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SCIENTIA SINICA Informationis, Volume 42, Issue 5: 648-662(2012) https://doi.org/10.1360/112011-850

Axiomatic theory of truth degree for a class of first-order formulas and its application

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  • AcceptedDec 23, 2011
  • PublishedMay 14, 2012

Abstract

Let φ be the set consisting of all closed first-order formulas containing no function symbols. Based on the finite model and uniformly distributed probability theory, the present paper analyses a classic example in non-monotone logic, and proposes the truth degrees of conjunctions of universal closures of literals. Then the present paper establishes an axiomatic theory of truth degree on φ and proves that truth degrees of formulas in φ are computable. Moreover, this paper proves that the set H of truth degrees of formulas in φ coincides with the set of truth degrees of propositional formulas, and especially, truth degrees of universal closures of literals are equal to 1/2. Lastly, the present paper introduces the concepts of similarity degree and pseudo-metric between formulas of φ, and proposes the theory of consistency degree for logic theories. As an application, the consistency degree of a kind of Horn type data base is calculated.

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