On the Multimodal Logic of Elementary Normative Systems

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, with standard Kripke semantics, for every modality 2i and normative system η, we expand the language adding a new modality 2ηi with the intended meaning of 2 η i φ 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 give a sound and complete axiomatisation for this logic, proving transfer results in the case that L and N are axiomatised by sets of Sahlqvist or shallow modal formulas.