Frequencies, histograms, and probabilities are one way to think about degrees of membership in a fuzzy subset, but there are cautions to be observed in doing that.
Consider the following triadic relations:
- x ∈r S = x is in S to the degree r
- x ∈j S = x is in S to the judge j
And consider their intensional counterparts:
- x isr P = x is P to the degree r
- x isj P = x is P to the judge j
These relations are all defined abstractly enough that you could probably fit them out with almost any reasonable statistical model.
One adds concreteness to these abstractions by specifying how the triadic relations combine under operations analogous to the usual set-theoretic and logical operations. It’s been a while since I studied it but I seem to recall that most of the fuzzy literature works with rules for combining degrees that cannot be interpreted as probabilities, rather as what they like to call possibilities.
How one combines interpreters is a really good question. This may be another one of those places where we have to turn Peirce’s trick of replacing interpreters with interpretants.