Tarski on Logical Notions

Transcription

1 TARSKI ON LOGICAL NOTIONS Tarski on Logical Notions LUCA BELLOTTI (Dipartimento di Filosofia - Università di Pisa - Italy) Address: Via E. Gianturco 55, I La Spezia, Italy Phone number: bellotti5@supereva.it

2 Tarski on Logical Notions ABSTRACT. We try to explain Tarski s conception of logical notions, as it emerges from a lecture of his, delivered in 1966 and published posthumously in 1986 (History and Philosophy of Logic, 7, pp ), a conception based on the idea of invariance. The evaluation of Tarski s proposal leads us to consider an interesting (and neglected) reply to Skolem in which Tarski hints at his own point of view on the foundations of set theory. Then, comparing the lecture of 1966 with Tarski s last work and with an earlier paper written with Lindenbaum, it is shown that Tarski s conception of logical notions, with its essentially type-theoretic character, did not undergo any significant modifications throughout his life. A remark on Tarski s prudential attitude on the topic in the famous paper on the concept of logical consequence (and elsewhere) concludes our paper. 1. What are logical notions? In his lecture What are Logical Notions?, delivered in London in 1966, repeated in Buffalo in 1973, and published posthumously in 1986 by John Corcoran in History and Philosophy of Logic (Vol. 7, pp ), Alfred Tarski proposed calling a notion logical if and only if it is invariant under all possible one-one transformations of the world onto itself (Tarski 1986, p. 149). 2

3 The proposal is obviously in the spirit of Klein s Erlanger Programm (see, e.g., Klein 1872), and after a few examples taken from geometry in order to explain Klein s idea, Tarski examines some consequences of his informal definition of logical notion. As Corcoran points out (Tarski 1986, fn. 6, p. 150), the term notion is applied primarily to sets, classes of sets, classes of classes of sets, etc., but also to quantifiers, truth functions, etc., in that the latter can be construed as notions in the strict sense (for example, by algebraic techniques). After remarking on the fact that the notions denoted by terms which can be defined within any of the existing systems of logic (ibid., p. 150) are logical in the proposed sense, Tarski examines a few semantical categories, in the sense of the Wahrheitsbegriff (Tarski 1935), searching for examples of logical notions. Among individuals there are no such examples; among classes there are two logical notions, the universal class and the empty class; among binary relations the logical notions are four: the universal relation, the empty relation, the identity relation, and the diversity relation. When we consider classes of classes, the non-triviality of the Tarskian definition becomes evident: the only properties of classes of individuals which we can call logical are properties concerning the number of elements in these classes (ibid., p. 151). Relations between classes are considered subsequently and inclusion, disjointness and overlapping of classes turn out to be logical in Tarski s sense, in accordance, the author remarks, with common usage. In general, as Gila Sher remarks (Sher 1996), Tarskian logical terms are those terms whose evaluation commutes with all isomorphisms of domains, so that they are 3

4 connected to the most general laws of formal structure. Sher cites the following examples: the intersection of universal classes is universal ; the union of a nonempty class with another (possibly identical) class is nonempty (Sher 1996, p. 674), and she explicitly refers to Tarski 1986 when she proposes to characterize formality as follows: A term is formal if and only if it is invariant under isomorphic structures (ibid., p. 677). Sher also points out that Mostowski s definition (Mostowski 1957) of a formal quantifier on a universe A (namely, a function from the power set of A to {T,F} which is invariant under permutations of the power set of A induced by permutations of A) follows the same criterion. The first systematic account of the operations on a given domain which should be considered as logical according to Tarski s definition has been given recently by Vann McGee (McGee 1996). These operations are identity, substitution of variables, negation, finite or infinite disjunction and existential quantification with respect to a finite or (any) infinite number of variables, plus the operations that can be obtained as combinations of them. The last part of Tarski s lecture is devoted to a brief discussion, in the light of his preceding remarks, on whether mathematical notions are logical notions. Tarski remarks that this question is separated from another one: whether mathematical truths are logical truths. But one might ask whether it is possible to separate the two items. At first glance, it seems that this would require making in some way notions (in Tarski s sense) independent of truths on them, and this is not at all a neutral philosophical issue. This problem shall not be addressed here, not only because it would take us too far, but also because it is left out of Tarski s discussion, in the interpretation of which we are mainly interested here. 4

5 The question whether mathematical notions are logical ones reduces to the following: is the membership relation logical (in the proposed sense)? Here two main different approaches to set theory give opposite answers. If we use the typetheoretic method of Principia Mathematica (Whitehead and Russell 1910), which starts from a fixed fundamental universe of discourse, the universe of individuals (Tarski 1986, p. 152), transformations on the universe of individuals induce transformations on the universal classes of higher types, and then the membership relation turns out to be a logical notion. In contrast, if we use the axiomatic approach, the first order method (ibid., p. 152), in the style, e.g., of Zermelo-Fraenkel (ZF) or Von Neumann-Bernays- Gödel (NBG) set theories, membership is not logical: it is a relation just like any other relation, implicitly defined by the axioms, and it is not one of the four logical two-place relations mentioned above. Tarski concludes that his suggestion does not imply by itself any answer to the question of whether mathematical notions are logical (ibid., p. 153). As Sher points out (Sher 1991), according to Tarski s definition in general, all predicates definable in standard higher-order logic are logical. Tarski emphasizes that, according to his definition, any mathematical property can be seen as logical when construed as higher-order. Thus, as a science of individuals, mathematics is different from logic, but as a science of higher-order structures, mathematics is logic (Sher 1991, p. 63). 5

6 2. A problem There are various problems, strictly connected with each other, arising from Tarski s proposal. Only a few shall be address here. It is necessary to keep in mind the informal character of Tarski s lecture, which allows only a rather informal and conjectural discussion. The apparent ease of Tarski s conclusion, and of Sher s comments just quoted, seem to understate a deep difference between two well-known philosophical conceptions of set theory, a difference of which Tarski himself was fully aware. Briefly, the first conception is an absolute conception: sets are certain well determined objects, the properties of which set theory has the task to investigate; the second one might be called abstract or axiomatic : it is the conception which emerges, typically, in Skolem s later works. Although it is possible to mention and analyse technical results to give arguments in favour or against one of these conceptions, their opposition is not technical, but rather has a genuine philosophical nature. This general problem bears on the evaluation of Tarski s proposal, because in order to evaluate it properly we need a previous assessment of the appropriate context in which the distinction between logical notions and non logical ones is set up. For instance, the context chosen by Sher for her treatment of logical consequence and of the logical/non-logical distinction is axiomatic set theory. She does not discuss her choice; she says: I will not attempt to decide between ZFC and other theories of formal structure here, but my view is that [...] ZFC is a reasonable candidate for the reduction of logical consequence (Sher 1996, p. 682). But the choice of a context in this connection is not without importance. Discussing 6

7 Tarski s definition of logical consequence, José M. Sagüillo remarks that, for instance, in Tarski s papers on the theory of models (Tarski 1954, 1955) interpretations are settheoretic objects, namely, elements of the universe of pure sets. This view presupposes an ontology of sets. Validity of a given argument-text amounts to the non-existence of a certain sort of set that provides for a countermodel (Sagüillo 1997, p. 238). As Corcoran remarks, in general the invalidity of an argument depends on the existence of a suitable domain and there might not be enough domains to provide counter interpretations for all invalid arguments (Corcoran 1972, p. 43, quoted by Sagüillo 1997, p. 238). Although the matter addressed in the passages just quoted is the characterization of logical consequence, not of logical notions, and although one could argue that the completeness theorem makes the problem raised by Corcoran much less dramatic in the specific case of first order logical consequence, these passages express clearly the fact that the context in which these definitions are given can make all the difference. In the 1966 lecture (Tarski 1986), Tarski seems to be utterly sympathetic with the absolute conception of set theory when he gives the examples of what follows from his proposal: he speaks of semantical categories, of individuals, of classes, of classes of classes, etc., considering the shift from lower to higher types with a definiteness that could hardly be adapted, or could not be adapted at all, to axiomatic first order set theories. Rather, the natural context of these examples seems to be some sort of type-theoretic approach, perhaps something not far from the theory of semantical categories, discussed in the Wahrheitsbegriff (Tarski 1935). 7

8 The main problem can now be stated: how to reconcile Tarski s absolutist attitude with his apparent acceptance on equal terms of both the type-theoretical and the axiomatic approach as alternative possibilities in the conclusion of the 1966 lecture. The difficulty pointed out emerges with clarity when classes of classes and cardinalities are discussed (Tarski 1986, p. 151): Tarski maintains that properties concerning cardinality are logical properties of classes (in fact, the only ones). This seems to be in contrast to the acceptance (conclusion, ibid., p. 153) of the possibility of developing set theory in first order axiomatic style, since, in the latter case, the notion of cardinality is not logical in Tarski s sense. In fact, a possible ambiguity of the term cardinality is evident on Tarski s proposal. In first order axiomatic set theories cardinalities are usually defined either (using the Axiom of Choice) as initial ordinals (i.e., ordinals which are not equinumerous with any smaller ordinal), or (in theories with classes) as proper classes (i.e., equivalence classes with respect to equinumerosity). In the first case, cardinalities are sets, hence elements of the domain; in the second case, they are classes of elements of the domain; in both cases, they are not logical notions in Tarski s sense. On the other hand, cardinalities as properties of classes are logical notions in that sense. Likewise, if cardinality is a logical notion, then it seems that assertions on cardinalities, such as Cantor s Theorem, should receive some status which accounts for their being on logical notions (even though in his lecture Tarski refuses to discuss the whole topic of mathematical truths and logical truths): but how this could be possible in a first order axiomatic theory is not at all clear. We are led, of course, to the extremely rich and difficult plexus of problems generated by the Skolem paradox. We do not want to touch on these problems here; 8

9 but recall that the concept of relativity of cardinality which emerges from the Löwenheim-Skolem Theorem puts every naïve absolutist notion of set in need of a decisive reformulation and refinement (to say the least). As Skolem puts it: Axiomatizing set theory leads to a relativity of set-theoretic notions, and this relativity is inseparably bound up with every thoroughgoing axiomatization (Skolem 1922, p. 296 of the English translation). The existence of a denumerable model of first order set theory (e.g. ZF or NBG) does not constitute a problem for the abstract approach: no bijection exists in the model between the natural numbers and the real numbers, and Cantor s Theorem holds; but denumerable models seem to be utterly problematic for Tarski s idea that cardinality is a logical notion. Even the definition itself of logical as invariant under all possible permutations of the world becomes highly problematic: what are all possible permutations, if functions are sets of ordered pairs, hence (for example) sets of sets of sets of individuals (according to Tarski 1986, p. 150), and the power set operation itself is subjected to the above mentioned Skolemian remarks? The notions of power set and of cardinality are just the first examples one meets, in the study of models of axiomatic set theories, of notions which are not absolute (we recall that a formula defining a notion, e.g. the formula P(x,y) defining the notion x is the power set of y, is absolute if it holds for elements of a transitive model in the model if and only if it holds tout court; intuitively, a notion is absolute if we can verify whether it holds for elements of the domain of a model without going outside the model). More generally, the basic distinction between mappings within a model and mappings outside it, which is decisive for any explanation of Skolem s paradox, seems far from being easily adaptable to the examples and the ideas Tarski develops in the lecture. 9

10 3. Tarski and Skolem Perhaps it is not so surprising to discover that it is only in a discussion of Skolem s ideas that Tarski reveals something which seems to point to a possible solution of the difficulties noticed above. In Le raisonnement en mathématiques et en sciences expérimentales (Tarski 1958), proceedings of a Colloquium held in Paris in 1955, discussing Skolem s lecture Une relativisation des notions mathématiques fondamentales (Skolem 1958), Tarski says (Tarski 1958, p. 18; our translation and emphasis): The Löwenheim-Skolem theorem itself is not true but for a certain particular interpretation of the symbols. In particular, if we interpret the symbol of a formalized theory of sets as a dyadic predicate analogue to any other predicate, then the Löwenheim-Skolem Theorem can be applied. But if instead we treat like the logical symbols (quantifiers, etc.), and we interpret it as meaning membership, we will not have, in general, a denumerable model. This seems to show that Tarski, in 1955, was strongly inclined to consider membership as a logical notion, as results from the analogous treatment he proposes for the membership symbol and for symbols such as the quantifiers, that he considers typically logical elsewhere (see, e.g., The Concept of Logical Consequence, Tarski 1936, p. 418). This would account for the treatment of classes in the main body of the lecture we are examining: the basic conception of set here is an absolute one, in which 10

11 Tarski s examples fit perfectly. From the text it is not clear what specific form this conception should have taken in Tarski s mind; however, the above quotation shows that in spite of the neutrality of the conclusion of the lecture Tarski was, against Skolem, utterly convinced of the necessity of a notion of membership on a par with logical notions. Could not the conclusion of the 1966 lecture be a symptom of a change in Tarski s ideas on this topic with respect to eleven years before? One would be inclined to think that the main body of the 1966 lecture shows that Tarski s position had not undergone any significant modifications; further confirmation to this thesis comes from a comparison with other Tarskian works. 4. Tarski s last work The last book written by Tarski (with Steven Givant), A Formalization of Set Theory without Variables (Tarski and Givant 1987), provides some interesting points for discussion. Corcoran himself concludes the note on his editorial treatment of Tarski s lecture inviting the reader to consult this book for further discussion and applications of the main idea of the lecture (Tarski 1986, p. 144). In section 1.2 of Tarski and Givant 1987, where the basic language L is introduced, it is said: In the prevailing part of our discussion, we assume that L contains only one nonlogical constant, the membership symbol E (Tarski s emphasis). In section 3.5 (ibid.) Tarski introduces precise stipulative definitions of logical object and logical constant (where object is clearly equivalent to notion ). We start from a basic universe U, and 11

12 thinking in the framework of the theory of types (ibid., p. 57) we can construct various derivative universes of higher types, on which every permutation P of the basic universe induces a uniquely determined permutation. An object is invariant under P if and only if it is carried onto itself by the suitable (i.e. corresponding to its type) induced permutation. Now, a member M of any derivative universe is defined a logical object (as a member of that universe) if and only if it is invariant under every permutation P of the basic universe. Correspondingly, a symbol S of Tarski s extended formalism L x (an extension of the above mentioned basic language, with four predicate functors, a firstorder predicate for class identity, a second-order predicate for relational identity and the nonlogical constant E of set membership; the details are not relevant here) is defined a logical constant if and only if for every realization of that formalism with universe U, S denotes a logical object in some derivative universe based on U. These definitions are followed by examples that correspond exactly to some of the examples given in Tarski 1986; it is of course no coincidence that this paper is explicitly mentioned on the same page (Tarski and Givant 1987, p. 57). On the preceding page (ibid., p. 56, bottom) Tarski recalls that E is the only nonlogical constant of his languages, a constant which is of course reinterpreted in any of their realizations <U,E>. The fact that E is nonlogical does not change in any way the type-theoretic spirit of the whole approach of the section, explicitly declared (see above): membership in the sense of E is simply a relation defined on U, subjected to reinterpretation, and it has nothing to do with membership in the sense of the type-theoretic construction of the derivative universes of higher types based on U. In the former case, we have a relation between elements of U; in the latter, membership is between elements of a (derivative) universe of a certain type 12

13 and elements of a universe of higher type. The distinction is expressed clearly by Sagüillo (although he refers to Tarski 1986, not to Tarski and Givant 1987): the logicality of the type-theoretic membership relation is justified under the criterion of invariance under all transformations of the universe in a hierarchy of types [...] it is easy to see that the membership relation is invariant between adjacent types and hence, it is a logical notion (Sagüillo 1997, pp ). We can find exactly the same distinction in Sher 1991 (pp ): the second-level set-membership relation on a domain A (in her terminology, a two-place quantifier over pairs of a singular term and a predicate), which is {<a,b>: a A B A a B}, is a logical term; the first-level membership relation, which is {<a,b>: a,b A b is a set a is a member of b} is a non-logical one. Perhaps it is not useless to remark here that when we talk of the type-theoretic spirit of Tarski s and Givant s work we refer specifically to the discussion on logical notions; the possibility, shown in the book itself, of a reconstruction in Tarski-Givant style of almost all the current theories of sets shows that axiomatic set theory, as such, is not in question and not even in discussion there. In his last work, then, Tarski seems to reaffirm the idea that the appropriate context in order to deal with logical notions is a type-theoretic approach, in the sense that only such a context seems to provide the tools for a precise stipulative definition of logical object. 13

14 5. The type-theoretic character of Tarski s definition The paper Lindenbaum and Tarski 1936 is not only mentioned in the 1966 lecture, but also explicitly indicated in Tarski and Givant 1987, p. 57, as the place in which the conception of logical symbols explained was suggested for the first time. And that paper shows the deep continuity of Tarski s thought on logical notions. Not only can one find here, in the form of theorems, precisely the same examples that would have been used in 1966, together with a broad discussion of geometry; but also, a theorem (the first one) stating that every relation between objects which can be expressed by purely logical means is invariant with respect to every one-one mapping of the world (i.e. the class of all individuals) onto itself and this invariance is logically provable (Lindenbaum and Tarski 1936, p. 385 of the English translation). This bears similarity to Tarski 1986 and Tarski and Givant 1987, but the order of ideas is reversed: Lindenbaum and Tarski (1936) show that the notions of simple type theory, which they independently consider as logical notions, are invariant under permutations; in the later works (1986, 1987) invariance is proposed by Tarski as a definition of logicality. But what is most important, for our present interests, in the paper by Tarski and Lindenbaum, is the notion of logic they assume. It is explained in the first paragraph: their logic is a system which includes as a subsystem the logic of Principia Mathematica modified in such a way that a simple theory of types and the axiom of extensionality are assumed (Lindenbaum and Tarski 1936, p. 384; our emphasis). There is plenty of evidence that this notion of logic was not chance for Tarski in the Thirties: this is one of the few unanimously accepted results of the recent debate on his 14

15 work in that period, a debate raised by Etchemendy s book The concept of logical consequence (Etchemendy 1990). Mario Gómez-Torrente, examining from a historical perspective Tarski s paper on the concept of logical consequence (Tarski 1936), recognizes that in several works of this period Tarski [...] uses a simple type theory as what he calls a logical basis for the formalization of several different mathematical disciplines (Gómez-Torrente 1996, p. 132). It can be amply documented that in the works of this period Tarski reserves his most inclusive use of the word logic for a system of logic based on the theory of types (ibid., p. 134). More precisely, Tarski s system is a simple theory of types with axioms of comprehension, extensionality and infinity; to include the theorems of this general theory of classes among the truths of logic was a widespread usage at the time. Sher agrees that the context of Tarski s researches is a Russellian type-theoretic logic (with simple types) (Sher 1996, p. 655), and she presents as evidence the reference to Tarski 1933 in Tarski 1936 (fn.1, p. 410 of the English translation) and, in general, the results of a comparison of Tarski 1936 with other articles from the same period (Sher 1996, p. 655, fn. 3). Also Sagüillo, in order to justify Tarski s attribution of logical validity to omega arguments (i.e., arguments represented by argument-texts having a universal sentence as their conclusion, whose numerical instances constitute the premise-set) in Tarski 1936, after observing that neither numerals nor the predicate natural number are logical constants in the standard first order formalization under Tarski s criterion of invariance (Sagüillo 1997, p. 228), refers to Tarski 1933, asserting that the system developed in the 1933 omega article is, in fact, a reformulation of the language of Principia Mathematica (see Tarski 1933, p. 279). So, Tarski s language is a language of types [...] It is important to recall 15

16 that in this framework, a natural number is not a member of the universe of discourse but it is a class of classes of individuals (ibid., p. 229), and that generally any true cardinality sentence is logically true. At this point it appears beyond any reasonable doubt that the logical framework of Tarski s investigations in the Thirties was type-theoretic, and that his conception of logical notions was in a sense coherent and constant throughout his life: it was always developed in that type-theoretic context, and nothing ever came to question the fundamental idea of invariance as the basic feature of logicality. 6. Tarski s scepticism A problem arises in the final pages of On the concept of logical consequence (Tarski 1936, pp ). It would seem that they do not contradict in the strict sense the picture delineated so far; yet it remains to be explained why Tarski shows here a rather sceptical attitude, e.g. in his comment (Tarski 1936, pp ): no objective grounds are known to me which permit us to draw a sharp boundary between the two groups of terms (sc. logical and extra-logical terms), or (ibid., p. 420): I consider it to be quite possible that investigations will bring no positive results [...] 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 extralogical (our emphasis). Why this scepticism in the same year (1935) in which the paper 16

17 written with Lindenbaum was read in Vienna? At first sight it could seem possible that Tarski simply changed his mind during his life, and that this could be the most natural explanation of the divergence between Tarski 1936 and 1986; but this would not explain why in Tarski and Givant 1987 the proposal of characterization of logical notions is explicitly traced back to the paper written with Lindenbaum. There is further evidence of Tarski s scepticism. At Harvard, perhaps in , Tarski gave the lecture On the Completeness and the Categoricity of Deductive Systems which has been recently presented in summary by Jan Tarski and Jan Wolenski as an Appendix to the other Tarskian lecture Some Current Problems in Metamathematics published in History and Philosophy of Logic (Vol. 16, 1995, pp ). In the discussion of three different notions of completeness, Tarski introduces the assumption that the constant terms of a given theory are divided into two classes, the logical and the non-logical; among the former are at least the constants of the sentential calculus and the quantifiers (Tarski 1995, p. 166; our emphasis). Correspondingly, sentences (of the language of the theory) containing only logical constants are called logical ; the others non-logical. A precise distinction cannot be found between logical and non-logical constants here; nevertheless, the following summarized discussion (ibid.) depends directly on the concepts of logical basis of a theory and of logical consequence, both requiring the distinction logical/non-logical. As the editors correctly remark, the following discussion is of interest primarily if the logical sub-theory includes a version of the theory of sets, and if some non-logical terms and axioms also occur (ibid., fn. 6, pp ). We have then a broad concept of logical notion; a concept that here, however, is not precisely delimited though its 17

18 distinction from the concept of non-logical notion is relevant and important for the definition of other concepts, in this case, of concepts of a metamathematical nature. But Tarski s sceptical attitude about the distinction between logical terms and non-logical ones is put most explicitly is his letter to Morton G. White (1944), published by the latter in the Journal of Philosophy in 1987 (Tarski 1987). After observing that we can simply define logical terms by enumeration, Tarski says: Sometimes it seems to me convenient to include mathematical terms, like the -relation, in the class of logical ones, and sometimes I prefer to restrict myself to terms of elementary logic (Tarski 1987, p. 29); then he concludes: Is any problem involved here?, showing a rather noncommittal attitude. This conclusion anticipates the refusal to decide between taking membership as logical or non-logical in the 1966 lecture, and it seems to support the impression that Tarski did not consider more than a matter of convenience, the problem of delimiting precisely the class of logical notions. In this connection, Sagüillo underlines Tarski s conventional attitude towards the membership relation as is seen in [the] letter to Morton White in which Tarski asks, showing a certain candor, whether the epsilon should be taken to be a logical or non-logical sign (Sagüillo 1997, p. 233, fn. 15). Thus there remains the problem of explaining Tarski s scepticism. There are not sufficient elements to give a definite answer to this problem. We can only guess that the context in which Tarski 1936 was presented was too deeply imbued with general philosophical questions on the language of natural science (e.g., analiticity) to allow proposals whose applicability to such questions was probably doubtful for Tarski. This seems to be confirmed by the fact that in the other works mentioned above the proposal 18

19 is always applied substantially to formal languages, and also by the well-known prudence shown by Tarski throughout his life toward any application of results and techniques regarding formal languages to non-formal ones. There is further (though not direct) evidence of Tarski s prudence in Carnap s autobiography (Carnap 1963), where Carnap reports Tarski s idea (still maintained by Tarski, Carnap says, when the autobiography itself was written, at the beginning of the Sixties) that the distinction between logical and factual assertions and notions is merely a matter of degree and not of nature. In a passage, Carnap says precisely that his own conception of semantics starts from the basis given in Tarski s work, but differs from his conception by the sharp distinction which I draw between logical and non-logical constants, and between logical and factual truth (Carnap 1964, p. 62; our emphasis). In another passage, quoted by Gómez-Torrente (1996, p. 146), Carnap reports that already in 1930, in contrast to our [i.e., the logical positivists ] view that there is a fundamental difference between logical and factual statements [...] Tarski maintained that the distinction was only a matter of degree (Carnap 1963, p. 30). Also Quine, discussing the meetings at Harvard of in a letter (1943) to Carnap, confirms that he and Tarski argued against Carnap that while the notion of logical truth could be precisely characterized, the notion of analytical implication was an unexplained notion that we were not committed hitherto (Quine 1990, p. 296, quoted by Gómez-Torrente, ibid., p. 147). Thus, the discussions among Carnap, Quine and Tarski are evidence of the fact that Tarski, at least during a long period in his life, regarded his own proposal of distinction between logical notions and non-logical ones as something applicable to formal mathematical theories, or even to other disciplines, but not to the comprehensive theories which 19

20 Carnap was interested in; in any case, not as a solution of general philosophical problems such as analiticity. This is perhaps the reason of Tarski s sceptical attitude. References Carnap, R.: 1963, Intellectual Autobiography, in Schilpp 1963, pp Corcoran, J.: 1972, Conceptual structure of classical logic, Philosophy and Phenomenological Research 33, Etchemendy, J.: 1990, The concept of logical consequence, Harvard University Press, Cambridge, Mass. Gómez-Torrente, M.: 1996, Tarski on logical consequence, Notre Dame Journal of Formal Logic 37, Klein, F.: 1872, Vergleichende Betrachtungen über neuere geometrische Forschungen, Deichert, Erlangen. Lindenbaum, A. and A. Tarski: 1936, Über die Beschränktheit der Ausdrucksmittel deduktiver Theorien, Ergebnisse eines mathematischen Kolloquiums 7, 15-22; English translation by J. H. Woodger in Tarski 1983, McGee, V.: 1996, Logical operations, Journal of Philosophical Logic 25, Mostowski, A.: 1957, On a generalization of quantifiers, Fundamenta Mathematicae 44,

21 Quine, W. V. O.:1990, Dear Carnap, Dear Van. The Quine-Carnap correspondence and related work, University of California Press, Berkeley. Sagüillo, J. M.: 1997, Logical consequence revisited, Bulletin of Symbolic Logic 3, Schilpp, P. A. (ed.): 1963, The Philosophy of Rudolf Carnap, Open Court, La Salle. Sher, G. Y.: 1991, The bounds of logic: a generalized viewpoint, MIT Press, Cambridge, Mass. Sher, G. Y.: 1996, Did Tarski commit Tarski s fallacy?, Journal of Symbolic Logic 61, Skolem, T.: 1922, Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Proceedings of the Fifth Scandinavian Mathematical Congress (Helsinki); English translation by S. Bauer-Mengelberg in van Heijenoort 1967, Skolem, T.: 1958, Une relativisation des notions mathématiques fondamentales, in Le raisonnement en mathématiques et en sciences expérimentales: Actes du Colloque de Logique Mathématique, Paris 1955, C.N.R.S., Paris, Tarski, A.: 1933, Einige Betrachtungen über die Begriffe der ω-widerspruchsfreiheit und der ω-vollständigkeit, Monatshefte für Mathematik und Physik 40, ; English translation by J. H. Woodger in Tarski 1983, Tarski, A.: 1935, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica (Lemberg) 1, ; English translation by J. H. Woodger in Tarski 1983,

22 Tarski, A: 1936, Über den Begriff der logischen Folgerung, Actes du Congrès International de Philosophie Scientifique 7, Hermann, Paris 1-11; English translation by J. H. Woodger in Tarski 1983, Tarski, A.: 1954, Contributions to the theory of models, I-II, Indagationes Mathematicae 16, Tarski, A.: 1955, Contributions to the theory of models, III, Indagationes Mathematicae 17, Tarski, A.: 1958, Intervention sur le rapport de M. le Professeur Skolem, in Le raisonnement en mathématiques et en sciences expérimentales: Actes du Colloque de Logique Mathématique, Paris 1955, C.N.R.S., Paris, Tarski, A.: 1983, Logic, semantics, metamathematics: papers from 1923 to 1938, translated into English and edited by J. H. Woodger, Clarendon Press, Oxford 1956; second edition, edited by John Corcoran, Hackett, Indianapolis. Tarski, A.: 1986, What are logical notions?, edited (with an introduction) by John Corcoran, History and Philosophy of Logic 7, Tarski, A.: 1987, A philosophical letter of Alfred Tarski, edited by M. G. White, Journal of Philosophy 84,

23 Tarski, A.: 1995 Some Current Problems in Metamathematics, edited by J. Tarski and J. Wolenski, History and Philosophy of Logic 16, Tarski, A. and S. Givant: 1987, A formalization of set theory without variables, American Mathematical Society Colloquium Publications, vol. 41, AMS, Providence, Rhode Island. van Heijenoort, J. (ed.) : 1967, From Frege to Gödel: a source book in mathematical logic, , Harvard University Press, Cambridge, Mass. Whitehead, A. N. and B. Russell: 1910, Principia Mathematica, vol. 1, Cambridge University Press, Cambridge. 23