Characterizations of maximal consistent theories in the formal deductive system (NM-logic) and Cantor space

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Authors:
Hongjun Zhou
Title: Characterizations of maximal consistent theories in the formal deductive system (NM-logic) and Cantor space
Journal: Fuzzy Sets and Systems
Volume 158
Number
Pages: 2591-2604
Year: 2007





Abstract

A maximal consistent theory is a maximal theory with respect to its consistency. The present paper is divided into two parts. The first one is devoted to characterize the maximality of a consistent theory in the formal deductive system L* (which is a logic system equivalent to the nilpotent minimum logic). It is proved that each maximal consistent theory in this logic must be the deductive closure of a collection of simple compound formulas. Hence, it follows that there is a one-to-one correspondence between the set of all maximal consistent theories and the set of evaluations e assigning to each propositional variable p its truth degree e(p) ∈ {0, 1 2 , 1}. The Satisfiability Theorem and Compactness Theorem of L* are obtained. The second part is to investigate the topological structure of the set of all maximal consistent theories over L*, and the results show that this topological space is a Cantor space