Fuzzy Set Theory: The Basics

Abstract

A set is fuzzy when an element can belong partly to it, rather than having to belong completely or not at all. Fuzzy set theory, therefore, begins with the assignment of membership values to elements which are not restricted to 0 (nonmembership) or 1 (full membership), but which may lie somewhere in the interval from 0 to 1. This verbal statement, and the mathematics which formalize it, have some intuitive appeals since ordinary common sense presents us with sets which fit this description. Fuzzy set theorists themselves have not been forthcoming about what fuzziness is, and some even eschew definitional issues altogether. Mathematicians, to some extent, can do this because they do not necessarily deal in substantive questions about empirical reality. We in the behavioral and social sciences, however, cannot afford to ignore these issues. After all, psychologists, philosophers, linguists, and social scientists have used terms which might be related to or even synonymous with fuzziness. Is fuzziness a type of ambiguity or vagueness? How is it related to uncertainty? Is it a kind of probability (and would probability do just as well)? But before moving into a full-blown discussion of terms, it is worth considering what the founders of fuzzy set theory have had to say about fuzziness. While they are reluctant to say what it is, they have been quite vocal on what it is not.