I have only small comment to the message
which is given below.
Let us consider the following problem.
There is a set of pupils X={John, George, Steven}
and there is a test which checks the knowledge of multiplication table.
Let us consider the following results of the test:
John - 0% (correct answers)
George - 30% (correct answers)
Steven - 60% (correct answers)
Robert - 100% (correct answers)
Now let's try to create the following sets:
A - a set of persons who knows multiplication table.
B - a set of persons who doesn't know multiplication table.
As we can see A={Robert}, B={John}.
Well but what about George and Steven?
Does George and Steven know the multiplication table or not?
It is difficult to answer to this question using "Yes" or "NO" answers.
Because of that at school teachers do not use only "Yes" and "No"
statements.
Teachers use "degree of knowledge" (i.e. degree).
This is quite natural solution and everybody knows this problem.
The sets A and B have no crisp boundary
or in other words it is more convenient
to use degree instead of "Yes" and "No" statements.
Except the situation when people
are drunk the knowledge of multiplication table is non-random
(in short period of time).
Then I have no idea how to define the sets A and B using probability
theory (and because of that also Bayesian methods).
I don't know how to call those phenomena.
Maybe this is fuzzy set maybe not.
However, if we follow this way of thinking
then it is very easy to prove that in general
t-norm cannot be apply to this kind of
"grade of membership".
Because of such problems
I don't know why t-norm can be applied to other grades of membership
(i.e. to completely subjective answer to the question
"How well x belong to the set A?".)
The final conclusion:
Non-crisp sets
are used in everyday life very often
and they are completely no-random
but I don't think that current fuzzy logic
describes the problem well.
Regards,
Andrzej Pownuk
---------------------------------
Ph.D., research associate at:
Chair of Theoretical Mechanics
Faculty of Civil Engineering
Silesian University of Technology
URL: http://zeus.polsl.gliwice.pl/~pownuk
- ---------------------------------
>
> "Lotfi A. Zadeh" wrote:
> > ... Third, is to interpret " approximately a" as a fuzzy set or,
> > equivalently, as a possibility distribution. Alternatively, the fuzzy
> > set may be interpreted as a random set or a conditional probability of
> > concept of maximum.
> > ...
> > In conclusion, of the three possible interpretations, the
> > one that is the best fit to the way in which humans form perceptions,
is
> > the fuzzy set interpretation. ...
>
> Lotfi,
>
> Long ago at a university far, far away (CMU), I did a thesis that
reported
> several medium scale experiments [O(1000) trials] on humans -- well,
> engineers
> with graduate degrees -- that bear directly on your statement. My
evidence
> showed that humans can be induced to overcome classic KST cognitive
biases
> and
> agree with Bayesian updating. My method was to generate linguistic
> explanations
> of applications of Bayes rule.
>
> If your assertion is supported by similar evidence, it may be that
> _any_ rule of combination can be made to appear to "best fit to the
> way in which humans form perceptions." Perhaps all one has to do is
> use linguistic terms.
>
> Can you provide a citation describing experiments supporting the
> conclusion you state above?
>
> Thanks.
>
> chris
>
> PS In related analysis in my thesis I found that sharp intervals worked
> just
> fine for selecting English language expressions of uncertainty. In
fact,
> for a
> certain set of terms ("certain" being one of them), there was remarkable
> structure in the intervals (e.g., bi-modal symmetry in the width of the
> intervals).
>
> --
> Christopher Elsaesser 703.883.6563 (office)
> Mail Stop W432
> The MITRE Corporation
> 7515 Colshire Drive
> McLean, Virginia 22102-7508
>
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