Combining intermediate propositional logics with classical logic

In [17], we introduced a modal logic, called L, which combines intuitionistic propositional logic IPC and classical propositional logic CPC and is complete w.r.t. an algebraic semantics. However, L seems to be too weak for Kripke-style semantics. In this paper, we add positive and negative introspection and show that the resulting logic L5 has a Kripke semantics. For intermediate logics I , we consider the parametrized versions L5(I) of L5 where IPC is replaced by I . L5(I) can be seen as a classical modal logic for the reasoning about truth in I . From our results, we derive a simple method for determining algebraic and Kripke semantics for some specific intermediate logics. We discuss some examples which are of interest for Computer Science, namely the Logic of Here-and-There, Gödel-Dummett Logic and Jankov Logic. Our method provides new proofs of completeness theorems due to Hosoi, Dummett/Horn and Jankov, respectively.