Re: Fuzzy logic compared to probability

Fred A Watkins (fwatkins@hyperlogic.com)
Fri, 1 Mar 1996 19:51:07 +0100


hrubin@b.stat.purdue.edu (Herman Rubin) wrote:

>In article <312B60FB.41C67EA6@colorado.edu>,
>Robert Dodier <dodier@colorado.edu> wrote:
>>Hello,

> ..................

>>I am aware that proponents of this Bayesian probability theory
>>claim that it is the only consistent generalization of binary
>>logic (according to Cox's theorems). Where does this leave
>>fuzzy logic?

Alone. Cox's Theorem needs twice-differentiable operators; fuzzy uses
min and max, which do not meet the requirement.

>One can even go a little farther, and not separate prior
>probability from utility. See my paper in _Statistics and
>Decisions_, 1987.

>>It would appear that if the fuzzy logic concept of `degree of
>>truth' is the same as the Bayesian `degree of belief,' then
>>either fuzzy logic is the same as probability or else less
>>powerful (as it would have to be inconsistent). One could salvage
>>fuzzy logic by interpreting `degree of truth' differently from
>>`degree of belief,' but does `degree of truth' then remain useful?

The trouble with probability as "degree of belief" is that if there
were no people, there would be no belief, hence no probability. And
it's likely that one's belief is ultimately grounded in experience,
anyway.

>>In particular, can one bet on a `degree of truth' ??

You tell me: Do you want to bet that an electron is a particle?

>The problem with a linear truth value system is that it is not
>enough. If one has two events, besides the truth values of A
>and B, there are the truth values of their union and intersection.
>For an extremely simple example to show the problem, let C be ~A,
>and let A have truth value 1/2. Then C does also. Now look at
>what happens in B = A or B = C.

If you mean the *proposition* "Either B is A OR B is C", that's in a
different universe from A, B, and C, and one might establish the truth
(or lack of it) in various ways. No particular way is etched in the
sky for us to always follow. Let's try always to compare the same
kinds of fruit.

>It is known that every consistent betting scheme corresponds to
>probability and conditional probability. If a "fuzzy" situation
>is expanded to include other bets, probability results.

I do hope you're not going to pull out Cox's Theorem again...

>Belief function approaches are not adequate to include actions,
>unless they are essentially probability.

.. but it sounds like you are.

>>I've asked this question before, but got no satisfying answers;
>>I think it's interesting enough to try again. Please, no homey
>>parables about murky water. :) (As an aside to S.F. Thomas, my
>>university's library doesn't have your book, and I would rather not
>>buy it just to answer this one question; surely there is a brief
>>answer.)
>>
>>I apologize in advance for any misconceptions and misstatements.
>>
>>Regards,
>>Robert Dodier
>>--
>>How Does a Person Decide Whom to marry??
>>"You flip a nickel, and heads means you stay with him and tails means
>>you try the next one." -- Kally, 9

>--
>Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
>hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558

Ultimately, probability is a property of *sets*, which are
well-defined objects of mathematics. Fuzzy things are more troublesome
in that they resist definition. One must always remember that once
*definitions* are absent, it's VERY hard to reason. But it can be
done, and it doesn't mean probability, necessarily.