Contradiction Degree Of Formulas In N-valued S-MTL System And The Average Truth Degree Of Logical Theory

Since the concept of the tautologies and contradictions is put forward, many experts and scholars made a thorough research about it, and obtained a series of theoretical achievements. So for most formulas which are neither tautologies nor contradictions, how to evaluate their degree of authenticity? From now existing literature, experts determine a formula’s authenticity by analyzing whether the formula is close to a tautology. Considering its dual aspects, why not estimate a formula’s authenticity by considering whether it is close to a contradiction, and we can introduce the basic methods of quantitative logic into this theory, we will establish a uniform theory of integral conflict degree of formulas by applying their ideas and the methods in logic system which is based on strong left continuous t-norm (short for S-MTL), then the existing research results about integral conflict degree of formulas can be incorporated into the broader system. So that we can study the divergence degree and consistency degree in another way.In quantitative logic, besides the quantitative research on single formula, scholars also made a lot of researches on the properties of the theory. They researched the properties of consistency degree and divergence degree of theories, by which to distinguish the extent of compatibility among different theories, and hence distinguish the extent of goodness and badness of different theories. Besides that, the truth degree of formula is introduced into the theory, through treating the infimum value of truth degree of formula in all logical conclusions as truth degree of logical theory. This method lost the information provided by these logical conclusions with larger truth degree values in the theory F; and when the theory have only one formula, in multiple valued logic, the truth degree of logic theory F is not equal to the truth degree of formula B. So in order to completely extend the concept of truth degree of formula into logical theory F, Concepts of the average truth degree and deviation of a finite theory are introduced into n-valued Lukasiewicz logical system in this paper, some important property of them are discussed. It shows that we can give an overall and comprehensive evaluation to the reliability of a given theory by combining with the average truth degree and deviation of a theory.The following is the main outcomes:1. Based on the general probability measure, the unified theory of contradiction degree of formulas is proposed in n-valued MTL propositional logic, some important properties of contradiction degree are discussed, and the integral representation of contradiction degree of formulas is given.2. The difference degree p*between formulas is defined by using the contradiction degree of formula, which is proved to be a pseudo-metric, and thus a pseudo-metric space (F (S), p*) in F (S) can be established.3. Concepts of the average truth degree and deviation of a finite theory are introduced into n-valued Lukasiewicz system, some important properties of them are discussed. It shows that we can give an overall and comprehensive evaluation to the reliability of a given theory by combining with the average truth degree and deviation of a theory.