Which classical correspondence is valid in intuitionistic modal logic?

Abstract

Modal logics reason about properties of relational structures, and such properties are often characterized by axioms of modal logics. This connection between properties of relational structures and axioms of modal logics are called correspondence, and has been investigated well in the classical setting. The problem we consider is an intuitionistic version of this correspondence. In particular, this paper considers which part of the correspondence results known in classical setting is true for intuitionistic one. We first define the notion of robustness of axioms so that an axiom is robust if and only if its corresponding properties in classical and intuitionistic semantics are the same. Next we give a syntactically defined class of axioms, and prove that all axioms in this class are robust. This result is an analogue of the classical result by Sahlqvist, and its proof is partly based on a known proof of his theorem.