Absolute Truth Degree And Radomize Theory In N-valued Proposition Logic Systems

The characteristic of mathematical logic lies in symbolization and formalization, which is quite different from computational mathematics. The former emphasises on formal reasoning and strict proof, while the later concerns with numberical computation and approximate solutions. Computational mathematics is more flexible than mathematical logic. It can build the connection between mathematical logic and computational mathematics by bringing the numberical calculation into mathematical logic.The mean of "designated truth values" in mathematical logic had reflected the idea of graded method. Also, graded method is the reflection of ideology. Professor Wang Guojun established the quantitative logic based on basic logic concepts. Consequently, numberical calculation was introduced in the mathematics logic. It makes the mathematic logic more flexible and useable.In Wang's paper[9], the truth degree was introduced.At the same time, similarity and pseudo-distance were also defined based on the concept of truth degree which provides a possible framework for approximate.The main work of this paper includes:Chapter One: The study of algebra framework in fuzzy logic systems is an important task. And it had become an embranchment in the research of non-classical logic systems. In this chapter, some preliminaries are recalled and the characteristics of t-norm and residuated lattices are investigated. And also put out the characteristic of t-norm and residuated lattices.Chapter Two: In [10], by means of infinite product measure of uniformly distributed probability spaces, the author made mention of the truth degree theory, which was based on measure theory. Later, the truth degree theory was extended to the n-valued Lukasiewicz logic system by using this method. But, all the above works run in the uniformly distributed probability spaces. This definition of truth degree was built on infinite product measure, therefore, the calculation of truth degree is very difficult and can not be realized by algorithm. This chapter will avoid the concept of infinite product measure, and give a new definition of truth degree-absolute truth degree. It's a reformative and laconic definition which makes it possible to compute truth degrees of formulas by computer .Chapter Three: In classical two-valued propositional logic system, truth degrees of formulas were defined by measure theory. Also the similarity and pseudo-distance were from the viewpoint of equal probability. The possibility that each atomic propositionholds is uniform and equals to (1/2). The randomization characteristic of probability was not reflected under this method. In this chapter, firstly, based on the method of probability measure, the concept of probability truth degrees and introduced its general expression in unevenly probability space is given. Secondly, the similarity degree between two formulas on a residuated lattice is defined in terms of L-fuzzy similarity. And the character properties are discussed in four types of n—valued logic systems based on the left-continuous t-norms. Lastly, a pseudo-distance was introduced by means of probability truth degree, which provides another possible framework for approximate reasoning in n-valued propositional logics. It can provide a kind of ingerence frame for the approximate reasoning theory.