In this section, we will introduce the concepts of the extension principles and fuzzy relations, which expand the notions and applicability of fuzzy sets introduced previously. Then we present the definition of linguistic variables and linguistic values, and explain how to use them in fuzzy rules, which are an efficient tool for quantitative modeling of words or sentences in a natural or artificial language. By interpreting fuzzy rules as appropriate fuzzy relations, we investigate different schemes of fuzzy reasoning, where inference provedures based on the concept of the compositional rule of inference are used to derive conclusions from a set of fuzzy rules and known facts.

Fuzzy rules and fuzzy reasoning are the backbone of fuzzy inference systems, which are the most important modeling toll based on fuzzy

set theory. They have been successfully applied to a wide range of areas, such as automatic control, expert systems, pattern recognition, time series prediction, and data clissification.