A certain type of inference rules in modal logics, generalizing Gabbay's Irreflexivity rule, is introduced and some general completeness results about modal logics axiomatized with such rules are proved.

ABSTRACT We consider the question of the complete axiomatization of a relative modal logic with composition and intersection.

Over the last twenty years, in all of these neighbouring fields, modal systems have been developed that we call multi-dimensional. (Our definition of multi ...

ABSTRACT We consider the question of the complete axiomatization of a relative modal logic with composition and intersection.

This paper is devoted to the completeness issue of RMLCI — the relative modal logic with composition and intersection— a restriction of the propositional dynamic logic with intersection. The trouble with RMLCI is that the operation of intersection is not modally definable. Using the notion of mosaics, we give a new proof of a theorem considered in a previous paper “Complete axiomatization of a relative modal logic with composition and intersection”. The theorem asserts that the proof theory of RMLCI is (...) complete for the standard Kripke semantics of RMLCI. (shrink)