Re: Thomas' Fuzziness and Probability

From: Earl Cox (earldcox1@home.com)
Date: Wed Aug 15 2001 - 12:53:13 MET DST

  • Next message: P. Sarma: "1Re: Fuzzy Practical Uses"

    Since the propositions you specify are either true or false, then, indeed,
    if you apply truth[1] or truth[0] to the compound propositions they will be
    similar if not equivalent. But in fuzzy logic we are discussing the degree
    to which something is a member of a class (that is, the semantics of the
    proposition). Now if we have fuzzy propositions,

    Bill is Tall
    Bill is Short
    Bill is Medium

    then if Bill is Tall to a degree [.12], is Short to a degree [.88] and
    Medium to a degree [.65] then the proposition AND or OR propositions will
    yield dissimilar truth functions. Because fuzzy logic does not obey the Law
    of the Excluded Middle, someone can be both Tall and Short and Medium at the
    same time -- just with different degrees of truth. This property of fuzzy
    set theory is important in fuzzy reasoning where fuzzy conditional and
    unconditional propositions (often expressed as if-then rules) are run in
    parallel and accumulate evidence. Thus,

    if height is TALL then weight is Heavy
    if height is Short then weight is Light
    if height is medium then weight is Moderate

    form an output fuzzy set (weight) based on the fusion of the consequent
    fuzzy sets (Heavy, Light, Medium) to the degree that the predicate of the
    proposition (or rule) is true. Fuzzy systems are highly sensitive to
    compound truth statements since (a) there is an often non-linear translation
    function between the antecedent and consequent of the proposition involving
    three distinct fuzzy spaces, (b) the rules are run in parallel and
    accumulate evidence that effectively AND's or OR's the propositions
    (depending on your choice of inference mechanism) and (c) the outcome of the
    fuzzy system reflects the amount of evidence in the supporting antecedents.

    You eye color logical metaphor might have been cast as,

    Consider the mayor of Ashtabula. Let A = "mayor's right eye is wide".
    Let B = "mayor's left eye is narrow". Let B' = "mayor's left eye is wide".
    What do you suppose is the truth value of A B ? What about A B' ?

    You should ask yourself, what is the truth value of B B'? These are mutually
    contradictory in Boolean logic, but simply represent a logical analysis of
    the degree to which the eye might be considered both wide and narrow (which
    is a real world state -- there is no point at which the width of an eye goes
    from narrow to wide, it is a gradual transition.)

    Anyway, reading introductions to fuzzy logic and then forming an opinion
    about the robustness of its representational power is a poor way to engage
    in serious debate.

    But you will have to take this up with other participants, I will check back
    in another two years to see if everyone is still debating the same issues.

    earl

    --
    Earl Cox
    VP, Research/Chief Scientist
    Panacya, Inc.
    134 National Business Parkway
    Annapolis Junction, MD 20701
    (410) 904-8741
    -------------------------------------------
    

    AUTHOR: "The Fuzzy Systems Handbook" (1994) "Fuzzy Logic for Business and Industry" (1995) "Beyond Humanity: CyberEvolution and Future Minds" (1996, with Greg Paul, Paleontologist/Artist) "The Fuzzy Systems Handbook, 2nd Ed." (1998) "Fuzzy Tools for Data Mining and Knowledge Discovery" (due Early Fall, 2001)

    "Robert Dodier" <robertd@athenesoft.com> wrote in message news:23af61c2.0108142014.6f210d4f@posting.google.com... > In response to the following example which I posted: > > > > Consider the mayor of Ashtabula. Let A = "mayor's right eye is blue". > > > Let B = "mayor's left eye is blue". Let B' = "mayor's left eye is brown". > > > What do you suppose is the truth value of A B ? What about A B' ? > > > > > > The difficulty is that rules of the kind applied in fuzzy logic > > > ignore relations between the elements of a compound proposition. > > "Earl Cox" <earldcox1@home.com> wrote: > > > [...] I also fail to see how your example about the eye color > > addresses anything about fuzzy logic. > > I see that I left too much implied. The truth values of B and B' in the > example above are more or less equal; I don't think you'll want to argue > that point. Yet then the truth values assigned to the compound > propositions "A and B" and "A and B'" would have to be similar also, > under the truth(A and B) = min(truth(A), truth(B)) or truth(A)*truth(B), > or any other definition of truth(A and B) which is solely a function > of truth(A) and truth(B). > > > And why do you suppose (erroneously) that fuzzy logic -- fuzzy set theory > > in particular -- ignores relationships between compound propositions?? > > Well, it's from reading introductions to fuzzy logic that give > definitions of the truth value of a compound proposition in terms of > truth values of its elements. > > 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. > > Regards, > Robert Dodier > -- > "Nature exists once only." -- Ernst Mach

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