关于严格蕴含系统的布尔值模型（续）

数学季刊 ›› 2004, Vol. 19 ›› Issue (4): 346-349.

关于严格蕴含系统的布尔值模型（续）

1.Department of Philosophy, Nankai University, Tianjin 300071, China; 2. College of Mathematics and Information Science, Henan University, Kaifeng 475001, China

收稿日期:2003-06-27 出版日期:2004-12-30 发布日期:2024-02-29
作者简介:LlNa(1958-),female,native of Kaifeng,Henan,a professor of Nankai University,M.S.D., engages in morden logic.
基金资助:
SupportedbytheNationSocialScienceFoundationofChina(04BZX047) SupportedbyTianjinCitySocialScienceFoundationofChina(TJ03-ZX009)；

On a Boolean-valued Model of the Strict Implication System(Continuous)

1.Department of Philosophy, Nankai University, Tianjin 300071, China; 2. College of Mathematics and Information Science, Henan University, Kaifeng 475001, China

Received:2003-06-27 Online:2004-12-30 Published:2024-02-29
About author:LlNa(1958-),female,native of Kaifeng,Henan,a professor of Nankai University,M.S.D., engages in morden logic.
Supported by:
SupportedbytheNationSocialScienceFoundationofChina(04BZX047) SupportedbyTianjinCitySocialScienceFoundationofChina(TJ03-ZX009)；

摘要/Abstract

摘要： The reference [4] proved the consistency of S₁and S₂ among Lewis’ five strict implication systems in the modal logic by using the method of the Boolean-valued model. But, in this method, the consistency of S₃, S₄ and S₅ in Lewis’ five strict implication systems is not decided. This paper makes use of the properties: (1) the equivalence of the modal systems S₃ and P₃, S₄and P₄; (2) the modal systems P₃ and P4 all contained the modal axiom T(□p → p); (3) the modal axiom T is correspondence to the reflexive property in V^B. Hence, the paper proves: (a) ‖As₃1‖ = 1; (b) ‖AS₄1‖ = 1; (c) ‖AS₅l‖ = 1 in the model <V^B, R, ‖ ‖>(where B is a complete Boolean algebra, R is reflexive property in VB). Therefore, the paper finally proves that the Boolean-valued model V^Bof the ZFC axiom system in set theory is also a Boolean-valued model of Lewis’ the strict implication system S³, S⁴ and S^5.

关键词: the , strict implication , system;Boolean-valued model;Boolean value

Abstract: The reference [4] proved the consistency of S₁and S₂ among Lewis’ five strict implication systems in the modal logic by using the method of the Boolean-valued model. But, in this method, the consistency of S₃, S₄ and S₅ in Lewis’ five strict implication systems is not decided. This paper makes use of the properties: (1) the equivalence of the modal systems S₃ and P₃, S₄and P₄; (2) the modal systems P₃ and P4 all contained the modal axiom T(□p → p); (3) the modal axiom T is correspondence to the reflexive property in V^B. Hence, the paper proves: (a) ‖As₃1‖ = 1; (b) ‖AS₄1‖ = 1; (c) ‖AS₅l‖ = 1 in the model <V^B, R, ‖ ‖>(where B is a complete Boolean algebra, R is reflexive property in VB). Therefore, the paper finally proves that the Boolean-valued model V^Bof the ZFC axiom system in set theory is also a Boolean-valued model of Lewis’ the strict implication system S³, S⁴ and S^5.

Key words: the , strict implication , system;Boolean-valued model;Boolean value

中图分类号:

O153.2

李娜, 刘华珂. 关于严格蕴含系统的布尔值模型（续）[J]. 数学季刊, 2004, 19(4): 346-349.

LI Na, LIU Hua-ke. On a Boolean-valued Model of the Strict Implication System(Continuous)[J]. Chinese Quarterly Journal of Mathematics, 2004, 19(4): 346-349.

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