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.

本文给出了有限宽度的传递逻辑的Fine完备性定理的推广，并证明了有限“ suc-eq-width”传递逻辑的Kripke完整性。在有根传递框架中，每个有限suc-eq-width公理的框架条件要求具有不同适当后继点的反链的基数的上限是有限的。本文还介绍了Rybakov完备性定理对有限宽度的传递逻辑的一般化，并证明了“ suc-eq-width”有限的传递逻辑的Kripke完整性。对于每个有限suc-eq-width公理的框架条件，在有根传递框架中，对于深度有限的下限且具有不同适当后继点的点的反链，要求基数的上限是有限的。我们将构造suc-eq-width有限但不具有宽度的传递逻辑的连续体，以及suc-eq-width有限但不具有宽度的传递逻辑的连续体。这表明我们的新完备性结果分别比Fine定理和Rybakov定理涵盖了更多的逻辑。