Re: Thomas' Fuzziness and Probability

From: Andrzej Pownuk (pownuk@zeus.polsl.gliwice.pl)
Date: Thu Aug 23 2001 - 07:55:26 MET DST

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

    > > Btw, I doubt that the fact that you know something is reflected by
    > > probabilistic logic, without your formalizing it in any way.
    >
    > That is =EXACTLY= what Bayesian probability theory is all about !!!
    > In Bayesian probability theory, _ALL_ probabilities are conditional
    > on the knowledge base one is willing to apply to the problem at hand.
    > The conditional probability P(A|{B}) in Bayesian theory is the degree
    > of confidence one has in the truth of Boolean proposition 'A', given
    > that the set of Boolean propositions {B} (the knowledge base one is
    > willing to apply to the problem) is assumed to be true. The Laws of
    > probability and Bayes theorem provide all the tools one needs to reason
    > about such `uncertain' Boolean propositions, as well as to incorporate
    > new data into one's knowledge base. See G. Larry Bretthorst's paper
    > ``An Introduction To Model Selection Using Probability Theory As Logic''
    > <http://bayes.wustl.edu/glb/model.ps.gz>, or the draft of E. T. Jaynes'
    > magnum opus ``Probability Theory: The Logic of Science''
    > <http://bayes.wustl.edu/etj/prob.html>.

    Probability theory is related
    with the question
    "how often something happened".
    When in each experiment we get the same results,
    then this problem is not related with theory of probability.
    We know the answers with probability one.

    When I see that knowledge
    of my students is related with probability,
    (For the same question I got
    different response.)
    then I doubt about their knowledge.

    Let us consider segment on the plane,
    which is divided into two parts

      T /_____ segment
      T \
      T
      T Top part of the plane
      T
    XXXXXXXXXXXXXXXXXX
      B
      B
      B Bottom part of the plane
      B
      B

    Now we can ask the following question.

    Is the segment at the bottom part of the plane?

    The answers "NO" or "YES" are not perfect in this case.

    We can build a non-probabilistic measure
    which describe this problem.

    m(segment | bottom)= Length(part B)/Length(part B + part T)

    In this case

    m( segments | bottom)=5/10=0.5

    I think that this problem
    is not related with probability.
    (In each experiment we get the same result.)
    Because of that in this case
    we can't apply probabilistic logic.

    I think that in real word
    exist uncertain problems
    which can't be described
    by probability theory.

            Andrzej Pownuk

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