Propositional Logic Systems Theory The Concept Of Truth Degrees Of Its Application

As is well known,symbolization and formalization are the essential characteris-tics of mathematical logic,which is quite distinct from computational mathematics. The former lays stress on formal deduction and rigorous argument,while the later concerns with numerical computation and permits approximate solving.Professor Wang Guojun established the theory of quantitative logic by grading the basic con-cepts in propositional logics,which was a bridge between artificial intelligence and computational mathematics.In quantitative Iogic,the concepts of truth degree of for-mulas was given,moreover,the similarity degree between two formulas and pseudo-metric among formulas were proposed.From then on, there are a series of research results in quantitative logic. However,all these results are obtained based only on truth degree of formulas,not considering truth degree among formulas.Based on this, The concept of truth degrees of logic theories is proposed firstly in the present pa-per,and this concept is generalized from individual to collective. The structure of the thesis and main contents are arranged as following:Chapter 1 Preliminaries.We mainly recall the basic knowledge commonly propo-sitional logic systems that will be used later.Chapter 2 Truth degree of a logic theory in two-valued propositional logic sys-tem L. The concept of truth degrees of logic theories is proposed firstly in the present paper, and this concept is generalized from individual to collective. The relation be-tween truth degree and divergency degree ofΓis proved:div(Γ)=1-T(Γ),and the concept of divergence degree can be simplified thereby.Moreover, the relationη(Γ)=1/2(1+T(Γ)) between truth degree and consistency degree of a given logic theoryΓis obtained whenΓis consistent.Finally, theoriesΓ1 andΓ2 are divided into six categories, in which the relation of truth degrees as well as divergency degree and consistency degree of logic theoriesΓ1,Γ2 andΓ1∪Γ2 is compared,respectively.Chapter 3 Truth degree of a logic theory in propositional logic system Cn. The relation between truth degree and divergency degree ofΓis given:div(Γ)= 1-T(Γ),which is established in propositional logic systems Ln,∏n and Ln*, and the concept of divergence degree can be simplified thereby. The relation of truth degrees as well as divergency degrees and consistency degrees of logic theoriesΓ1,Γ2 andΓ1∪Γ2 is compared,respectively. Chapter 4 Truth degree of a logic theory in propositional logic systems L* and Godel. The relation between truth degree and divergency degree ofΓis given: div(Γ)=1-T(Γ),the concept of divergence degree can be simplified.Chapter 5 Conditional truth degree of a logic theory in two-valued propositional logic system L.Firstly,based on the concept of truth degree of a logic theory and conditional probability,the concept of conditional truth degree of a theoryΓbased onΣis proposed,Σ-truth degree of a theoryΓfor short.Secondly,the relation betweenΣ-truth degree and E-divergency degree ofΓis given:div(Γ|Σ)=1-T(Γ| E).Finally,theoriesΓ1 and F2 are divided into six categories,in which the relation ofΣ-truth degrees,as well asΣ-consistency degrees and E-divergency degrees of logic theoriesΓ1;Γ2 andΓ1∪Γ2 is compared, respectively.