Second, isn't this just probability, which we already know and love? No, it isn't. If I say that "the Leaning Tower of Pisa is extremely short", I don't mean that I'm very, very sure that it's short. If I say "my mother was half Irish", I don't mean that I have no idea whether she was Irish or not, and might find evidence later on that she was completely Irish. Truth values are separate from probabilities.

Fuzzy logic is just sloppy probability, although Lofti Zadeh doesn't realize it. (I heard him give a talk on it at NIH, and my summary of his talk is: He invented fuzzy logic because he didn't understand how to use probabilities. He actually said: "What if you ask 10 people if Bill is tall, and 4 of them say yes, but 6 of them say no? Probabilities have no way of representing this.")

You can select your "fuzzy logic" functions (the set of functions used to specify a fuzzy logic, which say what value to assign A and B, A or B, and not A, as a function of the values of A and B) to be consistent with probability theory, and then you'll always get the same answer as probability theory.

The rules for standard probability theory are correct. But "sloppy" fuzzy-logic probability functions, like "A or B = max(A,B); A and B = min(A,B); not(A) = 1-A", have advantages when Bayesian logic gives lousy results. Here are 2 situations where fuzzy logic outperforms use of Bayes' law:

1. You have incomplete or inaccurate information. Say you are told that A and B have a correlation of 1: P(A|B) = P(B|A) = 1. By Bayes' law, P(A^B) = P(AvB) = P(A) = P(B). Then you're told that P(A) and P(B) are different. You're then asked to compute P(A^B). Bayes law fails you, because the facts you've been given are inconsistent. Fuzzy logic is a heuristic that lets you plow through the inconsistency: it enforces p(AvB) >= p(A^B), when Bayes' law just blows up.

2. You are a robot, making a plan. For every action you take, you have a probability of success that you always associate with that action. You assume that the probability of success for each step in a plan is independent of the other steps. But in reality, sometimes they are highly correlated. Because you assume probabilities are independent, you strongly favor short plans over long plans. Using fuzzy logic allows you to construct longer plans.

Fuzzy logic is just a pragmatic computational tool. Nothing that's going to help you get around a paradox, except in the sense that it will let you construct a model that's inaccurate enough that the paradox disappears from sight.

When you switch to using these numbers to differentiate between "short" and "extremely short", that's not probability. But then you're no longer talking about truth values. You're just measuring things. The number 17 is no more true than the number 3.

All that said, the approach you just described is interesting. I'm missing something, but it's very late, so I'll have to try to figure it out tomorrow.