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Pure Extensions, Proof Rules, and Hybrid Axiomatics

Patrick Blackburn 1 Balder ten Cate 2
1 LANGUE ET DIALOGUE - Human-machine dialogue with a significant language component
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity). We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to a wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic.
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Submitted on : Tuesday, December 12, 2006 - 1:24:43 PM
Last modification on : Friday, February 4, 2022 - 3:31:10 AM

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Patrick Blackburn, Balder ten Cate. Pure Extensions, Proof Rules, and Hybrid Axiomatics. Studia Logica, Springer Verlag (Germany), 2006, 84 (2), pp.277-322. ⟨10.1007/s11225-006-9009-6⟩. ⟨inria-00119855⟩



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