By the way, how would you incorporate probabilities into binary logic? Either you can include statements about probabilities in binary logic (“probability on top of logic”), or you can assign probabilities to binary logic statements (“logic on top of probability theory”). The situation is just analogous to that of fuzziness. If you do #1, that means binary logic is the most fundamental layer. If you do #2, I can also do an analogous thing with fuzziness.
The rules of probability reduce to the rules of binary logic when the probabilities are all zero or one, so you get binary logic for free just by using probability.
But under this approach the binary logic is NOT operating at a fundamental level—it is subsumed by a probability theory. In other words, what is true in the binary logic is not really true; it depends on the probability assigned to the statement, which is external to the logic. In like manner, I can assign fuzzy values to a binary logic which are external to the binary logic.
By the way, how would you incorporate probabilities into binary logic? Either you can include statements about probabilities in binary logic (“probability on top of logic”), or you can assign probabilities to binary logic statements (“logic on top of probability theory”). The situation is just analogous to that of fuzziness. If you do #1, that means binary logic is the most fundamental layer. If you do #2, I can also do an analogous thing with fuzziness.
The rules of probability reduce to the rules of binary logic when the probabilities are all zero or one, so you get binary logic for free just by using probability.
Yes, we all know that ;)
But under this approach the binary logic is NOT operating at a fundamental level—it is subsumed by a probability theory. In other words, what is true in the binary logic is not really true; it depends on the probability assigned to the statement, which is external to the logic. In like manner, I can assign fuzzy values to a binary logic which are external to the binary logic.