Fuzzy Set Theory: A Primer

In the Classical Logic, the ‘Law of the Excluded Middle’ states that out of two contradictory propositions (where one proposition is the negation of the other) one must be true, and the other false, or that every proposition must either be true or false: it will not be possible to be and not to be the same thing; these truth values with a sharp demarcation may be represented by ‘1’ for ‘true’ and ‘0’ for ‘false’ and form the basis of the traditional set theory. Thus it cannot consider cases of vagueness. However, there are many concepts and predicates that are surrounded by vagueness or uncertainty. For example, consider the predicates such as ‘tall’ and ‘nice’, adverbs such as ‘quickly’ and even quantifiers such as ‘many’, all of which can be vague. So are the predicates ‘developed’ and ‘poor’, or the concepts of ‘development’ and ‘poverty’. A well-known account of vagueness comes in terms of ‘degree theory’ that drops classical Logic. The degree theory proposes a gradual transition between ‘perfect falsity’ to ‘perfect truth’, so that there are more than two truth values, that is, an infinite number of truth values along a spectrum between perfect truth and perfect falsity. Thus truth comes in degrees. Fuzzy set logic, developed by the American mathematician Lotfi Aliasker Zadeh, seeks to quantify the degree of truth in borderline cases. Thus perfect truth may be represented by ‘1’ and perfect falsity by ‘0’, with borderline cases having a truth value anywhere between 0 and 1. The present paper provides an introduction to fuzzy set theory