Approximate Reasoning Theory In N-valued S-MTL Propositional Logic System And Reversibility Of Triple I Method

Monoidal T-norms based Logic, MTL for short, is a logic system based on left-continuous t-norms. It is also a logic system based on regular implication operators. The left-continuous t-norms is the semantic correspondence of the syntactic strong conjunction operators. The regular implication operators correspond to syntactic implication in MTL. As a class of fuzzy logic, it possess many fine properties such as monotonicity, exchange property and adjoint property of its implication operators and strong conjunction operators. The unified form of Triple I Method are also based on regular implication operators. It is the numerical implementation of the Triple I Principle. It is certain that there are connections between MTL and Triple I Method since they share same class of implication operators. But there still exists gap between the numerical implementation of Triple I pinciple and logical formal inference of MTL.For eliminating the incompatibility between the numerical implementation and logical formal inference, Professor Wang Guojun established the truth degree theory based on even probability measure in classical two valued propostional logic syestem. Similar theories were generalized into n-valued ?ukasiewicz and R0 propostional logic systems. And some problems in fuzzy reasoning were solved using these theories. But all the above truth degree theories are based on even probability measure. Since in practical environment there always some unevenness exists, it is necessary that generalizng the truth degree into general probability measure and monoidal t-norm based logic. Considering the connection between MTL and Triple I Method, it is worthwhile to investigate properties of Triple I Method in the same frame with Monoidal T-norms based Logic. Since the reversibility of fuzzy inference methods is an important criterion to judge the effect of implication operators matching inference methods, it is also one of the tasks of this thesis.The main work of this paper includes:Chapter One: The concepts of strong regluar implication operator and strong monoidal t-norm based logic are defined. It is proved that the ?ukasiewicz implcation is the largest strong regluar implication operator and Goguen implication is the least.Chapter Two: Based on general probability measure, the truth degree of formula is defined and its integral expression is given. The inference rules w.r.t the degrees of the truth is proved, the similarity degree and pseudo-distance between formulae are defined and their properties are proved. This paves the way for the further study on approximate reasoning.Chapter Three: A necessary and sufficient condition of reversibility of triple I method for FMP is obtained when the t-norm adjoint to the given implication operator is continuous. Along with other theorems, a necessary and sufficient condition of reversibility of triple I method for FMT is proved when the implication operator in FMT is continuous in the first argument and regular.