Łukasiewicz 命题逻辑中命题的 Choquet 积分真度理论

摘要

The notion of Choquet integral truth degrees of propositions in Łukasiewicz propositional logic is introduced ,by means of the Choquet integral of McNaughton functions with respect to uncertainty measures on the set of all valuations .When the involved uncertainty measures satisfy finite additivity property ,the notion of Choquet integral truth degrees can induce in a natural way a pseudo-metric ,with which the set of all propositions becomes a pseudo metric space and thus several graded reasoning meth-ods can be established .The notion of Choquet integral truth degrees will reduce to the existing notion of Borel probability truth de-grees in probabilistically quantitative logic when the uncertainty measures are Borel probability measures .This paper is a continuation of probabilistically quantitative logic and provides a possible framework for reasoning about non-linear uncertainty of propositions .%将已有的不确定性测度概念引入到了悲ukasiewicz命题逻辑中的全体赋值之集上，然后利用McNaughton函数关于该不确定性测度的Choquet积分定义了命题的Choquet积分真度概念。证明了当赋值空间上的不确定性测度满足有限可加性时Choquet积分真度函数就具有良好性质，由此可诱导出命题集上的一个伪距离，进而可建立逻辑度量空间并展开程度化推理，特别是证明了当赋值空间上的不确定性测度取为Borel概率测度时Choquet积分真度函数就退化为概率计量逻辑中的Borel概率真度函数。本文是已有命题逻辑概率计量化工作的继续与深入，为表示逻辑命题间不确定性的非线性关系提供了一种推理框架。