Non-normal modal logics : neighbourhood semantics and their calculi

Dans cette thèse, nous étudions la théorie de la preuve des logiques modales non-normales. Ces logiques ont été considérées dès les prémisses de la logique modale moderne, et ont été appliquées à de nombreux domaines, tels que le raisonnement déontique, le raisonnement épistémique, ainsi que le raisonnement dans les systèmes multi-agents. Nous définissons des systèmes de preuve pour des logiques modales non-normales ayant de bonnes propriétés calculatoires et sémantiques. Ces systèmes permettent de donner des procédures de décision et d'extraire des contre-modèles pour les formules non-valides qui sont en même temps adaptés à l'implémentation dans des démonstrateurs. Ils peuvent également être utilisés pour l'automatisation du raisonnement déontique et épistémique This thesis provides a proof-theoretical investigation of non-normal modal logics. Non-normal modal logics are modal logics that do not satisfy some of the modal principles of the weakest normal modal logic K. They have been studied since the very beginning of modern modal logic, and have found an interest in many areas, such as deontic, epistemic, and multi-agent reasoning. Moreover, they have been also studied as modal extensions of intuitionistic, rather than classical, logic. Non-normal modal logics have been mainly investigated from the point of view of the semantics. In contrast, their proof theory is not equally developed. In this thesis we pursue two general aims: concerning non-normal modal logic with a classical base, we aim to define proof systems for them with “good” computational and semantic properties, such as providing decision procedures, allowing for countermodel extraction, and being suited for theorem-proving. Concerning non-normal modal logic with an intuitionistic base, we aim to lay down a general framework for defining intuitionistic counterparts of classical non-normal modal logics, that should also capture relevant systems already studied in the literature. The thesis consists of three main parts. The first part contains an introduction to the semantics and the proof systems of both classical and intuitionistic non-normal modal logics. The second part presents our original results about classical non-normal modal logics: we introduce a new semantics and propose two new kinds of sequent calculi. In the third part we define a family of intuitionistic non-normal modal logics and provide both sequent calculi and a semantic characterisation