Dear Konrad,
> > The question is,
> > what the probability is that:
> > a) John know 0% of passwords,
> > b) Robert know 50% of passwords,
> > c) Michael know 100% of passwords?
> >
> > In other words,
> > how to calculate the numbers 0%, 50%, 100% by using probability
theory?
>
> Since we've already been told that these numbers hold, the
probabilities
> are 100% - there is no uncertainty involved (and no need for
calculations
> since we've been told the answers).
> If there is no uncertainty, there is
> no need for probability theory
> (and also it becomes off-topic for this
> list).
Let's try to answer the following questions:
1) Does John know the passwords?
2) Does Robert know the passwords?
3) Does Michael know the passwords?
We can say that John doesn't know,
we can say that Michael knows
but what about Robert?
Robert knows 50% of passwords.
He doesn't know all the passwords then we can reject
that person from the set of persons who know passwords.
Well, is that correct?
Let's consider the set of 100 passwords
and assume that Robert know 99 passwords (i.e. 99%).
I think that most people agree that he belong to the set of peoples who
know passwords "better" than to the set of person who doesn't know the
passwords.
The amount of percent is a measure of that fitting to particular sets:
set A - persons who know passwords,
set B - persons who doesn't know the passwords.
Why he fit to this set better?
I don't think that this is connected with some randomness.
The uncertainty is in the definition of the sets A and B.
In other word in the natural language there is not enough word to
describe this situation precisely.
The description will be precise when we apply percent of knowledge
instead just "knowledge" or "no knowledge" (i.e. two sets).
The same problem is with the set of full bottles.
Let us consider two sets:
a) set "full bottle" - which contain the bottle which are full of water,
b) set "empty bottle" - which contain the bottle which are empty,
What to do with the bottle which contains 80% of water?
Is this bottle full or empty?
I think that it is not possible to answer precisely
to that question directly.
The reason is imprecision of the natural language.
The terms "full" and "empty"
are not precise enough in order to describe this problem.
We need some more precise "description"
(some more precise language).
If we say that this bottle is full in 80% then that description is very
precise.
We have to use this degree (or percent)
in order to describe this situation.
As we can see we have some uncertainty.
Where is the probability?
Of course we can use another description of that situation which will be
also precise. For example we can use the total amount of water in the
bottle in liters (in some situation it is precise enough).
According to my experience this kind of description is commonly used
in order to describe continuous phenomena (height, volume, temperature
etc.).
How to describe precisely continuous phenomena using one word?
> However, most of the examples you gave earlier _do_ contain
uncertainty,
> so that probability theory _is_ an appropriate tool for talking about
> them. You have been arguing that they are not random, but the Bayesian
> interpretation of probability theory, along with several other
> interpretations, view the theory as describing uncertainty, not
> randomness.
Let us consider the set of "full bottles" and "empty bottles"
(or a set of persons who know the passwords if you want).
and the following bottles:
a) 0% of water,
b) 10% of water,
c) 90% of water,
d) 100% of water,
In classical logic "full bottles"={d}, "empty bottles"={a}.
However in real life it is better to say:
bottle "a" is empty,
bottle "b" is in 90% empty,
bottle "c" is full in 90%,
bottle "d" is full in 100% or simply full.
Where is randomness?
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Dear Alex,
>> In other words,
>> how to calculate the numbers 0%, 50%, 100% by using probability
>> theory?
>
> it appears that you had something else in mind. At a risk of arguing
with
>a strawman (i.e., ascribing to you a question for which I have an
answer),
>I will suggest how we can calculate 0%, 50% and 100% using probability
>theory. If we consider our universe of passwords consisting of just
two
>passwords: A and B, then 0, 0.5 and 1 are the probabilities that John,
>Michael, and Robert, respectively know a password randomly drawn from
the
>set {A,B} of the passwords (think of two pieces of paper put in a hat:
one >with password A and the other - with password B).
>
> The question is whether this is what you had in mind.
Well, that was good.
Thank you for this e-mail.
I see that I made some mistake.
I see that it is possible
to calculate the degree by using probabilistic methods.
However, we can also calculate the value of the integral by using
probabilistic methods (Monte Carlo method)
but that not mean that there
is something random or uncertain in the integral.
You apply the same trick to my example.
The amount of knowledge about the passwords is completely crisp
(under some assumption) but
I agree (unfortunately :))
that it is possible to create
some probabilistic Monte Carlo like method in order
to calculate that amount of knowledge.
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I discussed this problem also
with some fuzzy specialist and they say:
no uncertainty = no fuzziness
Additionally, they say that
there is no way of direct description of fuzziness.
I do not agree with that.
I explained that in my examples.
Additionally, as we can see prof. Zadeh said
that in my examples I described possibility
(if I understand his words correctly).
Lack of understanding of basic problems is very common
in fuzzy community.
The main conclusions (in my opinion)
are the following:
1) There is some no-random uncertainty
in the word description of continuous phenomena
(One word cannot describe finite or infinite many stages precisely
and that fact is not connected with probability).
2) If we accept that degree at school or percent of water in the bottle
are some degree of membership
(or degree of truth or I don't know how to call that...)
then it is very easy to show that in general
t-norm cannot be apply in this theory
(due to dependency problem).
Well, fuzzy set theory is a little fuzzy ...
What to do with that?
I don't know unfortunately.
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I see that my collection of discussion
about fuzzy set theory
without any positive conclusion
(http://zeus.polsl.gliwice.pl/~pownuk/fuzzy.htm)
has a new element :)
Thank you very much again for all comments.
Regards,
Andrzej Pownuk
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Ph.D., research associate at:
Chair of Theoretical Mechanics
Faculty of Civil Engineering
Silesian University of Technology
URL: http://zeus.polsl.gliwice.pl/~pownuk
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