Newsgroups: comp.ai.fuzzy
Path: cantaloupe.srv.cs.cmu.edu!nntp.club.cc.cmu.edu!goldenapple.srv.cs.cmu.edu!das-news2.harvard.edu!oitnews.harvard.edu!fas-news.harvard.edu!newspump.wustl.edu!news.ecn.bgu.edu!usenet.ins.cwru.edu!gatech!csulb.edu!hammer.uoregon.edu!news-xfer.netaxs.com!news.mathworks.com!uunet!in2.uu.net!uucp3.uu.net!decan!sthomas
From: sthomas@decan.com (S. F. Thomas)
Subject: Measuring the grade of membership was Re: Defining fuzzy descriptors (was  NOT and DIFF)
X-Newsreader: TIN [version 1.2 PL2]
Organization: DECAN Corp.
Message-ID: <E7Ar4p.Dw8@decan.com>
References: <331C5756.66A77D4C@decan.com> <19970305031200.WAA09923@ladder01.news.aol.com> <33243DEF.70E6@calvanet.calvacom.fr> <E7482s.8Hx@me.utoronto.ca>
Date: Wed, 19 Mar 1997 15:31:36 GMT
Lines: 265

Taner Bilgic (taner@ie.utoronto.ca) wrote:

<snip>

: The paper is:

: Bilgi and I.B. Turksen (1997) "Measurement of Membership Functions: 
: Theoretical and Empirical Work" to appear in D. Dubois and H. Prade (Eds.) 
: Handbook of Fuzzy Sets and Systems Vol. 1, Foundations, forthcoming.

: and it is available from my web address:
: http://www.ie.utoronto.ca/EIL/profiles/taner.html
<snip>
: I would love to hear about your comments/flames/omissions etc.

I downloaded the paper to which you referred and read it.
I have the following comments:

Overall, it provides a useful survey and bibliography.  
But it seems to me more inconclusive, even provocative and 
speculative, than is customary in something intended for
a handbook.  Also, it speaks more to the fundamental notion
of the grade of membership than to the topic of this thread,
which has to do with rules for the operators AND, OR and NOT.
Perhaps that is just as well, however, because the resolution
of derivative issues often becomes difficult when fundamentals
are forgotten, and the discussion drifts away from its conceptual
moorings.  Therefore, I welcome the opportunity to address the
fundamental issues addressed in your paper.

I have difficulty with the measurement-theoretic approach, 
as you well know, which I see as being fundamentally, and fatally, flawed 
in its treatment of error of measurement.  Moreover, I see the fuzzy
set paradigm as providing the missing concept that would
rescue measurement theory, which fails because it proceeds from
within a paradigm of point measurement, when we know that point
measurement, literally to an infinite number of decimal places,
is a priori impossible in practical measurement.  I believe I 
have laid out how, proceeding from within a paradigm of fuzzy 
measurement as the general case, this essential and fatal flaw 
of measurement theory can be overcome (Thomas, 1995, p. 36 ff.).  
What that means of course is that fuzzy theory in some sense 
may be made to ride to the rescue of measurement theory, not 
the other way around.  

But on an even more fundamental level, if we ask what is it
about fuzzy descriptors in natural language that makes them 
fuzzy, I think the answer is fairly clear, and fairly simple:
two competent speakers presented with the same (calibrational) 
proposition "x is F" need not agree that x is describable by 
F within the language convention.  "x is F" is asssumed 
calibrational in the sense that  x  is shown to or otherwise 
made known to speakers of the language to whom the proposition
is posed, so that the only question at issue is the appropriateness
of word usage, ie. of the descriptor F with respect to a 
referent value x, assumed known.  It is because of the imperfect 
convention of word usage that when faced with a fuzzy proposition 
in actual use--"x is F" when  x  is an unknown so far as the 
listener is concerned, eg. a jury being told by a witness that 
"the perpetrator was tall"--that a fuzzy range of possible values 
for x is permitted by such a description.  If usage were consistent 
across speakers of the language, or even for the same speaker at 
different times, then inferences about unknowns could be crisp.  But 
usage is not consistent, therefore the descriptors are fuzzy,
and the inferences drawn therefrom likewise are fuzzy.  In
other words, fuzziness in natural language may be traced to
an observable *chance* phenomenon, namely unpredictability of
word usage in a calibrational setting.

On this conceptualization, the membership function merely maps, 
directly, the degree of agreement, as x is varied over an 
appropriate universe of discourse.  (Numeric measurement is but 
a special case, not qualitatively different.  For example,
saying that John is tall is far more fuzzy than saying John is
70 inches tall, but even "70 in" is in principle fuzzy, when
allowance is made for inherent imprecision of the measurement
device used, as well as for errors of measurement tied to the
actual measurement procedure adopted.)  Therefore, the problem 
of measuring the grade of membership reduces to a simple matter 
of constructing an appropriate calibrational experiment, and 
counting the fraction concurring with the use of F to describe x.  

I appreciate that there is a point of view, exemplified in
your paper, opposed to the above, in which the notion of grade 
of membership is considered to have a subjective, psychological 
origin.  Proceeding from this view, it becomes a problematic 
matter how to measure the grade of membership.  The measurement-
theoretic approach to which you resort follows the well-worn
path already laid down in investigations into other inherently
subjective attributes having psychological origin, for example
subjective utility and subjective probability.  Following 
this approach it is readily established that the grade of membership 
may be established on at least an ordinal scale, and also, with 
little extra in the way of justifiable assumptions, it may
also be established on an interval scale.  But you have trouble
making the jump to a ratio scale, let alone an absolute scale.
Thus you say:

  "The formal existence of scales that are stronger than interval
   pose the problem of coming up with a _natural origin_.  However,
   since the ranking of properties of an individual is highly
   a subjective [sic] act of the observer, there cannot be 
   universally accepted bounds on the measurement scale 
   (Norwich and Turksen, 1984).  This makes all the scales resulting
   from the measurement, _relative_ to the observer and amounts
   to concluding that the formally attainable absolute or 
   ratio scales are not likely to arise in the framework of
   fuzzy set theory (Bilgis and Turksen, 1995a)." (p. 15)

This strikes me as being a highly unsatisfactory conclusion, and
when I reflect back on the questionable assumption at its
foundations, viz. that the notion of grade of membership is
inherently subjective and has a psychological origin, I have to
reject both the result, and the (unexamined) assumption which
gives rise to it.  Possibly it proves the fundamental inconsistency
in traditional fuzzy set theory in asserting at one and the same
time a calculus which clearly is based on an absolute [0,1]
scale for the notion of grade of membership, yet asserts that
this notion is subjective and has a psychological origin.

It seems far better to adopt the straightforward interpretation
earlier sketched where fuzziness may be traced to its origins
in the objectively observable chance phenomenon which is
the dominant characteristic of word usage in a natural language.
That is, for a given calibrational proposition "x is F" one
asks speakers of the language, would they affirm the calibrational
proposition, yes or no.  The fraction concurring provides the
grade of membership, and establishes immediately the absolute
scale on [0,1] which is fundamental to the fuzzy set theory.  
The linkage with probability strengthens our understanding of
the notion of fuzziness, and does not detract from it, as 
might first, defensively, be thought.  (But that's another
subject!)

Direct estimation is in principle a dodge, because it puts
the subject into the metalanguage, guessing at usage, rather 
than leaving the experimenter alone in the metalanguage, recording 
actual usage rather than guessing at it.  That, it seems to me,
is what empirical science demands.  (If we were studying the
metasemantic guesses about object language usage of a group
of subjects, it would of course be a different matter.  But why
ask people to guess how, for example, the word "tall" is used 
if we can set up an experiment to observe it directly?)

The group/individual distinction also offers less than meets
the eye.  There is no problem in principle with studying the
idiosyncratic usage within any given subpopulation of speakers,
including the ultimate subpopulation consisting of one individual.
In that case the experimenter must make sure that on any repetition 
of a trial, the subject has forgotten what his previous response was,
allowing genuine usage variation to reveal itself.  If he/she is 
truly perfectly consistent, ie. not fuzzy in repeated usage, 
then that's fine: crisp is a special case of fuzzy.

The subjective/objective distinction is also essentially a 
dodge.  The question is whether *usage* is observable or
not, notwithstanding perceptions, beliefs and feelings,
which may indeed be subjective (Thomas, 1995, p. 35).  If 
usage is observable, then it may be mapped for an
individual as easily as for a population, at least in priciple.
If usage is not observable, then the speaker is carrying on
conversations with herself, and it becomes difficult for a
listener to infer anything from what such a speaker might
say.  It's the Alice-in-Wonderland problem of words being
made to mean whatever the user wants them to mean, which 
strikes me as not really being the concern of a theory of semantics,
fuzzy set or otherwise.
 
Those are my general comments.  I also have a few comments 
specific to your reference to my work (p. 7):

First of all, thanks for the citation, I do appreciate it.

Your reference to "a likelihood semantics for fuzzy sets" seems
confusing to me in context, the notion of "semantic likelihood"
being different from the notion of "mathematical likelihood". 
A naive reader will likely already be familiar with the concept of
mathematical likelihood, which is defined only when an
associated probability model is defined.  As a probability model 
defined by Hisdal is on the preceding page, I think a naive
reader will seek to link the likelihood function associated
with that model to "a likelihood semantics" associated now
with me.  I can only see confusion resulting.  My suggestion:
make it clear that the notion of semantic likelihood advanced
by me for describing the membership function for descriptors
in natural language is an _analogue_ of the notion of mathematical 
likelihood well known from the statistical literature, not 
mathematical likelihood itself (See Thomas, 1995, p. 33.)
The (absolute) mathematical likelihood may also be construed
as the membership function of a fuzzy set, but only in a 
formal sense (but see Thomas, 1995, pp. 11, 12, 173) not 
corresponding with any descriptor in natural language.

You put in quotes attributing to me something I'm pretty sure
I've never said, viz: "meaning is essentially objective and is
a convention among the users of a language."  I would prefer 
that you quote me as saying "... the meaning of a word is 
essentially and necessarily a matter of convention, whether in 
natural or artificial language." (op. cit., p. 27).  

You also say "Thomas claims that measurement is essentially
a vague process".  I am pretty sure I have made no such 
claim.  I don't think there is anything vague about the
measurement process.  I don't even think that there is anything vague
about a measurement report.  What I believe is that measurement
is fuzzy in general, as a practical matter, it being possible 
to have crispness and/or precision fall out as special cases of fuzziness.  
In any case, a fuzzy measurement is not the same as a vague 
measurement, whatever the latter may mean.  I am clear in
my mind that one can be fuzzy without at all being vague,
and conversely one can be vague without being at all fuzzy.
For example, for a witness to say the perpetrator was tall
leaves a fuzzy restriction for the height of the perpetrator, but 
is not vague.  For an example of something that is vague
but not fuzzy, consider a typical politician's reply to the question
"do you agree with abortion, yes or no?".  There is no fuzziness
allowed in the response options yes or no, but the typical
politician has the verbal resources that permit a perfectly
vague response that would permit either or both a yes answer
or a no answer, depending on the listener.  Therefore the two 
words fuzzy and vague are not synonyms, although they are close 
enough in meaning that they are sometimes confused.  In any
event, I have nowhere claimed that measurement is essentially
a vague process, merely that measurement reports may be 
considered fuzzy in general, and include crispness and precision
as special cases. 

Then you say "these views lead Thomas to equate the likelihood
function to the membership function".  I would again object
(see earlier) that in context it is not clear to what likelihood 
function you refer.  I have also been careful to distinguish
absolute from relative likelihood, since the latter cannot 
correctly be "equated" to the membership function.

Finally, you say "like Hisdal, Thomas also finds out that
the min-max calculus as proposed by Zadeh is not sufficient
in this framework."  And earlier, in relation to Hisdal's
work you had said: "this should come as no surprise since 
in this probabilistic setting we know that joint probabilities 
are not always simply recovered from the constituent 
probabilities."  You make an achievement seem like a failing.
The point is that this framework allows a *range* of rules
to be *derived* for conjunction and disjunction, *including* the
Zadehian min-max rules, and moreover sheds some light on
the conditions under which the Zadehian rules are appropriate,
and when not, in relation to the natural language semantics
sought to be modelled by the fuzzy set theory. 

I hope these remarks are helpful.  

: Cheers,

: --
: Taner Bilgic                       taner@ie.utoronto.ca

Regards,
S. F. Thomas

PS. You give an incorrect PO Box number for ACG Press in your
reference listing for Thomas, 1995.  I suggest you leave that
information out altogether.  The following would be sufficient:

  S. F. Thomas (1995).  _Fuzziness and Probability_, ACG Press, 
    Wichita, Kansas. (ISBN 0-9649049-0-X).

