I sort axiomatic theories of truth into two large families, namely into typed and type-free theories of truth. Roughly speaking, typed theories prohibit a truth predicate's application to sentences with occurrences of that predicate, while type-free theories do not. I will not consider syntactically restricted theories, that is, theories in which the truth predicate cannot be combined with any term to form a sentence, but typed theories either impose no restriction on the truth of sentences with the truth predicate or they prove that all sentences with the truth predicate are not true. At any rate, in typed theories one cannot prove the truth of any sentence containing T. Type-free theories of truth are often also described as theories of self-applicable truth.
Making this distinction precise is not entirely straightforward, however, and I will postpone the discussion of the distinction until Chapter 10, as only then will I have some examples of the theories to hand.
Of course, axiomatic theories of truth can be classified in other ways as well. For instance, one can distinguish between compositional and non-compositional theories, or between disquotational and non-disquotational theories. By and large, I find it easier to treat typed theories together in one part as typed theories have more in common technically, than, for instance, compositional theories do. Similarly all disquotational theories, that is, theories based merely on disquotation sentences as their axioms, may initially look much alike, but as I will show, their formal properties vary wildly.