# Re: Thomas' Fuzziness and Probability

From: S. F. Thomas (sfrthomas@yahoo.com)
Date: Sat Aug 11 2001 - 12:50:09 MET DST

• Next message: Herman Rubin: "Re: Is there anything wrong with fuzzy inference?"

"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.
> 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
> > 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

############################################################################
This message was posted through the fuzzy mailing list.
(1) To subscribe to this mailing list, send a message body of
"SUB FUZZY-MAIL myFirstName mySurname" to listproc@dbai.tuwien.ac.at
(2) To unsubscribe from this mailing list, send a message body of
"UNSUB FUZZY-MAIL" or "UNSUB FUZZY-MAIL yoursubscription@email.address.com"
to listproc@dbai.tuwien.ac.at
(3) To reach the human who maintains the list, send mail to
fuzzy-owner@dbai.tuwien.ac.at
(4) WWW access and other information on Fuzzy Sets and Logic see
http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info
(5) WWW archive: http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html

This archive was generated by hypermail 2b30 : Sat Aug 11 2001 - 13:37:47 MET DST