Page 558 - Encyclopedia of Philosophy of Language

Key Figures
nected to the axiomatization of the well-ordered sets (1921). Throughout the 1920s set theory remained the main subject of his publications. His work on the foundations of mathematics, however, involved him in the study of metalogical notions like 'true sentence,' 'logical consequence/ and 'undefined concept.' As a result he embarked on a new series of papers of which the most important was the one published in 1933 under the title Pofccie prawdy w jfzikach nauk dedukcyjnych.
2. Truth
In this monograph Tarski seeks to provide a concept of truth suitable for scientific and mathematical pur- poses, that is, a concept of truth which, unlike its everyday counterpart, is free of inconsistencies. His analysis of'truth' sets out with the following example:
'It is snowing* is true if and only if it is snowing. (1)
A distinction is made between a language L, the one containing 'It is snowing,' and a language L', which contains sentence (1). It is supposed that the latter can be used to talk about L. In particular this means that L' is assumed to be rich enough to express the truth and falsity of sentences of L. A possible definition in L' of a true sentence of L can be called adequate if all of the (possibly infinitely many) sen- tences of L' having the same form as (1) are provable in L'. Given this, the question can now be asked in a rigorous fashion whether or not an adequate defi- nition of truth for sentences of L in L' can be given .
Tarski gives two answers to this question. Assume, to begin with, that L and L' are identical. That is, suppose a language is capable of expressing its own truth predicate, as is natural language. Then, for any proposed definition of truth, there is a sentence of the form of (1) that can be disproved in L. Hence, no adequate definition of truth can be given in L.
Second, suppose L' includes second-order logic over L and Peano-Arithmetic. Then an adequate definition of a 'true sentence' in L can be given in L'. To prove this, Tarski first tackled the more general problem of defining satisfaction. Using satisfaction, other semantic notions like 'definable relation' and 'logical consequence' are then easily defined.
3. Paradox
Tarski's first result pertains to a semantic paradox that had distressed philosophers of language as well as scientists for quite some time. Consider the following famous sentence, known as the liar sentence:
This sentence is false. (2)
In the languages of deductive sciences such sentences are really dangerous, because of the possibility of pro- ving anything from a single contradiction. Tarski's solution is based on the clear distinction between the language to be described, the object language, and the language of description, the metalanguage. It is obvious that a sentence is true or false only as a sen- tence of a language. One can eliminate the paradox above by observing that whatever language L the ref- erence of the expression 'this sentence' belongs to, the sentence that 'this is not a true sentence of L' does not belong to L but to its metalanguage, say L'.
An important appendix was added to the German translation of this monograph (entitled Der Wahrheitsbergiff in den formalisierten Sprachen) in which Tarski showed how to extend his method to languages not respecting the basic principle of the Lesniewski-Ajdukiewicz theory of semantic cate- gories, especially to the language of the Zermelo-
Fraenkel set theory.
4. Influence
Tarski's writings have had—and still have—a tremen- dous influence on various areas of fundamental research. His method of defining metalogical concepts, as well as the model-theoretical techniques he developed, are among the foundations of con- temporary logic. Tarski's famous book Logice mat- ematycznej i rnetodzie dedukcyjnej (1936) has been the basic textbook of logic for decades all over the world. It is translated into a dozen languages. The English translation appeared in 1941 under the title Intro- duction to Logic and to the Methodology of Deductive Sciences.
The importance of Tarski's work for what is, in the 1990s, known as formal semantics is also hard to overestimate. Contemporary research into truth-
conditional semantics, and model-theoretic semantics, as well as investigations into the analysis of truth and semantic paradox can be traced back directly to Tarski's work on metalogical concepts, especially his theory of truth.
See also: Formal Semantics; Paradoxes, Semantic; Truth.
Bibliography
Givant S R, McKenzie R N (eds.) 1981 The Collected Works of Alfred Tarski, vols. i-iv. University of California, Berkeley, CA
Tarski A 1956 Logic, Semantics. Metamathematics. Clar- endon Press, Oxford
536