Abstract: Based on vector representation of formula of classical propositional logic system, this paper gives a vector representation of formula by using assignment and assignment order of formula in three-valued Lukasiewicz propositional logic system. Firstly, this paper defines the truth degree of formula and gives the formula to calculate the similarity degree, pseudo-metric among formulas. Secondly, this paper obtains some simple properties of truth degree and similarity degree pseudo-metric among formulas. Thirdly, this paper proves that unary logical connection, binary operation logical connections→,∨,∧are continuous in pseudo-metric space (F(S),ρ). The related conclusions of this paper lay a solid foundation for further research of multi-valued Lukasiewicz propositional logic system.

Key words: three-valued Lukasiewicz propositional logic system, truth degree, similarity degree, pseudo-metric

摘要： 以经典命题逻辑中公式的向量表示形式为基础，利用三值Lukasiewicz逻辑系统中公式的赋值及赋值顺序给出了公式的向量表示形式。利用向量表示形式给出了公式真度的定义和公式间的相似度、伪距离的计算公式；得到了公式的真度和公式间相似度、伪距离的一些简单性质；证明了在伪距离空间(F(S),ρ)中，一元运算逻辑连接词和二元运算逻辑连接词→、∨、∧均关于ρ是连续的。相关结论为进一步研究多值Lukasiewicz逻辑系统中近似推理打下了夯实的基础。

关键词: 三值Lukasiewicz逻辑系统, 真度, 相似度, 伪距离