Throughout this book Peano arithmetic has been used as the base theory. Like many other philosophers, I see the theory of truth for the language of arithmetic as the starting point for developing a theory of truth for other, usually more comprehensive languages as base languages and perhaps eventually for natural languages.

When applying the axiomatic theories of truth discussed in this book to base theories other than Peano arithmetic, one is confronted with at least two kinds of problems: first one needs to settle on a sort of truth theory – choosing between one based on the disquotation sentences, or the compositional axioms, a typed or untyped theory, and so on – such that the chosen sort of theory is both suitable for the base theory in question and consonant with its underlying philosophical motivation; and second, once a kind of truth theory has been chosen, the formulation of this kind of truth theory with the new base theory may not be straightforward: there may be different ways of applying the chosen axiomatic conception of truth to a base theory; moreover, as I will show, some ways have unwanted consequences and may even lead to inconsistencies.

Both kinds of problems are beyond the scope of this book. But in this chapter I will show how some of the formal results about axiomatic theories of truth obtained in this book or closely related results can shed at least some light on the problems.