On the topic of fuzzy logic: Is there a semantics for fuzzy logic in which the fuzzy truth value of the statement “predicate P is true of object x” is the expected value of P(x) after marginalizing out a prior belief distribution over possible hidden crisp definitions of P?
I guess not. The point is that “matters of degree” are inherently different from probabilities, and the former cannot be reduced to the latter. To best clarify this point, we need a formal semantics of fuzzy logic (where fuzziness is treated as matters of degree). I’m not sure if there’s such research in the literature, I’ll have a look when I have time...
I’m not sure they are inherently different. I read Kosko’s popular book on Fuzzy Logic many years ago and can’t remember the details of the argument, but he claimed that probabilistic logic is a special case of fuzzy logic, as propositional logic is a special case of probabilistic logic (ie, with probabilities of 0 and 1).
Several things. First, you’re claiming “probabilistic is a special case of fuzzy” but that does not imply “fuzzy is a special case of probabilistic” which was the original point of contention.
Secondly, you probably have confused fuzzy logic with “possibility theory”. There can be many types of fuzzy logic, and the issue we’re currently debating is whether “truth” can be regarded as a matter of degree, ie, fuzziness as degree of truth. Possibility theory is a special type of fuzzy theory which results from giving up the probability axiom #3, “finite additivity”. That is probably what your author is referring to.
Yes, because probabilistic logic is a special case of fuzzy logic. (The phrase “special case” is odd, because you could argue that it’s simply the correct case, and all others are wrong.)
On the topic of fuzzy logic: Is there a semantics for fuzzy logic in which the fuzzy truth value of the statement “predicate P is true of object x” is the expected value of P(x) after marginalizing out a prior belief distribution over possible hidden crisp definitions of P?
I guess not. The point is that “matters of degree” are inherently different from probabilities, and the former cannot be reduced to the latter. To best clarify this point, we need a formal semantics of fuzzy logic (where fuzziness is treated as matters of degree). I’m not sure if there’s such research in the literature, I’ll have a look when I have time...
I’m not sure they are inherently different. I read Kosko’s popular book on Fuzzy Logic many years ago and can’t remember the details of the argument, but he claimed that probabilistic logic is a special case of fuzzy logic, as propositional logic is a special case of probabilistic logic (ie, with probabilities of 0 and 1).
Several things. First, you’re claiming “probabilistic is a special case of fuzzy” but that does not imply “fuzzy is a special case of probabilistic” which was the original point of contention.
Secondly, you probably have confused fuzzy logic with “possibility theory”. There can be many types of fuzzy logic, and the issue we’re currently debating is whether “truth” can be regarded as a matter of degree, ie, fuzziness as degree of truth. Possibility theory is a special type of fuzzy theory which results from giving up the probability axiom #3, “finite additivity”. That is probably what your author is referring to.
Yes, because probabilistic logic is a special case of fuzzy logic. (The phrase “special case” is odd, because you could argue that it’s simply the correct case, and all others are wrong.)