Suppose we have the fuzzy membership grade for a person $x$ with a set $S = \text{set of tall people}$ be $0.9$, i.e. $\mu_S(x)=0.9$.

Does this mean that the probability of person $x$ being tall is $0.9$?


No, you can't extract any probability from a fuzzy membership grade. The uncertainty expressed by fuzzy logic is about partial truth, not about probability. $ \mu_S(x) = 0.9 $ doesn't mean that "$ x $ is tall" is true with a probability of 0.9, but that "$ x $ is tall" is 90% true (notice the difference in semantics). You have to think about fuzzy logic as an extension of logic (as its name implies), rather than an extension of probability.

It's true, however, that fuzzy logic is flexible and lets you define how the membership grades are combined in logic formulae, to the extend you can replicate probability theory within the fuzzy logic framework. Wikipedia has a good overview on this: https://en.wikipedia.org/wiki/Fuzzy_logic#Comparison_to_probability.

However, please understand that, in general, fuzzy membership $ \neq $ probability. How we come up with the fuzzy membership grade is subjective and application-dependent. Conversely, probabilities have a well-defined and unambiguous interpretation. The point of being fuzzy is to replicate our reasoning process which, even if is not necessarily formal and rigorous, is often very accurate. To do so it needs a set of (admittedly arbitrary) rules on how to calculate the "truthfulness" of logical formulae. This may turn to be very useful in applications where manipulating probabilities or coming up with them in the first place is not tractable.


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