Elements of the theory of fuzzy sets

Abstract

Foundations of the theory of fuzzy sets and fuzzy logic were formulated in 1965 by L.A. Zadeh [293]. This theory was introduced as a means for representing, manipulating, and utilizing data and information that possess nonstatistical uncertainty. Fuzzy logic provides inference mechanisms that enable approximate reasoning and model human reasoning capabilities to be applied to knowledge-based systems. The theory of fuzzy sets provides a mathematical apparatus to capture and handle the uncertainty and vagueness inherently associated with human cognitive processes, such as perception, thinking, reasoning, decision making, etc. The most straightforward example is the linguistic uncertainty of a natural language [294]. Conventional approaches to knowledge representation lack the means for representing the meaning of concepts with unsharp boundaries (fuzzy concepts). As a consequence, approaches based on first order logic and classical probability theory do not provide an appropriate conceptual framework for dealing with the representation of commonsense knowledge, since such knowledge is by its nature both uncertain and lexically imprecise. The need for a conceptual framework which can address and formally implement the issue of uncertainty and lexical imprecision has been, in large measure, a motivating factor for the development of the theory of fuzzy sets and fuzzy logic [68].