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

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.