Measurement 39 (2006) 1–11

www.elsevier.com/locate/measurement

Measurements, evaluations and preferences: A schemeof classification according to the representational theory

P. Cecconi, F. Franceschini *, M. Galetto

Politecnico di Torino, Dipartimento di Sistemi di Produzione ed Economia dell’Azienda,

Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received 30 April 2004; received in revised form 18 April 2005; accepted 21 October 2005Available online 23 November 2005

Abstract

The paper traces out a scheme for classification of the three ways by which observable or non-directly observable prop-erties of an object can be judged/described. The definition of measurement as an empirical and objective operation is thestarting point. We compare the concept of representational measurement with those of evaluation and preference. Evaluationmaintains the empiricity but not the objectivity of measurement: there is no unanimously acknowledged reference for thedescription of latent constructs. Preference is neither empirical nor objective: every subject has his/her own relation toexpress the judgment and this relation is not exogenously known. Representational theory is effective in giving a precisedefinition for empiricity but it is not exhaustive in defining objectivity in the case of preferences and evaluations. Finally,we trace possible research paths to be undertaken for further analysis of the proposed problem.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Representational theory of measurement; Measurement; Evaluation; Preference empiricity; Objectivity; Quality

1. Introduction

The properties of an object, either directly ornon-directly observable, may essentially be judgedand described by three basic operations: measure-ments, evaluations and preferences. The aim of thepresent work is to propose a scheme for classifica-tion of these three operations.

The question has a particular scientific impor-tance and some possible repercussions involve many

0263-2241/$ - see front matter � 2005 Elsevier Ltd. All rights reserved

doi:10.1016/j.measurement.2005.10.012

* Corresponding author. Tel.: +39 011 5647225; fax: +39 0115647299.

E-mail address: fiorenzo.franceschini@polito.it (F. Fran-ceschini).

disciplines like metrology, decision-making, andquality measurement.

Considering this latter aspect, when we ask asample of people to express opinions about thequality of a good, we carry out an operation whichis in suspense among measurement, preference andevaluation. It is necessary to know what the concep-tual paradigms at heart of these three operationsare.

The representational theory of measurementbased on the properties of binary relations is theused instrument of investigation.

Given a set of objects or alternatives A, a binaryrelation R on A is a subset of the Cartesian productA · A. Binary relations arise very frequently from

.

2 P. Cecconi et al. / Measurement 39 (2006) 1–11

everyday language: for example, if A is the set of allpeople in a certain country, then the set:

F ¼ fða; bÞ : a 2 A; b 2 A and a is brother of bÞg

defines a binary relation on A, which we may call‘‘brother of’’.

The properties (reflexivity, symmetry, transitiv-ity, etc . . .) of a generic relation R must be definedon a specific reference set. For example, the relation‘‘brother of’’ is symmetric on the set of all males in acertain country, but it is not symmetric on the set ofall people in that country. Formally, rather than asimple relation, we speak of a relational system

(A, R), that is a relation applied to a set of objects.This concept, introduced by Tarski [26], has beenthe natural vehicle for the subsequent developmentof the representational theory of measurement[23,14].

When a subject judges or describes objects (phys-ical or abstract), he/she considers one or more com-parison relations. These relations may be tangible orintangible, uniformly or not uniformly interpret-able. For example, the relations ‘‘more beautifulthan’’, ‘‘more elegant than’’, ‘‘worthier than’’ ‘‘pre-ferred to’’ are relations intangible and arbitrarilyinterpretable by different subjects, whereas the rela-tions ‘‘heavier than’’, ‘‘longer than’’, ‘‘warmer than’’are not. These latter are observable relations, whichdo not enable a free interpretation by the subjects.There is a direct reference to the scales of the Inter-national System of measurement.

In this work, the analysis of the differencesamong the three operations takes place consideringonly a specific property of the objects. ‘‘The subjectsof measurement are properties. Of course, propertiesexist only in connection with empirical objects. Usu-

ally, one object shows various properties. In measur-

ing one property, we neglect all the other properties

the object in question may have’’ [19].Moreover, the considerations here referred to are

exclusively limited to rankings. We only considerthe cases in which the individuals are able to estab-lish a priority ranking, that is a hierarchy amongobjects.

2. Two criteria of discrimination between the threeoperations: Empiricity and objectivity

‘‘Measurement is the assignment of numbers to prop-

erties of objects or events in the real world by means of

an objective empirical operation, in such a way as to

describe them. The modern form of measurement the-

ory is representational: numbers assigned to objects/

events must represent the perceived relations between

the properties of those objects/events’’ [8].

The definition of a measurement refers to twofundamental concepts: empiricity and objectivity.This allow us to discriminate between the threeoperations: measurements, evaluations and prefer-ences.

The empiricity arises when a judgment of a rank-ing ‘‘is the result of observations and not, for exam-

ple, of a thought experiment. Further, the concept

of the property measured must be based on empiri-

cally determinable relations and not, say, on conven-

tion’’ [9].There is empiricity when the type of relation is

observable, that is the property of the object provesto be, in a precise moment, in a well-defined statecharacterized without ambiguity. Empiricity meansthat there is ‘‘an objective rule for classifying some

aspect of observable objects as manifestations of the

property’’ [9].The objectivity concerns the kind of results that

the judgment produces, ‘‘within the limits of errorindependent of the observer’’ [7]. ‘‘Experiments may

be repeated by different observers and each will get

the same result’’ [25]. Full objectivity means inde-pendence of the subjects. The result of the operationgives only information about the measuredproperty.

The measurement requires both empiricity andobjectivity. It is an operation of objective descrip-tion: the results of n different measurements, in thesame operating conditions, are univocal and inde-pendent by subjects. We suppose there is no ‘‘error’’and uncertainty in an ideal measurement process(the environmental and other influential variablesare considered non-existent). It is also an empiri-cal operation: ‘‘Measurement has something to

do with assigning numbers that correspond to orrepresent or ‘‘preserve’’ certain observed relations’’[20].

The preference is neither empirical nor objective.Preferences are, by definition, subjective and con-flicting. We are not able to know exogenously, indetail, the relation that each subject applies whenassigning a ranking. An outside observer will haveconsiderable difficulties in interpreting the resultsgenerated by this kind of operation. In other words,different subjects interpret the relation in differentways and can establish disagreeing orderings. In this

Table 1Scheme of classification of the three operations

Objective Empirical

Measurement Yes YesEvaluation No YesPreference No No

Q

qn

R P

N

F

qm

nn

nm

Ri Pi

M

Fig. 1. Homomorphism of L onto N . Two elements in Q may bemapped into the same number in N.

P. Cecconi et al. / Measurement 39 (2006) 1–11 3

case, the uncertainty concerns deeply the kind ofrelation applied by each subject.

The evaluation is somewhere between measure-ment and preference. It is not objective becauseevaluations are individual perceptions, performedwithout the use of an univocal instrument of mea-surement. Nevertheless, it is an operation that wantsto be empirical: the meaning of intangible relationsis circumscribed by means of an exogenous processof semantic definition from the outset. Subjects arecalled on to conform to this process. Operatively,there is uncertainty in the interpretation that sub-jects give to the provided dimension of observation.

The three operations can be classified as illus-trated in Table 1:

As explanation, it is convenient to make explicitthe meaning of the following terms [2]:

• Exogenous: The expression of a description/ordering is subordinated to a coactive, explicitand declared constraint. An outside observerimposes rules (concerning the dimension ofobservation, interpretation of the scales, etc . . .)to which subjects conform from the outset.

• Endogenous: The expression of a description/ordering happens according to a latent, implicit,non-declared point of view. Each subject decidesto adopt the rules he considers more convenient,without declaring them.

3. The representational definition of measurement

The assumption that the relations are observableis at heart of the representational point of view.

In a measurement, subjects are called on to‘‘judge’’ an observable relation, on which there areno doubts about meaning and interpretation. Rela-tions like ‘‘longer than’’, ‘‘heavier than’’, ‘‘warmerthan’’, etc. are some possible examples.

The representational theory of measurementrelated to a quality or a property of an object hasfour fundamental parts [9]:

(1) An empirical relational system. Consider somequality (for example the length of an object)and let qi represent an individual manifestationof the quality Q, so that we can define a set ofall possible manifestations as Q = {q1 . . .}. Letthere be a family of empirical relations Ri onQ, R = {R1, . . . ,Rn}. Then the quality can berepresented by an empirical relational systemL ¼ hQ;Ri.

(2) A numerical relational system. Let N represent aclass of numbers N = {n1 . . .}. Let there be afamily of relations P = {P1, . . . ,Pn} defined onN. Then N ¼ hN ; P i represents a numericalrelational system.

(3) A representation condition. Measurement isdefined as an objective empirical operation suchthat L ¼ hQ;Ri is mapped homomorphicallyinto (onto) N ¼ hN ; P i by M and F. Specifi-cally, M is the function mapping Q to N, sothat nh = M(qh) (M :Q ! N). F is the func-tion mapping one-to-one the relations of R onP (F :R! P).The above homomorphism is the representationcondition.Firstly, it implies that if qh is related to qm by anempirical relation Rk, that is Rk(qn,qm), Pk isthe numerical relation corresponding to Rk,nh = M(qh) is the image of qh in N under M thenRk(qh,qm) implies and is implied by Pk(nh,nm).Measurement is a homomorphism—not an iso-morphism—becauseM is not a one-to-one func-tion. It maps separate but indistinguishableproperty manifestations into the same number.S ¼ hL;N ;M ; F i constitutes a scale of measure-ment for L.The definition of a representational measure-ment is illustrated in Fig. 1.

4 P. Cecconi et al. / Measurement 39 (2006) 1–11

‘‘In measurement we start with an observed or

empirical relational system and we seek a map-

ping to a numerical relational system which ‘‘pre-

serves’’ all the relations and operations observedin the empirical one’’ [20].

‘‘Whatever inferences can be made in the numer-

ical relational system apply to the empirical one’’[5].

(4) Uniqueness condition. The representation condi-tion may be valid for more than one mappingfunction M. There are admissible transforma-tions from one scale to another scale withoutinvalidating the representation condition. Theuniqueness condition defines the class of trans-formations for which the representation condi-tion is valid. For ordinal scales, all monotoneincreasing functions are admissible transforma-tions [9].

3.1. An example of ordinal measurement

A measurement can be viewed as a representa-tion of some properties of the real world by sym-bols; nominal and ordinal scales are examples ofrepresentation by non-numeric symbols. The abilityto order a set of objects/events is related to thenotion of the ‘‘amount’’ of a property manifestedby each one. In ordinal measurement, the relation‘‘�’’ (‘‘it has got the property more than’’) is transi-tive and reflexive. Therefore,

q � r () MðqÞ > MðrÞConsider the hardness measurement of minerals(the relation R ‘‘harder than’’). It is a typical exam-ple of ordinal measurement.

The Mohs scale orders minerals from diamond totalc, on the basis of which scratches which. The abil-ity to scratch (i.e. to etch, to cut in surface) is theempirical relation and the ordering is the formalrelation.

The scale is built as follows: ten standard miner-als are arranged in an ordered sequence so that

“In order to improve the quality of a produ

strongly moderately slightly neither agdisagree disagree disagree nor disag

-3 -2 -1 0

Fig. 2. An example of a measureme

precedent ones in the sequence can be scratchedby succeeding ones and cannot scratch them. Thestandards are assigned numbers 1–10. The sequenceis talc 1, gypsum 2, calcite 3, fluorite 4, apatite 5,orthoclase 6, quartz 7, topaz 8, corundum 9, dia-mond 10. A mineral sample of unknown hardnesswhich cannot be scratched by quartz and cannotscratch it, is assigned measure 7 [7].

3.2. Some clarifications about the concept of

measurement

The homomorphism is a faithful representation ofempirical relations by symbolic relations. The con-dition of homomorphism, namely of an assignmentof symbols (in general numbers) to objectsneventsaccording to the degree of presence of a certainproperty, is considered by the followers of the repre-sentational theory as a necessary and sufficient con-dition to define a measurement.

Some authors [5] assert that not all possible rulesof assignment yield right measurements. The assign-ment of numbers is a representational measurementonly if three requirements are satisfied. ‘‘The assign-ment of numbers must:’’

(a) be orderly;

(b) represent meaningful attributes;(c) yield meaningful predictions.

According to the authors, the presence of thesethree conditions defines the ‘‘automatic consistency

check’’, which realizes the difference between arepresentational measurement and non-represen-tational one: ‘‘when mineral (a) scratches mineral

(b), then (a) is represented above (b) in the order’’[5].

If consistency check fails, we cannot speak aboutrepresentational measurements. Consider, for exam-ple, a subject expressing his/her own opinion abouta given product. The judgment is stated on a rat-

ing scale of the type indicated in Fig. 2. In this

ct we have to mark down its price”

ree slightly moderately stronglyree agree agree agree

+1 +2 +3

nt with no consistency check.

P. Cecconi et al. / Measurement 39 (2006) 1–11 5

particular case, suppose that the subject selects thelabel ‘‘+3’’.

This statement ‘‘does not represent a specific

behaviour’’ of the subject. In fact, we may wish tomake many inferences on the basis of this behav-iour. For example that he/she will turn his/herown attention to those firms adopting this policy,that he/she is deceiving himself/herself about some-thing that will not happen, or that he/she believes inpaying suppliers by manufacturing money. ‘‘But wecannot make a firm prediction about some other

response to this or another rating scale. There is noconsistency check, hence no representational mea-

surement’’ [5].Vice versa, measurements of hardness, mass,

length, etc. present a consistency check. This checkis performed by an appropriate measurement sys-tem realizing the homomorphism from the empiricalrelational system to the numerical one.

The measurement system guarantees the twofundamental components of ameasurement: ‘‘assign-ment and empirical determination’’ [16]. It circum-scribes without ambiguity the empirical relations(harder than, heavier than, longer than, etc . . .), andperforms the assignment of numbers to the objectsaccording to the rule of the corresponding homo-morphism.

The presence of a conventional, non-ambiguous,empirical reference is the reason at the base of theobjectivity of measurement results.

Property level

References for theproperty levels

level level level 1 2 3

Fig. 3. Basilar scheme of evaluation o

4. Evaluations

In general, the qualities of an object/event can beclassified as physical or non-physical, observable ornon-directly observable. The utility, the quality of aperformance, the customer satisfaction, the attitude,etc. are examples of non-physical and non-directlyobservable magnitudes. Attempts to measure theselatter cause many problems.

The first problem is ‘‘the difficulty of establishing

an adequate objective concept, or theoretical con-

struct, of these qualities based on empirical opera-tions’’ [7].

The description of these attributes or latent con-structs is not objective. It may origin a free interpre-tation of the meaning by the subjects.

In these situations, we cannot speak about realmeasurement, but just about evaluations or attribu-tions of values to individual judgments.

Descriptions about non-tangible qualities arepossible only by means of subjective judgmentsexpressed on adequate scales (Fig. 3).

Fig. 3 illustrates the concept of evaluation of anon-physical magnitude. The subject ideally com-pares the object (first scale pan) with the referenceterms on the scale (second scale pan). The evalua-tion consists in identifying the judgment on the scalewhich balances the two scale pans.

The first fundamental component of a measure-ment (empiricity) runs short. Consider, for example,

Reference X

Subject’sjudgments

level level 4 5

Term of comparison

f a non-physical magnitude [12].

6 P. Cecconi et al. / Measurement 39 (2006) 1–11

the aesthetical beauty of an object: in this case‘‘there is not an objective rule for classifying some

aspect of observable objects as manifestations of the

beauty. Similarly, there are no objective empiricalrelations such as indistinguishability or precedence,

in respect of beauty. The basis for the measurement

of beauty is thus absent from the outset’’ [7].The aim of an evaluation is that of ‘‘building’’

and ‘‘imposing’’ to evaluators, in some way, thisempiricity component.

This way, Pawson [18] asserts that: ‘‘First, evalu-ation deals with the real, that is we evaluate thingsand empirical relations about things. Secondly, evalu-ation should follow a realist methodology. Thirdly,evaluation, perhaps above all, needs to be realistic’’.

4.1. The evaluation of non-tangible qualities: The

exogenous imposition of rules for the subjects

Let A be a set of objects and R the relation ‘‘hea-vier than’’. That is, for any pair of object x, y in A

we define:xRy x is heavier than y

‘‘Note that R can be defined either by a balance or bypsychophysical experiments using an observer to com-

pare the weights. The two procedures yield similar

empirical relational systems with the same object

set. The interpretation of the relation ‘‘heavier than’’,

however, is physical in the former system and psycho-

logical in the latter’’ [4].

In the second case, we should speak ofevaluations.

The evaluation is typically a normative process.It is based on exogenous rules driving subjects inthe attribution of values to intangible and interpret-able qualities. Essentially, there are two kinds ofrules:

• clear and precise definition of non-physical attri-butes (abstract constructs);

• definition of operating evaluation scales.

The first fundamental step of an evaluation pro-cess consists in the definition of a reference axis. Theevaluation process wants to make the initial latentconstruct observable and less interpretable, givingto it ‘‘empirical substance’’, by specific empiricalrules. The second phase consists in providing suit-able evaluation scales.

An evaluation requires that there is ‘‘uniformity’’for subjects in accepting the rules provided. This

uniformity does not exist in preference judgments,where everyone is free to choose the interpretationwith which the judgment takes place.

4.2. Evaluation: A subjective homomorphism

We said that evaluation is a partially empiricaland subjective operation. We are going to justifythis position by means of the representationaltheory.

Psychological tests or Questionnaires for theevaluation of product or service quality are exam-ples of evaluation processes.

Paraphrasing measurement definition, the evalu-ation becomes the assignment of numbers, or labelsto properties or events of the real world by means ofan empirical subjective operation, in such a way asto describe them.

In the evaluation context, the mapped relation R

is not empirical. The relation becomes empirical bymeans of a set of semantic (what the constructmeans) and operating rules (evaluation scale). Weexogenously impose the dimension or relation thatsubjects have to observe.

What is perceived as ‘‘having the property morethan’’, in the empirical world, has an immediatetranslation in the ranking performed by the subject.In ordinal evaluation, there is a homomorphismfrom the empirical world onto the symbolic(numeric) one, that is to say, there is a faithful rep-resentation of empirical objects and their relationsin the numerical world. The representational formis, therefore, maintained. From this point of view,the homomorphism is not able to formalize the dif-ference between the operation of evaluation andmeasurement [17].

The question is that we deal with a subjectiveoperation. The ranking performed by a subject willnot coincide with that of another subject. It is notunivocal. Different subjects can observe differentdegrees of the property in the same object.

This dependence may cause some important for-mal consequences.

The concepts of indifference threshold d andthe distinction between the indifference relation I

(reflexive and symmetric) and the equivalence rela-tion E (reflexive, symmetric and transitive) arestrictly connected with each subjective operation.This latter distinction supports the differencebetween evaluation and measurement.

There is an indifference relation between twostimuli-objects (for example, two sounds of different

P. Cecconi et al. / Measurement 39 (2006) 1–11 7

intensity) every time each subject is not able to dis-criminate them.

The equivalence relation E is stronger than theindifference relation I. It does not confine itself sim-ply to asserting the non-difference between two ele-ments (for example the two auditory stimulimentioned before), but it establishes their equality[20].

The different significance of the two relations isfundamental in psychological evaluations. Imaginethree weigths a, b and c; if, weighing them by hands,aIb (a is indifferent to b) and bIc, it cannot be saidthat also aIc. A little difference between a and b

and between b and c can become noticeable whenthe conjoint effects of the single differences are con-sidered. The absence of transitive property in theindifference relation shows this possibility. Thisproperty, on the other hand, belongs to the equiva-lence relation E: if aEb (a is equivalent to b) andbEc, then aEc.

The problem of discrimination between the equiv-alence and indifference relation is connected to theconcept of ‘‘Just Noticeable Differences’’, originallyintroduced by Fechner [6], and to the definition ofsemiorders [15]. Just Noticeable Differences arisewhen different stimuli are judged indifferent from asubject who is not able to discriminate among them.These differences become noticeable using measure-ment instruments more accurate than a simple sub-jective evaluation. The homomorphism performedby the single subject will not coincide with thehomomorphism performed by means of an ade-quate measurement system.

4.3. Problems and questions still open

Some fundamental problems arise from ordinalevaluations.

4.3.1. Dimension of representation ‘‘compatible’’ with

the way of thinking of subjects

The priority aim of an evaluation consists inyielding empirical predictions, for example, aboutthe purchasing intentions of customers, the under-taking of an investment, etc. Therefore, it is neces-sary that the provided dimension of representationis compatible with the way of thinking of the sub-jects, otherwise the evaluations will be neither signif-icant nor useful.

Setting up a good evaluation scale means havingto know the dimension of representation which ismost important for the subjects. ‘‘Clearly, not all

rating scales are compatible with intuitive thought,nor does compatibility imply that rating scales are

isomorphic with such thought’’ [5].

4.3.2. The design of a scale reflecting the real

capacity of discrimination of subjects

This problem mostly arises from rating scaleswith enumerated categories, where the subjectexpresses himself on a verbal category and not ona linear continuum. In fact, problems of interpreta-tion of the categories meaning can subsist. A verballabel, ‘‘very satisfying’’, may not have the samemeaning for both a very demanding subject andone who is easy to please. This poses a real problemto codify information, being usually unknown theinterpretation of the scale adopted by each subject[12].

The response category can be interpreted by sub-jects as too wide or too narrow. When the categoryis too wide, the subject is forced to classify as equalobjects that he perceives as slightly different. In thiscase, there are no representational measurements,because a faithful representation of observed rela-tions in numerical relations does not occur: ‘‘it is

essential that the relations among the objects of the

world be properly reflected by the relations among

the numbers assigned to them’’ [4].The opposite problem can also appear. The

response categories could be too detailed and thesubject is confused in giving an answer. He may findhimself in more than one response category.

The adequate definition of the width and of themeaning of each scale category brings evaluationsto the stage of homomorphisms from an empiricalrelational system to a numerical one, that is tosay, representational evaluations. This resultremains an actionable target only after ‘‘numerous

interactions with evaluator subjects’’ [7].

4.3.3. The aggregation of evaluations expressed by

many subjects

This problem is connected with any subjectiveoperation. The aggregation of individual rankingsin a social global ranking is object of study of manydisciplines: social and behavioural sciences, opera-tional research and economics [11,13,21].

5. Preference

Preference is the act by which, in presence of twoor more possible objects, one of them can be chosen

8 P. Cecconi et al. / Measurement 39 (2006) 1–11

over the other, because it is considered more pleas-ant, more convenient, more conform to own tastes,interests, ideals, etc.

‘‘Preference is necessarily relative to a subject. A

preference is always somebody�s preference. A prefer-

ence, moreover, is relative not only to a subject but

also to a certain moment or occasion or situation in

the life of a subjects. Not only may have different peo-ple with different preferences, but one and the same

man may revise his preferences in the course of his

life . . . the concept of preference is related to the

notion of betterness’’ [27].‘‘Preferences, to a greater or lesser extent, govern

decisions . . . into our axiomatic system an individual�spreference relation on a set of alternatives enters as a

primitive or a basic notion. This means that we shallnot attempt to define preferences in terms of other

concepts . . .preferences between decision alternatives

might be characterized in terms of several factors

relating to the alternatives’’ [10].

When a subject says that he/she prefers alterna-tive a to b, he/she makes a relation between a andb which seems perfectly mouldable with the mathe-matic notion of a binary relation.

Suppose A is a collection of alternatives amongwhich you are choosing, and suppose

P ¼ fða; bÞ 2 A� A : youðstrictlyÞprefer a to bg

Then P is called (strict) preference relation.If A is a set of alternatives, aPb holds if and only

if you prefer (strictly) a to b, it is possible to assign areal number u(a) to each a 2 A, such that for all a,b 2 A

aPb () uðaÞ � uðbÞ

The function u is often called ordinal utilityfunction.

This assignment allows the relation ‘‘preferredto’’ of the single subject to be observed.

Nevertheless, as Roberts asserts, ‘‘often, ‘‘pre-ferred to’’ does not define a relation’’ (20, pp. 272–273). With these remarks, Roberts wants to under-line the absolute peculiarity of this relation thatenjoys neither the property of consistency nor thatof transitivity.

Consider the following example: a subject iscalled on to vote among three candidates A, B andC. If the subject prefers alternative A to alternativeB and B to C, he/she will not necessarily preferalternative A to C, as the transitivity property

requires. It can occur that C is preferred to A. Inthe background concerning the problems connectedwith the aggregation of preferences, the situationjust described is noted as Condorcet Paradox [3].

The subject assigning preferences could have animplicit model of preferences such that it can notbe mapped on any numerical structure (it can con-tain intransitivity chains).

The lack of transitivity often arises for intangiblerelations like ‘‘preferred to’’, ‘‘more beautiful than’’,‘‘more elegant than’’, etc. . . . relations arbitrarilyinterpretable, because non-directly observable.

5.1. The impossibility of the representational form

for preferences

What differences emerge comparing the defini-tion of preference with that of measurement?

In general, we are faced with a preference assign-ment every time the subject is called on to perform aranking among things without a ‘‘measurement sys-tem’’ or a set of predefined rules, such as in the eval-uation process. The preference ranking amongalternatives is the result of an endogenous activityof a subject who chooses, in an arbitrary way,how to represent the relation.

Preference becomes ‘‘an arbitrary measurement

conceived as a decision-making activity’’ [22].Examples of arbitrarily interpretable relations

are: ‘‘worthier than’’, ‘‘better than’’, ‘‘more beauti-ful than’’, etc.

The assignment of preferences is neither empiri-cal, nor objective. It does not deal with an empiricaloperation because the subject chooses arbitrarily therelation considered remarkable for the ranking. Theabove choice is completely endogenous, differentfrom subject to subject. We are not able to under-stand exogenously what is the dimension of repre-sentation selected and followed by the subject andthe interpretation he/she has given to it.

There is not a transfer of observable relationsinto numerical relations. We cannot speak of ahomomorphism as defined by the representationalpoint of view. There is no mapping onto a numericalrelational system ‘‘preserving both relations (andoperations) observed in the empirical relational sys-

tem’’ [20].Paraphrasing a Stevens� expression, preference

can be defined as ‘‘a measurement according to any

rule’’ [24]. Subjects may have chosen to observeone of the infinite possible relations on the objectsin order to perform a preference ranking.

P. Cecconi et al. / Measurement 39 (2006) 1–11 9

It is obvious that this operation is completelysubjective: ‘‘the measurement value is not so much a

property of the thing measured as something which

expresses an appreciation of the measurer towardsthe thing itself. What counts, does not count becauseit counts in itself, but because it is judged to count by

someone’’ [16].When subjects report their own opinions about

not adequately detailed constructs, an attributionof values to individual preference judgments takesplace.

Constructs like the utility of a service, the aes-thetic of a product, the guidablity of a vehicle, mustbe specified and detailed to transform preferencesinto evaluations. Otherwise, each subject will inter-pret the construct as he/she considers more con-venient. The reason for this is a ‘‘semantic

ambiguity’’ in the constructs [5]. As an example,consider the responses to the following two attitudequestions:

• ‘‘The adoption of a Certified Quality System is a

necessary burden’’.• ‘‘Advantages of a Certified Quality System are lar-

ger than disadvantages’’.

A positive answer to both items seems to indicatea favourable attitude of the management towardquality certification, but how should that responsebe interpreted and represented? Who answers‘‘yes’’ has a mildly positive attitude. Those witheither a strongly positive or strongly negative atti-tude would answer ‘‘no’’. In contrast, an affirmativeresponse to the latter item should be interpreted asmeaning that the responders� favorability surpassesneutrality, but how far we do not know. ‘‘The pointis that they involve choice based on semantic knowl-

edge, our semantic knowledge. There is nothing in

the observation of affirmative answers themselves that

dictates how they are to be interpreted and repre-

sented’’ [5].The presence of this ambiguity is the main differ-

ence between preference and evaluation. In an eval-uation process we will try to circumscribe thissemantic ambiguity, fully defining in this case theconcepts of ‘‘necessary burden’’, ‘‘advantage’’ and‘‘disadvantage’’.

We say that a preference has not the empiricityand objectivity requirements of a measurement.However, the representational theory considers thatrepresentational measurement of preference are pos-sible. It is important to concisely define this posi-

tion. When subjects assign a ranking to theelements of a certain set, they make their own pref-erences explicit and consequently their own relation‘‘preferred to’’ on that set is made observable.According to the representational theory, thisassignment is a real representational measurement.It is viewed as a homomorphic representation ofthe relation ‘‘preferred to’’ from the empirical worldof the objects into the numerical world (even if therelation is not explicit in the empirical domain).

Roberts [20] presents the case of preferencesamong classical music composers. The authorpoints out that it is possible to speak of representa-tional measurements of individual preferences whenthese satisfy the axioms of Cantor�s theorem. Givena set of objects and defined a relation of preferenceP on it, the relational system (A,P) has to satisfy thefollowing conditions:

Asymmetry: if aPb ) �bPa, "a,b 2 A (naturalproperty of preferences).

Negative transitivity: if �aRb & �bRc ) �aRc,"a,b,c 2 A.

Axioms of Cantor�s theorem are necessary andsufficient conditions to have ordinal representa-tional measurements.

However, enacting Roberts� position, ‘‘some

information can be obtained on the evaluating sub-

ject, about their way of seeing things, but surelynot on the elements of the set, i.e. the empirical

world’’ [17]. Roberts does not dwell on the analysisof the meaning of the relation but he/she simplyanalyses and interprets the results of the representa-tion. The author seems to neglect that the relation‘‘preferred to’’ is not empirical because it is inter-pretable and therefore observable in an arbitraryway.

The problem is that we do not know what is themeaning of the relation ‘‘preferred to’’, in terms ofempirical relations. Therefore, an empirical rela-tional system for the preference cannot be identified.

6. Conclusions

Preference, evaluations and measurements arethree basic operations by which it is possible todescribe objects. The difference between them arethe different way of treating relations amongobjects.

In the case of preference, relations are thought bysubjects and transferred directly to the final rankingunder an endogenous decision-making activity,different from subject to subject. The inability to

10 P. Cecconi et al. / Measurement 39 (2006) 1–11

exogenously know the way in which things havebeen interpreted by subjects leads to the conceptof non-representational measurements (the represen-tation is unknown).

Measurements set the empirical relations thatmust be represented without ambiguity by meansof a measurement system which realizes the ‘‘inter-nal consistency check’’ and makes feasible the funda-mental requirement of objectivity.

On the other hand, it is the aim of an evaluationprocess to provide all possible ‘‘tools’’ in order toreduce the problems of semantic interpretation ofthe items and of the scale categories. In this way,individual judgments are treated as representationalevaluations.

The notion of homomorphism, defining represen-tational measurements, is effective to mark the dif-ference between preference and measurement, butit does not define a clear border between the opera-tion of measurement and evaluation. The source ofthe difference between the two operations is thatevaluations give space to the possibility of choosingamong different dimensions of representation fromthe outset.

Representational theory allows giving a preciseand effective definition of empiricity in the threeoperations, but not that of objectivity. Someauthors have emphasized the inability to incorpo-rate the fundamental requirement of objectivity ofa measurement in the formalization of the approachas the limit of the representational point of view.This way, someone claims the necessity of an oper-ational-representational approach for the sake of anexhaustive definition of measurement, with theintroduction of a measurement system [17].

Particular attention should be posed to the prob-lem of aggregation of individual evaluations or pref-erences into ‘‘social’’ or group results. Problems ofaggregation do not appear in the field of measure-ments because of their objective nature. On the con-trary, an abundant literature about the aggregationof preferences is available. In this framework,renowned for its relevance, remains the ArrowImpossibility Theorem (1951) which represents themilestone of Social Choice Theory. By means ofthe theory of relations, Arrow proved the non-exis-tence of aggregation mechanisms of individual pref-erences into a social preference such that fourfundamental axioms are satisfied: UnrestrainedDomain of Preferences, Independence of irrelevantalternatives, Pareto weakness, and Non-dictatorship[1].

With the aim of tracing a possible path for thefuture research, the traditional problem of aggrega-tion of individual preferences can be reformulatedin the problem of the aggregation of individualevaluations in order to see if the imposition of exog-enous bonds relaxes Arrow�s conditions. In sub-stance, it is natural wondering if Arrow�s Theoremis still valid when the input is changed from prefer-ences into ‘‘canalized’’ preferences as evaluations.

Arrow�s theorem can be viewed as a second pos-sible instrument of investigation to highlight furtherthe difference between preferences and evaluations.

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