Yes, I have worked out a fuzzy logic with probability distributions over fuzzy values.
YKY
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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.
About the so-called “discontinuous truth values”, I think the culprit is not that the truth value is discontinuous (it doesn’t make sense to say a point-value is continuous or not), but rather that we have a binary predicate, “less-than”, which is a discontinuous truth functional mapping.
The statement “less-than(tv, 0.5)” seems to be a binary statement. If we make that predicate fuzzy, it becomes “approximately less than 0.5″, which we can visualize as a sigmoidal curve, and this curve intersects with the slope=1 line at 0.5. Thus, the truth value of the fuzzy version of that statement is 0.5, ie, indeterminate.
All in all, this problem seems to stem from the fact that we’ve introduced the binary predicate “less-than”.
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.
It’s good that you pointed out Kripke’s fixed point theory of truth as a solution to the Liar’s paradox. It seems to be an acceptable solution.
On the other hand, I also agree that “fuzziness as a matter of degree” can be added on top of a binary logic. That would be very useful for dealing with commonsense reasoning—perhaps even indispensable.
What is particularly controversial is whether turth should be regarded as a matter of degree, ie, the development of a fuzzy-valued logic. At this point, I am kinda 50-50 about it. The advantage of doing this is that we can translate commonsense notions easily, and it may be more intuitive to design and implement the AGI. The disadvantage is that we need to deal with a relatively new form of logic (ie, many-valued logic) and its formal semantics, proof theory, model theory, deduction algorithms, etc. With binary logic we may be on firmer ground.
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.
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...
A conjecture (seems easy to prove):
“If, in a fuzzy logic where truth values range from [0,1], we allow logical operators (which are maps from [0,1] to [0,1]) or predicates that does not intersect the slope=1 line, then we can always construct a Liar’s Paradox.”
An example is the binary predicate “less-than”, which has a discontinuity at 0.5 and hence does not intersect the y=x line.