Is Tarski's theory of truth widely accepted

Question

Tarski's truth theorem asserts that a truth definition for a (reasonably strong) theory cannot be formalized within that theory.

It seems that Tarski's theory of truth has met with a lot of criticism. For examples, Hilary Putnam commented in “A comparison of something with something else”: As a philosophical account of truth, Tarski’s theory fails as badly as it is possible for an account to fail; Hartry Field in an influential discussion and diagnosis of what is lacking in Tarski’s account, pointed out that Tarski’s theory does not offer an account of reference and satisfaction at all. Rather, it only offers a number of disquotation clauses such as "‘Snow’ refers to snow".

So I wonder if Tarski's theory of truth is widely accepted in philosophical logic.

Answer 1

Perhaps the main point of divergence (though still fundamentally a refinement of rather than a wholesale repudiation of Tarski's theory) in the analytic philosophy literature was provided by Saul Kripke in his Outline of a Theory of Truth. Kripke's main point was that our language winds up being prone to deep problems of paradoxes of self-reference in a host of different ways, and Tarski's restriction both does not go far enough (because even seemingly "harmless" restricted sentences might give rise to paradoxical forms of interpretation) and seems to miss natural language aspects of the use of truth in practice.

His proposed modification is to see Tarski's truth hierarchies as partial definitions, and to take the union of such definitions at limit ordinal stages in the sequence of refined definitions to give a Fixed Point construction for the complete definition of truth featuring self-reference. Kripke evaluates the resulting theory and acknowledges that the internal logic of such a theory of truth cannot be classical - some sentences featuring the truth predicate are left "ungrounded" by the process of revision - and this is often seen as cause of concern for the theory of compositional logical semantics, but at least as regards the possibility of formal self-referential Truth, he demonstrated that it could be done.