�� Of Formulas In Propositional Logic Implication Tautology Of Truth Degrees And The Relative ��-theory

As is well known,symbolization and formalization are the essential characteristics 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 concepts in propositional logics,which was a bridge between artificial intelligence and computational mathematics.In quantitative logic,the concept of truth degree of formulas was given,moreover,the similarity degree between two formulas and pseudo-metric among formulas were proposed.Based on the above concepts,three patterns of approximate reasoning theory were established in propositional logics. There are a series of research results on quantitative logic.However,all these results are obtained based only on the system axioms and the inference rules,not considering some possible existing inference premises.This is naturally unable to measure the extend to which a formula is a conclusion of a theoryΓ.The main contribution of the thesis is to extend absolute research to relative research in the classical two-valued logic and four important multi-valued logics,which is based on inference premiseΓ.The structure of the thesis is arranged as follows:Chapter 1 Preliminaries.We mainly recall the basic knowledge in five commonly propositional logic systems that will be used later.Chapter 2 The theory ofΓ-implication truth degree of formulas in classical propositional logic.Firstly,theΓ-implication truth degree of formulas in classical propositional logic is proposed,and its various related properties are discussed in detail.At the same time,the result is that the implication truth degree of all finite theories is dense in[0,1].Secondly,we defineΓ-implication similarity degree and pseudo-metric,and get some basic properties of them.Thirdly,on basis ofΓ-implication truth degree,three patterns of approximate reasoning are investigated. Furthermore,the problem of errors accumulation in approximate reasoning is solved.Fourthly,there is a question whether we can get the same conclusion by adopting different approximate reasoning patterns.About this question,we give affirmative answer and prove the equivalence of them.Lastly,we simplify the proof of operators' continuity,combining probability logic andΓ-implication truth degree. Chapter 3 The theory of relativeΓ-tautology degree of formulas in four logics. Firstly,the concept of relativeΓ-tautology degree of formulas in L ukasiewicz, L~*,G(o|¨)del and product logics,is proposed,and its basic properties arc obtained. Secondly,theΓ-similarity degree between formulas is defined,then a pseudo-metric between formulas is introduced.Moreover,approximate reasoning patterns is proposed. Thirdly,the characterization ofρг(A,D(Σ)) and the equivalent form of divergence degree are given in multi-valued and continuous logics.Lastly,inner relations of three patterns of approximate reasoning are obtained.