Making sense of higher order fuzzy sets : Uncertainty vs. bipolarity
Résumé
Interval-valued or type 2 fuzzy sets refer to fuzzy sets whose membership grades are fuzzy sets. One motivation for the latter was the idea that membership functions are sometimes difficult to identify and are thus ill-known. Insofar as type 2 fuzzy sets correspond to a lack of information, fuzzy membership grades are epistemic fuzzy sets and represent fuzzy restrictions on the unknown membership grade. Being an ill-known type 1 fuzzy set, we claim it makes little sense to apply a compositional calculus of set-valued membership grades to the set-theoretic combination of ill-known type 1 fuzzy sets. The same ambiguity applies to Atanassov intuitionistic fuzzy sets, which moreover convey an idea of bipolar information. We explain that bipolar membership and uncertain membership are clashing semantics leading to different ways of handling the basic connectives.