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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A preliminary version of this paper was presented at the fifth conference on Advances in Modal Logic (AiML 2004) in Manchester. We would like to thank Maarten Marx for his comments on an early draft and Agnieszka Kisielewska for help with the proof reading.
Special Issue Ways of Worlds II. On Possible Worlds and Related Notions Edited by Vincent F. Hendricks and Stig Andur Pedersen
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Blackburn, P., Cate, B.t. Pure Extensions, Proof Rules, and Hybrid Axiomatics. Stud Logica 84, 277–322 (2006). https://doi.org/10.1007/s11225-006-9009-6
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DOI: https://doi.org/10.1007/s11225-006-9009-6