NOTES ONω-INCONSISTENT THEORIES OF TRUTH IN SECOND-ORDER LANGUAGES

Abstract

It is widely accepted that a theory of truth for arithmetic should be consistent, butω-consistency is less frequently required. This paper argues thatω-consistency is a highly desirable feature for such theories. The point has already been made for first-order languages, though the evidence is not entirely conclusive. We show that in the second-order case the consequence of adoptingω-inconsistent truth theories for arithmetic is unsatisfiability. In order to bring out this point, well knownω-inconsistent theories of truth are considered: the revision theory of nearly stable truthT#and the classical theory of symmetric truthFS. Briefly, we present some conceptual problems withω-inconsistent theories, and demonstrate some technical results that support our criticisms of such theories.