That is...

That is, he showed that it is impossible, in any system strong enough to capture the axioms and results of arithmetic, to define within it a formula which holds of all and only the sentences within it that are true (on its standard interpretation).

The result, like Gödel’s, again turned upon the possibility of constructing a “self-referential” sentence, in this case one saying of itself (given any putative truth predicate) that it is not true; to demonstrate this possibility of construction, Tarski depended, as Gödel had, of arithmetization to represent the formal syntax of a language within the language itself.[^144] Both results undermined intuitively plausible assumptions about the ability of formal systems to capture the basis of ordinary judgments about the truth of mathematical propositions.

The results of Gödel and Tarski were to have a deep and determinative influence on the methodological assumptions of philosophers within the analytic tradition. Most decisive were their effects on the program, of which Frege, Russell, Carnap, Schlick, Wittgenstein and Hilbert had all been partisans, of seeking to clarify the logical structure of a language or a specialized portion thereof (for instance the language of arithmetic) purely through a syntactic description of its structure.

In a later paper, published in 1944, Tarski presented his own earlier result as demanding that the purely syntactic description of language structures be supplemented with what he called “semantic” concepts of truth and designation.

Semantics, he said, is a discipline which, speaking loosely, deals with certain relations between expressions of a language and the objects (or ‘states of affairs’) ‘referred to’ by those expressions As typical examples of semantic concepts we may mention the concepts of designation, satisfaction, and definition as these occur in the following examples: the expression ‘the father of his country’ designates (denotes) George Washington ; snow satisfies the sentential function (the condition) ‘x is white’ ; the equation ‘2 x = 1’ defines (uniquely determines) the number ½ .[^145] Because it is impossible, as was shown by Tarski’s own earlier result, to give a consistent purely syntactical definition of truth for a language within that language itself, the theorist who wishes to give an account of truth must avail himself also of the semantic or “referential” relationships between the language’s terms and the objects they stand for.