Abstract: This paper gives the vector representation of n-ary formula in classical propositional logic system based on random sequence and random mapping, defines the D-randomized truth degree of formulas and D-randomized similarity degree and D-randomized pseudo-metric among formulas based on the vector representation of formula, and explains that the definitions of truth degree and D-randomized pseudo-metric are equivalent to the original probability definition. Then, this paper proves that the D-randomized truth degree of n-ary formula is not more than 2²ⁿ cases. Finally, some simple properties of pseudo-metric among formulas are obtained based on the equivalent representation of D-randomized similarity degree and D-randomized pseudo-metric among formulas.

Key words:  classical propositional logic system, D-randomized truth degree, D-randomized similarity degree, D-randomized pseudo-metric

摘要： 给出了经典命题逻辑系统中n元命题公式基于随机数列和随机映射的向量表示形式，利用命题公式的基于随机数列的向量表示形式给出公式的D-随机真度、公式间的D-随机相似度和D-随机伪距离的等价表示形式。说明了一个具体的n元经典命题公式的D-随机真度最多只有 2²ⁿ种情况。利用命题公式间的D-随机相似度和D-随机伪距离的等价表示形式，给出了关于命题公式的D-随机真度、命题公式间的D-随机相似度和D-随机伪距离的一些性质的新的证明。

关键词: 经典命题逻辑系统, D-随机真度, D-随机相似度, D-随机伪距离