Logics with the universal modality and admissible consecutions

In this paper we study admissible consecutions (inference rules) in multi-modal logics with the universal modality. We consider extensions of multi-modal logic S4n augmented with the universal modality. Admissible consecutions form the largest class of rules, under which a logic (as a set of theorems) is closed. We propose an approach based on the context effective finite model property. Theorem 7, the main result of the paper, gives sufficient conditions for decidability of admissible consecutions in our logics. This theorem also provides an explicit algorithm for recognizing such consecutions. Some applications to particular logics with the universal modality are given.