Re: Thomas' Fuzziness and Probability

From: S. F. Thomas (sfrthomas@yahoo.com)
Date: Sat Aug 18 2001 - 09:24:14 MET DST

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

    "Earl Cox" <earldcox1@home.com> wrote in message news:<xJDe7.13461$L9.3523724@news1.rdc1.md.home.com>...
    > I really hate to continue this. But yes, the theorems of fuzzy logic,
    > *themselves* use multi-valued logic.

    Clearly you are mistaken, as I believe I can demonstrate in few words.
    Let A be a fuzzy term that fuzzily characterizes point clusters
    belonging to some domain U, say. Then its membership function mu[A]:U
    -> [0,1] is an object of ordinary (bivalent) mathematics, and there is
    nothing fuzzy about it; nor is there anything fuzzy about about the
    theorems which require manipulation of it. Note that something fuzzy
    with respect to U, namely A, has been rendered as something non-fuzzy
    with respect to [0,1]^U. Fuzzy objects in the object language have
    non-fuzzy counterparts in the meta=language, which proceeds in the
    usual manner of quite ordinary mathematics, where the rules of a
    bivalent logic apply.

    > I am finished with this thread.

    Too bad.

    > earl

    Regards,
    S. F. Thomas

    >
    >
    > "S. F. Thomas" <sfrthomas@yahoo.com> wrote in message
    > news:66b61316.0108100550.3a699afb@posting.google.com...
    > > "Earl Cox" <earldcox1@home.com> wrote in message
    > news:<%yKc7.52091$m8.16672957@news1.rdc1.md.home.com>...
    > > > I suppose the statements:
    > > >
    > > > >The important distinction is not
    > > > > between bivalent logic and multivalent logic, but between
    > > > > meta-language and object language. A bivalent logic in the
    > > > > meta-language is perfectly adequate for the purpose of modeling the
    > > > > fuzziness in the object language.
    > > >
    > > > must make sense to someone. But any metalanguage
    > > > that can convert two-valued logic into continuous valued
    > > > logic must be, at heart, fuzzy logic (since this is exactly
    > > > what fuzzy logic, via the extension Principle, does.)
    > >
    > > Clearly you have missed the point. I can put it more simply as
    > > follows: ALL the theorems of fuzzy set theory and of fuzzy logic, or
    > > whatever flavor, are stated and proved within a framework of bivalent
    > > logic. More broadly, to fix in the mind the distinction between
    > > meta-language and object-language, a fuzzy term such as "tall" will
    > > populate the object language, but its membership function mu[TALL]
    > > belongs to the meta-language, where the bivalent rules or ordinary
    > > mathemtics applies. There is nothing fuzzy about fuzzy set theory,
    > > just as there is nothing random about probability theory.
    > >
    > > > In any case, I beg to differ in very strong terms,
    > > > the important distinction is exactly that -- between the
    > > > concepts that can be modeled with bivalent and
    > > > those that can be modeled with multivalent logic.
    > > > Obscuring the problem with lots of mumbo-jumbo
    > > > about meta-languages and object languages
    > > > contributes nothing.
    > >
    > > You've missed the point, as illustrated above. Next time you look at a
    > > theorem of fuzzy logic, as whether the theorem *itself* uses a
    > > bivalent or multivalent logic. That should clear up your evident
    > > confusion.
    > >
    > > > earl
    > >
    > > Regards,
    > > S. F. Thomas
    > >
    > > > --
    > > > 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)
    > > >
    > > >
    > > >
    > > >
    > > > "S. F. Thomas" <sfrthomas@yahoo.com> wrote in message
    > > > news:66b61316.0108091708.7d6b9958@posting.google.com...
    > > > > robert@localhost.localdomain (Robert Dodier) wrote in message
    > > > > news:<9kt895$rs$1@localhost.localdomain>...
    > > > > > In the interest of brevity, I've indulged in wanton snippage,
    > > > > > but I hope what's left yields something comprehensible.
    > > > > >
    > > > > > S. F. Thomas <sfrthomas@yahoo.com> wrote:
    > > > > >
    > > > > > > Robert Dodier wrote:
    > > > > > > > [...] OK, now this is something I haven't heard about -- how
    > does
    > the
    > > > > > > > extended likelihood calculus take loss, risk, and action into
    > account?
    > > > > > >
    > > > > > > [...] Under the likelihood calculus, the same is possible, but the
    > > > > > > fact that likelihood is a point function, not a set function,
    > > > > > > renders the general rule for change of variable different --
    > > > > > > easier in fact -- from what it is under the probability calculus.
    > > > > > > As to issues of risk and action, the notion of expected loss
    > > > > > > consequent upon any given action, is rendered as a possibility
    > > > > > > (or likelihood) distribution, or in effect a fuzzy set.
    > > > > >
    > > > > > Suppose, then, that I have a possibility or likelihood for two
    > > > > > different actions. Can I say that one action is preferable to
    > > > > > the other? If so, how do I determine which is more preferable?
    > > > >
    > > > > Sometimes it is very clear which is preferable, sometimes less so. If
    > > > > you are minimizing loss then the smaller the centroid of the
    > > > > possibility set, the more preferred; however, the centroid is an
    > > > > insufficient measure of preference, for the largeness of spread also
    > > > > comes into the picture, with smaller spread (i.e lesser fuzziness)
    > > > > being in general preferable to larger spread. In a single-criterion,
    > > > > single decision-maker (DM) problem, this can go to the heart of the
    > > > > insufficiency of the Bayesian paradigm, and indeed explain why a
    > > > > (rational) decision-maker may choose neither to take, nor place, the
    > > > > Bayesian bets. Suppose for two actions, the corresponding possibility
    > > > > sets on expected utility have the same centroid, but one has greater
    > > > > fuzziness than the other, then the rational thing to do is to opt for
    > > > > the action with lesser fuzziness, no? Contemplating a Bayesian
    > > > > betting scenario, a decision-maker always has as an option the
    > > > > certainty of status-quo, i.e to neither take the bet nor place the
    > > > > bet, either of which options would presumably carry some residual
    > > > > possibilistic uncertainty deriving precisely from the modeling
    > > > > uncertainty which is sought to be illuminated by the Bayesian
    > > > > analysis. That is the single-criterion, single DM problem. In the
    > > > > more usual case, the optimal action in any given situation must be
    > > > > evaluated on more than one criterion. And quite often in practical
    > > > > decision-making, we have multiple decision-makers, and various ways
    > > > > of attempting to resolve differences among them. The utility calculus
    > > > > will take one nowhere fast in attempting to address these questions.
    > > > > With the likelihood/possibility calculus, it is in fact possible to
    > > > > address these questions, as inter-personal comparisons, both of
    > > > > belief and of preference, are far easier to address within such a
    > > > > framework. _Fuzziness and Probability_, in the part of it that
    > > > > elaborates an approach to decision analysis under uncertainty,
    > > > > attempts to do just that.
    > > > >
    > > > >
    > > > > > > > > [...] But probability (over sample space) gives rise to
    > likelihood
    > (over
    > > > > > > > > parameter space) and the calculi required to manipulate the
    > two
    > are
    > > > > > > > > different.
    > > > > > > >
    > > > > > > > (i) This betrays a very limited view of what a model can be:
    > apparently
    > > > > > > > there are but samples and parameters. Many interesting models
    > are
    > not so
    > > > > > > > simple.
    > > > > > >
    > > > > > > What do you mean? And how does it relate to what we are
    > discussing?
    > > > > >
    > > > > > In the world of models implicit in your statement above, there are
    > > > > > sampling distributions for observable variables and there are
    > > > > > parameters that govern those distributions. Some models are that
    > > > > > simple, yes. There are many models which don't fit into this neat
    > > > > > division of labor. Does every class of models require its own
    > > > > > reasoning calculus?
    > > > >
    > > > > At the very least, there is deductive reasoning, and
    > > > > associated logical calculus, and there is inductive reasoning, and
    > > > > associated logical calculus. These two suffice in my view to carry
    > > > > the burden of any discourse concerning any object phenomenon, or
    > > > > system of
    > > > > inter-acting phenomena, that may be of interest. However, there is a
    > > > > third
    > > > > kind of reasoning which must remain outside either of these. It is
    > > > > the reasoning that derives from *insight* and which leads us in a
    > > > > very mysterious fashion to posit the intension models and associated
    > > > > premises/hypotheses that may then be subject to various kinds of
    > > > > deductive and inductive massaging. There is also a fourth, which is
    > > > > what we are here engaged in, which is a kind of meta-reasoning.
    > > > >
    > > > > > > (ii) The likelihood calculus which you state above looks
    > > > > > > > suspiciously similar to a rule derived from laws of probability.
    > > > > > >
    > > > > > > It *is* derived from the laws of probability. You must have missed
    > > > > > > large parts of the thread while feigning sleep.
    > > > > >
    > > > > > Well, I have no problem with deriving fuzzy reasoning from
    > probability,
    > > > > > but I thought that was precisely what you were arguing against.
    > > > >
    > > > > Goodness, no. What I do argue however is that the semantics of
    > > > > likelihood do not just fall neatly out from the semantics of
    > > > > probability. Probability provides some of the underpinning, but not
    > > > > all. Otherwise Fisher would not have been led up a blind alley by
    > > > > asserting that the "likelihood of a or b is like the income of Peter
    > > > > or Paul, we don't know what it is until we know which is meant." This
    > > > > leads to a likelihood calculus in which set evaluation is of the form
    > > > >
    > > > > L( {a,b} ) = L(a OR b) = Max( L(a), L(b) )
    > > > >
    > > > > which rather quickly proves to be inadequate. Had it not been
    > > > > inadequate, I don't think classical statistics would have gone to all
    > > > > the trouble it has to develop indirect methods of describing the
    > > > > uncertainty in model parameters consequent upon sampling. Nor would
    > > > > there have been a neo-Bayesian revival intended to supplant the
    > > > > classicists precisely by offering a method of *direct*
    > > > > characterization. Indeed, Bayes offers a likelihood calculus in which
    > > > >
    > > > > L(a OR b) ~ (L(a) + L(b))
    > > > >
    > > > > where ~ is to indicate that some normalization, appropriate to the
    > > > > construction of likelihood as a metaphorical (belief) probability, is
    > > > > necessary. It is only with the fuzzy set theory that semantics
    > > > > suggests itself
    > > > >
    > > > > L(a OR b) = L(a explains the data OR b explains the data)
    > > > >
    > > > > where "explains the data" is a fuzzy predicate no different in
    > > > > principle from "is tall", and subject to calibration in conceptually
    > > > > the same way. This leads, albeit with some reworking of the Zadehian
    > > > > fuzzy set theory along the way, to
    > > > >
    > > > > L(a OR b) = L(a) + L(b) - L(a)*L(b)
    > > > >
    > > > > where indeed the laws of probability are invoked, and at that in a
    > > > > very simple way, but it is the fuzzy set semantics, and the device of
    > > > > the calibrational proposition, that provides the essential frame that
    > > > > Fisher overlooked.
    > > > >
    > > > > Btw, there is a school of fuzzy which is the mirror-image of what it
    > > > > is you seem to wish to maintain. They maintain that fuzzy is logically
    > > > > prior to "crisp". This misses an essential point in my view. And that
    > > > > is that a bivalent logic is perfectly capable of generating
    > > > > ever-higher levels of fuzziness, in exactly the same way that you can
    > > > > run fuzzy models on binary computers. The ultimate bivalence of the
    > > > > computer does not disable it when it comes to elaborating computable
    > > > > fuzzy models; in fact the reverse. The important distinction is not
    > > > > between bivalent logic and multivalent logic, but between
    > > > > meta-language and object language. A bivalent logic in the
    > > > > meta-language is perfectly adequate for the purpose of modeling the
    > > > > fuzziness in the object language. The essential notions of membership
    > > > > function, and of rules of combination (fuzzy union, intersection,
    > > > > etc.) all belong to the bivalent meta-language we must recall, and all
    > > > > our theorems are cast in the bivalent meta-language. And to even
    > > > > suggest casting it in a logically primitive multivalent meta-language
    > > > > would be hopelessly confusing in my view. Instead, from the vantage
    > > > > point of a bivalent meta-language, it is possible to see, at the level
    > > > > of the object language, a class of crisp terms which are very clearly
    > > > > a special case of fuzzy. But that is in the object language. And that
    > > > > observation does not render fuzzy prior to crisp; rather it is the
    > > > > crispness of the meta-language that permits us to bootstrap our way to
    > > > > the higher reaches of fuzziness in successions of object languages. I
    > > > > make this point because probability stands in a similar relation to
    > > > > fuzzy, and to likelihood. Just because we use probability to generate
    > > > > the membership or likelihood function, it does not follow that there
    > > > > is no value-added in making the leap from the one to the other. And I
    > > > > maintain that there is an essential duality between the two, with the
    > > > > distinctness, yet connectedness, that that implies.
    > > > >
    > > > > > > [...] First you claim boredom with the discussion, and say you're
    > > > > > > going to sleep, only to re-appear, apparently wide awake and
    > engaged.
    > > > > >
    > > > > > I always feel like a million bucks after a good nap.
    > > > >
    > > > > Well, in that case, Dodier, I hope the present offering again succeeds
    > > > > in putting you gently to sleep.
    > > > >
    > > > > > Regards,
    > > > > > Robert Dodier
    > > > >
    > > > > Regards,
    > > > > S. F. Thomas

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