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

Studia Logica 109 (6):1177-1200 (2021)
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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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Ming Xu
Wuhan University

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References found in this work

Modal Logic.Alexander Chagrov - 1997 - Oxford, England: Oxford University Press.
An incomplete logic containing S.Kit Fine - 1974 - Theoria 40 (1):23-29.
Logics containing k4. part I.Kit Fine - 1974 - Journal of Symbolic Logic 39 (1):31-42.
Logics containing k4. part II.Kit Fine - 1985 - Journal of Symbolic Logic 50 (3):619-651.

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