一类二值谓词逻辑中公式的准真度理论

Theory of quasi-truth degrees of formulas in two-valued predicate logic

在二值谓词逻辑中引入了一阶语言的一类特殊解释,该类解释中的解释域取为非空有限集.在此基础上基于有限均匀分布概率测度空间的可数无穷乘积引入了逻辑公式的相对真度与准真度概念,证明了关于准真度而言MP规则与HS规则成立,并基于准真度对全体谓词公式之集进行了分类.准真度理论虽然并不与逻辑有效公式以及矛盾式概念完全吻合,但可证明存在一类公式,对该类公式而言,逻辑有效性等价于准真度为1,矛盾性等价于准真度为0.所以准真度为1或0分别是逻辑有效公式或矛盾式概念的一种推广.

In two-valued predicate logic a class of special interpretations of first language with nonempty finite domains is introduced. Then, based on countable infinite product of a family of evenly distributed finite probability measure space, the concept of the relative truth degree as well as quasi-truth degrees of well formed formulas is proposed, and it is proved that the MP rule and HS rule hold for the quasi-truth degrees of wffs. Moreover the set of all predicate wffs can be classified by means of quasi-truth degree. Though the theory of quasi-truth degree doesn't completely coincide with the concepts of logically efficient formula and contradiction, there is a kind of formulas for which the efficiency in logic is equivalent to the quasi-truth degree being 1, and the character of contradiction is equivalent to the quasi-truth degree being 0. Therefore the quasi-truth degree being 1 and 0 is respectively a generalization of the concepts of logically efficient formula and contradiction.