As has been shown in Section 3, Tarski (1935, p. 257) rejected an axiomatization of truth like TB based solely on the T-sentences because the resulting theory ‘would lack the most important and most fruitful general theorems’. Moreover, he did not expect that adding some of those general theorems as axioms would lead to a satisfactory theory of truth because he thought that such an axiomatization would be somewhat arbitrary (see p. 20).

Ironically Tarski's definition of truth prepared the ground for the wide acceptance of a theory – or rather, a family of theories – that go beyond purely disquotational theories but are nevertheless seen as natural, and far from arbitrary. The inductive clauses from Tarski's definition of truth can be turned into axioms. The resulting theory is thought by many philosophers and logicians to be a theory of truth that is natural and, in a sense, complete: it proves generalizations of the kind Tarski had in mind. In particular, it proves the general principle of contradiction, the statement that a sentence and its negation cannot both be true.

Donald Davidson assigned an important role to this axiomatization of truth in his theory of meaning (see Davidson 1984c and Fischer 2008). He proposed to turn Tarski's clauses for defining truth into axioms. Although significant work was done by Davidson and his disciples to extend the theory from simple formal languages to natural languages containing adverbs and other phrases not dealt with by Tarski, Davidson never specified the axioms of the theory for a simple formal language like the one considered by Tarski. There are, however, some decisions to be taken in the formulation of this theory.