Dear Robert,
Then the degrees at school/university are random?
Well, I always suspected that :)
I agree that in general there is some random component
in determining the degree.
However when my students give me random answers then I know that their
knowledge is not very deep.
I choose so simple example in order to show the phenomena.
A lot of phenomena have some degree.
(1) fever
If we would like to be more precise we use degree of fever.
The term fever is imprecise.
We have to use degree in order to be more precise.
I don't thing that degree of fever are random.
(2) the terms full/empty
bottle A (volume of water)*100%/(volume of bottle)=100%
bottle B (volume of water)*100%/(volume of bottle)=80%
bottle C (volume of water)*100%/(volume of bottle)=20%
bottle D (volume of water)*100%/(volume of bottle)=0%
Let us consider the set F of full of bottles.
The term full is imprecise.
Bottle A belong the set F.
Bottle D doesn't belong the set F.
What about bottle B and C?
In order to answer more precisely instead of using the word full
we can use degree of fullness (for example in percent).
These degrees extend out language and have very precise meaning.
I think that the problem is in words description of the real world.
Language contains discrete number of words.
In order to describe the continuous phenomena
we HAVE TO use some degree
in order to be precise enough.
The non-crisp sets appears in the situation in which
we have to connect the words with their meaning.
That process has nothing to do with probability.
Let us consider the car.
Then let's remove one wheel.
Is that a car or not?
Then let's remove two wheels?
Is that a car or not?
Then let's remove engine?
Is that a car or not?
The question is how many parts I have to remove in order to get
something that is not a car?
The boundary of the set of cars is not sharp.
I don't see any random phenomena here.
The problem is weakness of natural language.
That weakness force as to use some degree, levels etc.
in order to be precise enough.
Well, I also don't think that this is the fuzzy logic.
However in this case fuzzy logic description is a little better:)
Regards,
Andrzej Pownuk
>
> Well, I can't even be sure that the set A includes Robert, since
Robert
> may not even use a multiplication table to decide on each question.
(He
> could be autistic and very quickly add the numbers together each time,
> never memorizing the table). Bayesians care a lot less about
membership
> in A than in the probability that Robert will get the next question
right,
> given he answered the test with X% accuracy.
>
> Bob Welch
>
> 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
> - ---------------------------------
>
