Editing Semantic theory of truth (section)

==Tarski's theory of truth<!--'Tarski's theory of truth' redirects here-->==

To formulate linguistic theories<ref>Parts of section is adapted from Kirkham, 1992.</ref> without semantic [[paradox]]es such as the [[liar paradox]], it is generally necessary to distinguish the language that one is talking about (the ''object language'') from the language that one is using to do the talking (the ''[[metalanguage]]'').  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's ''name'' for a sentence, such that this name is simply the sentence ''P'' rendered in the object language. In this way, the metalanguage can be used to talk about the object language; '''Tarski's theory of truth'''<!--boldface per WP:R#PLA--> ([[Alfred Tarski]] 1935) demanded that the object language be contained in the metalanguage.

{{anchor|Convention T}}
Tarski's '''material adequacy condition'''<!--boldface per WP:R#PLA-->, also known as '''Convention T'''<!--boldface per WP:R#PLA-->, 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 true [[if, 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 to [[formal language]]s, cf. also [[First-order_logic#Semantics|semantics of first-order logic]].  He gave a number of reasons for not extending his theory to [[natural language]]s, 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 is ''closed'' (that is, it can describe the semantic characteristics of its own elements).  But Tarski's approach was extended by [[Donald Davidson (philosopher)|Davidson]] into an approach to theories of ''[[meaning (linguistics)|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. (See [[T-schema]])

For a language ''L'' containing ¬ ("not"), ∧ ("and"), ∨ ("or"), ∀ ("for all"), and ∃ ("there exists"), Tarski's inductive definition of truth looks like this:

* (1) A primitive statement "''A''" is true if, and only if, ''A''.
* (2) "¬''A''" is true if, and only if, "''A"'' is not true.
* (3) "''A''∧''B''" is true if, and only if, "''A" is true'' and "''B" is true''.
* (4) "''A''∨''B''" is true if, and only if, "''A" is true'' or "''B" is true'' or ("''A" is true'' and "''B" is true'').
* (5) "∀''x''(''Fx'')" is true if, and only if, for all objects x, "Fx" is true.
* (6) "∃''x''(''Fx'')" is true if, and only if, there is an object ''x'' for which "Fx" is true.

These explain how the truth conditions of ''complex'' sentences (built up from [[Logical connective|connective]]s and [[Quantifier (logic)|quantifiers]]) can be reduced to the truth conditions of their [[constituent (linguistics)|constituent]]s. The simplest constituents are [[atomic sentence]]s. A contemporary semantic definition of truth would define truth for the atomic sentences as follows:

* An atomic sentence ''F''(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) is true (relative to an [[Assignment (mathematical logic)|assignment]] of values to the variables ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)) if the corresponding [[Value (mathematics)|value]]s of [[Variable (mathematics)|variables]] bear the [[Relation (mathematics)|relation]] expressed by the [[Predicate (logic)|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 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 in [[First-order logic|modern logic]] and also in contemporary [[philosophy of language]]. It is a rather controversial point whether Tarski's semantic theory should be counted either as a [[Correspondence theory of truth|correspondence theory]] or as a [[Deflationary theory of truth|deflationary theory]].<ref> Kemp, Gary. ''Quine versus Davidson: Truth, Reference, and Meaning.'' Oxford, England: Oxford University Press, 2012, p. 110.</ref>