Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets

This paper discusses basic notions underlying fuzzy sets, especially gradualness, uncertainty, vagueness and bipolarity, in order to clarify the significance of using fuzzy sets in practice. Starting with the idea that a fuzzy set may represent either a precise gradual composite entity or an epistemic construction refereeing to an ill-known object, it is shown that each of this view suggests a different use of fuzzy sets. Then, it is argued that the usual phrase fuzzy number is ambiguous as it induces some confusion between gradual extensions of real numbers and gradual extensions of interval calculations. The distinction between degrees of truth that are compositional and degrees of belief that cannot be so is recalled. The truth-functional calculi of various extensions of fuzzy sets, motivated by the desire to handle ill-known membership grades, are shown to be of limited significance for handling this kind of uncertainty. Finally, the idea of a separate handling of membership and non-membership grades put forward by Atanassov is cast in the setting of reasoning about bipolar information. This intuition is different from the representation of ill-known membership functions and leads to combination rules differing from the ones proposed for handling uncertainty about membership grades.