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YKY comments on Deleting paradoxes with fuzzy logic - Less Wrong
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The crisp portion of such a self-reference system will be equivalent to a Kripke fixed-point theory of truth, which I like. It won't be the least fixed point, however, which is the one I prefer; still, that should not interfere with the normal mathematical reasoning process in any way.
In particular, the crisp subset which contains only statements that could safely occur at some level of a Tarski hierarchy will have the truth values we'd want them to have. So, there should be no complaints about the system coming to wrong conclusions, except where problematically self-referential sentences are concerned (sentences which are assigned no truth value in the least fixed point).
So; the question is: do the sentences which are assigned no truth value in Kripke's construction, but are assigned real-numbered truth values in the fuzzy construction, play any useful role? Do they add mathematical power to the system?
For those not familiar with Kripke's fixed points: basically, they allow us to use self-reference, but to say that any sentence whose truth value depends eventually on its own truth value might be truth-value-less (ie, meaningless). The least fixed point takes this to be the case whenever possible; other fixed points may assign truth values when it doesn't cause trouble (for example, allowing "this sentence is true" to have a value).
If discourse about the fuzzy value of (what I would prefer to call) meaningless sentences adds anything, then it is by virtue of allowing structures to be defined which could not be defined otherwise. It seems that adding fuzzy logic will allow us to define "essentially fuzzy" structures... concepts which are fundamentally ill-defined... but in terms of the crisp structures that arise, correct me if I'm wrong, but it seems fairly clear to me that nothing will be added that couldn't be added just as well (or, better) by adding talk about the class of real-valued functions that we'd be using for the fuzzy truth-functions.
To sum up: reasoning in this way seems to have no bad consequences, but I'm not sure it is useful...
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.