MV代数的度量化研究及其在Lukasiewicz命题逻辑中的应用

Metrization on MV-Algebras and Its Application in Lukasiewicz Propositional Logic

设M是一个MV代数,Ω是从M到标准MV代数[0,1]_(MV)的全体同态之集,μ是Ω上的概率测度.基于μ在M中引入了元素(称之为元素命题)的真度概念以及元素命题间的相似度概念,并由此在M上建立了度量结构,从而在更广泛的框架下建立了度量理论.本文结果是已有的命题逻辑中逻辑公式的真度理论的一般化和代数化,思想也可应用到其他多值逻辑代数中.

Let M be an MV-algebra, Ω the set of all homomorphisms from M into the standard MV-algebra [0, 1] MY, and μ a probability measure on Ω. By means of the μ we introduce the concepts of truth degrees of elements of M （called element propositions） and similarity degrees between element propositions, and then define therefrom a metric on M. Thus we establish the metric theory in much wider framework. The results of the paper are generalization and algebraic counterpart of the existing theory of truth degrees of fomulas in propositional logics. The idea of the paper can be adapted to other many-valued logical algebras.