经典命题逻辑的概率语义及其应用  

Theory of Probability Semantics of Classical Propositional Logic and Its Application

作　　者：张家录[1] 陈雪刚[2] 赵晓东[3] 

机构地区：[1]湘南学院数学系,湖南郴州423000 [2]湘南学院计算机科学系,湖南郴州423000 [3]四川民族学院数学系,四川康定626000

出　　处：《计算机学报》2014年第8期1775-1785,共11页Chinese Journal of Computers

基　　金：湖南省重点建设学科;湖南省科技计划项目(2013FJ3032);湖南省教育厅科学研究重点项目(2014A135);湖南省社会科学基金项目(13YBA302)资助~~

摘　　要：文中将经典命题逻辑的赋值域由二值({0,1})推广到概率空间,引进了命题公式的概率赋值并建立命题逻辑的概率语义,证明了一个命题公式为重言式当且仅当其在每个概率赋值下的值都等于1.引入了命题公式的概率真度、不确定度、Λ-概率真度、Λ-不确定度等概念,并说明了Λ-概率真度是已有的二值命题逻辑各种真度概念的推广,通过讨论Λ-概率真度的性质,表明Λ-概率真度在全体公式集F(S)上满足Kolmogorov公理.证明在形式推演的一个有效推理中,结论的Λ-不确定度不超过各前提的Λ-不确定度与其必要度的乘积之和.利用公式的Λ-不确定度引进公式间的Λ-相似度和Λ-伪距离,证明了在一定条件下所建立的Λ-伪距离空间没有孤立点且通常的逻辑运算关于Λ-伪距离是连续的.在Λ-伪距离空间中,提出了F(S)上的两种不同近似推理模式,并通过实际应用例子说明所提出的近似推理模式是有效的.The concept of probability valuation was introduced in this paper by extending the value domain { 0,1} of classical propositional logic to a probability space,and also the probability semantics of propositional logic are established.The paper tries to prove that a formula is tautology if and only if its value equals 1 under each probability valuation.The concepts of probability truth degree,uncertainty degree,∧-probability truth degree and ∧-uncertainty degree of formulas are also introduced in the paper,and ∧-probability truth degree be served as generalization of all truth degrees in literature.The conclusion that the probability truth degree satisfies Kolmogorov axioms reached by discussing some of their properties.The paper proves that ∧-uncertainty degree of conclusion is less than or equal to the sum of the product of ∧-uncertainty degree of each premise and its essentialness degree in a formal inference.The ∧-similarity degree and ∧ pseudo metric between formulas are introduced by using ∧-uncertainty degree of formulas,and it indi cates that the ∧-pseudo metric space has not isolated point and that the logic operations are continuous in ∧-pseudo metric space.As an application,the proposals of two different approximate reasoning models in the ∧-pseudo metric space are raised as well as an example to illustrate the practical application of these approximate reasoning models.

关 键 词：概率语义 ∧-概率真度 ∧-不确定度 ∧-伪距离 概率逻辑 近似推理 

分 类 号：O142[理学—数学]