Richard Marshall interviews Volker Halbach in 3:AM Magazine:
3:AM: From the beginning you have been investigating the nature of truth. One of the big distinctions that you’ve been working with is approaches to truth that are axiomatic and approaches that are semantic. So can we begin by asking you to sketch what the distinction is, and what is at stake?
VH: The axiomatic approach is very simple: We stipulate axioms for the truth predicate that look plausible and that avoid the paradoxes. Truth is taken to be a primitive notion. The axiomatic approach differs from traditional definitional approaches such as the correspondence or coherence theory of truth that it is not assumed from the outset that truth is definable.
Semantic theories of truth provide methods for defining semantics or models for a language with a truth predicate. The semantic definition of a model is usually carried out in set theory for a language that is essentially weaker. Semantic theories cannot provide models for the language in which the model is defined. In philosophical logic this is a standard approach that has been applied to many other notions: Toy languages with symbols for necessity, knowledge, or the like are given a suitable semantics. In semantic theories of truth the same strategy is applied to the truth predicate. Kripke’s theory and the revision semantics of truth by Gupta and Herzberger belong into this category, but also Tarski’s extremely successful model-theoretic account of truth that is the starting point for all later accounts.
More here.