Abstract
We present an axiom system for basic hybrid logic extended with propositional quantifiers (a second-order extension of basic hybrid logic) and prove its (basic and pure) strong completeness with respect to general models.
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Notes
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There is more to general frames that this: they can also be viewed as representations of modal algebras; see Chap. 5 of [6] for details. Both lines of work stem from a classic paper by S. K. Thomason [21]. This links general frames and modal algebras, and shows that (a) there are Priorean tense logics that are not complete with respect to any class of frames, that (b) every Priorean tense logic is complete with respect to a class of general frames. That is: general frames “tame” frame validity.
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Note that on standard models we could drop the restriction prohibiting substitution of quantified formulas as standard models admit all subsets of the frame as propositions. That is, the rule which permits any soft substitution is sound on standard models. However this does not lead to a completeness result for standard models, as (thanks to Kit Fine’s results [11]) we know that the set of all standard validities on the class of all standard models is not recursively enumerable.
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Namely: “Tense logic and the logic of earlier and later”, and “Quasi-propositions and quasi-individuals”, both of which can be found in the first edition of Papers on Time and Tense [16]. The new edition [17] contains several more papers that build on these two, including “Egocentric logic”. See Kofod’s PhD thesis [14] for further discussion.
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Acknowledgements
We would like to thank Antje Rumberg and the three anonymous referees for their comments and corrections on earlier versions of this paper.
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Blackburn, P., Braüner, T., Kofod, J.L. (2023). An Axiom System for Basic Hybrid Logic with Propositional Quantifiers. In: Hansen, H.H., Scedrov, A., de Queiroz, R.J. (eds) Logic, Language, Information, and Computation. WoLLIC 2023. Lecture Notes in Computer Science, vol 13923. Springer, Cham. https://doi.org/10.1007/978-3-031-39784-4_8
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