Re: Thomas' Fuzziness and Probability

From: Robert Dodier (robertd@athenesoft.com)
Date: Wed Aug 22 2001 - 07:42:13 MET DST

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    I (Robert Dodier) wrote:

    > > Consider a less-extreme example, then: let A = "the mayor is tall",
    > > B = "the mayor is heavy", and B' = "the mayor is well-dressed". For the
    > > sake of argument suppose that truth(B)=truth(B'). Despite the fact that
    > > we know that there is some relation between height and weight, the
    > > truth value assigned to a compound containing A and B is just the same
    > > as what we get by putting B' in the place of B.

    Stephan.Lehmke@cs.uni-dortmund.de (Stephan Lehmke) wrote:

    > Again, i must ask whether the situation would change if two-valued
    > logic were employed?

    I'm assuming that these propositions are not definitely true nor false,
    so two-valued logic is not applicable here. Let's stick to the topic,
    shall we?

    > [...] Fuzzy logic is about vagueness, not telepathy (whatever I
    > know has to be reflected by the logical system, whether I care to
    > write it down or not).
    >
    > Btw, I doubt that the fact that you know something is reflected by
    > probabilistic logic, without your formalizing it in any way.

    True enough, but probabilistic logic has the advantage that the
    list of probabilities needed to answer a particular question can
    determined automatically. This, I believe, is extremely important
    in constructing systems for automated inference.

    For example, suppose I ask a Bayesian robot about p(A B). The robot
    says I need to tell it either p(A) and p(B|A) or p(B) and p(A|B).
    The relation between A and B doesn't need to be formalized to get this far.

    The situation is more interesting in problems with many variables,
    some of which have stated values, and the goal is to compute a
    distribution over one or more variables. In this case, it is non-trivial
    to determine which relationships come into play. However, the laws
    of probability always allow one to determine which formal relations
    are needed. Once these relations are stated, the computations can begin.

    > > There is a two-fold drawback of defining truth value of a compound
    > > strictly as a function of truth values of its parts. (i) You cannot
    > > exploit information about the relation between A and B; even if you
    > > know what it is, there is simply no place to put it in the computation
    > > of the truth value of a compound proposition. (ii) The rules for computing
    > > truth value of the compound don't tell you when you need to supply
    > > some information about the relation between the parts.
    >
    > I think we're talking cross purposes here. Logic is not about
    > computing truth values, but about drawing conclusions from assertions.

    Well, is it not the case that drawing conclusions, in fuzzy logic,
    is accomplished by computing truth values?
     
    > Of course, it is perfectly possible to state the relations between A
    > and B it the form of axioms, as I have pointed out before.

    No, this won't help. According to the usual rules of inference in
    fuzzy logic, the truth value of a compound is a function of the truth
    values of the parts alone, and therefore there is no place to enter
    information about relations between the parts.

    You can state all the axioms you want; there is no way to enter these
    into the usual truth-functional rules. Perhaps you have in mind some
    alternate rules -- if so, how do you know your rules take precedence?
    More to the point, how would an automated system know which rules
    take precedence?

    Regards,
    Robert Dodier

    --
    ``Socrates used to meditate all day in the snow, but Descartes'
    mind worked only when he was warm.'' -- Bertrand Russell
    

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