  Tarski’s aim in a series of works on semantical notions was precisely to propose mathematically acceptable definitions of those notions, and in particular acceptable in some one of the chief foundational systems (the system chosen by Tarski will change with time; see below). With appropriate definitions of those notions, the theorems about them would be susceptible of being reformulated making use of the defined notions, and the uneasiness described by Tarski in the text just quoted would be alleviated. The alternative of taking the semantical notions as primitive is also considered by Tarski in several places, but he clearly prefers to avoid it if possible. The reason is that in the axiomatic alternative there is no negligible risk that there remain ways of generating the semantical antinomies for the semantical primitives of the system (cf. Tarski 1983b, p. 255). With the definitional procedure, however, the consistency of the definition would depend exclusively on the consistency of the theory in which it is formulated, and this will be a theory we have reasons to think is consistent.