Re: Thomas' Fuzziness and Probability

From: Andrzej Pownuk (
Date: Tue Aug 21 2001 - 07:14:20 MET DST

  • Next message: Herman Rubin: "Re: Thomas' Fuzziness and Probability"

    >>I am employed at the Silesian Technical University as a university
    >>In my work I have to very often answer to the following question.
    >>"Does John know topic x?"
    >>"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.

    Fuzzy set is a function from a set X into the interval [0, 1].
    More general definition is the following.
    Fuzzy set is a function from a set X into the set L with linear order.
    I think that the interval [2, 5] is a set with linear order "<=".

    We can also transform the interval [2, 5] into the interval [0, 1].

    We can also define degree in the following way.

    m(John | Topic x) = (number of John's correct answer in the test)/(number of

    I know that this model is not perfect.
    In practice we have to apply
    more complicated definition.

    >>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
    >>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.

    Maybe I am wrong,
    but I thing that in simply cases this problem
    is not connected with probability.

    Let us consider that topic x can be fully
    described using 4 question.
    For example students have to
    learn 4 different definitions.
    (Which are very simply to understand.)

    In the next lesson I can ask John.

    Do you know the fist definition? (Let us assume that the answer is YES.)
    Do you know the second definition? (Let us assume that the answer is NO.)
    Do you know the third definition? (Let us assume that the answer is YES.)
    Do you know the forth definition? (Let us assume that the answer is NO.)

    After that I know that John know the topic x in 50%.

    m(John | Topic x)=2/4=0.5

    I know that with probability 1.
    I can repeat this test 1000 times and I get the same result.
    (I know that this method is not perfect,
    but I think that it is good enough.)

    Because of that I think that the definition of the number

    Number m(John | Topic x) is an objective measure of John's knowledge of the
    topic x.
    (This measure is non-probabilistic.)

    >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;

    I agree, but in simply cases it is very good measure.

    >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.

    I agree that this is very difficult job.
    What do you think about the following question?

    What is the probability that John know
    that the capital city of Poland is Warsaw?

    I think that this fact can be check with probability 1
    after 1 experiment.

    >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

    I fully agree.

    Let us consider that these are the results of John and Mary test.

         John Mary John AND Mary John OR Mary
    q1 - 1 - 1 1 1
    q2 - 0 - 1 0 1
    q3 - 1 - 0 0 1
    q4 - 0 - 1 0 1

    We can see that

    m( John | Topic x)=0.5
    m( Mary | Topic x)=0.75

    m(John and Mary | Topic x)=0.25
    m(John and Mary | Topic x)=1

    I think that the number m(John and Mary | Topic x)
    is not only a function of the numbers
    m( John | Topic x), m( Mary | Topic x).

    m(John and Mary | Topic x) <> f( m( John | Topic x) , m( Mary | Topic x) )

    I asked my question
    because fuzzy logic always use function in the following form.

    m(John and Mary | Topic x) = f1( m( John | Topic x) , m( Mary | Topic x) )
    m(John or Mary | Topic x) = f2( m( John | Topic x) , m( Mary | Topic x) )

    I don't understated haw it is possible.

    I think that we cannot calculate the number m(John and Mary | Topic x)
    using only the numbers m( John | Topic x) and m( Mary | Topic x).

    I think that some types of uncertainty cannot be described using probability
    In this mail I presented one of such examples.

    The relation between word "TALL" and people's height is also
    This relation is uncertain but non-random.
    (In this case I suspect that
    we have to apply both random and non-random
    measure of uncertainty, but I am not sure.)

    The relation between the term "full glass" and the amount of water in the
    glass is the next example of non-probabilistic types of uncertainty.

    I think that we can measure "knowledge of John"
    using some kind of non-probabilistic measure.
    I don't understated why we can apply t-norm or s-norm
    without additional information
    about properties of physical phenomena.
    Why we cannot apply more complicated functions?

               Andrzej Pownuk

    MSc. Andrzej Pownuk
    Chair of Theoretical Mechanics
    Silesian University of Technology

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