Semantic theory of truth

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

The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work published by Polish logician Alfred Tarski in the 1930s. Tarski, in "On the Concept of Truth in Formal Languages", attempted to formulate a new theory of truth in order to resolve the liar paradox. In the course of this he made several metamathematical discoveries, most notably Tarski's Indefinablity Theorem, which is similar to Gödel's Incompleteness Theorem. Roughly, this states that the concept of "truth" for the sentences of a given language cannot consistently be defined within that language.

To formulate theories about linguistic matters, it is generally necessary, in order to avoid semantic paradoxes like the liar paradox, to distinguish the language that one is talking about, the so-called object language, from the language that one is using, the so-called metalanguage. In the following, quoted sentences like "P" are always sentences of the object language. Everything not in quotation is part of the metalanguage. Tarski's material adequacy condition, also known as Convention T or the T-schema, holds that any viable theory of truth must entail, for every sentence P of a language, that:

(1) "P" is true if, and only if, p.

(where p abbreviates, in the metalanguage, the proposition expressed by the sentence "P" of the object language.)

For example,

(2) "Snow is white" is true if and only if snow is white.

The first half of (2) is about the sentence, "Snow is white". The second half is about snow. These sentences (1 and 2, etc.) have come to be called the "T-sentences". The reason they look trivial is that the object-language and the metalanguage are both English. But this would also be a T-sentence:

(3) "Der Schnee ist weiß" is true (in German) if and only if snow is white.

(It is important to note that as Tarski originally formulated it, this theory applies only to formal languages. He felt that natural languages were too complex and irregular to be suited to such formal treatment. But Tarski's approach was extended by Davidson into an approach to theories of meaning for natural languages, which involves treating "truth" as a primitive, rather than a defined concept. (See truth-conditional semantics.)

Tarski developed the theory, to give an inductive definition of truth, as follows.

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 if and only if A is not true.

(ii) A conjunction A&B is true if and only if A is true and B is true

(iii) A disjunction A v B is true if and only if A is true or B is true.

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

(v) An existential statement "there exists x A(x)" is true if and only if 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. A contemporary semantic definition of truth would define truth for the atomic sentences 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 expressed by the predicate F.

Tarski himself defined truth for atomic sentences in a variant way that does not use any technical terms from semantics, such as the "expressed by" above. This is because he wanted to define these semantic terms in terms of truth, so it would be circular were he to use one of them in the definition of truth itself. 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.