Having used the distinction between typed and type-free axiomatic theories of truth before, I shall now try to be more precise about what it means for a theory of truth to be typed. Typically, a system of truth will be classified as typed if it is based in some way on an object and metalanguage – or language level – distinction. Type-free theories of truth are often also called theories of self-referential or self-applicable truth.
The terminology typed and type-free comes from and has been originally applied to theories about sets, concepts, universals, and the like, of course.
Typing could be applied to theories of truth by imposing syntactic restrictions. The terms of the language would have to be classified by their types and the truth predicate restricted to terms of appropriate type. In particular, a truth predicate would only be applied to formulae containing variables if they only ranged over formulae not containing that truth predicate. Accordingly, variables of different types would have to be used. Such an approach seems incompatible with my approach here, as the language ℒ of the base language features only one sort of quantifier that ranges over natural numbers. One could achieve a restriction, however, by choosing a coding of formulae in the natural numbers that only codes sentences of the language ℒ without the truth predicate.