# Re: Thomas' Fuzziness and Probability

From: Herman Rubin (hrubin@odds.stat.purdue.edu)
Date: Tue Aug 21 2001 - 07:31:56 MET DST

• Next message: Andrzej Pownuk: "Re: Thomas' Fuzziness and Probability"

In article <9ljvvl\$1ao\$1@zeus.polsl.gliwice.pl>,
Andrzej Pownuk <pownuk@zeus.polsl.gliwice.pl> wrote:
>>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.

I am not sure what it is, but it does not look like that.

>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.)

>What is the correct answer?

What is the question? For one to do something meaningful,
one needs the probability model of the answers on the tests
given their state of knowledge. I have no idea what your
assumptions are, and I would need to know these to advise
you what to do.

As someone who has given many grades, I would not be willing
to accept that John knowing the topic x with degree 4 means
that John gets 4 on the class test; tests scores themselves
are not precise (and often not even good) measures of
knowledge, and the test design might cause the deviations
between the knowledge and scores to be dependent. In any
case, the inference question is how well John knows topic x
and how well Mary knows topic x. How well "John and Mary"
know topic x is meaningless unless they can collaborate in
making use of their knowledge, and then it is a question
which cannot be answered by information about their individual
capabilities.

> 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
>------------------------------------------------

```--
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
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