**Previous message:**Disgusted: "Earl Cox - fix the bugs in The Fuzzy Sytems Handbook 2nd edition"**Maybe in reply to:**Joe Pfeiffer: "Thomas' Fuzziness and Probability"**Next in thread:**S. F. Thomas: "Re: Thomas' Fuzziness and Probability"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

"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
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*> account?
*

*> > > >
*

*> > > > [...] Under the likelihood calculus, the same is possible, but the
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*> > > > fact that likelihood is a point function, not a set function,
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*> > > > renders the general rule for change of variable different --
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*> > > > easier in fact -- from what it is under the probability calculus.
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*> > > > As to issues of risk and action, the notion of expected loss
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*> > > > consequent upon any given action, is rendered as a possibility
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*> > > > (or likelihood) distribution, or in effect a fuzzy set.
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*> > >
*

*> > > Suppose, then, that I have a possibility or likelihood for two
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*> > > different actions. Can I say that one action is preferable to
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*> > > the other? If so, how do I determine which is more preferable?
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*> >
*

*> > Sometimes it is very clear which is preferable, sometimes less so. If
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*> > you are minimizing loss then the smaller the centroid of the
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*> > possibility set, the more preferred; however, the centroid is an
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*> > insufficient measure of preference, for the largeness of spread also
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*> > comes into the picture, with smaller spread (i.e lesser fuzziness)
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*> > being in general preferable to larger spread. In a single-criterion,
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*> > single decision-maker (DM) problem, this can go to the heart of the
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*> > insufficiency of the Bayesian paradigm, and indeed explain why a
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*> > (rational) decision-maker may choose neither to take, nor place, the
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*> > Bayesian bets. Suppose for two actions, the corresponding possibility
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*> > sets on expected utility have the same centroid, but one has greater
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*> > fuzziness than the other, then the rational thing to do is to opt for
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*> > the action with lesser fuzziness, no? Contemplating a Bayesian
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*> > betting scenario, a decision-maker always has as an option the
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*> > certainty of status-quo, i.e to neither take the bet nor place the
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*> > bet, either of which options would presumably carry some residual
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*> > possibilistic uncertainty deriving precisely from the modeling
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*> > uncertainty which is sought to be illuminated by the Bayesian
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*> > analysis. That is the single-criterion, single DM problem. In the
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*> > more usual case, the optimal action in any given situation must be
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*> > evaluated on more than one criterion. And quite often in practical
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*> > decision-making, we have multiple decision-makers, and various ways
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*> > of attempting to resolve differences among them. The utility calculus
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*> > will take one nowhere fast in attempting to address these questions.
*

*> > With the likelihood/possibility calculus, it is in fact possible to
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*> > address these questions, as inter-personal comparisons, both of
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*> > belief and of preference, are far easier to address within such a
*

*> > framework. _Fuzziness and Probability_, in the part of it that
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*> > elaborates an approach to decision analysis under uncertainty,
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*> > attempts to do just that.
*

*> >
*

*> >
*

*> > > > > > [...] But probability (over sample space) gives rise to likelihood
*

*> (over
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*> > > > > > parameter space) and the calculi required to manipulate the two
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*> are
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*> > > > > > different.
*

*> > > > >
*

*> > > > > (i) This betrays a very limited view of what a model can be:
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*> apparently
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*> > > > > there are but samples and parameters. Many interesting models are
*

*> not so
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*> > > > > 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
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*> > > sampling distributions for observable variables and there are
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*> > > parameters that govern those distributions. Some models are that
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*> > > simple, yes. There are many models which don't fit into this neat
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*> > > division of labor. Does every class of models require its own
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*> > > reasoning calculus?
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*> >
*

*> > At the very least, there is deductive reasoning, and
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*> > associated logical calculus, and there is inductive reasoning, and
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*> > associated logical calculus. These two suffice in my view to carry
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*> > the burden of any discourse concerning any object phenomenon, or
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*> > system of
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*> > inter-acting phenomena, that may be of interest. However, there is a
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*> > third
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*> > kind of reasoning which must remain outside either of these. It is
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*> > the reasoning that derives from *insight* and which leads us in a
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*> > very mysterious fashion to posit the intension models and associated
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*> > premises/hypotheses that may then be subject to various kinds of
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*> > deductive and inductive massaging. There is also a fourth, which is
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*> > what we are here engaged in, which is a kind of meta-reasoning.
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*> >
*

*> > > > (ii) The likelihood calculus which you state above looks
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*> > > > > suspiciously similar to a rule derived from laws of probability.
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*> > > >
*

*> > > > It *is* derived from the laws of probability. You must have missed
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*> > > > large parts of the thread while feigning sleep.
*

*> > >
*

*> > > Well, I have no problem with deriving fuzzy reasoning from probability,
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*> > > but I thought that was precisely what you were arguing against.
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*> >
*

*> > Goodness, no. What I do argue however is that the semantics of
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*> > likelihood do not just fall neatly out from the semantics of
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*> > probability. Probability provides some of the underpinning, but not
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*> > all. Otherwise Fisher would not have been led up a blind alley by
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*> > asserting that the "likelihood of a or b is like the income of Peter
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*> > or Paul, we don't know what it is until we know which is meant." This
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*> > leads to a likelihood calculus in which set evaluation is of the form
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*> >
*

*> > L( {a,b} ) = L(a OR b) = Max( L(a), L(b) )
*

*> >
*

*> > which rather quickly proves to be inadequate. Had it not been
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*> > inadequate, I don't think classical statistics would have gone to all
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*> > the trouble it has to develop indirect methods of describing the
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*> > uncertainty in model parameters consequent upon sampling. Nor would
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*> > there have been a neo-Bayesian revival intended to supplant the
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*> > classicists precisely by offering a method of *direct*
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*> > characterization. Indeed, Bayes offers a likelihood calculus in which
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*> >
*

*> > L(a OR b) ~ (L(a) + L(b))
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*> >
*

*> > where ~ is to indicate that some normalization, appropriate to the
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*> > construction of likelihood as a metaphorical (belief) probability, is
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*> > necessary. It is only with the fuzzy set theory that semantics
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*> > suggests itself
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*> >
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*> > L(a OR b) = L(a explains the data OR b explains the data)
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*> >
*

*> > where "explains the data" is a fuzzy predicate no different in
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*> > principle from "is tall", and subject to calibration in conceptually
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*> > the same way. This leads, albeit with some reworking of the Zadehian
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*> > fuzzy set theory along the way, to
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*> >
*

*> > 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
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*> > very simple way, but it is the fuzzy set semantics, and the device of
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*> > the calibrational proposition, that provides the essential frame that
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*> > 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
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*> > prior to "crisp". This misses an essential point in my view. And that
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*> > is that a bivalent logic is perfectly capable of generating
*

*> > ever-higher levels of fuzziness, in exactly the same way that you can
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*> > run fuzzy models on binary computers. The ultimate bivalence of the
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*> > computer does not disable it when it comes to elaborating computable
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*> > 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
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*> > point of a bivalent meta-language, it is possible to see, at the level
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*> > of the object language, a class of crisp terms which are very clearly
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*> > a special case of fuzzy. But that is in the object language. And that
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*> > observation does not render fuzzy prior to crisp; rather it is the
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*> > crispness of the meta-language that permits us to bootstrap our way to
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*> > the higher reaches of fuzziness in successions of object languages. I
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*> > make this point because probability stands in a similar relation to
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*> > fuzzy, and to likelihood. Just because we use probability to generate
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*> > 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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**Next message:**Herman Rubin: "Re: Is there anything wrong with fuzzy inference?"**Previous message:**Disgusted: "Earl Cox - fix the bugs in The Fuzzy Sytems Handbook 2nd edition"**Maybe in reply to:**Joe Pfeiffer: "Thomas' Fuzziness and Probability"**Next in thread:**S. F. Thomas: "Re: Thomas' Fuzziness and Probability"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

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