Fuzzy approximate reasoning: Logic, intuition or comparison?

From the early beginnings of approximate reasoning to its present use, numerous research have investigated the appropriateness of compositional rule of inference (CRI) for generalized modus ponens. Other approximate reasoning methods such as compatibility-modification, extensional inference, and analogical inference have been proposed as a means of performing deductive reasoning on fuzzy rules, i.e., “If u is A, then v is B” then given A' produce B'. A detailed presentation and comparison of several of these investigations are provided. A connection between the results of CRI and the use of comparison indices between fuzzy sets A and B is synthesized from numerous examples in the research literature and with additional examples. CRI is shown to satisfy extensional inference when a fuzzy extension of the logical definition of equivalence is used for the comparison index C. C(A', A) is always less than or equal to C(B', B) where B' is the result of CRI with the rule relation R(A(u),B(v)). Requirements for the rule relation R are discussed. Comparison indices are reviewed and examined to determine under what conditions extensional inference might apply to them. A recent proposal to generalize a standard compatibility-modification inference pattern based on Zadeh's consistency index is critiqued.