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

  • Authors:

  • Hongjun Zhou;Guojun Wang

  • Affiliations:

  • Institute of Mathematics, Shaanxi Normal University, Xi'an 710062, China;Institute of Mathematics, Shaanxi Normal University, Xi'an 710062, China and Research Center for Science, Xi'an Jiaotong University, Xi'an 710049, China

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2007

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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)@?{0,12,1}. 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.