Dear Konrad,
> > The uncertainty is in the definition of the sets A and B.
>
> Agreed. Why choose an ambiguous definition? Or: why are you asking for
> unique answers to questions which you admit are ambiguous?
>
Due to imprecision of our natural language
in real life we deal with more or less
ambiguous questions very often.
Let us consider the set of passwords
and let's assume that
John know 0% of passwords,
Robert know 50% of passwords,
and Michael 100% of passwords.
In such circumstances the question
"Does Robert know the passwords?"
is ambiguous.
It is not convenient to answer
"yes" or "no" to that question.
However in real life we can answer
to that question let's say indirectly.
We can say "Robert knows 50% of passwords".
I think that this answer is very precise.
The problem is
whether this is really the answer to the question.
Of course in traditional logic
(or in very rigorous way of thinking)
such answer is unacceptable.
Robert knows the passwords or doesn't know.
There is no other possibility.
However if we extend the "space"
of possible answers then
we can answer to that question very precisely
("Robert knows 50% of passwords" this is quite precise answer, isn't
it?).
Maybe I am wrong but I see some analogy to complex number theory. How to
calculate the solution of the following equation?
x^2+1=0
Of course it doesn't exist in the space of real numbers. However if we
extend the space of possible answers
then we can calculate the answer very precisely
(this is of course x1=I,x2=-I).
Presented way of thinking is rather "non-standard". However the
answers "Robert know 50% of passwords" or something similar are very
common in the natural language.
As we can see in real life there are some ways of giving the answer to
the ambiguous questions. However in order to do that we have to
extend a little the space of possible answers.
Example 2
Let us consider the car after very serious accident.
Is that a car?
Well, both answers "yes" or "no"
may not be very appropriate.
However we can say for example:
"this is a car but without the wheels, pane, damaged engine etc." (or
some more precise description).
By the way,
in this example we need
a lot of parameters in order to describe what is going on.
One parameter (or degree of membership, degree of truth or whatever ...)
is not precise enough.
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Is that multivalued logic?
I don't know.
This is simple a real life.
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Classical probability theory is based on "yes" or "no" answers
because of that I doubt that it is possible
to describe this kind of problem by using probability.
This is simple beyond of the definition of probability
(however I am not sure that).
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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