ABSTRACT
The objective of this section is to study the common inference techniques used in fuzzy reasoning. If the information is given through ‘if–then’ rules then the problem is stated in the following way: given a fact, X is A’, and a rule ‘If X is A then y is B’, we need to obtain a new fact, Y is B’ as a logical consequence. Unlike classical reasoning, where inference can be performed just by manipulating symbols, here the problem is set at the semantic level. Namely, inference is performed by manipulating membership functions of fuzzy sets by means of many-valued logic connectives. Our study is centered on Zadeh’s compositional rule of inference (CRI) as the basic mechanism of reasoning in a fuzzy context. This rule allows one to reduce the inference to the composition of fuzzy relations and membership functions and can be extended in different ways to obtain diverse fuzzy versions of the modus ponens inference rule and others. A discussion on these extensions is the central point of this section. There are implication-based and conjunction-based approaches to fuzzy reasoning, depending on the meaning and usage of the fuzzy rules. Furthermore, there are several many-valued conjunction and implication connectives that let us implement both approaches. The selection of such connectives is an important topic from an algebraic point of view. Other aspects are also studied, such as inference based on fuzzy truth values to deal with qualified fuzzy statements and the interpretation of fuzzy inference as approximate reasoning. Finallyt other models of inference that do not fit exactly into the CRI-type approaches, such as possibilistic logic or similarity reasoning, are also briefly discussed.
