Research For The Connections Of Several Familiar Multiple-valued Logic Systems' Tautology And The Derived Function From L₄ System

Multiple-valued logic theory was advanced independently by E.Post J. and Lukasiewicz in the 1920s. After a hundred years of development, it has been thriving as a subject. The famous multiple-valued logic systems are Lukasiewicz system,Standard sequence system,Godel system,Kleene system and Product system,W system etc. In order to explore the different relationships between logic systems, many scholars had paid a lot of effort.For example, basing on residual lattice theory, P.Hajek advanced the BL logical algebra which was matched with the BL logical system.Latter, in order to promote the BL system and formalize research triangle on the left-continuous mode of logic system, the MTL system was proposed by F. Esteva and others. And they provided a number of important MTL systems mode expansion,for example, IMTI,WNM,NM system etc. NM system and L* are the equivalent systems.Since Lukasiewicz, Godel, Product and Standard sequence logic system were advaced, there were a large number of scholars dedicated to the study of logic, and emerged in a large number of research results. Professor Wang guojun discussed systematically some three-valued logic theories of the Lukasiewicz system,Kleene system and Godel system, and made the theory of basing on the n-valued logic system tautology. In 1998, the fuzzy logic in generalized tautology theory was proposed. In a multiple-valued logic system, we generally research the formula of v(A)≥αand call itα-tautology.Multiple-valued logic also has the two studies of language structure and semantics. These two aspect's mutual connection is depending on the completeness theorem. So a research topic of many-valued logic is the function completeness. And the derived function is the basis for this issue. This article researches the tautology and the derived function of several multiple-valued logic systems.The following describes the structure of this document and the main content:Chapter one is introduction. The article mainly introduces the previous scholars' research and the achievement regarding the multiple-valued system theories, and summarizes the main research work and the conclusion in this article.Chapter two is the prerequisite knowledge and related concepts:I firstly introduced the concepts of Ln system, standard sequence logic system Sn, Gn,Kn systems and Wn system, and Lukasiewicz system,Godel system,Kleene system,W system which are based on the continuous value [0, 1].Then introduced the implication operator of these systems and the concepts of the tautology,α-tautology,α+-tautology and the definition of the logical equivalence. For the following research in chapter four, I also introduced the definitions of the formula set and derived function.Chapter three:I study respectively the tautology and generalized tautology of some common logical systems.And I have conducted the research and the summary of the tautology and generalized tautology's relationship between the identical system and the different systems.In these systems they have the conclusion:If they have the subalgebra relationship, their tautologies have the inclusion relation. But the inclusion relation is one-way. If not, the conclusion is not true. I have gived the counter-examples and explained concretely; Secondly, through appropriately constructing homomorphic mapping, multiple-valued logic system tautologies or generalized tautologies are the logical equivalent with the tautology of the classic logical system. If a formula is as long as not the tautology of C2 system,it is definitely not the generalized tautology of other multiple-valued logic systems, more is not a tautology. Especially, in the Kn system and Kleene system, I mainly have the conclusions:1/2-T(K2n+1)=1/2-T(K3)= T(C2);1/2-T(KL)=1/2-T(K3).;1/2-T(KL)=T(C2).In the W system,I have the conclusions: 1/2-T(W)=1/2-T(W3)=1/2-T(L3); (1/2)+-T(W)=T(W3)=T(L3) etcChapter four:On the basis of the derived method of the classic logic system's Boole function and combining with the previous scholar's research who has studied three-valued system's derived function, I continue studying the four-valued system's derived function. I have given the specific derived method, the proof of necessary and sufficient condition of the derived function. In the end I have given an example. That studying is a step forward for exploring the n-valued system. Finally, it is pointed out the deficiency that does not apply to other multiple-valued logic systems.