# Re: Thomas' Fuzziness and Probability

From: Andrzej Pownuk (pownuk@zeus.polsl.gliwice.pl)
Date: Fri Aug 24 2001 - 14:32:22 MET DST

• Next message: Wise: "Re Fuzzy and Datamining"

>I see your difficulty. You think that if A is a fuzzy term, and its
>membership function is denoted simply by a, let's say, then the
>one-minus rule of negation gives the membership function of NOT A as
>1-a. Hence the "middle" is included, so to speak, and LEM and LC
>should fail, as indeed it obviously does if the min-max rules are then
>applied. For we have A AND NOT A being modeled in the meta-language as
>min(a,1-a), which gives us the well-known middle with a peak at 0.5
>(assuming of course that a has its max at 1, its min at 0, and there

>Now let's try another rule of conjunction, in particular the
>Lukasiewicz bounded-sum rule, for which we have for two membership
>functions a and b, and their corresponding terms A and B,

> mu[A AND B] = a AND b = max(0, a+b-1).

>In the particular case where B is NOT A, and b=1-a, we have under this
>rule

> a AND b = max(0,a+1-a-1) = 0 everywhere

>and in accordance with the law of contradiction, the term A AND NOT A
>is rendered as the comstant absurdity whose membership value is
>everywhere 0. LC is upheld.

I am employed at the Silesian Technical University as a university teacher.
In my work I have to very often answer to the following question.

"Does John know topic x?"
or
"What is the relation between topic x and Mr John's knowledge?"

Sometimes it is very difficult to answer this question.
In order to answer to this question I use number between 2 and 5.

If John know topic x, then I use number 5.
If John don't know topic x, I use number 2.
If I am not sure that John know topic x, I use number between 2 and 5.

I think that this is a definition of fuzzy set.

For example.

John belong to the set of people which know topic x with degree 4=
= John get 4 at the class test.

Let's us consider the following situation?

John get 3 at the class test. ( m(John | Topic x)=3)
Marry get 4 at the class test. ( m(Marry | Topic x)=4)

Do John and Mary know topic x?

What is the answer to this question?

a) m(John and Marry | Topic x)=min{3, 4}=3
b) m(John and Marry | Topic x)=(3+4)/2 (I think that this is quite good
solution.)
c) m(John and Marry | Topic x)=max(2, 3+4-5)= 2 (I think this is cruel.)

Andrzej Pownuk

P.S.
I belong to the set of people
who know English language
with degree of membership 3=.
I apologise for that.

------------------------------------------------
MSc. Andrzej Pownuk
Chair of Theoretical Mechanics
Silesian University of Technology
E-mail: pownuk@zeus.polsl.gliwice.pl
URL: http://zeus.polsl.gliwice.pl/~pownuk
------------------------------------------------

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