Fuzzy logic is the logic of partial set membership, and it has gotten a lot of use in control systems. Here is the most accessible introduction that I've found for it: Fuzzy Logic course notes (PDF)

I'll describe a very simple example: a thermostat. It turns on a room heater if its room is too cold, and turns it off if the room is too hot. That is crisp-logic operation. If the heater can be throttled, then one can do fuzzy-logic operation, throttling it down with greater temperature. One can define two categories: "too hot" and "too cold". For instance, "too hot" can have membership 1 above 25 C, membership 0 below 15 C, and linear in between. "Too cold" can be a mirror image of that, 0 for above 25 C, 1 for below 15 C and linear in between. Both categories have membership 0.5 at 20 C.

So to decide how much to run one's heater, one does (full)*(too-cold membership) + (off)*(too-hot membership), and one has a nice linear decline from 15 C to 25 C.

But fuzzy logic has interesting mathematical properties, and I will be discussing them.