Fuzzy logic and fuzzy set theory

Fuzzy set theory and fuzzy logic are systems in which we don't have a black and white, true or false scenario. For example, in most universes, if X is your set, A is a subset of X, then if x is some element of X, x is either in A or x is not in A. What these fuzzy systems do is give a probability, i.e. a number from [0,1], to how much "x belongs in A." This is a crude description, but generally the idea.

My question for you all is, have any of you dealt with these types of systems? If so, how did you use them? What'd you think? I have never heard of class using them, but they are really interesting to me. Also, does anyone know of good books or sources on them? I know this has (somewhat?) to do with quantum mechanics/probability, so feel free to lecture me on just how so.