Deflationary Theory of Truth - Tarski and Deflationary Theories

Tarski and Deflationary Theories

Some years before Strawson developed his account of the sentences which include the truth-predicate as performative utterances, Alfred Tarski had developed his so-called semantic theory of truth. Tarski's basic goal was to provide a rigorously logical definition of the expression "true sentence" within a specific formal language and to clarify the fundamental conditions of material adequacy that would have to be met by any definition of the truth-predicate. If all such conditions were met, then it would be possible to avoid semantic paradoxes such as the liar paradox (i.e. "This sentence is false.") Tarski's material adequacy condition, or Convention T, is: a definition of truth for an object language implies all instances of the sentential form

(T) S is true if and only if P

where S is replaced by a name of a sentence (in the metalanguage) and P is replaced by a translation of that sentence in the metalanguage. So, for example, "La neve è bianca is true if and only if snow is white" is a sentence which conforms to Convention T; the object-language is Italian and the metalanguage is English. The predicate "true" does not appear in the object language, so no sentence of the object language can directly or indirectly assert truth or falsity of itself. Tarski thus formulated a two-tiered scheme that avoids semantic paradoxes such as Russell's paradox.

Tarski formulated his definition of truth indirectly through a recursive definition of the satisfaction of sentential functions and then by defining truth in terms of satisfaction. An example of a sentential function is "x defeated y in the 2004 US presidential elections"; this function is said to be satisfied when we replace the variables x and y with the names of objects such that they stand in the relation denoted by "defeated in the 2004 US presidential elections" (in the case just mentioned, replacing x with "George W. Bush" and y with "John Kerry" would satisfy the function, resulting in a true sentence). In general, a₁, ..., a_n satisfy n-adic predicate φ(x₁. ..., x_n) just in case substitution of the names 'a₁', ..., 'a_n' for the variables of φ in the relevant order yields "φ(a₁, ..., a_n)", and φ(a₁, ..., a_n). Given a method for establishing the satisfaction (or not) of every atomic sentence of the form A(...x_k...), the usual rules for truth-functional connectives and quantifiers yield a definition for the satisfaction condition of all sentences of the object language. For instance, for any two sentences A, B, A&B is satisfied if and only if A and B are satisfied (where '&' stands for conjunction), for any sentence A, ~A is satisfied if and only if A fails to be satisfied, and for any open sentence A where x is free in A, (x)A is satisfied if and only if for every substitution of an item of the domain for x yielding A*, A* is satisfied. Whether any complex sentence is satisfied is seen to be determined by its structure. An interpretation is an assignment of denotation to all of the non-logical terms of the object language. A sentence A is true (under an interpretation I) if and only if it is satisfied in I.

Tarski thought of his theory as a species of correspondence theory of truth, not a deflationary theory.