Fuzzy Logic Formal System Completeness Theorem

Spurred by the success in applications, especially in fuzzycontrol, fuzzy logic aroused the interest of many famous scholars and a series of important results have been created. In 1992,P.Perny and B.Roy introduced a type ofa t-norm in ref.[12].In ref[10],J.C.Fodor called it nilpoten minimum and gave its R implication.Since 1997, many creative works have been done starting with them asfollows: firstly,a formal system L^* for a type of fuzzy propositional calculus,based onan instance of I_min0.φ which was called R_o-implicator by author, was cortstructed inref.[4]; secondly, the generalized tautological theory of its corresponding semanticalsystem,which was called the revised Kleen system by author,established in ref.[5]: after that , the full implicational triple I method of fuzzy reasoning, which aimsto lay a theoretic foundation for fuzzy controllers developing, was prepsnted in ref.[7]etc.Recently, D.W.Pei has proved the completeness theorem for L^* with respect toW-semantic [3]. In this paper,I will present my following works:(i)several properties and theorms of L^* are to be given;(ii) the completeness theorem for L^* with respect to W-semantic is to be provedby Henkin's technique by which the classical two-valued completeness had been proved;(iii)the formal system K_L^* for the fuzzy predicate calculus corresponding to L^* isto be constructed;(iv) the completeness fOr the SChematic extension C; of C* with respect to Wi--semanticis to be solved.(v) the completeness theorem for K2 with respect to W--interpretation was proved;...