Tarski Alfred (1901–83), Polish-born American mathematician, logician, and philosopher of logic famous for his investigations of the concepts of truth and consequence conducted in the 1930s. His analysis of the concept of truth in syntactically precise, fully interpreted languages resulted in a definition of truth and an articulate defense of the correspondence theory of truth. Sentences of the following kind are now known as Tarskian biconditionals: ‘The sentence ‘Every perfect number is even’ is true if and only if every perfect number is even.’ One of Tarski’s major philosophical insights is that each Tarskian biconditional is, in his words, a partial definition of truth and, consequently, all Tarskian biconditionals whose right-hand sides exhaust the sentences of a given formal language together constitute an implicit definition of ‘true’ as applicable to sentences of that given formal language. This insight, because of its penetrating depth and disarming simplicity, has become a staple of modern analytic philosophy. Moreover, it in effect reduced the philosophical problem of defining truth to the logical problem of constructing a single sentence having the form of a definition and having as consequences each of the Tarskian biconditionals. Tarski’s solution to this problem is the famous Tarski truth definition, versions of which appear in virtually every mathematical logic text.
Tarski’s second most widely recognized philosophical achievement was his analysis and explication of the concept of consequence. Consequence is interdefinable with validity as applied to arguments: a given conclusion is a consequence of a given premise-set if and only if the argument composed of the given conclusion and the given premise-set is valid; conversely, a given argument is valid if and only if its conclusion is a consequence of its premise-set. Shortly after discovering the truth definition, Tarski presented his ‘no-countermodels’ definition of consequence: a given sentence is a consequence of a given set of sentences if and only if every model of the set is a model of the sentence (in other words, if and only if there is no way to reinterpret the non-logical terms in such a way as to render the sentence false while rendering all sentences in the set true). As Quine has emphasized, this definition reduces the modal notion of logical necessity to a combination of syntactic and semantic concepts, thus avoiding reference to modalities and/or to ‘possible worlds.’ After Tarski’s definitive work on truth and on consequence he devoted his energies largely to more purely mathematical work. For example, in answer to Gödel’s proof that arithmetic is incomplete and undecidable, Tarski showed that algebra and geometry are both complete and decidable. Tarski’s truth definition and his consequence definition are found in his 1956 collection Logic, Semantics, Metamathematics (2d ed., 1983): article VIII, pp. 152–278, contains the truth definition; article XVI, pp. 409–20, contains the consequence definition. His published articles, nearly 3,000 pages in all, have been available together since 1986 in the four-volume Alfred Tarski, Collected Papers, edited by S. Givant and R. McKenzie. See also GÖDEL’S INCOMPLETENESS THEO- REMS , LOGICAL CONSEQUENCE , TRUT. J.Cor.