Monotonic Modal Logics
Helle Hvid Hansen
Abstract:
Monotonic modal logics form a generalisation of normal modal logics in
which the additivity of the diamond modality has been weakened to
monotonicity: <>p \/ <>q --> <>(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.
Keywords: Non-normal modal logic, neighbourhood semantics,
definability, correspondence theory, algebraic duality, coalgebra,
Craig interpolation, simulation.