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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Re: Fuzziness and Fundamental Measurement (Was Re: Modeling  Was: Re: Fuzzy theory or probability theory?)
Message-ID: <1994Dec8.180032.3719@decan.gate.net>
Organization: Decision Analytics, Inc.
Date: Thu, 8 Dec 1994 18:00:32 GMT
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Herman Rubin (hrubin@b.stat.purdue.edu) wrote:
: In article <1994Dec8.043637.3916@decan.gate.net>,
: S. F. Thomas <sthomas@decan.gate.net> wrote:
: >Herman Rubin (hrubin@b.stat.purdue.edu) wrote:
: >: In article <1994Dec6.173831.11033@decan.gate.net>,
: >: S. F. Thomas <sthomas@decan.gate.net> wrote:
: >: >mackw@bytex.com wrote:
: >
: >: >: In article <94Nov29.133132edt.774@neuron.ai.toronto.edu>, 
: >: >: <radford@cs.toronto.edu> writes:

: 	[Lots of philosophical discussion deleted.]

: Much of this is an attempt to describe the inconsistent manner
: in which people actually use words, and jump to conclusions.

: Probability has given us a formal system in which we can put
: things in a consistent manner, and given all the information,
: evaluate courses of action.  Statistical decision theory 
: considers the same problem if the probabilities are unknown,
: and gets similar results for self-consistent action.

: What the fuzzicists need to do to attempt to convince me that
: they have more than philosophy is to put down their model in
: such a way that at least an infinitely large, infinitely fast,
: costless computer can come up with a clear procedure.  Then
: we can worry about making practical approximations.

I don't think I have said anything that disagrees with that...

: In probability, one has to consider not only probabilities
: of events and distributions of random variables, but JOINT
: probabilities.  The fuzzicists need to do this as well, and
: to do it in such a manner that it agrees with a careful
: principled approach.  We have the betting odds approach to
: probability, which provides an argument that the model is
: what is needed.  We have other approaches.  

: Let the fuzzicists produce such, and provide arguments to
: contradict the various probability approaches.  
  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Well, after all the philosophizing, and the formal axiomatization
advanced in _Fuzziness_and_Probability_ the conclusion reached is
that (from the Conclusion): 

   " ... One of the issues which has been of concern
   in the theory has been the relation between, and the line of
   demarcation that should separate, the respective applications of
   fuzzy and probabilistic calculi.  The present development puts
   forward an interpretive framework that mixes the two calculi
   quite intimately. The somewhat paradoxical result is that the
   differences between the two are made very clear, as also are the
   respective limits of application of the two concepts. Probability
   remains the more basic of the two, and stands in the same
   relation to the other, as it already does to the more familiar
   concept of likelihood."

Mr. Rubin, like the "fuzzicists" he decries, seems to _want_ fuzziness
and probability to be mutually contradictory concepts.  They are not.
In my investigations, they turn out to be mutually supporting.

: They have not, except to say that "people do not behave that way".

Yes, there is an unfortunate human tendency to filter out unfortunate
facts which undermine a belief system, and sometimes to react with
hostility rather than humility when these unfortunate facts are 
pointed out.  Much of that has gone on with the defense of fuzzy-theoretic
fundamentals, which, at least in the "accepted" version, remains
inadequate.  A similar attitude prevails among (some) statisticians
of the Bayesian persuasion, who use subjectivity to rationalize
the indefensible, much like (some) fuzzicists, who use fuzziness to
rationalize the indefensible.  

: With this I agree; one of the important things which we
: have found is that our intuition needs to be checked by
: the hard rules of mathematics and logic.

Quite!  It is well to remember this especially when our own ox is 
the one being gored! :-)

: Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399

Cheers!
S.F.Thomas
