Approximate Reasoning In Quantitative Logic And The Theory Of Conditional Truth Degree Of Formulas In Classical Logic

Symbolism and formalism are characteristic features of mathemat-ical logic,which is quite distinct from computation mathematics.The former em-phasises on formal reasoning and accurate proof while the latter concerns with nu-merical calculation and permits approximate solutions. The quantitative logic byWang brings numerical calculation into mathematical logic with the fundamentalidea as follows: In classical logic system L(also the Lukasiewicz fuzzy logic systemL_n and Luk,fuzzy logic systems L_n~* and L~*),the concept of a formula's truth de-gree which describes the extent to which a formula can be regarded as a tautologyis introduced;then the concepts such as similarity degree,pseudo-metric, divergencydegree,consistency degree are proposed;finally, an approximate reasoning theory inF(S) which contains three approximate reasoning models is brought in.There have been lots of articles about the quantitative logic,and there arestill lots of questions that need discussing about,such as the characterization ofρ(A, D(Γ))i.e.the distance from the formula A to the set D(Γ) of all logical con-clusions ofΓ,the inter-relations among the three approximate reasoning models,accumulation of errors in approximate reasoning and the characterization of the ex-tent to which a formula A can be semantically implied byΓand so on.This papermainly deals with the questions above.Chaper One:Basic knowledge.In this paper a brief account of the quantitativelogic,integral semantics theory and the concept of L inclusion degree is given.Thischapter is a basis of the following chapters.Chapter Two:The approximate reasoning in quantitative logic.This chapter con-sists of three parts. The first part concerns with the characterization ofρ(A, D(Γ)),the equivalent form of divergency degree and the three approximate reasoning mod-els' inter-relations. In the second part the problem of accumulation of errors inapproximate reasoning in classical logic system L is discussed about.At last,threeapproximate reasoning models in the integral semantics theory are proposed andtheir inter-relations are given out. Chapter Three:Conditional truth degree in classical propositional logic.Thereare two parts in this chapter.The first part proposes the concept of a formula'sconditional truth degree in classical logic system.In the second part,with respect tothe extent to which a formula A can be semantically implied byΓ,the concept ofa formula's conditional truth degree is generalized to a method of lattice-degreedsemantical implication by using the L inclusion degree theory.