6 - Comparing axiomatic theories of truth

Summary

The main technical results of this book compare axiomatic theories of truth but also compare such theories of truth with other theories like the base theory or, in some cases, second-order theories of arithmetic. These results establish that certain axiomatic theories of truth are reducible to certain other theories of truth. Philosophers and logicians have defined and discussed many different notions of reducibility, and I will employ different notions here as well.

Which notion of reducibility is appropriate depends on the purpose of the comparison and one's philosophical stance on truth. For instance, an instrumentalist about truth might want to compare truth theories on the basis of their truth-free consequences alone; the theory of truth itself is seen merely as a means to an end. However, if one is investigating whether the paradoxes are adequately resolved in certain theories of truth, then one cannot focus exclusively on truth-free consequences: one will need to compare what the different theories of truth prove about the liar sentence, for instance. To compare the conceptual strength of truth theories, one might not be so concerned about their behaviour with respect to the paradoxes, but one must still take into account the truth-theoretic consequences of the theories; one might compare the theories by examining whether one theory can define the truth predicate of the other theory.

In the final part of the book I will return to the philosophical significance of various reducibility results and the notions of reducibility employed in them and look at applications such as ontological reductions, but here I first introduce various technical notions.