Reddit mentions of Axiomatic Theories of Truth

Ooh, I'm studying this stuff right now! Check out this paper by Kripke for a semantic approach to truth inside a theory. That sparked a lot of interest in the area, so check out this book or just read this page, both by Volker Halbach, for a great outline of the different theories that have emerged to capture truth in a theory.

Basically, the whole idea depends on your notion of truth. Someone correctly mentioned below that you cannot define truth inside a theory (a predicate T such that for every statement P in the language you have the following equivalence: T(P) <-> P). This said, people have developed different axiomatizations of a truth predicate that enables certain qualities we think truth should have (that it is compositional for example, so that T(p&q) <-> (T(p)&T(q))). These each have their pitfalls, though; this paper by Hannes Leitgeb outlines nice qualities that a truth predicate should have (but cannot have simulataneously).

There are some really interesting results in the literature and I'd be happy to link you to further papers/answer questions if you're interested.