TARSKI ON LOGICAL NOTIONS

Tarski on Logical Notions

LUCA BELLOTTI

(Dipartimento di Filosofia - Università di Pisa - Italy)

Address: Via E. Gianturco 55, I-19126 La Spezia, Italy

Phone number: +39-187-524673

Email: bellotti5@supereva.it

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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. 143-154), 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. 143-154), 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).

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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

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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.

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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 type-

theoretic 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).

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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

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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 “set-

theoretic 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).

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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;

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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.

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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

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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

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“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 Lx (an

extension of the above mentioned basic language, with four predicate functors, a first-

order 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

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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. 232-233). We can find exactly the same distinction

in Sher 1991 (pp. 57-58): 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’.

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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

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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

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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. 418-420). 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. 418-419): “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 extra-

logical” (our emphasis). Why this scepticism in the same year (1935) in which the paper

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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 1939-

1940, 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.

159-168). 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. 166-167). We have then a ‘broad’ concept of

logical notion; a concept that here, however, is not precisely delimited though its

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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 non-

committal 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

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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 1940-1941 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

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. 3-85.

Corcoran, J.: 1972, “Conceptual structure of classical logic”, Philosophy and

Phenomenological Research 33, 25-47.

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, 125-151.

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, 384-392.

McGee, V.: 1996, “Logical operations”, Journal of Philosophical Logic 25, 567-580.

Mostowski, A.: 1957, “On a generalization of quantifiers”, Fundamenta Mathematicae

44, 12-36.

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,

216-241.

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, 653-686.

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,

290-301.

Skolem, T.: 1958, “Une relativisation des notions mathématiques fondamentales”, in Le

raisonnement en mathématiques et en sciences expérimentales: Actes du

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Tarski, A.: 1933, “Einige Betrachtungen über die Begriffe der ω-Widerspruchsfreiheit

und der ω-Vollständigkeit”, Monatshefte für Mathematik und Physik 40, 97-112;

English translation by J. H. Woodger in Tarski 1983, 279-295.

Tarski, A.: 1935, “Der Wahrheitsbegriff in den formalisierten Sprachen”, Studia

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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, 409-420.

Tarski, A.: 1954, “Contributions to the theory of models”, I-II, Indagationes

Mathematicae 16, 572-588.

Tarski, A.: 1955, “Contributions to the theory of models”, III, Indagationes

Mathematicae 17, 56-64.

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, 17-18.

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Tarski, A.: 1986, “What are logical notions?”, edited (with an introduction) by John

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Tarski, A.: 1987, “A philosophical letter of Alfred Tarski”, edited by M. G. White,

Journal of Philosophy 84, 28-32.

23

Tarski, A.: 1995 “Some Current Problems in Metamathematics”, edited by J. Tarski and

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Tarski, A. and S. Givant: 1987, A formalization of set theory without variables,

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van Heijenoort, J. (ed.) : 1967, From Frege to Gödel: a source book in mathematical

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