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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Fuzziness and Fundamental Measurement (Was Re: Modeling  Was: Re: Fuzzy theory or probability theory?)
Message-ID: <1994Dec8.043637.3916@decan.gate.net>
Organization: Decision Analytics, Inc.
Date: Thu, 8 Dec 1994 04:36:37 GMT
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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:

(( cuts ))

: >: 1) Correct me if I'm wrong, but I don't believe that any probability 
: >: measure of what heights are considered tall exists or are in use by 
: >: police departments.  This would seem to contradict the statement that
: >: the police officer needs this information to understand the description 
: >: of "tall."

: You are partially right and partially wrong.  There is no agreement
: about what "tall" means.  This means that, in practice, the police
: officer does NOT understand the description.  A five-foot third grader
: would be tall, and in those tribes which have bred themselves for height, 
: a six-footer would be short.

You are merely pointing out what is well known among fuzzy-set
theorists, which is that terms in natural language can "mean"
different things depending on context.  As an extreme example,
"tall" in the context of modern office buildings means something
quite different from "tall" in the context of adult males.  You have
added two more examples ... the context of third graders, and 
the context of "tribes bred for height", whatever that is.

: >The fact is every competent speaker of the language already has this
: >information.

: This is false.  It is doubtful that different people can communicate
: well in this manner.  This is why precise terminology and models are
: needed.

Well, people do communicate in this manner, and quite "well" as a 
matter of fact.  This is not to deny that additional precision is 
sometimes required.  When the perpetrator is "booked", a mug shot
will be taken (that beats any verbal picture that a witness might
paint) and his height is measured to within about an inch.  I
purposely use the fuzzy hedge "about" to convey the idea that 
even if you have measured height to within an inch, it is still not
absolutely precise.  Let me give an excerpt from
_Fuzziness_and_Probability_, which addresses the question of
measurement and precision (from Chap. II, Sect. 2, "Philosophy
of Measurement"):


   "  '...Now if we are able, at least in some relevant aspects,
      to understand the significance of uncertain knowledge, we 
      can proceed from that to incorporate the case of near-certain
      knowledge, merely by diminishing the degree of uncertainty.
      If, however, we take the case of near-certain knowledge as our
      typical case (especially if we allow ourselves to neglect
      the \*(EM perhaps very small \*(EM degree of uncertainty
      which accompanies it) we are bound to find it much harder to
      proceed to uncertain knowledge, which then appears as 
      radically different.' John Hicks, _Causality_in_Economics_.
   
   
   As has previously been
   pointed out in Chap. I, ... there is little conceptual
   difference between calibrating the meaning of a word in a natural
   language, and calibrating reports of a measuring instrument.  We
   have no difficulty accepting the fact that a word such as "tall"
   may possibly refer to many points in the space of heights.  By
   contrast, most theories of science proceed from the assumption
   that the attributes with which they deal may be measured with
   precision. This is useful and appropriate in the "exact"
   sciences, where measurement may be made to a sufficient number of
   decimal places for most purposes and for most
   attributes of concern. But the assumption of precision fails very often in
   the social and economic sciences,
   where, it would seem,
   imprecision and fuzziness of measurement constitute the
   general case.  Not surprisingly, the special, limiting case of precise
   measurement falls out rather easily from this general case; in
   contrast, an attempt to deal with 'error of measurement', when
   one proceeds from a paradigm of precision raises what Krantz,
   Luce, Suppes and Tversky (1971, p.13) call 'difficult conceptual
   problems ... poorly understood'.
   
   The need for a paradigm of imprecision and 
   fuzziness in measurement therefore seems to be great. The
   practical measurement problems which are faced in utility theory,
   subjective probability theory and psychometric scaling in
   general, are really problems that derive from the inherent
   imprecision of the human judgements required. These problems
   belie the first of the measurement axioms that is usually
   involved in the development of these theories -- the ordering axiom.
   This axiom simply says that for any two stimuli  A  and  B , one of
   the following three relations hold:  A  is perceived to be greater
   than  B ,  A  is perceived as equal to  B , or  A  is perceived to be
   less than  B . Where imprecision rears its head and invalidates
   this kind of axiom is when intransitivities of equality judgments
   arise, yielding for example contradictory judgments such as  A 
   equals  B ,  B  equals C , but instead of  A  equals  C  as expected, we
   find  A  greater than  C  or vice versa. This problem is ubiquitous
   in psychometric scaling of judgmental attributes, and it arises
   when  A  and  B  are sufficiently close in attribute value as to be
   indistinguishable to the human judge,
   and similarly  B  and  C , but  A  and  C  are
   perceptibly different. The conception of data as fuzzy sets,
   already discussed in Chap. I,  allows a resolution of
   this kind of problem.
 
      .....(cuts related to basics of classical approach to measurement)...
   
   The inherent deficiency of this approach is 
   that it is not practically possible to construct --
   from purely empirical relations -- a function
   mapping whose range is the real line. The reason 
   is quite simply that to specify uniquely a point 
   in the real line requires an infinite number of 
   decimal places, which no measurement device in 
   general can provide. The result is that our 
   representation mapping  f  at best associates 
   elements of  A  not with points in  Re  in general, 
   but with subsets of  Re  representing bands of 
   achievable precision. For example, when we say of 
   an object that its length is  10.1cm  we are really 
   at best saying that its length lies somewhere on 
   the interval  10.1 +/- .05 cm . The premise of point 
   measurement on which classical measurement theory 
   proceeds is therefore at best a convenient 
   idealization.... 

      .......
   
   The crux of the problem lies in the treatment 
   of error. The problem of equivalence intransitivity stems from
   the intrinsic accuracy or sensitivity limitations of the judge or
   measurement device -- what may be called intrinsic error. Errors
   of observation (Topping, 1955) may include as well accidental and
   systematic elements. The classical formulation of the problem of
   measurement runs into difficulty because it does not allow,
   within the formulation itself, for the treatment of error.
   Intrinsic error is not allowed because it is inconsistent with
   the fundamental ordering axiom which is usually assumed -- a
   practical necessity within a paradigm of point representation.
   Systematic error is not allowed because the formulation deals in
   principle with primary measurement, whereas systematic error is a
   problem only for secondary measurement devices defined by
   reference to a primary standard -- more a practical problem than
   one of principle. Accidental error is treated, not as part of the
   problem of measurement in principle, but as an extra-theoretical
   problem of statistics. Once again, the premise of point
   measurement is too rigid an ideal to allow variations of
   measurement to be considered within the theoretical framework;
   variations must be explained through the invocation of random or
   accidental occurrences, when the real culprit is more usually
   intrinsic accuracy limitations [of the judge or measurement device].
   
   Within the fuzzy set framework being proposed 
   for the problem of measurement, it is possible to treat all three
   forms of error. Systematic error and intrinsic error would be
   revealed by means of the kind of calibration exercise already
   discussed (Chap. I, pp. ... ). What would be regarded as
   accidental error in the classical framework is treated not so
   much as "error", but as variations in measurement reports wholly
   consistent with the intrinsic imprecision inherent in those
   reports. To illustrate, suppose the true value of a measured
   quantity to be  10.14 , to two decimal places. Two fuzzy
   measurement reports  10.1  and  10.2  may
   conceivably be used to describe this quantity if the fuzzy bands
   of precision respectively associated with these reports both
   include the point  10.14  among the points to which they may refer
   (see Fig. 1.2).
   True accidental error is that which obtains when not even the
   fuzzy term in question may properly be applied as a descriptor of
   the measured quantity in question, and thus should be discarded
   altogether.  Misreading of instruments, transcription errors and
   the like fall into this category. Such "true" errors typically
   give rise to "outliers", which, as every schoolboy knows, need to
   be carefully investigated before being accepted as genuine data.
   In any event, the present fuzzy set framework treats genuine
   measurement variation as a problem in the combination of
   evidence, which is discussed later (Chap. IV, Sect. 4.7).
          
         ........

   What I want to address here is the philosophical
   problem already alluded to 
   regarding the reconciliation of the need
   on the one hand, to retain the points constituting 
   the universe of discourse, with the decision, on 
   the other hand, to eschew the premise of point 
   measurement in preference for fuzzy measurement. 
   The reconciliation lies essentially in making an 
   explicit acknowledgement of the notion of the 
   attribute as a primitive concept in the formulation of
   the problem of measurement. In this way, 
   the essential properties of the attribute-space or 
   universe of discourse could be elaborated, in the 
   abstract, prior to the perceptional issues of 
   describing or comparing objects, in necessarily 
   fuzzy terms, with respect to degree of possession 
   of the attribute in question. Let me attempt to 
   explain.
   
   Recall that in the classical formulation, the 
   primitive concepts are a set of objects  A , say, 
   and a tuple of relations,  R1 ,..., Rn   among
   the set of objects. Thus, in the development of an 
   ordinal hardness scale, the attribute of hardness 
   is not acknowledged as a primitive concept, but 
   derived from the relation "is harder than" which 
   is what is taken as primitive. This is a matter of 
   philosophical choice, no doubt. It follows from 
   the empiricist's position that all we really 
   'know' about the attribute 'hardness' is what is embodied
   empirically, and testably, in the relation 'is harder than'.
   Given any two materials we can empirically determine which one
   'is harder than' the other by the simple
   empirical rule -- for example -- of seeing
   which one scratches the other. The attribute of hardness is not
   defined primitively within this philosophical system, but serves
   only as a convenient label for the scale which results after the
   measurement exercise is undertaken. This would be fine, were it
   not for the empirical nastiness that relations are not always
   clear-cut -- the equivalence intransitivity problem. The fuzzy set
   solution to this empirical nastiness is to admit the attribute as
   a primitive concept, alongside the empirical relations of the
   classical system, but separate and distinct from them. Moreover,
   the concept of the attribute, although it must have empirical
   significance -- since empirical relations themselves have meaning
   only through invocation of an attribute -- is nevertheless an
   abstraction. To explain, consider the two equivalent statements:
   
      'John is taller than James.'                              (A)
   
      'The height of John is greater than the height of James.' (B)

   Statement A contains the relation "is taller 
   than." Statement B contains the relation "is greater than", _and_
   the explicit introduction of the attribute of height. The
   attribute of height is an empirical one in the obvious sense that
   objects (John and James) may be empirically compared with respect
   to the attribute of height. However, height itself -- the quality
   - is an abstraction. Moreover, the essential properties of the
   attribute are determined conceptually as the result of the act of
   abstraction, not _perceived_ by the aggregation of individual
   empirical relations between individual objects according to the
   attribute. We must in some sense know, or understand, the
   quality, height, before we can devise an empirical procedure to
   determine whether an arbitrary "John" is taller than an arbitrary
   "James". How else can one make the connection between separate
   performances of the experiment, as somehow being conceptually
   identical?
   
   If we admit the attribute as a primitive abstraction within our
   philosophical measurement system, we are freed to consider the
   properties of the abstraction, as an abstraction. This frees us
   of the essential problem of the classical formulation, which lies
   in the attempt to elaborate, at one and the same time, the
   properties of the attribute space, as well as the error
   characteristics that derive from the empirical idiosyncrasies or
   sensitivity limitations of the judge or measurement device. To
   separate the two, we are faced first with the task of showing the
   sense in which the properties of an abstract attribute space may
   be said to be elaborated without recourse to idiosyncratic
   measurement devices, and second, with the problem of showing how
   fuzzy sets may be used to represent the error characteristics
   inherent in the empirical idiosyncracies of measurement devices.
  
   First of all, it needs to be recognized that 
   measurement is but a form of description -- numerical description,
   and therefore more precise than ordinary non-numerical
   description, or so we hope. That being the case, we ought to pay
   some attention to numbers, and just what it is about numbers that
   enable them to play such a pivotal role in any discussion of
   measurement. Numbers are themselves just words -- descriptors of
   attribute-value of the attribute of cardinality
   (in the first instance -- where the so-called 
   'whole' numbers are concerned),
   which may be used to characterize discrete populations. For
   example, in describing a flock of sheep in a field, we may -- in
   an obvious ideogram -- describe its cardinality as
   ' { o o o } ' if there are "three" sheep in the flock. This notion of
   cardinality is an abstraction which we may as easily apply to a
   fleet of ships or bags of rice, or whatever. And descriptors
   of attribute-value for this attribute of cardinality may be
   designated { o }, { o o }, { o o o },
   "one", "two", "three", or "1", "2", "3", etc. What is noteworthy
   about these descriptors of cardinality is that (i) they are
   exact, and (ii) we have through our numbering system a descriptor
   for every possible value of the attribute -- as every schoolboy
   knows, there is no whole number beyond which we cannot count.

   Consider now the notion of the linear continuum: The Euclidean
   concept of the linear continuum is that of an abstract entity
   consisting of points, distinguishable from each other only by
   relative position in the continuum. At the time the notions of
   point and line were elaborated and investigated by Euclid, there
   existed no convention by which every point in a continuum could
   in principle be described in a way that would distinguish it from
   every other point in the continuum. Now, of course, we know --
   thanks to Cantor, Frege, Dedekind and the other founders of real
   analysis -- how to construct conceptually the set of real numbers
   starting from the whole numbers, and moreover we know that there
   is a one-to-one correspondence between a linear continuum and the
   set of real numbers. It is by virtue of this correspondence that
   we could claim to be able -- in principle -- to distinguish every
   point in a continuum. But as we have already seen, this ability
   in principle does not quite extend to an ability in practice.

   The concept of the linear continuum is nevertheless important,
   for the following reason: In 
   the same way that we conceive of the attribute of 
   cardinality, and its corresponding set of
   attribute values -- the whole numbers -- we conceive of 
   the attribute of position in a line and
   its corresponding set of attribute values -- 
   the real numbers. Moreover, if we perform
   the act of abstraction by which we conceive of any attribute and its 
   corresponding set of (abstract) attribute values, 
   we by the act of abstraction determine the fundamental nature of
   that attribute-space: whether it is discrete or continuous;
   whether it is ordered or unordered; whether it possesses a metric
   or not; whether it is one-or many-dimensioned, etc. For example,
   the attribute of sex has a corresponding set of attribute values
   consisting of the two values called respectively "male" and
   "female". The attribute-space  { male , female } 
   is of course
   discrete, unordered, possesses no metric and is one-dimensional.
   The attributes of length, mass, temperature, etc. we conceive of
   as having a set of attribute values which form a linear
   continuum, that is to say are continuous, totally ordered,
   possess metrics and are one-dimensional. Some of these
   (length, mass) we conceive of as having in their linear continua of
   attribute values a 'natural zero' corresponding for example to no
   length or no mass. The attribute of shape we conceive of as
   having a space of attribute values which form a continuum,
   although not a linear continuum. We have a relatively small
   handful of terms such as 'square', 'round', 'rectangular', etc.,
   which describe certain planar shapes, but we recognize as part of
   our abstraction that there is indeed a continuum of shapes, an
   exhaustive characterization of which would require the use of a
   vector space of high dimension, perhaps of infinite dimension.
 
   The point of all this is quite simply the 
   contention that the character of an attribute space is determined
   fundamentally by an act of abstraction that precedes subsequent
   description (measurement) of objects with respect to the
   attribute, or comparison of objects according to the degree of
   possession of the attribute. This effectively achieves a
   separation between the issues of representation and uniqueness,
   on the one hand, and the issues pertaining to the empirical
   idiosyncracies of a particular judge or measurement device, on
   the other hand. The representation and uniqueness issues are now
   purely abstract, and therefore clear-cut (although not necessarily simple)
   while problems of error and equivalence intransitivity
   which reside essentially in the perceptional domain may be dealt
   with by the use of fuzzy descriptors, which allow for imprecision
   -- hence error and occasional intransitivity -- in judgment.
   
   This leaves the second task of showing how 
   fuzzy sets may be used to represent the error 
   characteristics inherent in the empirical idiosyncracies of
   measurement devices.
  
   For purposes of exposition, it is useful to 
   discuss how the problem of error is usually 
   treated. 

    ... (description of statistical approach to measurement error deleted)...

  
   All of this seems quite unexceptionable. But 
   it is all based on a point idealization of data which ignores
   intrinsic imprecision in the individual measurements. What if we
   were to treat data like the fuzzy sets they may be more
   realistically conceived to be? As has already been pointed out
   the report ' 10.1 ' may be conceived of as a fuzzy descriptor
   no different in principle from 'tall'. It has already been
   discussed (Chap. I), making use of the concept of the
   calibrational proposition, how such a fuzzy descriptor may be
   calibrated.
   
   Such a calibration process lies outside of 
   the traditional (statistical) analysis. Where such 
   a calibration process may detect both systematic 
   error, either in the device itself or in the 
   person using the device, the statistical analysis 
   must in practice assume that such error has been 
   eliminated. Further, where such a calibration process actively
   attempts to elaborate the precision and sensitivity limitations
   (intrinsic error) of the measurement device or process, the
   statistical analysis, by its very nature, can shed no light on
   intrinsic error. To illustrate, suppose the measurements
   previously used as an example were reported to two significant
   figures, rather than three, we would have eight identical
   observations  10 , 10, ... , 10 etc. Treating these as point
   observations, as the statistical analysis requires, the sample
   variance is zero. Within the strict logic of the procedure, we
   may conclude that the measured quantity is simply the point  10 ,
   correct to an infinite number of decimal places. More correctly,
   common sense forces us to recognize that the true value of the
   measured quantity could in fact lie, at best, anywhere on the
   interval  10 +/- .5 , depending on the intrinsic error of the
   process. The traditional analysis sheds no light on this; by
   contrast the calibration process has the _objective_ of yielding a
   (fuzzy set) characterization of the intrinsic error.
   
   Suppose now that the measurement reports have 
   been properly calibrated in fuzzy-set terms. In 
   place of the assumed point observations
   10.1, 10.0, 10.0, 10.2, etc. of a measured quantity, we 
   would have [these construed as fuzzy observations].
   We would then be faced with the question how to
   combine these measurement reports into a single overall
   representation of the uncertainty surrounding the value of the
   quantity being measured. In the present framework, this is a
   problem in the "combination of evidence", which 
   will be elaborated further, in Chap. IV, Sect. 4.7."


: A model is a formal system with a dictionary connecting the objects
: in the real world with those in the formal system.  Generally, there
: are many models for a given situation.  Those texts which talk about
: THE probability space are contributing to the confusion.  

I agree with you here.  

: Probability modeling sets up this formal system.  I have yet to see
: a formulation by the fuzzicists which does this; saying that the
: truth of a statement is not 0 or 1 does not provide enough to work
: with.

Here too I agree.  _Fuzziness_and_Probability_ attempts to correct
this lacuna in the fuzzy set fundamentals.

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

Cheers!
S.F.Thomas
