Monotonic Modal Logics

Monotonic modal logics form a generalisation of normal modal logics in which the additivity of the diamond modality has been weakened to monotonicity: 3p∨3q → 3(p∨q). This generalisation means that Kripke structures no longer form an adequate semantics. Instead monotonic modal logics are interpreted over monotonic neighbourhood structures, that is, neighbourhood structures where the neighbourhood function is closed under supersets. As specific examples of monotonic modal logics we mention Game Logic, Coalition Logic and the Alternating-Time Temporal Logic. This thesis presents results on monotonic modal logics in a general framework. The topics covered include model constructions and truth invariance, definability and correspondence theory, the canonical model construction, algebraic duality (for monotonic neighbourhood frames), coalgebraic semantics, Craig interpolation via superamalgamation, and simulations of monotonic modal logics by bimodal normal ones. The main contributions are: generalisations of the Sahlqvist correspondence and canonicity theorems, a detailed account of algebraic duality via canonical extensions, an analogue of the Goldblatt-Thomason theorem on definable frame classes, results on the relationship between bisimulation and coalgebraic notions of structural equivalence, Craig interpolation results, and a simulation construction which preserves descriptiveness of general frames.