Unified theory of integral true degrees in NM theory

NM理论中积分真度的统一理论

Mathematics and Computer Science College, Yan'an University, Yan'an 716000, China

Received: 30 June 2022

Abstract

Based on the NM propositional logic system of nilpotent minimum logic, the concept of the integral truth degree of the formula is firstly proposed by integrating the function induced by the formula, and the MP rule and HS rule of the integral truth degree are proved by means of the integral invariance. Secondly, in NM propositional logic the integral similarity and integral pseudo-distance are introduced into the set of general formulas of the system, and some good properties about the similarity and pseudo-distance are proved. Finally, in terms of the concept of divergence degrees and diameter, a new membership function for reflecting the consistency degrees of theories in NM propositional fuzzy logic is proposed, which is proved that of inconsistent theories are equal to 0 and that of completely consistent theories are equal to 1.

摘要

基于在幂零极小逻辑NM(nilpotent minimun)命题逻辑系统中, 通过将公式诱导的函数进行积分的方法提出了公式的积分真度概念, 利用积分不变性证明了积分真度MP规则、HS规则; 在NM命题逻辑系统的全体公式之集上引入了积分相似度和积分伪距离, 并且证明了关于相似度和伪距离的一些性质; 通过发散度以及直径的概念, 提出了NM命题模糊逻辑中反映理论相容程度新的隶属函数, 利用隶属函数给出了相容度的概念, 证明了不相容理论的相容度为0, 完全相容理论的相容度为1。