1 Compare Tarski [2] (see bibliography at the end of the paper). This work may be consulted for a more detailed and formal presentation of the subject of the paper, especially of the material included in Sections 6 and 9-13. It contains also references to my earlier publications on the problems of semantics (a communication in Polish 1930; the article Tarski [1] in French, 1931; a communication in German, 1932; and a book in Polish, 1933). The expository part of the present paper is related in its character to Tarski [3]. My investigations on the notion of truth and on theoretical semantics have been reviewed or discussed in Hofstadter [1], Juhos [1], Kokoszynska [1] and [2], Kotarbinski [2], Scholz [1], Weinberg [1], et al.

2 It may be hoped that the interest in theoretical semantics will now increase, as a result of the recent publication of the important work Carnap [2].

3 This applies, in particular, to public discussions during the I. International Congress for the Unity of Science (Paris, 1935) and the Conference of International Congresses for the Unity of Science (Paris, 1937); cf., e.g., Neurath [1] and Gonseth [1].

4 The words "notion" and "concept" are used in this paper with all of the vagueness and ambiguity with which they occur in philosophical literature. Thus, sometimes they refer simply to a term, sometimes to what is meant by a term, and in other cases to what is denoted by a term. Sometimes it is irrelevant which of these interpretations is meant; and in certain cases perhaps none of them applies adequately. While on principle I share the tendency to avoid these words in any exact discussion, I did not consider it necessary to do so in this informal presentation.

5 For our present purposes it is somewhat more convenient to understand by "expressions," "sentences," etc., not individual inscriptions, but classes of inscriptions of similar form (thus, not individual physical things, but classes of such things).

6 For the Aristotelian formulation see Aristotle [1], gamma, 7, 27. The other two formulations are very common in the literature, but I do not know with whom they originate. A critical discussion of various conceptions of truth can be found, e.g., in Kotarbinski [1] (so far available only in Polish), pp. 123 ff., and Russell [1], pp. 362 ff.

7 For most of the remarks contained in Sections 4 and 8, I am indebted to the late S. Lesniewski who developed them in his unpublished lectures in the University of Warsaw (in 1919 and later). However, Lesniewski did not anticipate the possibility of a rigorous development of the theory of truth, and still less of a definition of this notion; hence, while indicating equivalences of the form (T) as premises in the antinomy of the liar, he did not conceive them as any sufficient conditions for an adequate usage (or definition) of the notion of truth. Also the remarks in Section 8 regarding the occurrence of an empirical premise in the antinomy of the liar, and the possibility of eliminating this premiss, do not originate with him.

8 In connection with various logical and methodological problems involved in this paper the reader may consult Tarski [6].

9 The antinomy of the liar (ascribed to Eubulides or Epimenides) is discussed here in Sections 7 and 8. For the antinomy of definability (due to J. Richard) see, e.g., Hilbert-Bernays [1], vol. 2, pp. 263 ff.; for the antinomy of heterological terms see Grelling-Nelson [1], p. 307.

10 Due to Professor J. Lukasiewicz (University of Warsaw).

11 This can roughly be done in the following way. Let S be any sentence beginning with the words "Every sentence." We correlate with S a new sentence S* by subjecting S to the following two modifications: we replace in S the first word, "Every," by "The"; and we insert after the second word, "sentence," the whole sentence S enclosed in quotation marks. Let us agree to call the sentence S "(self-)applicable" or "non-(self-)applicable" dependent on whether the correlated sentence S* is true or false. Now consider the following sentence:

Every sentence is non-applicable.
It can easily be shown that the sentence just stated must be both applicable and non-applicable; hence a contradiction. It may not be quite clear in what sense this formulation of the antinomy does not involve an empirical premiss; however, I shall not elaborate on this point.

12 The terms "logic" and "logical" are used in this paper in a broad sense, which has become almost traditional in the last decades; logic is assumed here to comprehend the whole theory of classes and relations (i.e., the mathematical theory of sets). For many different reasons I am personally inclined to use the term "logic" in a much narrower sense, so as to apply it only to what is sometimes called "elementary logic," i.e., to the sentential calculus and the (restricted) predicate calculus.

13 Cf. here, however, Tarski [3], p. 5 f.

14 The method of construction we are going to outline can be applied -- with appropriate changes -- to all formalized languages that are known at the present time; although it does not follow that a language could not be constructed to which this method would not apply.

15 In carrying through this idea a certain technical difficulty arises. A sentential function may contain an arbitrary number of free variables; and the logical nature of the notion of satisfaction varies with this number. Thus, the notion in question when applied to functions with one variable is a binary relation between these functions and single objects; when applied to functions with two variables it becomes a ternary relation between functions and couples of objects; and soon. Hence, strictly speaking, we are confronted, not with one notion of satisfaction, but with infinitely many notions; and it turns out that these notions cannot be defined independently of each other, but must all be introduced simultaneously.

To overcome this difficulty, we employ the mathematical notion of an infinite sequence (or, possibly, of a finite sequence with an arbitrary number of terms). We agree to regard satisfaction, not as a many-termed relation between sentential functions and an indefinite number of objects, but as a binary relation between functions and sequences of objects. Under this assumption the formulation of a general and precise definition of satisfaction no longer presents any difficulty; and a true sentence can now be defined as one which is satisfied by every sequence.

16 To define recursively the notion of satisfaction, we have to apply a certain form of recursive definition which is not admitted in the object-language. Hence the "essential richness" of the meta-language may simply consist in admitting this type of definition. On the other hand, a general method is known which makes it possible to eliminate all recursive definitions and to replace them by normal, explicit ones. If we try to apply this method to the definition of satisfaction, we see that we have either to introduce into the meta-language variables of a higher logical type than those which occur in the object-language; or else to assume axiomatically in the meta-language the existence of classes that are more comprehensive than all those whose existence can be established in the object- language. See here Tarski [2], pp. 393 ff., and Tarski [5], p. 110.

17 Due to the development of modern logic, the notion of mathematical proof has undergone far-reaching simplification. A sentence of a given formalized discipline is provable if it can be obtained from the axioms of this discipline by applying certain simple and purely formal rules of inference, such as those of detachment and substitution. Hence to show that all provable sentences are true, it suffices to prove that all the sentences accepted as axioms are true, and that the rules of inference when applied to true sentences yield new true sentences; and this usually presents no difficulty.

On the other hand, in view of the elementary nature of the notion of provability, a precise definition of this notion requires only rather simple logical devices. In most cases, those logical devices which are available in the formalized discipline itself (to which the notion of provability is related) are more than sufficient for this purpose. We know, however, that as regards the definition of truth just the opposite holds. Hence, as a rule, the notions of truth and provability cannot coincide; and since every provable sentence is true, there must be true sentences which are not provable.

18 Thus the theory of truth provides us with a general method for consistency proofs for formalized mathematical disciplines. It can be easily realized, however, that a consistency proof obtained by this method may possess some intuitive value -- i.e., may convince us, or strengthen our belief, that the discipline under consideration is actually consistent -- only in case we succeed in defining truth in terms of a meta-language which does not contain the object-language as a part (cf. here a remark in Section 9). For only in this case the deductive assumptions of the meta-language may be intuitively simpler and more obvious than those of the object-language -- even though the condition of "essential richness" will be formally satisfied. Cf. here also Tarski [3], p. 7.

The incompleteness of a comprehensive class of formalized disciplines constitutes the essential content of a fundamental theorem of K. Gödel [1], pp. 187 ff. The explanation of the fact that the theory of truth leads so directly to Gödel's theorem is rather simple. In deriving Gödel's result from the theory of truth we make an essential use of the fact that the definition of truth cannot be given in a meta-language which is only as "rich" as the object-language (cf. note 17); however, in establishing this fact, a method of reasoning has been applied which is very closely related to that used (for the first time) by Gödel. It may be added that Gödel was clearly guided in his proof by certain intuitive considerations regarding the notion of truth, although this notion does not occur in the proof explicitly; cf. Gödel [1], pp. 174 f.

19 The notions of designation and definition lead respectively to the antinomies of Grelling-Nelson and Richard (cf. note 9). To obtain an antinomy for the notion of satisfaction, we construct the following expression:

The sentential function X does not satisfy X.
A contradiction arises when we consider the question whether this expression, which is clearly a sentential function, satisfies itself or not.

20 All notions mentioned in this section can be defined in terms of satisfaction. We can say, e.g., that a given term designates a given object if this object satisfies the sentential function "x is identical with T" where 'T' stands for the given term. Similarly, a sentential function is said to define a given object if the latter is the only object which satisfies this function. For a definition of consequence see Tarski [4], and for that of synonymity -- Carnap [2].

21 General semantics is the subject of Carnap [2]. Cf. here also remarks in Tarski [2], pp. 388 f.

22 Cf. various quotations in Ness [1], pp. 13 f.

23 The names of persons who have raised objections will not be quoted here, unless their objections have appeared in print.

24 It should be emphasized, however, that as regards the question of an alleged vicious circle the situation would not change even if we took a different point of view, represented, e.g., in Carnap [2]; i.e., if we regarded the specification of conditions under which sentences of a language are true as an essential part of the description of this language. On the other hand, it may be noticed that the point of view represented in the text does not exclude the possibility of using truth-tables in a deductive development of logic. However, these tables are to be regarded then merely as a formal instrument for checking the provability of certain sentences; and the symbols 'T' and 'F' which occur in them and which are usually considered abbreviations of "true" and "false" should not be interpreted in any intuitive way.

25 Cf. Juhos [1]. I must admit that I do not clearly understand von Juhos' objections and do not know how to classify them; therefore, I confine myself here to certain points of a formal character. Von Juhos does not seem to know my definition of truth; he refers only to an informal presentation in Tarski [3] where the definition has not been given at all. If he knew the actual definition, he would have to change his argument. However, I have no doubt that he would discover in this definition some "defects" as well. For he believes he has proved that "on ground of principle it is impossible to give such a definition at all."

26 The phrases "p is true" and "p is the case" (or better "it is true that p" and "it is the case that p") are sometimes used in informal discussions, mainly for stylistic reasons; but they are considered then as synonymous with the sentence represented by 'p'. On the other hand, as far as I understand the situation, the phrases in question cannot be used by von Juhos synonymously with 'p'; for otherwise the replacement of (T) by (T') or (T'') would not constitute any "improvement."

27 Cf. the discussion of this problem in Kokoszynska [1], pp. 161 ff.

28 Most authors who have discussed my work on the notion of truth are of the opinion that my definition does conform with the classical conception of this notion; see, e.g., Kotarbinski [2] and Scholz [1].

29 Cf. Ness [1]. Unfortunately, the results of that part of Ness' research which is especially relevant for our problem are not discussed in his book; compare p. 148, footnote 1.

30 Though I have heard this opinion several times, I have seen it in print only once and, curiously enough, in a work which does not have a philosophical character -- in fact, in Hilbert-Bernays [1], vol. II, p. 269 (where, by the way, it is not expressed as any kind of objection). On the other hand, I have not found any remark to this effect in discussions of my work by professional philosophers (cf. note 1).

31 Cf. Gonseth [1], pp. 187 f.

32 See Nagel [1], and Nagel [2], pp. 471 f. A remark which goes, perhaps, in the same direction is also to be found in Weinberg [1], p. 77; cf., however, his earlier remarks, pp. 75 f.

33 Such a tendency was evident in earlier works of Carnap (see, e.g., Carnap [1], especially Part V) and in writings of other members of Vienna Circle. Cf. here Kokoszynska [1] and Weinberg [1].

34 For other results obtained with the help of the theory of truth see Gödel [2]; Tarski [2], pp. 401 ff.; and Tarski [5], pp. 111 f.

35 An object -- e.g., a number or a set of numbers -- is said to be definable (in a given formalism) if there is a sentential function which defines it; cf. note 20. Thus, the term "definable," though of a meta- mathematical (semantic) origin, is purely mathematical as to its extension, for it expresses a property (denotes a class) of mathematical objects. In consequence, the notion of definability can be re-defined in purely mathematical terms, though not within the formalized discipline to which this notion refers; however, the fundamental idea of the definition remains unchanged. Cf. here -- also for further bibliographic reference -- Tarski [1]; various other results concerning definability can also be found in the literature, e.g., in Hilbert-Bernays [1], vol. I, pp. 354 ff., 369 ff., 456 ff., etc., and in Lindenbaum-Tarski [1]. It may be noticed that the term "definable" is sometimes used in another, meta-mathematical (but not semantic), sense; this occurs, for instance, when we say that a term is definable in other terms (on the basis of a given axiom system). For a definition of a model of an axiom system see Tarski [4].