.

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This interpretation can be upheld on logical grounds as a Tarskian definition, if not Tarski’s own analysis. The strategy makes it important to review the historical question of Tarski’s intentions concerning the restriction or nonrestriction of the set of fixed terms F in implementing the definition. The answer seems to be that Tarski’s own formulation of the definition of logical consequence by no means lends itself to unrestricted specification of fixed terms, and that, properly understood, Tarski’s criterion confers logical necessity on inferences that meet its conditions. This proposal at once refutes Etchemendy’s counterexamples and the modal fallacy he ascribes to Tarski’s argument for the equivalence between his analysis and the intuitive concept of logical consequence. The obvious remedy for the difficulties Etchemendy raises against Tarski’s definition of consequence is to restrict the set of fixed terms F relative to which Tarski’s model set satisfaction-preservation criterion is applied to all and only the logical terms of an argument ‹K, S ›, and not to permit fixing extralogical terms. This extricates Tarski’s argument for the equivalence of his definition with the intuitive concept from the modal fallacy Etchemendy attributes to it, by upholding principle P Š P (where P specifically is ‘S is a Tarskian consequence of K (for F = the set of all and only logical terms in ‹ K, S ›)’). If not all logical terms are fixed, then Tarski’s definition has no chance of succeeding even against much less sophisticated counterexamples than those Etchemendy’s describes, as when, for example, F = {‘ P’, ‘ Q’}, and the argument to be evaluated is ‹{ P Š Q, P}, Q›. If not only logical terms are fixed, then counterexamples like Etchemendy’s obtain. These, as already observed, fall into two general nonexclusive categories, based on Tarski’s disjunctive definition of consequence by satisfaction-preservation. They occur when certain extralogical terms in K of ‹ K, S › are fixed in F , and some sentence in K happens contingently to be false, but S (intuitively) is not logically true, or when certain extralogical terms in S are fixed in F , and S happens contingently to be true, but the sentences in K (intuitively) are not logically false. The case Etchemendy mentions, where set F is maximal relative to the terms in the argument being evaluated, entails a counterexample of a Tarskian consequence that intuitively is not a logical consequence, as a special instance of one or both of the two kinds of cases described. What

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Etchemendy is unable to produce is a counterexample to Tarskian logical consequence in which F contains all and only decidedly logical terms.6 Are there any positive reasons for supposing that the Tarskian necessary consequencehood principle is true? Here is a straightforward demonstration reductio ad absurdum. Suppose (for purposes of indirect proof) that the necessity of Tarskivalidity principle in P Š P is false. Then there is an argument ‹K, S › that is Tarski-valid in some logically possible world Wi , but not in some other logically possible world Wj . The only difference in model set theoretical terms between logically possible worlds is the instantiation or noninstantiation of particular variables, and the truth or falsehood of corresponding predications. It follows that if the Tarskian necessary consequencehood principle is not true, then Tarski-validity is determined on a world-by-world basis by the instantiation or noninstantiation of particular variables in particular logically possible worlds. Tarski-validity is not determined in this way, as the following considerations show. Argument ‹K, S › is Tarski-valid in world Wi if and only if all sequences modulo F (restricted to all and only logical terms, here and throughout) are satisfaction-preserving for ‹K*, S *› in Wi . If this depended on whether or not a particular variable is instantiated in Wi , then there would be a sequence in which variables are substituted for variable terms in ‹K*, S *›, for which all the members of K* but not S * are satisfied in Wj . To see that this is not the case, suppose that the Tarski-validity of ‹K, S › in Wi depends on the instantiation of subjectpredicate relation Fa , and consider the sequence that substitutes for variable terms ‘Fa ’ the complementary subject-predicate relation ‘ F a ’, uninstantiated in world Wj (or similarly for contingently uninstantiated objects). If the Tarski-validity of ‹ K, S › depended on the instantiation of F a in Wi , then the Fa substitution sequence Tarski-invalidates the argument. But by hypothesis the argument is Tarski-valid, and is so if and only if it is satisfaction-preserving on all sequences, including the Fa substitution sequence, consisting of items contingently instantiated in Wj , but not in Wi . It follows that Tarski-validity is not determined by the instantiation or noninstantiation of particular variables in different logically possible worlds, that there is no argument ‹K, S › that is Tarski-valid in some but not all logically possible worlds, and therefore that Tarski6

It is noteworthy that Etchemendy refers to Tarski’s distinction between logical and extralogical terms only in mounting objections against the reduction principle, that logical truths are implications of ordinary true universal closures (1990, pp. 109-110). See notes 8 and 9 below.

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consequencehood is logically necessary in the sense that an argument Tarski-valid in any logically possible world is Tarski-valid in all logically possible worlds.7 The Tarskian necessary consequencehood principle, that P Š P, is thereby proved, where P is ‘ S is a Tarskian consequence of K (for some F )’, and this, together with the valid modal inference from (( P & Q) Š ~R) to P Š ( Q Š ~R), entirely avoids Etchemendy’s modal fallacy objection. It places logical necessity in Tarski’s analysis exactly where Etchemendy claims it intuitively belongs, governing the relation between Q (‘All members of K are true’) and ~R (‘ S is true (not false)’).

6. Uncertain Logical or Extralogical Status of Identity

In his criticism of Tarski’s reduction principle for quantificational semantics, Etchemendy advances additional problems about the logical truth of sentences ostensibly containing all and only logical terms, such as the sentence,  x  y ( x z y ). If this is understood under extensional quantificational semantics as asserting the substantive claim that at least two things exist, then it appears contingent at best, if not, from the standpoint of a Parmenidean or Spinozistic metaphysics, necessarily false. But on the assumption that it is true in the actual world that there exist at least two things, then any argument having this presumably contingent sentence as its conclusion will turn out to be Tarski-valid, but not intuitively valid, provided that all the terms in the construction are logical and none extralogical, and all and only logical terms are fixed in F for purposes of determining Tarskian consequence. The argument with this contingent sentence as consequence will then be Tarski-valid in some but not all logically possible worlds, for it will be valid by Tarski’s criterion in just those worlds that contain at least two existents, and false in those that contain one or none. This is an interesting objection. But it contains several ontologicallogical confusions that should be untangled. The ontology or semantic 7

It may appear that this proof of the necessity of Tarski-validity and consequencehood assumes that sequence items, especially object term referents, need not exist. Etchemendy might avoid the conclusion, then, by restricting all sequence intersubstitutions to terms for actually existent objects. But the argument holds with or without such a restriction. Note that in order for Fa to be true in world W i but not in W j , it need not be the case that a exists in W i but not in W j . It is sufficient for a to exist, and for property F to be instantiated, both in W i and W j , but for a to have F in W i , and not to have F in W j , where some existent object other than a has F in W j instead.

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domain of quantificational logic is rarely restricted to spatiotemporal entities. If a plurality of abstract objects is included, as even W.v.O. Quine requires in his austere semantics of spatiotemporal entities and classes, and if sets as abstract entities belong to every logically possible world, then the proposition  x  y ( x z y ) is, after all, in some sense, a logically necessary if not a necessary logical truth (see Etchemendy 1990, p. 168, Ch. 8, note 2). In that case, Tarski’s criterion correctly judges any argument with the proposition as conclusion to be model set theoretically as well as intuitively valid. It will be impossible for the argument’s assumptions to be true and the conclusion false, because it will be impossible for the conclusion to be false. The counterexample has plausibility if at all only on the presupposition that the existential quantifier limits the domain of existents to spatiotemporal entities, which can vary contingently from world to world. But in neither case, as Etchemendy seems to agree, can the proposition as a substantive assertion express a truth of pure logic. It may be necessary therefore to introduce special predicates for spatiotemporal and abstract existents, by which means the ontological and logical meanings of quantified statements can be disambiguated. The above proposition might better be written in any of the following forms, using ‘E!’ for spatiotemporal, and ‘ A!’ for abstract existence: x x x x

y y y y

(E!x & E!y & x z y) (E!x & A!y & x z y) (A!x & A!y & x z y) (( E! x › A! x ) & ( E! y › A! y ) & x z y )

Etchemendy seems to mean the first of these by his use of  x  y ( x z y ). But if the expression is reformulated in any of these four suggested more perspicuous ways, then what is intuitively at most a contingent, or, in any case, extralogical truth also turns out to be so by Tarski’s criterion, and hence does not automatically confer Tarskivalidity on every argument that takes the proposition as its conclusion. This is because, regardless of the logical or extralogical status of ‘=’ and ‘z ’, ‘ E!’ and ‘A!’ evidently are not logical terms, so that substitutions of variables for the only variable terms ‘E!’ and ‘A!’ in the expression are not satisfaction-preserving for all Tarski sequences, and in particular not when the substituted variable is any uninstantiated predicate, such as ‘is a winged horse’ (instantiated contingently in some but not all logically possible worlds), or ‘is entailed by the completeness of first-order arithmetic’ (uninstantiated in every logically possible world). If the existential generalizations are properly spelled out, then

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they are seen merely to appear to involve only logical terms in their incompletely articulated forms. In fact, when intended as substantive ontic assertions, statements like those in Etchemendy’s illustrations implicitly combine fixed logical with variable extralogical terms. When Tarski’s criterion is applied to any of these disambiguated propositions, the expected evaluations of the propositions as logically contingent obtain, disarming Etchemendy’s counterexample. Alternatively, the difficulty can also be easily surmounted if only the relational term ‘z ’ is classified as extralogical, thereby exempting it on the proposed restriction from being fixed in F . When this is done, the sentence  x  y ( x z y ) can no longer properly be regarded as expressing a logical truth. This is perhaps as it should be, since the hypothesis by which the sentence poses a problem for Tarski’s definition is that the sentence makes a substantive assertion of what is at best a contingent ontic truth, and as such is not purely logical. The solution meets with resistance only when it is considered that by this principle expressions of what are often understood to be logically necessary truths, such as the reflexivity, symmetry, and transitivity of identity, are, even if logically necessary , not necessary logical truths.8 x (x = x) x y (x = y Š y = x) x y z (( x = y & y = z ) Š x = z )

This is awkward because it is customary to regard such propositions as necessary truths, true in every logically possible world. If they are not logical truths, but are nevertheless necessary, then they are extralogical logically necessary truths. The most reasonable way to think about these sentences is that they are not after all logical truths, on the grounds that, superficial symbolic appearances notwithstanding, ‘=’ and ‘z ’ are not logical terms. Identity is a metaphysical rather than logical concept, but the above sentences can be understood as signifying logically necessary analytic or Kantian synthetic a priori truths, like those expressed in the strictly extralogical (nontautological) necessary sentences, ‘All bachelors are unmarried’ and ‘ 2 2 > 2’. The familiar slogan, ‘Identity is self8

Etchemendy otherwise regards identity ‘=’ and nonidentity ‘z’ as logical terms (1990, pp. 111-124). The position is found in his (1988, p. 69): “However, as long as the quantifiers are treated as logical constants, Tarski’s analysis always leaves the domain of quantification fixed. Because of this, sentences like (15) [(x) (y) (x z y)] will come out logically true on Tarski’s account [. . .]. This [is] simply because on the present selection of logical constants, there are no nonlogical constants in the sentence to replace with variables. Thus, such sentences are logically true just in case they happen to be true; true, of course, on the intended interpretation.”

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identity’, may unpack part of the meaning of relational identity terms, and as such suggest that the reflexivity, and, by extension, symmetry and transitivity, of identity is analytic, if indeed it is not, as Kant would have it, synthetic a priori .9 Etchemendy considers and rejects the suggestion that the identity and nonidentity predicates ‘=’ and ‘z ’ be treated as extralogical. He argues that if identity is made extralogical and excluded from the set of fixed terms, the same type of problem emerges by way of other predicates. He considers these propositions: D E J

x y z ( x is taller than y š y is taller than z o x is taller than z ) x ™( x is taller than x ) y  x ( x is taller than y )

The negation of the conjunction of these in (7) ™( D š E š J ), he notes, is true (on the reasonable assumption that J is false), but not logically true, necessarily true, a priori true, or true by virtue of meaning. He claims: Now according to the present strategy, we are not treating the identity predicate as a logical expression. But since the identity predicate does not appear in (7) this will have no effect one way or the other, on our assessment of this sentence. Here, Tarski’s account equates the logical truth [sic?!] of (7) with the ordinary truth of (8) R [™(x y z (xRy š yRz o xRz) š x ™(xRx) š y x (xRy))] [. . .] the truth of (8)  and so our assessment of (7)  depends at the very least on the actual size of the universe, and perhaps on additional nonlogical facts as well. (p. 118)

Yet this is certainly not the evaluation of a correct application of Tarski’s criterion. In unpacking (7) in terms of D, E , J , or in (8) just as it is stated, there are extralogical terms other than ‘=’ or ‘z ’, such as ‘taller 9

The difficulties of distinguishing between logical and extralogical terms are considerable, and I do not mean to downplay the problems involved. For purposes of my countercriticism of Etchemendy’s attack on Tarski, however, I have no need to offer an exact criterion. The burden of proof presumably lies with Etchemendy to demonstrate that Tarski commits a modal fallacy. To show that his argument to this effect is inconclusive, I need only fall back on Tarski’s intuitive characterization of the distinction between logical and extralogical terms, and the grey area in between where Etchemendy’s best problem cases occur, as disallowing definitive counterexamples. My reflections on these topics have benefited from Gila Sher’s (1991) study of the concept of logical terms, especially her Chapter 3, pp. 36-66, To Be a Logical Term. Etchemendy’s criticism of Tarski is mentioned on pp. 45-46. See also Peacocke (1976); McCarthy (1981); Moreno (1990).

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than’, to confirm the intuitive evaluation that these propositions do not express Tarskian logical truths, and do not automatically confer Tarskivalidity on arguments in which they occur as conclusions. Etchemendy rightly observes that Tarski’s criterion has no extralogical identity or nonidentity terms to treat as variable in (7) or (8). But he overlooks the other extralogical terms in the expressions by which Tarski’s criterion again evaluates the sentences so as to coincide exactly with Etchemendy’s intuitive classifications of these statements as nonlogical truths.10 Finally, if we were to decide, perhaps for historical or theoretical reasons, that it is more appropriate to consider identity and nonidentity as logical rather than extralogical terms, or that Tarski himself preferred such a classification, the necessity of Tarskian consequencehood can still be upheld in this alternative way by revising the restriction on fixed terms to all and only logical terms except identity and nonidentity. If other problematic terms are identified, exceptions can also be made for these. Nor would such exclusions need to be considered ad hoc , because of the pretheoretically quasi-logical character of identity and nonidentity relations, and whatever other terms may affect the validity status of arguments.

10

Etchemendy (1990, pp. 123-124), offers another counterexample involving the counterintuitive provability by Tarski’s criterion of the continuum hypothesis in secondorder logic. But it is hard to see how the objection could apply to correct applications of Tarski’s validity criterion. On p. 169, n. 11, Etchemendy offers as an “equivalent to the continuum hypothesis” the closure, X Y Z [N(x) š R(Y ) š Card(X ) < Card(Z ) o Card(Y ) d Card(Z )], where N and R are the properties of being isomorphic to the set of natural and real numbers, respectively. But if sequence substitution of variables for variable terms (in Etchemendy’s sense, relative to an arbitrarily fixed set F ) in Tarski’s criterion is limited as it should be to extralogical terms, then the continuum hypothesis is rightly judged non-satisfaction-preserving for certain substitutions of variables for the extralogical higher-order predicate terms ‘N ’, ‘R’, and ‘Card’. The counterexample holds only if Tarski sequence substitution is not restricted to logical terms, but lets arbitrary terms be fixed for purposes of determining validity or logical truth. Other countercriticisms of Etchemendy’s objections to Tarski appear in Hart (1991). Priest (1995) defends Tarski by allowing that the sentence ‘There are n things’ is always a logical truth. Moreno (1990) by contrast argues that for Tarski set-theoretical constants are extralogical. McGee (1992) largely agrees with Etchemendy’s criticisms, and upholds Tarski’s analysis qualifiedly. McGee writes, p. 292: “[. . .] if we think of Tarski’s (1956 [1936]) paper, not as giving a logical analysis, but as announcing a programme, the programme of investigating logical systems by looking at their models, we shall regard it as an enormous success.”

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7. Historical Defence of Tarski’s Analysis

When Tarski’s paper “On the Concept of Logical Consequence” is considered, it is clear that Tarski himself at no point countenances fixing extralogical terms in implementing his satisfaction-preservation theory of logical consequence. The analysis is presented in terms of the concept of a model, defined in terms of an arbitrary sequence of objects that satisfies a sentential function obtained from a sentence in an argument by uniformly substituting variables for its exclusively extralogical terms. Tarski offers the following explanation: One of the concepts which can be defined in terms of the concept of satisfaction is the concept of model. Let us assume that in the language we are considering certain variables correspond to every extra-logical constant, and in such a way that every sentence becomes a sentential function if the constants in it are replaced by the corresponding variables. Let L be any class of sentences. We replace all extra-logical constants which occur in the sentences belonging to L by corresponding variables, like constants being replaced by like variables, and unlike by unlike. In this way we obtain a class Lƍ of sentential functions. An arbitrary sequence of objects which satisfies every sentential function of the class Lƍ will be called a model or realization of the class L of sentences [. . .]. (Tarski 1956, pp. 416-417; selected emphases)

Far from thinking he has thereby presented a water-tight materially adequate definition of logical consequence, Tarski goes on to acknowledge the difficulties he perceives in the extensional adequacy of the analysis. These recapitulate his cautionary words at the beginning of the essay that there is unlikely ever to be an entirely satisfactory formal analysis of the intuitive concept of consequence. 11 The most important problem Tarski recognizes has to do with the essential distinction between logical and extralogical terms, without which the definition cannot properly function. This clearly indicates that Tarski intends the criterion to work only by means of fixing specifically logical terms, to the exclusion of extralogical, though he concedes that

11 Tarski (1956), p. 409: “With respect to the clarity of its content the common concept of consequence is in no way superior to other concepts of everyday language. Its extension is not sharply bounded and its usage fluctuates. Any attempt to bring into harmony all possible vague, sometimes contradictory, tendencies which are connected with the use of this concept, is certainly doomed to failure. We must reconcile ourselves from the start to the fact that every precise definition of this concept will show arbitrary features to a greater or less degree.”

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there may be no hard and fast distinction between the two, due primarily to a lack of precision in ordinary language. At first, Tarski is cautious about the extensional adequacy of his analysis, noting as a particular weak spot its reliance on the distinction between logical and extralogical terms. I am not at all of the opinion that in the result of the above discussion the problem of a materially adequate definition of the concept of consequence has been completely solved. On the contrary, I still see several open questions, only one of which  perhaps the most important  I shall point out here. (p. 418) Underlying our whole construction is the division of all terms of the language discussed into logical and extra-logical. This division is certainly not quite arbitrary. If, for example, we were to include among the extra-logical signs the implication sign, or the universal quantifier, then our definition of the concept of consequence would lead to results which obviously contradict ordinary usage. (p. 418)

Then Tarski admits that he knows of no satisfactory way to draw a sharp distinction between these two categories of terms. He allows that, in the limiting case, as for purely formal systems, all terms of a language could be regarded as logical, for which the satisfaction-preservation analysis would entail the coextension of “material” and “formal” logical consequence. On the other hand, no objective grounds are known to me which permit us to draw a sharp boundary between the two groups of terms. It seems to be possible to include among logical terms some which are usually regarded by logicians as extra-logical without running into consequences which stand in sharp contrast to ordinary usage. In the extreme case we could regard all terms of the language as logical. The concept of formal consequence would then coincide with that of material consequence. The sentence X would in this case follow from the class K of sentences if either X were true or at least one sentence of the class K were false. (pp. 418-419)

Finally, Tarski leaves open the question whether the distinction between logical and extralogical terms can be philosophically justified. This has the result of relativizing the meaning of ‘logical consequence’ and its cognates to potentially arbitrary stipulations of the distinction between logical and extralogical terms, say, as afforded by the syntax and formation rules of a formal language. Perhaps it will be possible to find important objective arguments which will enable us to justify the traditional boundary between logical and

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extra-logical expressions. But I also consider it to be quite possible that investigations will bring no positive results in this direction, so that we shall be compelled to regard such concepts as ‘logical consequence’, ‘analytical statement’, and ‘tautology’ as relative concepts which must, on each occasion, be related to a definite, although in greater or less degree arbitrary, division of terms into logical and extra-logical. (p. 420)

But Tarski never sanctions the determination of logical consequence by fixing extralogical terms as nonvariable for criteriological purposes, regardless of how the two categories of terms are distinguished. This makes Etchemendy’s criticism of the proof for the equivalence between Tarskian and intuitive consequence irrelevant, since Tarski for good reasons prohibits any such application of the definition. Etchemendy’s reaction to Tarski’s clear-sighted avowal of his theory’s limitations is to divide all terms into four grammatical categories for objects, predicates, connectives, and operators, and then to allow variable and name-domain substitutions under Tarski’s criterion within all four cross-term-restricted categories, relative to some (i.e., any ) set of fixed terms F . From this starting place it is not surprising to find counterexamples to Tarski’s analysis. But Tarski nowhere says that logical consequence by his definition is to be relative to some or just any set of fixed terms, but specifically only of logical terms, however these are determined or defined. Etchemendy’s criticisms as a result are based on a significant distortion of Tarski’s definition of logical consequence. The identification of a modal fallacy in Tarski’s argument for the coincidence of intuitive and model set theoretical concepts of logical consequence, like the counterexamples to the extensional adequacy of the analysis, is without proper foundation in Tarski’s original statement of the criterion.12

12

George, “Editor’s Introduction,” in Bolzano (1972), p. xxxv: “At first sight, the principal difference between [Bolzano’s and Tarski’s] definitions [of logical consequence] is the presence of the term ‘truth’ in Bolzano’s definition, but since he says that ‘our judgments are true if we combine with our idea of a certain object an idea which this object really has’, the word ‘truth’ can be eliminated from the definition. One can then see that the main difference lies in the absence of the notion of a function in Bolzano and the fact that he does not draw a sharp dividing line between logical and extralogical parts of propositions [. . .]. Tarski, by contrast, asks us to turn all extralogical constants into variables [. . .].” Etchemendy (1988, p. 68) recognizes the importance of the distinction between logical and extralogical terms in Tarski’s definition of logical consequence: “[Tarski] then assumes that we have, perhaps arbitrarily, divided the primitive constants of the language into two classes, the logical signs and the nonlogical signs.” See Simons (1987).

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If problems involving fixed terms of ambiguous logical-extralogical status of the sort Tarski worries about are advanced in an Etchemendystyle criticism, it is likely that the intuitive concept of consequence will no more be settled in its assessment of whether a valid inference obtains than the evaluation Tarski’s technical definition provides, and as such cannot constitute a decisive counterexample to Tarski’s analysis. If we can agree that a term is intuitively logical and not extralogical, then we can fix it for purposes of applying Tarski’s criterion. If we cannot, then to that degree we will probably lack confidence in whether or not the argument is valid or the conclusion a logical consequence in the ordinary sense. The objections Etchemendy raises to Tarski’s analysis of logical consequence, including the intuitive counterexamples and the modal fallacy with which he charges Tarski’s argument for the coincidence of intuitive and model set concepts of logical consequence, are avoidable by restricting the fixed terms relative to which sequence satisfaction is determined to all and only an argument’s purely logical terms. There is no need to do so in offering a “charitable” historical reading of Tarski’s concept, because his formulation of the criterion in the first place does not authorize fixing extralogical terms.13

The Pennsylvania State University Department of Philosophy University Park, PA 16802 USA e-mail: [email protected] http://www.personal.psu.edu/dlj4/

REFERENCES Bolzano, B. ([1837] 1972). Theory of Science: Attempt at a Detailed and in the Main Novel Exposition of Logic with Constant Attention to Earlier Authors. Edited and translated by R. George. Berkeley: University of California Press. Etchemendy, J. (1988). Tarski on Truth and Logical Consequence. The Journal of Symbolic Logic 53, 51-79. Etchemendy, J. (1990). The Concept of Logical Consequence. Cambridge: Harvard University Press.

13

I am grateful to the J. William Fulbright Commission for Cultural, Educational and Scientific Exchange Between Italy and the United States of America for supporting this research in Spring 1996 during my tenure as J. William Fulbright Distinguished Lecture Chair in Contemporary Philosophy of Language at the University of Venice.

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Gottlob, F. ([1918] 1977). Thoughts. In: P.T. Geach (ed.), Logical Investigations, translated by P.T. Geach and R.H. Stoothoff, pp. 1-130. Oxford: Basil Blackwell. Gomez-Torrente, M. (1996). Tarski on Logical Consequence. Notre Dame Journal of Formal Logic 37, 125-151. Hart, W.D. (1991). Critical Notice on The Concept of Logical Consequence by J. Etchemendy. The Philosophical Quarterly 41, 488-493. Hughes, G.E. and M.J. Cresswell (1972). An Introduction to Modal Logic. London: Methuen & Co. Ltd. Jacquette, D. (1994). Tarski’s Quantificational Semantics and Meinongian Object Theory Domains. Pacific Philosophical Quarterly 75, 88-107. McCarthy, T. (1981). The Idea of a Logical Constant. The Journal of Philosophy 78, 499-523. McGee, V. (1992). Two Problems with Tarski’s Theory of Consequence. Proceedings of the Aristotelian Society 92, 273-292. Moreno, L.F. (1990). Tarski and the Concept of Logical Constant. Logique et Analyse, 131-132, 203-214. Peacocke, C. (1976). What is a Logical Constant? The Journal of Philosophy 73, 221-240. Priest, G. (1995). Etchemendy and Logical Consequence. Canadian Journal of Philosophy 25, 283-292. Sher, G. (1989). A Conception of Tarskian Logic. Pacific Philosophical Quarterly 70, 341-368. Sher, G. (1991). The Bounds of Logic: A Generalized Viewpoint. Cambridge, MA: The MIT Press. Simons, P. (1987). Bolzano, Tarski and the Limits of Logic. Philosophia Naturalis 24, 378-405. Suppes, P. (1988). Philosophical Implications of Tarski’s Work. The Journal of Symbolic Logic 53, 80-91. Tarski, A. and J.C.C. McKinsey (1944). The Algebra of Topology. Annals of Mathematics (series 2) 45, 141-191. Tarski, A. ([1936] 1956). On the Concept of Logical Consequence. In: Logic, Semantics, Metamathematics: Papers from 1923 to 1938, translated by J.H. Woodger, pp. 409-420. Oxford: The Clarendon Press.

PART VI TRUTHS AND FALSEHOODS

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Arianna Betti SEMPITERNAL TRUTH. THE BOLZANO-TWARDOWSKI-LEĝNIEWSKI AXIS*

1. Introduction In 1913 Stanisáaw LeĞniewski published his article on the sempiternity of truth, “Is Truth Only Eternal or Is It both Eternal and Sempiternal?” (LeĞniewski 1913a). 1 The paper, directed against KotarbiĔski’s “The Problem of the Existence of the Future” (KotarbiĔski 1913), made an important contribution to the debate on the excluded middle current in the Lvov circle in those years.2 The discussion involved at the same time absoluteness, eternity and sempiternity of truth, i.e. truth for ever and truth since ever, and had as ideal reference point Twardowski’s “On the So-Called Relative Truths” (1900), 3 where the founder of the LvovWarsaw School had attacked the relativity of truth. Contrasting KotarbiĔski’s positions, LeĞniewski defended “absolutism,” consequently

*

This paper was submitted in 1996. Until now versions of it have circulated in various forms. A Polish translation of it appeared in Filozofia Nauki, VI, Nr. 2 (22), 1998, pp. 51-75. Having done much more work on the subject in the meantime, I have added in proof the changes which allow this paper to appear in print. The present version is to be considered the final and official one. 1 Warning: the English translation contains mistakes which alter the text, especially at p. 109. 2 To the discussion belonged also LeĞniewski (1913b). 3 See Twardowski (1900), labeled henceforth in the text Relative Truths. I should warn the reader that the German translation of the latter omits some parts of the text. See infra, nn. 39, 45. This paper and Twardowski (1911) have finally a good translation by Arthur Szylewicz in Kazimierz Twardowski – On Actions, Products and other Topics in Philosophy, J. Brandl and J. WoleĔski (eds.), Rodopi, Atlanta/Amsterdam, 1999, resp. pp. 147-168 and 103-132.

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 371-399. Amsterdam/New York, NY: Rodopi, 2006.

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taking sides with Twardowski.4 Twardowski had revived Bernard Bolzano’s ideas on the subject, and, mainly thanks to him, these became known in the Lvov-Warsaw School (see, for instance, Jadacki 1993, p. 191). There is no doubt that LeĞniewski knew Twardowski’s ideas and it seems evident that the latter influenced him: LeĞniewski’s results are mostly compatible with the “absolutistic” content of Twardowski’s 1900 article. And, similarly, no doubts can be raised about the Bolzanian origin of the aspects of eternity and sempiternity of truth defended by Twardowski in Relative Truths (see, for instance, WoleĔski and Simons 1988, p. 430, n. 24; and Simons 1992, Ch. 2, p. 15, n. 11; see also Smith 1988, p. 325): though his name is not quoted, traces of Bolzano’s legacy can be found even in the examples given by Twardowski, some of which are the same as used by Bolzano in his Wissenschaftslehre.5 Yet, since Bolzano, Twardowski and LeĞniewski supported different theories of meaning with different ontological presuppositions, “sempiternity of truth” actually stands for three different conceptions. This paper is a survey of these three conceptions. I suggested elsewhere a comparison between Bolzano and the early LeĞniewski as to their theories of meaning and truth, claiming the possibility of a (direct or indirect) influence of Bolzano upon LeĞniewski. The analysis presented here is also meant as a contribution to the picture sketched there. 6

4

Twardowski himself seems to have attacked KotarbiĔski, see WoleĔski (1990a), p. 194. The latter contains also a discussion of the LeĞniewski-KotarbiĔski controversy. 5 One of them is “The fragrance of this flower is pleasant,” see Twardowski (1900, p. 316 [Germ. transl., p. 416]), see also Bolzano WL §147. The parallel is quoted also by Peter Simons. The German translation, however – both at pp. 416 and 438 – has not “diese Blume riecht angenehm” as reported in Simons (1992, p. 15 n. 11), but “der Duft dieser Blume ist angenehm.” See also the discussion of the demonstrative ‘ten’ (this), Twardowski (1900, p. 323 [Germ. transl., p. 428]). See also Bolzano (1972), pp. 75-76 and 193-197 [WL §59 and §147]. 6 See Betti (1998b). I should remark that the Austro-Polish tradition of absolutistic theories of truth – eternity and sempiternity aside – has its roots also in the school of Franz Brentano, Twardowski’s teacher. See most of all WoleĔski and Simons (1989). See also the volume Albertazzi, Libardi and Poli (1996).

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2. Bernard Bolzano in the Wissenschaftslehre (1837) In Bolzano’s Wissenschaftslehre truth-bearers are propositions-inthemselves, or simply propositions. They are objects with the following features 7: x they are non-existing objects, that is they do not enter into the causal chain, nor do they exist in any time or place, but subsist in the universe as a certain something (for such objects I will employ henceforth the label “lektological”) (see Bolzano 1972, p. 21 [WL §19]); x they are the matter or content of mental acts as well as the sense or the meaning in a restricted sense of linguistic expressions8 and they subsist independently of their being thought or expressed linguistically; mental acts and linguistic expressions, which are real, do exist; x they are complex objects, composed of parts called ideas-inthemselves; ideas may refer to objects; as a result of the fact that an idea may refer to one, or more than one object or not refer at all, it is singular objectual, common objectual or non-objectual (empty). Objects may be subdivided into qualities (of which relations are a special kind), i.e. any object that belongs to at least one other object, and pure objects, objects which are not qualities. Every quality is an object, while not every object is a quality; x they have always the form ‘a has (b)-ity’ where ‘a’ is an idea, ‘(b)-ity’ is an idea (of quality), and moreover, also ‘has’ is an idea.9 Now, given the ideas ‘a’, ‘has’, ‘(b)-ity’, the proposition ‘a has (b)-ity’ is true if and only if

7

I shall consider briefly only those elements which are essential to the discussion. See Casari’s papers in the references for a more exhaustive picture of Bolzano’s views, and the indispensable Berg (1962). 8 Note that the relation between linguistic expressions and propositions-in-themselves is here oversimplified, as no sentence expresses a proposition directly, but via an idea of that proposition. 9 The issue is connected with the so-called “Bolzano’s Conjecture,” see Casari (1992), p. 75. As it is by now common in Bolzano scholarship, I will employ expressions surrounded by square brackets to designate lektological objects.

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‘a’ is non-empty; ‘(b)-ity’ is an idea of quality (and is non-empty); At least one of the qualities which ‘(b)-ity’ refers to belongs to any object which ‘a’ refers to. 10

While in Bolzano truth-bearers are propositions, truth-makers are the relationships (Verhältnisse) that a proposition enunciates, where a relationship is the belonging of a quality (which ‘(b)-ity’ refers to) to an object (which ‘a’ refers to).11 A true proposition-in-itself is called a truth-in-itself (see Bolzano 1972, p. 32 [WL §25]). 12 Truths are objective: The number of blossoms that were on a certain tree last spring is a statable, if unknown, figure. Thus, the proposition which states this figure I call an objective truth, even if nobody knows it. (Bolzano 1972, p. 32 [WL §25])

Any proposition is either true or false, always and everywhere (see Bolzano 1972, p. 172 [WL §125]). Bolzano’s position may be summed up in the following statement (which holds also for falsity): (B)

For any proposition-in-itself p, if p is true at a time t, then it is true also at an arbitrary time t 1 past or future with respect to t.13

Consequently, when it is said that a proposition is neither true nor false, it is not a proposition-in-itself which is meant, but a linguistic expression:

10

See Casari (1992), pp. 73-75. See also Bolzano WL §28, §131, §196. For B2, which, however, may be derived as a theorem, see, for instance, Bolzano (1972), p. 121 [WL §80, 2]. 11 See most of all the dense Casari’s (1992) Appendix on the “still open” semantic value of propositions. I propose a different interpretation in my “Bolzano’s Universe: Metaphysics, Logic and Truth,” which is to appear in L. Haaparanta & H.J. Koskinen (eds.), Categories of Being (Oxford University Press). 12 English translations from the Wissenschaftslehre are from Bolzano (1972), unless otherwise indicated. 13 The apparent oddity of saying that an atemporal truth-bearer ‘is true in t’ requires additional explanation, which I cannot satisfactorily give here. The issue involves a discussion of the thesis: (AT) atemporal truth does not follow from atemporal truthbearers. Since truth is a relational quality (B1-B3) of a proposition, it does not follow only from the atemporal status of the latter that truth is atemporal. This means also that to be a ‘sempiternalist’ you do not need such truth-bearers, as clear from Twardowski’s and LeĞniewski’s position. See also Simons (1991; 1994). There could be several ways to better reformulate (B) by taking (AT) into account.

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What is meant is that this linguistic expression admits one interpretation on which it has a true sense and another on which it has a false sense, or that it is so indefinite that we do not find ourselves justified in either the one or the other of these interpretations. (Bolzano 1973, p. 169 [§125])

Discussing some fundamental logical laws, like Į o Į, Bolzano claims that while affirming “if an object has a certain property, then it has such a property” it is not necessary to add “at the same time.” Some propositions state instead a merely transient relationship, linguistically expressed by sentences like, for instance, “it is raining.” In order to completely express a truth, such sentences require the addition of a time determination and of a location, such as “on August 18, 1996, in Leiden, The Netherlands, it is raining.” 14 This follows from the fact that for Bolzano any real thing – with the possible exception of God – is located in time (see Bolzano 1972, p. 110 [WL §79, 5]). If we want to say, then, that a quality truly belongs to some real object, we must always specify the time in which the quality belongs to the object, and this is of such universal validity that we may even say of the attributes of God that they belong to Him at a certain time, namely at all times. (see Bolzano 1972, p. 32 [WL §25]) 15

Hence no sentence of the form “The real object A has (the attribute) b” expresses a complete truth, unless we include in the idea-subject a temporal specification. For according to Bolzano, the time in which we may truly attribute a certain property to an object belongs to the idea of the object, and not to the idea of the copula of the proposition. (Bolzano 1972, pp. 57-58 [WL §45])

Any time-indication pertains thus to the subject-idea of the proposition. The sentence “Caius is now learned” is not to be looked at as if it expressed a proposition in which the part corresponding to (a deindexicalized) “now” belonged to the copula, because the temporal determination belongs to the subject. A more correct expression would then be

14 “And so I hope no one will take it seriously that the truth or falsity of propositions is a property of them that varies with time and place” (Bolzano 1973, p. 169 [WL §125]). 15 For Bolzano God is one of the real objects, which in the causal chain all undergo the effects of external causes. Yet, being placed at the beginning of the causal chain, God is not subject to any effect, causa prima. See, for instance, Bolzano (1972), p. 248 [WL §168].

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“Caius in his present state is learned” (or: “the present Caius is learned”).

Sentence (1) and the sentence (2)

“Caius was ignorant ten years ago,”

therefore, express propositions which have different subjects. The propositions respectively expressed by (1) and (2) are the following: (1*) (2*)

[Caius 2006 has learnedness] [Caius 1996 has ignorance] (see Bolzano 1972, pp. 57-58 [WL §45]).

The sentence (3)

“I have a sensation of pain”

does not completely express a true proposition unless we transform it (with the proviso that it is uttered by Kurt Weill on April 3, 1950 at 00.01) into (4)

“Kurt Weill, on April 3, 1950 at 00.01, has a sensation of pain.”

Thus if times – conceived of as particular determinations of real objects – are different, many contradictory attributes may be attributed to the same thing. Propositions with predicate-ideas contradicting each other may be true if and only if their subject-ideas are different, so if two contradictory qualities (as for instance ignorance and learnedness) are correctly asserted of the same substance (i.e. for Bolzano an existing or real object), it follows that two different time specifications are present in that substance. Bolzano has a reason for not attaching the time determination to predicates: any quality is a determination, but the converse does not hold (see Bolzano 1972, pp. 121-122 [WL §80]). Time (and space) determinations are among those which are not qualities: Not all determinations of objects require a predicate idea in a proposition where this object is the subject. Rather, there are ideas that serve as determinations of objects without being attributes of them. These ideas have the peculiarity that they can never occur in the place of the predicate idea (b) but only as parts of the subject idea (A) itself. Of this sort are especially the determinations of time and space of existing things, because the time in which the existing thing is located and during which certain attributes can in truth be attributed to it is not an attribute of this thing. For this reason the idea of this time does not occur in the predicate, but in the subject idea of the proposition. This holds analogously also of the spatial determinations of things. (Bolzano 1972, pp. 121-122 [WL §80])

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When speaking of the “Parts which the Author Takes all Propositions to Have” (§127), Bolzano says that language allows us to express through the copula of the canonical form of propositions [A has b] not only person and number of the subject, but also time determinations. But since language connection is not an essential connection, from this we cannot conclude, as Bolzano has already said in §45 and §79, that the copula ‘has’ is formed also by the determination of a time at which something is had (see Bolzano 1972, p. 177 [WL §127, 5]). 16 There are cases in which this is particularly evident: Bolzano says that it seems that the proposition ‘Every truth has an object with which it deals’ says that the belonging of such-and-such a quality to truths takes place at the present time, even if truths are objects which are not in time at all. But if the parts of the proposition expressed by “the object A – has at time t – the attribute b” are to be clearly indicated, they must be expressed in the following way: “the object A at time t – has – (the attribute) b.” For it does not happen at time t that the attribute b is claimed for the object A; but the object A, inasmuch as it is thought to exist at time t (hence to have this determination) is claimed to have the attribute b. (Bolzano 1972, p. 177 [WL §127, 5])

As to Bolzano, one of the reasons why it seems that the same proposition is sometimes true and sometimes false in accordance with different times, places and objects, follows from the fact that we may look at some parts of it as variable. If we vary those parts, however, we do not have the ‘same proposition’ changed, because variation on a proposition yields not the same, but another proposition than the original one. For instance it is not the case that the proposition (5)

‘This flower has a pleasant fragrance’

is sometimes true and sometimes false: we are faced with many propositions which are obtained by the same proposition if we consider certain parts of it as variable and we replace in it first one idea and then another (see Bolzano 1972, p. 194 [WL §147]). 17 Bolzano gives some examples to explain the concept better. Consider the propositions

16 I write (b)-ity to stress that with b Bolzano means an idea of quality: he writes “A has b.” As regards the form “A has b-ity” as the primary form of truth-bearers in Bolzano, see Betti (1998b). 17 The theory of variation (Veränderung) of ideas in a proposition is one of Bolzano’s most celebrate. See also §69 and §108 for the concept of variation of parts in ideas, which

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‘The man Caius has mortality’ ‘The man Caius has omniscience’ ‘The being Caius has mortality’.

If in (i) we consider the idea (part of (i)) ‘Caius’ as variable, the new propositions thus obtained are – for admissible substitutions – all true.18 If we do the same in (ii), we obtain all false propositions, whatever we substitute for ‘Caius’. By repeating the procedure in (iii), we obtain propositions some of which are true and some of which are false. By these means, Bolzano introduces then the concept of the validity (Gültigkeit) of a proposition-in-itself,19 which is defined as “the concept of the relation of all true propositions to the total of all propositions which can be generated by treating certain ideas in a proposition as variables and replacing them with others according to a certain rule” (Bolzano 1972, p. 196 [WL §147]).

3. Kazimierz Twardowski: On the So-Called Relative Truths (1900) and Actions and Products (1911) Many of the considerations already found in Bolzano on truth and time we find again in Twardowski’s Relative Truths. According to Twardowski when philosophers say that truth is relative relying on various examples of elliptical sentences, sentences with indexicals, sentences of general form, and sentences about ethical principles, they make a mistake. They confuse judgements as actions (czynnoĞci) (Twardowski 1900, p. 335 [Germ. transl., p. 446]) or mental products (wytwory) (Twardowski 1900, p. 317 [Germ. transl., p. 418]), with the sentences (powiedzenia) which express them. Therefore relativists replace erroneously the proper truth-bearer, the judgement, by the (type)

is fundamental to extend the definitions of some relations among non-empty ideas to all ideas. See also Berg (1962), pp. 92 ff. 18 In this case Bolzano restricts the substitution procedure to ideas referring (distributively) to men: ‘Gino’, ‘Wojciech’, ‘Franz’, etc. A proposition like ‘The man flower has mortality’ would be empty (i.e. has an empty subject, in Bolzano), and therefore false. See Bolzano 1972, p. 195 [WL §147]. 19 See p. 393, below. Rolf George translates ‘Gültigkeit’ as ‘satisfiability’ (see Bolzano 1972, p. 193), Burnham Terrell correctly as ‘validity’ (see Bolzano 1973 p. 187).

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sentence.20 Yet sentences are only the external expression of judgements, and often they do not express all which the one who judges has in mind. Therefore: We can always convince ourselves very easily of the fact that the conditions placed by relativists are not satisfied, by integrating the sentences given by relativists in order that they become exhaustive expressions of judgements, and by freeing them from any ambiguity by means of an exact definition of the expressions contained in them. (Twardowski 1900, p. 323 [Germ. transl., p. 428])

For instance, if standing in Lvov on the castle mountain I claim that it is raining, “I have not in mind anyever rain falling in anyever place and at anyever time, but I utter a judgement on the rain falling here and now” (Twardowski 1900, p. 319 [Germ. transl., p. 421]). Relativists claim that a true sentence ‘it is raining’, may become false. But for Twardowski this is not the case. The sentence (6)

‘it is raining here and now’,

when uttered on the 1st of March according to the Gregorian calendar at 12.30 p.m. according to Central Europe time on the castle mountain in Lvov, contains the same judgement as (6*)

‘on the 1st of March according to the Gregorian calendar at 12.30 p.m. according to Central Europe time on the castle mountain in Lvov it is raining’.

The sentence (7)

‘it is raining here and now’,

when uttered on the 1st of March according to the Gregorian calendar at 4.00 p. m. according to Central Europe time on the castle mountain in Krakow, contains the same judgement as (7*)

‘on the 1st of March according to the Gregorian calendar at 4.00 p.m. according to Central Europe time in Krakow on the castle mountain it is raining’.

20 Twardowski actually means type sentences, although not stating a distinction comparable to our type/token one. Twardowski’s target is here mainly Franz Brentano.

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According to Twardowski, (6) and (7) are the same sentence containing two different judgements, (6*) and (7*). Therefore it is not tenable that the same judgement goes from being true to being false: It is evident that [the judgement expressed by (6) and (6*)] which asserts in accordance with a real state of affairs (stan rzeczy) that it is raining, is not only true in a certain place and time, but always. 21 (Twardowski 1900, p. 321 [Germ. transl., p. 424])

Thus the salient claim in Relative Truths, where by ‘truth’ Twardowski understands ‘true judgement’ (as Bolzano with Wahrheit an sich understands a true proposition-in-itself) (see Twardowski 1900, p. 315, the first five lines of p. 315 are omitted in the German translation) can be rewritten as follows: (TW) For any judgement g, if g is true at a time t, then it is true also at an arbitrary time t´ past or future with respect to t.22 This can be easily applied also to falsity. Twardowski recognizes that between judgements and sentences there is a very precise link, but he denies that it is an identity relationship, just as a concept or a presentation is not identical with its external sign, the substantive (see Twardowski 1900, p. 317 [Germ. transl., p. 418]). Twardowski gives also a definition of truth for sentences: Now truth and erroneousness, considered as properties of a sentence, can themselves possess further properties, which they do not possess if they are considered to be in the proper and first sense properties of judgements. This further property [. . .] is exactly their relativity. Of sentences one can perfectly say that they are only relatively true. Yet the truth of a sentence depends on the fact that the judgement expressed by means of that sentence is true; nevertheless, since usually a given sentence can express some judgements which are partly true and partly false, it is relatively true because it expresses a true judgement only under a certain condition, i.e. if we consider it as an expression of a true judgement. (Twardowski 1900, p. 335 [Germ. transl., p. 446])

21

The Polish word for ‘state of affairs’ translates the German ‘Sachverhalt’, even if the Polish-into-German translator of Relative Truths, Wartenberg, chose to write “[. . .] Urteil, welches in Ueberinstimmung mit der Wirklichkeit feststellt [. . .].” 22 I modified in this way WoleĔski’s (1990a) formulation (p. 191) where we find ‘proposition’ in place of the Twardowskian ‘judgement’ (sąd).

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For Twardowski the relativity of the truth of a sentence is therefore a second-level property (property of property) which cannot be ascribed to truth as a property of judgements. An example of a relatively true sentence is “The father lives,” because – just as in Bolzano (see p. 375, above) – such a sentence is ambiguous and it may express either true or false judgements. This will depend on the person who utters or hears it (see Twardowski 1900, p. 335 [Germ. transl., p. 447]). According to Twardowski we have the same judgement when we have, “to say it in the language of traditional logic” (Twardowski 1900, p. 317),23 the same subject, predicate, quantity and quality, etc. of the judgement (Twardowski 1900, p. 317 [Germ. transl., p. 418)]. Note that Twardowski thinks of judgements as objects of a propositional structure. 24 Twelve years divide Relative Truths from Actions and Products. The detailed examination that some considerable problems of interpretation in Actions and Products would deserve, chiefly on the concept of meaning (see WoleĔski 1989, p. 41), will not be possible here. It is however possible to claim with a reasonably safe margin that in this work Twardowski revises the more psychologistic views of his On the Content and Object of Presentations (1894). In marking the line of demarcation between logic and psychology on the basis of the distinction act/product which underlies the theory of meaning, he proposes, Twardowski writes: And so the exact separation of products from acts has already decisively contributed to free logic from the influence of psychology. (Twardowski 1911, p. 31 [§45]) 25

Twardowski’s mature theory of meaning is connected with the rigorous definition of the distinction between actions and products of the acts, which in 1900 were still interchangeable terms to denote judgements as mental objects (see p. 378, above). On the basis of a grammatical approach, Twardowski says that there is a basic distinction between physical, psychical (i.e. mental), and psychophysical acts and their products. The relationship between an act and what results from it is exemplified in the relationship between a verb and the corresponding

23

The sentence is omitted in the German translation. “[. . .] the term ‘subject’ may denote a term in the sentence, but also a concept in the judgement, and things are no different with the terms ‘predicate’ and ‘copula’,” (Twardowski 1900, p. 335 [Germ. transl., p. 446)]). 25 Not translated in Pelc (1979). 24

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substantive as internal complement (see Twardowski 1911, p. 6 [§8] [Engl transl., p. 15]): Physical Mental Psychophysical

Act running judging speaking

Product run judgement speech

Note that Bernard Bolzano is listed as one of the philosophers who have clearly separated actions and products (see Twardowski 1911, §10, p. 6, n. 2). 26 A psychophysical product differs from a mental product by being perceptible to senses, from a physical product because in the corresponding action consciousness is involved. In some cases a psychophysical product becomes expression of a mental product, for instance a sentence is a psychophysical product which expresses a mental product, the judgement. The sentence in this case is the sign of the judgement, while the latter is the meaning of the sentence. According to Twardowski the term ‘judgement’ may have only four meanings: the act of uttering a judgement, the product of such an uttering, the disposition of uttering judgements and the enuntiatio or propositio or Aussage, of which he writes, referring to Relative Truths, “that in Polish I proposed to call sentence (powiedzenie)” (Twardowski 1911, §15, p. 10, n. 1 and n. 2 [Engl. transl., p. 25, n. 2 and n. 3]).27 The judgement exists in the period of time in which someone performs the corresponding act of judging, and, for this reason, is called a non-durable product (cf. Twardowski 1911, §23, p. 14 [Engl. transl., p. 17]). Products which last longer than the act which produces them are called durable products. In any case, a non-durable mental product like a judgement may be fixed in a durable psychophysical product. In this case such a fixing is not direct, but is the result of the fixing of an obligatory go-between, the verbal sentence (cf. Twardowski 1911, §37, p. 25 [Engl. transl., p. 22]), which is the non-durable psychophysical product fixed by the written sentence, that is, on the contrary, a durable psychophysical product. In this way the judgement, which is the meaning of the written sentence, “survives” in it (cf. Twardowski 1911, §33, p. 22 [Engl. transl., p. 20]), and has in it an

26

In the Polish original it is much more evident than in the English translation (p. 25, n. 1) that Twardowski ascribes to Bolzano a clear and correct position on the subject. Other philosophers quoted here by Twardowski are Bergmann, Meinong and Stumpf. 27 The use of ‘judgement’ in this fourth sense is ascribed to àukasiewicz, see id., §44, p. 28, n. 1.

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existence called potential. The fixed sign may at any moment cause the formation of an identical or similar judgement, and hence it lasts as long as a (partial) potential cause of it exists (cf. Twardowski 1911, §34, p. 23 [Engl. transl., pp. 20-21]).28 Non-durable products do not exist in the actual sense separately from the corresponding act, but only in connection with them; separately from the corresponding acts we may only examine them. (Twardowski 1911, §27, p. 18, n. 1 [not translated in Pelc (1979), my emphasis])

But, once they are fixed, judgements assume not only the appearance of durable products, but also of products which possess a certain degree of independence from the acts which produce them. Twardowski explains that this is due to the fact that we tend to attribute to the sign only one meaning, although it causes many judgements in many people. The unique meaning so conceived is no longer a mental product, but the set of the characteristics common both to all the individual judgements caused by the sign and to the judgement which belonged to whoever has fixed it in the sign. Twardowski makes it clear, in a fairly explicit manner, that he considered such sets to be (at least akin to) Husserl’s ideal meanings.29 And the reference to Husserl and to his Logische Untersuchungen is obviously of great importance to us. Given the 10-year distance which divides Actions and Products from Husserl’s work one should not ascribe Twardowski’s anti-psychologistic turn exclusively to Husserl.30 However, in the rather chequered story of Twardowski’s anti-psychologism, Husserl surely played an influential role. 31 Whatever relationship

28 Compare Bolzano: “An object [. . .] through whose idea we want to stimulate in a thinking being some other, associated, idea, is called a sign” (Bolzano 1973, §285, p. 308); cf. also Bolzano WL §285: “So the sight of those signs [of which the word ‘God’ is composed] will awaken at first only the idea of the word: God; but then also the idea of the object which this word denotes.” 29 Cf. Twardowski (1911), §39, p. 26 n. 3 [Engl. transl., p. 26 n. 12]. The reference is to Logische Untersuchungen, II, p. 97. 30 Twardowski is quoted in Husserl (1900-01), Third investigation, p. 287, but mainly Fourth Investigation p. 305 and Fifth Investigation pp. 527-528. In general, for the influence of Husserl upon Twardowski, see, for instance, Ingarden (1938). See also WoleĔski (1989), p. 41 and my (2006). 31 See for instance the long paper by Barry Smith, cf. Smith (1988), chiefly pp. 338-339. A certain influence may have been played by àukasiewicz’ anti-psychologism (cf. Jadacki 1993, p. 191), who may be influenced by Husserl, cf. Simons (1996), p. 319. Cf. also WoleĔski (1989), p. 41. See Schumann (1993) for further remarks on the HusserlTwardowski relationships.

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Twardowski’s in specie meanings may have to Husserl’s ideal meanings, the introduction of this theme leads Twardowski to make a further distinction between substitutive (artefacta) and non-substitutive judgements. Substitutive judgements are those which are not real judgements, but fictitious ones. Twardowski applies the concept to logic: the sentences uttered or written by the logician are not sentences which express or have as meanings judgements which are really uttered by him, but only presented judgements, produced by different acts from actual judging acts. Such is the case of the logician who, to give examples of correct inferences, constructs a correct syllogism made up of false sentences (cf. Twardowski 1911, §44, p. 29 [Engl. transl., p. 24]). In this case the logician does not actually judge: “All triangles are square,” “All squares are round,” All triangles are round,” but only presents the corresponding judgements. The meanings of those sentences, judgments in the “objective” sense just described, as Twardowski says, which have the character of artefacta because they are only presented judgements, are the real subject-matter of logic (see Twardowski 1911, §44, p. 29).32 Twardowski quotes Bolzano once more: The first person to argue this view of the object of logic in detail was Bernard Bolzano. He called the judgements that are rendered independent from the act of judgement in the way defined above ‘Sätze an sich’. Beside the ‘Sätze an sich’ Bolzano also knows the ‘Vorstellungen an sich’, that is presentations thus rendered independent from the act of presentation [Bolzano, Wissenschaftslehre, §§19-23 and §§48-53]. (Twardowski 1911, §44, p. 30, n. 1 (partial Engl. transl., p. 27, n. 16)

Note that the ontological status of Bolzanian propositions-in-themselves is truly different from Twardowski’s judgements. For Bolzano a judgement is the acceptance (Fürwahrhalten) of a proposition-in-itself, while Twardowski’s act of judging is the production of judgements. Unlike Twardowski’s product-judgements, propositions-in-themselves are lektological objects, subsisting in the universe as a certain something, and they are not made independent, nor do they assume the appearance of independent objects. They are independent, not produced 32

The English translation of the paragraph 44, p. 23, lines 32-22 is incomprehensible: “A proposition (actually: judgement) as a product of the action of judging, i.e., of making judgements, is expressed in propositions (actually: sentences) [. . .]. Such sentences thus express propositions (actually: judgements), so that propositions (actually: judgements) are meanings of such propositions (actually: sentences),” and, unfortunately, it goes on like this (my emphases).

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by acts, but rather contained in them as matter or content. They are not judgements abstracted from their psychological context, as seems to be the case in Twardowski. Yet one may think that Twardowski here is overcoming the ontological differences with Bolzano because he is much more interested in pointing out, first, that also Bolzano keeps the two spheres apart (acts/products in Twardowski), secondly that he postulates a difference between judging and the mere presenting of a judgement (or a proposition in Bolzano), as well (see Bolzano 1972, p. 43 [WL §34]; see also Casari 1985, p. 358). As regards Bolzano, such a difference is radical, because a judgement has as matter or content a proposition-initself, which is a lektological object, always complex, enunciating (a) relationship(s) between objects, of which at least one is a quality. A presentation/subjective idea of a proposition, on the contrary, has as matter or content an idea-in-itself of such a proposition, which refers to that proposition. As regards Twardowski, things are more or less the same, if we adopt a Twardowskian terminology, changing relationship into state of affairs and so on, as in the following scheme (where I is the act level, II – the content/product level, III – the object level):

Twardowski says very clearly that the act of presenting a judgement is a different act from the act of judging, and that their products are different (see Twardowski 1911, §44, p. 30, n. 1 [not translated in Pelc 1979]).

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4. Stanisáaw LeĞniewski: Is Truth only Eternal or is it both Eternal and Sempiternal? (1913) LeĞniewski’s fundamental thesis in (1913a) is that a true sentence – i.e. a truth as in Bolzano and in Twardowski – is true both eternally and sempiternally. Three elements are important in LeĞniewski’s paper: a clear formulation of the absoluteness of truth of Twardowskian and Bolzanian origin, transposed here in a nominalist key; the “discharge” of temporality from the verb to the predicate of sentences in a time different from the present, which is considered atemporal (as already seen in Bolzano); and the very remarkable distinction which today we claim is between tokens and types.33 LeĞniewski’s conclusions are: everything which is past does not exist at present; it is not the case that objects accepted by an affirmative sentence s exists only when s is true, and, conversely, it is not the case that s is true only when the object accepted by it exists; every truth is eternal and sempiternal; from the circumstance that we cannot create truth does not follow that we cannot create anything, as the sempiternity of truth does not make free creativity superfluous. To prove the eternity and sempiternity of truth LeĞniewski assumes strong ontological and semantical premises, which reduce the proofs to rather trivial exercises.34 LeĞniewski assumes the logical principle of contradiction and the principle of bivalence – which turns out here to be a special case of the ontological tertium non datur – consequently denying the existence of sentences which are neither true nor false (see LeĞniewski 1913b, §8, Remark I, pp. 350-351 [Engl. transl., pp. 83-85]). Moreover, LeĞniewski assumes that it is always possible to de-indexicalize indexical sentences (of which temporal indexes are a special case) (see p. 388, below). LeĞniewski’s (1913a) allows to add further elements to the theory of truth already drawn up in his previous papers (see LeĞniewski 1912, §§5-16, Remark II, pp. 212-217 [Engl. transl., pp. 31-37] and LeĞniewski 1913c). 35 For LeĞniewski a sentence s is a concrete linguistic object (with meaning!), and as such it has an existence with definite spatio-temporal boundaries. Since an object a may possess a property b if and only if it is present, then that special object which is a sentence may be true only when it is present, that is only in such a case may it symbolize a relation 33

The issue is present in LeĞniewski (1913b), too. However, there is nothing fallacious in them, contra WoleĔski (1993), p. 193. 35 For LeĞniewski’s early semantics, see my papers quoted in the bibliography. 34

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of inherence. Therefore in LeĞniewski’s universe truth-bearers are present objects made of concrete signs or sounds. To the question of how to give a meaning to the expression ‘eternal truth’ in such a perspective, LeĞniewski answers saying that the eternity at issue is metaphorical, just as two sentences uttered respectively today and tomorrow are the ‘same sentences’ only in a metaphorical sense. Hence (8)

For any time t, the sentence ‘Caesar crossed the Rubicon’ could be true at t if someone uttered it, wrote it, etc. in t. (Cf. LeĞniewski 1913a, p. 506 [Engl. transl., p. 97].)

The claim that ‘Caesar crossed the Rubicon’ is eternally true should be understood according to (8). LeĞniewski’s remarks on the meaning to be attributed to the expression ‘to be the same sentence as s’ are linked to the token/type distinction already mentioned. LeĞniewski was to use in his formal systems the expression ‘expression equiform to s’ to denote what is now generally called “a token of s.” I use here the token/type distinction for sake of simplicity, but LeĞniewski’s choice avoids the ontological problems which arise with the token/type terminology. For LeĞniewski, a type would be a very undesirable general object. Here below are summed up the conditions which a sentence of the form “a is b” must satisfy to be true according to LeĞniewski: L1 L2 L3

a is denotative; b is connotative; The object (objects) denoted by a possesses (possess) the properties connoted by b.

To L1-L2 one should add (or, better, put before), in the light of what has been said above, a condition of availability of truth-bearers, so one has (L)

For any sentence “a is b” and any time t, if “a is b” is uttered, written, etc. at t, “a is b” is true at t if and only if the conditions L1, L2, L3 are satisfied.

And therefore the eternity and sempiternity of truth may be re-formulated as follows: (L*) For any sentence s, if s is true at a time t, then is also true any sentence equiform to s uttered, written, etc., at an arbitrary time t´ past or future with respect to t. In one word, in LeĞniewski truth is omnitemporal: if s is true, any time a sentence s´ equiform to s is expressed, s´ is true. The relation of inherence that a sentence symbolizes exists independently from the

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moment in which the sentence is uttered. As was the case in Bolzano, the structure “a is b” is canonically a tenseless structure. For LeĞniewski the present tense in which the copula is expressed does not denote a present subsisting of the relation symbolized by the sentence “a is b,” but it is used as a substitute for a tenseless form which grammars do not contemplate. Consider now the trasformation of an expression in subjectpredicate form uttered (written, etc.) at t: (9)

(a was/will be b) uttered at t œ (a is b in a past/future time with respect to t)

(with b not already expressing or involving a temporal property). The future/past time indicated by ‘will be’/‘was’ is attached salva significatione to b in the sentence in canonical form on the right, where t is the moment in which the sentence on the left is uttered. The need to regard ‘is’ as a tenseless sign follows from the fact that not every sentence expresses a temporal property: temporal properties are on the whole similar to any other property, and they may be predicated or not. 36 Otherwise we would have the paradoxical consequence of getting from the sentence “In June nights are short” uttered in January the nonsensical “In June nights are short in January” (see LeĞniewski 1913a, §4, pp. 512513 [Engl. transl., pp. 101-102]). For LeĞniewski any sentence with indexicals like ‘I’, ‘my’, ‘he’ and so on, of which temporal determinations are special cases, are to be de-indexicalized in a similar fashion as (9) (see LeĞniewski 1913a, §4, p. 509 [Engl. transl., p. 99]). Consider now the examples: (i) (ii) (iii) (iv)

“Caesar “Caesar “Caesar “Caesar

crossed the Rubicon in 49 B.C.” uttered in 1996 will cross the Rubicon in 49 B.C.” uttered in 55 B.C. will cross the Rubicon” uttered in 1913 will cross the Rubicon” uttered in 55 B.C.

Their transformations are: (i*)

(Caesar is crossing the Rubicon in 49 B.C.), uttered in 1996

36 Unlike WoleĔski (1990a), p. 193, I have chosen to put the temporal index on b instead of putting it on the subject: LeĞniewski, unlike Bolzano, does not consider time to be a determination which is not a property; rather he takes the time to be a property just like any other property. I would keep the temporal index on the subject for LeĞniewski’s mature four-dimensional ontology in which (presumably) time-slices appear. See my Logic and Existence in Stanisáaw LeĞniewski (in Italian) MA thesis, University of Florence, 1994/5, chap. iv. On this point see also Smith (1990), §9, pp. 160 ff.

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(ii*) (Caesar is crossing the Rubicon in 49 B.C.), uttered in 55 B.C. (iii*) (Caesar is crossing the Rubicon in a future time with respect to 1913), uttered in 1913 (iv*) (Caesar is crossing the Rubicon in a future time with respect to 55 B.C.), uttered in 55 B.C. The sentences (i*)-(iv*) are different expressions uttered in different times: (i*)-(ii*) are equiform (i.e. they are the “same” sentence in a metaphorical sense), whereas (iii*)-(iv*) are non-equiform (they are not the “same” sentence, not even in a metaphorical sense). In the sentences (i)-(ii) it is not important when they are uttered, while in (iii)-(iv) the moment of utterance is a piece of information which must be supplemented in the transformed sentences (iii*)-(iv*), which are respectively false and true. Consequently, LeĞniewski denies that the “same” sentence from true (iv*) becomes false (iii*). The two sentences (iii)-(iv) are apparently equiform but semantically different sentences, one being true and the other false, and symbolizing different relations of inherence, i.e. R(ab) >55 B.C. and R(ab) >1913 , where a stands for ‘Caesar’ and b for ‘crossing the Rubicon’.37 Now take the sentence (i*): it symbolizes the relation R(ab) 49 B.C.. The situation may be presented schematically as follows:

37 I write ‘R(ab) ’ for ‘R(ab)-which-is-a-future-object-with-respect to-t’, etc. The index >t here should not be seen as a part of the relationship asserted by ‘a is b’. We can look at it as a linguistic means to express a semantic-procedural indication as for ‘where in time’ we have to look for R(ab), like in Scheme 1.

390 55 B.C. Token1 ‘a is b in 49 B.C.’

Arianna Betti

49 B.C. Token2 ‘a is b in 49 B.C.’

1996 Token3 ‘a is b in 49 B.C.’

Token4 ‘a is b in 49 B.C.’

R ( a, b) 49 B. C. In 49 B.C. Caesar crosses the Rubicon. The tokens 1-4 of the true sentence “Caesar is crossing the Rubicon in 49 B.C” all symbolize the relation R(ab) 49 B.C., but are expressed at different times, hence R(ab) 49 B.C. is a future object with respect to tokens 1 and 2 while it is a past object with respect to tokens 3 and 4. That the truth of “a is b” is eternal and sempiternal “metaphorically” (8) means that it is supposed that along the temporal line, beginning from a moment t, a token of “a is b” is expressed at any t´ such as t t t´ (eternity) and at any t˝ such as t˝ d t (sempiternity). Thus the transformations at (9) are to guarantee the (metaphorical) eternity and sempiternity of truth, that is it makes it possible to treat all sentences as if they were of the form “a is b at t,” in this case “Caesar is crossing the Rubicon in 49 B.C.” True transformed sentences of the form “a is b at t” symbolize in any case an object R(ab) t, and the fact that the latter is past, present or future with respect to the moment in which “a is b at t” is uttered does not have any influence on the truth of the token-sentence.

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5. Historical Remarks According to Barry Smith, Twardowski’s view of scientific disciplines in Actions and Products in terms of the durable products of judging acts “finds echoes in LeĞniewski’s view of his own logical systems as collections of concretely existing marks” (Smith 1988, p. 345). Yet Twardowski’s view, simply because the theory of meaning which nourishes it, does not seem to be so near to the nominalism of LeĞniewski’s systems, the origins of which can be more easily discerned in LeĞniewski’s own earlier works (see Betti 1998a). On the contrary Twardowski’s conception might appear as multiplicatio entium sine necessitate for a nominalist, for apparently it adds to the act, the product and the object, also the meaning in specie as abstract product (see Brandl 1998, p. 30). One may rather wonder whether Twardowski’s concept of the judgement as non-durable product, which lasts as long as the action of judging lasts, might have had an influence on LeĞniewski’s position regarding truth-bearers as non-durable concrete objects. Such a hypothesis may seem convincing in the light of passages like the following in Twardowski: Also of some convictions we actually say that they remain for several centuries, and of the thoughts of the wise that they may outlive him. However, what is at issue here is not the actual durable existence of products, but rather the fact that similar actions and products are repeated for many generations [. . .]. Likewise, we say that there are in us concepts, convictions, desires, even if at a given moment the corresponding acts do not occur in us. This means only – as is well known – that there are in us dispositions due to which in the future may occur in us products which are just like the previous ones. So if we speak of the durable existence of products of this kind, what is in question is the repetition of identical actions and products or their potential existence. (Twardowski 1911, §23, pp. 14-15 [the English translation in Pelc omits the footnote])

The same holds for the footnote that follows the words quoted above: This potentiality can be grasped [. . .] if for instance it the “existence” of truths which no one knows yet is spoken about, that is of the “existence” of true judgements which no-one has uttered. Of course what is at issue here is the possibility of uttering such judgements, and what exists are not the judgements, but the possibility of uttering them. (Twardowski 1911, p. 15, n. 1 [the English translation in Pelc omits the footnote])

Nevertheless, at least two remarks should be made.

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1) Twardowski’s position is not clear. If one considers the case of logic, it seems we can reasonably exclude that in Twardowski truth-bearers are token judgements. Even in Relative Truths, although Twardowski says that only judgements, mental expressed acts/products, may be true, his definition of ‘being the same judgement’ could make us doubt that he is thinking of token judgements. In spite of the fact that Twardowski in Actions and Products discusses the expression ‘to be the same’, saying for example that, if we say that the ‘same thing’ happened to two persons, what is interesting in this case is the characteristics common to the two events, “because “the same” cannot take place twice” (Twardowski 1911, §39, p. 26, n. 1 [not translated in Pelc 1979]), the object of logic are, however, the judgements which are rendered independent from the acts which produce them. Whether or not that are meanings similar to ideal objects in Husserl’s sense, 38 according to LeĞniewski they would be general objects, objects that have all the characteristics common to the individual objects with respect to which they are general. One need not remark how much LeĞniewski disagreed with these positions, which he explicitly polemicized against more than once (see LeĞniewski 1913c, §3, Remark V, pp. 140-142; LeĞniewski 1913b, §1, Remark II, pp. 318-320 [Engl. transl., pp. 50-53]). 2) Another point to be noticed which weakens the hypothesis of Twardowski’s influence on the development of LeĞniewksi’s nominalism regards the possibility of fixing a judgement as a non-durable mental product in a durable psychophysical product (sentence). For LeĞniewski, Twardowski’s fixing of truth-bearers would not guarantee at all – either temporarily, or “apparently” – a change in ontological status: in Twardowski’s terminology, we would not be fixing anything, because signs, too, are non-durable products; concerning the judgement 39 uttered yesterday which today is not present and which therefore cannot be true, LeĞniewski writes: I say in my text “I utter a judgement;” if, however, someone would prefer that the “judgements” were “written,” or even “proved,” “felt” or “lived”

38

A negative answer is favoured by Brandl (1998) and Smith (1989). But it seems only BuczyĔska-Garewicz (1980) discusses to any extent the matter. She argues for a rather radical dissimilarity. Though leaving the issue aside here, I incline rather towards the opposite view. 39 I warn the reader that in LeĞniewski (1913a) LeĞniewski uses ‘judgement’ only to adhere to KotarbiĔski’s terminology. We should understand ‘sentence’ any time LeĞniewski writes ‘judgement’.

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– this would not have any influence on the result of my argumentations. (LeĞniewski 1913a, 3, p. 503, n. 1 [Engl. transl., p. 95]) 40

The influence that Actions and Products may have played on LeĞniewski did not regard, I believe, his nominalism, be it the nominalism of his post-1920 logical systems or his nominalistic take on the ontological status of truth-bearers in this period. Twardowski’s work could have instead influenced the transition from LeĞniewski’s 1911 Marty-style theory of truth to his 1912-1913 theory (cf. Betti 1998a) in a negative sense, that is, he might have caused LeĞniewski’s reactions against Twardowski’s notion of meaning, and driven him to abandon judgmentcontents as truth-makers in 1911. At the same time it may have driven LeĞniewski to compare his ideas on truth and theory of meaning with Bolzano’s or with Bolzano-oriented views. Moreover, in Actions and Products Twardowski speaks highly of Bolzano: With the enucleation and the employment of the concepts of ‘Sätze an sich’ and ‘Vorstellungen an sich’ Bolzano surpassed significantly the logicians contemporary to him, as he did with the introduction of the concept of the logical variable and (using the name of Gültigkeit) of the concept of logical value [sic] that play such a great role in modern symbolic logic. Among other matters Bolzano uses these concepts to set up logical relationships and to define the concept of probability. (Twardowski 1911, p. 30, n. 1 [this part of the footnote is not translated in Pelc 1979]) 41

This quotation immediately reminds one of the concept of variation of ideas in Bolzano’s propositions-in-themselves mentioned above. Twardowski calls the concept of Gültigkeit “the concept of logical value,” which Jan àukasiewicz, Twardowski’s pupil and LeĞniewski’s colleague, was to call in Die logischen Grundlagen der Wahrscheinlichkeitsrechnung (1913) more properly “the logical concept of validity of a proposition.” In §24 “Bolzano’s concept of validity,” àukasiewicz says that Bolzano was an [. . .] author whose works have at present acquired great importance, as they well deserve, and who developed opinions that come quite close to mine. (àukasiewicz 1913, §24, p. 52)

40 Therefore, in opposition to what Smith seems to suggest, written sentences are not durable at all for LeĞniewski, nor are matters different in his formal systems. 41 Twardowski writes wartoĞü (value) where àukasiewicz, speaking of the same concept, writes waĪnoĞü (validity). Cf. àukasiewicz (1913), §24, p. 52, n. 20.

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In the note that follows, àukasiewicz writes: For the reference to Bolzano I am indebted to Professor Twardowski; although Bolzano’s principal work [i.e. Wissenschaftslehre, ab] was long known to me, I had previously paid no attention to his remarks on the concept of the ‘validity’ of a sentence. (àukasiewicz 1913, §24, p. 52, n. 20)

Then àukasiewicz goes on to comment Bolzano’s examples, dedicating many pages to the difference between his own concept of logical value and Bolzano’s concept of validity.

6. Three Sempiternities of Truth The similarity of Bolzano, Twardowski and LeĞniewski’s positions is undoubtedly a very close one. If we consider, however, the difference in the objects which play the role of truth-bearers, LeĞniewski’s nominalist position on sempiternity (L*), called here omnitemporal, turns out to be certainly different from those of Bolzano’s and Twardowski’s (B), (TW), which I would call atemporal. By combining Twardowski’s positions in Relative Truths and in Actions and Products, we obtain a sufficiently systematic picture that may be used to compare his views with those of the others. However we may interpret Twardowski’s abstract meanings, they have ontological features which do not sit well with Bolzano’s platonistic absolutism. Twardowski’s absolutism, I would say, may be labelled “Aristotelian,” the term being suggested by the characteristics of meanings which are the object of logic, artefacta-judgements/in specie meanings. Hence we have propositions-in-themselves (lektological beings) in Bolzano, judgements (species inferred by abstraction) in Twardowski and concrete occurrences of sounds uttered (or of signs written) at different times in LeĞniewski. For LeĞniewski, as already seen, neither type sentences nor the “judgements” that a sentence expresses (be they tokens or types), but only token expressions may fulfil such a function. It is not enough for two sentences to have the same truth-maker to be “the same sentence,” they would be so only in a metaphorical sense. For LeĞniewski there is no “judgement” as the meaning of a sentence s. Tthe solution he gives to the problem of the ontological status of his truth-bearers allows him to keep a wholly consistent position regarding the sempiternity of truth. His position is different from Bolzano’s and Twardowski’s, since it is an original interpretation of sempiternalism in a nominalistic vein.

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7. Created Truth? As said above, LeĞniewski’s Is Truth . . . was a reply to KotarbiĔski’s The Problem of the Existence of the Future. In it KotarbiĔski claims that all truths are eternal, while only some of them are sempiternal (see KotarbiĔski 1913). This means that: (a)

judgements (in KotarbiĔski’s terminology) which are true at a time t are also true at every time t´ such that t < t´ (eternity).

But this does not mean that: (b)

all of them are likewise true at every time t˝ such that t˝ > t (sempiternity).

(a) and (b) apply to judgements false at a time t, too. KotarbiĔski’s definition of a true judgement is the following: (K)

A judgement g which accepts an object P is true at t if and only if P exists at t.

For KotarbiĔski, non-sempiternal truths are those regarding events in which free human creativity plays a fundamental role. Suppose that P is an object created by a human action; P does not exist until t´, the time at which it is created; if g is the judgement which accepts P, g is not true until t´. But if g is false, that g is false is eternally true (a), and it would not be possible to create P. But since we supposed that it was possible to create P in t´, in a time t such that t < t´, g must be neither true nor false. KotarbiĔski’s real premise appears to be the principle that the creation of the object R(ab) involves the creation of the truth of “a is b” (and vice versa). Such an assumption is similar to the so-called “bridgeprinciple” which Morscher points out when writing on Marty’s theory of truth. Marty infers temporality of judgements starting from the temporality of the objects which the judgements refer to.42 Now a question arises. Since for LeĞniewski the availability of a token of “a is b” is a necessary condition for this token to be true, should we say that he introduces a ‘creative’ element in his theory of truth, given that the truth-

42 I am not able to justify the possible hypothesis of Marty’s influence on KotarbiĔski in that period, and most of all of the possible role of Marty’s doctrine of irrealia which go in and out of time: KotarbiĔski had suggested the reading of Marty’s Untersuchungen, but this happened in 1920 (from Jan WoleĔski’s letter of 26.1.1996). For Marty’s theory of truth, see Morscher (1990).

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bearer must be present and therefore created? Actually the token may possess any property, truth included, only starting from the moment it is created. Therefore it seems we need to understand ‘creating a truth’ in another, modified, sense, which LeĞniewski does not discuss thoroughly. As to LeĞniewski, one can say that s is true continuously only in a metaphorical sense, as previously observed, because a true sentence is an object with a determined spatio-temporal dimension: there is not one sentence s uttered many times, but many true sentences equiform to s as uttered. But then, one may say, if truth-bearers must be created, they must be created together with their truth. As a matter of fact, (see scheme 1) truth as a property of the token 2 (truth 2 ) is by no means truth as a property of the token 3 (truth 3 ): truth, for LeĞniewski, is a relationship between a sentence and an object, called symbolic relation, and “to be true” for a sentence, is to have (in LeĞniewski’s terminology) symbolic function, the property of symbolizing something. Truth 2 is a relation between the token2 and the object R(ab), whereas truth3 is a different relation between the token 3 and the same object R(ab). Therefore, truth is only metaphorically uncreated, just as truth is metaphorically eternal and sempiternal. But one can also invoke a definition of ‘created truth’ according to which creating a truth would mean in LeĞniewskian terms to deny the claim (L*). Hence it would mean to have the token p t come out false, but the token p tƍ true (see LeĞniewski 1913a, 4, p. 506 [Engl. transl., p. 97]). When we create p tƍ, we cannot make it possess the property of truth if p t was or will be false. Thus it follows that what we really cannot create in such a situation is a token p tƍ with symbolic function. The truth (symbolic function) of the truth-bearer “a is b” is to be brought back to the existence of the relation of inherence between the object a and the property b with a definite spatio-temporal dimension. Coming back to the example of Caesar and the Rubicon of the scheme 1, what one cannot do is create a token 5 which does not symbolize R(ab) 49 B.C.. The creation of the object R(ab) is therefore an act which is completely independent of the creation of a token of “a is b.” The object R(ab) remains also the sole object independent of the creation of a token of “a is b” and its truth: neither the symbolic function (property of tokens) nor the symbolic relation (between tokens and truth-makers) are, properly speaking, independent of the time of utterance (contra WoleĔski 1990a, p. 196). Thus in LeĞniewski the truth of a sentence and the objectivity and omnitemporality of that truth depend in a very strong sense uniquely on the existence of the truth-maker of the sentence. On the contrary, one could say that in Bolzano the existence of the truth-maker guarantees the

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truth of a proposition, but objectivity and atemporality seem guaranteed rather by the ontological status (objective and atemporal) of the truthbearer as can be deduced from the argument that when it is said that a proposition is relatively true, what is actually meant by ‘proposition’ is the (elliptical) linguistic sentence which expresses the proposition-initself. The same seems to apply also to Twardowski.

Vrije Universiteit Amsterdam Faculteit der Wijsbegeerte De Boelelaan 1105 1081 HV Amsterdam e-mail: [email protected]

REFERENCES Albertazzi, L., M. Libard and R. Poli, eds. (1996). The School of Franz Brentano. Dordrecht: Kluwer. Berg, J. (1962). Bolzano’s Logic. Stockholm: Alqvist & Wiksell. Betti, A. (1998a). Il rasoio di LeĞniewski. Rivista di Filosofia 79 (1), 89-117. Betti, A. (1998b). De Veritate: Another Chapter. The Bolzano-LeĞniewski Connection. In: K. Kijania-Placek and J. WoleĔski (eds.), The Lvov-Warsaw School and the Contemporary Philosophy, pp. 115-137. Dodrecht: Kluwer. Betti, A. (2004). àukasiewicz and LeĞniewski on Contradiction. Reports in Philospohy 22, 247-271. Betti, A. (2006). The Strange Case of Savonarola and the Painted Fish – On the Bolzanization of Polish Thought. In: A. Chrudzimski and D. àukasiewicz (eds.), Actions, Products, and Things. Brentano and the Polish Philosophy, pp. 115-136. Frankfurt: Ontos. Bolzano, B. (1972). Theory of Science. Edited and translated by R. George. Berkeley & Los Angeles: University of California Press. (Partial English translation based on Wissenschaftslehre, Leipzig, 1929-1931). Bolzano, B. (1973) Theory of Science. Edited by J. Berg. Translated by B. Terrell. Dordrecht: Reidel. Bolzano, B. (1985-1994; referred to as “WL” in text). Wissenschaftslehre. Edited by J. Berg. In: Bernard-Bolzano-Gesaumtausgabe, vol. I, pp. 11-14. Stuttgart and Bad Cannstadt: Frommann Verlag/Günther Holzboog, 1985-2000. Brandl, J. (1998). Twardowski’s Distinction between Actions and Products. In: K. Kijania-Placek and J. WoleĔski (eds.), The Lvov-Warsaw School and the Contemporary Philosophy, pp. 23-34. Dodrecht: Kluwer. BuczyĔska-Garewicz, H. (1980). Twardowski’s Idea of Act and Meaning. Dialectics and Humanism 3, 153-164. Casari, E. (1985). L’universo logico bolzaniano. Rivista di Filosofia 76 (3), 339-366.

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Casari, E. (1992). An Interpretation of Some Ontological and Semantic Notions in Bolzano’s Logic. In: Bolzano’s Wissenschaftslehre 1837-1987, pp. 55-105. Firenze: Olschki. Coniglione, F., R. Poli and J. WoleĔski, eds. (1993). Polish Scientific Philosophy: The Lvov-Warsaw School. PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 28. Amsterdam: Rodopi. Husserl, E. ([1896] 1979). Besprechung von H. Cornelius, Versuch einer Theorie der Existentialurteile (München: Rieger, 1894). In: Aufsätze und Rezensionen (1890-1910) (Husserliana, vol. XXII), pp. 357-380. The Hague: M. Nijhoff Publ. Husserl, E. (1900-1901). Logische Untersuchungen. Halle a.d.S.: Max Niemeyer. (Critical edition: Husserliana, vol. XVIII, XIX/1, XIX/2). Ingarden, R. (1938). DziaáalnoĞü naukowa Kazimierza Twardowskiego [Scientific Activity of Kazimierz Twardowski]. In: Kazimierz Twardowski. Nauczyciel  uczony  obywatel [Kazimierz Twardowski. Teacher  Scholar  Citizen], pp. 13-30. Lwów: PTF. Jadacki, J.J. (1993). Kazimierz Twardowski’s Descriptive Semiotics. In: Coniglione et al., eds. (1993), pp. 191-206. KotarbiĔski, T. (1913). Zagadnienie istnienia przyszáoĞci [The Problem of the Existence of the Future]. Przegląd Filozoficzny 16, 74-92. English translation: Prolegomena to Three-Valued Logic, The Polish Review 13 (1968), 7-22. LeĞniewski, S. (1912). Próba dowodu ontologicznej zasady sprzecznosci [An Attempt at a Proof of the Ontological Principle of Contradiction]. Przegląd Filozoficzny 15, 202-226. English translation: An Attempt at a Proof of the Ontological Principle of Contradiction. In: LeĞniewski (1991), vol. 1, pp. 20-46. LeĞniewski, S. (1913a). Czy prawda jest tylko wieczna czy teĪ i wieczna i odwieczna? [Is Truth Only Eternal or Is It Both Eternal and Sempiternal?] Nowe Tory 8, 493-528. English translation: Is All Truth Only True Eternally or Is It also True without a Beginning? In: LeĞniewski (1991), vol. 1, pp. 86-114. LeĞniewski, S. (1913b). Krytyka logicznej zasady wyáączonego Ğrodka [A Critique of the Logical Principle of the Excluded Middle]. Przegląd Filozoficzny 16, 315-352. English translation: The Critique of the Logical Principle of the Excluded Middle, in: LeĞniewski (1991), vol. 1, pp. 47-85. LeĞniewski, S. (1913c). Logiþeskie rassuždenija [Logical Studies]. St. Petersburg: A. SmoliĔski. Polish translation of the first part: Próba dowodu ontologicznej zasady sprzecznoĞci, Filozofia Nauki 2 (2) (1994), 117-147. (Page references are to the Polish translation.) LeĞniewski, S. (1991). Collected Works. 2 vols. Dordrecht & Warszawa: Kluwer & PWN. àukasiewicz, J. (1913). Die logische Grundlagen der Wahrscheinlichkeitsrechnung. Kraków. English translation in: Selected Works, ed. L. Borkowski (Amsterdam & Warszawa: North-Holland Publishing Company & PWN, 1970), pp. 16-63. (Page references are to the English translation.) Pelc, J., ed. (1979). Semiotics in Poland 1894-1969. Dordrecht & Warszawa: D. Reidel & PWN. Morscher, E. (1990). Judgement-Contents. In: K. Mulligan (ed.), Mind, Meaning and Metaphysics: The Philosophy and Theory of Language of Anton Marty, pp. 181-196. Dordrecht: Kluwer. Schumann, K. (1993). Husserl and Twardowski. In: Coniglione et al., eds. (1993), pp. 41-58. Simons, P.M. (1991).Verità atemporale senza portatori di verità atemporali. Discipline filosofiche 2, 33-47.

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Simons, P.M. (1992). Philosophy and Logic in Central Europe from Bolzano to Tarski. Dordrecht: Kluwer. Simons, P.M. (1994). Absolute Truth in a Changing World. In: J. Hintikka, T. Czarnecki, K. Kijania-Placek, T. Placek, A. Rojszczak (eds.), Philosophy and Logic – In Search of Polish Tradition – Essays in Honour of Jan WoleĔski on the Occasion of His 60th Birthday, pp. 37-54. Dordrecht: Kluwer. Simons, P.M. (1996). Logic in the Brentano School. In: Albertazzi et al., eds. (1996), pp. 305-321. Smith, B. (1988). Kasimir Twardowski: An Essay on the Borderlines of Ontology, Psychology and Logic. In: Szaniawski, ed. (1988), pp. 313-375. Smith. B. (1990). On the Phases of Reism. In: WoleĔski, ed. (1990b), pp. 137-183. Szaniawski, K., ed. (1988). The Vienna Circle and the Lvov-Warsaw School. The Hague: Nijhoff. Twardowski, K. ([1894] 1982). Zur Lehre vom Inhalt und Gegenstand der Vorstellungen – Eine psychologische Untersuchung. Wien: Philosophia Verlag. Twardowski, K. (1900). O tak zwanych prawdach wzglĊdnych [On the So-Called Relative Truths]. In: KsiĊga pamiątkowa Uniwersytetu Lwowskiego ku uczczeniu piĊüsetnej rocznicy fundacji JagielloĔskiej uniwersytetu krakowskiego [Commerative Book of Lvov University in Commemoration of the 500 th Anniversary of the Jagiellonian Foundation of Cracow University]. Lwów: Senat Akademicki Uniwersytetu Lwowskiego. Reprinted in: Wybrane pisma filozoficzne [Selected Philosophical Writings] (Warszawa: PWN, 1965), pp. 315-336. German translation: Über sogenannte relative Wahrheiten. Archiv für systematische Philosophie 8 (1902), 415-447; reprinted in: D. Pearce and J. WoleĔski (eds.), Logische Rationalismus: Ausgewählte Schriften der Lemberg-Warschauer Schule (Frankfurt: Athenäum, 1988), pp. 38-58. Twardowski, K. (1911). O czynnoĞciach i wytworach: Kilka uwag z pogranicza psychologii, gramatyki i logiki [On Actions and Products: Some Remarks on the Border Area of Psychology, Grammar and Logic]. Kraków: Gubrynowicz i syn. English translation of some fragments: Actions and Products: Comments on the Border Area of Psychology, Grammar and Logic, in: Pelc, ed. (1979), pp. 13-27. WoleĔski, J. (1989). Logic and Philosophy in the Lvov-Warsaw School. Dordrecht: Kluwer. WoleĔski, J. (1990a). àukasiewicz, KotarbiĔski and Many-Valued Logics. In: WoleĔski, ed. (1990b), pp. 191-197. WoleĔski, J., ed. (1990b). KotarbiĔski: Logic, Ontology and Semantics. Dordrecht: Kluwer. WoleĔski, J. (1993). Metamatematyka a epistemologia [Metamathematics and Epistemology]. Warszawa: PWN. WoleĔski, J. and P.M. Simons (1988). De veritate: Austro-Polish Contributions to the Theory of Truth from Brentano to Tarski. In: Szaniawski, ed. (1988), pp. 391-443.

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Artur Rojszczak FROM THE ACT OF JUDGING TO THE SENTENCE: THE TRUTH-BEARER AND THE OBJECTIVISATION OF TRUTH

1. An Introduction 1. It seems that the following questions have not yet been answered satisfactorily. The first question is: “How it is possible to go from the act of judging as a truth-bearer to the sentence as a truth-bearer?” And the second is: “Why should sentences and not, for example, judgments or something else play the role of truth-bearers?” Choosing sentences as truth-bearers, as Tarski did in his (1933) definition of truth, has a deeper historical and philosophical basis than is commonly believed. Tarski’s choice is not only the result of, so to speak, a “referential theory of meaning,” which is always formulated in a definition of a sentence as “an expression of language with its meanings.” These deepest grounds I would see as lying in the tradition of the Lvov-Warsaw School, specifically in Twardowski’s views about meaning and judgment, but also in the views on language and truth held by Twardowski’s pupils and Tarski’s teachers: Jan àukasiewicz, Tadeusz KotarbiĔski and Stanisáaw LeĞniewski. Tarski himself refers to these figures directly (see e.g. Tarski 1933, footnotes: 1, 3, 5; p. 17) and this entitles me to state that the tradition which I mention goes even further back in time – namely to the tradition of the Brentano School (or, if one will, to the tradition of Brentanian philosophy), to Franz Brentano himself and to Bernard Bolzano, whose ideas have been rediscovered only in our own century (e.g. the idea of variation in Husserl’s phenomenology or in Tarski’s semantics). What I am interested in, therefore, is the problem of the truth-bearer from Bolzano to Tarski. 2. To paraphrase Jan WoleĔski’s statement that the Brentanian tradition has been the natural context for research in semantics in Poland (see e.g. WoleĔski 1994, p. 85), I propose that the Brentanian tradition has

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 401-419. Amsterdam/New York, NY: Rodopi, 2006.

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been the natural context for research in the philosophy of the entire Lvov-Warsaw School and especially for the theory of truth and for the above-mentioned turn from mental acts to language as the locus of truthbearers. I see the natural context of Brentanian philosophy for research in truth theory in Poland not only in Brentanian and post-Brentanian theories of intentionality, but also:

• in theories of judgment (from Brentano, through Meinong, • • • • •

Husserl, and Reinach to Twardowski, àukasiewicz, CzeĪowski and Ajdukiewicz); in the criticism of psychologism in post-Brentanian philosophy (Husserl, Twardowski, àukasiewicz, KotarbiĔski); in the philosophy of language (Brentano, Marty, Twardowski, LeĞniewski) including the theory of meaning (Bolzano, Brentano, Twardowski, Reinach, àukasiewicz, Tarski, Ajdukiewicz); in truth theories and in commentaries on these theories (on the part of Brentano, Bolzano, Husserl, KotarbiĔski, LeĞniewski, àukasiewicz, Tarski); in reforms and in the very discovery of logic in the modern sense (which includes not only the establishment of mathematical logic, but also in reforms such as that of Brentano); in the problem of the objectivisation of the knowledge of the empirical subject (Bolzano, Brentano, Husserl, Twardowski).

This last issue is in fact the main topic of the present paper. 3. It is not my aim to engage in a purely historical analysis or exegesis of published and unpublished texts relevant to my central problem. Rather, what I am going to do is to collect and put in order arguments for an account of the nature of the truth-bearer. The ordering of arguments sometimes has a historical basis. It may also, however, lie in the intrinsic rationality of given modes of philosophical speculation. The collection of arguments is arrived at by raising different questions in reference to the truth-bearer problem. These were in some cases raised by authors within the Brentanian tradition – as, for example, the question: “How is objective knowledge on the part of an empirical subject possible?” (see Husserl 1891; see also Dallas 1984). The present text is more a summary of research carried out in the relevant contexts of Brentanian philosophy than a rational-historical analysis of sources. 4. Note that the assumption of the relevance of the problem of the truth-bearer in the theory of truth served as a ground and presupposition for the kind of research I tried to do. As a result of this, reference to

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Brentano’s work on the truth-bearer problem appears all the more relevant and perhaps decisive. I thus reject Lorenz Puntel’s view expressed in the following passage: Der Ausdruck ‘Wahrheitsträger’, der bekanntlich eine wörtliche Wiedergabe des englischen Ausdrucks ‘truth bearer’ und ein geläufiger Terminus auch der deutschsprachigen wahrheitstheoretischen Diskussion ist, sei hier nur mit dem Hinweis verwendet, daß er eigentlich aufgrund seiner zu Scheinfragen verleitenden Konnotationen besser vermieden werden sollte. (Lorenz 1987, pp. 14-15)

I think that “redundant questions” in the Brentano School often led to a decision which further built up the ground for a theory of truth (Brentano’s reism should be mentioned here).1 I hold the same view with respect to the problem of the truth-maker. The classical concept of truth is based somehow on the belief that truth has something to do with an “extra-linguistic reality.” That belief was expressed in the Brentano School not only in the semantics of the school but also, for example, in the problem of the object, content and subject of mental acts, as well as in the problem of states of affairs or Sachverhalte.

2. The Objectivity of Knowledge: Some Remarks and Definitions I will not enter into a detailed explanation of the concept of objective knowledge. I would merely like to show some ways of understanding this concept depending on its definition only. 1. Objectivity is to be understood in a narrow sense as independence of a subject (consciousness). Firstly, if we speak about the objectivity of knowledge, we think about the objectivity of products (results) of certain knowing acts, where by the knowing act we understand a physical event or a physical process which depends on the subject (i.e. the act is an act of some subject and stands in a real causal relation to this psychological subject). A product of knowledge is objective if the product is independent of each and every subject (and of each and every consciousness). This means either a) that different subjects with cognitive disposition have access to the objective products of cognition (this is what is supposed by K.R. Popper2 in his 1 For the dependence between Brentano’s reism and his theory of truth, see e.g. Rojszczak (1994). 2 The appearance of K.R. Popper’s name seems to be in contradiction with what I said above, that this text refers to the period from 1837 (first edition of Wissenschaftslehre of

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concept of the third world and by B. Bolzano in his account of the world of sentences in themselves), or b) that the products of cognitive acts become independent e.g. they live independently with respect to consciousness (as in Popper’s third world). Secondly, and this is Brentano’s view, to guarantee the objectivity of an act of knowledge (as contrasted with the product of the act) we need some objective condition of knowledge. In Brentano, the role of such an objective condition is played by the identity of the knowing subject with the object of knowledge (at least in the case of empirical knowledge). Thirdly, objectivity in the sense of subject independence may also be called intersubjectivity. In reference to the act of cognition this means that different subjects can repeat in exactly similar fashion the same acts of cognition in such a way as to yield the same or similar results. And what is more, they each should be in a position to give a recipe for a given kind of act and to give information about these results. An example in this regard is provided by Brentano’s “experience of evidence” (Evidenz) or by Husserl’s “notion of insight” (Einsicht). In reference to the results of knowledge, we can understand objectivity as the equiaccessibility of the products to all relevantly situated knowing subjects. 2. In the broadest sense, we can understand the concept of objectivity as meaning absoluteness (as independence of time). In regard to the products of acts of cognition, absolute knowledge is either timeless (like Bolzano’s sentences “an sich”) or unchangeable in the course of time. Although absolute acts of knowledge are usually attributes of a god or of some other “objective” or “transcendental” consciousness, the demand of absoluteness for the products of cognitive acts seems more appropriate to us.3 3. It must be said that in the works of the above authors objective knowledge is in the above-given senses not always, or perhaps even never, the same concept as true knowledge. On the one hand, not every Bernard Bolzano) to 1933 (the year of the first edition of Alfred Tarski’s work). Despite this fact and regarding a lot of convergences in interpretations of Popper’s third world with Bolzano’s world of sentences in themselves; and first of all regarding the fact that Popper’s theory is better known than Bolzano’s, I decided to mention Popper’s ideas (Popper 1972) for clarity. 3 It is possible, of course, to reject the thesis that Bolzano’s sentences in themselves can be interpreted as products of cognitive acts. This possibility  and the possibility of rejecting some other theses  does not change the core of our problem. Usually, in a discussion about the theory of truth one asks if the truths are according to the theory absolute or changeable in the course of time. But where we are talking about the cognition of truth or true knowledge, we ask for absolute products of cognitive acts.

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case of objective knowledge is a case of true knowledge. In Bolzano, there are absolute false sentences (Falschheiten an sich). In Popper’s third world there are also hypotheses and theories that are false. On the other hand, not every case of true knowledge is a case of objective knowledge. Brentano assumes that there can be judgments which are true about the external world but which are such that these judgments will never be correct, i.e. able to be proved by deduction or by the experience of evidence. A condition for the appearance of the latter is the identity of subject with object, and this cannot take place in the context of our cognition of the external world. 4. In the above-mentioned senses, the concept of objectivity is opposed to that of relativity (i.e. relativity either to the knowing subject or to time). With reference to this opposition and to the philosophical tradition, we might also mention the issue of independence in relation to space. It is hard to say what such a metaphor might mean, but one can demand that our knowledge and the corresponding truths remain unchangeable not only with respect to a subject of cognition and a point in time, but also under “different (spatial) circumstances.” So, the metaphorical expression ‘independent of the circumstances’ will give us at the moment the explanation of, so to speak, space-independence. It could be that space-independence is less relevant to our concerns than are the other ones. For example, if we take psychological acts as truthbearers, then the acts are always described as non-spatial. On the other hand, however, some philosophers, like Twardowski (1900), insisted on the spatial indexicality of the judgment in the logical sense (“I think it is raining in Kraków today”).

3. How Knowledge Gets Objectivised By the ‘objectivisation of knowledge’ I understand viewing a given case of knowledge as possessing one of the conditions specified above. Similarly, by the objectivisation of truth I will understand assuming any condition which refers to the knowing subject, the truth-bearer, the truth-maker and other conditions which give us the reasons to describe the truth of a given theory of truth as objective in the above senses. Knowledge can be objectivised by assuming the following conditions (see Table 1):

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1. Objectivisation as absolutisation; 1.1. with regard to the subject of knowledge: (i)

the assumption of a timeless subject (one which does not exist in time), as for instance God or some transcendental consciousness whose knowledge-acts, or the results of such acts, are outside time; this is a less interesting possibility for us because we are most interested in empirical subjects; (ii) the assumption that we do not need any knowing subject at all who performs an act at a special point in time, because of the existence of knowledge without an empirical subject (Popper, Bolzano); 1.2. with regard to the act of knowledge: (i)

the assumption of timeless media of knowledge (tools): these may be ideal contents of judgment, meanings, senses or logical laws – tools which guarantee from the beginning of the act of knowledge its independence of time; so, the media of knowledge which by nature exist in time express something outside of time; (ii) the conception of the media of knowledge as being temporally instantiated counterparts of timeless media; especially in the case of so-called indirect knowledge there is a danger of using tools of knowledge which by their nature lead us to false results; despite the fact that we use judgments and language which are in time, they exemplify something outside of time; (iii) in another case it could be the assumption that we have at our disposal permanent or eternal products of the act of knowledge, such as sentence-types, propositions or other similar ideal meanings, theories and so on; these exist in time but we can recognize them and use them at each point in time (see Smith 1989); 1.3. with regard to the object of knowledge: (i)

the assumption of timeless objects of knowledge, such as numbers, logical objects, sentences themselves; if we “really” know such objects then it is impossible that at another point in time they could be recognized in

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another way; even if they are never known, knowledge of them is guaranteed to be time-independent; (ii) the assumption that our knowledge has validity at any point in time (a kind of a priori knowledge); 1.4. with regard to the conditions of knowledge: (i)

the assumption of a timeless condition of knowledge, such as Brentano’s identity of the subject and the object of knowledge; (ii) the assumption of a condition which is valid at any point in time, such as Husserl’s insight, as this is seen to be involved in our knowledge of mathematical inferences; (iii) the assumption of intersubjectivity which can be understood in such a way that at any point in time it is possible to repeat the act of knowing in such a way that it results in the same product; 1.5. combinations of 1.1. to 1.4. 4 2. Similar possibilities for the objectivisation of knowledge with regard to space look as follows: 2.1. assumptions in relation to the subject of knowledge, as mentioned above, are less interesting because the subject of knowledge is always described as non-spatial; 2.2. with regard to the act of knowledge: (i)

the assumption of the timelessness of the media of knowledge; it should be mentioned here that we can add to the set of physical acts, which are described always as non-spatial by their very nature, such entities as Bolzano’s sentences in themselves, logical laws and so on; (ii) the assumption of instances (as in the case of absolutisation  see 1.2.(ii) above), and (iii) the assumption of permanent products as media of knowledge, such as theories and type-sentences; 4

It should be mentioned here that the list given above is incomplete. It does not mention which of the assumptions can refer only to acts of knowledge and which only to the products of these acts. Also not all of the various combinations are possible. Some solutions of the problem may be seen as archaic. But this does not change the fact that such treatments were arrived at in the past. Or one can see it as they were accepted.

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2.3. regarding objects of knowledge: (i)

the assumption of objects which are outside of space, such as numbers, sentences in themselves and  something which seems to be important  the mind or psyche; (ii) the assumption of some kind of knowledge that is valid a priori for any object under any circumstances; 2.4. with regard to the conditions of knowledge: (i)

timeless conditions, such as psychological conditions characterized in terms of evidence (for example, the statement that the presence of evidence makes an act of knowledge objective), insight and so on; (ii) a kind of condition of intersubjectivity in the above sense; 2.5. combinations of 2.1.-2.4. 3. Objectivisation in the narrow sense, e.g. independence of a subject (or of consciousness): 3.1. with regard to the subject of knowledge: (i)

the assumption that there is no empirical subject (but only a transcendental one, or something similar); (ii) the monistic assumption, which is more interesting because it refers to an empirical subject, assumption that there is only one consciousness; (iii) the assumption of solipsism; then the problem of the independence of knowledge from a subject does not arise at all. 3.2. Objectivisation as the independence of a subject (or of consciousness) with regard to the object of knowledge: (i)

the introduction of entities which can be used as epistemic tools and which do not depend on any sort of consciousness for their existence; examples would be Bolzano’s sentences in themselves, the laws of logic and so on; (ii) the assumption of epistemic tools which are products of acts of knowledge and which continue to exist without any subject, such as sentence-types or theories.

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3.3. Objectivisation with respect to the object of knowledge: (i)

the assumption of objects which are independent of consciousness or which are completely outside the realm of consciousness; it should be mentioned here that by such an assumption we should decide the problem of physical objects; (ii) the assumption of the intelligibility of the object of knowledge; then these objects are cognizable for any subject; maybe it is worth mentioning that this is one of the most commonly held assumptions in the theory of knowledge; 3.4. Objectivisation with regard to other conditions of knowledge: (i)

a condition which is independent of any consciousness, such as Brentano’s identity of the subject and the object of knowledge (so, for Brentano, for any species which has the ability to make judgments there is a class of judgments which fulfils this condition) or like Husserl’s epoche; (ii) the assumption that every individual of a species with the ability to gain knowledge has the same abilities, such as, for instance, Husserl’s assumption to the effect that each of us might, after appropriate training, be capable of achieving phenomenological insight; 3.5. combinations of 3.1.-3.4.

4. Ways of Objectivising Truth As we can see in what follows, the possibilities of objectivising truth are analogous to those for objectivising. The objectivisation of truth can be achieved with the help of the following assumptions (see Table 2): 1. Absolutisation: 1.1. with respect to the subject: (i)

the assumption of a timeless subject who cognizes a given truth (again, this is uninteresting if we are interested in cases where an empirical subject is involved);

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(ii) the assumption that truth can exist even without any epistemological subject or an assumption of truths which are able to be grasped by any subject (for example, sentences in themselves or true products of cognitive acts that belong to Popper’s third world). 1.2. Absolutisation with regard to the truth-bearer: (i)

the timelessness of the truth-bearer (for instance, sentences in themselves in Bolzano’s theory); (ii) the assumption that at any point in time we can exemplify timeless truth-bearers by means of an act of knowledge; (iii) the assumption of the existence of permanent products such as, for example, sentence-types, which can be exemplified at any point in time; (iv) the indexicality of the truth-bearer with respect to time. 1.3. Absolutisation with respect to truth-makers can be achieved by: (i)

the assumption of the timelessness of that in the world which decides the truth value of the truth-bearer (for instance, numbers, logical objects in Husserl and Frege or sentences in themselves in Bolzano – conceived as truth-makers); (ii) the thesis to the effect that, although what decides the truth value of the truth-bearer is in time, it is unchangeable (such as physical laws - this is Brentano’s and Marty’s view) (see Smith 1994, especially Ch. 4); (iii) the assumption that there are no truth-makers of whatever truth-bearers one likes; for instance, the assumption that the truth value of analytical sentences is independent of the world, or if something is true at one point in time, then independently of everything that happens, it will remain true; (iv) time-indexicality of the truth maker (“Today it is raining in Kraków”). 1.4. Other conditions of the time-independence of truth are, for example: (i)

the time-independence of the criteria of truth (such as non-contradiction) and

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(ii) criteria which are valid at any point in time (such as those mentioned above, the identity of subject and object in Brentano); 1.5. And, of course, combinations of 1.1.-1.4. 2. The possibilities of objectivising truth with regard to space: 2.1. the assumption with respect to the subject, for instance a nonspatial subject (e.g. the Brentanian psyche); 2.2. assumptions with respect to the truth-bearer: (i)

a truth-bearer that is outside of space (because it belongs to the domain of what is mental in something like the Cartesian sense, or because the truth-bearer is something like a sentence in itself in Bolzano’s sense); (ii) the assumption of the possibility of exemplifying timeless truth-bearers – for instance, an act of judging (which exemplifies) and a sentence in itself (which is exemplified); (iii) products which are independent of space  such as, for example, sentence-types; (iv) space indexicality; 2.3. Assumptions with respect to the truth-maker: (i)

truth-makers outside of space (numbers, sentences in themselves, logical objects, mental acts); (ii) any truth-makers (the idea of variation or the idea of fulfilling with any sequence of objects from the universe can be examples); (iii) the space indexicality of truth-makers (for instance, by quantification); (iv) truth-makers which fill all of space (like maximal situations); 2.4. other conditions: (i) outside of space, or (ii) criteria valid under any circumstances consistency, coherence and so on);

(identity,

2.5. combinations of 2.1.-2.4. 5 5

Some of the expressions used are very clearly metaphorical, for instance “a truth-maker which fills all of space.” I think, however, that if they were reformulated as expressions

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3. Objectivisation of truth as independence of a subject: 3.1. with respect to the subject: (i)

the assumption of a transcendental subject (uninteresting); (ii) the assumption that we do not need a subject of knowledge at all (truth exists in and of itself); (iii) monism; (iv) solipsism; 3.2. with regard to the truth-bearer: (i)

the truth-bearer is independent of consciousness (it is a matter of ideal contents, sentences in themselves); (ii) the assumption of the exemplification of truth-bearers which are independent of a subject; (iii) the assumption of permanent products, such as sentence-types; (iv) the indexicality of the relation to a subject. 3.3. Objectivisation (as independence of a subject) with regard to the truth-maker: (i)

the assumption that truth-makers are independent of consciousness (this includes not only objects like sentences in themselves or numbers but, above all, objects from the external world; this is one of the most commonly held assumptions in classical theories of truth); (ii) the assumption of the intelligibility of truth-makers (this is quite a strong assumption: what decides the truth value of the truth-bearer is capable of being known by everyone); (iii) the indexicality of truth-makers with regard to the subject (“His token about what he sees is true”). 3.4. Other conditions, such as, for example, (i) subject independence, or (ii) criteria of truth valid for any subject (coherence, correspondence and so on);

of a given theory in which they were more understandable (like a “maximal situation” or “each sequence of objects from a given universe”), then they could be defended.

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3.5. combinations of 3.1-3.4.

5. Remarks and Conclusions 1. From what has been said above one can arrive at the conclusion that for the epistemological problem of the objectivisation of knowledge and of truth there are (or there have been in the post-Brentanian tradition) more metaphysical answers than epistemological ones. Most of the above ways of objectivising truth are metaphysical solutions. Only where we speak about indexicality, intelligibility, a priori knowledge and about criteria do we come close to proposing an epistemological answer to the problem of the objectivisation of truth. Our answers are either directly metaphysical (when they deal, for instance, with sentences in themselves), or they refer to metaphysical problems indirectly (where they deal with ideal meanings, type-sentences and propositions). 2. Since the ways of objectivising knowledge and truth are very closely analogous, one can expect that attempts to objectivise knowledge will have their expressions in the area of truth theory. The truth-bearer is always defined in one of two ways: (DO) (The ontological definition of the truth-bearer): The truth-bearer is an entity that possesses a truth value (see e.g. Morscher and Simons 1982, p. 214). whereby truth values are attributes. (DL) (The linguistic definition of the truth-bearer): The truth-bearer is that about which we say that it is true (that of which we predicate “true”). Now, if we look at the analogies between ways of objectivising knowledge and truth and if we do it as seriously as possible, then we can propose another definition of the truth-bearer, which we will call the epistemological definition of the truth-bearer: (DE) The truth-bearer is an epistemological tool to which we attribute the epistemic evaluation of being true. Here the expression “epistemological tool” should be understood broadly enough to include all that is useful for an act of cognition. An additional assumption is that not truth but true knowledge is understood as the primary notion. It is possible that the epistemological notion of the

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truth-bearer leaves a choice as to what the truth-bearer is. It can be such entities as: a judging thing (Brentano, KotarbiĔski), an act of judging (early Brentano), a judgment as the product of an act of judging (statement), an utterance as the psycho-physical product of an act of judging (àukasiewicz), a sentence as a psycho-physical product which expresses a judgment (Twardowski), a sentence as a psycho-physical product which expresses the content of a judgment (Ajdukiewicz), a proposition or some other ideal meaning as a psycho-physical product which is fixed or as something which is exemplified in such a product. 3. If the truth-bearer can be seen as an epistemological tool, then, in Bolzano-type theories, it has implications for the redundance problem of truth. This can be seen as follows: The redundancy theory of truth says that there is no difference between such utterances as: “It is true that p” and “p.” This theory rejects truth-bearers as useless and understands p as, let us say, a state of affairs. Then the sentences “It is true that Caesar was murdered” and “Caesar was murdered” mean the same. But if we say something about reality indirectly, as when we say something about a cognitive act or about epistemological tools such as psychological acts, then if p refers to such an act, we are interested in an epistemic evaluation of this act. Ramsey tried to reduce sentences which refer to epistemological tools to sentences which refer to reality directly. The sentence “He always makes correct (true) judgments” should be reformulated as “For any p: if he makes the judgment that p, then p.” But in such a case, and this is an argument against this theory, sentences which are supposed to be logically equivalent turn out to have distinct senses. Instead of an utterance about epistemological tools we get a sentence about states of affairs. Now, let us assume that the following sentence: (S1)

He always makes correct judgments.

in a Bolzano-type theory means the same thing as the following: (S2)

He always presents true sentences in themselves.

According to Ramsey, we can reformulate sentence (S2) in two ways: (R1) For any p: if he presents a sentence in itself about p, then p. (R2) For any p: if he presents a true sentence in itself about p, then p. Sentence (R1) is false because if someone presents (makes an instance of) a false sentence in itself, then non-p (or we assume a theory of

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negative states of affairs). In sentence (R2), the word ‘true’ is not eliminated. 4. In the choice of the truth-bearer in the sense of definition (DE) one can either

• choose the tools of knowledge (those which serve best for our •

knowledge), or choose the way of objectivisation (where the truth-bearer guarantees the greatest independence of time, circumstances and subject).

It seems that the choices are (or at least were in the past) seen as contradictory: what at the turn of 19th and 20th century was seen as the proper tools of knowledge (e.g. judgments or their contents) did not offer many possibilities for the objectivisation of truth. 5. With reference to 4, it seems to be “the golden mean.” This regards expressions of language as truth-bearers:

• on the one hand, one chooses a language because it can be a good •

6

and sufficient expression for the tools of knowledge (for acts and their products); on the other hand, one chooses a language because it makes possible the objectivisation of truth in many ways. 6

I would like to express my thanks to the people from the Institute of Philosophy at Salzburg University, especially to Dr Johannes Brandl, for their help during my stay in Austria where I wrote an earlier version of this paper.

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TABLE 1. objectivisation \ assumptions

with regard to a subject S

of knowledge

time absolutisation

space absolutisation

objectivisation in a narrow sense

timelessness

for any t

outside of space

timeless S

knowledge without a subject of knowledge

S outside of space

for any x1, x2, x3

each S

without a knowing subject

transcendental S

for any subject

knowledge without a knowing subject monism solipsism

with regard to the media of knowledge MK

presentation of MK independent independent of MK outside of subject timeless MK of subject space MK permanent permanent permanent products as products as MK products as MK MK

with regard to the objects of knowledge OK

timeless OK

each OK

with regard to the conditions of timeless CK knowledge CK

valid CK for any t

presentation of MK outside of space

presentation of timeless MK

OK outside of space

CK outside of space

combinations

ĺ...

independent intelligible OK of subject OK

valid CK for any independent circumstances of subject CK

ĺ...

ĺ...

cognitive abilities intersubjectivity

intersubjectivity

intersubjectivity combinations

each OK

ĺ...

ĺ...

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From the Act of Judging to the Sentence

TABLE 2. objectivisation / assumptions

with regard to a subject S

of truth

time absolutisation

space absolutisation

objectivisation in the narrow sense

timelessness

for any t

outside time

timeless S

truth without a subject of knowledge

S outside of time

for any x1, x2, x3

any S

without empirical subject

for any subject

truth without a subject of transcendental knowledge S monism solipsism

presentation of TB outside of space

presentation of a timeless TB with regard to the truth-bearer TB

timeless TB

permanent products as TB

TB outside of time

indexicality of TB

presentation of ideal TB

TB permanent products as TB independent of subject indexicality of TB

permanent products

indexicality

psyche as TB

with regard to the truth-maker TM

TM unchangeable in time timeless TM

any TM

TM outside of space

indexicality of TM

other conditions timeless criteria

criteria valid for any t identity of TB with TM

combinations

combinations

ĺ...

TM which “fills all of space” any TM

TM independent of subject

indexicality of TM criteria valid for any criteria outside circumstances of space identity of TB with TM ĺ...

ĺ...

intelligibility of TM

indexicality of TM

criteria criteria valid independent for any S of S identity of identity of TB with TM TB with TM ĺ...

ĺ...

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REFERENCES Ajdukiewicz, K. (1927). O stosowaniu kryterium prawdy [On the Application of the Criteria of Truth]. Przegląd Filozoficzny 30, 280-283. Ajdukiewicz, K. (1930). Reizm [Reism]. Przegląd Filozoficzny 33 (1-2), 140-160. English translation in: KotarbiĔski (1966), pp. 515-536. Ajdukiewicz, K. (1931). O znaczeniu wyraĪeĔ [On the Meaning of Expressions]. In: KsiĊga Pamiątkowa Polskiego Towarzystwa Filozoficznego we Lwowie [Commemorative Book of the Polish Philosophical Society in Lvov], pp. 31-77. Lwów: PTF. Bolzano, B. (1937). Wissenschaftslehre. 2 vols. Reprinted in: Grundlegung der Logik, Hamburg: Felix Meiner, 1978. Brentano, F. (1930). Wahrheit und Evidenz. Hamburg: Felix Meiner. Husserl, E. (1891). Philosophie der Arithmetik. Reprinted in: Husserliana 12. Haag: Nijhoff, 1970. Husserl, E. (1900). Logische Untersuchungen. Erster Band. Reprinted in: Husserliana 18. Haag: Nijhoff, 1975. Husserl, E. (1901). Logische Untersuchungen. Zweiter Band. Reprinted in: Husserliana 19. Haag: Nijhoff, 1984. KotarbiĔski, T. (1913). Zagadnienie istnienia przyszáoĞci [The Problem of the Existence of the Future]. Przegląd Filozoficzny 16 (1), 74-92. KotarbiĔski, T. (1929). Elementy teorii poznania, logiki formalnej i metodologii nauk [The Elements of Epistemology, Formal Logic and the Methodology of Science]. Lwów: Zakáad Narodowy im. OssoliĔskich. English translation: KotarbiĔski (1966). KotarbiĔski, T. (1966). Gnosiology: The Scientific Approach to the Theory of Knowledge. Oxford: Pergamon. LeĞniewski, S. ([1913] 1992). Czy prawda jest tylko wieczna, czy teĪ wieczna i odwieczna [Is Truth only Eternal, or both Eternal and Sempiternal]? Nowe Tory 8 (10), 493-528. English translation: Is all Truth only True Eternally or Is It also True Without Beginning?, in: Collected Papers, edited by S. Surma, S.J. Srzednicki, D.I. Barnett and V.F. Ranett (Dordrecht: Kluwer Academic Publishers), pp. 86-114. àukasiewicz, J. (1907). Logika a psychologia [Logic and Psychology]. Przegląd Filozoficzny 10 (4), 489-491. àukasiewicz, J. (1910). O zasadzie sprzecznoĞci u Arystotelesa: Studium krytyczne [On the Principle of Contradiction in Aristotle. A Critical Study]. Kraków: PAU. Marty, A. (1908). Untersuchungen zur Grundlegung der allgemeinen Grammatik und Sprachphilosophie. Halle: Niemeyer. Morscher, E. and P. Simons (1982). Objektivität und Evidenz. In: E. Morscher, J. Seifert and F. Wenisch (eds.), Vom Wahren und Guten, pp. 205-222. Salzburg: Verlag St. Peter. Popper, K.R. (1972). Objective Knowledge. Oxford: University Press. Puntel, B.L., ed. (1987). Der Wahrheitsbegriff. Darmstadt: Wissenschaftliche Buchgesellschaft. Rojszczak, A. (1994). Wahrheit und Urteilsevidenz bei Franz Brentano. Brentano Studien 5, 187-218. Simons, P. (1992). Philosophy and Logic in Central Europe from Bolzano to Tarski. Dordrecht / Boston / London: Kluwer.

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Smith, B. (1989). Logic and Formal Ontology. In: J.N. Mohanty and W. McKenna (eds.), Husserl: An Introductory Survey of His Thought, pp. 29-67. Lanham: University Press of America. Smith, B. (1994). Austrian Philosophy: The Legacy of Franz Brentano. La Salle, IL: Open Court. Tarski, A. ([1933] 1956). PojĊcie prawdy w jezykach nauk dedukcyjnych [The Concept of Truth in Formalized Languages]. Warszawa: Towarzystwo Naukowe Warszawskie. English translation: The Concept of Truth in Formalized Languages, in: Logic, Semantics, Metamathematics: Papers from 1923 to 1938 (Oxford: Clarendon Press), pp. 152-278. Twardowski, K. (1900). O tak zwanych prawdach wzglĊdnych [On So-Called Relative Truths]. In: KsiĊga Pamiątkowa Uniwersytetu Lwowskiego ku uczczeniu piĊüsetnej rocznicy fundacji JagielloĔskiej Uniwersytetu Krakowskiego [Commemorative Book of Lvov University for the Celebration of the Five-Hundredth Anniversary of the Foundation Cracow University], pp. 1-25. Lwów: Senat Akademicki Uniwersytetu Lwowskiego. Twardowski, K. ([1912] 1979). O czynnoĞciach i wytworach. Kilka uwag z pogranicza psychologii, gramatyki i logiki [Actions and Products. Comments on the Border Area of Psychology, Grammar and Logic]. In: KsiĊga Pamiątkowa ku uczczeniu 250-tej rocznicy zaáoĪenia Uniwersytetu lwowskiego przez króla Jana Kazimierza [Commemorative Book for the Celebration of the Two-Hundred-Fiftieth Anniversary of Foundation Lvov University by King John Casimir], vol. 2, pp. 1-33. Lwów: Uniwersytet Lwowski. English translation: Actions and Products: Comments on the Border Area of Psychology, Grammar and Logic [fragments], in: J. Pelc (ed.), Semiotics in Poland 1894-1969 (Dordrecht: Reidel), pp. 13-27. WoleĔski, J. (1985). Filozoficzna Szkola Lwowsko-Warszawska [The Lvov-Warsaw Philosophical School]. Warszawa: PWN. WoleĔski, J. (1988). Logic and Philosophy in the Lvov-Warsaw School. Dordrecht: Kluwer. WoleĔski, J. (1994). Szkola Lwowsko-Warszawska: miĊdzy brentanizmem a pozytywizmem [The Lvov-Warsaw School: Between Brentanism and Positivism]. Principia 8-9, 69-89.

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Wojciech ĩeáaniec WHAT DOES “TRUTH IN VIRTUE OF MEANING” REALLY EXPLAIN?

1. Introduction Since Quine’s influential article “Two Dogmas of Empiricism” philosophers of Anglo-Saxon persuasion have taken to thinking that there is no problem of analytic sentences, because the category of analytic sentence is indefinable. The indefinability has been believed to reside in this: The class of analytic sentences consists of two sub-classes: sentences that instantiate truths of logic, and sentences that could be turned into instantiations of truths of logic by replacing some of their constituent expressions by other, synonymous, expressions; however, there is no concept of synonymity (it is believed), because the only one that could have been available would have to be defined in terms of analyticity itself, thereby giving rise to a circular definition of the category of analytic sentences. I shall not dwell on the question whether the putative circularity would be all that bad  perhaps it would not.1 I shall, instead, show that it is possible to define, without any circularity at all, a category of sentences that can be called “analytic sentences” and that are not instantiations of truths of logic. However, this success will turn out to have a price. Part of this price will be easy to pay: this will be the admission that many analytic sentences have what could be called a “non-analytic component,” which is, quite often, an empirical sentence or a class of such sentences (see

1

See, e.g., Saito (1962). In Gupta and Belnap (1993), you find a theory of truth that makes heavy use of a certain kind of circularity. In Professor PaĞniczek’s opinion, the theory of non-well-founded sets would help to deal with the problem of circular definitions (oral communication).

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 421-444. Amsterdam/New York, NY: Rodopi, 2006.

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ĩeáaniec 2003). This non-analytic component makes for analytic sentences’ not being necessarily true, sometimes being false, depending on whether the analytic component itself is true or false. This is certainly the most unpleasant consequence of my theory of the analytic; it also provokes the question in the title of this article: “What does ‘truth in virtue of meaning’ actually explain?” if it does not explain the truth purely and simply of whatever is true “in virtue of meaning”; it does not, for sometimes there is nothing to explain. Truth in virtue of meaning will turn out to be something like “Truth according to Larry,” which may or may not be truth sans phrase.2 On reflection, however, we see that this unpleasant consequence is not at all unexpected: if ‘analytic’ means, roughly, ‘true in virtue of logical form and the meaning of constituent expression’, then the truth value of whatever is analytic depends on the meaning of expressions. The meaning of expressions, now, is a very worldly affair whatever else it is, because so are expressions themselves, as part of language, which is part of the world of what happens and is the case as well as part of what people think that happens and is the case. This is quite along the lines traced out by Carnap (1958).3 No empiricist should shrink back in horror at that. Where I part company with Carnap is that I shall be talking about sentences of common language (common English, in this case), as distinct from more disciplined, or “regimented,” variants of English employed in science and scholarship, law, literary criticism, cookery or other areas of life in which exactness is called for. But then, there will be a further part of the price I will ask you to pay for my success, namely a revival of the problem of “synthetic a priori” sentences. I shall not urge that there is any such category (but, see ĩeáaniec 1996 on the “synthetic a priori”); and least of all do I have any terminological agenda. (I am not even interested in my definition of the analytic being accepted as “correct” – whatever that means, conform to current usage, for instance.) I shall only show that among what I shall here call analytic sentences there are such that have a certain property, and such that do not. I shall not insist that either should be carved out from the category of (what I shall call) analytic sentences, as a special category of sentences, called “synthetic a priori” or anything else, on a 2 In this, I go a bit further than Smith in (1991): Smith countenances a priori sentences that are false without making it clear whether such sentences would be analytic or synthetic according to those who believe in the existence of these categories. 3 The thrust of Carnap’s proposal has been assessed as the “most promising avenue” in the study of analyticity, see WoleĔski (1993).

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par with the category of the analytic. I shall only say: Within the category of analytic sentences there are these two sub-categories to be distinguished, and that is that.

2. Logical Truths and Sentences Called “Analytic” There are, for all we know, sentences of common language for which we can decide that they are instances of truths of logic. ‘John was born in Berlin or he wasn’t’ may be a sentence of this sort. How this should be possible for this one or any other sentences is not relevant here. Nor is it relevant that such should be instances of truths of any particular system, say, the first-order predicate logic or any other. I shall call such sentences, for short, “logical truths” (as distinct from “truths of logic”, of which the former are instances). Deciding if a sentence is an instance of a truth of logic presupposes, in any case, that the logical form of the sentences has been established, and this, as we have known at the latest since Russell’s “On Denoting,” is by no means a trivial matter. However, in this article I shall pass over this non-trivial matter in silence. There are other sentences, however, that are not instances of truths of logic, but, nevertheless, can be turned into such, namely, by replacing some of their constituent expressions with other expressions. 4 The sentence ‘If John is a father then he is male’ belongs, most likely, here, because it can be turned into an instance of a truth of logic, to wit, ‘If John is a male parent, then he is male’ by substituting ‘male parent’ for ‘father’. The sentence ‘If John has bought a new motor-car, he has purchased a new automobile’ belongs into the present category, too, because it can be turned into an instance of a truth of logic, namely, by substituting ‘automobile’ for ‘motor-car’ and ‘purchased’ for ‘bought’ (or the other way round). Some people would contend that the sentence ‘Nothing is red and green all over at the same time’ goes into this category, too, because it can (they would say) be transformed into an instance of a truth of logic by substituting ‘is not green’ for ‘is red’. I shall have things to say on this sentence later. The problem with this second category of sentences is that it does not seem quite clear why, exactly, the substitutions that turn them into logical truths are legitimate. The six sentences directly or indirectly mentioned in the foregoing paragraph are, clearly, pairwise quite 4

The distinction between these two categories of sentences has been inaugurated by Bolzano in (1972), section 148.

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different, and if someone were to claim that they are pairwise equivalent with respect to truth-value,5 then he or she would owe us an explanation what relation underwrites this equivalence. An obvious approach to this issue would be to say that the sentences are pairwise equivalent because the expressions whose mutual substitution turns the one sentence into the other  namely, ‘father’ and ‘male parent’, ‘buy’ and ‘purchase’, ‘motorcar’ and ‘automobile’, ‘is red’ and ‘is not green’  are pairwise equivalent. However, what does this alleged equivalence itself rest on? One possible answer would be: On the interchangeability of the expressions at issue – except in intensional contexts – salva veritate. This, alas, would be crassly circular, because initially, it was the interchangeability that was supposed to be explained by some kind of equivalence, not the other way round. This is the kind of problems that, as we know, Quine put his finger on. Another possible answer would be: because both expressions in each pair  ‘father’ and ‘male parent’, etc.  have the same meaning, mean the same. But this answer can be misunderstood and taken to imply that there are entities of some otherworldly, or at least non-physical (and not mental) nature, called “meanings,” attached to expressions, such that the entity attached to ‘father’ is identical with (or just exactly the same as?) the one attached to ‘male parent’. Unfortunately, however, such entities are not only unobservable, but even qua theoretical entities quite elusive, so explanations of anything are best not couched in terms of meanings-asentities. In this article, nonetheless, I shall attempt to show that this plain answer: “‘Father’ and ‘male parent’ have the same meaning” can be understood in a way which does not commit us to any meanings-asentities or any other “queer entities” of their ilk.

3. Explaining What Expressions Mean What happens when we come across an expression that we do not understand? What happens when someone in our environment does? In situations of this sort claims to the effect that one does not know what the expressions in question mean are made, and sometimes also offers to explain what those expressions mean. If I do not understand an expression, I say that I do not know what it means; whereupon someone might explain to me what the expression means, or, in other words, tell 5

Except in some clearly defined types of contexts, e.g., intensional contexts.

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me what the meaning of this expression is. I can also say that I do not know the meaning of an expression. Such things happen frequently, with children, foreigners, laymen in a special field or people with poor education. (A student of mine, a native speaker of American English, did not know what the words ‘recalcitrant’, ‘perfunctory’, ‘prevaricate’, and ‘schoolman’ meant; he asked me what they meant and I told him. Neither of us felt a kind of “Quinean remorse” about meanings, and least of all were we disposed to say “Why, these words don’t mean anything, because, as Quine has shown, words don’t have meanings . . .”. Although his education was very good  by American standards  he obviously was not in the habit of reading “antiquated stuff,” i.e., literature from the 1970s or earlier, where such words as ‘recalcitrant’ sometimes occur.) In none of such practices is any ontological commitment to “intensional entities” involved.6 The fact that claims and offers of this sort would be made deserves attention. For it is not logically impossible that things should be quite different: We could have a world where people sometimes do not understand expressions and yet do not realize this or at least have not developed that sort of interests and vocabulary that allows them to say that they do not know what an expression means or that they are not able to explain the meaning of an expression. We all know situations when someone does not understand an expression and yet does not see that and reacts to an utterance in which the expression occurs as if he understood it: Why does this type of situation not occur each time someone does not understand an expression? Why aren’t situations in which a person who has not understood an expression and yet does not realize that he or she has not vastly exceptional? Or else, why is it not the case that each time a person does not understand an expression he or she is only bewildered, stupefied, but stops short of asking what the expression means? (This happens sometimes too, but not always.) Why are we in most cases able to say what an expression means if we understand it? Why are we not 6

The same point is urged by Austin (1961, pp. 26-28). He there says that asking “What is the meaning of a word?” in the sense of “What is the thing called ‘the meaning of a word’?” is similar to asking “What is the point of doing a thing?” in the sense of “What is the thing called ‘the point of doing a thing’?”, which is absurd because there are no such entities as “points of doing a thing.” He says bluntly “There is no simple and handy appendage of a word called the ‘meaning of the word «x»’” (p. 30). We should add that if someone asks what the meaning of an expression is he or she does not ask what the entity called “the meaning of the expression” looks like or what it is made of; and neither are offers to explain what the meaning of an expression to be taken in the sense of producing a strange entity, called “the meaning of the expression.” Like ‘sake’ and ‘behalf ’, ‘meaning’ is a syncategorematic expression.

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puzzled when asked what an expression that we seem to understand means? – in the way in which most people are puzzled by the question why ships do not sink or why aeroplanes take off, for all their confidence with which they go aboard of ships or aeroplanes . . . No a priori considerations alone can answer that. This is a role of psychology, linguistics, cognitive science. However, there are more striking facts to be noticed. Suppose that someone does not understand an expression and asks us what the expression means. If we understand the expression in question, we are, in most cases, able to tell him what the expression means. In so doing, we say various things, point to various objects, or point to objects and say various things. For instance, if someone asks us what ‘the Sun’ means, we can either point to the Sun, or say something like “the star that is closest to the Earth” or both point to the Sun and say “this celestial body.” 7 The meaning of colour words is, perhaps, best explained by pointing to suitable examples, but in many cases, as shown by Hume’s famous “missing shade of blue” thought-experiment, a purely verbal explanation will do. Suppose, for instance, that someone asks what ‘glâs’ means in Welsh: we can say that it means a colour intermediate between blue and green. In most cases, the things that we say, or do, or say and do, in response to the question what a certain expression means are quite different from what other people say, or/and do, when challenged to explain the meaning of the expression in question. They are also fragmentary, inchoate, and ad hoc, not as measured against the standard of some “ideal meaning” inspected by an “inward gaze,” but in the sense that they are parts of what other people would offer when asked to explain the meaning of the expression in question – or what we ourselves would offer when so asked on another occasion; that they are always susceptible of paraphrase and/or improvement. 8 However, a striking fact about that  a fact that can fully be borne out only by sufficient empirical research  is that for most expressions of a language most competent speakers of the language are in agreement as to

7

This is better than just pointing to the Sun, for reasons given by Wittgenstein in his Philosophical Investigations, section 33. 8 Few expressions of common language, English or any other, have “canonical formulas” by means of which their meaning is best explained. Such canonical formulas – called “definitions” – exist for terms of art of various disciplines, but such terms are hardly part of common language. Once they have become part of common language, their definitions are stripped of their “canonicity” and become just one, among others, way of explaining the meaning of expressions.

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which ways of explaining the expressions’ meaning are correct, no matter how fragmentary and ad hoc they are. This fact deserves much more attention than it has so far received. If t is a term of art of a scientific or scholarly discipline, then, almost certainly, there exists a definition or a family of definitions of t, such that most students of the discipline in question have learnt it/them; small wonder, therefore, that they explain the meaning of t by quoting the definition or one of the definitions in question. However, as noted above, most expressions of common language have no definitions. On the other hand, since competence in using a language will, now and then, be compared to competence in using a tool or a device, it is useful to reflect on this: Among people who are competent in using, say, TV sets or cellular facsimile machines, anything but a near-perfect agreement prevails as to what counts as a correct way of explaining how these devices work. Yet more strikingly, most people are fairly competent in “using” their bodies, although here we are even farther  than with respect to cellular fax machines  from a situation where most people would be able to explain how their bodies work in a way that is acceptable to most other people. The same  mutatis mutandis  is true with respect to social and legal institutions, economies, lotteries, electronic networks and many other types of things. It is, therefore, at least conceivable that we could have a situation where linguistic communication flourishes but for most expressions of common language most of its competent users disagree on what ways of explaining the expressions’ meaning are correct, or have no opinion on this issue at all. And yet, this is not so. It might, it is true, appear as if with respect to the ways of explaining the meaning of expressions various kinds of recondite knowledge had a role to play, too, as is the case with the ways of explaining how machines, bodies, or economies work. Do we not need to know various things about stars to agree that the meaning of ‘the Sun’ is correctly explained by saying that the Sun is the star that is closest to the Earth? Yes, we do, and yet the situation is not the same as with machines and economies. For imagine that someone does not know enough astronomy to know that the Sun is, in fact, the star that is closer to the Earth than any other star is. If this person has so far had no problems with more usual ways of explaining the meaning of ‘the Sun’, such as pointing to the Sun and saying “this celestial body” or the like, then he (or she) (1) will at once accept the correctness of explaining the meaning of ‘the Sun’ as ‘the star that is closest to the Earth’ if he (or she) is provided the piece of astronomy information about stars, and (2) will accept it as

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equivalent to ways that he has so far been ready to provide or accept as correct. Note that with respect to machines, bodies, economies, lotteries, etc. (1) might be true under the condition that suitable information has been provided, but (2) is usually not true, for people cherish wildly different  and quite often wildly erroneous  theories about how machines etc. work. If someone knows what ‘the Sun’ means, yet is not in the position to recognize ‘the star that is closest to the Earth’ as a correct way of explaining the meaning of ‘the Sun’, it suffices to tell him a few things about stars and he will realize that ‘the star that is closest to the Earth’ is not only a correct way of explaining the meaning of ‘the Sun’, but also one that is equivalent ceteris paribus to the ways he has judged correct so far. If someone, by contrast, holds an erroneous theory as to, for instance, how thermostats work, or on why ships float, an exposition of a correct theory, no matter how convincing, will not convince him except at the cost of having him give up the theory he has held so far. The point I am urging here can be summarized this way: For most expression of common language, most competent speakers of the language know how to explain what the expressions in question mean, and, although they give differently-looking explanations, they are consonant as to which ones among these are correct. To this, it should be added that in most cases people also know that most of the correct ways of explaining the expressions’ meaning are fragmentary, inchoate, ad hoc . . . as noted above, and that other people know this too. This is the reason why they judge even most fragmentary ways of explaining the meaning of many an expression correct. For instance, all agree that ‘man who has never been married’ is a correct way of explaining the meaning of ‘bachelor’, even though this is far from all that there is to bachelors: Not only have they to never have been married, but, among other things, they ought to be capable of marrying, to be of a certain age, and the like. Yet all these things are so universally known – and universally known to be universally known – that they are seldom, if ever, mentioned when it comes to explaining what ‘bachelor’ means. My hypothesis is that most people count various ways of explaining the meaning of expressions as correct despite their manifest incompleteness because they assume that those who offer them as ways of explaining the meaning of expressions know that these ways are incomplete and also know what should have been added to them. For instance, if someone tells me that the expression ‘monarch’ means ‘someone like the Queen’, I can count this answer as correct because I

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assume that he knows just in what respects a person has to be “like the Queen” in order to be called a “monarch.” Knowing such things involves more than just “competence in common language” of the sort that anyone who does not produce odd-sounding sentences can be credited with. Some people who are competent in common language do not have this knowledge, which is one of the reasons why the agreement as to which ways of explaining the meaning of expressions are correct prevails only among most speakers and with respect to most expressions. Another reason is that certain expressions are rarely used. Clearly, expressions like ‘perfunctory’ are part of common English, yet it is possible for a young, educated, white middle-class Anglo-Saxon to get away with not knowing what such expressions mean. More interestingly, there are whole families of expressions that are not seldom used and yet for which most of their competent users are uncertain as to which ways of explaining their meaning are correct. Many semantic expressions belong here, for instance ‘truth’, ‘meaning’, ‘sign’, ‘sense’, ‘proposition’ and the like. It is difficult to say what ‘mean’ in “What does ‘the Sun’ mean?” means; and it is even more difficult to command anything like broad consensus for any answer to this question, no matter how plausible. Words like ‘time’ or ‘space’ may have a similar property, as the well-known Augustinian difficulties might be taken to demonstrate. But such expressions, and families of such expressions, are exceptional. One more qualification is necessary: although most competent speakers agree which ways of explaining the meaning of most expressions are correct, they, clearly, do not think that for most expressions most ways of explaining their meaning are equivalent. Most speakers of English would agree that ‘a sort of device for lifting large and heavy objects, used on construction sites’ is a correct way of explaining the meaning of ‘crane’, and that so is ‘a kind of water-bird’, yet most speakers of English would also agree that these two are two non-equivalent ways of explaining the meaning of ‘crane’. Most speakers are aware that such non-equivalent ways of explaining the meaning of the same expression can be grouped into various mutually exclusive families that render “different meanings” (note the plural form) of the expression in question. Some speakers might be unfamiliar with some of such families, though most are familiar with most. Now another fact that has to be noted here is that such mutually exclusive ways of explaining the meaning of an expression find application in various contexts in which the given expression occurs. (As

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we have known at least since Frege and Wittgenstein, expressions have meaning in certain contexts only.) If we say: “The chairs were hard, the questions were hard, too” (say, after an examination) the word ‘hard’ does not mean the same thing. All persons that understand this sentence would agree that the meaning of ‘hard’ in the first occurrence could not be explained the same way as that in the second. It could not; which is why the Quinean zeugma “The chairs were hard, and so were the questions” sounds so funny (Quine 1960, p. 130). To take another example: If someone says that ‘purchase’ explains the meaning of ‘buy’ quite well one has to add that it does not for ‘buy’ occurring in such contexts as ‘I don’t buy into this nonsense’. Questions as to what a certain expression means are often raised in situations where a context for the expression is given. John has asked what ‘prevaricate’ means; it is because John has just found this word in a book he read. Quite often, too, persons to whom such questions are directed ask back: “In what context, though?” Only then can the question “What does this expression mean?” be answered correctly, i.e. only then can, from among all families of correct ways of explaining the meaning of the expression at issue, the one that is correct in the given context be singled out. For certain expressions, there might be something like a “dominating” family of ways of explaining the meaning of the expression in question, so that one of its ways is provided if no context is mentioned. This may be the case with ‘the Sun’. In any case, it has always been the (indispensable) role of lexicographers  and their heroic work  to collect a body of texts in which all types of contexts for any given expression can be found, contexts in which various ways of explaining the meaning of the expression are adequate. The point I am making here must by no means be confused with another, namely, that competent speakers of common language have a (possibly naive) semantic theory, or the like. This may, or may not, be the case; I have no opinion. Interestingly, in certain languages, such as German, the natural way of asking what an expression means does not take any of the expressions that linguists or semanticians have erected into terms of art of their disciplines (in German, “Was heißt das?”). A semantician or logician (unless he is a Quinean) makes the distinction between extension and intension, but answers to questions of the type “What does this expression mean?” do not always clearly fall into one and only one of these categories. For the present purpose, the question of what kind of cognitive processing is required for anyone to realize that they do, or do not, know what a certain expression means, or to come up with an explanation of what an expression means (what its meaning is)

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can be left unanswered and passed on to cognitive scientists. Least of all do we need the grotesque imagery of “queer entities,” called “meanings,” that are inspected in a kind of “inner vision” or “inward gaze.” For most expressions of common language, correct answers to the question “What does this expression mean?” are so diverse that there is no way in which some of them could be said to render the “intension” of the expression in question, as distinct from others that do not render the “intension.” Intensions, if they exist, cannot be gathered just from the way in which people explain the meaning of expressions. From what, then? This is irrelevant for the present purposes. For instance, if someone explains the meaning of ‘the Sun’ as ‘the star that is closest to the Earth’, this latter formula can be said to express a piece of scientific knowledge about the Sun, rather than “what everyone knows about the Sun” or “what everyone identifies the Sun by,” if there are any such things and if they are any part of the “intension” of ‘the Sun’. But this, to repeat, is irrelevant, as long as a consensus prevails as to this way of explaining the meaning of ‘the Sun’ being correct. As noted above, this consensus may at first be less than nearly-universal, as some people may not know enough astronomy to understand about stars, suns and the like. But, as noted above, too, what would be sufficient for reaching a nearlyuniversal consensus about that way of explaining the meaning of ‘the Sun’ being correct is just a few pieces of basic astronomy information. Similarly, if I chose to explain the meaning of ‘green’ by saying that it means “the colour of the banner of the Seal of the Prophets,” many people would be at a loss judging whether I have explained the meaning of ‘green’ correctly. The reason for this perplexity would not be that I use standards of correctness different from theirs, but that they simply do not know what colour the said banner is supposed to be. After they have found this out, they would all agree that I had explained the meaning of ‘green’ correctly.

4. Analytic Sentences Suppose we have a sentence, s, and suppose that the meaning of one of its constituent expressions has been explained in an at least partly verbal way. It might be  and more often than not it is  the case that the verbal part of the explanation of the meaning of the expression can be inserted into the sentence in question with, or without, a little syntactical restructuring which leaves the logical form of the sentence untouched.

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If, now, the resulting sentence, s, turns out to be an instance of a truth of logic, the original sentence can be said to be true in virtue of its logical form and the meaning of its constituent expressions, or, for short, analytic.9 If it does not, the whole procedure: explaining the meaning of some of its constituent expressions, inserting verbal parts of such explanations into it, restructuring it syntactically in a suitable way can be applied to s. If at this stage, or after a finite number of such stages, a logical truth is arrived at, the original sentence may still be said to be true in virtue of its logical form and the meaning of its constituent expressions or analytic. For example, the sentence ‘If John has furze in his garden, he has gorse in his garden’ is analytic (in the sense defined in this article, here as henceforth), because ‘gorse’ is a correct way of explaining the meaning of ‘furze’, and the substitution of ‘gorse’ for ‘furze’ yields the logical truth ‘If John has gorse in his garden, he has gorse in his garden’. Similarly, ‘If John is proud of his automobile, he is proud of his motorcar’.10 The sentence ‘Every father is male’ is analytic, because ‘male parent’ is a correct way of explaining the meaning of ‘father’, and the substitution of ‘male parent’ for ‘father’ yields the logical truth ‘Every male parent is male’. The sentence ‘Every bachelor is unmarried’ is analytic, because ‘man who has always been, and still is, unmarried’ is a correct way of explaining the meaning of ‘bachelor’, and after the appropriate substitution we get the logical truth ‘Every man who has always been, and still is, unmarried is unmarried’. The sentence ‘Nothing is red and green all over at the same time’ is analytic. This sentence, a classical putative example of a category called ‘the synthetic a priori’, is somewhat more complex than the foregoing four.11 I shall explain, later on, why it is also more difficult in other 9

Once again I have to stress that I do not claim any kind of superiority for my definition of ‘analytic’ over other definitions of this term. Least of all do I want to contend that it does not matter if we talk about analytic sentences in the sense of ‘analytic’ defined here or any other; it does. In this article, I never use the word ‘analytic’ in any other sense than the one introduced here. 10 Such examples are nice and very “well-behaved” but also misleading because they give the impression that analyticity of sentences is in some inextricable way bound up with synonymy of expressions: hence Quine’s attacks on the concept of synonymy in “Two Dogmas of Empiricism” and Strawson and Grice’s defence of this concept in “In Defence of a Dogma.” Unfortunately, true synonyms are very rare in common language. 11 The idea of the following argument is an indirect loan from Putnam (1956). In ĩeáaniec (1996-1997), you will find a more straightforward elaboration on that Putnamian theme as well as a criticism of Putnam’s argument.

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respects and why it does give rise to the problem of the synthetic a priori. Yet, as I shall also show, it is analytic in the sense defined in this section. First, a bit of syntactical restructuring is necessary. I transform the sentence into: ‘Nothing is red all over and is green all over at the same time’. I hope it will be granted on all hands that the new sentence is semantically and logically equivalent to the previous one. Then I note that a correct way of explaining the meaning of ‘is red all over’ is this: ‘is the colour of blood all over, and thereby is not even spotwise the colour of grass’; and a correct way of explaining the meaning of ‘is green all over’ is: ‘is the colour of grass all over’ 12  as long, at least, as blood and grass have the colour they have so far had. After appropriate substitutions we get from the original sentence ‘Nothing is red and green all over at the same time’ the sentence ‘Nothing is the colour of blood all over, and thereby is not even spotwise the colour of grass, and is the colour of grass all over at the same time’. I hope it will be agreed that this is an instantiation of a truth of logic, for instance, of ~ x [F(x) š ~ G(x) š G(x)] Two problems may arise here: (1) if building in the mention of the colour of grass into our way of explaining the meaning of ‘is red all over’ was at all necessary, and (2) what role the mention of surface (‘all over’) and of time (‘at the same time’) in the original sentence had to play. For obviously, I was playing fast and loose with both ‘all over’ and ‘at the same time’. To the first question the answer is “no”; saying that ‘is red all over’ means ‘is the colour of blood all over’ would explain the meaning of ‘is red all over’ sufficiently. However, adding of ‘and thereby not that of grass’ does not make this way of explaining the meaning of ‘red’ incorrect: it only makes it a bit long-winded and redundancy-laden. The redundancy in question is, however, sufficient for turning the initial sentence into a truth of logic. Redundancy of this sort is useful in the same way as artifices employed by mathematicians often are for proofs: mathematicians add and subtract at the same time a certain number or an 12 Well, should someone insist of taking ‘colour’ here in the sense of “the specific shade,” the above ways of explaining the meaning of expressions should be turned into: ‘is a colour that is more like the colour of blood than the colour of grass all over, and thereby is not even spotwise any colour that is more like the colour of grass than the colour of blood’ and ‘is a colour that is more like the colour of grass than the colour of blood all over’, respectively. The reader will easily find that even so the original sentence comes out as a logical truth in the end.

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algebraic expression  seemingly a futile procedure  for certain transformations to become possible. The second question is more interesting and I should venture the following answer. In the original sentence, ‘Nothing is red and green all over at the same time’, the function of the phrase ‘all over at the same time’ is to indicate that if a thing happened to be red and green, it would not be red in the same way as that in which it would be green. For obviously, there are things that are both red and green, but in different ways, say, now green and then red, or green on one side and red on another. Yet the role of the phrase ‘all over and at the same time’ is much more significant, and much more clear, in the case of the transformed sentence, i.e., ‘Nothing is the colour of blood all over, and thereby is not even spotwise the colour of grass, and is the colour of grass all over at the same time’. For in this sentence, the addition of ‘even spotwise’ after ‘thereby not’, ‘all over’ after ‘grass’, and ‘at the same time’ at the end of the sentence give us the assurance that the expression ‘is . . . the colour of grass’ as occurring twice over in this sentence, expresses in reality one predicate, so that the logical form of the sentence is, as stated above, that of a truth of logic. But this is not all, for had it not been for ‘all over’ after ‘blood’, and ‘even spotwise’ after ‘thereby’, the way of explaining the meaning of ‘is red’ would not have been correct, as there are things that are red, i.e., the colour of blood, and yet spotwise green at the same time. The assurance of which I am speaking here can, however, be given only under certain conditions that I shall mention towards the end of this article. This is all. However, there remain a few problems to discuss.

5. Messing Up in the Right Way How can we know that inserting into a sentence verbal parts of explanations of the meaning of some of the sentence’s constituent expressions is legitimate? or that it turns the sentence into another sentence which is equivalent to it with respect to truth-value? I have no answer to this. It is better to admit that there is a problem here than to pass over it in silence. For present purposes this replaceability may well be taken for granted. In regimented languages certain privileged forms of definitions are such that the definiendum is always replaceable by the definiens. I surmise that the idea that a certain expression renders the meaning of some other expression correctly implies that the latter can be replaced by the former in all but intensional

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contexts; but  as I noted above  it is difficult to say what it means that a certain expression means this or that. When we say that ‘furze’ means gorse (or ‘gorse’13) or that ‘to commence’ means ‘to begin’ or that ‘motor-car’ means ‘automobile’ we seem to be claiming quite straightforwardly that the expression in question can be replaced, in all but intensional contexts, with the expression employed in saying what the former expression means. Apparently, there is some kind of cognitive processing at work here which renders ‘to commence’ and ‘to begin’, as well as ‘furze’ and ‘gorse’ equivalent. There is a further difficulty. As I said above, ways of explaining the meaning of expression are next to never complete. If ‘furze’ explains the meaning of ‘gorse’  to take one exceptional example  completely, then it is only one, clearly demarcated, puzzle in the fact that ‘furze’ can be replaced by ‘gorse’ in all but intensional contexts salva veritate. This puzzle would have been the only one to deal with if turning analytic sentences into logical truths had been a matter of, in a Quinean phrase, “putting synonyms for synonyms.” But it is not: there are far too few synonyms to be put for each other. However, ‘unmarried man’ does not explain the meaning of ‘bachelor’ without remainder; yet still, it appears legitimate to replace the latter with the former in the sentence ‘All bachelors are unmarried’. Why? My answer is this: Operating with a less than complete way of explaining the meaning of an expression that is part of a sentence that one seeks to turn into a logical truth is legitimate as long as the logical form of the sentence as well as that of complete ways of explaining the meaning of the expression in question are such that whatever would have to be added to the way employed would not turn the sentence into something else than a logical truth. If this is the case for the sentence ‘All bachelors are unmarried’ and ‘unmarried man’ remains to be seen, but it appears that any complete way of explaining the meaning of ‘bachelor’ would have the logical form of a conjunction with ‘unmarried man’ as one of the conjuncts; now, given that the sentence ‘All bachelors are unmarried’ has the logical form of this sort: x [F(x) o G(x)] and if it can be broken down to: x {[G(x) š H(x)] o G(x)} 13

No clarity seems to prevail as to whether the expression ‘. . . means . . .’ operates on the level of object language or that of metalanguage.

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then it is a logical truth, no matter what ‘H(. . .)’ stands for. The adequate grasp of the logical structure of the sentence as well as of that of a complete way of explaining the meaning of an expression is important here and may not always come easy, or be “taken for granted.” For instance, there is a German saying “Deutsch sein heißt Charakter zu haben,” loosely: To be German means to have guts. On the face of it, this saying is either the claim that all Germans have guts, or a way of explaining the meaning of ‘German’. Now, some people have tried to interpret this saying in the sense of: Only Germans have guts; which is, clearly, unwarranted by either way of taking the saying in question. How could this interpretation be conceived? I think a plausible explanation is that people took the saying as a way of explaining the meaning of ‘German’, substituted ‘persons who have guts’ for ‘Germans’ in the sentence ‘Only Germans have guts’ and happily arrived at a truth of logic. What escaped their attention was that the sentence ‘Only Germans have guts’ has such a logical structure that adding (by means of conjunction) various things to ‘persons who have guts’ in the way of explaining the meaning of ‘German’ employed could very well turn it into a something else than a logical truth.

6. Primitive and Derivative Ways of Explaining the Meaning of Expressions Ways of explaining the meaning of expressions can be grouped along different lines, but one that is particularly interesting and important here is the one that divides such ways that can be justified and such that cannot. If someone asks me “What does ‘the Sun’ mean?” and if I point to the Sun saying “that shining thing up there,” there is little I can say in order to justify this way of explaining the meaning of ‘the Sun’, should it be challenged. What I can do is assure that the message has got through, i.e. the person has realized what it was that I pointed to, that he has understood the words ‘that shining thing up there’, but once these conditions have been satisfied, there is hardly anything more I can do. Would anyone go on voicing his or her doubts about the correctness of this way of explaining the meaning of ‘the Sun’, I should most likely get impatient and say something like “Well, this is how this expression is used in English, and that is that” or the like. Similarly for ‘father’. If anyone challenged me to defend the correctness of ‘male parent’ as a way of explaining the meaning of ‘father’, I should say something like “This is all nitpicking and being

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fussy; ‘father’ simply does mean ‘male parent’; there is nothing you can do about it.” Similarly for ‘bachelor’ in the meaning of ‘unmarried man’. Should someone question the correctness of this way of explaining the meaning of ‘father’ or ‘bachelor’, we can only say: “Well, this is how these expressions are used, please get some exposure to common English.” This, too, is a sort of defence, or justification, yet one that does not invoke any other facts than just that the expressions in question mean what we claim they do. Let us call such ways of explaining the meaning of expressions primitive. In other cases, however, ways of explaining the meaning of expressions can very well be defended or justified, and that not just by insisting that they are correct. We have seen this already: on the example of the way of explaining the meaning of ‘the Sun’ that mentioned stars and their distance to the Earth. The defence of such a way of explaining the meaning of an expression consists in supplying certain information. This information, once absorbed, will suffice to convince someone who has a different-looking way of explaining the meaning of the same expression that the two ways are, in fact, equivalent, at least assuming that, and for as long as, the information in question is true. Similarly, should someone explain the meaning of ‘bachelor’ by saying that this expression means ‘a male human being who has never taken marital vows’, a person who knows nothing about ways and manners in which people get married might ask why this way of explaining the meaning of ‘bachelor’ should be correct, if ‘a male human being who has never been married’ is. This challenge can be met much more easily than the previous one, where the starhood of the Sun was at stake; namely, by explaining that people get married as a result of taking appropriate vows (rather than, say, inheriting the status of being married, or having it conferred upon themselves by some authority that did not ask their opinion). Similarly, ‘the colour of blood’, ‘the colour of grass’ are correct ways of explaining the meaning of ‘red’ and, respectively, ‘green’, but someone who for some unusual reasons is not familiar with either blood or grass may question their correctness. This challenge can easily be met, for instance by producing representative samples of grass and of blood, as long as blood and grass actually are the colours they have been so far. Let us call such ways of explaining the meaning of expressions derivative. This term is intended to convey the idea that derivative ways of explaining the meaning of expressions are derived from other, primitive or derivative, ways of explaining the meaning of expressions,

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as well as from certain premises assumed true (such as for instance, that the Sun is a star, or that people get married by taking certain vows, that blood and grass still are the colour they have been for millions of years). What impact has this “primitive” vs. “derivative” distinction on the issue of sentences true in virtue of their logical form and the meaning of their constituent expressions? Answer: derivative ways rest, as just explained, on certain assumptions that can be expressed in sentences. Now, some among these sentences are empirically true (see the examples above); others might be true in virtue of their logical form alone. Yet others, however, may not fall clearly in either of these categories, and share, instead, certain characteristic features with sentences that are true in virtue of their logical form and the meaning of their constituent expressions. Suppose, now, that we have a sentence, s, that is true in virtue of its logical form and the meaning of its constituent expressions, or analytic in the sense of this article. We examine all sentences that are necessary for turning s into a logical truth. Suppose that all of them are true in virtue of their logical form alone, i.e. they are logical truths. Then everything is well with s, it is not only analytic but also true and cannot be false except if the ways of explaining the meaning of some of its constituent expressions employed in turning it into a logical truth become incorrect. Suppose, for a change, that at least one of the sentences necessary for turning s into a logical truth is only empirically true. (Call it e.) The situation starts being different: the “empirical component” of s  as e might be called  influences by its truth value the truth value of s. Hence, the latter, although analytic, can very well turn out to be false. Suppose, thirdly and finally, that among sentences necessary for turning s into a logical truth we find one that is analytic, too; call it sƍ. We examine all sentences that are necessary for turning sƍ into a truth of logic; suppose, furthermore, that among these, too, there is one that is analytic; call it sƎ. For sƎ we repeat the procedure, and find an analytic sentence once again . . . Suppose this goes on indefinitely. Now compare this situation with one in which a process like this, started off for the sentence t, stops at some stage, because all sentences necessary to turn all analytic sentences encountered on any previous stage have been found to be either logically or empirically true. In the latter case, the truth of t can be said to rest on the joint foundation of logic and experience, and no matter how shaky the experience part of this foundation is, the foundation is at least epistemologically “wellbehaved.” In the former case, the truth of s, its analyticity

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notwithstanding, cannot be said to rest on a foundation like that. If logic and experience is all that the truth of a sentence can rest on, then s must be said to be without foundation. Depending on how bearable or otherwise this state of affairs appears to you, you might or might not be tempted to postulate a new category of (true) sentences for s, call it “synthetic a priori” or what have you, and theorize about what sort of cognitive faculties are necessary for getting hold of such truths. The fruits of succumbing to this temptation might appear highly objectionable to soberly-minded philosophers, but it seems to me that one has to face squarely the problem here in the first place. I do not know how the problem here described could actually arise with respect to common language, since the latter is not formalized or in any other way “regimented.” Failing this, we can seldom know that the process should go on indefinitely. However, in certain cases we can at least go through a few stages and see that at each of them a new analytic, or at least analytic-looking, sentence appears. For a study of examples, see ĩeáaniec (1992). Even worse, in certain other cases we see immediately that the process will go on indefinitely because one of the analytic sentences run into at a later stage is the initial sentence itself. For illustration, take the sentence ‘Nothing is red all over and green all over at the same time’ once again. As I established above, this sentence is analytic; part of my argument for this was that the expression ‘is the colour of blood and thereby not that of grass’ is a correct way of explaining the meaning of ‘is red’. Is this a primitive or a derivative way of explaining the meaning of ‘is red’? For all its obviousness, it is derivative. Because it is possible to say: For a thing to be red all over it is enough that it should be the colour of blood all over. If you add ‘and thereby not even spotwise that of grass’ you introduce a restriction that for most red things may be quite harmless since, as it happens, they are not, not even spotwise, the colour of grass at the same time; however, this restriction would force us not to call ‘red’ a rare thing that is, strange as this seems, both red all over and (at least spotwise) green, if there ever be any such. In other words, your way of explaining the meaning of ‘is red’ would be correct if, but only if, we could be certain that there are no things that are red and green all over. Which is what the sentence just proven analytic claims. Expressed in the symbolism employed above: s = sƍ. To this, one could be tempted to retort: Surely, we can be certain that there are no things red and green all over.

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This may be right, but the certainty in question has to be drawn from something other than the finding that the sentence ‘Nothing is red and green all over at the same time’ is analytic. For what is at stake here is exactly the question if what is analytic is also true; what we have arrived at is that the sentence ‘Nothing is red all over and green all over at the same time’ is true if it is true; more precisely, that, if it is true, it is true if it is analytic. And this is far too little.

7. Saying the Unsaid What makes sentences look  but sometimes just look  like instantiations of a truth of logic is that certain expressions are arranged in them in a certain order, for instance, ‘male’ occurs twice over in the sentence ‘Every male parent is male’, the syntactical form of the sentence being such that it looks as if it is an instance of this truth of logic: x {[F(x) š G(x)] o G(x)}. For this to be more than just misleading appearance of the sentence’s being an instantiation of the above truth of logic, it has to be assumed that, among other things ‘male’ occurs both times in the same meaning; otherwise we should have a simple case of equivocation. Now does it?, i.e., would most competent speakers of common English agree that all correct ways of explaining the meaning of ‘male’ that apply to the word’s first occurrence in the sentence ‘Every male parent is male’ apply also to its second occurrence? My hypothesis is that they would, if they understood this question at all. I shall try to make this hypothesis plausible by propping it up with another, namely: When a person who has next to no theoretical knowledge of grammar, semantics and related issues but who is a competent speaker of common English is asked if ‘hard’ means the same in both of its occurrences in the sentence: ‘The chairs were hard, the questions, too, were hard’, he or she understands that what is at stake here is whether ‘hard’ changes its meaning according to which word: ‘chairs’ or ‘questions’ it is an attribute of. However, in ‘Every male parent is male’ there are no obvious hints as to whether we should explain the meaning of ‘male’ in a different, not just differentlooking, way both times it occurs in the sentence. More importantly, however, no facts are known, or at least widely known, that might provoke the thought that the meaning of ‘male’ in the sentence under consideration should be explained in two different ways for both of its occurrences in that sentence. Suppose, however, that one

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day gender change should become a frequent affair, so that most people knew that there were many people who had changed gender and that quite a few of them might have become parents before the gender change. In a world like that, it may not be quite obvious that both occurrences of ‘male’ in the sentence ‘All male parents are male’ should carry the same meaning. For it might appear that a correct way of explaining the meaning of ‘male parent’ is something like ‘a person parented a child or children and was male at the time when he or she did so’. The sentence ‘A person who parented a child or children and was male at the time when he or she did so is male’ may, however, not be true, and least of all is it an instance of a truth of logic. This holds for the imaginary world here introduced as it does for the actual world, but the difference is that in the actual world no-one would see the need to explain the meaning of ‘male parent’ with reference to time. As a result, in the actual world the sentence ‘All fathers are male’ would come out as true in virtue of its logical form and the meaning of the constituent expressions, just as it would not in the imaginary world  not so remote from ours, perhaps – here characterized. It might seem that the sentence ‘Every father is male’ could have remained what it is in our world if certain things had been said and added to it. For instance, ‘at the time when they become fathers’ could be added, so as to yield ‘All fathers are male at the time when they become fathers’. This is right; however, such additions and modifications will not be done as long as there does not appear to be any need for them. And as it happens, AD 1997 there is no need (yet) to bother about the time when someone has become a father. Besides, even if this particular modification were made, there would still be no guarantee that there are not various other ways in which the sentence ‘Every father is male’ might turn out to be false, or possibly false. A catalogue of all possible ways in which the sentence in question could and should be protected against reality does not exist. What all of this amounts to is that sentences that are true in virtue of meaning may cease to be so, and to be true purely and simply, not because the ways of explaining the meaning of their constituent expressions turned out to be false, but the sentences themselves do not say certain things that appear redundant. This is a second way, after the first one set forth in the previous subsection, that analytic sentences turn out to be vulnerable to all kinds of empirical objections, as well as to invite various kinds of speculations about the “synthetic a priori.” Now the truth of these redundant things, as well as the truth that they are redundant (if truth it be), might be of purely logical nature, in which case

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all is well. It might, however, be of empirical nature (as it seems to be in the case just discussed), in which case we have, again, to say that the analytic sentence under consideration has an empirical component. Or else, it can fall within a third, “trouble-making,” category, analogous to the third case discussed in the previous subsection. This is where the spectre of the “synthetic a priori” comes in again. For illustration, let us revert to the colour sentence ‘Nothing can be red and green all over at the same time’. As we saw above, given certain ways of unpacking the meaning of ‘red’ and ‘green’, judged, as I hope, correct, this sentence can be turned into an instance of a truth of logic, namely, into the sentence ‘Nothing is the colour of blood all over and is not even spotwise the colour of grass and is the colour of grass all over at the same time’. As we also saw, the phrase ‘all over at the same time’ had a job to do in this transformation, because without it, certain expressions could not be taken to express the same predicates, without which the logical form of the sentence would not have been that of a truth of logic. As I indicated, the expression ‘all over at the same time’ does this job only under certain conditions. The condition that is of particular interest here is this: It is to be assumed that there are no other ways, except differences in time and in distribution in space, that things can be red in a different way than they are green. For imagine a world in which this were not true and each thing at least could, and some even did, have two colours: one warm one and one cold one. In this world, the sentence just quoted would not necessarily be a logical truth, since there could be things that are both red all over and green all over at the same time, if only their redness were a warm colour and their greenness a cold one, or the other way round. Such things would be the colour of blood all over and not even spotwise the colour of grass (whereby the warm variant of the colour of grass would be meant) and yet they would also be the colour of grass (whereby the cold variant of the colour of grass would be meant). However, even in a world like that the way of explaining the meaning of ‘is red’ employed here would not be incorrect since everyone knew that what was meant by ‘is not even spotwise the colour of grass’ was the warm variant of the colour of grass. The sentence ‘Nothing is the colour of blood all over and is not even spotwise the colour of grass and is the colour of grass all over at the same time’ would be empirically false, the appearance of being a logical truth being due to the fact that it was only an abbreviated form of the sentence ‘Nothing is the warm colour of blood all over and is not even spotwise the warm colour of grass and is the cold colour of grass all over at the same time’.

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Things would have been different if the original sentence had been, not ‘Nothing is red and green all over and at the same time’, but ‘Nothing is red and green in exactly the same ways and respects, whatever these might be’. The latter, however, would be a different sentence, nonequivalent with the former. It can be argued that what people mean by ‘all over and at the same time’ is more adequately expressed in ‘in all ways and respects, whatever these might be’. This may be true: People who say the original sentence might mean the latter sentence. But the latter sentence is no correct way of explaining what the former means, and neither is ‘in all ways and respects, whatever these might be’ a correct way of explaining the meaning of ‘all over at the same time’. And even if people mean that nothing can be red and green in exactly the same ways and respects, whatever these might be, when they say ‘Nothing is red and green all over at the same time’, they go far beyond the limits of what is epistemologically defensible, because they simply do not know, due to their finitude, just what those “other ways and respects” might be. As a conclusion: analytic sentences can very well be defined without circularity, but the definition I have proposed here cannot serve as a warrant that they are also true. Establishing that they are must remain a matter of further, in part empirical, study. Moreover, the spectre of the “synthetic a priori” cannot be chased away by appeal to the analyticity of certain sentences, because it is nothing else than that analyticity itself that gives rise to the spectre. Of the two evils, delving into the empirical study of the world of what happens and is the case is the lesser one; the role of philosophy is to be among those who say where to delve.

Acknowledgements This work has been written thanks to the support of the Fonds zur Förderung der wissenschaftlichen Forschung of Vienna (Austria), grant number: P8661-HIS, and the Alexander-von-Humboldt Foundation of Bonn (Germany). The author wishes to express his gratitude to Prof. Barry Smith of SUNY at Buffalo (USA) for helpful comments on, and to dr Agnieszka Lekka-Kowalik of the University of Neuchâtel (Switzerland) for a penetrating criticism of an earlier version of this paper. Also, he has greatly profited from remarks and comments supplied by his audience at a talk on a topic related to that of the present paper delivered at the University of Ljubljana (Slovenia) in December 1995, especially by

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professors: Matjaž Potrþ, Andrej Ule, and Marko Uršiþ. The essay was finished on the day on which the writer’s son, Kazimierz Franziskus Kilian was born (and so the author knows what he is talking about whenever he mentions ‘male parentage’).

Uniwersytet Zielonogórski Department of Philosophy al. Wojska Polskiego 71a 65-762 Zielona Góra, Poland e-mail: [email protected]

REFERENCES Austin, J.L. (1961). Philosophical Papers. Oxford: Clarendon Press. Bolzano, B. (1972). Theory of Science. Translated and edited by R. George. Oxford: Blackwell. Carnap, R. (1958). Beobachtungssprache und theoretische Sprache. Dialectica 12, 235-248. Gupta, A. and N. Belnap (1993). The Revision Theory of Truth. Cambridge, MA: The MIT Press. Putnam, H. (1956). Reds, Greens, and Logical Analysis. The Philosophical Review 65, 206-217. Quine, W.v.O. (1960). Word and Object. New York / London: The MIT Press and John Wiley. Saito, S. (1962). Circular Definitions and Analyticity. Inquiry 5, 158-162. Smith, B. (1991). Grundlegung eines fallibilistischen Apriorismus. In: N. Leser, J. Seifert and K. Pflitzner (eds.), Die Gedankenwelt Sir Karl Poppers: Kritischer Rationalismus im Dialog, pp. 393-411. Heidelberg: Carl Winter. WoleĔski, J. (1993). Metamatematyka a epistemologia [Metamathematics versus Epistemology, in Polish]. Warszawa: Wydawnictwo Naukowe PWN. ĩeáaniec, W. (1992). Fathers, Kings, and Promises. Husserl Studies 9, 147-177. ĩeáaniec, W. (1996). The Recalcitrant Synthetic A Priori. Lublin: Artom. http://www.unige.ch/lettres/philo/enseignants/fab/recalcitrant.zelaniec.pdf ĩeáaniec, W. (1996-1997). Red and Green After All These Years. Acta Analytica 16, 223-239. ĩeáaniec, W. (2003). Some Peculiarities of Synthetic Components of (Meaning) Polstulates. Justification, Truth, and Belief, http://www.jtb-forum.pl/jtb/ papers/wz_sposc.pdf

Józef Misiek DO WE NEED A DEFINITION OF TRUTH?

0. Introduction Plato maintained that truth belongs to the trinity of the highest values. After Plato the concept of truth lost some of its significance. Maybe, under the influence of logic, it was decided that truth is merely a truth value, one of the two truth values of statements, just one of the numerous signs used in logic. In this way, the ground for the more common treatment of truth was prepared. Nowadays, the concept of truth attracts the attention of many a philosopher. It is not the aim of this article to join in the ongoing debates. It is, rather, to attempt an analysis of the tacit assumptions adopted by analytic philosophers in the discussion of this problem. The starting point for our considerations will be the problem of truth as it was posed by Ajdukiewicz in his influential Zagadnienia i kierunki filozofii (Warszawa [1949] 1983). Our goal consists in emphasizing that truth is a value which cannot be defined. This is quite obvious for every philosopher who treats Plato’s philosophy seriously.

1. The Issue of Truth According to Ajdukiewicz, the question of truth is the problem of the truthfulness of statements or thoughts. He identifies this problem with that of defining the concept of truthfulness. Ajdukiewicz does not explain why the concept of truth should require a definition. He begins with the classical definition of truth and presents various objections against it. He comes to the conclusion that the classical definition is vague and, therefore, a precise reformulation is necessary. Having discussed nonclassical definitions Ajdukiewicz gives his own formulation of the

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 445-465. Amsterdam/New York, NY: Rodopi, 2006.

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classical one. In my opinion it displays the same degree of precision as the original formulation. However, I am not going to criticize this version of the definition of truth. I would rather emphasise the fact hat he does not even try to explain why any definition of truth is necessary. Ajdukiewicz’s acceptance of the necessity of such a definition, as well as the lack of explanations in this matter, will be the object of further considerations.

2. Aristotle’s Definition The need for defining truth is usually derived from Aristotle. While trying to find arguments to prove the law of the excluded middle, this eminent philosopher claimed in Metaphysics (IV, vii, 1, in Tredennick translation; Greek text ī, 1011b, 27) that: To say that what is is not or that what is not is, is false; but to say that what is is and what is not is not, is true.

While we must agree with this opinion, the question arises whether it is a definition or rather a statement. Aristotle, himself, calls it a definition or at least such is the standard translation of the Greek term ȩȡȚȗȦ used by him. It is interesting that this term has about ten different linguistic equivalents in modern languages. What is surprising is the fact that it is always translated as a definition. This agreement requires second thought. One point seems to be evident: if Aristotle’s formula is to be taken as a definition then we have to accept two kinds of definitions. The first one comprises definitions which serve the purpose of explaining definiendum by means of definiens. Definitions of the first kind are useful for those who do not understand definiendum but they do understand definiens. The second kind requires understanding of both definiens and definiendum. Such a demand is obvious in the case of Aristotle’s definition of truth and even more in the definition proposed by Ajdukiewicz. One may wonder who can benefit of definitions of this kind.

3. Aristotle and the Analytic Tradition Let us associate two facts. Analytic philosophy, represented here by Ajdukiewicz, claims that the concept of truth should be defined. This,

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according to him, is the proper formulation of the problem of truth. On the other hand, it is well known that Aristotle is the spiritual father of scholastic and neo-scholastic philosophy. It seems that analytic philosophy is a modern branch of this philosophical current. This is why it is not surprising that Aristotle’s translators, when interpreting the works of this ancient author, use the terminology elaborated by (neo)scholastic or analytic philosophy. Particularly when a dictionary gives them too many options. In this way, we have established that the agreement among Aristotle’s translators and the agreement among analytic philosophers regarding the need for defining truth may be two sides of one coin. We are not going to examine here whether all translations of Aristotle’s Metaphysics are actually in agreement with the newer ones as regards the term ‘definition’. This is the task for historians of philosophy. Neither are we going to investigate whether the philosophers remaining under the influence of Aristotle’s philosophy have always claimed that the problem of truth means searching for a definition of truth. We shall be satisfied with a much weaker conclusion: Aristotle’s authority does not provide or, at least, does not need to provide any additional evidence for the opinion, popular among analytic philosophers, that the problem of truth is the problem of defining this concept. His authority does not provide additional evidence because there is no certainty that it is quite independent, for it may reflect the opinions of translators influenced by analytic or (neo)scholastic philosophy rather than that of Aristotle himself.

4. Reasons for Defining Concepts We should ask why the concept of truth ought to be defined. Ajdukiewicz does not give any reasons apart from the one that the classical definition is not clear. In order to answer the question one has to analyse the fundamental presuppositions of analytic philosophy. Analytic philosophy is convinced that the main obstacle in the progress of philosophy lies in imprecise language. Therefore, in order to avoid language traps and secure the advancement of philosophy, one has to make philosophical concepts more precise. We can agree with such a thesis even though it is not absolutely clear which concepts require further elaboration, to what extent they should be made precise, what instruments should be used for that purpose and finally, what concepts should be adopted as the basis for this task. By the “basis” we understand

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the set of concepts which are recognized as precise enough to be used for making all other concepts precise. Also, a method of making concepts precise by isolating the different shades of meaning of one term or by defining a term seems to be important for such work. Analytic philosophy in its logical empiricist version as well as, it seems, the Lvov-Warsaw school supports the above mentioned conviction. Sometimes analytic philosophers go even further: they claim that only formal languages are precise enough while natural language is not. This means that according to both schools natural language is not appropriate for the sciences, that is, for the precise communication of cognitive results. If a philosopher uses it, he either wants to popularise knowledge or does not yet have at his disposal a proper formal language, which is the only appropriate instrument for the communication of thought. In connection with the above presuppositions of analytic philosophy we should note that although one can strive for the precision of all concepts, definitions can at best be only formulated for those concepts that are already very precise. This is why, if it were true that the concept of truth was vague we would have to reject the possibility of defining this concept. In our further considerations we shall try to show – contrary to analytic philosophy – that the concept of truth is clear and, at the same time, that no definition of this concept is either necessary or possible.

5. The Reason for Defining the Truth Let us start with the simplest question – can the concept of truth be defined at all? Obviously, each analytic philosopher knows perfectly well that not all concepts can be defined. If someone tries to, they are threatened with vicious circularity. Thus, if not every concept can be defined, we should consider which concepts can be defined and are worth defining and which are not. I have not yet come across a work in which this problem is be posed. Yet, I do not claim that it has never been posed or even solved. Judging from the common tendency to define the concept of truth, such work does exist and, maybe, even includes a promise that finding the definition of truth is within reach. The author of this paper must admit humbly that he has never come across it. On the other hand, however, it can be noticed that the course which the debates on the definition of truth have been taking suggests something quite contrary: more and more “theoreticians of truth” come to nihilistic conclusions. This seems to indicate that before someone finally

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finds such a definition the concept of truth will be banished from philosophy. Let us ask further questions. Is the concept of truth less clear than other philosophical concepts? It seems to be the opposite. This concept is quite clear provided that the criteria of clarity imposed on it are not so demanding that even mathematics would find them too difficult to comply with. Here we shall use an argument ad hominem which, contrary to argument ad personam, is not a logical error. If a philosopher demands a definition of the concept of truth, it means that he does not understand this concept or understands it less than other concepts. Can such a philosopher evaluate the correctness of the proposed definition? I do not think so. Therefore, he should not take part in the debate on the concept of truth. Besides, if someone does not understand basic philosophical concepts, he is not a philosopher. It is of no use to discuss philosophy with such a person. The discussion would be futile. The above arguments certainly lead to the following conclusion. Even if someone believes that the concept of truth is vague, they must bear in mind that it is vague to some extent only. In other words, the concept of truth must be clear for him in some contexts and vague in others. In this situation, the decision whether the concept of truth should be defined must depend on the degree of its vagueness. If the degree is high, there is no sense in seeking the definition; one should rather look for some nondefinitional way of making this concept more precise. Even discriminating between the contexts in which the concept of truth becomes vague may bring about considerable progress.

6. Radicalism of the Analytic Philosophy In general, analytic philosophers are not inclined to pose the problem of truth in such a way. This might be caused by the influence of Wittgenstein, who promised the ultimate solutions to all philosophical problems. As can be seen from the subsequent text, radical attempts at solving the problem of truth may only achieve anything through the elimination of the concept of truth from philosophy. In just this way, the radical and ultimate solution of all philosophical problems consists in the extermination of these problems.

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7. The Liar Paradox There is, however, one real reason why the concept of truth may be considered as apparently clear. This is provided by the liar paradox. The very fact of its existence may be recognized as proof that the concept of truth is inherently obscure. There are many different versions of this paradox. Yet, in each of them the problem is to formulate a sentence that claims its own falsity or to formulate a sequence of sentences which, referring to one another, lead to the deduction that at least one of them is both true and false. For instance, let us imagine a sheet of paper which on one side has the following inscription: The sentence on the other side is true. while on the other side we read: The sentence on the other side is false. If the sentence on the front is true, it is as it says and therefore the sentence on the back is true. Thus, things are as the sentence on the back says they are, that is, the sentence on the front is false. If, however, we assume that the sentence on the front side is, then the sentence on the back is false and therefore the sentence on the front must be true. In this way we have come to the conclusion that if the sentence on the front is true then it is false, while if it is false, it is true. In other words, the sentence on the front is true if and only if it is false. The same may be proved for the sentence on the back. In this way, making use of the sense of the predicate true and assuming ordinary laws of logic, we come to the conclusion that the sentences considered are both true and false.

8. Questions Implied by the Paradox The liar paradox used as an argument to prove the vagueness of the concept of truth and thus to justify the need for defining truth provokes certain questions. Where does the paradox come from? Is the vagueness of the concept of truth its actual source? It seems that vagueness of truth is not important here. The whole reasoning that leads to the paradox is perfectly clear. If there is something vague, it is just the sentences that lead to the paradox. We are not going to claim here that in everyday speech we never use sentences

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that refer to themselves – directly or through mediation of other sentences. Undoubtedly, such sentences are sometimes very useful. The vagueness we have in mind regards another problem: does the paradox threaten only the person who has used the paradox-generating sentences or does it threaten any person who uses the same language as that person? In other words, it is a question about the nature of natural language: Do there exist only definite acts of speech whose multiplicity is called natural language or can natural language be treated as a system which is governed by its own internal rules and which can be considered independently of definite speech acts, that is, independently of persons who use this language?

9. The Conception of Language in Analytic Philosophy Analytic philosophy can be characterized as a philosophical approach that admires the strictness of formal languages and therefore perceives natural language as an imperfect version of a formal language. Additionally, analytic philosophy is interested in one aspect of language only: it recognizes language exclusively as an instrument for transmitting information  objective information and therefore not connected with this or any other person. It is a very narrow conception of natural language, clearly modelled on formal languages. This is why analytic philosophy treats grammatical rules of natural languages as if they were analogous to syntactic rules governing formal languages. According to such a view, the issue of paradoxes in natural language depends only on the decision whether the sentence (or sentences) that leads to contradiction is (are) in accord with the grammatical rules governing the natural language or not. Many persons believe that expressions which generate paradoxes are apparent sentences because, as Wittgenstein put it in Tractatus, thesis 3.332: No proposition can make a statement about itself, because a propositional sign cannot be contained in itself (that is the whole of the “theory of types”).

Logic and later analytic philosophy have rejected such a solution to the problem of paradoxes. It turned out too restrictive because it forbids all diagonal ways of reasoning which are indispensable in mathematics. It has not, however, been substituted with any other solution – as far as natural language is concerned. This is so, because analytic philosophers never paid serious attention to natural language. Neither have logicians.

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10. A Possible Solution of the Paradox The above indicates why the discussion of the liar paradox in natural language requires a different approach. We must not claim that these or other expressions of natural language are not proper sentences or, at least, we must not do so on the basis of the fact that they are not in accord with the rules allegedly governing the syntax of this language. We should rather notice that natural language consists of speech acts and these are connected with particular persons who use certain expressions in a particular linguistic and extra-linguistic context. They use speech acts in order to convey the particular sense they have in mind and doing this they rely upon linguistic skills they have, not on grammatical or syntactic rules. In other words, we must not imagine that natural language is just a less perfect version of any formal language. Then we realize that in natural language we never have to do with abstract sentences but always with definite speech acts. Thus, if a person uses sentences that lead to a semantic paradox like, e.g., the liar paradox, it is a contradiction in which a particular person is involved. This contradiction pertains to other persons inasmuch as they themselves use such expressions. Therefore, the statement that a paradox has occurred in natural language makes no sense; the contradiction does not pertain to a language understood in an abstract way, because such an understanding of natural language stems from the erroneous identification of such a language with formal language. The contradiction pertains to the speech acts of a particular person. The above claim will be illustrated in the next section. Here, to reject the temptation of perceiving natural language as if it were a formal language, we just wish to stress that formal language is not a language (in the sense in which we understand natural language) because it is just a mathematical structure called a free semi-group with a finite or infinite number of generators (Tarski 1983, p. 174). And this must be kept in mind, even if one accepts the view that formal languages model certain properties of natural languages. Similarly, physical space is not the same as three-dimensional Euclidean space. This remains true even when all the properties of Euclidean space are ascribed to physical space.

11. The Crocodile Paradox Let us now consider another case of reasoning, just to illustrate the fact that in some cases contradiction need not be recognized as a genuine

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paradox (see Ajdukiewicz 1985, vol. 1, p. 137). It is a well-known story dating back to antiquity. A crocodile snatched a child. Answering the mother’s plea to give her child back, the crocodile gave her the following promise “I shall give the child back to you if and only if you give me a true answer as to whether I give it back.” The woman said, “You will not give me my child back.” The crocodile responded that in this case he would not give the child back because she had said so herself. The mother replied to this that he should give the child back because her answer to his question was true. Further discussion between these two showed that the crocodile had made a promise which he could not keep – he should give the child back if and only if he did not give it back. We may suspect that the mother knew the liar paradox well and used it as a weapon against the crocodile. We do not know whether she managed to recover the child but she certainly proved the crocodile’s stupidity. And in antiquity – unlike today – reason was highly valued. This is why such an injury inflicted on the crocodile had to be lethal. Therefore, we may hope that the mother recovered her child. Thus, the crocodile story is justly recognized as not being a paradox. If the crocodile had been wiser he would not have made such a foolish promise. So why should we believe that there is a liar paradox in natural language? Is there any difference between the liar’s statement and the crocodile’s promise which allows us to call one contradiction a genuine paradox of great importance for the theory of knowledge and recognize the other, very similar contradiction, as a mere puzzle? Is there really nothing to stop us from using contradiction generating sentences like those quoted above? Surely, we possess common sense which allows us to realize that the very use of certain expressions, grammatically similar to other propositions, leads to contradiction. We need not count any expression as a proposition only because it is constructed like propositions. And if we have used such proposition-like expressions we can always recognize our mistake and correct it. The more so that expressions of this kind do not carry any content apart from claiming the truth or falsity of themselves or other sentences from the contradiction generating set. Thus they are not propositions communicating any thought concerning anything but themselves. They are not necessary to communicate anything. They do not communicate anything but they suggest something, e.g. that the person who has used them is either as intelligent as a crocodile or, just the opposite, is a sophist who wants to have fun at our expense. Therefore, we can see no reason why the liar paradox should be treated in a different way than the crocodile story.

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12. A Didactic Argument The view defended here can be reduced to the following thesis: we must not say that the liar paradox occurs in natural language. The liar paradox may occur in the speech acts of people who lack common sense, are absent-minded or want to make a joke. Defective or malicious use of common language by some people does not make other people use it in a similar way. To prove such an obligation we would have to make an additional assumption: that natural language is not an instrument for communicating thoughts between individual people but it is an object which exists like formal languages do and does not depend on the people who use it. In other words, we would have to assume that it is not the user’s linguistic competence but mechanical syntactic rules that decide on the acceptability of certain expressions. To put it short, natural language would have be conceived as a (corrupt) version of a formal language. If the above is true, we should say that the liar paradox constitutes a problem in formal languages only. This is why the presented propositions lead to a paradox on condition that they are understood as paraphrases of formal language sentences in natural language. Such diagnosis seems to be confirmed by the students’ standard response to this paradox. If they do not have sufficient training in formal logic, i.e. when they have not yet assimilated the concept of formal language, they usually are unable to see any problem when confronted with the paradox. But students who have mastered the concept of formal language and therefore are able to understand an expression in natural language as a paraphrase of a formal language expression, can see the difficulty at once. A philosopher preoccupied with logic, in a vain hope that he will find in it a universal instrument for solving philosophical problems, is tempted to think about natural language as if it were a formal one. He is even encouraged to commit such a mistake. For instance, competent logicians who discuss the problems of formal languages usually apply paraphrases of formal expressions in natural language. This habit has certain didactic advantages, because it facilitates a discussion of the heart of the matter without considering the intricate details concerning of formal language. It can also claim a long tradition: from Aristotle, through Frege and Russell, until today. Nevertheless, it may be and often is, a cause of misunderstandings. And we claim that such a misunderstanding is the source of the conviction that natural language is just a corrupted formal language and consequently that the liar paradox is a serious problem in both kinds of languages.

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13. Conclusions Following from the Discussion of the Paradox We have tried to show that the liar paradox causes no problems in natural language unless this language is assumed to have the properties of formal languages. This is why the paradox cannot be used as an argument to prove the lack of clarity of the concept of truth in natural language. Thus, we have undermined the justification of the thesis that the concept of truth is vague. Though we have not yet shown that the concept of truth is clear, we shall do it in further in this paper.

14. An Argument from Tarski’s Definition So far we have tried to demonstrate that the argument which indicates vagueness of the concept of truth in natural language and which is based on the liar paradox issues from the confusion of natural language with formal language. This argument is usually supported with another one, taken from Tarski’s works. It is well-known that in his analysis of the concept of truth Tarski established criteria for the material adequacy and formal correctness of the definition of truth. Using those criteria he managed to formulate the definition of truth for expressions of some formal languages. The philosophers who are in favour of defining the concept of truth claim that Tarski solved the problem of truth for formal languages. Following him they want to solve the philosophical problem of truth, that is, to formulate a definition of truth for propositions of natural language. Thus they conceive such solution as a natural completion of Tarski’s work. It seems to be the gist of their argument: the problem of truth posed by Tarski have found only partial solution. This partial solution is a kind of guarantee that general solution is possible. The supporters of the latest argument usually prefer to forget that Tarski also proved that the concept of truth in natural language could not be defined. Let us also forget it for a while, too. In this way our further argumentation will be easier. We shall make one more simplification. When analysing the argument from Tarski’s works we shall accept the interpretation of his definition which is accepted in modern logic. Only later shall we try to get rid of both simplifications.

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15. What has Tarski Achieved? Tarski has undoubtedly constructed the definition of truth for formal languages. He constructed it in the same language in which these formal languages had been defined. This means that he constructed a definition of truth in natural language which, in logicians’ terminology, is referred to by a misnomer: the name of metalanguage. The very fact that Tarski formulated such a definition shows that he understood the concept of truth. That is, he understood the concept of truth referring to the propositions of natural language and he understood the concept of truth referring to sentences of formal languages: he recognized both these concepts as clear – if not on the level of declarations, then on the level of hidden assumptions which were decisive for the effectiveness of his efforts. What is more, even the Tarski’s procedure of defining the concept of truth can be perceived as a procedure reducing the latter concept to the former one, i.e., reducing the concept of truth which refers to sentences of a formal language to the concept of truth which refers to propositions of natural language. Therefore, if the concept of truth referring to natural language were vague, then the concept of truth defined by Tarski would also have to be vague.

16. A Clash of Arguments We can now clearly see that the claim that the concept of truth in natural language is vague clashes with the argument from Tarski’s definition. Everyone who maintains that the concept of truth in natural language is vague must maintain that Tarski did not solve the problem of truth in logic. Everyone who believes that Tarski solved the problem of truth in logic must agree that the concept of truth in natural language is clear. Thus, we have shown that the two arguments which seemingly lead to the vagueness of the concept of truth at closer scrutiny turn out to be not only incompatible with each other but contradictory. We have also shown that the argument from the liar paradox can be explained as a mistake issuing from confusing formal language with everyday speech. And, if we reject this argument, there is nothing to stop us from recognizing the argument from Tarski’s definition as the best proof of the perfect clarity of the concept of truth in natural language: if Tarski’s definition of truth is grounded in the concept of truth in natural language then that concept has to be as clear as the concept he defined.

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This is why, if the concept of truth referring to natural language propositions is clear, no analytic philosopher should demand a definition of this concept in order to make the very concept clear. Philosophers should not forget, that too many concepts in natural language (and particularly in philosophical language) are vague and it is those concepts that demand more precise description. Thus, striving for a definition of the concept which is clear and which would probably have to be defined using some of those less clear concepts would be a waste of time and energy. It is much more reasonable to use clear concepts to define those which are less clear. Discussing the interpretation of Tarski’s achievements we have said that Tarski must have understood the concept of the truth of formal language sentences first and only then could he have attempted to define this concept. One could ask what is the need for a definition when the concept is clear and comprehensible. This is why we must cope with an additional problem: why did Tarski decide to define the concept of truth in formal languages if this concept was clear enough, also to him, even before it was defined?

17. Why Should Clear Concepts Be Defined? The solution to the above problem is quite simple, though not trivial. We must stress the fact that even clear concepts may display a different degree of clarity and that the degree of a concept’s clarity depends on a context in which it is used. We are not going to claim that the clarity of the concept of truth depends, for instance, on the weather. We only want to indicate that an ordinary context in which a philosopher considers this concept is connected with individual propositions. Indeed, I have not yet encountered a philosophical work which would discuss the problem of truth in reference to a set of propositions. Yet, in logic it is essential to speak of the truth of sets of sentences and, what is more, of infinite sets of sentences. It is a general rule that what is clear for finite sets may not be clear for infinite ones. Hence the need for a definition of the concept of truth which would refer to infinite sets of sentences. Tarski’s definition is an answer to this need. The above considerations should perhaps be completed with a more general reflection – in fact, only clear concepts can be defined. Vague concepts may be only analysed and explained. But explaining is something different from defining. It is a subtle activity, consisting of

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several stages. And only in exceptional cases might it happen that the scholar will reach the last stage  a correct definition. This shows, in particular, that the concept of truth in natural languages does not require defining because philosophy, at least at the present moment, does not need to consider infinite sets of propositions in natural language. Not only does this concept not require defining, it does not even allow defining. The only thing that can be done is defining the concept of truth with the help of other semantic concepts belonging to natural language. Such a definition could probably satisfy the perverse taste analytic philosophers have for seeking definitions of concepts with a zeal equal to that with which medieval alchemists searched for the philosopher’s stone. We cannot see any other use of such a definition.

18. A Historical Complication The argumentation presented above may easily raise one objection regarding the sense of Tarski’s definition of truth. For the sake of the presentation we have accepted the sense preferred by modern logicians. We must admit that Tarski himself interprets his definition in a slightly different way and it is easy to see his hesitation in his subsequent papers on this matter. Thus, to complete our argumentation we must explain why Tarski did not immediately accept the interpretation preferred by modern logicians. The question posed in this way is, in a sense, quite trivial. It is a rule in science that the discoverer shows other researchers new horizons, but his interpretation need not be correct in every detail. For instance, Christopher Columbus discovered America but was convinced that he had merely reached the Indies. There is another thing, which is perhaps less trivial: to see in what way the philosophy accepted by the scholar was an obstacle upon his way to discovery and how he succeeded in overcoming it. The opening paragraph of Tarski’s work “Concept of Truth in Formalized Languages” (1956, p. 152) ends with the following statement (p. 165): If these observations are correct, then the very possibility of a consistent use of the expression ‘true sentence’ which is in harmony with the laws of logic and the spirit of natural language seems to be very questionable and consequently the same doubt attaches to the possibility of constructing a correct definition of this expression.

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In a later work “Semantic Conception of Truth and the Foundations of Semantics” (Tarski 1952, p. 13) Tarski’s approach is more cautious. Having emphasised the fact that all languages which (I) are semantically closed and (II) yield to the ordinary laws of logic, must be inconsistent, he says that (p. 21): The problem arises as to the position of natural language with regard to this point. At first blush it would seem that this language satisfies both assumptions (I) and (II) and that therefore it must be inconsistent. But actually the case is not so simple. Our natural language is certainly not one with an exactly specified structure. We do not know precisely which expressions are sentences and we know even to a smaller degree which sentences are to be taken as assertible. Thus the problem of consistency has no exact meaning with respect to this language.

It is clear that Tarski’s approach to the question of the consistent use of the concept of truth, i.e. the very problem of definability of the concept of truth in natural language, differs. But more important is another point. In the previous section we stated that Tarski’s definition is, in a sense, a reduction of the concept of truth pertaining to formal languages to the concept of truth as referring to the propositions of natural language. Therefore, if Tarski indeed recognized the concept of truth referring to natural language propositions as vague, a serious difficulty arises. How can his definition of truth explain the concept of truth of formal language propositions if the explanation he gives is based on the vague concept of truth referring to natural language propositions? It seems that Tarski did not notice this difficulty. We shall now try to explain how this might have happened.

19. An Explanation of the Complications To better understand the situation in which Tarski formulated his definition and to comprehend his hesitation in interpreting this discovery, we must realize that between the 1920s and 1940s the term logic meant something completely different from first order logic. In those days it was the theory of types, i.e. either the original version of this theory, that is the so-called ramified theory of types, or the simple theory of types elaborated by Ramsay. Both these systems can be recognized as logical calculi of infinite order. Besides, they are both characterized by the fact that they can be conceived as an infinite sequence of logical calculi of nth order, where n may be any natural number. Each of these calculi is expressed in its own formal language which is included in all languages

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of a higher order and includes all languages of a lower order. Tarski himself worked within a theory called the algebra of classes, which is a further simplification of the simple theory of types and which can also be regarded as a logical calculus of infinite order. On the other hand, Tarski was subject to the influence of the nominalist philosophy proclaimed by the Lvov-Warsaw school (LeĞniewski and KotarbiĔski) as well as by logical empiricism. We must stress, however, that he never treated nominalism as a dogma: when he saw that the acceptance of certain existential assumptions incompatible with nominalism was necessary for a particular proof, he acted like a Platonist. This is the crux of difference between the nominalism of mathematicians and the nominalism of philosophers. Philosophers perceive nominalism as a sober and elegant philosophical view while mathematicians, as soon as they stop philosophising and start to deal with mathematics, perceive that nominalism exerts a devastating influence on their work. This is why a philosopher can be a dogmatic nominalist while a mathematician can not. Tarski’s central conviction is that expressions of formal languages may be conceived as non-abstract objects, e.g. as definite inscriptions. It is clearly a nominalist interpretation. Moreover, he was convinced that only formal languages offer the means for precise expression of scientific statements. Thus, when he discovered the need for discerning language and metalanguage he additionally assumed that metalanguage also had to be formalized. He probably believed that only in such a case theorems of logic would become sufficiently precise (here we speak of such theorems as Tarski’s theorem of non-definability of the concept of truth). For this reason Tarski was careful to indicate in what formal language his definition of truth could be expressed. He stressed that if the concept of truth were to be defined for languages of nth order, the definition itself would have to be expressed in a language of the n th +1 order. As we mentioned above, the theory of types provides a sequence of formal languages of higher and higher order. Such a sequence was almost perfectly suited for a nominalist interpretation of his discovery: for any formal language of 4 th order it offers a language of n th +1 order, in which the definition of truth pertaining to expressions of the former language may be formulated. In this way, the postulate that the definition of truth should be expressed in a formal language, could be satisfied. And it was probably the nominalist philosophy which made Tarski focus his attention on formal languages only, which meant he overlooked the difficulty indicated above. The theory of types might also have had some influence here, because it offered formal languages of an arbitrary

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high order. Dealing with these languages one could overlook the fact that the whole theory of types (together with its formal languages) was defined in natural language – the foundation of all other languages.

20. A Change of Perspective in Logic The real problem with the formalization of a metalanguage occurs when we want to formulate a definition of truth for the language of the full theory of types. In order to do this we have to use an informal (meta)language, but even then there is a way to satisfy the metalanguage formalization postulate. It is enough to notice that the hierarchy of logical types can be extended so that it reaches some definite ordinal number. Then, we can construct a formal language (also in natural language) which will facilitate speaking of the whole extended hierarchy. In such a language we can also express the definition of truth for the propositions of the language of the (full) theory of types. Such a solution, however, is just an unnecessary complication introduced in order to satisfy the metalanguage formalization postulate. It is unnecessary, because the definition of truth in a formal metalanguage cannot be constructed until a definition of truth in natural language is found. What is more, the formal metalanguage must be constructed in natural language. Thus, natural language is indispensable anyway. Another circumstance which fostered the change of perspective was the decline of the theory of types. It soon became clear that the hierarchy of logical types coincides with the world described by set theory or rather with a fragment of the latter. In this way logicians discovered that in fact they did not need the theory of types in order to prove their theorems. Set theory is much better for this aim. In other words, they discovered that they need ordinary mathematical language which, in fact, they had always used, even when they declared the necessity and/or possibility of expressing their results in a formal language. When the theory of types was superseded by set theory (on the metalevel), the next stage was reached: they realised that first order logic is not just one among many logical systems but simply the logic. Such a logic is not strong enough for reconstructing in it all mathematics. Therefore, it become clear that such a logic or, rather, logical calculus should be an object of research in a mathematical discipline called logic. And this very discipline as well as other mathematical disciplines contends itself with an ordinary mathematical language which is a specimen of natural language. Thus, they discovered that the declared

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philosophical faith concerning necessity of formalisation was not a condition of scientific precision but just a burden. On the other hand, they realised that formalisation is just a means of constructing the objects of their research. The objects constructed in this way are just regular mathematical objects like real numbers defined via Dedekind’s construction or ordinal numbers constructed as von Neuman sets. In other words, they discovered that the distinction between logic and meta-logic while being temporarily useful is misleading because formal theories are not (and have never been) theories nor formal languages are languages. They are just mathematical objects of research in a sober mathematical discipline called logic. All this necessitated a new interpretation of Tarski’s opinions concerning his definition of truth. In logic such a reinterpretation took place in due course. But in analytic philosophy some technical developments were accepted while old interpretations were retained. In this way analytic philosophy become obsolete in matters of logic.

21. Conclusions In this paper, we have posed the question whether definition of truth is necessary. We started with the conviction, characteristic of analytic philosophy, that the concept of truth in a natural language is vague like all other concepts of such a language. We have also revealed the tacit assumption of analytic philosophy that vague concepts should be defined because only in this way can their vagueness be eliminated. We have considered two standard arguments which could indicate the need for defining the concept of truth. At closer inspection, it turned out that one of them – the argument from the paradox of the liar  could be abolished. First of all, the liar paradox, itself, could be constructed only in formal languages. This is why this argument does not prove that the concept of truth is vague. If an analytic philosopher claims that the concept of truth is vague, it is because he has learned to believe that only defined concepts are clear. Besides, their argument is in conflict with another one, taken from Tarski’s definition. What is more, the second argument supports the opposite thesis: if the explanation of the concept of truth given by Tarski for formal languages is correct then the concept of truth in natural language must be clear because it is this very concept that constitutes the foundation on which Tarski’s definition rests. We have also explained why in Tarski’s works we can find opinions concerning the interpretation of his definition of truth which are not fully

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compatible with the interpretation accepted today. Our explanation refers to certain historical circumstances connected with the state of logic in the 1930s and the philosophical views accepted by Tarski. To put things bluntly, we have presented here the arguments which corroborate the thesis that the problem of the definition of truth in natural language is just an artefact of the tacit philosophical assumptions adopted by analytic philosophy. Such a thesis may be astonishing, for it seems to lead to the conclusion that the criticism of non-classical definitions of truth becomes groundless. This is not our conclusion. We are only going to claim that criticism of the non-classical definitions of truth must be transferred to another chapter named The Problem of the Object of Cognition. This is because all the so called non-classical definitions of truth univocally assume that cognition does not pertain to any actually existing reality. This is why those who adopt such an assumption must do something with the concept of truth which surely serves to evaluate the correspondence between cognition and reality. In this situation, the rejection of the concept of truth would be the simplest solution, but from the point of view of sociotechnique it would be suicidal: Who could accept a philosophy that got rid of the concept of truth? Let us be more cautious: who would have been able to do it before postmodernism come into being? This is why it was best to pursue another, politically beneficial strategy: to accept the term ‘truth’ and give it a meaning different from the one that it had always possessed. And there is no better instrument for manipulating the sense of a commonly understood concept than a precise (or allegedly precise) definition.

22. A Reflection concerning Analytic Philosophy Concluding, we wish to express some more general reflections. In its strivings to make philosophy scientific, analytic philosophy has silently assumed that clarity is the highest cognitive value. Striving for clarity it has adopted a radical thesis that all concepts of natural language are vague; only concepts belonging to formal languages are recognized as clear. This is why this philosophy has become fascinated by formal languages and this is why it has tried to conceive natural language as an imperfect version of a formal one. And, finally, this is why it has postulated defining all important philosophical concepts. No wonder that, having accepted so many false assumptions, it had to adopt another false thesis: that the liar paradox is one more proof of the vagueness of the concept of truth. It did not notice that the paradox

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causes no problems in natural language just because this language is not formal. Neither did it notice that vague concepts could not be defined at all and, therefore, started to pursue the mirage of constructing a definition of truth for natural language. Tarski’s achievements undoubtedly encouraged the pursuit of such a definition because they were understood as a partial realisation of this aim. Such an understanding of Tarski’s achievements issues logically from the previously adopted assumptions but is as false as the assumptions, themselves. In particular, assuming that the concept of truth in natural language is vague undermines the definition of truth constructed for formal languages. Logic has arrived at a proper interpretation of Tarski’s achievements fairly soon, but analytic philosophy still contemplates the logic of 60 years ago or, in other words, its own philosophical views expressed by eminent logicians of those days. For this reason analytic philosophy ceased to inspire logical research a long time ago. What is more, this research may bring interesting results only if logicians reject the views proclaimed by analytic philosophers. On the other hand, the present state of analytic philosophy exerts an inspiring influence on another domain, namely, on so-called postmodernism. The advocates of the latest current, which I only hesitantly would call a philosophy, have come to the conclusion (and not without a reason) that, if the most scientific philosophy is not able to define the concept of truth, this concept should be included in the category of myths which were not exterminated by modernism. As a conservatist hostile to progressive ideas proclaimed by modernism (to say nothing of postmodernism) and as a philosopher once sympathetic to analytic philosophy, I can only despair over its present condition. No doubt, as is usual in philosophy, some people will find the arguments presented above unsound. Therefore, I would like to ask my prospective disputants to declare whether, according to them, the concept of truth in natural language is clear enough/vague to the degree that a definition of this concept is necessary/unnecessary/permissible/impermissible or, perhaps, the other way round.

Acknowledgements I would like to express my cordial gratitude to all those who have read earlier versions of this paper. Their comments and critical remarks were

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instrumental in making essential improvements. They are: prof. dr hab. Jerzy Perzanowski, prof. dr hab. Andrzej WroĔski, dr hab. Jerzy Szymura, dr Marek Suwara, dr Janusz Páazowski. All errors which remain in the text must be counted as my own. Special words of gratitude are due to prof. dr hab. Maria Dzielska for having explained to me the secrets involved in the translation of Aristotle’s works from Greek. I would also like to thank prof. dr hab. ElĪbieta Tabakowska, dr Piotr Mróz and mgr Eva Klimontowicz for their invaluable help in translating the Polish version of this paper into English. The paper benefited from results obtained during the work on research program no. 10204/90/01 Podstawy filozofii analitycznej a filozofia matematyki. The paper is a chapter of a book which is a kind of contemporary commentary to Ajdukiewicz’s manual quoted in the introduction.

Uniwersytet JagielloĔski Department of Philosophy ul. Grodzka 52 31-044 Kraków, Poland e-mail: [email protected]

REFERENCES Ajdukiewicz, K. ([1949] 1983). Zagadnienia i kierunki filozofii [Problems and Trends of Philosophy]. Warszawa: Czytelnik. Ajdukiewicz, K. (1985). Paradoksy staroĪytnych [Paradoxes of the Ancients]. In: JĊzyk i poznanie [Language and Knowledge], vol. 1, pp. 137-144. Warszawa: PWN. Aristotle (1908). Metaphysics. Translated by W.D. Ross. Oxford: Clarendon Press. Tarski, A. (1952). The Semantic Conception of Truth and the Foundations of Semantics. In: L. Linsky (ed.), Semantics and the Philosophy of Language: A Collection of Readings, pp. 13-47. Urbana, IL: University of Illinois Press. Tarski, A. (1956). The Concept of Truth in Formalized Languages. In: A. Tarski, Logic, Semantics, Metamathematics, pp. 152-278. Indianapolis: Hackett, 1983. Wittgenstein, L. (1981). Tractatus Logico-Philosophicus. Edited by D.F. Pears. London: Routledge & Kegan Paul.

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PART VII RATIONALITY: ITS CRITERIA AND DEFINITION

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Ryszard Kleszcz CRITERIA OF RATIONALITY

0. Introduction The problem of rationality and its criteria has been explored in many philosophical disputes in which, as it is not difficult to notice, the very term ‘rationality’ is vague. This is connected with the fact that there exist many fields in which this term may be applied. Moreover, the very notion is both descriptive and prescriptive. Thus, it is obvious that there are many conceptions of rationality. Keeping that in mind I will confine my considerations to the framework of the rationality of beliefs treated as propositional attitudes and as such not susceptible to truth valuation but qualified in terms of rationality. To be rational, the beliefs stemming from different sources should be somehow justified; they should satisfy certain criteria. These criteria are here of our main interest.

1. Remarks on the Traditional Approach Numerous attempts to characterise rationality dwell merely on its conditions (criteria). Their satisfaction is the necessary and sufficient condition of rationality. On the traditional perspective these conditions were rather general and, sometimes, not unequivocal. Traditional rationalism makes recourse to manifest ideas, intellectual intuitions and evidence while traditional empiricism to experience and inductive reasoning. 20th-century representatives of these views formulated precise criteria although they (logical empiricists, for example) did not escape the general frameworks laid down by their predecessors. According to the adherents of logical empiricism scientific knowledge constitutes the ideal and the only feasible model of rationality, and scientific study is the realisation of human rationality. A human can try

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 469-483. Amsterdam/New York, NY: Rodopi, 2006.

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to construct a theory that would pretend to be universally valid for all rational beings. Hence, only in science we could be entirely rational. 1 The criteria of rationality adopted here concern, on the one hand, language (empirical sense) and, on the other hand, empirical justification. We come across similar conditions of rationality in the analytical current of the Lvov-Warsaw School. On metaphilosophical grounds they can not be reduced to logical empiricism. However, the criteria of rationality formulated not by Twardowski himself but by his disciples are similar – they concern language and justification. Ajdukiewicz’s account, which is representative, points at two requirements: 1) intersubjective communicability and 2) intersubjective verifiability.2 Only those beliefs which can be expressed precisely enough by means of words and whose legitimacy is confirmed by everybody (in principle of course and not de facto) may be treated as beliefs worthy of common acceptance.

2. Standard Model of Rationality The postulates of rationality accepted by logical empiricism and particularly within the Lvov-Warsaw School can be treated as the standard approach to the problem of the criteria of rationality. It is not difficult to notice that the approach used by contemporary advocates of rationality and rationalism is similar. K. Szaniawski, a philosopher and disciple of philosophers from the Lvov-Warsaw School who, at the same time, was influenced by an empiricism close to the logical version of it, proposes three conditions of rationality: 1) proper (precise) articulation; 2) respecting the requirements of logic; 3) sufficient justification (Szaniawski 1994, p. 532; 1994a, p. 524). This set of conditions shall be further referred to as the standard conditions of rationality. It should be noticed that this model of rationality runs up against serious difficulties.3 And so in the case of the postulate of proper

1

As S. Amsterdamski notices, this ideal of knowledge is consistent as long as the development of knowledge (and technology) is treated as an absolute good. See Amsterdamski (1989), p. 318. 2 These two conditions appear in K. Ajdukiewicz, and I. Dąmbska mentions expressibility, communicability and inter-subjective controllability. If we assume that what is communicable must be expressible then we can say that both authors have the same point of view regarding criteria of rationality. See Ajdukiewicz (1949), p. 72; Dąmbska (1938), p. 101. 3 It should be stressed that the difficulties analysed further are considered by some contemporary representatives of standard model, for example by K. Szaniawski.

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articulation the question would be if it is possible to distinguish any catalogue of commonly accepted criteria of precision. This problem seems simpler in science; it is more complicated in humanist and philosophical fields where language is natural and where the most basic methodological questions are not decided. Generally, the possibilities to find unique and unequivocal criteria of language precision seem to be inconceivable. As far as respecting the requirements of logic we can see prima facie minor difficulties. For the sake of simplicity we shall call the postulate of respecting the requirements of logic logical rationality.4 Let us now clarify that this kind of rationality is understood as two requirements: one stipulating the avoidance of contradiction; the other presupposing the ability to deduce. This explanation again creates difficulties. The postulate of rigorously avoiding the principle of contradiction would result in making it unacceptable (i.e. beyond the limits of rationality) for any pair of contradictory sentences to appear within a system. Similarly, the requirement that all inferences, even only possible inferences, should be made given a rational system of beliefs, would make non-rational the beliefs that do not satisfy this requirement. Of course, beliefs entertained by real people often contain such contradictions5 since real people are not able to comply with the idealised conception of logical rationality. The requirements forbidding a contradictory set of sentences and demanding the performance of all (at least all realisable) inferences cannot be referred to real people but to creatures possessing solely intellectual aims, a prodigious memory and applying exclusively infallible methods of reasoning. The possibility of the occurrence of contradictions among human beliefs is connected with the above-mentioned features and their finitistic status. This situation calls for giving up the idealised model of logical rationality. The third condition postulates empirical justification. If a statement is to express rational beliefs, it should be properly justified. In this context, Russell’s view seems obvious – beliefs which have no justification cannot be treated as rational (1912, p. 111). It is also known that the

4

I thoroughly discuss this problem in the dissertation “Logical Rationality” (unpublished). Its main ideas were presented in Kleszcz (2001), passim. 5 If beliefs are divided into potential (not activated) and actualised (activated) ones then, mainly among the former, we come across beliefs such that when verbalised yield contradiction. But we become aware of this only after their activation. It is worth mentioning that contradiction in the activated part is sometimes caused by the fact that relations between beliefs are complicated and not clear.

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degree of trust one has in a given statement should not exceed the degree of its justification. However, this problem becomes more complicated since it is disputable even whether science can provide a paradigmatic model of justification. The discussion is devoted to fundamental issues such as, for example, inductive reasoning. And the whole problem is still more complicated when we consider the other dimensions of culture – primarily I mean the way philosophy is understood as irreducible to science. In the standard model one can say that the beliefs which do not fulfil any one of these requirements must be considered irrational. But the lack of clear-cut criteria makes the decision whether the criteria are fulfilled or not, difficult. K. Szaniawski, as it was already mentioned, presents the so-called standard model. However, perceiving the difficulties it raises he states that development in our knowledge of language and in our metascientific consciousness destroy the illusion that we might be able to formulate a catalogue of criteria which can be entirely satisfied. Hence, at present we have some indication of what rationality is rather than having precisely codified instructions which might be applied in a mechanical way (Szaniawski 1994, p. 534). While this thesis is basically correct it bears further elaboration.

3. Critical Remarks on the Standard Model As I have already mentioned, implementation of the standard model comes up against a wide range of problems connected with a rigid system of rules for rationality. The critical remarks come from various directions and point at various obstacles that emerge when the model is applied. Sometimes, the criticism the problem of rationality evokes is much broader and all the approaches that aspire after universal rationality are subjected to it. Critics of that kind can be referred to as relativists. Among them there are some philosophers but it is mainly sociologists and cultural anthropologists who maintain that all criteria of rationality are relative because they are imposed by culture. Hence, when we ask about the rationality of a belief or a theory we refer to some general system in which the given culture is embedded. The same sort of conclusions are found in the late Wittgenstein. He rejects the thesis that there exists a unique language system susceptible to any formal analysis. What follows from such an approach to philosophy of language is that criteria of rationality should vary according to the type of language game

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being played.6 The program of methodological anarchy also shows a similar tendency. Its main exponent, P.K. Feyerabend denies that proceedings (criteria) are rational (or not) independently of the circumstances, thus repudiating the thesis about the existence of a universal rationality that would comply with absolute rules and criteria (Feyerabend 1978, p. 339). It appears that such a perspective is philosophically destructive. Norms of a given culture can never be used as support to make an argument legitimate, sound or plausible. Undoubtedly, the authors who emphasise self-destruction of these approaches are right (Putnam 1981, p. 113). Without entering into details, we shall only mention the incoherence of relativism which is visible in discussions since the times of Plato’s Protagoras. Moreover, Putnam while referring to Wittgenstein shows that a relativist is not able to tell the difference between the following expressions: “being right” and “thinking he is right” (Putnam 1981, p. 122). If we put aside relativism in the version presented above as being invalid, there remains the problem of the standard approach. At this point it would be useful to explore H. Putnam’s ideas. His account goes against the so-called criterial conception according to which the methods of rational justification should provide a list (a canon or catalogue) of rules of rationality. The list would elucidate whether a given statement had any cognitive import or not. Here, the institutional norms decide about what is rational; it is assumed that what is rationally accepted can be criterially verified (Putnam 1981, p. 110). But this very thesis cannot be criterially verified and thus cannot be rationally accepted. Putnam shows that commonly accepted norms do not give grounds to determine if a philosophical argument is rationally justified and every discussion devoted to the nature of rationality subscribes to the concept of rational justification which is much wider than the concept of criterial rationality.7

6

According to Wittgenstein, in many “language games” when we use language it plays different roles that cannot be reduced to one only (1972, Part One, §7). Language games differ from each other in aims, rules, etc., and can be treated as a family of “language games” (1972, Part One, §108). On this view there is no single meaning that corresponds to a given word, meaning can be established only on the basis of the analysis of the “language game.” 7 It should be emphasised that the view of reason in positivism, and thus logical empiricism, is too narrow. See Kleszcz (1981), passim.

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It is apparent that such a criterial conception comprises not only the logical empiricist conceptions but also those of Wittgenstein and of the philosophers of everyday language.8 It can be justly assumed that the objections raised against this conception are justified also in relation to the standard approach. Therefore, regardless the evident advantages it possesses, it must undergo some reinterpretation. According to Putnam it is difficult to determine the concept of rationality. He says that one way to come to a better understanding of the nature of rationality is to develop philosophical conceptions of “rationality.” It is also legitimate to borrow some ideas from ethics. All these may result in the formulation of conditions applicable in every methodology or system of rational procedures. They can be stated as follows: 1) Desire that basic assumption should have wide appeal; 2) warranting that the system is able to withstand rational criticism; 3) the recommended system should be applicable in the sense in which a proposed morality is liveable in our lives. Although Putnam’s considerations cannot be treated as conclusive for the problem of rationality, they are perspicacious and open up new, challenging possibilities to deal with this question.

4. Two-Tier Model of Rationality The foregoing disputes lead to the conclusion that the proposals which provide a set of criteria of rationality are not satisfactory, especially when this set is reduced to a catalogue (a canon) with strictly defined rules of application (Putnam’s critical remarks concerning all criterial conceptions may not be amiss here). The concept of “rationality” they adhere to is too narrow and does not correspond to the one widely accepted in the literature on the subject. Moreover, such a (unitary) model cannot be adequately applied to various types of knowledge and activities. All these can make us arrive at the view that apart from unitary model there should be constructed some “weaker” model (Vermeersch 1974, p. 75) which would maintain the requirements of criticism and at the same time postulate weaker conditions for language precision, weaker standards of effective control of data etc. This approach gives undeniable

8

Wittgenstein maintains that shared norms accepted by a group play the role of institutional norms. All members of the group obey them thus forming a lifestyle. What is disputable is the question whether the institutional norms Putnam speaks about correspond to Wittgenstein’s “forms of life.”

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advantages. However, it should be noticed that when this line is closely followed it turns out necessary to create not one but several “weakened” models adequate in different domains. In virtue of the above considerations I intend to propose a slightly different solution in which two levels of criteria of rationality are to be distinguished. On the first level – the level of meta-principles – we find some general and universal principles which, however, are not “strict” rules. Their application would require that the principles be made more specific by taking the given circumstances into account. Establishing such meta-principles does not imply that we presuppose the existence of an “eternal,” unchanging, transhistorical model of rationality independent of philosophical reflections and of the progress of knowledge (certainly including its formal branches). Moreover, it does not imply the acceptance of the existence of an idealised model of a human, as is the case in the constructs originating from epistemic logic. The second level would differentiate particular types of knowledge, building for them appropriate models of rationality. The fact that beliefs may follow not only detailed principles but also meta-principles satisfies the need to have some universal criteria of rationality. At this point the question arises if it is possible to defend the universal principles against the attacks of those who put special emphasis upon the historical, relativist character of every criterion. However, that attitude, especially in its most radical version, is unjustified. Nevertheless, it should be noticed that even if we say that criteria have a historical character this does imply the conclusion that there are no criteria at all. Also, an advocate of the thesis that the history-transcending point of view does not exist may agree to the existence of meta-principles which would need to be made more specific to apply in concrete situations. I shall now focus my attention on the meta-principles only. The question of their specification calls for more careful study.

5. Meta-Principles of Rationality Meta-principles are general rules relevant to all domains of knowledge and activity. Below, I spell out four such principles not denying that their number is disputable. In this respect I share Davidson’s view who, when speaking about similar notion of the so-called “basic principles of rationality,” recognizes the fact that they do not constitute a complete list. At the same time, he takes notice of the fact that there are principles accepted by all beings possessing propositional grounds or acting

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intentionally. So, every thinking being accepts certain basic standards or norms of rationality (Davidson 1985, p. 351). Thus, being aware that this very matter is replete with possibilities, I shall provide four such metaprinciples that seem to be essential. They are: 1) precision of language; 2) compliance with logical requirements; 3) criticism; 4) ability to solve problems. 5.1. Precision of Language (Articulation) The requirement of the precision of language (articulation) shall be referred to as ‘conceptual rationality’. Let us assume that this requirement implies the elimination of all defective expressions of language (incoherent, vague, unsound expressions). The language should be made precise relatively to the particular aims that are pursued in a given domain. Aristotle’s remark is worth mentioning here, “[. . .] for it is the mark of an educated man to look for precision in each class of things just so far as the nature of the subject admits” (Ethica Nicomachea, 1094b). The fact that it is impossible to reach complete unequivocality within natural languages should not constitute an obstacle. Similarly, the fact that the very concept of “language precision” is rather vague should not undermine its importance; on the contrary, it should make us more careful about its use since when it is not respected our utterance may be unintelligible. The postulate to make language sufficiently precise or even as precise as possible is vital because rationality and rationalism take the existence of common criteria for granted, and this requires a common language. What is more, these tendencies require that communication occurs as it is only through communication that common criteria may be found. On the level of language it can be said that “rationality” and “discursivity” are interconnected. In addition, they are semantically close. Rationality presupposes a language as a necessary condition. Hence, we must admit that Davidson is right when he claims that only beings who have a language can be acknowledged as rational (Davidson 1982, passim). For a living creature to be rational means, according to Davidson’s view, to show [adopt] propositional attitudes. And this is possible only when one has intentions and desires and, finally, a language. Rationality is constituted within the communication process, which emphasises the need to make the language precise.

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5.2. Compliance with Logic The second meta-principle postulates respecting the rules of logic and thus logical rationality. Logical rationality is here connected with two requirements: consistency and suitable capacity for making deductions. Bearing in mind the difficulties emerging when idealised rationality is considered we favour the concept of minimal rationality (Cherniak 1986, p. 8, p. 81; Kleszcz 2001, pp. 68-70). The postulate of minimal rationality requires the fulfilment of a number of conditions. First of all, contradictions appearing in the set of beliefs should be eliminated. That we have a mechanism of contradictionelimination does not mean that every contradiction without exception should be effectively eliminated. If we adopt the condition of minimal rationality, we shall skirt the difficulties and be able to preview the behaviour of a human entertaining a belief. All the same it is useful in the context of disputes occurring in the philosophy of science. 9 Apart from the above considerations the postulate of compliance with logic requires that one should have adequate capacity for making deductions. What is meant here is the requirement of minimal inference. With regard to deducing abilities this requirement is that a subject entertaining a certain set of beliefs should make only valid inferences from this set (Cherniak 1986, p. 12). Minimal rationality (in the sense of minimal inference) would not presuppose logical omniscience. At the same time, having it would constitute a necessary condition for having any beliefs. Here the question arises: to what criteria should one keep if one wants to be rational and to meet the requirement of minimal inference? We would not consider as rational someone who infers in an entirely accidental way. Therefore, some preferences regarding the inferences made are necessary. With this respect it should be mentioned that someone should include not require that the inferences be accurate and feasible but also consider their pragmatic aspects. In other words, we should consider the usefulness of the inferences made relative to the aims that are being striven towards. So, we dealing here with the pragmatic context which settles the question of the usefulness of choice (Foley 1991, p. 366, p. 389).

9

The postulate of minimal consistency could be applied, e.g. to scientific theories in which we meet anomalies. Minimal consistency which does not demand the total elimination of all inconsistencies allows us to treat such theories as rational. As time goes on, the anomalies can be removed.

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5.3. Principles of Criticism At the basis of this meta-principle is the belief that as human beings we are all fallible and what we can know is not definitive or final. Central here is the role of self-criticism i.e., subjecting our own beliefs and statements to criticism. A critical attitude toward our own beliefs ought to be accompanied by a readiness to accept the critical remarks made by others and to consider them objectively. The specification of the postulate of criticism will result in often making use of disparate means within the frameworks of such various domains as science, humanities, philosophy, debates in ethics and religion. In more formal disciplines the criticism will centre on elucidating possible infringement of logical validity; in (empirical) sciences the essential factors are to have recourse to experience (experiment) and to confront the experience with the theoretical theses. In the humanities, broadly understood, interpretation and specific hermeneutics play the central role. Criticism should always be wellfounded, reasonable and justified by suitable arguments. However, the question appears if every legitimate criticism should provide its own assumptions and, moreover, justify them. A positive answer to this question is linked with the belief that any criticism involves certain assumptions and those ought to be justified. The lack of justification would make the criticism illegitimate. But it can be showed, as for example Popper does, that such a requirement raises real doubts (Popper 1980, p. 378). Even if we suppose that every criticism entails holding certain assumptions, we must admit that some of them are commonly accepted. And then the criticism may aim at showing that certain beliefs are not compatible to the commonly accepted ones. Also, it may happen that the criticism is based upon assumptions which belong to the criticised theory. In such cases we can dispense with justification of the assumptions as well. Criticism on principle does not conform to all “immunizing” strategies that tend to raise certain spheres above criticism or to delimit it. Openness to criticism, that is the readiness to let our beliefs go through a “baptism of fire,” is a good method of keeping us safe from such strategies. It is also essential to specify what renders the theory (or a belief) unjustified. Criticism is treated as an especially vital condition in some philosophical trends, for example, critical rationalism (Radnitzky 1981, passim). It is rare to come across opinions questioning it, although the importance of criticism is much more often only declared.

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5.4. Solvability of Problem The last meta-principle points at the solvability of a problem as a distinct characteristic of rationality. In practice, this meta-principle is often combined with the previously mentioned meta-principle of criticism. Criticism of a belief (theory) is tantamount, among others, to the endeavour of proving that belief vacuous and cannot be stipulated for solving those problems which it seems to solve. The intrinsic property of every problem is that it calls for a solution. When we have a theory at hand, we understand this theory if we understand the problem it solves. The solutions offered by a theory (relatively better, more adequate) are an indication of its rationality for a human facing a problem. The beliefs (theories) we refer to may concern moral, political or religious questions. And thus they are not susceptible to strictly scientific solutions. A theory evaluated in view of this meta-principle invites manifold questions. The answers to them may permit us to determine if the theory can de facto solve the problem. The questions would deal with the following matters: 1) Does the theory solve problems or only move them aside? 2) If the theory solves problems, does it do so better than other competing theories? 3) Does the theory create any additional difficulties when the problem is being solved? 4) Is the theory fruitful as far as new, unexpected aspects of the problem are concerned? Problem solvability may sometimes play the role of a supreme test revealing whether a theory (system of beliefs) is rational or not.10 5.5. Hierarchy of Meta-Principles The above presentation of meta-principles gives rise to the question of their interrelation. There seems nothing untoward in supposing that all of them are essential and indispensable, albeit it may be ascertained that compliance with logic deserves special attention. 11 This, combined with the universality of this requirement, can be comprehended as truth value in all models and as objective neutrality. W.W. Bartley’s view is worth

10 Kekes, for example, thinks that problem-solving is merely an external criterion of rationality. The others would be internal to the forms of life, norms, tradition, etc. See Kekes (1987), p. 268. 11 J. BocheĔski argues that it is logic that provides the ideal model of rationality. It is manifested by the facts that: 1) expressions are used whose meaning can be justified; 2) only justified theses are put forward (BocheĔski 1988, p. 8). This opinion, in its general version, may be accepted if we add that the ideal model must be really adapted to a given sphere.

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mentioning in this context. He names four ways of eliminating an error through criticism of our beliefs. As the most important he mentions the logic test, that is the analysis of the consistency of a given theory. 12 In order to emphasise the specific role of this meta-principle, I would like to draw attention to another problem stemming from the question whether the principles of logic can be treated as sound. This problem appears in connection with the content of the meta-principle of criticism demanding that everything should undergo criticism (Bartley 1984, p. 132). As it is well known, a number of different systems of logic exist and can be chosen. But these systems share the feature that if we start with true premises we arrive at true conclusion no matter which rule of logic we apply. Can this trait be changed as well? By taking recourse to Quine’s statements we could suggest a positive answer (Quine 1953, p. 42). 13 At the same time, however, this modification may concern the sentences determining logical relations, which seems quite essential. The laws of logic constitute merely an element of the whole. As Quine points out, one may, for example, postulate revision of the law of the excluded middle to simplify quantum mechanics. In his opinion such an operation would not differ from the change which occurred when Einsteinian physics superseded those of Newton. On the other hand, Bartley underlines that critical argumentation cannot be maintained when we refute the idea that true premises lead to true conclusion and thus he rejects the thesis that logic constitutes part of our system of beliefs which can be revised in the above sense. But when we imagine that logic may be changed, by this very act we presuppose a logic. The idea of critical argument assumes the concept of deductibility and this allows us to put forward a thesis concerning the principal difference between revision or correction of the beliefs other than logic and rejecting logic. 12

Bartley puts these principles in the following order: 1) logical test, i.e. the analysis of the consistency of the theory; 2) observational test, which allows the possibility of refuting the theory by empirical methods [empiria]; 3) test of scientific theory, where possible conflict with other scientific hypotheses is investigated; 4) test of the problem, where problem solvability is checked. I think that the principles proposed by Bartley enter the sphere of specific principles of rationality characteristic for different areas. See Bartley (1984), pp. 126-136. 13 Quine in slightly metaphorical words refers to science as to a whole consisting of different parts from somehow accidental truths, e.g. of geography or history, to deep truths of physics, mathematics and logic. This whole can be treated by analogy as a field of force for which the sphere of experience is only a boundary condition. Change in experience results in the modification in the whole “body of science.” This whole is not determined by the boundary conditions, which leave us free to choose the statements that are to be modified.

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6. Axiology of Rationality Hitherto presented discussions on the problem of the criteria of rationality lead us to the conclusion that particular doctrines that support different criteria usually possess axiological bases expressed explicitly or implicitly. It is impossible to neglect the role of the axiological factor. When we postulate a certain attitude which is to be critical, open to discussion with others, respect facts, independent of judgement, antidogmatic, we not only give general methodological indications but also refer to some hierarchy of values, some vision of a man and of a society. Therefore, it is difficult not to accept K. Szaniawski’s view that rationality cannot be axiologically neutral. The same axiological tenor is there when we find ourselves in the domain of science. It is apparently present when we consider what is scientific and what is not scientific, which is especially visible in discussions concerning the problem of demarcation between science and metaphysics (WoleĔski 1991, p. 73). On the whole, if we accept that an ethos is determined by the lifestyle of a given society, by its hierarchy of values and cultural orientation then the milieu of scientists certainly has such an ethos. Let us stress the fact that the lack of entirely objective criteria of rationality augments the role of this ethos. Thus, vagueness of borders and of criteria of rationality does not overshadow the very postulate of rationality. The significance of the interference of axiological factors has been noticed by various philosophers who emphasise the role of reason and of what is rational. Szaniawski’s views have been already mentioned. It should be added that another Polish philosopher, M. Heller, also speaks about the relation between rationality of thought and axiological decisions. According to him this relationship is so close that it is legitimate to say that “rationality is the ethics of thinking which is part of ethics” (Heller 1993, pp. 18-25 passim). The axiology discussed here assumes criticism and refers to the fallibility of a human being. The existing ethos of research would be the condition of revision of the way in which the rules of rationality are applied. It is especially clear in case of the principle of criticism that cannot be employed beyond the limits of ethical requirements. Any criticism should comply with at least two requirements: 1) justification; 2) loyalty. However, quite often it must face much more strict conditions that can be found in the distinction made by T. KotarbiĔski between protective criticism and police (oppressive) criticism. The former, as opposed to the latter, tries to find not only the weak points of a theory but also what is implicitly valuable.

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7. Conclusions The proposed conception of the criteria of rationality must be further elaborated to gain precision, especially in the case of the “second level” of the meta-principles. However, it seems to have a few advantages: (1) (2)

(3)

(4) (5)

It does not attempt to formulate a rigid system of criteria of the type that has been criticised. This does not imply that shared, general meta-principles have been given up. They are not rigid rules but constitute a framework for further specification. At the same time, it assumes that further specification takes place in some definite context. This involves communication with others and takes place through the process of discourse. There is an emphasis upon reference to axiology. It does not take for granted any solutions in a dogmatic way.

Although far from being satisfactory, the conception manages to avoid two pitfalls: the Scylla of dogmatic criteria and the Charybdis of anarchism and relativism.

Uniwersytet àódzki Department of Philosophy ul. dr. S. KopciĔskiego 16/18 90-232 àódĨ, Poland e-mail: [email protected]

REFERENCES Ajdukiewicz, K. (1949). Zagadnienia i kierunki filozofii [Problems and Trends in Philosophy]. Warszawa: Czytelnik. Amsterdamski, S. (1989). Ideaá nauki Koáa WiedeĔskiego [The Vienna Circle’s Ideal Of Science]. Archiwum Historii Filozofii i MyĞli Spoáecznej 34, 311-320. Aristotle (1931), Ethica Nicomachea. Translated by W.D. Ross. Oxford: Oxford University Press. Bartley, W.W., III (1984). The Retreat to Commitment. La Salle & London: Open Court. BocheĔski, J. (1988). Co logika daáa filozofii? [What has Logic Given Philosophy?]. Studia Filozoficzne 6-7, 7-13. Cherniak, Ch. (1986). Minimal Rationality. Cambridge, MA: The MIT Press. Davidson, D. (1982). Rational Animals. Dialectica 36, 317-327. Davidson, D. (1985). Incoherence and Rationality. Dialectica 39 (4), pp. 345-354.

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Dąmbska, I. (1938). Irracjonalizm a poznanie naukowe [Irrationalism versus Scientific Knowledge]. Kwartalnik Filozoficzny 14, 85-118, 185-212. Foley, R. (1991). Rationality, Belief and Commitment. Synthese 89 (3), 365-392. Feyerabend, P.K. (1978). Die Rationalität der Forschung. In: Ausgewählte Schriften, Band 1: Der Wissenschaftstheoretische Realismus und die Autorität der Wissenschaften. Braunschweig / Wiesbaden: Friedrich Vieweg und Sohn. Heller, M. (1993). MoralnoĞü myĞlenia [Ethics of Thinking]. Tarnów: Biblos. Kekes, J. (1987). Rationality and Problem Solving. In: J. Agassi and L.C. Jarvie (eds.), Rationality: The Critical View. Dordrecht: Martinus Nijhoff Publishers. Kleszcz, R. (1994). Krytyka rozumu pozytywistycznego [Critique of Positivist Reason]. In: Honoris Causa: KsiĊga pamiątkowa dla uczczenia aktu nadania profesorowi Leszkowi Koáakowskiemu honorowego doktoratu Uniwersytetu àódzkiego [Commemorative Book for the Celebration of Conferring Professor Leszek Koáakowski the title of the doctor honoris causa], pp. 131-146. àódĨ: Wydawnictwo àódzkie. Wydawnictwo Uniwersytetu àódzkiego. Kleszcz, R. (2001). Rationality and Requirements of Logic. Logica Trianguli vol. 5, pp. 63-71. Popper, K. (1980). The Open Society and its Enemies. vol 2. London / Henley: Routledge Kegan Paul. Putman, H. (1981). Two Conceptions of Rationality. In: Reason, Truth and History, pp. 103-126. Cambridge: Cambridge University Press. Quine, W.v.O. (1953). Two Dogmas of Empiricism. In: From a Logical Point of View, pp. 20-46. Cambridge, MA: Harvard University Press. Radnizky, G. (1981). Du positivisme logique au rationalisme critique en passant par la théorie critique. Archives de Philosophie vol. 44, pp. 99-115. Russell, B. (1912). The Problems of Philosophy. London: Williams and Norgate. Szaniawski, K. (1994). RacjonalnoĞü jako wartoĞü [Rationality as Value]. In: O nauce, rozumowaniu i wartoĞciach [On Science, Reasoning and Values], pp. 531-539. Warszawa: Wydawnictwo Naukowe PWN. Szaniawski, K. (1994a). Plus ratio quam vis. In: O nauce, rozumowaniu i wartoĞciach [On Science, Reasoning and Values], pp. 523-530. Warszawa: Wydawnictwo Naukowe PWN. Vermeersch, E. (1974). Rationality: Some Preliminary Remarks. Philosophica (Philosophia Gandensia) 2, 73-82. Wittgenstein, L. (1972). Dociekania Filozoficzne [Philosophical Investigations]. Warszawa: PWN. WoleĔski, J. (1991). Aksjologia i metodologia [Axiology and Methodology]. In: J. Goükowski and K. PigoĔ (eds.), Etyka zawodowa ludzi nauki, pp. 73-78. Wrocáaw: Ossolineum.

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Mieszko Taáasiewicz ON THE CONCEPT OF RATIONALITY

1. Types of Rationality Many authors try to distinguish types of rationality but very few of them provide a clear principle for such distinctions. In many cases, they do not even care whether the distinction is a logical division or not. In this part of my paper, I will reconstruct the principles of the distinctions aiming to minimise possible misunderstandings. Particular types of rationality can be distinguished (1) according to the kind of objects that can be rational (in which case the set that is divided is the range1 of the concept of rationality), (2) according to the kind of situation in which rationality is attributed to something (in which case we divide the set of these situations), and (3) according to the kind of criteria of rationality (in which case we divide the set of these criteria). (1) First, in dividing the concept of rationality according to the kind of objects that can be rational, many authors distinguish the rationality of thinking and the rationality of acting. It is easy to agree that this is a logical division of the range of the concept of rationality. The distinction is, of course, based on an enthymematic premise that acting is completely different from thinking; acting is understood here as doing something physical. It is also understood that any human activities other than thinking and acting (like mystical experiences) either do not exist or at least cannot be rational in any sense. Then, and only then, the ranges of both types of rationality are separate and complementary to the range of rationality in general.

1 The range of a concept is the set of possible designates of this concept. It is thus to be distinguished from an extension of a concept, which is the set of all actual designates of the concept.

In: J.J. Jadacki and J. PaĞniczek (eds.), The Lvov-Warsaw School – The New Generation (PoznaĔ Studies in the Philosophy of the Sciences and the Humanities, vol. 89), pp. 485-503. Amsterdam/New York, NY: Rodopi, 2006.

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Second, we can distinguish the rationality of values and the rationality of knowledge. However, this is not a logical division of the range of the concept of rationality and these types of rationality are used just to indicate that values can be rational and that the rationality of values differs somehow from the more usual rationality connected with science. Third, rationality can be divided into ontic, epistemic and pragmatic rationality. Ontic rationality is said to be the rationality of the world as an entity. It is unlike other types of rationality, which are always connected with some human activities. This shows that ontic rationality differs from other types of rationality much more essentially than these other types differ among themselves. Due to the fact that the inclusion of the whole world in the range of rationality seriously affects the meaning of ‘rationality’, changes the proportions among the types of rationality and causes so many doubts and difficulties in the theory of rationality, it is only reasonable to set ontic rationality outside the scope of this work. This is the first reason why the distinction of ontic, epistemic and pragmatic rationality is not a logical division of the range of rationality. The second reason is that the ranges of epistemic and pragmatic rationality are not separate. Epistemic rationality concerns all objects connected with knowledge, especially scientific knowledge, such as statements, theories, methods, but also cognitive acts (like accepting theories, performing experiments, etc.). Pragmatic rationality concerns all objects connected with practical activity, such as the standards of effective action and, of course, cognitive acts, which belong to epistemic rationality as well. Pragmatic rationality covers the scope of rationality of action but is broader, having in its range also thoughts connected with practical activity. (2) If the concept of rationality is divided according to the kind of situations in which we attribute rationality to something, there is scientific rationality and common-sense rationality. Here we probably have a logical division of the set of these situations into scientific activity and everyday life, but the ranges (in the technical sense introduced above) of both types of rationality are identical. (3) The set of criteria of rationality is usually divided in two ways. First, it can be divided according to the relation between the criteria and values (considered axiologically). As a result of such a division, we can distinguish two kinds of rationality: the rationality of aims (its criteria refer to values in an essential and non-instrumental way) and the rationality of means (in which case criteria are only instrumental because the same sort of things can be judged as rational from the scientific and

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the common-sense point of view). The division of the set of criteria based on this principle is certainly logical. The question of the ranges of these types is usually not discussed. We can assume that the ranges (possible aims and means) can differ in just two elements: the ultimate aim, which, if it exists, cannot be a means to anything, and the primary means, which, if it exists, cannot be an aim. It is likely to be true that, with the exception of these two elements, all aims can be means to other aims and all means can be aims to other means. Second, the set of criteria of rationality can be divided according to whether a criterion requires support by empirical evidence (an object that meets such a criterion is materially rational) or whether it requires a proper logical form without reference to empirical basis (an object that meets such a criterion is formally rational). This distinction is a logical division of the set of criteria. The ranges of both types of rationality are identical (statements and theories) and the extension of material rationality is a subset of the extension of formal rationality.

2. Issues Regarding the Concept of Rationality 2.1. Defining Rationality 2.1.1. Three Methods of Defining Rationality 2.1.1.1. Sometimes rationality is defined by mentioning particular methodological rules that must be followed by theories (hypotheses, statements, methods) that are to be rationally acceptable. Sets of such rules are numerous and usually valid only for some specific fields of science. A great number of such sets of methodological rules are given in the physical sciences. The classic one is the Hempel-Oppenheim nomological-deductive model of scientific explanations. In order to be rationally acceptable, a hypothesis: should have a rich empirical content (i.e. many different testable consequences), should give new predictions, should be well established in a more general theory considered to be true, should be falsifiable, as simple as possible, etc. The sets of the rules of rationality given above are surely not complete; one can find many more other rules, which are more or less specific. Often they are just handbook principles that are called “standards of methodological correctness” rather than “rules of rationality,” but all of them are somehow delimited by the rules of a

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higher level. The second way to define rationality is to formulate these higher-level rules. 2.1.1.2. One of the most representative definitions of this kind is the following: A person x rationally accepts statement p = df (1) (2) (3)

x knows the methods of justification for statements like p; x has evidence that supports p and the evidence is adequate according to these methods; x believes that p to the extent that he has good evidence for p.

This is a very general formula and it says relatively little about what is rational unless it is supported by more specific rules like those given in §2.1.1.1. The second definition is equally general but more informative: Rationally acceptable knowledge = df knowledge: (1) (2) (3) (4) (5)

gained according to proper methods; formulated in an intersubjectively understandable language, which has only an informative (not persuasive) function; put in a logical system; justified in an intersubjectively testable way; separated from emotional and volitional states (as wishful thinking, e.g., is not).

Many authors give some more detailed approximations of certain points of these characteristics but what is common and essential here is formal rationality; what is not logical cannot be rational either. (However, it is not easy to establish how logical standards can be applied to nonlinguistic objects in some types of rationality.) 2.1.1.3. The third way to define rationality  rationality itself  is in terms of what “stands to reason.” It is sometimes said that only reason does not need ratio for itself or that only these statements are scientifically acceptable that can be sufficiently justified by the “natural light of reason” or that rationality is an intellectual responsibility for one’s claims and beliefs. Another definition of this kind says that rationality is just readiness to change one’s mind. However, most authors are reluctant to formulate definitions like these, since the concept of reason is vague and hazy and, in their opinion, the informative value of such definitions is null. Nevertheless, I do believe that such definitions of rationality have certain value and are not unimportant.

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2.1.2. Models of Rationality  Dispute on Historical Development We shall concentrate now on the second kind of definitions of rationality. As it was said, the rules of rationality mentioned in §2.1.1.1 and §2.1.1.2 are hierarchical; the more general rules delimit those that are less general. For example, a demonstration of the better testability of a given theory is an appropriate method of empirical justification of this theory. A significant change of the more general rules would cause a parallel change among the less general rules. To stress this connection, many authors call the sets of such general rules the models of rationality. All variants mentioned in §2.1.1.2 are very similar and they constitute only one model of rationality, which is commonly accepted in contemporary Western culture. What is subject to dispute is the question whether this model is relative: does it change by developing over time, and is it different in other cultures? An argument for the changeability of models of rationality is simple. One can just give examples of how, at different times or in different cultures, something was regarded as rational while it is not regarded as rational here and now. Admittedly, there is a problem since ancient or exotic peoples did not use the word ‘rationality’ and it can be quite difficult to decide whether what they would have called rationality is really parallel to our rationality and not to our superstitions, for example. However, this doubt can be avoided, especially in the historical context. Since the beginning of Greek philosophy, there has been a distinction between common belief, on the one hand, and scientific knowledge, which is subject to special methodological standards, on the other. There is no serious obstacle to regarding these standards as standards of rationality.2 Historical relativity is often illustrated by the sequence of the following models of rationality: (1)

2

Ancient mythological model. This model provides inter alia the principle that the external world exists objectively and that it can be known, the principle of supranaturalism and even some logical rules (although given implicitly) such as the principle of consistency.

In the case of exotic cultures such as the culture of Azanda, which is often mentioned by philosophers, this problem is more complicated. However, taking the postulates of comparison charitably, we can distinguish «common reasoning» and the special procedures and ways of reasoning used by a shaman, which can be regarded as «shaman’s rationality».

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Ancient scientific model. According to this model, scientific justification can refer only to empirical and logical facts (logical rules are given explicitly). Physics is to investigate the world qualitatively not quantitatively and is based on common experience. Explanation in terms of final causes is admissible. Modern scientific model. It includes the principle of empirical justification and logical form (like the previous model) as well as the principles of causalism and of determinism. Physics is to investigate quantitative relations in the world. It is based on experiments, which use technical equipment, and has a welldeveloped methodology (models, thought experiments, idealisation). The world should be described objectively. Contemporary scientific model. Some people maintain that this model has already been applied in quantum mechanics. According to this model, one should not expect an entirely objective description because every description of an object is a description of our knowledge as well (we must take into account the interference between our experimental procedures and the experimental results). Moreover, all explanations should be probabilistic rather than deterministic (to coincide with the probabilistic nature of investigated regularities). Explanations may include elements of indeterminism and non-classical logic.

Since there is good historical evidence for the existence of such models one can hardly deny that models of rationality change. On the other hand, many authors can not agree that rationality is totally relative and that all standards of scientific rationality originate from nonscientific processes (like changes in the social structure of a community). The thesis of this work is that a consensus between these views is possible. In light of the distinctions made above, it is plain that the models of rationality are in fact criteria of rationality, not definitions strictly speaking. Admittedly, many concepts are definable only by giving some criteria that are to be met by objects falling under these concepts. However, in the case of rationality we need more. First, we should try to formulate a non-criterial definition that gives not only the denotation (via criteria) but also the connotation of the concept of rationality, i.e. a set of the essential properties of rational objects. Second, we should find some properties of rationality itself (not just of rational objects), such as whether rationality admits of degrees. Attempts to propose a definition of the third kind (§2.1.1.3) are a good expression of this need. Saying that something is rationally acceptable we not only say that it meets certain standards but we attribute

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some essential properties to it as well. These properties are relative in the sense that nothing has them forever. These properties are related to our cognitive skills, our acceptance, in general  to the state of our mind. It will be convenient for us to have a phrase that captures those so far mysterious properties. Let us say that something that has such properties “stands to reason” – whatever exactly this would mean. An object, for example a theory, can be rationally acceptable at one time because it stands to reason at that time and, then, it can cease to be rationally acceptable because we change our mind and the theory does not stand to reason anymore. This is in agreement with the view that models of rationality are relative. However, the concept of rationality defined as standing to reason does not change. At all times and places, within different models of rationality, we call different theories rationally acceptable as long as they meet appropriate criteria. But what is rationally acceptable is always what stands to reason. In the concept of rationality, there is something more than relative criteria, something that distinguishes rationality from the rules of performing sorcery in the Azanda culture, and that allows us to regard ancient Greeks as just as rational as we are. It is extremely hard to say what this “something” exactly is, i.e. what properties we attribute to that which is rationally acceptable (the only thing we can say so far is that it stands to reason). However, from the fact that it is hard to make this “something” explicit, it certainly does not follow that we should not try. These non-criterial unchangeable properties given in the definition of the third kind, whatever they are, play a significant role in the evolution of the models of rationality. All of a sudden, we realize that the standards that belong to the present model have become irrational. But we still do not have new standards, so we cannot recognize this irrationality by comparing our standards to the new ones. We can realize that the standards have become irrational only by referring to something outside the model. I think that, in that case, we just refer to these properties, i.e. to the connotation of ‘rationality’. Of course, since the connotation is vague, the change in the model of rationality is not clear and distinct either. Such a change takes years to complete. Thus, the consensus is: models of rationality are relative and change; non-criterial definition (however vague) is not relative and constant. It is noteworthy that the concept of truth finds itself in a very similar situation. Since Aristotle we have had the same definition: veritas est adequatio rei et intellectus. On the other hand, the theories of the criteria of truth are numerous and extremely unstable.

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To formulate a good non-criterial definition of rationality, i.e. to find the properties which something that stands to reason has, is a goal that is very ambitious and hard to achieve. I do not undertake such a task in this work.3 Instead, I will show some properties of rationality, itself.

3. Problem of Decidability 3.1. The Function of the Concept of Rationality The starting point in a philosophical analysis should always be to find a basis, a criterion, for proposed solutions. A criterion for solutions in this work is the function played by the concept of rationality. This criterion is chosen due to the great practical importance of the concept of rationality in meta-scientific analysis. The main function of the concept of rationality is to provide a criterion of appraisal of scientific theories. In the past, the concept of truth had such a function. The real aim of science was true knowledge. People believed that using proper methods one can gain knowledge which was necessarily true and certain. Theories were accepted not because they were justified but because people believed that they were true. Of course, the reasons why a statement was regarded as true were to be given but a sufficient way to do this was to give a criterion of truth and show that this statement met this criterion. A procedure like that might have been quite complicated, it could have required creating a huge epistemological system, but it had nothing to do with rationality. Only the new belief that every piece of knowledge is questionable, that the criteria of truth in science do not work (for example, a pure experiment with inductive inferences or, on the contrary, a pure deduction from a priori laws), created the need to find out why we accept some theories and reject others when we do not know which are true. Only then did rationality become an essential concept in meta-scientific analysis. Now we do not call a theory “true” but consider it to be rationally acceptable. Science is distinct from other kinds of knowledge in that its claims are rationally acceptable, not in that they are true. K. Popper even maintains that false scientific theories may be rationally acceptable.

3

I have made an attempt to move in this direction in my book PojĊcie racjonalnoĞci nauk empirycznych [The concept of rationality in empirical sciences] (2000), which is also summarized in English (2002).

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Nevertheless, the concept of truth has not become immaterial. The aim of science is still truth. The awareness that this aim is impossible to reach does not diminish the role of truth as a regulative idea – we will always have truth in view. This tie to the truth already belongs to rationality. Apart from the enigmatic «standing to reason», there is another property associated with the concept of rationality: what is rationally acceptable is most likely to be true. Thus, there is a close connection between truth and rationality but also a significant difference. Truth is an irrelative property that belongs or does not belong to statements and theories. It does not depend on people, on human knowledge, human will or, of course, on historical and cultural contexts. Rationality, understood as what “stands to reason,” depends on the state of human knowledge and on its historical and cultural context. Statements and theories, as was said above, are not rationally acceptable for ever. It may be rational to accept something given one state of knowledge and entirely irrational to accept it given another. What is rationally acceptable is closest to the truth. If, thanks to development of science, we get closer, something new will become “the closest to the truth” and, therefore, rationally acceptable; it will become irrational to insist on old statements. 3.2. Decidability As was said, the concept of truth has lost its significant role as a tool for appraising theories because of the lack of a universal criterion of truth. It has become obvious that we could not decide in all cases whether a theory is true or false. Some authors maintain that we cannot decide that in any case. Thus, since the question of truth and falsity is undecidable, we cannot compare and appraise theories with respect to their truthvalue. The concept of rationality was said to have replaced the concept of truth in this function. So that this replacement can be functional and profitable, rationality, in contrast to truth, must be decidable. ‘Decidability of rationality’ is understood here as ‘the existence of effective procedures to establish whether a given statement/theory is rationally acceptable or not’. Suppose that, like truth, rationality is not decidable. Such a view may seem intuitively correct. Often we are not quite sure whether what we are doing is rational or not. In many cases, we cannot decide ultimately. However, the consequences of such a supposition are very undesirable. The concept of rationality loses all its advantages over the concept of truth. We cannot explain why we want to make our theories rationally

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acceptable since we cannot be certain about what is rationally acceptable just as we cannot be certain about what is true. In such a situation, it would be better to strive just for the truth, not for rational acceptability, since it is the truth that is the final aim of science. Moreover, we would not need to investigate the still vague and unclear relations between truth and rationality since our concept of truth is much better defined than our concept of rationality. In fact, however, rationality is somehow closer to us than truth. We want our theories to be true but, first, we try to make them rationally acceptable. We want to be sure that, after meeting certain conditions, our theory is already rationally acceptable, that it is not rational acceptability that it may lack. We may not be sure whether the theory is true, but we must know that it may well be true, that it is rationally acceptable. Can we take for granted, then, that rationality is always decidable? How are we to interpret the examples of doubt and hesitation mentioned above? They can be explained as follows. Our doubts, in fact, do not concern the rational acceptability of a theory or statement but they concern its truth. Rationality is always related to actual knowledge. Sometimes this knowledge may be insufficient to make a decision with a satisfactory degree of rightness. It is rational to choose the most probable variant but if the probability is still very low, many doubts will arise. In case of theories, if a theory meets the standards of rationality in a greater degree than all other theories, it is rational to accept just this theory but it is still possible that we will believe it to be false. A rational choice must be distinguished from an optimal choice. The optimal choice is the best choice from among all possibilities. The rational choice is the best choice from among all known possibilities. Sometimes, when we are aware that our criteria are imperfect, we may sense a difference between the rational and the optimal choice. At such a moment, we may feel doubts and hesitate, which may lead to a belief that rationality is undecidable. However, it is important to recognise that this belief itself is irrational. After all, it may also happen that the conclusions we reach by reasoning differ from our subconsciously held superstitions. In such a case, we may have trouble with the correct choice, too. Our hesitation is likewise irrational, however. In real situations, we do not ask whether it is rationality which is doubted or whether it is something else (correctness or truth). Rationality is mixed with other aspects of decisions, acts and choices and, especially in this field, it is a vague concept. What can introduce order into this issue is the recognition that rationality is always decidable. The

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regulative aspect of this analysis is evident here. While taking into account the function of the concept of rationality it is useful (one could say: rational) to regard rationality as decidable in all circumstances. Let rationality be decidable because, then, the concept of rationality will be practically useful. 3.3. Controversies within Rationality Another property of rationality which can be inferred from accepted premises is that logically incompatible theories can be rationally acceptable at the same time. At the beginning suppose that it is not so, that only one of such theories can be rationally acceptable. Since rationality is decidable, we can know, after a more or less complicated procedure, which one is rationally acceptable (unless both are rationally unacceptable). As a result, the controversies among adherents of these theories would be only short-lasting because one party has a guaranteed victory, the one that represents the only rationally acceptable theory. As a matter of fact, however, science has witnessed great apparently never-ending controversies. One of such great controversies in physics was between N. Bohr and A. Einstein on the interpretation of quantum mechanics. A contemporary example of such a controversy is the dispute about the beginning of the world and the speed of its expansion in cosmology. The existence of such controversies is explicable only if either we accept the claim that rationality is essentially undecidable or if we assume that there may be circumstances in which incompatible theories can be rationally acceptable at the same time. I favour the latter option, in which case there are three possibilities in case of a controversies between two theories: first, neither theory is rationally acceptable, second, just one of the theories is rationally acceptable, third, both theories are rationally acceptable and we need to choose between them using a criterion other than rationality. The third is the most interesting and probably the most frequent situation in science. Although this conclusion may seem not intuitive or even discordant with the so-called “common” use of the word ‘rationality’, it does have considerable support from some other intuitions. It is often said that rationality is the basis for all possible arguments and controversies, that no discussion and no agreement is possible without respect for certain initial premises and assumptions (criteria of rationality). In this way, the field of controversies is limited a priori to rationally acceptable theories and the thesis of this work is implicitly assumed.

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Many controversies arise among adherents of theories, which are for some reasons not testable. It may seem that, in this case especially, the concept of rationality is unusually important. It may really be so but it is quite correct to regard all essentially untestable theories simply as irrational and to suspend opinion about theories that are untestable, for some insignificant but technical reasons, until they become testable. The situation is slightly different in case of theories that are testable in principle (the appropriate experiment is logically and physically possible) but it is known that they cannot be tested in the near predictable future. If these theories are of great emotional or philosophical importance, it is extremely inconvenient to suspend any opinion about them. Good examples are contemporary cosmological theories, which have significant emotional content since many people try to use them to justify their, among others, religious (or non-religious) points of view. This struggle to compare such theories leads those people to compare the degree of rational acceptability of such theories.

4. Does Rationality Admit of Degrees? 4.1. Some Views in the Literature (1) J. ĩyciĔski (1985-1988): if alternative interpretations I 1 and I 2 are proposed in a certain research program P then the interpretation that is justified to a higher degree within program P is the more rationally acceptable; an interpretation based on events of zero probability or on events that are a priori excluded in program P is rationally unacceptable (it would be irrational to accept it). This definition may be illustrated by an example of dice. Assume that: according to interpretation (i), the die will come up one particular side (e.g. with three dots); according to interpretation (ii), the die will come up one of two particular sides (e.g. the one with three dots or the one with four dots); according to interpretation (iii), the die will come up one of three particular sides; according to interpretation (iv), the die will levitate in the air (contravening the law of gravitation). On ĩyciĔski’s view, interpretation (i) is minimally rationally acceptable since the event predicted by it has a probability of 1/6. Interpretation (ii) is rationally more acceptable than interpretation (i) since it predicts an event with a probability of 1/3. Interpretation (iii) is rationally the most acceptable since it predicts an event with a probability of 1/2. Interpretation (iv) is rationally unacceptable.

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(2) J. Such (1983): the evolution of science is possible only because rationality admits of degrees. “More rational” means “reaching the truth in a higher degree.” Rationality consists in respecting the principles that enable us to gain knowledge. The more numerous, more exact, better awarded, more strictly fulfilled are the principles, the more rational is the knowledge gained. (3) J. WoleĔski (1983): undoubtedly, irrationality admits of degrees; Plotinus’ mysticism is much more irrational than Descartes’ rationalism (although rationalism is irrational, too). WoleĔski does not decide whether rationality admits of degrees. Of course, the fact that irrationality admits of degrees does not entail that rationality does. (4) R.A. Sorensen (1991): irrationality admits of degrees but rationality does not. Rationality is defined as the absence of irrationalities. Sorensen says: “Just as an empty cup cannot be more empty than another empty cup, a rational person cannot be more rational than another rational person” (p. 480). Irrationality is not well defined by Sorensen: “we are free to plug in different theories of irrationality” (p. 476). Let us consider these views in turn. (4) Sorensen’s thesis is easily acceptable because irrationality (rationality) is attributed to humans, not to human actions or their products (like statements, theories, etc.). This helps us to omit problems such as deciding what is more irrational: vicious circle or contradiction; inconsistency or incompleteness. We can just say that the more irrational person commits more irrational acts. However, Sorensen’s claim that rationality does not admit of degrees is much more doubtful. If we accept the claim that whoever is rational is free from irrationalities then rationality will indeed not admit of degrees. But irrationality may be defined according to different standards; some of them may be more severe, some may be less severe. It is quite reasonable to say that someone who is free of irrationalities defined according to more severe standards is more rational than someone who is free from irrationalities defined according to less severe standards. (3) Irrationality admits of degrees, which can be interpreted in a way similar to Sorensen’s. It is more irrational to accept doctrine d than to accept doctrine dc when d contains more statements that are rationally unacceptable than dc. As in Sorensen’s view, the rational unacceptability of statements does not admit of degrees. (2) Rationality does not need to admit of degrees to make the evolution of science possible. If we have two procedures and one enables us to reach the truth in a greater degree and the other in a lesser degree,

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then we can just say that the first is rational and the second is not rational at all. It is irrational to choose a worse procedure if we can choose a better one. (1) Taking into consideration the rationality of the four interpretations described in (1) above, we can see that apart from ĩyciĔski’s solution according to which it would be the most rational to accept interpretation (iii), the least rational (though still rational) to accept interpretation (i), and altogether irrational to accept interpretation (iv), we can accept quite a different solution. We can regard only the interpretation that involves the highest probability, i.e. interpretation (iii), as rationally acceptable, while treating all the other interpretations as rationally unacceptable. Such a solution is justified if we are to make a prediction of the dice throws and minimize the risk of error. If we are to say just what is possible in such a situation, all interpretations (i) to (iii) can be regarded as rationally acceptable (because all are possible) without differentiating them with respect to probability; it will, of course, be irrational to accept interpretation (iv). Evidently, the view that rationality admits of degrees is more commonly shared among authors, but neither is it the only one possible view (vide Sorensen’s view) nor is it the most justified view. 4.2. Does Rationality Admit of Degrees? I will claim that rationality does not admit of degrees. It is either rational or irrational to accept theories (statements, etc.) but it cannot be more rational to accept one theory (statement, etc.) than another. I offer three reasons in support. First, many definitions of rationality tacitly assume that rationality does not admit of degrees though they do not explicitly exclude the possibility that rationality does admit of degrees. On the other hand, some definitions do exclude this possibility. Second, the presumption that rationality admits of degrees can, in some cases, cause unbearable practical difficulties in establishing a hierarchy (in setting a proper order of increasing rationality within the set of competing theories). If, on the other hand, we agree that rationality does not admit of degrees, such problems (and any other problems of comparable importance) will not arise. Third (this reason will be considered in the next section), all differences in the degrees to which theories are accepted, can be explained without reference to degrees of rationality. All these reasons have an evidently pragmatic character, which coincides with the general attitude exhibited in this paper. Thus, I do not say that it is logically impossible, for example, for rationality to admit of

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degrees. Rather, I claim that if we regard rationality as not admitting of degrees, we avoid many difficulties and problems that we encounter otherwise. If we assume that rationality does not admit of degrees, the concept of rationality will become more useful in practice. Let us consider the above reasons in more detail. First, all definitions of rationality that refer to the best (most effective) act, to maximizing the profit or minimizing the loss, presume that rationality does not admit of degrees. A choice is rational if it is the best according to certain criteria; a choice that is even very slightly worse, even though still very good, is not rational anymore. When faced with a choice between two options, we must choose the better one to be rational. If in a given situation we cannot find the best option but many equally good ones then a fortiori we cannot attribute degrees to rationality. Furthermore, I have not encountered any definition of rationality that would exclude the possibility that rationality does not admit of degrees. Even the situations in which it might appear that options are exceptionally susceptible to be ordered accordingly to the “degree of rationality” (like the example of dice in §4.1), can be interpreted without the presumption that rationality admits of degrees. Second, many problems arise when comparable theories meet different criteria of rationality. There are three possible situations (suppose, for simplicity, that we are comparing only two theories): (a) the sets of criteria that are met by particular theories are distinct; (b) the set of criteria met by one of the theories is a subset of the set of criteria met by the other; (c) these sets intersect. Case (a) is probably very rare because at least one of these theories does not meet the criteria of formal rationality (which means that it is inconsistent). Since formal rationality is considered a necessary condition of epistemic (or any other) rationality, this theory will be automatically regarded as rationally unacceptable. Thus, here we do not need to use degrees of rationality in any case. Case (b) is relatively easy to judge. Some authors simply say that it is more rational to accept a theory that meets more criteria than to accept a theory that meets fewer criteria. However, other authors say that it is just irrational to accept a theory that meets fewer criteria if there is possibility of accepting a theory that meets more criteria. We shall remember that what is rational is relative to our knowledge. So the question whether it is rational to accept a theory cannot be considered outside of the actual context, which in this case is that there is a better theory to choose.

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Case (c) is the most common: both theories meet some basic criteria (i.e. criteria of formal rationality) and we have to compare them on the basis of other criteria, some of which are met by one theory while others are met by the other. There are some people who believe that they can establish a hierarchy of criteria and that they can decide which criteria are more important and which are less important. If that were so, one could say that it is more rational to accept a theory that meets the more important criteria. Unfortunately, it is not possible to decide which criteria are the more important and which are the less important. Even if such a decision can be made individually, it is impossible to get a consensus on it. As an example, we can consider the never-ending controversies on whether it is more rational to accept a theory that is predictively «fertile» or, perhaps, one that is not so fertile but much simpler and more economically formulated. It is often the case that if we want a theory to meet a given criterion we must reject some other criteria. In such a case, even those who think that rationality admits of degrees, acknowledge that, in certain cases, it is impossible to decide which theory it would be more rational accept. On the other hand, if we assume that rationality does not admit of degrees, such problems do not arise. If we accept the claim that there may be circumstances in which it will be rational to accept logically incompatible theories, we can say that all theories that meet minimal criteria accepted in a given scientific environment, are rationally acceptable. Further distinctions must concern other properties of such theories, not rational acceptability. The awareness that theories can be justified in different degrees does not affect the problem of degrees of rationality. We are rational if we justify our claims with respect to the appropriate methodological rules and if the degree of our belief is positively related to the degree of justification. Then, even if this justification is not very good, we can remain fully rational having the proper distance to such justified claims. 4.3. How do We Appraise Theories? The last issue concerns the question how we appraise theories, how doubts can arise about which of the given theories is to be accepted. If, as I argued, rationality is decidable then the question which theory is rationally acceptable cannot be subject to constant hesitation. If, as I argued, rationality does not admit of degrees then we cannot engage in endless disputes over which theory is more rational. However, in science and especially in philosophy of science, there are often controversies which theory is more worthy of acceptance and some of these

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controversies seem to be undecidable. The question is: why do some theories seem correct and acceptable to some people while they seem incorrect or unacceptable to other people, who are similarly educated and adhere to the same standards of rationality? There does seem to be a third criterion of evaluating theories, other than truth and rationality, which – in contrast to them  does admit of degrees. This criterion is the similarity to what we believe to be true. Nevertheless to accept this as a criterion is not to give up on rationality. In every form of rational argument, we finally come to the fundamental level where arguments are in balance and to make a decision we need a commitment to one of the beliefs based on something else than rationality. It is irrational to tend to rationalise all parts of commitment, because it is impossible. In such situations, we must rely on unjustifiable beliefs although it does not mean we are always free to do so. In many cases, we have just the one rational possibility to choose.

5. Summary The aim of this work was to analyse some issues connected with the concept of rationality. However, it should be taken into consideration that many authors distinguish many types of rationality and the aim was to investigate whether these types are just subconcepts of one concept of rationality or, rather, quite different concepts that have only the word in common. The result was that only ontic rationality seriously differed from other types; these other types were just subconcepts of rationality itself, delimitation of the understanding of the concept of rationality, or of the situations in which the concept is used or, finally, of the kind of criteria of rationality. The main problem connected with the concept of rationality is the problem of definition. Three ways of defining are distinguished. The first is to give some highly specific criteria of rationality; the second is to give a set of general criteria (a model of rationality); the third way is to relate the concept of rationality to the concept of reason. This distinction becomes a basis for the formulation of the following claims: (1)

(2)

Giving criteria of rationality should be distinguished from formulating a non-criterial definition, just as it is done in the case of truth. To reach a better understanding about the nature of rationality, it is necessary to concentrate (more than is usual) on this non-criterial definition.

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The above distinction (1) can be a starting point for the solution of the controversy between those who believe that rationality depends on historical and cultural circumstances and those who oppose this view. The models of rationality vary but some properties connoted by the concept of rationality do not. These properties are relative in the sense that they do not belong to a certain object independently but are related to the state of our knowledge. Thus, the extension of the concept of rationality varies but its content remains the same.

However, the aim of this work is not to establish this content but rather some properties of rationality itself. The main criterion for the proposals put forth is the function and usefulness of the concept of rationality as a basis for appraising scientific theories. The following claims are derived from this standpoint: (1)

(2)

The concept of rationality should be used in such way that it would be always possible to decide whether a given theory is rationally acceptable or not (rationality should be decidable). One should agree that there may be circumstances in which it logically incompatible theories are both rationally acceptable at the same time.

These two postulates make the concept of rationality an efficient tool for the preliminary selection of theories. The cost of such a proposal is that rationality is not a tool for the ultimate selection of theories (scil. we could not single out only one theory using the concept of rationality; we would have to make a decision based on some different criteria, making a choice among theories all of which are rationally acceptable). Unfortunately, we cannot accept postulate (1) and reject postulate (2) because then it would be impossible to reconstruct undecidable controversies that in fact arise. The concept of rationality prepared in such way would be obviously inadequate. Finally, I have claimed that rationality does not admit of degrees. To explain on what basis we make decisions of acceptance or rejection, a third criterion other than truth or rationality is needed: a commitment based on non-rational beliefs. Aside from the context of discovery and the context of justification there is a third context, the context of acceptance.

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Uniwersytet Warszawski Department of Philosophy ul. Krakowskie PrzedmieĞcie 3 00-047 Warszawa, Poland e-mail: [email protected]

REFERENCES Sorensen, R.A. (1991). Rationality as an Absolute Concept. Philosophy 66, 473-486. Such, J. (1983). Modele racjonalnoĞci w fizyce [Models of Rationality in Physics]. Studia Filozoficzne 5-6, 207-216. Taáasiewicz, M. (2000). PojĊcie racjonalnoĞci nauk empirycznych [The Concept of Rationality in Empirical Sciences]. Warszawa: Wydawnictwo UW. Taáasiewicz, M. (2002). The Concept of Rationality of Empirical Sciences. In: M. Taáasiewicz (ed.), Logic, Methodology and Philosophy of Science at Warsaw University, pp. 165-169. Warszawa: Wydawnictwo UW. WoleĔski, J. (1983). Racjonalizm i pewnoĞü wiedzy [Rationalism and the Certainty of Knowledge]. Studia Filozoficzne 5-6, 165-172. ĩyciĔski, J. (1985-1988). Teizm i filozofia analityczna [Theism and the Analytical Philosophy]. 2 vols. Kraków: Znak.

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POZNAē STUDIES IN THE PHILOSOPHY OF THE SCIENCES AND THE HUMANITIES

Contents of back issues of the Polish Analytical Philosophy subseries

VOLUME 67 (1999) Kazimierz Twardowski ON ACTIONS, PRODUCTS AND OTHER TOPICS IN PHILOSOPHY (Edited by Johannes L. Brandl and Jan WoleĔski) J. Brandl, J. WoleĔski, Introduction; A. Szylewicz, Translator's Note; SelfPortrait (1926/91); Biographical Notes. I. ON MIND, PSYCHOLOGY AND LANGUAGE — Psychology vs. Physiology and Philosophy (1897); On the Classification of Mental Phenomena (1898); The Essence of Concepts (1903/24); On Idio- and Allogenetic Theories of Judgment (1907); Actions and Products (1912); The Humanities and Psychology (1912/76); On the Logic of Adjectives (1923/27). II. ON TRUTH AND KNOWLEDGE — On So-Called Relative Truths (1900); A priori, or Rational (Deductive) Sciences and a posteriori, or Empirical (Inductive) Sciences (1923); Theory of Knowledge. A Lecture Course (1925/75). III. ON PHILOSOPHY — Franz Brentano and the History of Philosophy (1895); The Historical Conception of Philosophy (1912); On Clear and Unclear Philosophical Style (1920); Symbolomania and Pragmatophobia (1921); Address at the 25th Anniversary Session of the Polish Philosophical Society (1929/31); On the Dignity of the University (1933).

VOLUME 68 (2000) Tadeusz CzeĪowski KNOWLEDGE, SCIENCE AND VALUES A PROGRAM FOR SCIENTIFIC PHILOSOPHY

(Edited by L. GumaĔski) L. GumaĔski, Introduction. PART 1: LOGIC, METHODOLOGY AND THEORY OF SCIENCE — Some Ancient Problems in Modern Form; On the Humanities; On the Method of Analytical Description; On the Problem of Induction; On Discussion and Discussing; On Logical Culture; On Hypotheses; On the Classification of Sentences and Propositional Functions; Proof; On Traditional Distinctions between Definitions; Deictic Definitions; Induction and Reasoning by Analogy; The Classification of Reasonings and its Consequences in the Theory of Science; On the so-called Direct Justification and Self-evidence; On the Unity of Science; Scientific Description.. PART 2: THE WORLD OF HUMAN VALUES AND NORMS — On Happiness; How to Understand "the Meaning of Life" ?; How to Construct the Logic of Goods?; The Meaning and the Value of Life; Conflicts in Ethics; What are Values?; Ethics, Psychology and Logic.. PART 3: REALITY-KNOWLEDGE-WORLD — Three Attitudes towards the World; On Two Views of the World; A Few Remarks on Rationalism and Empiricism; Identity and the Individual in Its Persistence; Sensory Cognition and Reality; Philosophy at the Crossroads; On Individuals and Existence; J.J. Jadacki, Trouble with Ontic Categories or Some Remarks on Tadeusz CzeĪowski's Philosophical Views; W. Mincer, The Bibliography of Tadeusz CzeĪowski.

VOLUME 74 (2000) POLISH PHILOSOPHERS OF SCIENCE AND NATURE IN THE 20TH CENTURY (Edited by Wáadysáaw Krajewski) W. Krajewski, Introduction. I. PHILOSOPHERS — J. Wolenski, Tadeusz KotarbiĔski – Reism and Science; A. Jedynak, Kazimierz Ajdukiewicz – From Radical Conventionalism to Radical Empiricism; L. GumaĔski, Tadeusz CzeĪowski – Our Knowledge though Uncertain is Probable; M. Taáasiewicz, Jan àukasiewicz – The Quest for the Form of Science; I. Szumilewicz-Lachman, Zygmunt Zawirski – The Notion of Time; A. Jedynak, Janina Hosiasson-Lindenbaumowa – The Logic of Induction; T. Bigaj, Joachim Metallmann – Causality, Determinism and Science; J. WoleĔski, Izydora Dąmbska – Between Conventionalism and Realism; A. Koterski, Henryk Mehlberg – The Reach of Science; I. Nowakowa, Adam Wiegner’s Nonstandard Empiricism; W. Krajewski, Janina KotarbiĔska – Logical Methodology and Semantic; M. Taáasiewicz, Maria KokoszyĔska-Lutmanowa – Methodology, Semantics, Truth; T. Batóg, Seweryna àuszczewska-Romahnowa – Logic and Philosophy of Science; M. Omyáa, Roman Suszko – From Diachronic Logic to Non-Fregean Logic; J. WoleĔski, Klemens Szaniawski – Rationality and

Statistical Methods; A. Jedynak, Halina Mortimer – The Logic of Induction; K. Zamiara, Jerzy Giedymin – From the Logic of Science to the Theoretical History of Science; J.M. Dolega, B. J. Gawecki – A Philosopher of the Natural Sciences; A. Bronk, Stanisáaw KamiĔski – A Philosopher and Historian of Science; Z. Hajduk, Stanisáaw Mazierski – A Theorist of Natural Lawfulness. II. SCIENTISTS — W. Krajewski, Marian Smoluchowski – A Forerunner of the Chaos Theory; A. Motycka, Czesáaw Biaáobrzeski’s Conception of Science; M. Tempczyk, Leopold Infeld – The Problem of Matter and Field; M. Czarnocka, Grzegorz Biaákowski – Science and Its Subject; J. Plazowski, Jerzy Rayski – Physicist and Philosopher of Physics; J. Misiek, Zygmunt ChyliĔski – Physics, Philosophy, Music; W. Sady, Ludwik Fleck – Thought Collectives and Thought Styles. III. GENERAL SURVEYS — K. Ajdukiewicz, Logicist AntiIrrationalism in Poland; K. Szaniawski, Philosophy of Science in Poland; I. Nowakowa, Main Orientations in the Contemporary Polish Philosophy of Science.

VOLUME 77 (2003) KNOWLEDGE AND FAITH (Edited by J. Jadacki and K. ĝwiĊtorzecka) Editorial Note; J.J. Jadacki, K. ĝwiĊtorzecka, On Jan Salamucha’s Life and Work. PART I: LOGIC AND THEOLOGY — On the «Mechanization» of Thinking; On the Possibilities of a Strict Formalization of the Domain of Analogical Notion; The Proof ex motu for the Existence of God. Logical Analysis of St. Thomas Aquinas’ Arguments. PART II: HISTORY OF LOGIC — The Propositional Logic in William Ockham; The Appearance of Antinomial Problems within Medieval Logic; From the History of Medieval Nominalism. PART III: METAPHYSICS AND ETHICS — From the History of One Word (‘Essence’); The Structure of the Material World; Faith; The Relativity and Absoluteness of Catholic Ethics; The Problem of Force in Social Life; A Vision of Love. COMMENTS AND DISCUSSIONS — J.M. BocheĔski, J. Salamucha, "The Proof ex motu for the Existence of God. Logical Analysis of St. Thomas Aquinas' Arguments"; J.F. Drewnowski, The Mathematical Logic and the Metaphysics; H. Scholz, J. Salamucha, "The Appearance of Antinomial Problems within Medieval Logic"; J. Bendiek, On the Logical Structure of Proofs for the Existence of God; K. Policki, On the Formalization of the Proof ex motu for Existence of God; J. Herbut, Jan Salamucha's Efforts Towards the Methodological Modernization of Theistic Metaphysics; F. Vandamme, Logic, Pragmatics and Religion; E. NieznaĔski, Logical Analysis of Thomism. The Polish Program that Originated in 1930s; Bibliography.

VOLUME 87 (2005) Adam Wiegner OBSERVATION, HYPOTHESIS, INTROSPECTION (Edited by Izabella Nowakowa) I. Nowakowa, Introduction: Adam Wiegner's Nonstandard Empiricism. Adam Wiegner, OBSERVATION, HYPOTHESIS, INTROSPECTION — Translator's Note; List of Translational Decisions. HOLISTIC EMPIRICISM — A Note on Holistic Empiricism (1964); The Problem of Knowledge in light of L. Nelson's Critical Philosophy (1925); The "Proton Pseudos" in Wundt's Criticism of R. Avenarius' Philosophy (1963); Philosophical Significance of Gestalt Theory (1948); The Idea of a Logic of Knowledge (1934). OTHER EPISTEMOLOGICAL AND METHODOLOGICAL CONTRIBUTIONS — Remarks on Indeterminism in Physics (1932); A Note on the Concept of Relative Truth (1964); On the so-called "Relative Truth" (1963); On Abstraction and Concretization (1960). PHILOSOPHY OF MIND AND PHILOSOPHY OF PSYCHOLOGY — On the Nature of Mental Phenomena (1933); On the Debate about Imaginative Ideas (1932); On the Subjective and Objective Clarity in Thought and Word (1959); References; Original Sources; J. Kmita, Wiegner's Conception of Holistic Empiricism.