Bl ~ * System An Expansion Of The System And The System Of Propositional Calculus Formula Contradictions Theory

Fuzzy logic.as an important branch of non-classical logic .is broadly used in manyfields such as artificical intelligence,information science and computer science. Zadeh distiguished fuzzy logic in broad and narrow sense. In narrow sense,fuzzy logic is a logicsystem that takes fuzzy language as truth value,and it is an extension of classical logic.It seems that he only has interest in the use of these logic systems in artificical intelligenceand expert systems etc..And he does not care about whether it can be axiomatic.However,the mathematical logic experts care about the questions of logic system's axiomaticness,completeness, etc.. In order to solve the axiomaticness of fuzzy logic underZadeh's viewpoint,Hajek x M.Bazz had done massive work.Professor Wang degrees logic concepts in semantics and gives the truth degree offormulars in two-valued logic and multi-valued logic systems. Then the measurementlogic is founded . This provides the possibility for the study of two-valued logic andmulti-valued logic systems from numerical calculations.This is the theoretic bases of this thesis.The main points and primary coverage ofthis thesis is as follows:In the first chapter ,for convenience of the research,the definitions of BC* system,MTLsystem,Bi?o-algebra, MTL-algebra etc.,as well as their related propositions are given.The second chapter consists of four parts.In the first part , we add a new unaryconnective□to the BL* system . And the following four axioms are added to the axiomset of BL* system: (Dl) D(A -* B) -> (D.4 -> UB), (D2) UA -+ UUA, (D3) UA -+ A,(□4) D(.4 VB)-> UA V UB. These four axioms conform to the meaning of " very "in natural languge,thus we obtain an axiomization fuzzy logic system—BC* systemunder Zadeh's viewpoint. In the sencond part ,we add a new unary connective□and fiveaxioms which are (UBRol) Dl = 1, {UBRq2) Ua < a, {UBR03) U(a V b) < Ua V Ub,{UBR04) U(a -? b) < Ua -> Ub , (UBRob) Ua = UUa into the Bi?0-algebra.Then weobtain algebra structure of BC* system—QB.Ro-algebra and discuss some properties ofDSi?o-algebra and its subalgebra .In order to discuss the completeness of BC* system ,thethird part gives□filter and prime□filter in the D5i?o-algebra . The fourth part provesthe BC* system's Weak Completeness Theorem and Strong Completeness Theorem basedon□filter and prime D filter in the D5i?0-algebra .Finally,we prove the BC* system'sStandard Completeness Theorem using method of construction.The third chapter consists of two parts. In the first part, the concept of contradictiondegree of formulars is given in two-valued logic system L,then the difference degree betweenformulars is introduced based on the contradiction degree in two-valued logic system L. Weprove that the difference degree is pseudo metric.So we found logic metric space (F(S),p'L) in two-valued logic system L. Finally,we prove that pseudo metric pL is continuous about-i.—\ V. A. In the second part .the concept of contradiction degree of formulars and integralexpression of contradiction degree are given in n-valued Lukasiewicz logic system. Thenthe difference degree between logic formulars is introduced based on the contradictiondegrees. We also prove that the difference degree is pseudo metric. So we obtain logicmetric space (F(S),p-r ) in n-valued Lukasiewicz logic system.