Re: Thomas' Fuzziness and Probability

From: Andrzej Pownuk (pownuk@zeus.polsl.gliwice.pl)
Date: Tue Aug 21 2001 - 07:50:12 MET DST

  • Next message: Keller, Paul E: "Applications and Science of Computational Intelligence V (or15)"

    > >> Any such definition must ignore the relation between elements in a
    > >> compound: if truth(B')=truth(B), then in any proposition containing
    > >> A and B, I can swap in B' in place of B, and get exactly the same
    > >> truth value for the compound; whether the elements are redundant,
    > >> contradictory, or completely unrelated doesn't enter the calculation.
    > >
    > >>
    > This is only true if we have truth-functional logic. I know that
    > truth-functionality has been accepted as a given in the fuzzy-math
    > literature. However, we pay a terrible price for truth functionality. That
    > price is the loss of excluded middle and contradiction, which underlies
    > Elkan's paper of some years ago. The hard fact is that in the real world,
    > this loss often makes no sense.
    >
    > Consider, for example, Earl's fuzzy set in which Short had membership
    0.15,
    > Medium had membership 0.85, and Tall had membership 0.65 (or something
    like
    > that). Common sense tells us that Short OR Medium OR Tall should be 1,
    > instead of .85, and that Short AND Medum AND Tall should probably be zero
    > instead of 0.15.
    >
    > This is easily made reasonable if we see that Short, Medium and Tall
    areall
    > negatively associated. In my system, if A and B are not semantically
    > inconsistent, we can use any multivalued logic we please including
    Zadeh's;
    > but if A and B are semantically inconsistent, we MUST use A OR B = min(1,
    a +
    > b), and A AND B = max(0, 1 - (a + b). Applying this to Earl's example,
    Short
    > OR Medium OR Tall = 1, and Short AND Medium AND Tall = 0.
    >
    > This logic may not be considered truth functional, I suppose; we have to
    > parse the (complex) proposition to see what logic we should use. But the
    > results give us a multivalued logic which makes sense both mathematically
    and
    > to the layman.
    >
    > In your example, suppose that the proposiition we wish to evaluate is A
    AND
    > B. You state that if truth(B') = truth(B), that A AND B' has the same
    truth
    > value as A AND B. Sounds reasonable, and seems to be in accordance with
    the
    > rule of substitution. (let lower case letters be truth values.) We take
    the
    > Zadeh logic as a default. Suppose that B and A are semantically unrelated,
    > and that the truth value of A, B and B' are all 0.5. Then A AND B = A AND
    B'
    > = 0.5.
    >
    > Now suppose that A is actually B, and that B' is actually NOT B. Let
    truth(A)
    > =0.5, truth(B) is 0.5 and truth(B') = 0.5 With your method truth(B AND NOT
    B)
    > is 0.5, as is the truth(B OR NOT B). If, however, we parse these
    expressions
    > and use the appropriate logic operators, we get truth(B AND NOT B) = 0,
    and
    > truth(B OR NOT B) = 1, which makes perfect sense to a biologist like
    myself.
    >
    > If we parse Elkan's two expressions and use the appropriate logic above
    with
    > any multivalued logic as the default, we find that the two expressions are
    > perfectly equivalent.
    >
    > Conclusion: we better be careful how we apply the classical logic rules by
    > which we derive one logical proposition from another when using
    multivalued
    > logics.
    >
    > William Siler
    >

    Let us consider the set of people
    who know the answers in some test.

    I think that this is the example of fuzzy set
    with membership function
    which is equal to the score.

    The class-test score can be measured using the following function.

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

    I think that THIS IS NOT PROBABILITY.
    (John know answer or not.
    This is objective fact that can be measured.
    In each test we get the same result.)

    Let us consider that the test contain four questions.

    TEST={question 1, question 2, question 3, question 4}

    Now we assume that the result of John's test is the following.

    John
    q1 - know (1)
    q2 - don't know (0)
    q3 - know (1)
    q4 - don't know (0)

    m(John | TEST)=0.5

    We can do the same with Michael.

    Michael

    q1 - know (1)
    q2 - know (1)
    q3 - don't know (0)
    q4 - know (1)

    m(Michael | TEST)=0.75

    Now we can ask the following question.
    What are the results of John and Michael.

         John Michael John AND Michael John OR Michael
    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 AND Michael | TEST)=0.25
    m(John OR Michael | TEST)=1

    Now we can compare other persons

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

    In this case

    m(John | TEST)=0.5
    m(Harry | TEST)=0.75

    m( John AND Harry | TEST)=0.5
    m( John OR Harry | TEST)=0.75

    We can see that the result of logical operations "AND" and "OR"
    depend on the data and can not be assumed and constant

    but

    m(John AND (NOT John) | TEST) =0
    m(John OR (NOT John)| TEST) =1

    Unfortunately I don't know if this example is "semantically inconsistent".

               Andrzej Pownuk

    P.S.
    I am not going to criticise fuzzy set theory.
    I would like to understand
    what fuzzy logic really is.

    ------------------------------------------------
    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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