(From Desktop 10 help.)
There's a video example from an esri conference here: http://www.youtube.com/watch?v=Hd13H0XO0LU
A very good article from esri for ArcGis and Fuzzy logic. With a good example and some bibliography.
There is a dissertation on:
Watershed water quality modeling using integrated fuzzy modeling approach with HSPF model and radar rainfall data Author: Madhusudhan Narayana
"Watershed water quality modeling is very essential for watershed management activities. These modeling activities and resulting simulations were used as the main resource for several decision making processes. In this research study, uncertainties associated with the watershed water quality modeling were handled with fuzzy modeling approach in three different phases"
Unfortunately there are very few examples of fuzzy logic using ArcGIS 10 on the internet. Youtube will show you this one http://www.youtube.com/watch?v=Hd13H0XO0LU about bald eagle habitat.
Fuzzy logic can be used in all kinds of different scenarios from mineral exploration, site selection, basically anything that has multiple inputs that each meet a criteria. In site selection you may use slope, distance from water, distance from roads, etc and put it into a fuzzy logic model. Selection of membership will mostly be linear but you can play with those classifications to get a better idea of how they work. The fuzzy membership plays into the fuzzy overlay in how you combine layers with AND, OR, GAMMA (mostly AND/OR). ESRI has a site selection tutorial for $30 which shows how the above youtube video is applied.