Quantitative Method Based on Cotangent Similarity Degree in Three-Valued Ł ukasiewicz Logic

The main purpose of this paper is to establish a type of quantitative model by using the contangent similarity function in the three-valued Ł ukasiewicz propositional logic system \textŁ_3. We introduce the concepts of the cotangent similarity degree, cotangent pseudo-distance and cotangent truth degree of the propositions, together with their basic properties in \textŁ_3. We investigate the relationship between the cotangent truth degree and contangent pseudo-distance, and prove the continuity of the logical connectives \neg, \vee and \rightarrow in the \textŁ_3 logical metric space. We propose a graded reduction method and three types of graded reasoning frameworks on the propositions set F(S), and provide several examples and basic properties of it.