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From: Taner Bilgic <taner@ie.utoronto.ca>
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 16:44:01 GMT
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Dear Sydney,

Thanks for your comments on our paper. It has been very useful.
I will have to make a couple of quick remarks: 

When you say:
    Overall, [the paper] 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.
you are right and it is intentionally so. We, fuzzy set researchers
have a unique problem in our hands. We have different interpretations
of the membership function which in essence prescribe a different
semantics for fuzzy set theory and depending on the interpretation
even the underlying calculus changes! An analogous situation is true
for probability theory but although there are various interpretations
of probability values, at least everyone agrees on the calculus to 
use.

The first aim of our review is to lay out all different interpretations
and their consequences on the table. The second aim is to summarize the
empirical studies that are carried out mainly in the psychology
literature. Anybody doing fuzzy modelling has to be extremely careful on
the intended semantics of the calculus s/he is using.
And the question of operator selection is inevitably dependent on the
interpretation of the membership function.

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.

Similarly, for fuzzy set theory we *can* attain ratio and absolute
scales (see our 1995 FSS, 1997 LNAI papers, or the appendix of the
review) at the cost of accepting highly restrictive structural axioms.
But, then again, these axioms *may* be satisfied for certain
interpretations of the membership function and we have to be aware of
them.

I believe it is important to be aware of various interpretations of
membership functions which we summarized in the paper as: the likelihood
view, the random set view, the similarity view and the utility view.
These interpretations of the membership function are distinct and
sometimes competing.

I commend approaches that set an interpretation of the membership at the
outset and then start exploring an appropriate calculus for it.

-- 
Taner Bilgic                       taner@ie.utoronto.ca
