Can someone explain how this is different than bayesian statistics?
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.
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?
How useful is it for modeling real world problem?
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".
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”.
Instead you could put a distribution (but NOT a probability distribution) for belonging in the set of early that would look something like this.
If you wake up before 6:30 it would definitely 100% be early and at 8:30 it would. Or at 8:30 it would not at all, 0%, be early. Between there we would put some kind of partly belonging to Early rises.
In probability it is either or, but lack of knowledge makes us, but put a probability on what it is. In fuzzy logic, it is a bit of both at the same time.
Does not make it much easier to understand? You bet, which is why symbolic logic is much more useful.
You could say Bayesian statistics is a subset of Fuzzy logic.
Given how informal people have to be in Bayesian statistics to come up with reasonable priors (e.g. uniform), and how well it works by just guessing reasonable values, it could be argued that the power of Bayes is not in the inference but from the slack in the system it permits. Fuzzy logic is pure slack.
I think modern neural networks with activations like leaky relu look more at home in a fuzzy logic textbook than in a statistics text book.
I don't think that's accurate. Concepts like conditional probability and independence have no analogue in fuzzy logic.
Arguably not. Mathematicians have teased out differences between different many-valued logics and systems. A critical one between probability and fuzziness is that probability includes the axiom of the excluded middle and fuzziness does not. In probability the values of mutually exclusive events must sum to 1.0, in fuzziness they need not, because it doesn't require events to be mutually exclusive in the sense that probability requires.
Assuming that the die is fair, this is not possible in the real world.
Bayesian probability [1] states that this is not possible as expected.
``` P(land1 ^ land6) = P(land1) x P(land6 | land1) = 1/6 x 0 = 0 ```
However, fuzzy logic [1] results in an unintuitive result.
``` T(land1 ^ land6) = min(land1, land6) = min(1/6, 1/6) = 0 ```
[1] https://www.mathsisfun.com/data/bayes-theorem.html [2] http://www.sfu.ca/~jeffpell/papers/FuzzyLogic77.pdf