You can select your “fuzzy logic” functions (the set of functions used to specify a fuzzy logic, which say what value to assign A and B, A or B, and not A, as a function of the values of A and B) to be consistent with probability theory, and then you’ll always get the same answer as probability theory.
How do you do this? As far as I understand, it is impossible since probability is not truth functional. For example, suppose A and B both have probability 0.5 and are independent. In this case, the probability of ‘A^B’ is 0.25, while the probability of ‘A^A’ is 0.5. You can’t do this in a (truth-functional) logic, as it has to produce the same value for both of these expressions if A and B have the same truth value. This is why minimum and maximum are used.
Calling fuzzy logic “truth functional” sounds like you’re changing the semantics; but nobody really changes the semantics when they use these systems. Fuzzy logic use often becomes a semantic muddle, with people making the values simultaneously mean truth, probability, and measurement; interpreting them in an ad-hoc manner.
You can tell your truth-functional logic that A^A = A. Or, you can tell it that P(A|A) = 1, so that p(A^A) = p(A).
Calling fuzzy logic “truth functional” sounds like you’re changing the semantics;
‘Truth functional’ means that the truth value of a sentence is a function of the truth values of the propositional variables within that sentence. Fuzzy logic works this way. Probability theory does not. It is not just that one is talking about degrees of truth and the other is talking about probabilities. The analogue to truth values in probability theory are probabilities, and the probability of a sentence is not a function of the probabilities of the variables that make up that sentence (as I pointed out in the preceding comment, when A and B have the same probability, but A^A has a different probability to A^B). Thus propositional fuzzy logic is inherently different to probability theory.
You might be able to create a version of ‘fuzzy logic’ in which it is non truth-functional, but then it wouldn’t really be fuzzy logic anymore. This would be like saying that there are versions of ‘mammal’ where fish are mammals, but we have to understand ‘mammal’ to mean what we normally mean by ‘animal’. Sure, you could reinterpret the terms in this way, but the people who created the terms don’t use them that way, and it just seems to be a distraction.
At least that is as far as I understand. I am not an expert on non-classical logic, but I’m pretty sure that fuzzy logic is always understood so as to be truth-functional.
You might be able to create a version of ‘fuzzy logic’ in which it is non truth-functional, but then it wouldn’t really be fuzzy logic anymore.
Eep, maybe I should edit my post so it doesn’t say “fuzzy logic”. Not that I know that non-truth-functional fuzzy logic is a good idea; I simply don’t know that it isn’t.
How do you do this? As far as I understand, it is impossible since probability is not truth functional. For example, suppose A and B both have probability 0.5 and are independent. In this case, the probability of ‘A^B’ is 0.25, while the probability of ‘A^A’ is 0.5. You can’t do this in a (truth-functional) logic, as it has to produce the same value for both of these expressions if A and B have the same truth value. This is why minimum and maximum are used.
Calling fuzzy logic “truth functional” sounds like you’re changing the semantics; but nobody really changes the semantics when they use these systems. Fuzzy logic use often becomes a semantic muddle, with people making the values simultaneously mean truth, probability, and measurement; interpreting them in an ad-hoc manner.
You can tell your truth-functional logic that A^A = A. Or, you can tell it that P(A|A) = 1, so that p(A^A) = p(A).
‘Truth functional’ means that the truth value of a sentence is a function of the truth values of the propositional variables within that sentence. Fuzzy logic works this way. Probability theory does not. It is not just that one is talking about degrees of truth and the other is talking about probabilities. The analogue to truth values in probability theory are probabilities, and the probability of a sentence is not a function of the probabilities of the variables that make up that sentence (as I pointed out in the preceding comment, when A and B have the same probability, but A^A has a different probability to A^B). Thus propositional fuzzy logic is inherently different to probability theory.
You might be able to create a version of ‘fuzzy logic’ in which it is non truth-functional, but then it wouldn’t really be fuzzy logic anymore. This would be like saying that there are versions of ‘mammal’ where fish are mammals, but we have to understand ‘mammal’ to mean what we normally mean by ‘animal’. Sure, you could reinterpret the terms in this way, but the people who created the terms don’t use them that way, and it just seems to be a distraction.
At least that is as far as I understand. I am not an expert on non-classical logic, but I’m pretty sure that fuzzy logic is always understood so as to be truth-functional.
Eep, maybe I should edit my post so it doesn’t say “fuzzy logic”. Not that I know that non-truth-functional fuzzy logic is a good idea; I simply don’t know that it isn’t.