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

摘要

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 ￡~* (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 ￡~* are obtained. The second part is to investigate the topological structure of the set of all maximal consistent theories over ￡~*, and the results show that this topological space is a Cantor space.