Making sense of fuzzy sets : Ontic vs. epistemic representations
Résumé
As acknowledged for a long time, sets may have a conjunctive or a disjunctive reading. In the conjunctive reading a fuzzy set represents an object of interest for which a gradual (rather than Boolean) description makes sense. In contrast disjunctive fuzzy sets refer to the use of sets as a representation of incomplete knowledge. They do not model objects or quantities but partial information about an underlying object or a precise quantity. In this case the fuzzy set captures uncertainty, and its membership function is a possibility distribution. This is because a fuzzy set is a set of possible values, while its fuzziness only brings shades of uncertainty. We call epistemic such fuzzy sets, since they represent states of incomplete knowledge. Epistemic uncertainty is the realm of possibility theory, with applications such as computing with fuzzy intervals, approximate reasoning, or imprecise regression and kriging. Distinguishing between ontic and epistemic fuzzy sets is important in information-processing tasks because there is a risk of misunderstanding basic notions and tools, such as distance between fuzzy sets, variance of a fuzzy random variable, fuzzy regression, fuzzy models, fuzzy equations, conditioning fuzzy information, etc. We discuss several examples where the ontic and epistemic points of view yield different approaches to these concepts.