Mtl-algebra On The Degree Of Quantitative Research

The logic systems based on t norms have been dominated in fuzzy logic ever since along time ago. Taking t norms as the interpretation of conjunction, these logic systemsmirror many logic characteristics of human reasoning. As a result, they are widely used infuzzy reasoning and artificial intelligence. Among them, the logic system MT L based onleft-continuous t norms catches people's eyes. Many famous logics, such as Lukasiewiczsystem, Go¨del system, product system and L system are the extensions of MT L. Thealgebras of them are all the subclasses of MTL algebra. Scholars have focused on algebrastructures of MTL algebra and its subclasses, but have not payed more attentions on themeasure study of them.The quantitative logic combines mathematical logic with probability calculation, pro-vides a graded approach to basic concepts, builds logic metric spaces and gives approximatereasoning methods. Based on this way of research, we carry on measure study over MTLalgebra in this thesis. The main work is as follows:1. Quantitative algebra is proposed and several important standard MTL algebras turnout to be quantitative algebras. We build quantitative space over MTL algebra and provethe continuity of operators in the space. Meanwhile, we build quantitative logic theoreticalframe for MT L logic.2. The properties of truth degree of theory are discussed and the conditional truthdegree of theory in proposition logic is proposed on the ideal of conditional probability.With the conditional truth degree of theory, a new approximate reasoning mode betweenformula sets is proposed, which can be used to search the best reasoning conclusion andpremise. The harmonious degree of theory is proposed for characterizing the extent of theinternal harmony of theories.3. The logic properties of the system Lnare studied. Satisfiability theorem, compact-ness theorem and decidability theorem for the system Lnare proved, which enriches thetheory of the system. Moreover, necessary and sufcient conditions for ΣΓ-truth degree andconditional truth degree existing are given by the satisfiability theorem.