Compensatory Fuzzy Logic Inference

Abstract

This chapter deals with specific way of inferences possible in the framework of Compensatory Fuzzy Logic (CFL). It introduces new and useful inference systems based on CFL. They will be called Compensatory Inference Systems (CIS). It is a generalization of the deduction like in mathematical logic, with implication operators found in the literature regarding fuzzy logic. CIS is a combination of one Compensatory Logic and an implication operator. Every CIS is a logically rigorous inference system with easy application. In addition CIS may be coherently associated with the methods of deduction of the mathematical logic. The theoretical basis of this association is proved in this chapter. Valid formulas and right deductive structures according CFL are the new concepts introduced here. The formulas of the propositional calculus are valid in the bivalent logic if and only if they are valid according CFL. The same result is introduced for deductive right structures. The relevance of these results for approximate reasoning and knowledge discovery are illustrated. Further more, probabilistic properties expressed in a theorem allow applying statistical inference in the framework of CFL. This theorem expresses that the universal proposition over a sample can be a statistic estimator of the corresponding universal proposition over the entire universe. Thus CFL joins logical and statistical inferences and gives logical models of automated learning with properties of a statistical estimator.