Transitive Logics of Finite Width with Respect to Proper-Successor-Equivalence

Abstract

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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Acknowledgements

I would like to give thanks to the two anonymous referees provided by this journal for their comments and suggestions, which are of great help to improve this paper.

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Correspondence to Ming Xu.

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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

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Keywords

  • 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