Sahlqvist Via Translation

Abstract

In recent years, unified correspondence has been developed as a generalized Sahlqvist theory which applies uniformly to all signatures of normal and regular (distributive) lattice expansions. A fundamental tool for attaining this level of generality and uniformity is a principled way, based on order theory, to define the Sahlqvist and inductive formulas and inequalities in every such signature. This definition covers in particular all (bi-)intuitionistic modal logics. The theory of these logics has been intensively studied over the past seventy years in connection with classical polyadic modal logics, using versions of Gödel-McKinsey- Tarski translations, suitably defined in each signature, as main tools. In view of this state-of-the-art, it is natural to ask (1) whether a general perspective on Gödel-McKinsey- Tarski translations can be attained, also based on order-theoretic principles like those underlying the general definition of Sahlqvist and inductive formulas and inequalities, which accounts for the known Gödel-McKinsey-Tarski translations and applies uniformly to all signatures of normal (distributive) lattice expansions; (2) whether this general perspective can be used to transfer correspondence and canonicity theorems for Sahlqvist and inductive formulas and inequalities in all signatures described above under Gödel-McKinsey-Tarski translations. In the present paper, we set out to answer these questions. We answer (1) in the armative; as to (2), we prove the transfer of the correspondence theorem for inductive inequalities of arbitrary signatures of normal distributive lattice expansions. We also prove the transfer of canonicity for inductive inequalities, but only restricted to arbitrary normal modal expansions of bi-intuitionistic logic. We also analyze the difficulties involved in obtaining the transfer of canonicity outside this setting, and indicate a route to extend the transfer of canonicity to all signatures of normal distributive lattice expansions.