# Re: Fuzzy logic compared to probability

Herman Rubin (hrubin@b.stat.purdue.edu)
Fri, 1 Mar 1996 20:27:23 +0100

In article <4gn6re\$r5u@watt.electriciti.com>,
Fred A Watkins <fwatkins@hyperlogic.com> wrote:
>hrubin@b.stat.purdue.edu (Herman Rubin) 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?

I am unaware of Cox's papers. There are other papers which show
that probability is the only consistent formulation of betting
odds which only use algebra.

>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.

Probability is many things. The usefulness of it is that it is
generally not necessary, or even useful, to know which.

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

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

If you define what you mean, of course. The basis for rational
behavior under uncertainty is that one MUST act.

>>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.

What I am saying is much simpler, and all logics recognize it. If
A and B are propositions, so is (A & B). The truth value of this
proposition, in a truth-value logic, depends only on the truth
values of the propositions. Probability is inadequate as a truth-value
system; while it can be a construct in a Boolean system, it is not
enough to serve as one.

>>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...

This is the "Dutch Book" theorem. It only uses algebra. It does
not require knowing anything other than algebra.

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

>... but it sounds like you are.

I am starting from the point that one is trying to advise the
"rational" person what action to take, using that person's
value system and belief system. The axioms are quite weak.

>>>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

>>>I apologize in advance for any misconceptions and misstatements.

................

>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.

A REPRESENTATION of probability takes events into sets, and random
variables into functions. It is generally a mistake to use definitions
instead of representations and characterizations, even in formal systems.

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