Modal logics with functional alternative relations (Q1094406)

The main purpose of this paper is to investigate the lattice of extensions of the normal modal logic \(KD_ c\), where the schema \(D_ c\) is \(\diamond A\supset \square A\), the converse of the deontic schema D. D! is the system \(KDD_ c\). The author begins with an axiomatic characterization of the normal logics between \(KD_ c\) and KD!, which proves to be the sequence of logics \(KD_ c\square^ n\diamond \top\). He then proves that every normal extension of \(KD_ c\) has the finite model property, from which there emerges a picture of the lattice of normal extensions of KD! as an inverted cone of denumerably many isomorphic slices converging downwards to \(KD_ c\) at its apex. Finally, he investigates the quasi-normal extensions of \(KD_ c\), i.e. the extensions not closed under necessitation, and proves that there are non- denumerably many of these. By contrast, all extensions of KD! are normal.