This paper presents a generalization of Fine’s completeness theorem for transitive logics of finite width, and proves the Kripke completeness of transitive logics of finite “suc-eq-width”. The frame condition for each finite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points with different proper successors. The paper also presents a generalization of Rybakov’s completeness theorem for transitive logics of prefinite width, and proves the Kripke completeness of transitive logics of prefinite “suc-eq-width”. The frame condition for each prefinite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points that have a finite lower bound of depth and have different proper successors. We will construct continuums of transitive logics of finite suc-eq-width but not of finite width, and continuums of those of prefinite suc-eq-width but not of prefinite width. This shows that our new completeness results cover uncountably many more logics than Fine’s theorem and Rybakov’s theorem respectively.
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Chagrov, A., and M. Zakharyaschev, Modal logic, vol. 35 of Oxford Logic Guides, Oxford University Press, Oxford, 1997.
Fine, K., An incomplete logic containing S4, Theoria 40(1):23–29, 1974.
Fine, K., Logics containing K4, Part I, The Journal of Symbolic Logic 39(1):31–42, 1974.
Fine, K., Logics containing K4, Part II, The Journal of Symbolic Logic 50(3):619–651, 1985.
Kracht, M., Tools and techniques in modal logic, vol. 142 of Studies in Logic and the Foundations of Mathematics, Elsevier Science B. V., Amsterdam, Lausanne and New York, 1999.
Rybakov, V. V., Completeness of modal logics with prefinite width, Mathematical notes of the Academy of Sciences of the USSR 32(2):591–593, 1982.
Segerberg, K., An essay in classical modal logic, Philosophical Studies published by the Philosophical Society and the Department of Philosophy, University of Uppsala, Uppsala, 1971.
Zhang, Y., and M. Xu, Some results concerning finite axiomatizability of transitive logics of finite depth, Manuscript, Department of Philosophy, Renmin University; Department of Philosophy, Wuhan University, 2018.
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Presented by Yde Venema
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Xu, M. Transitive Logics of Finite Width with Respect to Proper-Successor-Equivalence. Stud Logica (2021). https://doi.org/10.1007/s11225-021-09943-4
- Modal logic
- Kripke completeness
- Transitive logics of finite width
- Transitive logics of finite suc-eq-width
- Transitive logics of prefinite width
- transitive logics of prefinite suc-eq-width