The Quantitative Study In First-Order Logic System

Study on quantitative logic is an important research direction of grading in logical system. In accordance with different subjects of study, theory of quantitative logic can be classified into theory of quantitative propositional logic and theory of quantitative first-order logic. Up to now, there are a large number of research results about the theory of quantitative propositional logic, however only a minimal amount of researches are about quantitative first-order logic.In this paper, we first investigate the validity degrees of first-order logical formulae in different ways, local and overall, and then propose the corresponding measure of validity degrees of first-order formulae. Furthermore, based on the corresponding validity degrees of first-order formulae, the similarity degrees and pseudo-metric between first-order formulae are studied. Moreover, the study on approximate reasoning in the corresponding logical metric space is developed. The contents of the dissertation include the following aspects:Part 1, based on the definition of the relative validity degrees of first-order formulae on an interpretation with a finite domain in the form of measure, the one in the form of equation is introduced, which simplifies the calculation and proof of the problem about the relative validity degrees of formulae. And the definition of the relative validity degrees of non-closed formulae in the form of equation is simplified even further. Moreover, it is proved that the relative validity degree of the disjunction of two formulae on an interpretation with a finite domain is actually the product of the relative validity degrees of the two formulae. Finally, it shows the change of the quasi-truth degrees of first-order formulae after the Gen rule is applied in the process of logical reasoning, and it is proved that the quasi-truth degrees of a big class of special first-order formulae are all equal to 1/2.Part 2, based on the quasi-truth degrees of first-order formulae, the quasi-similarity degree between first-order formulae is proposed, a kind of pseudo-metric ρ between first-order formulae is introduced, and furthermore the logical metric space (F.ρ) is established. Moreover, it is proved that the similarity relation based on the quasi-similarity degree between first-order formulae is a equivalence relation, and the relation between the quasi-truth degrees of formulae and the pseudo-metric between first-order formulae is studied. Furthermore, it is proved that the logical operators "(?)", "(?)", "∨" and "∧" are all continuous, and there doesn’t exist any isolated point in the logical metric space (F.ρ). Finally, based on the pseudo-metric between first-order formulae, we propose three different types of approximate reasoning patterns, and it is proved that they are equivalent under certain conditions.Part 3,all the finite interpretations of the first-order language are stratified according to the cardinalities of their domains, and the average value of all the relative validity degrees of a first-order formula on the interpretations with the cardinality n of their domains is defined as the n-validity degrees of the first-order formula. Then the definition of the validity degree vector of a first-order formula is introduced. Moreover, it is proved that, for an arbitrary positive integer n,the n-validity degrees of all the atomic formulae are equal to 1/2, and the symmetry theorem about the n-validity degree and the validity degree vector of a first-order formula is put forward. Furthermore, the n-validity degree of a first-order formula is unchanged when the individual constant in the formula is substituted with a new variable symbol.Part 4, when the expression ability of a first-order formula is degraded into the one of a propositional formula, all the n-validity degrees of the first-order formula are equal to the validity degree of the propositional formula, so that the theory of the n-validity degrees of first-order formulae reaches harmony with the theory of validity degrees in classical propositional logic.Part 5, based on the n-validity degrees of first-order formulae, the n-similarity degree is proposed, and furthermore another kind of pseudo-metric between first-order formulae is studied. Moreover, the corresponding n-logical metric space (F,ρn) is established, and furthermore it is proved that there doesn’t exist any isolated point there. Finally, we propose three different types of approximate reasoning pattern and prove their equivalence to each other under certain condition.