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

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 S(α)={φ1,φ2,…} satisfying φi∈{pi,■pi,(■pi2)(■(■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 all 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*.