The Quintuple Implication Principle of fuzzy reasoning

Abstract

Fuzzy Modus Ponens (FMP) and Fuzzy Modus Tollens (FMT) are two fundamental patterns of approximate reasoning. Suppose A and B are fuzzy predicates and “IF A THEN B” is a fuzzy rule. Approximate reasoning often requires to derive an approximation B∗ of B from a given approximation A∗ of A, or vice versa. To solve these problems, Zadeh introduces the well-known Compositional Rule of Inference (CRI), which models fuzzy rule by implication and computes B∗ (A∗, resp.) by composing A∗ (B∗, resp.) with A→B. Wang argues that the use of the compositional operation is logically not sufficiently justified and proposes the Triple Implication Principle (TIP) instead. Both CRI and TIP do not explicitly use the closeness of A and A∗ (or that of B and B∗) in the process of calculating the consequence, which makes the thus computed approximation sometimes useless or misleading.In this paper, we propose the Quintuple Implication Principle (QIP) for fuzzy reasoning, which characterizes the approximation B∗ of B (A∗ of A, resp.) as the formula which is best supported by A→B,A∗→A and A∗ (A→B,B→B∗ and B∗, resp.). Based upon Monoidal t-norm Logic (MTL), this paper applies QIP to solve FMP and FMT for four important implications. Most importantly, we show that QIP, when using Gödel implication, computes exactly the same approximation as Mamdani-type fuzzy inference does. This is surprising as Mamdani interprets fuzzy rules in terms of the minimum operation, while CRI, TIP and QIP all interpret fuzzy rules in terms of implication.