A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences.[1]
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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 undefinability theorem using the same formal technique as Kurt Gödel used in his incompleteness theorems. Roughly, this states that a truth-predicate satisfying convention-T for the sentences of a given language cannot be defined within that language.
To formulate linguistic theories[2] without semantic paradoxes like the liar paradox, it is generally necessary 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 names of sentences whereas the unquoted sentences are the sentences in the metalanguage which are translations of sentences in the object language. Tarski demanded that the object language was contained in the metalanguage. Tarski's material adequacy condition, also known as Convention T, holds that any viable theory of truth must entail, for every sentence P of a language, a sentence of the form (T):
(1) 'P' is true if, and only if, P.
(where 'P' is the name of the sentence P in the metalanguage which in turn is a translation of the corresponding sentence in the object language.)
For example,
(2) 'Snow is white' is true if and only if snow is white.
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 gave a number of reasons for not extending his theory to natural languages, including the problem of there being no systematic way of deciding whether a given sentence of a natural language is well-formed, and that natural languages are 'closed'; that is, they can describe the semantic characteristics of their own elements. 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:
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:
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 a rather controversial matter whether Tarski's semantic theory should be counted as either a correspondence theory or as a deflationary theory.
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