Axiomatizing hybrid logic using modal logic

Abstract

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. We also extend this definition to 'fairly nice frames', to capture their closure under disjoint union. 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.

Paraconsistency in hybrid logic

“…In [27], Blackburn and ten Cate approach several extensions of basic hybrid logic, and provide them with Hilbert axiomatizations. Also, Ian Hodkinson's work on axiomatizing hybrid logics using modal logics is quite interesting [62]. And finally, Torben Braüner worked on defining first-order hybrid logic [32], intuitionistic hybrid logic [33] and many-valued hybrid logic [60].…”

Sahlqvist-Type Completeness Theory for Hybrid Logic with Binder

“…Correspondence and completeness theory for hybrid logic Existing literature on correspondence and completeness theory for hybrid logic is abundant, see [1,4,6,10,11,12,13,14,18,20,22]. Gargov and Goranko proved that any extension of K H with pure axioms (formulas that contain nominals only but no propositional variables) is strongly complete.…”

Algorithmic Correspondence for Hybrid Logic with Binder

“…Correspondence Theory for Hybrid Logic In the literature, there are many existing works on the correspondence theory for hybrid logic [1,3,6,11,12,13,14,15,18,19]. In particular, ten Cate et al [19] showed that any hybrid logic obtained by adding modal Sahlqvist formulas to the basic hybrid logic H is strongly complete.…”