Axiomatizing hybrid logic using modal logic

We study hybrid logics with nominals and ‘actuality’ operators @i. We recall the method of ten Cate, Marx, and Viana to simulate hybrid logic using modalities and ‘nice’ frames, and we show that the hybrid logic of a class of frames is the modal logic of the class of its corresponding nice frames. Using these results, we show how to axiomatize the hybrid logic of any elementary class of frames. Then we study quasimodal logics, which are hybrid logics axiomatized by modal axioms together with basic hybrid axioms common to any hybrid logic, using only orthodox inference rules. We show that the hybrid logic of any elementary modally definable class of frames, or of any elementary class of frames closed under disjoint unions, bounded morphic images, ultraproducts and generated subframes, is quasimodal. We also show that the hybrid analogues of modal logics studied by McKinsey–Lemmon and Hughes are quasimodal.