Asemantic theory of truth is atheory of truth in thephilosophy of language which holds that truth is a property of sentences.[1]
Thesemantic conception of truth, which is related in different ways to both thecorrespondence anddeflationary conceptions, is due to work byPolishlogicianAlfred Tarski. Tarski, in "On the Concept of Truth in Formal Languages" (1935), attempted to formulate a new theory of truth in order to resolve theliar paradox. In the course of this he made several metamathematical discoveries, most notablyTarski's undefinability theorem using the same formal techniqueKurt Gödel used in hisincompleteness theorems. Roughly, this states that a truth-predicate satisfyingConvention T for the sentences of a given language cannot be definedwithin that language.
To formulate linguistic theories[2] without semanticparadoxes such as theliar paradox, it is generally necessary to distinguish the language that one is talking about (theobject language) from the language that one is using to do the talking (themetalanguage). In the following, quoted text is use of the object language, while unquoted text is use of the metalanguage; a quoted sentence (such as "P") is always the metalanguage'sname for a sentence, such that this name is simply the sentenceP rendered in the object language. In this way, the metalanguage can be used to talk about the object language;Tarski's theory of truth (Alfred Tarski 1935) demanded that the object language be contained in the metalanguage.
Tarski'smaterial adequacy condition, also known asConvention T, holds that any viable theory of truth must entail, for every sentence "P", a sentence of the following form (known as "form (T)"):
(1) "P" is trueif, and only if, P.
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; here is an example where the object language is German and the metalanguage is English:
(3) 'Schnee ist weiß' is true if and only if snow is white.
It is important to note that as Tarski originally formulated it, this theory applies only toformal languages, cf. alsosemantics of first-order logic. He gave a number of reasons for not extending his theory tonatural languages, including the problem that there is no systematic way of deciding whether a given sentence of a natural language is well-formed, and that a natural language isclosed (that is, it can describe the semantic characteristics of its own elements). But Tarski's approach was extended byDavidson into an approach to theories ofmeaning for natural languages, which involves treating "truth" as a primitive, rather than a defined, concept. (Seetruth-conditional semantics.)
Tarski developed the theory to give aninductive definition of truth as follows. (SeeT-schema)
For a languageL containing ¬ ("not"), ∧ ("and"), ∨ ("or"), ∀ ("for all"), and ∃ ("there exists"), Tarski's inductive definition of truth looks like this:
These explain how the truth conditions ofcomplex sentences (built up fromconnectives andquantifiers) can be reduced to the truth conditions of theirconstituents. The simplest constituents areatomic 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 the context of truth. Therefore it would be circular to use one of them in the definition of truth itself. Tarski's semantic conception of truth plays an important role inmodern logic and also in contemporaryphilosophy of language. It is a rather controversial point whether Tarski's semantic theory should be counted either as acorrespondence theory or as adeflationary theory.[3]
Kripke's theory of truth (Saul Kripke 1975) is based on partial logic (a logic of partially definedtruth predicates instead of Tarski's logic of totally defined truth predicates) with thestrong Kleene evaluation scheme.[4]
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