[js8]

Bayesian statistics is one approach to modelling fuzziness and uncertainty, but fuzzy logic explores other approaches. For example, see https://en.wikipedia.org/wiki/Possibility_theory

Also fuzzy logic builds a "traditional" logical framework for deduction on terms, again this is a bit different from Bayesian approach, which is more abductive reasoning.

[tr352]

Apart from the other answers, there is also one important technical difference. Fuzzy logic is truth functional but probability is not. That is, in fuzzy logic, if you know the fuzzy truth value of A and of B, you can calculate the fuzzy truth value of “A and B” “A or B” and so on. Not in probability. If you know, let’s say, A and B both have probably 0.9, you don’t know enough to calculate the probability of “A and B”, which lies somewhere between 0.8 and 0.9, or “ A or B”, which lies between 0.9 and 1.

[elcomet]

Yeah but in the real world it doesn't matter does it?

you cannot compute the probability A and B for a reason: they might be related (what if they're mutually exclusive for instance? )

So how does fuzzy logic deals with this?

[tr352]

Fuzzy logic deals with this in the exact same way as classical logic. Note that in classical logic, the truth value of "A and B" and "A or B" is also a function of the truth values of A and of B. When using fuzzy logic you have to make a choice as to which function you use. Typically these functions generalise the classical logic ones in the sense that they behave like the classical ones when using 0 and 1. These functions are defined by a so called T-norm.

[lonelappde]

That's great in toy theory, like the formal real number system.

How useful is it for modeling real world problem?

[jacques_chester]

Very useful. Many problems are more easily represented as fuzzy sets or fuzzy relations than in other terms. It's particularly good at encoding linguistic variables, such as "very fast", "too cold", "accelerate hard" and so on in a way that it can smoothly overlap them.

It's also useful for encoding uncertainties that are not yet mutually exclusive. There are other logics too (eg Dempster-Shafer evidence theory), often grouped together as "monotone measures".

[dragonwriter]

> So how does fuzzy logic deals with this?

It doesn't, because truth values are not probabilities. They are answers to questions more like “How tall is X” than “How likely is it that X is sufficiently tall”.