Re: fuzzy inference

On 26 Sep 2003 10:58:22 -0700, [EMAIL PROTECTED] (William Siler) wrote:

>Dmitry A. Kazakov <[EMAIL PROTECTED]> wrote in message news:<[EMAIL PROTECTED]>...
>> 
>Hi, Dmitry! Good to talk to you again.

Hi!

>Here is your posting and my comments:
>
>>>Would anyone seriously propose that the fuzzy and probability 
>>>formulas have no relation to each other?
>
>>But similarity in formulae tells nothing about the reasons of this
>>similarity.
>
>True. But consider this:
>
>The formulas are not similar - they are identical. This would lead one
>to ask - what is the reason for this identity?

Per chance? (:-))

>I think that the answer is fairly obvious. In fuzzy, a (the truth
>value of A) is  a measure of the degree to which A is true; in
>probability, p(A) (the probability of A) is the probability that A is
>true. What is A? A simple proposition: "A is true." The only
>difference is that fuzzy is more general, where probabillity considers
>A to have only zero-one values. So I think that fuzzy has generalized
>probabiity so that the two states of A, true or false, are no longer
>mutually exclusive.
>
>>>Or that there is no relation between
>>>the fuzzy truth value of A and the probability that A is true?
>
>>No longer they are 0 or 1. But 0.5 probability has in my view nothing
>>to do with 0.5 of a fuzzy truth value. Or else, why should it have? 
>
>The reason they should have something to do with each other is that 
>both truthvalue(A) and p(A) deal with the same proposition: "A is
>true".

IMO they do, but no longer "A is true" is either 1 or 0. Once "A is
true" becomes something in-between, they head in different directions.

>>As I see it, probability and fuzzy truth values are different
>measures,
>>which have different properties and are useful *because* both are
>>different. Otherwise, there were absolutely no reason in having fuzzy
>>truth values. 
>
>Of course probability and truth values are different. But what is the 
>difference? I claim that the difference is that states in probability
>are
>mutually exclusive, and in fuzzy they are not necessarily mutually
>exclusive.

Note that this does not tell about the nature of either the
probabilites or truth values. It is a requirement imposed on some
objects which probability / truth value we are going to measure. Then
even if outcomes are dependent, they still have probabilites, which
still can be evaluated, but in a more complex way. So this cannot be
the difference. The difference is in the set of operations we define
for both measures. These sets are quite different, if we consider Pos
vs. Pr. Using some tricks (as Dubois & Prade do) one can build an
artificial construction where Pos = Pr. So what?

>The important thing to me about my argument is that it leads to the
>conclusion that one cannot use min/max logic to combine A and NOT
>A, and one must use min/max logic when combining A and A. This 
>restores excluded middle and non-contradiction, and the resulting
>logic obeys ALL laws of classical logic, while incidentally blowing up
>Elkan's proof that min/max logic is only valid for crisp logic.

If you do so, then the truth values are no more possibilities, but
some other measure, say W(). That is no problem. The problem is that
to evaluate W(A&B) you have to know the history of A and B, because
the outcome would depend on whether B is NOT A. This would be the same
swamp the probability theory strayed in. But OK, let's put this
question aside.

So you are starting with the requirements:

1. for all A: W (A & NOT A) = 0
2. for all A: W (A & A) = W (A)

This is the boundary conditions. But the crux is what would be W for
all intermediate cases?

>Note: no-one has yet found anything wrong with my math.

Consider this:

Let A = NOT A, then according to 1. and 2.:

0 = W (A & NOT A) = W (A & A) = W (A)

Now you have a problem with the definition of NOT. Clearly you have to
discard 1-x, because then all half-true sets will be impossible. Thus
it has to be something else. Probably you will find a solution for
this, let's see.

---------
BTW, philosophically, I have a doubt on Pos(A & NOT A)=0. If A is not
crisp, and thus uncertain, then A & NOT A should be also uncertain.
Then how can we be sure whether it is certainly false? In my view, NOT
A is not an inversion of A. It is a pseudo-inversion, i.e. the most
close approximation to a *non-existing* ideal ~A, which will satisfy
Pos(A & ~A)=0. We have such situation all over mathematics, where may
objects have no or to many inverses. It is not a catastrophe, so I
take it easy. (:-))

---
Regards,
Dmitry Kazakov
www.dmitry-kazakov.de