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We define a family of intuitionistic non-normal modal logics; they can beeseen as intuitionistic counterparts of classical ones. We first considermonomodal logics, which contain only one between Necessity and Possibility. Wethen consider the more important case of bimodal logics, which contain bothmodal operators. In this case we define several interactions between Necessityand Possibility of increasing strength, although weaker than duality. For alllogics we provide both a Hilbert axiomatisation and a cut-free sequentcalculus, on its basis we also prove their decidability. We then give asemantic characterisation of our logics in terms of neighbourhood models. Oursemantic framework captures modularly not only our systems but also alreadyknown intuitionistic non-normal modal logics such as Constructive K (CK) andthe propositional fragment of Wijesekera's Constructive Concurrent DynamicLogic.

Intuitionistic Non-Normal Modal Logics: A general framework