- May 2017
A unified approach to truth and implication
The truth predicate is commonly thought to be symmetric, at least in the sense that for every sentence A, A and “A is true” should be inter-substitutable salva veritate. In this paper, we study an object-linguistic predicate for implication obeying similar symmetry requirements. While several non-classical logics are compatible with symmetric truth, theories of symmetric implication can only be formulated in a small class of substructural logics. We present an axiomatic theory of symmetric implication and truth over Peano Arithmetic, called SyIT, formulated in a non-reflexive logic, and we study its semantics and proof-theory. First, we show that SyIT axiomatizes a class of fixed-point models that generalize Saul Kripke’s (1975) fixed points for symmetric truth. Second, we compare SyIT with the theory PKF_ _(Halbach and Horsten 2006), an axiomatization of Kripke’s fixed points for truth in strong Kleene logic, and we show that SyIT and PKF are proof-theoretically equivalent. The latter result shows that going substructural and adding a symmetric implication predicate to a theory of symmetric truth comes at no proof-theoretical costs.
(Joint work with Carlo Nicolai, LMU Munich).