Playing with Modalities

Abstract

In this work, we will explore modalities through dialogical game lenses. Games provide a powerful tool for bridging the gap between intended and formal semantics, often offering a more conceptually natural approach to logic than traditional model-theoretic semantics. We begin by exploring substructural calculi from a game semantic perspective, driven by intuitions about resource-consciousness and, more specifically, cost-sensitive reasoning. The game comes into full swing as we introduce cost labels to assumptions and a corresponding budget. Different proofs of the same end-sequent are interpreted as strategies for a player to defend a claim, which vary in cost. This leads to a labelled calculus, which can be viewed as a fragment of subexponential linear logic. We conclude this first part with a discussion of cut-admissibility for the proposed system. In the second part, we show that our games offer an interesting insight also into modal logics. More precisely, we will focus on the modal logic PNL, characterised by Kripke frames with two types of disjoint and symmetric reachability relations. This framework is motivated by the study of group polarisation, where the opinions or beliefs of individuals within a group become more extreme or polarised after interaction. Our approach to reasoning about group polarisation is based on PNL and highlights a different aspect of formal reasoning about the corresponding models - using games and proof systems. We conclude by outlining potential directions for future research.