Theory of Probability Semantics of Classical Propositional Logic and Its Application
Jia Zhang · Chinese Journal of Computers · 2014
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 1under each probability valuation.The concepts of probability truth degree,uncertainty degree,Λ-probability truth degree andΛ-uncertainty degree of formulas are also introduced in the paper,andΛ-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 thatΛ-uncertainty degree of conclusion is less than or equal to the sum of the product ofΛ-uncertainty degree of each premise and its essentialness degree in a formal inference.TheΛ-similarity degree andΛ-pseudometric between formulas are introduced by usingΛ-uncertainty degree of formulas,and it indicates that theΛ-pseudo-metric space has not isolated point and that the logic operations arecontinuous inΛ-pseudo-metric space.As an application,the proposals of two different approximate reasoning models in theΛ-pseudo-metric space are raised as well as an example to illustrate the practical application of these approximate reasoning models.