Semantic theory of truth

The semantic theory of truth is the theory that belief that any claim that a proposition is true can be made only as a formal requirement regarding the language in which the proposition itself is expressed.

In some ways related to both the Correspondence Conception and the Deflationary Conception is the Semantic Conception of Truth, due to Alfred Tarski, a Polish logician who published his work on truth in the 1930s. Part of Tarski's motivation in developing this conception of truth was to resolve the Liar paradox and this led Tarski to several interesting mathematical discoveries. In particular, Tarski's Indefinablity Theorem, which is similar to Goedel's Incompleteness Theorem. Tarski took the T-sentences not to give the theory of truth itself, but to be a constraint on defining the notion of truth. That is, in Tarski's view, any adequate definition or theory of truth must imply all of the T-sentences (this constraint is known as Convention T). Tarski developed a rather complicated theory, involving what is known as an inductive definition of truth and introduced further ideas, such as the distinction between object language and meta-language (which is important in avoiding the semantic paradoxes such as the Liar Paradox).

For a language L containing ~ ("not"), & ("and"), v ("or") and quantifiers ("for all" and "there exists"), Tarski's inductive definition of truth looks like this:

(i) A negation ~A is true iff A is not true.

(ii) A conjunction A&B is true iff A is true and B is true

(iii) A disjunction A v B is true iff A is true or B is true.

(iv) A universal statement "for all x A(x)" is true iff each object satisfies "A(x)".

(v) An existential statement "there exists x A(x)" is true iff there is an object which satisfies "A(x)".

These explain how the truth conditions of complex sentences (built up from connectives and quantifiers) can be reduced to the truth conditions of their constituents. The simplest constituents are atomic sentences, and Tarski defined truth for these as follows:

(vi) An atomic sentence F(x1,...,xn) is true (relative to an assignment of values to the variables x1, ..., xn)) if the corresponding values of variables bear the relation denoted by the predicate F.

Tarski's semantic conception of truth plays an important role in modern logic and also in much contemporary philosophy of language. It is rather controversial matter whether Tarski's semantic theory should be counted as either a correspondence theory or as a deflationary theory. Tarski himself seems to have intended his account to be a refinement of the classical correspondence theory.