NOTES ON ω -INCONSISTENT THEORIES OF TRUTH IN SECOND-ORDER LANGUAGES
Eduardo Alejandro Barrio, Lavinia Picollo · The Review of Symbolic Logic · 2013
It is widely accepted that a theory of truth for arithmetic should be consistent, but ω -consistency is less frequently required. This paper argues that ω -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 adopting ω -inconsistent truth theories for arithmetic is unsatisfiability. In order to bring out this point, well known ω -inconsistent theories of truth are considered: the revision theory of nearly stable truth T # and the classical theory of symmetric truth FS . Briefly, we present some conceptual problems with ω -inconsistent theories, and demonstrate some technical results that support our criticisms of such theories.