Semantical study of intuitionistic modal logics

We investigate several intuitionistic modal logics (IMLs), mainly from semantical viewpoint. In these decades, in the area of type theory for programming language, various type systems corresponding (via Curry-Howard correspondence) to IMLs have been considered. However, IMLs arising from such a context are not extensively studied before. We try to understand the logical meaning of such IMLs, and what kind of structure are behind IMLs motivated from computation. The technical materials appearing in the thesis are divided into three parts: (1) proof systems and Kripke semantics for an intuitionistic version of linear-time temporal logic (LTL), which corresponds to a lambda-calculus for binding-time analysis; (2) correspondence theory in a certain Kripke semantics for IML; and (3) a new representation of existing Kripke semantics for IML by using neighborhood semantics (also called Scott-Montague semantics), and investigations of the relationship between them. Through these technical developments, the following observation arises: there are actually two possible meanings of the assertion “a formula A is true at a possible world x.” This explains the difference between the traditional modal logic and IMLs considered in this thesis, and several difficulties that emerge in establishing meta-theoretical results such as completeness and cut-elimination. In the thesis we also show that the existing Kripke model can be transformed into an equivalent neighborhood model, and each neighborhood model satisfying a certain condition can be transformed into an equivalent Kripke model. This mutual translation clearly explains how “non-standard” behavior of modalities (captured well by neighborhood semantics) has been simulated in the existing Kripke semantics.