golden@FRODO.STANFORD.EDU (Richard Golden) (02/20/88)
I'm sorry but I do not find Bruce D'Ambrosio's arguments convincing (although I would be happy to be convinced!!!) As I noted before, the AXIOMS of probability theory can be justified from constraints upon rational decision making (i.e., standard deductive logic). I have not seen (I would like to see) similar arguments constructed for the AXIOMS of fuzzy set theory. In response to point 1 that fuzzy logic is appropriate in cases where we do not have exact probabilities I would argue that probability theory is still applicable since we can do conditioning. That is, suppose we know that the probability of event A, p(A), lies in the interval [0.3,0.4]. We can model our uncertainty associated with p(A) by rewriting p(A) as p(A|t) where t is a dummy random variable uniformly distributed over the interval [0.3,0.4]. In response to point 2, that Zadeh's intuition was set theoretic, and not frequentist or subjective I would like to emphasize that all of my arguments break down if one takes the frequentist view of probability theory --- it is absolutely essential that the subjectivist view of probability theory is taken. The subjectivist view simply says that some number is associated with a particular event in the environment and this number reflects one's belief that the event occurs. Thus, there is no reason why we can't interpret this number as an indicator of approximate correctness. The final comment that there can be "no real conflict between fuzzy set theory and probability theory" is valid as long as you are not concerned with making inferences which are always consistent with the symbolic logic (i.e., Boolean Algebra). If you are concerned with making inferences consistent with symbolic logic...then you are right...there is no conflict... probability theory wins (unless you can justify the axioms of fuzzy set theory with respect to rational decision making for me). A final caveat upon the limitations of probability theory AND fuzzy logic. Both of these approaches assume one can represent belief as a single real-valued function -- this critical assumption should not be ignored. But if one does accept this assumption and one wants to make logically consistent inferences, probability theory is the way to go. Cc:
ok@quintus.UUCP (Richard A. O'Keefe) (02/22/88)
I don't like "fuzzy logic". The basic reason for that is very simple: it puts the fuzziness in the wrong place. The standard example is "John is very tall" where "very" is interpreted as a degree-of-belief in the proposition "John is tall". This fails to make it clear whether - there is some doubt about the height or - someone is tall, but there is some doubt about whether it's John. This ambiguity does not exist in the original statement. Putting the fuzziness on the truth-values instead of the functions seems wrong. In probability, there is a clear distinction between distributions (relating to functions) and probabilities (relating to propositions).