Theory Of Truth Degrees Of Formulas In Lukasiewicz Propositional Logic And Logic Metric Spaces

In recent years, the applications of fuzzy control has obtained a great success. But fuzzy reasoning, the core of fuzzy control is suspected because of lacking firm mathematics foundation. In order to solve this problem, a new research field has been built up and more scholars devoted themselves to this field. Applications of fuzzy logic range from the field of computer science, including automated theorem proving or consistency proofs or the theory of particularly algebraic structures(e.g. MV-algebras, BL- algebras and R₀-algebras).The theoretic foundation of fuzzy control is fuzzy reasoning. The variety of implication operators is one of the most important mathematical tools for fuzzy reasoning. It is well known that the choice of implication is very important to the fuzzy logical systems, for example the Lukasiewicz system use the Lukasiewicz implication, Godel system use the Godel implication and L^* system use R₀ implication etc. The great character of this implication is that they are adjoined with some definition of triangular norms. We call them regular implication. Much of the research has been done on the implications' applications. However, among the reaserch only a little is related to semantic properties. The present paper tries to find some semantic properties of the logic systems corresponding to regular implication from two different aspects. The arrangement of this paper is as follows:Chapter One: Basic knowledge. In this chapter, the concept of the set of all formed formulas is enlarged, and the corresponding concept of valuational region is also redefined. Moreover, some important properties of the regular implications and integral semantic are introduced. All above is the basis of the later chapters.Chapter Two: The concept of truth degrees of formulas in Lukasiewicz n-valued propositional logic is proposed. A limit theorem is obtained, which says that the truth function induced by truth degrees converges to the integrated truth function when n converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Lukasiewicz logic and the continuous valued Lukasiewicz logic.Chapter Three: Towards to the class of logic syetems which could construct logic pseudo-metric spaces, we do some research on their semantic properties wholely. First of all, we prove that all of the formed formulas are measurable on the valuational region. Therefore, in these logics, we can define a pseudo-metric between every two formulas conformably using the graded idea naturally. Then we discuss only part situation of thedistribution of isolated points in the spaces. At last, we obtain the fact that in logic metric space all of the logic operators are continuous and the regulations of integral reasoning hold, which are important conclusions. So, it is made possible to construct an unified framework of approximate reasoning in these logic systems.