Strong Completeness of Coalgebraic Modal Logics

Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics often present subtle difficulties – up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong completeness. We apply this method to reconstruct canonical model theorems that are either known or folklore, and moreover instantiate our method to obtain new strong completeness results. In particular, we prove strong completeness of graded modal logic with finite multiplicities, and of the modal logic of exact probabilities. In modal logic, completeness proofs come in two flavours: weak completeness, i.e. derivability of all universally valid formulas, is often proved using finite model constructions, and strong completeness, which additionally allows for a possibly infinite set of assumptions. The latter entails recursive enumerability of the set of consequences of a recursively enumerable set of assumptions, and is usually established using (infinite) canonical models. The appeal of the first method is that it typically entails decidability. The second method yields a stronger result and has some advantages of its own. First, it applies in some cases where finite models fail to exist, which often means that the logic at hand is undecidable. In such cases, a completeness proof via canonical models will at least salvage recursive enumerability. Second, it allows for schematic axiomatisations, e.g. pertaining to the infinite evolution of a system or to observational equivalence, i.e. statements to the effect that certain states cannot be distinguished by any formula. In the realm of Kripke semantics, canonical models exist for a large variety of logics and are well understood, see e.g. [2]. But there is more to modal logic than Kripke semantics, and indeed the natural semantic structures used to interpret a large class of modal logics go beyond pure relations. This includes e.g. the selection function semantics of conditional logics [4], the semantics 1998 ACM Subject Classification: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic — modal logic; I.2.4 [Artificial Intelligence]: Knowledge Representation Formalisms and Methods — modal logic, representation languages.