Some Connections Between Elementary and Modal Logic

Abstract

Publisher Summary This chapter outlines that a common way of proving completeness in modal logic is to look at the canonical frame. It discusses that the method is applicable to any complete logic whose axioms express a ΣΔ-elementary condition or to any logic complete for a Δ-elementary class of frames. The chapter also proves two mild converses to this result. The first is that any finitely axiomatized logic has axioms expressing an elementary condition if it is complete for a certain class of natural subframes of the canonical frame. The second result is obtained from the first by dropping finitely axiomatized and weakening elementary to Δ-elementary. Classical logic is used in the formulation and proof of these results. The proofs are not hard, but they show that there may be a fruitful and non-superficial contact between modal and elementary logic. The chapter also outlines some basic notions and results of modal logic. For simplicity, this is taken to be mono-modal. However, the results can be readily extended to multi-modal logics and, in particular, to tense logic.