Concerning a quantum-like uncertainty relation for pairs of complementary fuzzy sets

AbstractIt appears far more natural and rewarding to consider “fuzziness” spread over the category of closed sets and semicontinuous set-valued mappings than over a set of points and point-to-point functions considered hitherto; for, none will argue that we actually “see” points (and not sets and singletons) when we study fuzzy sets and systems. This change of attitude merely necessitates an extension of the membership function to set-valued mappings and—with regard to dynamics—a transcription of ordinary differential equations to differential inclusions (i.e., orientor field equations in the case of control problems). However, the proper perspective now requires topology in the spaces under consideration and a shift from Boolean algebra to Brouwerian algebra since fuzziness disobeys some of the fundamental laws, notably tertium-non-datur and transitivity. Also, in faithful adherence to intuitionistic thinking, we have to consider the cardinality of sets as being, at the most, potentially infinite, never actually infinite. This emphasizes the constructivity as well as the subjectivity aspects of any fuzzy set theory. Both hypotheses (concerning topology and logic), taken jointly, already program (in the limit) a fundamental uncertainty relation in fuzzy set theory, very much like the celebrated Heisenberg uncertainty relation in quantum mechanics, if we make the fuzzy quantum (the greatest negligible set) and the physical quantum correspond to one another. These ideas will be introduced into the formalism of orientor field control in a companion paper to follow