Isomorphisms Of T-norms And Implication Operators And Theory Of Generalized Tautologies

In 1997, based on RQ implication operator professor Wang Guojun proposed revised Kleene system. Again in 1998, professor Wang proposed the concept of generalized tautology and discussed the classes of generalized tautologies deeply in revised Kleene system. So the theory of generalized tautologies was built, which gave a new direction in fuzzy logic research.An implication operator is residuated to a special left-continuous t-norm, and R0 algebra proposed by professor Wang can be seen as the algebra which is built by the special left-continuous t-norm and implication operator.In 1998, the theorem that characterizes all continuous t-norms was introduced after Hajek gave some properties of continuous t-norms in detail. But the study on left-continuous t-norms was few. It was well known that RQ t-norm which RQ implication operator residuated to was left-continuous. In fact, any left-continuous t-norm has its own residuum-implication operator. And many-valued system could be obtained from implication operator.Left-continuous t-norm was connected with implication operator and logic system could be built by implication operator. The aim of this paper is to give new logic systems by constructing implication operators which are different from RQ operator. The generalized tautology was discussed in those systems.In the first part, as preparatory knowledge, the paper gives the concept of t-norrn and the representation theorem of continuous t-norms.In the second part, a class of left-continuous isomorphic to R0 t-norm are given. The concept of isomorphism among implication operators is introduced and it is proved that two implication operators are isomorphic if and only if the two t-norms residuated to them respectively are isomorphic. Moreover, modifications of classes of a-tautologies under isomorphisms are investigated.In the third and fourth part, a new class of t-norms and corresponding residuated implication operators Ha with respected to a parameter a in [0,1] are introduced, and the many-valued systems Ha were defined.It is proved that Ha turns to be R0 implication operator in case a is 1, and turns to be Godel implication operator in case a is 0. So R0 operator and Godel operator are united in the systems Ha The negation-a with respect to parameter a is defined in Ha, The many-valued system H1/2=([0,1]-1/2,1/2) is discussed in detail. The classification theorem of tautologies in F(S) is obtained in H1/2. The classfication of tautologies is defined on HQ. Because of the traits of H0, theclassfication of tautologies of HO is enlarged properly.Finaly, the operators P,P,P with respect to a parameter p in [0,1] is introduced. The system Ip and relations of classification of generalized tautologies among the systems Ip and its 3-valued system and the system C2 is investigated. It is proved that generalized tautologies are decidable in the system Ip.