Measurement 46 (2013) 2921–2926
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Measurement
journal homepage: www.elsevier.com/locate/measurement
The role of fuzzy scales in measurement theory
Eric Benoit ⇑, Laurent Foulloy
Laboratoire d’Informatique, Systèmes, Traitement de l’Information et de la Connaissance, Université de Savoie, B.P. 806, F-74016 Annecy cedex, France
a r t i c l e
i n f o
Article history:
Available online 15 May 2013
Keywords:
Measurement theory
Fuzzy scale
Group theory
Metrical scale
Scale classification
a b s t r a c t
The introduction of the representational theory of measurement by Stevens initiated a new
way to understand what measurement is and was followed by an intense scientific activity.
Ludwik Finkelstein mainly contributed to this activity through several synthetic surveys
and his formalisation of this theory includes a generalisation of the representation of measurement values to non-numerical sets. The role of group theory in the measurement theory was suggested by Stevens in his seminal paper. Such a role was explored by Narens and
Luce for the ordered scales. The studied groups are homomorph to groups acting on real
numbers, and other possible scales remain unexplored. For example, the metrical scales,
introduced by Coombs, are built on distances and do not fit the classic classification of
scale. Initially devoted to psychophysical measurement, metrical scales now appear in various fields, such as colour measurement or software measurement and need to be studied
in more detail. The purpose of this paper is to revisit the group-based classification of
scales and to show how such a classification includes metrical scales and more specifically
fuzzy scales.
Ó 2013 Elsevier Ltd. All rights reserved.
1. Preface
We met Ludwik Finkelstein for the first time twenty years
ago. During my Phd examination, Professor Finkelstein, as a
jury member, asked me how his studies on measurement
scales could be improved to help me. Ten years later, I finally
understood his encouragement and decided to re-open this
field with Laurent. This paper, as an instantaneous state of
our studies and inspired by the enthusiasm and kindness of
Ludwik Finkelstein, is dedicated to his memory.
2. Introduction: The representational theory of
measurement
Scales were introduced to model the link between physical quantities and information entities created by the
measurement process. The representational theory of measurement was proposed by Stevens in 1946 as a classification of scales based on their mathematical properties [1].
⇑ Corresponding author. Tel.: +33 450 096 544; fax: +33 450 096 559.
E-mail address: [email protected] (E. Benoit).
0263-2241/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.measurement.2013.04.043
After controversial approaches, this classification is now
commonly accepted as the most significant for scale-type
analysis. In 1975, Ludwik Finkelstein proposed a general
formal approach where the limitation of the representation
of measurement results by numbers is worked around by a
generalisation to a symbolic representation [2].
2.1. Introduction of the theory
In his seminal paper, Stevens proposed to classify scale
into four types: nominal, ordinal, interval and ratio scales
(see Table 1). This classification is driven by the definition,
for each scale type, of a set of relations on representations.
Stevens also associated a group structure to each scale
type. The set of mathematical transformations which leave
the scale-form to be invariant has a group structure that
characterises the scale type. Such a mathematical transformation modifies the measurement result given by a scale
into another measurement result.
A basic empirical operation is a relation on physical
quantities that has a representation in the representational
space. An admissible transformation is a function on scales
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E. Benoit, L. Foulloy / Measurement 46 (2013) 2921–2926
Table 1
First proposal for scale classification.
Scale
Basic empirical operation
Mathematical group structure
Admissible transformation
Nominal
Ordinal
Interval
Ratio
Determination of equality
And determination of greater or less
And determination of equality of intervals or differences
And determination of equality of ratios
Permutation group
Isotonic group
General linear group
Similarity group
y = f(x), f is a bijection
y = f(x), f is a monotonic increasing function
y = ax + b, a > 0
y = ax, a > 0
that preserves the link between the basic empirical relations and their representation. Such a transformation is defined on the set of measurement results. The mathematical
group structure of a scale is defined by the group of admissible transformations. These four scale types are ordered
by the «sub-group of» relation between groups.
2.2. Formalisation of the theory
The first contribution of Ludwik Finkelstein to the representational theory of measurement was to propose a formal approach of the definition of measurement. He
suggested generalising the universe of discourse of measurement values from sets of numerals to sets of symbols.
He then defined all scales by a symbolism expressed as:
C ¼ hX; S; M; RX ; RS ; Fi;
The relational structures hX, RXi and hS, RSi respectively
denote an empirical relational system and a representational relational system. The set RX is the set of relations
that are supposed to exist on the empirical set of quantity
manifestations. In fact this knowledge comes from the theory chosen to abstract the quantity. We are reminded that
a theory is made of entities and affirmations, i.e. axioms or
theorems, linking these entities [3]. The set RX is then part
of the theory according to the fact that the set RS is chosen
such that F is a bijection. The mapping M is then a homomorphic mapping or homomorphism in the sense that it preserves the relational structure. This constraint is expressed
by the representation theorem:
8x1 ; . . . ; xn 2 X; Fðr X Þ ¼ r S ;
ð2Þ
where rX is a relation of empirical relational system hX, RXi,
and rS is a relation of representational relational system hS, RSi: rXeRX, rseRS.
The term homomorphism must be interpreted in its
wider sense, that is as a morphism that preserves a set of
relations, and not as a synonym of group homomorphism.
The mapping M respecting (2) is not unique, and any
application f such that foM = f(M) respects (2) is an admissible transformation.
8x1 ; . . . ; xn 2 X; Fðr X Þ ¼ r S ;
r X ðx1 ; . . . ; xn Þ () r s ðf ðMðx1 ÞÞ; . . . ; f ðMðxn ÞÞÞ
2.3. The role of groups in scale classification
ð1Þ
where
– X refers to a set of quantity manifestations and RX is a
set of relations on X.
– S refers to a set of information entities, and RS is a set of
relations on S.
– M, called the representation, is a mapping from X to S.
– F is an injective mapping with domain RX and range RS.
r X ðx1 ; . . . ; xn Þ () r s ðMðx1 Þ; . . . ; Mðxn ÞÞ
The scales stay ordered within four types, each type
being associated to a class of admissible transformations.
As mentioned by Finkelstein the concept of scale is a
bridge between reality, i.e. the empirical world, and our
abstract representation of this reality driven by a theory
[4]. For any kind of measurement, that is strongly,
weakly or widely defined [5], a theory is needed, even
if it is a poor theory, i.e. with only few theorems. Actually, a scale is defined by the theory chosen for the abstract world that may represent the empirical world.
This theory fixes the class of admissible transformations,
and the validity of the scale is directly linked to the
validity of the theory.
ð3Þ
The representational theory of measurement has inspired several studies, especially for ordinal scales that
represent a central interest in psychophysics (see [6,7]
for surveys). The link between scale types and group structures remained unexplored until 1987, when Luce and Narens studied measurement scales on continuous spaces [8].
More recently, Narens showed the importance of an approach based on group theory to analyse the representational theory of measurement [9]. This tendency confirms
the need to take into account, at a higher level of abstraction, the incidence of the theory on the property of the
measurement scale.
The representation theorem induces a class of admissible transformations. Admissible transformations f are in
fact automorphisms that preserve the relational structure
hS; RS i. The classification of scale types is clearly defined
with the classification of admissible transformations. As
admissible transformations are automorphisms, their
class can be studied within the context of the group
theory.
In the case of nominal scales, the representational relational system is a relational system hS, = i and the set F of
admissible transformations defines the group ðF; oÞ on S
where o is the composition of functions. This group is the
symmetric group Sym(S), i.e. the group of all permutations
on S which is also the group of all bijections on S. Indeed,
by definition, any bijection on S preserves the equality
relation:
8a; b 2 S; a ¼ b () f ðaÞ ¼ f ðbÞ
ð4Þ
The set of admissible transformations of any other scale
type defines a subgroup of this general group.
Let F be a set of admissible transformations preserving
the equality on S and r another relation on S. As seen before, the preservation of the equality implies that elements
of the set F are bijections. The closure of F by the composition operator o is verified as follows:
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E. Benoit, L. Foulloy / Measurement 46 (2013) 2921–2926
Let f and g be two admissible transformations. We have
the properties
8s1 ; . . . ; sn 2 S; rðs1 ; . . . ; sn Þ () rðf ðs1 Þ; . . . ; f ðsn ÞÞ
ð5Þ
and
8s1 ; . . . ; sn 2 S; rðs1 ; . . . ; sn Þ () rðgðs1 Þ; . . . ; gðsn ÞÞ
ð6Þ
The composition of two bijections is also a bijection,
therefore the closure by the composition operator is immediately deduced:
8s1 ; . . . ; sn 2 S; rðs1 ; . . . ; sn Þ () rðfogðs1 Þ; . . . ; fogðsn ÞÞ
ð7Þ
Stevens identified the order-preserving group as the socalled isotonic group that is actually a semi-group [10].
Since his initial paper, this field has been studied, and
many ordered or partially-ordered structures have been
explored [11,12].
Stevens also identified the general linear group and the
similarity group as the respective structures of the set of
admissible transformations of interval scales and ratio
scales. Indeed, the set of the admissible transformations
of interval scales is the set of invertible affine transformations f(x) = ax + b, and is known as the affine group of degree 1 on the real numbers field R : GAð1; RÞ. The set of
admissible transformations of ratio scales is the set of
invertible linear transformations f(x) = ax which is the
general linear group of degree 1 on the real numbers field:
GLð1; RÞ. The main difference between GAð1; RÞ and
GLð1; RÞ is the inclusion of translations into GAð1; RÞ. Finally, the historical classification of scales is preserved but
now presented as a group-oriented classification (see
Table 2).
2.4. The isometry groups
Introduced by Coombs [13], the metrical scales are
scales that propagate a metric from the empirical world
to the representation world.
8x1 ;x2 2 X; 8a 2 R; dX ðx1 ;x2 Þ ¼ a () dðMðx1 Þ;Mðx2 ÞÞ ¼ a
structure is questionable due to its closeness to the interval
scale. The set of basic empirical relations of this scale are
the basic empirical relations of the interval scales and the
determination of equality of distances, but not the equality
of ratio.
The Euclidean group E(2) is more instructive. Actually
E(2) is a subgroup of GAð2; RÞ that has GLð2; RÞ as a subgroup, having itself GAð1; RÞ as subgroup. Indeed, GLð2; RÞ
is also the group of rank 2 invertible matrices of real numbers. With this remark, the extensions to 2-dimensional
interval scales, with the general affine group GAð2; RÞ, and
2-dimensional ratio scales, with the linear group GLð2; RÞ,
naturally take their place in the classification.
More generally, E(n) is a subgroup of GAðn; RÞ that has
GLðn; RÞ as a subgroup, having itself GAðn 1; RÞ as a subgroup. This simply generalises the interval scales and the
ratio scales to n-dimensional spaces. The Fig. 1 gives a synthesis of the generalisation of the group-oriented scale
classification.
Once again, these isometry groups are subgroups of
general affine groups. It means that the theory uses an affine space to represent the empirical world. In a more general case, non-Euclidean distances are not necessarily
defined on affine spaces. A theory may consider a metric
on a set of quantity manifestations with a representation
of the manifestations in a non-affine space. This is the case
for quantities for which measurements produce only distances. For example the Levenshtein distance is used to
measure distance between words. Poels presented a software measurement based on a distance based on the summation of elementary values, called length, associated to
elementary transformation of the software [14]. Within
some measurement contexts, the distance between quantity manifestations appears coherent. The associated quantity values are then supposed to be represented into a
space holding a metric. However, identifying the metric
ð8Þ
In such scales, the set of admissible transformations is
the set of transformations f that preserve the distance on S.
8s1 ; s2 2 S; 8a 2 R; dðs1 ; s2 Þ ¼ a () dðf ðs1 Þ; f ðs2 ÞÞ ¼ a
ð9Þ
In order to insert such scales into the scale classification
it can be first instructive to find how the most popular
isometry group, i.e. the Euclidean group, can be inserted
into the classification. In uni-dimensional spaces, the usual
distance is defined by d(x, y) = |x y|, and isometries are
translations. The group of translations on this space, denoted E(1), is a subgroup of GAð1; RÞ, the group of dilatations and translations. The utility of a scale with such a
Fig. 1. Relational graph of the mathematical group structures of scales
linked with the ‘‘subgroup of’’ relation.
Table 2
New proposal for group oriented scale classification.
Scale
Basic empirical operation
Mathematical group structure
Admissible transformation
Nominal
Ordinal
Interval
Ratio
Determination of equality
And determination of greater or less
And determination of equality of intervals or differences
And determination of equality of ratios
Symmetric group Sym(S)
Order-preserving group
Affine group GAð1; RÞ
Linear group GLð1; RÞ
y = f(x), f is a bijection
y = f(x), f is a monotonic increasing function
y = ax + b, a > 0
y = ax, a > 0
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E. Benoit, L. Foulloy / Measurement 46 (2013) 2921–2926
is not necessarily easy, and in many cases the metric is
chosen regarding to the goal of the measurement.
3. Fuzzy scales
The concept of fuzzy scale was introduced with the
definition of fuzzy nominal scales in [15]. They were introduced to formalise the fact that some quantity manifestations can be linked by fuzzy relations.
3.1. Definition
For the sake of simplicity, the notation of any fuzzy subset u will also denote the associated membership function.
Then the membership degree of a value x to a fuzzy subset
u is denoted u(x). A n-ary relation on a set U of fuzzy subsets is represented by a mapping from Un into [0, 1].
A measurement with a fuzzy scale is defined by a /symbolism hX; S; Ri where
– X is the set of quantity manifestations.
– S refers to a set of terms dedicated to the expression of
measurement values, it is often known as the universe
of discourse.
– R, called the representation, is a fuzzy mapping from X
to S.
In the context of measurements, a /-symbolism must
be complete and non-contradictory.
Let FS(X) be the set of fuzzy subsets of X.
Let FS(S) be the set of fuzzy subsets of S, i.e. the set of
fuzzy subsets of terms.
Let M, called fuzzy meaning, be the mapping from S to
FS(X) associated with R. Given a term s, the grade of
membership of the fuzzy set M(s) is denoted M(s)(x)
and we have the following equality:
8s 2 S; 8x 2 X; MðsÞðxÞ ¼ Rðx; sÞ
ð10Þ
Let D, called fuzzy description, be the associated mapping
from X to FS(S):
8s 2 S; 8x 2 X; DðxÞðsÞ ¼ Rðx; sÞ:
ð11Þ
A /-symbolism is complete and non contradictory when
M(S) is a f-partition of X [16]:
8x 2 X;
X
/ðMðsÞðxÞÞ ¼ 1:
ð12Þ
The main difference with non-fuzzy scales is that information entities are fuzzy subsets of terms, also called lexical fuzzy subsets (LFS), instead of being simply terms.
Actually, LFS are elements of a unit hypercube whose
dimension is |S|, where |S| is the cardinal of S. Due to the
constraint (12), the set D(X), image of X by D, is included
in a hypersurface such that (14) is verified. If / is the identity function then D(X) is included in the unit (|S| 1)-simplex. By using a fuzzy scale, the manifestations are
represented into a space whose dimension depends on
the cardinality of the universe of discourse.
8a 2 DðXÞ;
X
/ðaðsÞÞ ¼ 1
The other difference is that a fuzzy relation rX on X is
mapped to an intermediate fuzzy relation r0S on D(X). Actually the relation rX is not known and its link with the relation r0S is the expression of a theoretical knowledge about
the quantity.
8ðx; yÞ 2 X 2 ; r 0S ðDðxÞ; DðyÞÞ ¼ r X ðx; yÞ
ð15Þ
The definition of a fuzzy scale needs more parameters than
a non-fuzzy scale. Its depends on:
– The definition of the universe of discourse S.
– The definition of the relations rS.
– The definition of the bijective mapping /.
– The definition of the fuzzy relations r0S .
The definitions of the relations rS leads to the selection
of the type of scale and is based on the knowledge of quantities. The mapping / is related to the fuzziness of the symbol meanings on the set of manifestations. Finally the
previous definitions of {rS} and / produce enough constraints to limit the choice for the fuzzy relations.
3.2. Fuzzy nominal scale
A fuzzy nominal scale is defined by the symbolism
hX; S; R; fX g; f¼g; fðX ; S Þgi where X is a fuzzy equivalence relation on X and = is the equality on S. The determination of the scale is performed by the definition of a fuzzy
equivalence relation S on S which respects the singleton
coincidence with the operator = . In [16] we have proposed
P
a family of fuzzy equivalences relation denoted ( T)
and defined by:
8ða; bÞ 2 DðXÞ2 ; ðR TÞða; bÞ ¼
X
TðaðsÞ; bðsÞÞ
ð16Þ
s2S
s2S
where / is a bijective monotonic increasing function from
[0, 1] to [0, 1].
A fuzzy scale is then defined by the symbolism
hX, S, R, RX, RS, Fi where:
– RX is a set of fuzzy relations on X.
– RS is a set of crisp relations on S.
– F is a fuzzy mapping from RX to RS , where RS is a set of
fuzzy relations on D(X) which respects the singleton
coincidence with a relation of RS:
8ða; bÞ 2 S2 ; r 0S 2 R0S () 9r S 2 RS ; r0S ðfag; fbgÞ ¼ r S ða; bÞ:
ð14Þ
s2S
ð13Þ
where T is a t-norm.
These relations are fuzzy (17), reflexive (18), symmetric
(19), T#-transistive (20), and respect the singleton coincidence with =.
8ða; bÞ 2 DðXÞ2 ; ðR TÞða; bÞ 6 1
8a 2 DðXÞ; ðR TÞða; aÞ ¼ 1
ð18Þ
8ða; bÞ 2 DðXÞ2 ; ðR TÞða; bÞ ¼ ðR TÞðb; aÞ
ð19Þ
ð17Þ
3
8ða; b; cÞ 2 DðXÞ ; T ] ðdðR TÞða; bÞ; ðR TÞðb; cÞÞ
6 ðR TÞða; cÞ
where T# is a t-norm
ð20Þ
E. Benoit, L. Foulloy / Measurement 46 (2013) 2921–2926
It can be easily shown that a /-partition can be deduced
from the reflectivity condition (18) with /(a) = T(a, a). The
definition of a fuzzy nominal scale is then reduced to the
choice of the t-norm T or the bijective mapping /.
Fuzzy nominal scales clearly preserve fuzzy equivalence
relations between the empirical world and the representational world. One can object to this statement by saying
that nothing proves the empirical world to be fuzzy. Actually, as indicated in Section 1, the preserved relations do
not directly depend on the reality but on the theory used
to abstract this reality. The fact is that psychophysical approaches are increasingly modelling entities with fuzzy
classes. Practically, the fuzzy equivalence relations are
similarity relations.
In previous studies, we proposed to use the t-norm
TM(x, y) = min(x, y) [15]. In this case, the equivalence
P
relation ( TM) is TL-transitive where TL is the
Lukasiewicz t-norm TL(x, y) = max(x + y 1) [16]. A disP
tance d(x, y) = 1 ( TM)(x, y) is simply defined from this
equivalence relation. It can be verified from the TL-transitivity that the triangle inequality is verified by this
distance.
This scale has a new interesting property which is to
preserve the previous distance. Unfortunately the given
distance has a short-range effect. As many couples are
not at all equivalent, their distance is equal to 1. Actually
this distance cannot reflect a new empirical relation and
is only a consequence of the similarity.
The definition of a fuzzy nominal scale can be performed with an intermediate numerical space as a support
for the definition of the fuzzy meaning of each symbol of
the universe of discourse. For example, for colour measurement with a fuzzy scale, a colourimetric space can be used
to define the fuzzy meaning. In the following example, the
colours of a painting (see Fig. 2) are measured and represented in the Cie Lab space with their a and b coordinates.
Each fuzzy meaning is a fuzzy subset piecewise-defined on
a triangulation of the colourimetric space [15]. Fig. 3 shows
the fuzzy meaning of each symbol as follows: the membership function of a fuzzy meaning of a given symbol has a
degree 1 for the coordinate labelled with the associated la-
2925
Fig. 3. Example of fuzzy nominal scale defined by the fuzzy meaning of
its symbols. The colour histogram appears in the background.
bel, and a degree 0 for the other labelled coordinates. The
membership degrees of non-labelled coordinates are
interpolated.
3.3. Fuzzy metrical scales
Some quantities like colours, smells or software complexities are usually measured with inappropriate scales.
The theories chosen to abstract such quantities usually define a metric on the abstract space. In fact the abstract
space is an affine space even if this choice is not justified.
For example, colours are represented in many different
spaces like RGB, XYZ, Lab, HSV and the transformation
from one to another is not always an affine transformation.
Conversely, the metric, defined with psychophysic experiments, remains stable and is the most famous relation on
colours. The consequence is that theories including colours
must use scales with isometry groups as a structure but
not subgroups of an affine group.
We propose to build such metrical scales on the
basis of fuzzy scales. It is defined by the symbolism
0 hX; S; R; fX ; dX g; f¼; dS g; ðX ; S Þ; dX ; dS i where:
dS is a distance defined on symbols s of S by the theory.
0
dS is a distance defined on D(X) where the singleton
coincidence with dS is respected.
8ða; bÞ 2 S2 ; d0S 2 R0S ()
0
9dS 2 RS ; dS ðfag; fbgÞ ¼ dS ða; bÞ:
ð21Þ
The transportation distance dtp proposed in [17] can be defined at least on fuzzy scales whose bijective mapping f is
the identity function. It verifies also:
– The continuity property.
– The precision property that imposes that the distance
between two LFSs must be a positive real number.
– The consistency property that is usually verified by distances on crisp subsets.
Fig. 2. Van Gogh Irises painting as a source of colour measurement with a
fuzzy nominal scale.
This distance is computed as the solution for a mass
transportation problem where the masses are membership
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E. Benoit, L. Foulloy / Measurement 46 (2013) 2921–2926
degrees. Sources and destinations are symbols of the lexical set S and the unit cost from a source to a destination
is given by the distance dS. More generally, the transportation distance is applicable to measure the difference between the states of a system, the transformation of
which can be modelled into the mass transportation
problem.
3.3.1. Derived relations and operators
Given that the basic empirical relation is the determination of the equality between distances, it can be useful to
derive this relation in order to create a set of relations
and operators that can be used on measurement values obtained with a metrical scale. For example, a medium operator can be defined as:
mðA; BÞ ¼ fCjdðA; CÞ ¼ dðC; BÞ ¼ dðA; BÞ=2g
ð22Þ
Generally, the cardinal of m(A, B) depends on the metric
and could be equal to 0.
In the case of the transportation distance dtp(A, B), the
computation of the solution of the mass transportation
problem gives a set of shipments where each shipment
can be written as a triplet (source, destination,
quantity). The distance is then given by the total cost
of shipments:
dtp ðA; BÞ ¼
X
qij :cij
ð23Þ
ðsrci ;dstj ;qij Þ
where smt is the set of shipments, srci is the source symbol,
dstj is the destination symbol, qij is the quantity of transported material, i.e. the membership degree, and cij is the
unit cost that is actually the distance dS(srci, dstj). An element of m(A, B) is found by applying the same set of shipments to A, but with a total cost divided by 2. It guarantees
that m(A, B) is not empty, but generally this set is not a
singleton.
The definition of the vector median filters is based on
the distance between samples [18]. This operator is then
an interesting candidate to create a filter on LFSs given
by a metrical scale:
Let W = {a1, ..., an} be a set of elements of F(S), denoted
window, and Di be an aggregation of the distances between
ai and the other elements of the window.
Di ¼
n
X
dtp ðai ; aj Þ
ð24Þ
j¼1
The output of the median filter denoted MF is defined
by MF(W) = am such that Dm is the smallest Di.
The metric allows to define more or less specific sets of
measurement values and then to use statistical tools for
the management of uncertainty. However this field is too
large to be included in this paper.
4. Conclusion
As a consequence of Ludwik Finkelstein’s studies, we
explored the field of fuzzy scales and have provided a tentative inclusion concerning the representational theory of
measurement. Following his initiative to formalise this
theory with an extension to non-numerical representational spaces, in this paper we have turned to the group
theory-oriented classification suggested by Stevens and
we have provided a more recent view based on the recent
group theory. In this approach, we have included in the
scale classification the scales whose results are in multidimensional numerical spaces. We have also suggested
extending the study of scales to metrical scales expressing
their result in non-numerical spaces. These scales then appear as a third branch in the scale classification. The fuzzy
scales are presented as good candidates for this branch,
especially because they are able to hold relations such as
similarity relations or distance preservations.
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