Glivenko Type Theorems for Intuitionistic Modal Logics

In this article we deal with Glivenko type theorems for intuitionistic modal logics over Prior’s MIPC. We examine the problems which appear in proving Glivenko type theorems when passing from the intuitionistic propositional logic Int to MIPC. As a result we obtain two different versions of Glivenko’s theorem for logics over MIPC. Since MIPC can be thought of as a one-variable fragment of the intuitionistic predicate logic Q-Int, one of the versions of Glivenko’s theorem for logics over MIPC is closely related to that for intermediate predicate logics obtained by Umezawa [27] and Gabbay [15]. Another one is rather surprising.