What’s the Name of the Game? Dialogue Game Semantics for Intuitionistic Modal Logic with Strict Implication

Abstract

In this thesis we will develop dialogue game semantics for intuitionistic modal logic with strict implication. We begin by introducing Kripke se- mantics and proof systems for intuitionistic modal logic. Afterwards, we define a dialogue semantics for classical and intuitionistic propositional logic similar to the one found in [1] and a dialogue semantics for modal logic based on the work in [11]. We then discuss a suitable way to include strict implication in this framework. However, when adding strict impli- cation to the existing dialogue semantics for intuitionistic modal logic, the interdefinability of strict implication and the box operator is valid, even though it does not hold in intuitionistic modal logic. We suggest two ways of fixing this issue. First, we adapt the dialogue semantics for intu- itionistic modal logic to mirror the approach that is taken for the Kripke semantics. This involves adapting the dialogue semantics for classical modal logic into dialogue semantics for intuitionistic propositional logic and then creating a Bi-Modal dialogue semantics for intuitionistic modal logic. We then add strict implication to this system and show that the resulting semantics is sound and complete with respect to intuitionistic modal logic. The second approach that we explore is to restrict the way players may use previous information in our dialogue games. We then show that these two approaches are equivalent. Finally we discuss the benefits of both systems and discuss directions for further research.