The axioms for truth will be added to what is called the base theory. In the main part of this book I will use Peano arithmetic as the base theory, but applications to other more comprehensive base theories are intended, and the base theory may contain empirical or mathematical or still other axioms together with the appropriate vocabulary. At any rate, a base theory must contain at least a theory about the objects to which truth can be ascribed.

Truth theories have been proposed where the need for objects to which truth can be ascribed and for a theory of these objects seems to be avoided. If truth were analysed in terms of special quantifiers as in the so-called prosentential theory of truth by Grover et al. (1975), for instance, it might initially appear that such objects are avoided, but it is not at all clear that the new quantifiers avoid any ontological commitment.

I have no ambition to avoid ontological commitment to objects that can be true. If the axiomatic theories of truth I am going to discuss are intertranslatable with an approach without such ontological commitment, so be it. If such a translation is not possible, then I suspect that something is wrong with the approach. Here I will stick to the usual approach that takes truth to be a predicate.

In almost all cases, the axioms for truth can only serve their purpose when combined with a suitable base theory. If the truth axioms, on which the theories in this book are based, are separated from the base theory, the result is a very weak theory.