Re: Thomas' Fuzziness and Probability

From: Andrzej Pownuk (pownuk@zeus.polsl.gliwice.pl)
Date: Fri Aug 24 2001 - 14:32:22 MET DST

  • Next message: Wise: "Re Fuzzy and Datamining"

    >I see your difficulty. You think that if A is a fuzzy term, and its
    >membership function is denoted simply by a, let's say, then the
    >one-minus rule of negation gives the membership function of NOT A as
    >1-a. Hence the "middle" is included, so to speak, and LEM and LC
    >should fail, as indeed it obviously does if the min-max rules are then
    >applied. For we have A AND NOT A being modeled in the meta-language as
    >min(a,1-a), which gives us the well-known middle with a peak at 0.5
    >(assuming of course that a has its max at 1, its min at 0, and there
    >is gradation in-between).

    >Now let's try another rule of conjunction, in particular the
    >Lukasiewicz bounded-sum rule, for which we have for two membership
    >functions a and b, and their corresponding terms A and B,

    > mu[A AND B] = a AND b = max(0, a+b-1).

    >In the particular case where B is NOT A, and b=1-a, we have under this
    >rule

    > a AND b = max(0,a+1-a-1) = 0 everywhere

    >and in accordance with the law of contradiction, the term A AND NOT A
    >is rendered as the comstant absurdity whose membership value is
    >everywhere 0. LC is upheld.

    I am employed at the Silesian Technical University as a university teacher.
    In my work I have to very often answer to the following question.

    "Does John know topic x?"
    or
    "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.

    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 good
    solution.)
    c) m(John and Marry | Topic x)=max(2, 3+4-5)= 2 (I think this is cruel.)

    What is the correct answer?

               Andrzej Pownuk

    P.S.
    I belong to the set of people
    who know English language
    with degree of membership 3=.
    I apologise for that.

    ------------------------------------------------
    MSc. Andrzej Pownuk
    Chair of Theoretical Mechanics
    Silesian University of Technology
    E-mail: pownuk@zeus.polsl.gliwice.pl
    URL: http://zeus.polsl.gliwice.pl/~pownuk
    ------------------------------------------------

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