The extensions of the modal logic K5
The extensions of the modal logic K5
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模态逻辑K5的扩展
DOI：
10.2307/2273793
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发表时间：
1985
影响因子：
0.6
通讯作者：
S. K. Thomason
中科院分区：
文献类型：
--
作者：
Michael C. Nagle;S. K. Thomason
我们的目的是描绘命题模态逻辑K5的扩张（正规的和非正规的）。我们与每个扩展K5的逻辑关联一个有限性指数，以这种方式，逻辑的属性（例如，包含，正规性和表性）成为指数的有效可判定属性。此外，我们得到明确的有限公理化的所有扩展K5和抽象的表征格的这种扩展。本文对Ph.D.论文[2]的第一个命名的作者，谁愿意承认他的债务布赖恩F。Chellas在指导研究方面做出了相当大的努力，最终在[2]和[3]中达到了顶峰。我们也感谢W。J. Blok和Gregory Cherlin的观察大大简化了定理3和推论10的证明。所谓逻辑，我们指的是在命题变量的可数无限集合Var中的一组公式和联结词、→和□（其他联结词被用来表示），它包含了所有经典重言式，并且在分离和替换下是封闭的。一个逻辑是经典的，如果它在RE下也是封闭的（从A参与B推断□A参与B），如果它是经典的，并且包含□ n和□（P → q）→（□p → □q）。一个逻辑是准经典的，如果它包含一个经典逻辑;准正规的，如果它包含一个正规逻辑。因此，一个拟正规逻辑是正规的当且仅当它是经典的，并且当且仅当它在RN下是闭的（从A推断□A）。
Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K5. We associate with each logic extending K5 a finitary index, in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K5 and an abstract characterization of the lattice of such extensions. This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10. By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A↔B infer □A ↔□B) and normal if it is classical and contains □ ⊤ and □ (P → q) → (□p → □q). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □A).