Neighbourhood structures: bisimilarity and basic model theory

Neighbourhood structures are the standard semantic tool used to reason about non-normal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 22. We use this coalgebraic modelling to derive notions of equivalence between neighbourhood structures. 22-bisimilarity and behavioural equivalence are well known coalgebraic concepts, and they are distinct, since 22 does not preserve weak pullbacks. We introduce a third, intermediate notion whose witnessing relations we call precocongruences (based on pushouts). We give back-and-forth style characterisations for 22-bisimulations and precocongruences, we show that on a single coalgebra, precocongruences capture behavioural equivalence, and that between neighbourhood structures, precocongruences are a better approximation of behavioural equivalence than 22-bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with definability and image-finiteness. We prove a Hennessy-Milner theorem for modally saturated and for image-finite neighbourhood models. Our main results are an analogue of Van Benthem's characterisation theorem and a model-theoretic proof of Craig interpolation for classical modal logic.