Qualitative fuzzy sets: homotopy and perception

What should be the correct perception of a fuzzy set? We believe a fuzzy set should be able to tolerate "small amounts" of "physical" perturbations. There are three approaches: roughly, type II fuzzifies the grades, Lin fuzzifies the membership functions, and Thiele captures Kripke style semantics. However, it is not immediate these approaches could tolerate "physical" perturbation. In this paper, we propose a homotopy approach that clearly relates the notion of qualitative fuzzy set with "physical" deformation.