Many-valued Logic System H_α And Partial Implication Algebra

In 1997, based on R₀ implication operator professor Wang Guojun proposed the revised Kleene system. Again in 1998, professor Wang proposed the concept of generalized tautology and discussed the classes of generalized tautologies, which gave a new research direction in fuzzy logic. Subsequently, professor Wu Hongbo generalized the concept of generalized tautologies and proposed generalized semantic MP rules and generalized semantic HS rules in revised Kleene system, so we had more clear understanding about this system.Among the three kind of logic operations in many-valued logic system: negtion(→), union(V) and implication(→). It is very important to choose the implication operator because different implication operator forms different many-valued logic system. The residuated operators of left-continuous t-norms have better properties among many implication operators. It was well known that R₀ t-norm which R₀ implication operatior residuated to was left-continuous. Based on R₀ t-norm, Lan Rong proposed a new class of t-norms *_αand corresponding resituted implication operatiors H_αwith a parameterαin [0,1] and obtained many-valued logic system H_α.Many-valued logic system H_α(0≤α≤1) generalized revised Kleene system. Arbitray one system of H_αis ismorphic with another, so we mainly study the logic system H_1/2. In the logic system H_1/2, a new partion of F(S) can be obtained by utilizing the concept of generalized tautologies, moreover, a variety of generalized semantic MP and generalized semantic HS rules are obtained in H_1/2. After that, the correspoding resluts in H_1/2 are generalized to the systems H_α(0＜α＜1). At last, based on the implication operators of systems H_α, Triple I method and reversibility properties were discussed.Effect algebras were introduced as a mathematical model of quantum logic by American mathematicians Foulis and Bennett in 1994, which generalized orthomodular lattices. Many researchers, in the fields of mathematics and physics, were attracted to study them. In the third chapter the relation of effect algebra and many fuzzy logic algebra systems are discussed. Through the study of effect algebra, the partial implication algebra was proposed from the point of implication. It is proved that partial implication algebra and effect algebra are equivalent. Then we talk about the relation of partial implication algebra and MV-algebra.