# Lorenzo Rossi

- 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).