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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Measuring the grade of membership was Re: Defining fuzzy descriptors (was  NOT and DIFF)
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Date: Mon, 24 Mar 1997 23:27:23 GMT
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Taner Bilgic (taner@ie.utoronto.ca) wrote:

(( cuts ))

: As for measurement theory: measurement theory is founded on an
: *objective* account of meaning even when its subject is subjective
: probabilities or fuzziness. Furthermore, it assumes an *ideal* world. 
: We might choose to weigh a given set of objects in pairs using our 
: hands and rank them ordinally but measurement theory tells us that we
: can do better (ratio scale) *and* the conditions under which we can do
: better. Then it is up to a genious engineer to come up with a better
: scaling device.

Er..., somehow I think your ingenious engineer would be 
underwhelmed by the contribution of the measurement theorist.
The ancient Egyptians were measuring weight at least 6000 
years ago, presumably without modern measurement theory to 
guide them.  

Nor do I see any particular virtue in assuming
an *ideal* world.  To the extent that a model in some sense
approximates key features and relationships in the real 
world it is an idealization, but that is different from 
assuming an *ideal* world.  In the case of measurement theory,
I embrace the idealization that says each point in a linear
continuum can be uniquely labelled using the so-called 
real numbers--each correct to an infinity of decimal places--
as the labelling scheme.  It is a very good idealization
that has served us well in the physical sciences and 
engineering, where three significant figures is often 
an attainable level of precision, and moreover a good
approximation in practice to an infinity of decimal places.
In the social sciences, however, and certainly where the 
measurement of psychological or psychophysical attributes 
are concerned, it seems to me that the idealization breaks 
down.  The ordering axiom fails, along with the transitivity 
property of the equality relation, since human judges may 
often judge two stimuli A and B as equal, B and C as equal, 
but insist that A and C are perceptibly different.  Now I 
could continue to retain the idealization of the linear 
continuum for such attributes as, say, subjective utility, 
but it seems clear to me that any realistic measurement with 
respect to such a scale must remain essentially fuzzy.  
So I see a separation conceptually between, on the one hand, 
the underlying universe and the scale appropriate to it, 
which in the ideal could be conceived of as a totally ordered 
linear continuum, and, on the other hand, the gross imperfection
of the human judge relied upon for the scaling exercise who would 
inevitably fail to meet the requirements of (at least) the
total ordering axiom.  Fuzzy set theory, precisely, is
supposed to assist in mediating between the conceptual ideal
of a perfectly continuous underlying universe of discourse
and the practical reality that "measurement" with respect
to such a scale would at best consist of discrete, overlapping
fuzzy terms, which even if "numerical", would nonetheless
be essentially fuzzy.  (This is true also even if measurement
is to a few decimal places, although in such cases it becomes
more feasible in practice to pretend that the theoretical ideal 
of point measurement is satisfied.)
Thus fuzzy set theory provides some conceptual
tools that can come to the aid of measurement theory, which--
let's face it--is hardly needed in the physical sciences, and
in the areas where it is deployed, starts out usually with
an ordering axiom which simply is beyond the capability of 
the measuring instruments, usually human judges, to obey.  
So... rather than assume an "ideal" world, I would say get a 
better model.  

In any case, so far as the grade of membership is concerned, 
I think its conception as having an essentially psychological
origin is seriously flawed.  Certainly, as I have shown,
if denotative semantics of adjectives/descriptors provides
the empirical point of reference, no recourse need be had
to an ill-defined psychological origin for the notion of
grade of membership, and moreover a calculus can be *derived*,
as opposed to being merely assumed, which contains as special
cases all the results of the original Zadehian calculus.
Therefore, it would seem to me, the psychological interpretation
of the grade of membership is at the very least *not needed*,
if what we are after is the elucidation of certain features
of natural language semantics that are not captured by 
bivalent set theory.  If proposed as capturing some self-evident
psychological reality, that is of course another matter, but
then I would have to confess that the psychological reality,
if so it is, of the notion of grade of membership, has 
escaped at least my psyche.  Now, that doesn't stop me from
giving a subjective estimation of, say, the grade of membership
of 76 inches in "tall", but such a subjective estimation for
me is of a quantity that exists independently of my psyche.
I could also see a person walking down the street, and give
a subjective estimation of his height in inches, but that
would not make such a person's height an attribute having
psychological origin within my psyche.  I sometimes think that
it is because for most everyday fuzzy terms most of us could
make fairly good guesses as to population usage, that we
mistakenly ascribe such ability to something of intrinsically
psychological origin.  IMO, it is not.

But what do I know.

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

Regards,
S. F. Thomas
