系统L^＊中极大相容理论的结构刻画和紧致性定理

Structural Characterizaions of Maximal Consistent Theories over L~* and Compactness Theorem

为给模糊推理建立严格的逻辑基础,本文第二作者在1997年提出了一种新型的模糊命题演绎系统L*。本文基于系统L*的强完备性定理给出了极大相容理论的结构刻画,证明了每一个极大相容理论必然具有形式D({φ1,φ2,…}),这里φi∈{pi,■pi,(■p2i)&(■(■pi)2)}(i=1,2,…),p1,p2,…是系统L*中全体命题变元,进而给出了极大相容理论的若干刻画条件。本文还证明了系统L*的满足性定理和紧致性定理。至此,系统L*的基本定理包括完备性定理、强完备性定理、可判定性定理、满足性定理和紧致性定理已被我们所掌握,所以本文的结果完善了系统L*的理论体系。

To provide a logic foundation for fuzzy reasoning, the second author proposed in 1997 a new formal deductive system L^＊, Based on the strong completeness theorem of L^＊ , the present paper gives a characterization of maximal consistent theories over L^＊ . It is proved that each maximal consistent theory must be the deductive closure of some set with the form D（{φ1,φ2,…}）,satisfying φ1∈{pi,→pi,（→pi^2）＆（→（→pi）^2）} for all i = 1,2,…, where p1,p2, … are the propositional variables of L^＊. Several necessary and sufficient conditions for a consistent theory to be maximal are obtained. The Satisfiability Theorem and Compactness Theorem for L^＊, saying that a consistent theory has a model and a consistent theory has a model if and only if every finite subset has a model respectively, are also obtained. Hence al foundamental theorems for L^＊ including standard completeness, strong completenss theorem, decidable theorem, satisfiability theorem and compactness theorem are known to us, and in this sense the results of the present paper improve the theoretical system for L^＊.