Properties of independently axiomatizable bimodal logics

Abstract

In monomodal logic there are a fair number of high-powered results on completeness covering large classes of modal systems; witness for example Fine [74], [85] and Sahlqvist [75]. Monomodal logic is therefore a well-understood subject in contrast to polymodal logic, where even the most elementary questions concerning completeness, decidability, etc. have been left unanswered. Given that in many applications of modal logic one modality is not sufficient, the lack of general results is acutely felt by the “users” of modal logics, contrary to logicians who might entertain the view that a deep understanding of one modality alone provides enough insight to be able to generalize the results to logics with several modalities. Although this view has its justification, the main results we are going to prove are certainly not of this type, for they require a fundamentally new technique. The results obtained are called transfer theorems in Fine and Schurz [91] and are of the following type. Let L ∌ ⊥ be an independently axiomatizable bimodal logic and L _⎕ and L _∎ its monomodal fragments. Then L has a property P iff L _⎕ and L _∎ have P. Properties which will be discussed are completeness, the finite model property, compactness, persistence, interpolation and Halldén-completeness. In our discussion we will prove transfer theorems for the simplest case when there are just two modal operators, but it will be clear that the proof works in the general case as well.