An interesting new paper by Kentaro Fujimoto in the Bulletin of Symbolic Logic 16(3) discusses logical "theories of truth". For this precis, I will liberally plagiarize Fujimoto's paper, I doubt he'll mind. Start with B, a basic underlying logic which is a first-order classical system of arithmetic such as Peano arithmetic. Then try to introduce a truth predicate T, so that T(x) means "x is true" for a proposition x. This will get you a theory of logical truth, but it turns out there are a million or so ways of doing it, and none of them are currently accepted as terribly good.
Disquotational theories of truth.
Truth is axiomatized by Tarskian T-biconditionals, which are statements of the form:
T("P") is true iff P. If we allow P to range over sentences that may contain the truth predicate, the resulting theory becomes inconsistent due to the Liar paradox. Nice!
So, a disquotational approach must impose a restriction on the class of sentences available for P in the biconditionals. The obvious restriction is to just not allow the truth predicate in P at all. Problem is, the resulting logic TB has lost the ability to deduce anything about truth. It is a theorem that TB is proof-theoretically equivalent to the basic underlying logic without any truth statements.
Tarski provided the most widely known truth theory, the original hierarchical theory. His definition entails the T-biconditionals of the above form for the target language; however, the definition cannot be carried out within the target language. This is Tarski's theorem on the "undefinability of truth" in any one language. To incorporate this into a working theory of truth, he proposed a hierarchy of languages in which truth in one language could be defined in another language at a higher level.
Iterative compositional theories
As described in Feferman 1991 [J. Symb. Logic 56:1-49] "truth or falsity is grounded in atomic facts from the base language, i.e. can be determined from such facts by evaluation according to the rules of truth for the connectives and quantifiers, and where statements of the form T("A") are evaluated to be true (false) only when A itself has already been verified (falsified)." Such an approach is iterative in that a truth statement T("A") is true only if A is true, and it is compositional because a compositional sentence is determined only by its components according to the logical evaluation rules for connectives and quantifiers.
Feferman's Determinate truth theory
In this approach we limit the truth predicate, so there are two predicates T, and D its domain of significance. Feferman wants D to consist of just the meaningful and determinate sentences, ones which are true or false, in other words. Feferman further requires that D is "strongly compositional," so that a compound sentence is D iff all the substitution instances of its subformulae by meaningful terms belong to D.
The last two kinds of theories have the advantage of being "type-free," since they allow the proof of of the truth of sentences which contain the truth predicate. But, they have consistency problems remaining. Fujimoto's paper jumps off from here, and proposes a new improved truth theory which is also type-free, and then shows how to compare disparate truth theories using the new notion of "relative truth-definability."
It's heady stuff. I had only read Tarski's famous truth theory, and hadn't realized there were all these other proposals and complicated issues.
I think all this relates to linguistics, because let's face it, how can we have a theory of semantics without a theory of truth? How do people decide what's true? It's kind of mind-boggling.