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From: bilgic@PROBLEM_WITH_INEWS_GATEWAY_FILE (Taner Bilgic)
Subject: Re: Measuring the grade of membership 
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Date: Sat, 29 Mar 1997 06:04:13 GMT
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S. F. Thomas (sthomas@decan.com) wrote:
> 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 genius 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.  

We built steam engines before we had a theory about them but that did not
invalidate the theory now did it?

I will like to comment on your treatment of (representational) measurement
theory. I will use both your writings in this thread and Section 2.2 from 
your book, _Fuzziness and Probability_ on Philosophy of Measurement.

You find representational theory of measurement (henceforth RTM) inadequate 
for social sciences because, you claim:

- the linear ordering is too restrictive to model the "inherent imprecision 
  of the human judgments". The intrinsic errors cannot be handled in the 
  idealized framework of RTM.
- the relation (e.g. is harder than) which is taken as primitive in RTM
  is inappropriate. One should take the attribute (e.g hardness) as primitive.

Hence you find the idealization put forth by the RTM "useful" for exact
sciences but inadequate for social sciences.

I will briefly comment on each point you raise.

I agree that the linear ordering axiom, that is the basis of *most* 
representations in the RTM, is a strict one. Not only transitivity but 
even completeness (for a relation R on AxA, A={a,b,c,..}, for all a,b in A
either aRb or bRa) is problematic. However, RTM is fully aware of this 
problem. Within the RTM body of work there is a myriad of ordering axioms that
relax the linear ordering axiom (semiorders, interval orders, biorders,
valued-relations, fuzzy relations etc.) And these have been applied 
to model the intrinsic errors in various domains. Hence to disregard 
RTM on the basis of the linear ordering axiom and to claim that RTM
cannot handle intrinsic errors is not tenable. In your book you cite 
Krantz et al. (1971) Foundations of Measurement Vol. 1
as the main source on RTM. That is still the best source today along
with several other texts. Furthermore Krantz and company came up with 
Volumes 2 and 3 of their undertaking  in 1989 and 1990, respectively.
See Vol. 2 on various relaxations of the linear ordering axiom and the
fascinating chapter on color measurement, a subject considered to be 
"inherently fuzzy". Also in Volume 3, you will find answers to the 
"difficult, poorly understood" question of errors in measurement.

On the second issue that you raise against RTM, the attribute being 
primitive rather than the relation, here is what a linguist has to
say:
   ... when we learn a language like English we learn the meanings of 
   individual adjectives and, moreover, the semantic function which this
   comparative-forming operation performs, in general, so that 
   we have no difficulty in understanding, on first hearing, the meaning 
   of the comparative of an adjective which we had thus far only encountered 
   in the positive. If this is so then the meaning of an adjective must be 
   such that the comparative can be understood as a semantic transformation 
   of that meaning into the right binary relation. (Kamp 1975, p. 127)

Hence, I think you have the wrong reasons to oppose RTM as a proper 
foundational theory for fuzzy sets. RTM has tackled many phenomena, from 
physical sciences, cognitive sciences and social sciences alike.

What might be challenged is the philosophy that legitimizes RTM as a 
scientific theory and licenses it to prescribe meaning. This is the
objectivist account of meaning I was alluding to in my earlier post.
But I have yet to figure out a worthy replacement to objectivism in
this context.

Until that happens, I still value RTM as valuable for it provides
valuable insight into the semantics of fuzzy set theory. I hope anybody
interested in this thread can come to Barcelona for the FUZZIEEE'97
conference where in a special session on Practical Determination of
Membership Functions we will have more time to discuss these issues.

Session: Practical Determination of Membership Functions
Organizer: J.L. Verdegay

- Data Structure for a Fuzzy Machine Learning Algorithm
  T.P. Hong and C.Y. Lee
- Determination of Membership Values Using Adjectivalmeter
  R. Biswas
- Elicitation of Membership Functions: How Far can Theory Take us?
  T. Bilgic
- Empirical Determination of Membership Functions for Stimuli Comparison
  A. Sancho and J.L. Verdegay
- Fuzzy Sets as the Aggregation of Weighted Observations
  V. Torra


Cheers, Taner

Works cited:

@incollection{Kamp75,
    author  = {J. A. W. Kamp},
    title   = {Two theories about adjectives},
    booktitle = {Formal Semantics of Natural Language},
    year    = {1975},
    editor  = {E. L. Keenan},
    publisher = {Cambridge University Press},
    address = {London},
    pages   = {123--155}
}

@book{Kranetal71,
    author  = {D. H. Krantz and R. D. Luce and P. Suppes and A. Tversky},
    title   = {Foundations of Measurement},
    year    = {1971},
    publisher = {Academic Press},
    address = {San Diego},
    volume  = {1}
}

@book{Suppetal89,
    author  = {P. Suppes and  D.H. Krantz and R.D. Luce and A. Tversky},
    title   = {Foundations of Measurement},
    year    = {1989},
    publisher = {Academic Press},
    address = {San Diego},
    volume  = {2}
}

@book{Luceetal90,
    author  = {R.D. Luce and D.H. Krantz and P. Suppes and A. Tversky},
    title   = {Foundations of Measurement},
    year    = {1990},
    publisher = {Academic Press},
    address = {San Diego, USA},
    volume  = {3}
}

@book{Thom95,
    author  = {S. F. Thomas},
    title   = {Fuzziness and Probability},
    year    = {1995},
    publisher = {ACG Press},
    address = {Wichita Kansas USA}
}
