On the Multimodal Logic of Normative Systems

Abstract

We introduce Multimodal Logics of Normative Systems as a contribution to the development of a general logical framework for reasoning about normative systems over logics for Multi-Agent Systems. Given a multimodal logic L, for every modality $\Box_{i}$ and normative system η, we expand the language adding a new modality $\Box_{i}^{\eta}$ with the intended meaning of $\Box_{i}^{\eta}\phi$ being “φ is obligatory in the context of the normative system η over the logic L”. In this expanded language we define the Multimodal Logic of Normative Systems over L, for any given set of normative systems N, and we give a sound and complete axiomatisation for this logic, proving transfer and model checking results. The special case when L and N are axiomatised by sets of Sahlqvist or shallow modal formulas is studied.