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In this paper we present a duality between nonmonotonic consequence relations and well-founded convex geometries. On one side of the duality we consider nonmonotonic consequence relations satisfying the axioms of an infinitary variant of System P, which is one of the most studied axiomatic systems for nonmonotonic reasoning, conditional logic and belief revision. On the other side of the duality we consider well-founded convex geometries, which are infinite convex geometries that generalize well-founded posets. Since there is a close correspondence between nonmonotonic consequence relations and path independent choice functions one can view our duality as an extension of an existing duality between path independent choice functions and convex geometries that has been developed independently by Koshevoy and by Johnson and Dean.

We discuss the use of relation lifting in the theory of setbased coalgebra. On the one hand we prove that the neighborhood functor does not extend to a relation lifting of which the associated notion of bisimilarity coincides with behavorial equivalence. On the other hand we argue that relation liftings may be of use for many other functors that do not preserve weak pullbacks, such as the monotone neighborhood functor. We prove that for any relation lifting L that is a lax extension extending the coalgebra functor T and preserving diagonal relations, L-bisimilarity captures behavioral equivalence. We also show that if T is finitary, it admits such an extension iff there is a separating set of finitary monotone predicate liftings for T .

In this paper we introduce a game semantics for System P, one of the most studied axiomatic systems for non-monotonic reasoning, conditional logic and belief revision. We prove soundness and completeness of the game semantics with respect to the rules of System P, and show that an inference is valid with respect to the game semantics if and only if it is valid with respect to the standard order semantics of System P. Combining these two results leads to a new completeness proof for System P with respect to its order semantics. Our approach allows us to construct for every inference either a concrete proof of the inference from the rules in System P or a countermodel in the order semantics. Our results rely on the notion of a witnessing set for an inference, whose existence is a concise, necessary and sufficient condition for validity of an inferences in System P. We also introduce an infinitary variant of System P and use the game semantics to show its completeness for the restricted class of well-founded orders.

Game logic was introduced by Rohit Parikh in the 1980s as a generalisation of propositional dynamic logic (PDL) for reasoning about outcomes that players can force in determined 2-player games. Semantically, the generalisation from programs to games is mirrored by moving from Kripke models to monotone neighbourhood models. Parikh proposed a natural PDL-style Hilbert system which was easily proved to be sound, but its completeness has thus far remained an open problem.In this paper, we introduce a cut-free sequent calculus for game logic, and two cut-free sequent calculi that manipulate annotated formulas, one for game logic and one for the monotone µ-calculus, the variant of the polymodal µ-calculus where the semantics is given by monotone neighbourhood models instead of Kripke structures. We show these systems are sound and complete, and that completeness of Parikh's axiomatization follows. Our approach builds on recent ideas and results by Afshari & Leigh (LICS 2017) in that we obtain completeness via a sequence of proof transformations between the systems. A crucial ingredient is a validity-preserving translation from game logic to the monotone µ-calculus.