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CHAPTER 1

Representational Measurement Theory

R. DUNCAN LUCE AND PATRICK SUPPES

CONCEPT OF REPRESENTATIONALMEASUREMENT

Representational measurement is, on the onehand, an attempt to understand the nature ofempirical observations that can be usefullyrecoded, in some reasonably unique fashion,in terms of familiar mathematical structures.The most common of these representing struc-tures are the ordinary real numbers ordered inthe usual way and with the operations of ad-dition, +, and/or multiplication, ·. Intuitively,such representations seems a possibility whendealing with variables for which people havea clear sense of “greater than.” When data canbe summarized numerically, our knowledgeof how to calculate and to relate numbers canusefully come into play. However, as we willsee, caution must be exerted not to go beyondthe information actually coded numerically. Inaddition, more complex mathematical struc-tures such as geometries are often used, forexample, in multidimensional scaling.

On the other hand, representational mea-surement goes well beyond the mere construc-tion of numerical representations to a carefulexamination of how such representations re-late to one another in substantive scientific

The authors thank Janos Aczel, Ehtibar Dzhafarov, Jean-Claude Falmagne, and A.A.J. Marley for helpful com-ments and criticisms of an earlier draft.

theories, such as in physics, psychophysics,and utility theory. These may be thought ofas applications of measurement concepts forrepresenting various kinds of empirical rela-tions among variables.

In the 75 or so years beginning in 1870,some psychologists (often physicists or phy-sicians turned psychologists) attempted toimport measurement ideas from physics, butgradually it became clear that doing this suc-cessfully was a good deal trickier than wasinitially thought. Indeed, by the 1940s a num-ber of physicists and philosophers of physicsconcluded that psychologists really did notand could not have an adequate basis for mea-surement. They concluded, correctly, that theclassical measurement models were for themost part unsuited to psychological phenom-ena. But they also concluded, incorrectly, thatno scientifically sound psychological mea-surement is possible at all. In part, the theoryof representational measurement was the re-sponse of some psychologists and other socialscientists who were fairly well trained in thenecessary physics and mathematics to under-stand how to modify in substantial ways theclassical models of physical measurement tobe better suited to psychological issues. Thepurpose of this chapter is to outline the highpoints of the 50-year effort from 1950 to thepresent to develop a deeper understanding ofsuch measurement.

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2 Representational Measurement Theory

Empirical Structures

Performing any experiment, in particular apsychological one, is a complex activity thatwe never analyze or report completely. Thepart that we analyze systematically and re-port on is sometimes called a model of thedata or, in terms that are useful in the the-ory of measurement, an empirical structure.Such an empirical structure of an experimentis a drastic reduction of the entire experi-mental activity. In the simplest, purely psy-chological cases, we represent the empiricalmodel as a set of stimuli, a set of responses,and some relations observed to hold betweenthe stimuli and responses. (Such an empiricalrestriction to stimuli and responses does notmean that the theoretical considerations areso restricted; unobservable concepts may wellplay a role in theory.) In many psychologicalmeasurement experiments such an empiricalstructure consists of a set of stimuli that varyalong a single dimension, for example, a setof sounds varying only in intensity. We mightthen record the pairwise judgments of loud-ness by a binary relation on the set of stimuli,where the first member of a pair representsthe subject’s judgment of which of two soundswas louder.

The use of such empirical structures inpsychology is widespread because they comeclose to the way data are organized for subse-quent statistical analysis or for testing a theoryor hypothesis.

An important cluster of objections to theconcept of empirical structures or models ofdata exists. One is that the formal analysisof empirical structures includes only a smallportion of the many problems of experimen-tal design. Among these are issues such asthe randomization of responses between leftand right hands and symmetry conditions inthe lighting of visual stimuli. For example, inmost experiments that study aspects of vision,having considerably more intense light on the

left side of the subject than on the right wouldbe considered a mistake. Such considerationsdo not ordinarily enter into any formal de-scription of the experiment. This is just thebeginning. There are understood conditionsthat are assumed to hold but are not enumer-ated: Sudden loud noises did not interfere withthe concentration of the subjects, and neitherthe experimenter talked to the subject nor thesubject to the experimenter during the collec-tion of the data—although exceptions to thisrule can certainly be found, especially in lin-guistically oriented experiments.

The concept of empirical structures is justmeant to isolate the part of the experimentalactivity and the form of the data relevant tothe hypothesis or theory being tested or to themeasurements being made.

Isomorphic Structures

The prehistory of mathematics, beforeBabylonian, Chinese, or Egyptian civiliza-tions began, left no written record but none-theless had as a major development the con-cept of number. In particular, counting ofsmall collections of objects was present. Oralterms for some sort of counting seem to existin every language. The next big step was theintroduction, no doubt independently in sev-eral places, of a written notation for numbers.It was a feat of great abstraction to developthe general theory of the constructive opera-tions of counting, adding, subtracting, multi-plying, and dividing numbers. The first prob-lem for a theory of measurement was to showhow this arithmetic of numbers could be con-structed and applied to a variety of empiricalstructures.

To investigate this problem, as we do inthe next section, we need the general no-tion of isomorphism between two structures.The intuitive idea is straightforward: Twostructures are isomorphic when they exhibitthe same structure from the standpoint of


Concept of Representational Measurement 3

their basic concepts. The point of the formaldefinition of isomorphism is to make this no-tion of same structure precise.

As an elementary example, consider abinary relational structure consisting of anonempty set A and a binary relation R de-fined on this set. We will be considering pairsof such structures in which both may be empir-ical structures, both may be numerical struc-tures, or one may be empirical and the othernumerical. The definition of isomorphism isunaffected by which combination is beingconsidered.

The way we make the concept of havingthe same structure precise is to require the ex-istence of a function mapping the one struc-ture onto the other that preserves the binaryrelation. Formally, a binary relation structure(A, R) is isomorphic to a binary relation struc-ture (A′, R′) if and only if there is a functionf such that

(i) the domain of f is A and the codomainof f is A′, i.e., A′ is the image of Aunder f,

(ii) f is a one-one function,1 and

(iii) for a and b in A, aRb iff2 f (a)R′ f (b).

To illustrate this definition of isomorph-ism, consider the question: Are any two finitebinary relation structures with the same num-ber of elements isomorphic? Intuitively, itseems clear that the answer should be neg-ative, because in one of the structures all theobjects could stand in the relation R to eachother and not so in the other. This is indeedthe case and shows at once, as intended, thatisomorphism depends not just on a one-onefunction from one set to another, but alsoon the structure as represented in the binaryrelation.

1In recent years, conditions (i) and (ii) together havecome to be called bijective.2This is a standard abbreviation for “if and only if.”

Ordered Relational Structures

Weak Order

An idea basic to measurement is that the ob-jects being measured exhibit a qualitative at-tribute for which it makes sense to ask thequestion: Which of two objects exhibits moreof the attribute, or do they exhibit it to the samedegree? For example, the attribute of havinggreater mass is reflected by placing the twoobjects on the pans of an equal-arm pan bal-ance and observing which deflects downward.The attribute of loudness is reflected by whichof two sounds a subject deems as louder orequally loud. Thus, the focus of measurementis not just on the numerical representation ofany relational structures, but of ordered ones,that is, ones for which one of the relations is aweak order, denoted �∼, which has two defin-ing properties for all elements a, b, c in thedomain A:

(i) Transitive: if a �∼ b and b �∼ c, then a �∼ c.

(ii) Connected: either a �∼ b or b �∼ a or both.

The intuitive idea is that �∼ captures the order-ing of the attribute that we are attempting tomeasure.

Two distinct relations can be defined interms of �∼:

a � b iff a �∼ b and not (b �∼ a);a ∼ b iff both a �∼ b and b �∼ a.

It is an easy exercise to show that � is transi-tive and irreflexive (i.e., a � a cannot hold),and that ∼ is an equivalence relation (i.e.,transitive, symmetric in the sense that a ∼ biff b ∼ a, and reflexive in the sense thata ∼ a). The latter means that ∼ partitions Ainto equivalence classes.

Homomorphism

For most measurement situations one reallyis working with weak orders—after all, twoentities having the same weight are not in gen-eral identical. But often it is mathematically



easier to work with isomorphisms to the or-dered real numbers, in which case one mustdeal with the following concept of simple or-ders. We do this by inducing the preferenceorder over the equivalence classes defined by∼. When ∼ is =, each element is an equiva-lence class, and the weak order is called asimple order. The mapping from the weaklyordered structure via the isomorphisms ofthe (mutually disjoint) equivalences classesto the ordered real numbers is called a ho-momorphism. Unlike an isomorphism, whichis one to one, an homomorphism is many toone. In some cases, such as additive conjointmeasurement, discussed later, it is somewhatdifficult, although possible, to formulate thetheory using the equivalence classes.

Two Fundamental Problemsof Representational Measurement

Existence

The most fundamental problem for a theory ofrepresentational measurement is to constructthe following representation: Given an empir-ical structure satisfying certain properties, towhich numerical structures, if any, is it iso-morphic? These numerical structures, thus,represent the empirical one. It is the existenceof such isomorphisms that constitutes therepresentational claim that measurement ofa fundamental kind has taken place.

Quantification or measurement, in thesense just characterized, is important in someway in all empirical sciences. The primarysignificance of this fact is that given the iso-morphism of structures, we may pass from theparticular empirical structure to the numericalone and then use all our familiar computa-tional methods, as applied to the isomorphicarithmetical structure, to infer facts about theisomorphic empirical structure. Such passagefrom simple qualitative observations to quan-titative ones—the isomorphism of structures

passing from the empirical to the numerical—is necessary for precise prediction or controlof phenomena. Of course, such a representa-tion is useful only to the extent of the precisionof the observations on which it is based. A va-riety of numerical representations for variousempirical psychological phenomena is givenin the sections that follow.

Uniqueness

The second fundamental problem of repre-sentational measurement is to discover theuniqueness of the representations. Solving therepresentation problem for a theory of mea-surement is not enough. There is usually aformal difference between the kind of assign-ment of numbers arising from different pro-cedures of measurement, as may be seen inthree intuitive examples:

1. The population of California is greater thanthat of New York.

2. Mary is 10 years older than John.

3. The temperature in New York City thisafternoon will be 92 ◦F.

Here we may easily distinguish three kindsof measurements. The first is an example ofcounting, which is an absolute scale. Thenumber of members of a given collection thatis counted is determined uniquely in the idealcase, although that can be difficult in prac-tice (witness the 2000 presidential electionin Florida). In contrast, the second example,the measurement of difference in age, is aratio scale. Empirical procedures for mea-suring age do not determine the unit of age—chosen in the example to be the year ratherthan, for example, the month or the week.Although the choice of the unit of a per-son’s age is arbitrary—that is, not empiri-cally prescribed—that of the zero, birth, isnot. Thus, the ratio of the ages of any two peo-ple is independent of its choice, and the ageof people is an example of a ratio scale. The


A Brief History of Measurement 5

measurement of distance is another exampleof such a ratio scale. The third example, thatof temperature, is an example of an intervalscale. The empirical procedure of measuringtemperature by use of a standard thermometeror other device determines neither a unit noran origin.

We may thus also describe the second fun-damental problem for representational mea-surement as that of determining the scale typeof the measurements resulting from a givenprocedure.

A BRIEF HISTORYOF MEASUREMENT

Pre-19th-Century Measurement

Already by the fifth century B.C., if not before,Greek geometers were investigating problemscentral to the nature of measurement. TheGreek achievements in mathematics are all ofrelevance to measurement. First, the theory ofnumber, meaning for them the theory of thepositive integers, was closely connected withcounting; second, the geometric theory of pro-portion was central to magnitudes that we nowrepresent by rational numbers (= ratios of in-tegers); and, finally, the theory of incommen-surable geometric magnitudes for those mag-nitudes that could not be represented by ratios.The famous proof of the irrationality of thesquare root of two seems arithmetic in spiritto us, but almost certainly the Greek discov-ery of incommensurability was geometric incharacter, namely, that the length of the di-agonal of a square, or the hypotenuse of anisosceles right-angled triangle, was not com-mensurable with the sides. The Greeks wellunderstood that the various kinds of resultsjust described applied in general to magni-tudes and not in any sense only to numbersor even only to the length of line segments.The spirit of this may be seen in the first def-inition of Book 10 of Euclid, the one dealing

with incommensurables: “Those magnitudesare said to be commensurable which are mea-sured by the same measure, and those incom-mensurable which cannot have any commonmeasure” (trans. 1956, p. 10).

It does not take much investigation to de-termine that theories and practices relevant tomeasurement occur throughout the centuriesin many different contexts. It is impossibleto give details here, but we mention a fewsalient examples. The first is the discussionof the measurement of pleasure and pain inPlato’s dialogue Protagoras. The second isthe set of partial qualitative axioms, character-izing in our terms empirical structures, givenby Archimedes for measuring on unequal bal-ances (Suppes, 1980). Here the two qualitativeconcepts are the distance from the focal pointof the balance and the weights of the objectsplaced in the two pans of the balance. Thisis perhaps the first partial qualitative axiom-atization of conjoint measurement, which isdiscussed in more detail later. The third ex-ample is the large medieval literature giving avariety of qualitative axioms for the measure-ment of weight (Moody and Claggett, 1952).(Psychologists concerned about the difficultyof clarifying the measurement of fundamen-tal psychological quantities should be encour-aged by reading O’Brien’s 1981 detailed ex-position of the confused theories of weight inthe ancient world.) The fourth example is thedetailed discussion of intensive quantities byNicole Oresme in the 14th century A.D. Thefifth is Galileo’s successful geometrization inthe 17th century of the motion of heavenlybodies, done in the context of stating essen-tially qualitative axioms for what, in the ear-lier tradition, would be called the quantity ofmotion. The final example is also perhaps thelast great, magnificent, original treatise of nat-ural science written wholly in the geometricaltradition—Newton’s Principia of 1687. Evenin his famous three laws of motion, conceptswere formulated in a qualitative, geometrical



way, characteristic of the later formulation ofqualitative axioms of measurement.

19th- and Early 20th-CenturyPhysical Measurement

The most important early 19th-century workon measurement was the abstract theory ofextensive quantities published in 1844 byH. Grassmann, Die Wissenschaft der Exten-siven Grosse oder die Ausdehnungslehre. Thisabstract and forbidding treatise, not properlyappreciated by mathematicians at the timeof its appearance, contained at this earlydate the important generalization of the con-cept of geometric extensive quantities ton-dimensional vector spaces and, thus, to theaddition, for example, of n-dimensional vec-tors. Grassmann also developed for the firsttime a theory of barycentric coordinates in ndimensions. It is now recognized that this wasthe first general and abstract theory of exten-sive quantities to be treated in a comprehen-sive manner.

Extensive Measurement

Despite the precedent of the massive workof Grassmann, it is fair to say that the mod-ern theory of one-dimensional, extensive mea-surement originated much later in the cen-tury with the fundamental work of Helmholtz(1887) and Hölder (1901). The two funda-mental concepts of these first modern at-tempts, and later ones as well, is a binaryoperation ◦ of combination and an orderingrelation �∼, each of which has different inter-pretations in different empirical structures.For example, mass ordering �∼ is determinedby an equal-arm pan balance (in a vacuum)with a◦b denoting objects a and b both placedon one pan. Lengths of rods are ordered byplacing them side-by-side, adjusting one endto agree, and determining which rod extendsbeyond the other at the opposite end, and ◦means abutting two rods along a straight line.

The ways in which the basic axioms can bestated to describe the intertwining of these twoconcepts has a long history of later develop-ment. In every case, however, the fundamentalisomorphism condition is the following: Fora, b in the empirical domain,

f (a) ≥ f (b) ⇔ a �∼ b, (1)

f (a ◦ b) = f (a) + f (b), (2)

where f is the mapping function from theempirical structure to the numerical structureof the additive, positive real numbers, that is,for all entities a, f (a) > 0.

Certain necessary empirical (testable)properties must be satisfied for such a rep-resentation to hold. Among them are for allentities a, b, and c,

Commutativity: a ◦ b ∼ b ◦ a.

Associativity: (a ◦ b) ◦ c ∼ a ◦ (b ◦ c).Monotonicity: a �∼ b ⇔ a ◦ c �∼ b ◦ c.Positivity: a ◦ a � a.

Let a be any element. Define a standardsequence based on a to be a sequence a(n),

where n is an integer, such that a(1) = a,and for i > 1, a(i) ∼ a(i – 1) ◦ a. An exampleof such a standard sequence is the centimetermarks on a meter ruler. The idea is that theelements of a standard sequence are equallyspaced. The following (not directly testable)condition ensures that the stimuli are com-mensurable:

Archimedean: For any entities a, b,

there is an integer n such that a(n) � b.

These, together with the following struc-tural condition that ensures very small ele-ments,

Solvability: if a � b,

then for some c, a � b ◦ c,

were shown to imply the existence of the rep-resentation given by Equations (1) and (2).By formulating the Archimedean axiom dif-ferently, Roberts and Luce (1968) showed thatthe solvability axiom could be eliminated.



Such empirical structures are called exten-sive. The uniqueness of their representationsis discussed shortly.

Probability and Partial Operations

It is well known that probability P is an addi-tive measure in the sense that it maps eventsinto [0, 1] such that, for events A and B thatare disjoint,

P(A ∪ B) = P(A) + P(B).

Thus, probability is close to extensive mea-surement—but not quite, because the opera-tion is limited to only disjoint events. How-ever, the theory of extensive measurement canbe generalized to partial operations having theproperty that if a and b are such that a ◦ b isdefined and if a �∼ c and b �∼ d, then c ◦ d isalso defined. With some adaptation, this canbe applied to probability; the details can befound in Chapter 3 of Krantz, Luce, Suppes,and Tversky (1971). (This reference is subse-quently cited as FM I for Volume I of Foun-dations of Measurement. The other volumesare Suppes, Krantz, Luce, & Tversky, 1990,cited as FM II, and Luce, Krantz, Suppes, &Tversky, 1990, cited as FM III.)

Finite Partial Extensive Structures

Continuing with the theme of partial opera-tion, we describe a recent treatment of a finiteextensive structure that also has ratio scalerepresentation and that is fully in the spirit ofthe earlier work involving continuous models.Suppose X is a finite set of physical objects,any two of which balance on an equal-armbalance; that is, if a1, . . . , an are the objects,for any i and j, i �= j, then ai ∼ a j . Thus, theyweigh the same. Moreover, if A and B are twosets of these objects, then on the balance wehave A ∼ B if and only if A and B have thesame number of objects. We also have a con-catenation operation, union of disjoint sets. IfA ∩ B = ∅, then A ∪ B ∼ C if and only ifthe objects in C balance the objects in A

together with the objects in B. The qualitativestrict ordering A � B has an obvious opera-tional meaning, which is that the objects inA, taken together, weigh more on the balancethan the objects in B, taken together.

This simple setup is adequate to establish,by fundamental measurement, a scheme fornumerically weighing other objects not in X.First, our homomorphism f on X is reallysimple. Since for all ai and a j and X, ai∼ a j ,

we have

f (ai ) = f (a j ),

with the restriction that f (ai ) > 0. We extendf to A, a subset of X, by setting f (A) = |A| =the cardinality of (number of objects in) A.The extensive structure is thus transparent:For A and B subsets of X, if A ∩ B = ∅ then

f (A ∪ B) = |A ∪ B| = |A| + |B|= f (A) + f (B).

If we multiply f by any α > 0 the equationstill holds, as does the ordering. Moreover,in simple finite cases of extensive measure-ment such as the present, it is easy to prove di-rectly that no transformations other than ratiotransformations are possible. Let f ∗ denoteanother representation. For some object a, setα = f (a)/ f ∗(a). Observe that if |A| = n, thenby a finite induction

f (A)

f ∗(A)= n f (a)

n f ∗(a)= α,

so the representation forms a ratio scale.

Finite Probability

The “objects” a1, . . . , an are now interpretedas possible outcomes of a probabilistic mea-surement experiment, so the ai s are the possi-ble atomic events whose qualitative probabil-ity is to be judged.

The ordering A �∼ B is interpreted as mean-ing that event A is at least as probable as eventB; A ∼ B as A and B are equally probable;A � B as A is strictly more probable than B.



Then we would like to interpret f (A) as thenumerical probability of event A, but if f isunique up to only a ratio scale, this will notwork since f (A) could be 50.1, not exactly aprobability.

By adding another concept, that of theprobabilistic independence of two events, wecan strengthen the uniqueness result to thatof an absolute scale. This is written A ⊥ B.Given a probability measure, the definition ofindependence is familiar: A ⊥ B if and only ifP(A ∩ B) = P(A)P(B). Independence can-not be defined in terms of the qualitative con-cepts introduced for arbitrary finite qualitativeprobability structures, but can be defined byextending the structure to elementary randomvariables (Suppes and Alechina, 1994). How-ever, a definition can be given for the spe-cial case in which all atoms are equiproba-ble; it again uses the cardinality of the sets:A ⊥ B if and only if |X | · |A ∩ B| = |A| · |B|.It immediately follows from this definitionthat X ⊥ X , whence in the interpretation of⊥ we must have

P(X) = P(X ∩ X) = P(X)P(X),

but this equation is satisfied only if P(X) = 0,which is impossible since P(∅) = 0 andX � ∅, or P(X) = 1, which means that thescale type is an absolute—not a ratio—scale,as it should be for probability.

Units and Dimensions

An important aspect of 19th century physicswas the development, starting with Fourier’swork (1822/1955), of an explicit theory ofunits and dimensions. This is so common-place now in physics that it is hard to be-lieve that it only really began at such a latedate. In Fourier’s famous work, devoted tothe theory of heat, he announced that in or-der to measure physical quantities and expressthem numerically, five different kinds of unitsof measurement were needed, namely, thoseof length, time, mass, temperature, and heat.

Of even greater importance is the specifictable he gave, for perhaps the first time in thehistory of physics, of the dimensions of vari-ous physical quantities. A modern version ofsuch a table appears at the end of FM I.

The importance of this tradition of unitsand dimensions in the 19th century is to beseen in Maxwell’s famous treatise on electric-ity and magnetism (1873). As a preliminary,he began with 26 numbered paragraphs onthe measurement of quantities because of theimportance he attached to problems of mea-surement in electricity and magnetism, a topicthat was virtually unknown before the 19thcentury. Maxwell emphasized the fundamen-tal character of the three fundamental unitsof length, time, and mass. He then went onto derive units, and by this he meant quanti-ties whose dimensions may be expressed interms of fundamental units (e.g., kinetic en-ergy, whose dimension in the usual notation isM L2T –2). Dimensional analysis, first put insystematic form by Fourier, is very useful inanalyzing the consistency of the use of quan-tities in equations and can also be used forwider purposes, which are discussed in somedetail in FM I.

Derived Measurement

In the Fourier and Maxwell analyses, the ques-tion of how a derived quantity is actually to bemeasured does not enter into the discussion.What is important is its dimensions in terms offundamental units. Early in the 20th centurythe physicist Norman Campbell (1920/1957)used the distinction between fundamental andderived measurement in a sense more intrinsicto the theory of measurement itself. The dis-tinction is the following: Fundamental mea-surement starts with qualitative statements(axioms) about empirical structures, such asthose given earlier for an extensive structure,and then proves the existence of a representa-tional theorem in terms of numbers, whencethe phrase “representational measurement.”



In contrast, a derived quantity is measured interms of other fundamental measurements. Aclassical example is density, measured as theratio of separate measurements of mass andvolume. It is to be emphasized, of course, thatcalling density a derived measure with respectto mass and volume does not make a funda-mental scientific claim. For example, it doesnot allege that fundamental measurement ofdensity is impossible. Nevertheless, in under-standing the foundations of measurement, itis always important to distinguish whetherfundamental or derived measurement, inCampbell’s sense, is being analyzed or used.

Axiomatic Geometry

From the standpoint of representational mea-surement theory, another development ofgreat importance in the 19th century was theperfection of the axiomatic method in geom-etry, which grew out of the intense scrutinyof the foundations of geometry at the be-ginning of that century. The driving forcebehind this effort was undoubtedly the dis-covery and development of non-Euclidean ge-ometries at the beginning of the century byBolyai, Lobachevski, and Gauss. An impor-tant and intuitive example, later in the cen-tury, was Pasch’s (1882) discovery of the ax-iom named in his honor. He found a gap inEuclid that required a new axiom, namely, theassertion that if a line intersects one side of atriangle, it must intersect also a second side.More generally, it was the high level of rigorand abstraction of Pasch’s 1882 book that wasthe most important step leading to the mod-ern formal axiomatic conception of geometry,which has been so much a model for repre-sentational measurement theory in the 20thcentury. The most influential work in this lineof development was Hilbert’s Grundlagen derGeometrie, first edition in 1899, much of itsprominence resulting from Hilbert’s positionas one of the outstanding mathematicians ofthis period.

It should be added that even in one-dimensional geometry numerical representa-tions arise even though there is no orderrelation. Indeed, for dimensions ≥2, no stan-dard geometry has a weak order. Moreover, ingeometry the continuum is not important forthe fundamental Galilean and Lorentz groups.An underlying denumerable field of algebraicnumbers is quite adequate.

Invariance

Another important development at the endof the 19th century was the creation of theexplicit theory of invariance for spatial prop-erties. The intuitive idea is that the spatialproperties in analytical representations are in-variant under the transformations that carryone model of the axioms into another modelof the axioms. Thus, for example, the ordi-nary Cartesian representation of Euclideangeometry is such that the geometrical prop-erties of the Euclidean space are invariant un-der the Euclidean group of transformationsof the Cartesian representation. These are thetransformations that are composed from trans-lations (in any direction), rotations, and re-flections. These ideas were made particularlyprominent by the mathematician Felix Kleinin his Erlangen address of 1872 (see Klein,1893). These important concepts of invariancehad a major impact in the development of thetheory of special relativity by Einstein at thebeginning of the 20th century. Here the invari-ance is that under the Lorentz transformations,which are those that leave invariant geomet-rical and kinematic properties of the space-time of special relativity. Without giving thefull details of the Lorentz transformations, it isstill possible to give a clear physical sense ofthe change from classical Newtonian physicsto that of special relativity.

In the case of classical Newtonian me-chanics, the invariance that characterizes theGalilean transformations is just the invarianceof the distance between any two simultaneous



points together with the invariance of any tem-poral interval, under any permissible changeof coordinates. Note that this characterizationrequires that the units of measurement for bothspatial distance and time be held constant. Inthe case of special relativity, the single in-variant is what is called the proper time τ12

between two space-time points (x1, y1, z1, t1)and (x2, y2, z2, t2), which is defined as

τ12 =√(t1 − t2)2 − 1

c2

[(x1 − x2)

2 + (y1 − y2)2 + (z1 − z2)

2],

where c is the velocity of light in the givenunits of measurement. It is easy to see theconceptual nature of the change. In the caseof classical mechanics, the invariance of spa-tial distance between simultaneous points isseparate from the invariance of temporal in-tervals. In the case of special relativity, theyare intertwined. Thus, we properly speak ofspace-time invariance in the case of specialrelativity. As will be seen in what follows,the concepts of invariance developed so thor-oughly in the 19th and early 20th century ingeometry and physics have carried over andare an important part of the representationaltheory of measurement.

Quantum Theory and the Problemof Measurement

Still another important development in thefirst half of the 20th century, of special rel-evance to the topic of this chapter, was thecreation of quantum mechanics and, in par-ticular, the extended analysis of the problemof measurement in that theory. In contrast withthe long tradition of measurement in classicalphysics, at least three new problems arose thatgenerated what in the literature is termed theproblem of measurement in quantum mechan-ics. The first difficulty arises in measuring mi-croscopic objects, that is, objects as small asatoms or electrons or other particles of asimilar nature. The very attempt to measure a

property of these particles creates a distur-bance in the state of the particle, a disturbancethat is not small relative to the particle itself.Classical physics assumed that, in principle,such minor disturbances of a measured ob-ject as did occur could either be eliminated ortaken into account in a relatively simple way.

The second aspect is the precise limitationon such measurement, which was formulatedby Heisenberg’s uncertainty principle. For ex-ample, it is not possible to measure both posi-tion and momentum exactly. Indeed, it is notpossible, in general, to have a joint probabilitydistribution of the measurements of the two.This applies not just to position and momen-tum, but also to other pairs of properties of aparticle. The best that can be hoped for is theHeisenberg uncertainty relation. It expressesan inequality that bounds away from zero theproduct of the variances of the two proper-ties measured, for example, the product of thevariance of the measurement of position andthe variance of the measurement of velocityor momentum. This inequality appeared reallyfor the first time in quantum mechanics and isone of the principles that separates quantummechanics drastically from classical physics.An accessible and clear exposition of theseideas is Heisenberg (1930), a work that fewothers have excelled for the quality of itsexposition.

The third aspect of measurement in quan-tum mechanics is the disparity between theobject being measured and the relatively large,macroscopic object used for the measure-ment. Here, a long and elaborate story can betold, as it is, for example, in von Neumann’sclassical book on the foundations of quan-tum mechanics, which includes a detailedtreatment of the measurement problem(von Neumann, 1932/1955). The critical as-pect of this problem is deciding when a mea-surement has taken place. Von Neumann wasinclined to the view that a measurement hadtaken place only when a relevant event had



occurred in the consciousness of some ob-server. More moderate subsequent views aresatisfied with the position that an observationtakes place when a suitable recording has beenmade by a calibrated instrument.

Although we shall not discuss further theproblem of measurement in quantum mechan-ics, nor even the application of the ideas tomeasurement in psychology, it is apparent thatthere is some resonance between the difficul-ties mentioned and the difficulties of measur-ing many psychological properties.

19th- and Early 20th-Century Psychology

Fechner’s Psychophysics

Psychology was not a separate discipline untilthe late 19th century. Its roots were largely inphilosophy with significant additions by med-ical and physical scientists. The latter broughta background of successful physical measure-ment, which they sought to re-create in sen-sory psychology at the least. The most promi-nent of these were H. Helmholtz, whose workamong other things set the stage for extensivemeasurement, and G. T. Fechner, whoseElemente der Psychophysik (Elements ofPsychophysics; 1860/1966) set the stage forsubsequent developments in psychologicalmeasurement. We outline the problem facedin trying to transplant physical measurementand the nature of the proposed solution.

Recall that the main measurement deviceused in 19th-century physics was concatena-tion: Given two entities that exhibit the at-tribute to be measured, it was essential to finda method of concatenating them to form a thirdentity also exhibiting the attribute. Then oneshowed empirically that the structure satisfiesthe axioms of extensive measurement, as dis-cussed earlier. When no empirical concatena-tion operation can be found, as for examplewith density, one could not do fundamentalmeasurement. Rather, one sought an invari-ant property stated in terms of fundamentally

measured quantities called derived measure-ment. Density is an example.

When dealing with sensory intensity, phys-ical concatenation is available but just recov-ers the physical measure, which does not atall well correspond with subjective judgmentssuch as the half loudness of a tone. A newapproach was required. Fechner continued toaccept the idea of building up a measure-ment scale by adding small increments, asin the standard sequences of extensive mea-surement, and then counting the number ofsuch increments needed to fill a sensory in-terval. The question was: What are the smallequal increments to be added? His idea wasthat they correspond to “just noticeable dif-ferences” (JND); when one first encountersthe idea of a JND it seems to suggest a fixedthreshold, but it gradually was interpreted tobe defined statistically. To be specific, sup-pose x0 and x1 = x0 + ξ(x0, λ) are stimulusintensities such that the probability of identi-fying x1 as larger than x0 is a constant λ, thatis, Pr(x0, x1) = λ. His idea was to fix λ and tomeasure the distance from x to y, x < y, interms of the number of successive JNDs be-tween them. Defining x0 = x and assumingthat xi has been defined, then define xi+1 as

xi+1 = xi + ξ(xi , λ).

The sequence ends with xn ≤ y < xn+1.Fechner postulated the number of JNDs fromx to y as his definition of distance without,however, establishing any empirical proper-ties of justify that definition. Put another way,he treated without proof that a sequence ofJNDs forms a standard sequence.

His next step was to draw on an empiricalresult of E. H. Weber to the effect that

ξ(x, λ) = δ(λ)x, δ(λ) > 0,

which is called Weber’s law. This is some-times approximately true (e.g., for loudnessof white noise well above absolute threshold),but more often it is not (e.g., for pure tones).



His final step was to introduce, much asin extensive measurement, a limiting processas λ approaches 1

2 and δ approaches 0. Hecalled this an auxiliary mathematical prin-ciple, which amounts to supposing withoutproof that a limit below exists. If we denoteby ψ the counting function, then his assump-tion that, for fixed λ, the JNDs are equally dis-tant can be interpreted to mean that for somefunction η of λ

η(λ) = ψ[x + ξ(x, λ)] − ψ(x)

= ψ([δ(λ) + 1]x) − ψ(x).

Therefore, dividing by δ(λ)x

ψ([δ(λ) + 1]x) − ψ(x)

δ(λ)x= η(λ)

δ(λ)x= α(λ)

x,

where α(λ) = η(λ)

δ(λ).

Assuming that the limit of α(λ) exists, onehas the simple ordinary differential equation

dψ(x)

dx= k

x, k = lim

λ→ 12

α(λ),

whose solution is well known to be

ψ(x) = r ln x + s, r > 0.

This conclusion, known as Fechner’s law,was soon questioned by J. A. F. Plateau(1872), among others, although the empricalevidence was not conclusive. Later, Wiener(1915, 1921) was highly critical, and muchlater Luce and Edwards (1958) pointed outthat, in fact, Fechner’s mathematical auxil-iary principle, although leading to the correctsolution of the functional equation η(λ) =ψ[x + ξ(x, λ)] − ψ(x) when Weber’s lawholds, fails to discover the correct solutionin any other case—which empirically reallyis the norm. The mathematics is simply moresubtle than he assumed.

In any event, note that Fechner’s approachis not an example of representational mea-surement, because no empirical justificationwas provided for the definition of standardsequence used.

Reinterpreting Fechner Geometrically

Because Fechner’s JND approach using in-finitesimals seemed to be flawed, little wasdone for nearly half a century to constructpsychophysical functions based on JNDs—that is, until Dzhafarov and Colonius (1999,2001) reexamined what Fechner might havemeant. They did this from a viewpoint ofdistances in a possible representation calleda Finsler geometry, of which ordinary Rie-mannian geometry is a special case. Thus,their theory concerns stimuli of any finite di-mension, not just one. The stimuli are vec-tors, for which we use bold-faced notation.The key idea, in our notation, is that for eachperson there is a universal function � suchthat, for λ sufficiently close to 1

2 , �(ψ[x +ξ(x, λ)] − ψ(x)) is comeasurable3 with x.This assumption means that this transformeddifferential can be integrated along any suffi-ciently smooth path between any two points.The minimum path length is defined to bethe Fechnerian distance between them. Thistheory, which is mathematically quite elab-orate, is testable in principle. But doing socertainly will not be easy because its assump-tions, which are about the behavior of in-finitesimals, are inherently difficult to checkwith fallible data. It remains to be seen howfar this can be taken.

Ability and Achievement Testing

The vast majority of what is commonly called“psychological measurement” consists of var-ious elaborations of ability and achievementtesting that are usually grouped under the la-bel “psychometrics.” We do not cover any ofthis material because it definitely is neithera branch of nor a precursor to the representa-tional measurement of an attribute. To be sure,a form of counting is employed, namely, the

3For the precise definition, see the reference.



items on a test that are correctly answered, andthis number is statistically normed over a par-ticular age or other feature so that the count istransformed into a normal distribution. Again,no axioms were or are provided. Of the psy-chometric approaches, we speak only of a por-tion of Thurstone’s work that is closely relatedto sensory measurement. Recently, Doignonand Falmagne (1999) have developed an ap-proach to ability measurement, called knowl-edge spaces, that is influenced by representa-tional measurement considerations.

Thurstone’s Discriminal Dispersions

In a series of three 1927 papers, L. L.Thurstone began a reinterpretation ofFechner’s approach in terms of the then newlydeveloped statistical concept of a random vari-able (see also Thurstone, 1959). In particu-lar, he assumed that there was an underlyingpsychological continuum on which signal pre-sentations are represented, but with variabil-ity. Thus, he interpreted the representation ofstimulus x as a random variable �(x) withsome distribution that he cautiously assumed(see Thurstone, 1927b, p. 373) to be normalwith mean ψx and standard deviation (whichhe called a “discriminal dispersion”) σx andpossibly covariances with other stimulus rep-resentations. Later work gave reasons to con-sider extreme value distributions rather thanthe normal. His basic model for the probabil-ity of stimulus y being judged larger than xwas

P(x, y) = Pr[�(y) − �(x) > 0], x ≤ y.

(3)

The relation to Fechner’s ideas is really quiteclose in that the mean subjective differencesare equal for fixed λ = P(x, y).

Given that the representations are assumedto be normal, the difference is also normalwith mean ψy – ψx and standard deviation

σx,y = (σ 2

x + σ 2y − 2ρx,yσxσy

)1/2

so if zx,y is the normal deviate correspond-ing to P(x, y), Equation (3) can be expressedas

ψy − ψx = zx,yσx,y .

Thurstone dubbed this “a law of comparativejudgment.” Many papers before circa 1975considered various modifications of the as-sumptions or focused on solving this equationfor various special cases. We do not go intothis here in part because the power of mod-ern computers reduces the need for specia-lization.

Thurstone’s approach had a natural one-dimensional generalization to the absoluteidentification of one of n > 2 possible stimuli.The theory assumes that each stimulus has adistribution on the sensory continuum and thatthe subject establishes n − 1 cut points to de-fine the intervals of the range of the randomvariable that are identified with the stimuli.The basic data are conditional probabilitiesP(x j |xi , n) of responding x j when xi , i, j =1, 2, . . . , n, is presented. Perhaps the moststriking feature of such data is the follow-ing: Suppose a series of signals are selectedsuch that adjacent pairs are equally detectable.Using a sequence of n adjacent ones, abso-lute identification data are processed througha Thurstone model in which ψx,n and σx,n areboth estimated. Accepting that ψx,n are in-dependent of n, then the σx,n definitely arenot independent of n. In fact, once n reachesabout 7, the value is independent of size, butσx,7 ≈ 3σx,2. This is a challenging finding andcertainly casts doubt on any simple invari-ant meaning of the random variable �(x)—apparently its distribution depends not onlyon x but on what might have been presentedas well. Various authors have proposed alter-native solutions (for a summary, see Iverson& Luce, 1998).

A sophisticated treatment of Fechner,Thurstone, and the subsequent literature isprovided by Falmagne (1985).



Theory of Signal Detectability

Perhaps the most important generalization ofThurstone’s idea is that of the theory of sig-nal detectability, in which the basic change isto assume that the experimental subject canestablish a response criterion β, in generaldifferent from 0, so that

P(x, y) = Pr[�(y) − �(x) > β], x ≤ y.

Engineers first developed this model. It wasadoped and elaborated in various psycho-logical sources, including Green and Swets(1974) and Macmillan and Creelman (1991),and it has been widely applied throughoutpsychology.

Mid-20th-Century PsychologicalMeasurement

Campbell’s Objectionto Psychological Measurement

N. R. Campbell, a physicist turned philoso-pher of physics who was especially concernedwith physical measurement, took the verystrong position that psychologists, in partic-ular, and social scientists, in general, had notcome up with anything deserving the name ofmeasurement and probably never could. Hewas supported by a number of other Britishphysicists. His argument, though somewhatelaborate, actually boiled down to assertingthe truth of three simple propositions:

(i) A prerequisite of measurement is someform of empirical quantification that canbe accepted or rejected experimentally.

(ii) The only known form of such quantifi-cation arises from binary operations ofconcatenation that can be shown empir-ically to satisfy the axioms of extensivemeasurement.

(iii) And psychology has no such extensiveoperations of its own.

Some appropriate references are Campbell(1920/1957, 1928) and Ferguson et al. (1940).

Stevens’s Response

In a prolonged debate conducted before asubcommittee of the British Association forthe Advancement of Sciences, the physicistsagreed on these propositions and the psychol-ogists did not, at least not fully. They accepted(iii) but in some measure denied (i) and (ii),although, of course, they admitted that bothheld for physics. The psychophysicist S. S.Stevens became the primary spokesperson forthe psychological community. He first formu-lated his views in 1946, but his 1951 chapterin the first version of the Handbook of Exper-imental Psychology, of which he was editor,made his views widely known to the psycho-logical community. They were complex, andat the moment we focus only on the part rele-vant to the issue of whether measurement canbe justified outside physics.

Stevens’ contention was that Proposition(i) is too narrow a concept of measurement,so (ii) and therefore (iii) are irrelevant. Rather,he argued for the claim that “Measurement isthe assignment of numbers to objects or eventsaccording to rule. . . . The rule of assignmentcan be any consistent rule” (Stevens, 1975,pp. 46–47). The issue was whether the rulewas sufficient to lead to one of several scaletypes that he dubbed nominal, ordinal, inter-val, ratio, and absolute. These are sufficientlywell known to psychologists that we need notdescribe them in much detail. They concernthe uniqueness of numerical representations.In the nominal case, of which the assignmentof numbers to football players was his exam-ple, any permutation is permitted. This is notgenerally treated as measurement because noordering by an attribute is involved. An or-dinal scale is an assignment that can be sub-jected to any strictly increasing transforma-tion, which of course preserves the order andnothing else. It is a representation with infinite


Representational Approach after 1950 15

degrees of freedom. An interval scale is one inwhich there is an arbitrary zero and unit; butonce picked, no degrees of freedom are left.Therefore, the admissible transformation isψ �−→ rψ +s, (r > 0). As stated, such a rep-resentation has to be on all of the real numbers.If, as is often the case, especially in physics,one wants to place the representation on thepositive real numbers, then the transforma-tion becomes ψ+ �−→ s ′ψr

+, (r > 0, s ′ > 0).Stevens (1959, pp. 31–34) called a represen-tation unique up to power transformations alog-interval scale but did not seem to recog-nize that it is merely a different way of writ-ing an interval scale representation ψ in whichψ = ln ψ+ and s = ln s ′. Whichever one uses,it has two degrees of freedom. The ratio caseis the interval one with r = 1. Again, thishas two forms depending on the range of ψ .For the case of a representation on thereals, the admissible transformations are thetranslations ψ �−→ ψ + s. There is a differ-ent version of ratio measurement that is inher-ently on the reals in the sense that it cannotbe placed on the positive reals. In this case,0 is a true zero that divides the representa-tion into inherently positive and negative por-tions, and the admissible transformations areψ �−→ rψ, r > 0.

Stevens took the stance that what was im-portant in measurement was its uniquenessproperties and that they could come aboutin ways different from that of physics. Theremaining part of his career, which is sum-marized in Stevens (1975), entailed the de-velopment of new methods of measurementthat can all be encompassed as a form of sen-sory matching. The basic instruction to sub-jects was to require the match of a stimu-lus in one modality to that in another so thatthe subjective ratio between a pair of stim-uli in the one dimension is maintained in thesubjective ratio of the matched signals. Thisis called cross-modal matching. When oneof the modalities is the real numbers, it is

one of two forms of magnitude matching—magnitude estimation when numbers are to bematched to a sensory stimuli and magnitudeproduction when numbers are the stimuli to bematched by some physical stimuli. Using geo-metric means over subjects, he found the datato be quite orderly—power functions of theusual physical measures of intensity. Much ofthis work is covered in Stevens (1975).

His argument that this constituted a formof ratio scale measurement can be viewed intwo distinct ways. The least charitable is thatof Michell (1999), who treats it as little morethan a play on the word “ratio” in the scaletype and in the instructions to the subjects. Hefeels that Stevens failed to understand the needfor empirical conditions to justify numericalrepresentations. Narens (1996) took the viewthat Stevens’ idea is worth trying to formal-ize and in the process making it empiricallytestable. Work along these lines continues, asdiscussed later.

REPRESENTATIONAL APPROACHAFTER 1950

Aside from extensive measurement, the repre-sentational theory of measurement is largelya creation by behavioral scientists and math-ematicians during the second half of the20th century. The basic thrust of this schoolof thought can be summarized as accept-ing Campbell’s conditions (i), quantificationbased on empirical properties, and (iii), thesocial sciences do not have concatenation op-erations (although even that was never strictlycorrect, as is shown later, because of probabil-ity based on a partial operation), and rejectingthe claim (ii) that the only form of quantifica-tion is an empirical concatenation operation.This school disagreed with Stevens’ broaden-ing of (i) to any rule, holding with the physi-cists that the rules had to be established onfirm empirical grounds.



To do this, one had to establish the exis-tence of schemes of empirically based mea-surement that were different from extensivemeasurement. Examples are provided here.For greater detail, see FM I, II, III, Narens(1985), or for an earlier perspective Pfanzagl(1968).

Several Alternativesto Extensive Measurement

Utility Theory

The first evidence of something different fromextensive measurement was the constructionby von Neumann and Morgenstern (1947) ofan axiomatization of expected utility theory.Here, the stimuli were gambles of the form(x, p; y) where consequence x occurs withprobability p and y with probability 1 – p. Thebasic primitive of the system was a weak pref-erence order �∼ over the binary gambles. Theystated properties that seemed to be at leastrational, if not necessarily descriptive; fromthem one was able to show the existence of anumerical utility function U over the conse-quences and gambles such that for two binarygambles g, h

g �∼ h ⇔ U (g) ≥ U (h),

U (g, p; h) = U (g)p + U (h)(1 − p).

Note that this is an averaging representation,called expected utility, which is quite distinctfrom the adding of extensive measurement(see the subsection on averaging).

Actually, their theory has to be viewed as aform of derived measurement in Campbell’ssense because the construction of the U func-tion was in terms of the numerical probabil-ities built into the stimuli themselves. Thatlimitation was overcome by Savage (1954),who modeled decision making under uncer-tainty as acts that are treated as an assignment

of consequences to chance states of nature.4

Savage assumed that each act had a finite num-ber of consequences, but subsequent gener-alizations permitted infinitely many. Withoutbuilding any numbers into the domain and us-ing assumptions defended by arguments ofrationality, he showed that one can constructboth a utility function U and a subjective prob-ability function S such that acts are evaluatedby calculating the expectation of U with re-spect to the measure S. This representationis called subjective expected utility (SEU).It is a case of fundamental measurement inCampbell’s sense. Indirectly, it involved apartial concatenation operation of disjointunions, which was used to construct a sub-jective probability function.

These developments led to a very ac-tive research program involving psycholo-gists, economists, and statisticians. The basicthrust has been of psychologists devisingexperiments that cast doubt on either a repre-sentation or some of its axioms, and oftheorists of all stripes modifying the theoryof accommodate the data. Among the keysummary references are Edwards (1992),Fishburn (1970, 1988), Luce (2000), Quiggin(1993), and Wakker (1989).

Difference Measurement

The simplest example of difference measure-ment is location along a line. Here, some pointis arbitrarily set to be 0, and other points aredefined in terms of distance (length) from it,with those on one side defined to be positiveand those on the other side negative. It is clearin this case that location measurement formsan example of interval scale measurement

4Some aspects of Savage’s approach were anticipated byRamsey (1931), but that paper was not widely knownto psychologists and economists. Almost simultane-ously with the appearance of Savage’s work, Davidson,McKinsey, and Suppes (1955) drew on Ramsey’s ap-proach, and Davidson, Suppes, and Segal (1957) tested itexperimentally.



that is readily reduced to length measurement.Indeed, all forms of difference measurementare very closely related to extensive measure-ment, but with the stimuli being pairs of ele-ments (x, y) that define “intervals.” Axiomscan be given for this form of measurementwhere the stimuli are pairs (x, y) with bothx, y in the same set X. The goal is a numericalrepresentation ϕ of the form

(x, y) �∼ (u, v)

⇔ ϕ(x) − ϕ(y) ≥ ϕ(u) − ϕ(v).

One key axiom that makes clear how a con-catenation operation arises is that if (x, y) �∼(x ′, y′) and (y, z) �∼ (y′, z′), then (x, z) �∼(x ′, z′).

An important modification is called abso-lute difference measurement, in which the goalis changed to

(x, y) �∼ (u, v)

⇔ |ϕ(x) − ϕ(y)| ≥ |ϕ(u) − ϕ(v)|.This form of measurement is a precursorto various ideas of similarity measurementimportant in multidimensional scaling. Herethe behavioral axioms become considerablymore complex. Both systems can be found inFM I, Chap. 4.

An important generalization of absolutedifference measurement is to stimuli with nfactors; it underlies developments of geomet-ric measurement based on stimulus proximity.This can be found in FM II, Chap. 14.

Additive Conjoint Measurement

Perhaps the single development that mostpersuaded psychologists that fundamentalmeasurement really could be different fromextensive measurement consisted of two ver-sions of what is called additive conjoint mea-surement. The first, by Debreu (1960), wasaimed at showing economists how indiffer-ence curves could be used to construct car-dinal (interval scale) utility functions. It was,

therefore, naturally cast in topological terms.The second (and independent) one by Luceand Tukey (1964) was cast in algebraic terms,which seems more natural to psychologistsand has been shown to include the topologi-cal approach as a special case. Again, it wasan explanation of the conditions under whichequal-attribute curves can give rise to mea-surement. Michell (1990) provides a carefultreatment aimed at psychologists.

The basic idea is this: Suppose that an at-tribute is affected by two independent stim-ulus variables. For example, preference for areward is affected by its size and the delayin receiving it; mass of an object is affectedby both its volume and the (homogeneous)material of which it is composed; loudnessof pure tones is affected by intensity and fre-quency; and so on. Formally, one can thinkof the two factors as distinct sets A and X,so an entity is of the form (a, x) wherea ∈ A and x ∈ X. The ordering attribute is�∼ over such entities, that is, over the Cartesianproduct A × X. Thus, (a, x) �∼ (b, y) meansthat (a, x) exhibits more of the attribute inquestion than does (b, y). Again, the order-ing is assumed to be a weak order: transitiveand connected. Monotonicity (called indepen-dence in this literature) is also assumed: Fora, b ∈ A, x, y ∈ X

(a, x) �∼ (b, x) ⇔ (a, y) �∼ (b, y).

(a, x) �∼ (a, y) ⇔ (b, x) �∼ (b, y).(4)

This familiar property is often examined inpsychological research in which a dependentvariable is plotted against, say, a measure ofthe first component with the second compo-nent shown as a parameter of the curves. Theproperty holds if and only if the curves do notcross.

It is easy to show that this condition isnot sufficient to get an additive representa-tion of the two factors. If it were, then any setof nonintersecting curves in the plane couldbe rendered parallel straight lines by suitable



nonlinear transformations of the axes. Moreis required, namely, the Thomsen condition,which arose in a mathematically closelyrelated area called the theory of webs. Let-ting ∼ denote the indifference relation of �∼,

the Thomsen condition states

(a, z) ∼ (c, y)

(c, x) ∼ (b, z)

}⇒ (a, x) ∼ (b, y).

Note that it is a form of cancellation—of c inthe first factor and z in the second.

These, together with an Archimedeanproperty establishing commensurability andsome form of density of the factors, areenough to establish the following additiverepresentation: There exist numerical func-tions ψA on A and ψX on X such that

(a, x) �∼ (b, y)

⇔ ψA(a) + ψX (x) ≥ ψA(b) + ψX (y).

This representation is on all of the real num-bers. A multiplicative version on the positivereal numbers exists by setting ξi = exp ψi .The additive representation forms an inter-val scale in the sense that ψ ′

A, ψ ′X forms

another equally good representation if andonly if there are constants r > 0, sA, sX suchthat

ψ ′A = rψA + sA,

ψ ′X = rψX + sX ⇔ ξ ′

A = s ′Aξ r

A, ξ ′X = s ′

Xξ rX ,

s ′i = exp si > 0.

Additive conjoint measurement can begeneralized to finitely many factors, and it issimpler in the sense that if monotonicity isgeneralized suitably and if there are at leastthree factors, then the Thomsen condition canbe derived rather than assumed.

Although no concatenation operation is insight, a family of them can be defined in termsof ∼, and they can be shown to satisfy theaxioms of extensive measurement. This is thenature of the mathematical proof of the repre-sentation usually given.

Averaging

Some structures with a concatenation opera-tion do not have an additive representation, butrather a weighted averaging representation ofthe form

ϕ(x ◦ y) = ϕ(x)w + ϕ(y)(1 − w), (5)

where the weight w is fixed. We have alreadyencountered this form in the utility system ifwe think of the gamble (x, p; y) as definingoperations ◦p with x◦p y ≡ (x, p; y), in whichcase w = w(p). A general theory of such op-erations was first given by Pfanzagl (1959). Itis much like extensive measurement but withassociativity replaced by bisymmetry: For allstimuli x, y, u, v,

(x ◦ y) ◦ (u ◦ v) ∼ (x ◦ u) ◦ (y ◦ v). (6)

It is easy to verify that the weighted-averagerepresentation of Equation (5) implies bisym-metry, Equation (6), and x ◦ x ∼ x . The eas-iest way to show the converse is to show thatdefining �∼′ over X × X by

(a, x) �∼′ (b, y) ⇔ a ◦ x �∼ b ◦ y

yields an additive conjoint structure, fromwhich the result follows rather easily.

Nonadditive Representations

A natural question is: When does a concatena-tion operation have a numerical representationthat is inherently nonadditive? By this, onemeans a representation for which no strictlyincreasing transformation renders it additive.Before exploring that, we cite an example ofnonadditive representations that can in fact betransformed into additive ones. This is helpfulin understanding the subtlety of the question.

One example that has arisen in utility the-ory is the representation

U (x ⊕ y) = U (x) + U (y) − δU (x)U (y), (7)

where δ is a real constant and U is the SEUor rank-dependent utility generalization (see



Luce, 2000, Chap. 4) with an intrinsic zero—no change from the status quo. Because Equa-tion (7) can be rewritten

1 − δU (x ⊕ y) = [1 − δU (x)][1 − δU (y)],

the transformation V = −κ ln(1 − δU ),

δκ > 0, is additive, that is, V (x ⊕ y) =V (x) + V (y), and order-preserving. The mea-sure V is called a value function. The form inEquation (7) is called p-additive because it isthe only polynomial with a fixed zero that canbe put in additive form. The source of thisrepresentation is examined in the next majorsection. It is easy to verify that both the ad-ditive and the nonadditive representations areratio scales in Stevens’ sense. We know fromextensive measurement that the change ofunit in the additive representation is some-how reflecting something important about theunderlying structure. Is that also true of thechanges of units in the nonadditive represen-tation? We will return to this point, which canbe a source of confusion.

It should be noted that in probability theoryfor independent events, the p-additive formwith δ = 1 arises since

P(A ∪ B) = P(A) + P(B) − P(A)P(B).

An earlier, similar example concerningvelocity concatenation arose in Einstein’stheory of special relativity. Like the psycho-logical one, it entails a representation in thestandard measure of velocity that forms aratio scale and a nonlinear transformation toan additive one that also forms a ratio scale.We do not detail it here.

Nonadditive Concatenation

What characterizes an inherently nonadditivestructure is the failure of the empirical prop-erty of associativity; that is, for some elementsx, y, z in the domain,

x ◦ (y ◦ z) /∼ (x ◦ y) ◦ z.

Cohen and Narens (1979) made the then-unexpected discovery that if one simply dropsassociativity from any standard axiomatiza-tion of extensive measurement, not only canone still continue to construct numerical rep-resentations that are onto the positive realsbut, quite surprisingly, they continue to forma ratio scale as well; that is, the representa-tion is unique up to similarity transformations.They called this important class of nonaddi-tive representations unit structures. For a fulldiscussion, see Chaps. 19 and 20 of FM III.

A Fundamental Accountof Some Derived Measurement

Distribution Laws

The development of additive conjoint mea-surement allows one to give a systematic andfundamental account of what to that pointhad been treated as derived measurement. Forclassical physics, a typical situation in whichderived measurement arises takes the form〈A × X, �∼, ◦A〉. For example, let A denote aset of volumes and X a set of homogeneoussubstances; the ordering is that of mass asestablished by an equal-arm pan balance ina vacuum. The operation ◦A is the simpleunion of volumes. For this case we know thatm = Vρ, where m is the usual mass measure,V is the usual volume measure, and ρ is aninferred measure of density.

Observe that 〈A × X, �∼〉 forms an addi-tive conjoint structure. By the monotonicityassumption of conjoint measurement, Equa-tion (4), �∼ induces the weak order �∼A on A.It is assumed that 〈A, �∼A, ◦A〉 forms an ex-tensive structure. Thus we have the extensiverepresentation ϕA of 〈A, �∼A, ◦A〉 onto thepositive real numbers and a multiplicativeconjoint one ξAξX of 〈A× X, �∼〉 onto the pos-itive real numbers.

The question is how ϕA and ξA relate.Because both preserve the order �∼A, there



must be a strictly increasing function F suchthat ξA = F(ϕA). Beyond that, we can saynothing without some assumption describinghow the two structures interlock. One thatholds for many physical cases, including themotivating mass example, is a qualitative dis-tribution law of the form: For all a, b, c, d inA and x, y in X ,

(a, x) ∼ (c, y)

(b, x) ∼ (d, y)

}⇒ (a ◦A b, x) ∼ (c ◦A d, y).

Using this, one is able to prove that, forsome r > 0, s > 0, F(z) = r zs . Because theconjoint representation is unique up to powertransformations, we may select s = 1, that is,choose ξA = ϕA.

Note that distribution is a substantive, em-pirical property that in each specific case re-quires verification. In fact, it holds for manyof the classical physical attributes. From thatfact one is able to construct the basic structureof (classical) physical quantities that under-lies the technique called dimensional analysis,which is widely used in physical applicationsin engineering. It also accounts for the factthat physical units are all expressed as prod-ucts of powers of a relatively small set of units.This is discussed in some detail in Chap. 10of FM I and in a more satisfactory way inSection 22.7 of FM III.

Segregation Law

Within the behavioral sciences we have asituation that is somewhat similar to distri-bution. Suppose we return to the gamblingstructure, where some chance “experiment”is performed, such as drawing a ball from anurn with 100 red and yellow balls of whichthe respondent knows that the number of redis between 50 and 80. A typical binary gam-ble is of the form (x, C; y), where C denotesa chance event such as drawing a red ball, andthe consequence x is received if C occurs andy otherwise, that is, x if a red ball and y if ayellow ball. A weak preference order �∼ over

gambles is postulated. Let us distinguish gainsfrom losses by supposing that there is a spe-cial consequence, denoted e, that means nochange from the status quo. Things preferredto e are called gains, and those not preferredto it are called losses. Assume that for gains(and separately for losses) the axioms leadingto a subjective expected utility representationare satisfied. Thus, there is a utility function Uover gains and subjective probability functionS such that

U (x, C; y) = U (x)S(C) + U (y)[1 − S(C)]

(8)

U (e) = 0. (9)

Let ⊕ denote the operation of receiving twothings, called joint receipt. Therefore, g ⊕ hdenotes receiving both of the gambles g and h.Assume that ⊕ is a commutative5 and mono-tonic operation with e the identity element;that is, for all gambles g perceived as a gain,g ⊕ e ∼ g. Again, some law must link ⊕ tothe gambles. The one proposed by Luce andFishburn (1991) is segregation: For all gainsx, y,

(x, C; e) ⊕ y ∼ (x ⊕ y, C; y). (10)

Observe that this is highly rational in the sensethat both sides yield x ⊕ y when C occurs andy otherwise, so they should be seen as rep-resenting the same gamble. Moreover, thereis some empirical evidence in support of it(Luce, 2000, Chap. 4). Despite its apparentinnocence, it is powerful enough to show thatU (x ⊕ y) is given by Equation (7). Thus,in fact, the operation ⊕ forms an extensivestructure with additive representation V =−κ ln(1 − δU ), δκ > 0. Clearly, the sign of δ

greatly affects the relation between U and V:it is a negative exponential for δ > 0, propor-tional for δ = 0, and an exponential for δ < 0.

5Later we examine what happens when we drop thisassumption.



Applications of these ideas are given inLuce (2000). Perhaps the most interestingoccurs when dealing with x ⊕ y where x isa gain and y a loss. If we assume that V isadditive throughout the entire domain, thenwith x �∼ e �∼ y, U (x ⊕ y) is not additive. Thiscarries through to mixed gambles that nolonger have the simple bilinear form of binarySEU, Equation (8).

Invariance and Meaningfulness

Meaningful Statistics

Stevens (1951) raised the following issues inconnection with the use of statistics on mea-surements. Some statistical assertions do notseem to make sense in some measurementschemes. Consider a case of ordinal measure-ment in which one set of three observationshas ordinal measures 1, 4, and 5, with a meanof 10/3, and another set has measures 2, 3,and 6, with a mean of 11/3. One would saythe former set is, on average, smaller than thesecond one. But since these are ordinal data,an equally satisfactory representation is 1, 5,and 6 for the first set and 2, 2.1, and 6.1 forthe latter, with means respectively 12/3 and10.2/3, reversing the conclusion. Thus, thereis no invariant conclusion about means. Putanother way, comparing means is meaning-less in this context. By contrast, the median isinvariant under monotonic transformations. Itis easy to verify that the mean exhibits suitableinvariance in the case of ratio scales.

These observations were immediately chal-lenged and led to what can best be described asa tortured discussion that lasted many years.It was only clarified when the problem wasrecognized to be a special case of invarianceprinciples that were well developed in bothgeometry and dimensional analysis.

The main reason why the discussion wasconfused is that it was conducted at the levelof numerical representations, where two kindsof transformations are readily confused, rather

than in terms of the underlying structure itself.Consider a cubical volume that is 4 yards ona side. An appropriate change of units is fromyards to feet, so it is also 12 feet on a side.This is obviously different from the transfor-mation that enlarges each side by a factor of 3,producing a cube that is 12 yards on a side. Atthe level of numerical representations, how-ever, these two factor-of-3 changes are all tooeasily confused. This fact was not recognizedwhen Stevens wrote, but it clearly makes veryuncertain just what is meant by saying thata structure has a ratio or other type of repre-sentation and that certain invariances shouldhold.

Automorphisms

These observations lead one to take a deeperlook into questions of uniqueness and invari-ance. Mapping empirical structures onto nu-merical ones is not the most general or funda-mental way to approach invariance. The keyto avoiding confusion is to understand what itis about a structure that corresponds to correctadmissible transformations of the representa-tion. This turns out to be isomorphisms thatmap an empirical structure onto itself. Suchisomorphisms are called automorphisms bymathematicians and symmetries by physicists.Their importance is easily seen, as follows.Suppose α is an automorphism and f is a ho-momorphism of the structure into a numericalone, then it is not difficult to show that f ∗ α,where ∗ denotes function composition, is an-other equally good homomorphism into thesame numerical structure. In the case of a ra-tio scale, this means that there is a positive nu-merical constant rα such that f ∗ α = rα f . Theautomorphism captures something about thestructure itself, and that is just what is needed.

Consider the utility example, Equation (7),where there are two nonlinearly related rep-resentations, both of which are ratio scales inStevens’ sense. Thus, calculations of the meanutility are invariant in any one representation,



but they certainly are not across representa-tions. Which should be used, if either? It turnsout on careful examination that the one setof transformations corresponds to the auto-morphisms of the underlying extensive struc-ture. The second set of transformations cor-responds to the automorphisms of the SEUstructure, not ⊕. Both changes are important,but different. Which one should be used de-pends on the question being asked.

Invariance

An important use of automorphisms, first em-phasized for geometry by Klein (1872/1893)and heavily used by physicists and engineersin the method of dimensional analysis, is theidea that meaningful statements should beinvariant under automorphisms. Consider astructure with various primitive relations. Itis clear that these are invariant under the au-tomorphisms of the structure, and it is naturalto suppose that anything that can be mean-ingfully defined in terms of these primitivesshould also be invariant. Therefore, in partic-ular, given the structure of physical attributes,any physical law is defined in terms of the at-tributes and thus must be invariant. This def-initely does not mean that something that isinvariant is necessarily a physical law. In thecase of statistical analyses of measurements,we want the result to exhibit invarianceappropriate to the structure underlying themeasurements.

To answer Stevens’ original question aboutstatistics then entails asking whether the hy-pothesis being tested is meaningful (invariant)when translated back into assertions about theunderlying structure. Doing this correctly issometimes subtle, as is discussed in Chap. 22of FM III and much more fully by Narens(2001).

Trivial Automorphisms and Invariance

Sometimes structures have but one automor-phism, namely the function that maps each

element of the structure into itself—the iden-tity function. For example, in the additivestructure of the natural numbers with the stan-dard ordering, the only automorphism is theone that simply matches each number to itself:0 to 0, 1 to 1, and so on.

Within the weak ordering �∼ of a structure,there are trivial automorphisms beyond theidentity mapping, namely, those that just mapan element a to an equivalent element b; thatis, the relation a ∼ b holds.

Consider invariance in such structures. Wequickly see that the approach cannot yield anysignificant results because everything is in-variant. This remark applies to all finite struc-tures that are provided with a weak ordering.Thus, the only possibility is to examine theinvariant properties of the structure of the setof numerical representations.

Let a finite empirical structure be givenwith a homomorphism f mapping the struc-ture into a numerical structure. We have al-ready introduced the concept of an admissi-ble numerical transformation ϕ of f , namely,a one-one transformation of the range of fonto a possibly different set of real numbers,such that ϕ ∗ f is a homomorphism of the em-pirical structure. In order to fix the scale typeand thus the nature of the invariance of theempirical structure, we investigate the set ofall such homomorphisms for a given empiricalstructure. In the case of weight, any two homo-morphisms f1 and f2 are related by a positivesimilarity transformation; that is, there is apositive real number r > 0 such that r f1 = f2.In the qualitative probability case with inde-pendence, r = 1, so the set of all homomor-phisms has only one element. With r �= 1 inthe general similarity case, invariance is thencharacterized with respect to the multiplica-tive group of positive real numbers, each num-ber in the group constituting a change of unit.A numerical statement about a set of numer-ical quantities is then invariant if and only ifits truth value is constant under any changes of



unit of any of the quantities. This definition iseasily generalized to other groups of numeri-cal transformations such as linear transforma-tions for interval scales.

In contrast, consider a finite differencestructure with a numerical representation ascharacterized earlier. In general, the set of allhomomorphisms from the given finite struc-ture to numerical representations has no natu-ral and simple mathematical characterization.For this reason, much of the general theoryof representational measurement is concernedwith empirical structures that map onto thefull domain of real numbers. It remains true,however, that special finite empirical struc-tures remain important in practice in settingup standard measurement procedures usingwell-defined units.

Covariants

In practice, physicists hold on to invarianceby introducing and using the concept of co-variants. Typical examples of such covari-ants are velocity and acceleration, neither ofwhich is invariant from one coordinate frameto another under either Galilean or Lorentziantransformations, because, among other things,the direction of the velocity or accelerationvector of a particle will in general change fromone frame to another. (The scalar magnitudeof acceleration is invariant.)

The laws of physics are written in termsof such covariants. The fundamental idea isconveyed by the following. Let Q1, . . . , Qn

be quantities that are functions of the space-time coordinates, with some Qi s possibly be-ing derivatives of others, for example. Then, ingeneral, as we go from one coordinate systemto another (note that ′ does not mean deriva-tive) Q′

1, . . . , Q′n will be covariant, rather than

invariant, so their mathematical form is dif-ferent in the new coordinate system. But anyphysical law involving them, say,

F(Q1, . . . , Qn) = 0, (11)

must have the same form

F(Q′1, . . . , Q′

n) = 0

in the new coordinate frame. This same formis the important invariant requirement.

A simple example from classical mechan-ics is the conservation of momentum of twoparticles before and after a collision. Let vi

denote the velocity before and wi the velocityafter the collision, and mi the mass, i = 1, 2,of each particle. Then the law, in the form ofEquation (11), looks like this:

v1m1 + v2m2 − (w1m1 + w2m2) = 0,

and its transformed form will be, of course,

v′1m1 + v′

2m2 − (w ′1m1 + w ′

2m2) = 0,

but the forms of vi and wi will be, in general,covariant rather than invariant.

An Account of Stevens’ Scale-TypeClassification

Narens (1981a, 1981b) raised and partiallyanswered the question of why the Stevens’classification into ratio, interval, and ordinalscales makes as much sense as it seems to.His result was generalized by Alper (1987),as described later. The question may be castas follows: These familiar scale types have,respectively, one, two, and infinitely many de-grees of freedom in the representation; arethere not any others, such as ones havingthree or 10 degrees of freedom? To a first ap-proximation, the answer is “no,” but the pre-cise answer is somewhat more complex thanthat.

To arrive at a suitable formulation, a spe-cial case may be suggestive. Consider a struc-ture that has representations onto the reals—continuous representations—that form an in-terval scale. Then the representation has thefollowing two properties. First, given num-bers x < y and u < v, there is a positive affinetransformation that takes the pair (x, y) into



(u, v). It is found by setting u = r x + s, v =r y + s, whence r = v−u

y−x and s = yu−xv

y−x . Thus,in terms of automorphisms we have theproperty that there exists one that maps anyordered pair of the structure into any other or-dered pair. This is called two-point homogene-ity. Equally well, if two affine transforma-tions map a pair into the same pair, then theyare identical. This follows from the fact thattwo equations uniquely determine r and s. Interms of automorphisms, this is called 2-pointuniqueness. The latter can be recast by say-ing that any automorphism having two fixedpoints must be the identity automorphism.

In like manner, the ratio scale case is1-point homogeneous and 1-point unique. Thegeneralizations of these concepts to M—pointhomogeneity and N—point uniqueness areobvious. Moreover, in the continuous case itis easy to show that M ≤ N . The question ad-dressed by Narens was: Given that the struc-ture is at least 1-point homogeneous and N—point unique for some finite N , what are thepossibilities for (M, N )? Assuming M = Nand a continuous structure, he showed thatthe only possibilities are (1, 1) and (2, 2),that is, the ratio and interval scales. Alper(1987) dropped the condition that M = Nand showed that (1, 2) can also occur, but thatis the only added possibility. In terms ofnumerical representations on all of the realnumbers, the (1, 2) transformations are ofthe form x �−→ r x + s where s is any realand r is in some proper, nontrivial subgroupof the multiplicative, positive real group.One example is when r is of the form kn ,where k > 0 is fixed and n ranges over thepositive and negative integers.

This result makes clear two things. First,we see that there can be no continuous scalesbetween interval and ordinal, which of courseis not finitely unique. Second, there are scalesbetween ratio and interval. None of thesehas yet played a role in actual scientificmeasurement. Thus, for continuous structures

Stevens’ classification was almost complete,but not quite.

The result also raises some questions. First,how critical is the continuum assumption?The answer is “very”: Cameron (1989)showed that nothing remotely like the Alper-Narens result holds for representations on therational numbers. Second, what can be saidabout nonhomogeneous structures? Alper(1987) classified the M = 0 case, but the re-sults are quite complex and apparently notterribly useful. Luce (1992) explored empir-ically important cases in which homogeneityfails very selectively. It does whenever thereare singular points, which are defined to bepoints of the structure that remain fixed underall automorphisms. Familiar examples are 0in the nonnegative, multiplicative real num-bers and infinity if, as in relativistic velocity,it is adjoined to the system. For a broad classof systems, he showed that if a system hasfinitely many singular points and is homoge-neous between adjacent ones, then there areat most three singular points—a minimum,an interior, and a maximum one. The detailednature of these fixed points is somewhat com-plicated and is not discussed here. One spe-cific utility structure with an interior singularpoint—an inherent zero—is explored in depthin Luce (2000).

Models of Stevens’ Magnitude Methods

Stevens’ (1975) empirical findings, whichwere known in the 1950s, were a challengeto measurement theorists. What underlies thenetwork of (approximate) power functionrelations among subjective measures? Luce(1959) attempted to argue in terms of repre-sentations that if, for example, two attributesare each continuous ratio scales,6 with typi-cal physical representations ϕ1 and ϕ2, then

6Scale types other than ratio were also studied by Luceand subsequent authors.



a matching relation M between them shouldexhibit an invariance involving an admissi-ble ratio-scale change of the one attributecorresponding under the match to a ratio-scale change of the other attribute, that is,M[rϕ1(x)] = α(r)ϕ2(x). From this it is notdifficult to prove that M is a power func-tion of its argument. A major problem withthis argument is its failure to distinguish twotypes of ratio scale transformations—changesof unit, such as centimeters to meters—andchanges of scale, such as increasing the lineardimensions of a volume by a factor of three.Rozeboom (1962) was very critical of this fail-ure. Luce (1990) reexamined the issue fromthe perspective of automorphisms. Suppose Mis an empirical matching relation between twomeasurement structures, and suppose that foreach translation (i.e., an automorphism withno fixed point) τ of the first structure therecorresponds to a translation στ of the sec-ond structure such that for any stimulus x ofthe first structure and any s of the second,then x Ms holds if and only if for each auto-morphism τ of the first structure τ(x)Mστ (s)also holds. This assumption, called transla-tion consistency, is an empirically testableproperty, not a mere change of units. Assum-ing that the two structures have ratio scalerepresentations, this property is equivalent toa power function relation between the repre-sentations.

Based on some ideas of R. N. Shep-ard, circulated privately and later modifiedand published in 1981, Krantz (1972) devel-oped a theory that is based on three prim-itives: magnitude estimates, ratio estimates,and cross-modal matches. Various fairly sim-ple, testable axioms were assumed that onewould expect to hold if the psychophysicalfunctions were power functions of the cor-responding physical intensity and the ratiosof the instructions were treated as mathe-matical ratios. These postulates were shownto yield the expected power function repre-

sentations except for an arbitrary increasingfunction. This unknown function was elimi-nated by assuming, without a strong rationale,that the judgments for one continuum, suchas length judgments, are veridical, therebyforcing the function to be a simple multi-plicative factor. This model is summarized inFalmagne (1985, pp. 309–313). A somewhatrelated approach was offered by Falmagne andNarens (1983), also summarized in Falmagne(1985, pp. 329–339). It is based not on beha-vioral axioms, but on two invariance princi-ples that they call meaningfulness and dimen-sional invariance. Like the Krantz theory, ittoo leads to the form G(ϕ

rii ϕ

r j

j ), where G isunspecified beyond being strictly increasing.

Perhaps the deepest published analysis ofthe problem so far is Narens (1996). UnlikeStevens, he carefully distinguished numbersfrom numerals, noting that the experimen-tal structure involved numerals whereas thescientists’ representations of the phenomenainvolved numbers. He took seriously the ideathat internally people are carrying out theratio-preservation calculations embodied inStevens’ instructions. The upshot of Narens’axioms, which he carefully partitioned intothose that are physical, those that are behav-ioral, and those that link the physical and thebehavioral, was to derive two empirical pre-dictions from the theory. Let (x, p, y) meanthat the experimenter presents stimulus x andthe numeral p to which the subject producesstimulus y as holding the p relation to x. So if2 is given, then y is whatever the subject feelsis twice x. The results are, first, a commutativ-ity property: Suppose that the subject yields(x, p, y) and (y, q, z) when done in that orderand (x, q, u) and (u, p, v) when the numeralsare given in the opposite order. The predictionis z = v. A second result is a multiplicativeone: Suppose (x, pq, w), then the predictionis w = z. It is clear that the latter property im-plies the former, but not conversely. Empiricaldata reported by Ellemeirer and Faulhammer



(2000) sustain the former prediction and un-ambiguously reject the latter.

Luce (2001) provides a variant axiomatictheory, based on a modification of some math-ematical results summarized in Luce (2000)for utility theory. The axioms are formulatedin terms of three primitives: a sensory order-ing �∼ over physical stimuli varying in inten-sity, the joint presentation x ⊕ y of signals xand y (e.g., the presentation of pure tones ofthe same frequency and phase to the two ears),and for signals x > y and positive number pdenote by z = (x, p, y) the signal that the sub-ject judges makes interval [y, z] stand in pro-portion p to interval [y, x]. The axioms, suchas segregation, Equation (10), are behavioraland structural, and they are sufficient to ensurethe existence of a continuous psychophysicalmeasure ψ from stimuli to the positive realnumbers and a continuous function W fromthe positive reals onto the positive reals and aconstant δ > 0 such that for ⊕ commutative

x �∼ y ⇔ ψ(x) ≥ ψ(y), (12)

ψ(x ⊕ y) = ψ(x) + ψ(y) + δψ(x)ψ(y)

(δ > 0), (13)

ψ(x, p, y) − ψ(y) = W (p)[ψ(x) − ψ(y)].

(14)

We have written Equation (14) in this fashionrather than in a form comparable to the SEUequation for two reasons: It corresponds tothe instructions given the respondents, andW (p) is not restricted to [0, 1]. Recent, cur-rently unpublished, psychophysical data of R.Steingrimsson showed an important case of ⊕(two-ear loudness summation) that is rarely,if ever, commutative. This finding motivatedAczel, Luce, and Ng (2001) to explore thenoncommutative, nonassociative cases on theassumption ⊕ has a unit representation (men-tioned earlier) and assuming Equations (12)and (14) and that certain unknown functionsare differentiable. To everyone’s surprise, theonly new representations replacing (13) are

either

ψ(x ⊕ y) = αψ(x) + ψ(y), (α > 1)

when x ⊕ 0 � 0 ⊕ x, or

ψ(x ⊕ y) = ψ(x) + α′ψ(y), (α′ > 1)

when x ⊕ 0 ≺ 0 ⊕ x . These are called left- andright-weighted additive forms, respectively.These representations imply that some fixeddB correction can compensate the noncom-mutativity. Empirical studies evaluating thisare underway.

One invariance condition limits the form ofψ to the exponential of a power function of de-viations from absolute threshold, and anotherone limits the form of W to two parametersfor p ≥ 1 and two more for p < 1.

The theory not only is able to accommo-date the Ellemeirer and Faulhammer data butalso predicts that the psychophysical functionis a power function when ⊕ is not commuta-tive and only approximately a power functionfor ⊕ commutative. Over eight or more or-ders of magnitude, it is extremely close to apower function except near threshold and forvery intense signals. Despite its not being apure power function, the predictions for cross-modal matches are pure power functions.

Errors and Thresholds

To describe the general sources of errors andwhy they are inevitable in scientific work, wecan do no better than quote the opening pas-sage in Gauss’s famous work on the theoryof least squares, which is from the first partpresented to the Royal Society of Gottingenin 1821:

However much care is taken with observationsof the magnitude of physical quantities, they arenecessarily subject to more or less considerableerrors. These errors, in the majority of cases,are not simple, but arise simultaneously fromseveral distinct sources which it is convenientto distinguish into two classes.



Certain causes of errors depend, for each ob-servation, on circumstances which are variableand independent of the result which one obtains:the errors arising from such sources are calledirregular or random, and like the circumstanceswhich produce them, their value is not suscep-tible of calculation. Such are the errors whicharise from the imperfection of our senses and allthose which are due to irregular exterior causes,such as, for example, the vibrations of the airwhich make vision less clear; some of the errorsdue to the inevitable imperfection of the best in-struments belong to the same category. We maymention, for example, the roughness of the in-side of a level, the lack of absolute rigidity, etc.

On the other hand, there exist causes whichin all observations of the same nature producean identical error, or depend on circumstancesessentially connected with the result of the ob-servation. We shall call the errors of this cate-gory constant or regular.

It is evident that this distinction is relative upto a certain point and depends on how broad asense one wishes to attach to the idea of obser-vations of the same nature. For instance, if onerepeats indefinitely the measurement of a sin-gle angle, the errors arising from an imperfectdivision of the circular scale will belong to theclass of constant errors. If, on the other hand,one measures successively several different an-gles, the errors due to the imperfection of thedivision will be regarded as random as long asone has not formed the table of errors pertainingto each division. (Gauss, 1821/1957, pp. 1–2)

Although Gauss had in mind problems oferrors in physical measurement, it is quiteobvious that his conceptual remarks apply aswell to psychological measurement and, infact, in the second paragraph refer directlyto the “imperfection of our senses.” It wasreally only in the 19th century that, even inphysics, systematic and sustained attentionwas paid to quantitative problems of errors.For a historical overview of the work pre-ceding Gauss, see Todhunter (1865/1949). Ascan be seen from the references in the sectionon 19th- and early 20th-century psychology,

quantitative attention to errors in psycholog-ical measurement began at least with Fech-ner in the second half of the 19th century.Also, as already noted, the analysis of thresh-olds in probabilistic terms really began in psy-chology with the cited work of Thurstone.However, the quantitative and mathematicaltheory of thresholds was discussed earlierby Norbert Wiener (1915, 1921). Wiener’streatment was, however, purely algebraic,whereas in terms of providing relatively di-rect methods of application, Thurstone’sapproach was entirely probabilistic in char-acter. Already, Wiener (1915) stated veryclearly and explicitly how to deal with thefact that with thresholds in perception, therelation of indistinguishability—whether weare talking about brightness of light, loud-ness of sound, or something similar—is nottransitive.

The detailed theory was then given in the1921 paper for constructing a measure up to aninterval scale for such sensation-intensities.This is, without doubt, the first time that theseimportant psychological matters were dealtwith in rigorous detail from the standpoint ofpassing from qualitative judgments to a mea-surement representation. Here is the passagewith which Wiener ends the 1921 paper:

In conclusion, let us consider what bearing allthis work of ours can have on experimental psy-chology. One of the great defects under whichthe latter science at present labours is its propen-sity to try to answer questions without first try-ing to find out just what they ask. The experi-mental investigation of Weber’s law7 is a casein point: what most experimenters do take forgranted before they begin their experiments isinfinitely more important and interesting thanany results to which their experiments lead. Oneof these unconscious assumptions is that sensa-tions or sensation-intervals can be measured,

7Wiener means what is now called Fechner’s logarithmiclaw.



and that this process of measurement can becarried out in one way only. As a result, eachnew experimenter would seem to have devotedhis whole energies to the invention of a methodof procedure logically irrelevant to everythingthat had gone before: one man asks his sub-ject to state when two intervals between sensa-tions of a given kind appear different; anotherbases his whole work on an experiment wherethe observer’s only problem is to divide a givencolour-interval into two equal parts, and so onindefinitely, while even where the experimentsare exactly alike, no two people choose quitethe same method for working up their results.Now, if we make a large number of comparisonsof sensation-intervals of a given sort with refer-ence merely to whether one seems larger thananother, the methods of measurement given inthis paper indicate perfectly unambiguous waysof working up the results so as to obtain somequantitative law such as that of Weber withoutintroducing such bits of mathematical stupid-ity as treating a “just noticeable difference” asan “infinitesimal,” and have the further merit ofalways indicating some tangible mathematicalconclusion, no matter what the outcome of thecomparisons may be. (pp. 204–205)

The later and much more empirical work ofThurstone, already referred to, did not, how-ever, give a representational theory of mea-surement as Wiener, in fact, in his own waydid.

The work over the last few decades on er-rors and thresholds from the standpoint of rep-resentation theory of measurement naturallyfalls into two parts. The first part is the al-gebraic theory, and the second is the proba-bilistic theory. We first survey the algebraicresults.

Algebraic Theory of Thresholds

The work following Wiener on algebraicthresholds was only revived in the 1950s andmay be found in Goodman (1951), Halphen(1955), Luce (1956), and Scott and Suppes(1958). The subsequent literature is reviewed

in some detail in FM II, Chap. 16. We fol-low the exposition of the algebraic ordinaltheory there. We restrict ourselves here to fi-nite semiorders, the concept first introducedaxiomatically by Luce and in a modifiedaxiomatization by Scott and Suppes.

Let A be a nonempty set, and let � be abinary irreflexive relation on A. Then, (A �)is a semiorder if for every a, b, c, and d in A

(i) If a � c and b � d, then either a � d orb � c.

(ii) If a � b and b � c, then either a � d ord � c.

For finite semiorders (A, �) we can prove thefollowing numerical representational theoremwith constant threshold, which in the presentcase we will fix at 1, so the theorem assertsthat there is a mapping f of A into the positivereal numbers such that for any a and b in A,

a � b iff f (a) > f (b) + 1.

A wealth of more detailed and more delicateresults on semiorders is to be found in Sec-tion 2 of Chap. 16 of FM II, and researchcontinues on semiorders and various gener-alizations of them, such as interval orders.

Axioms extending the ordinal theory ofsemiorders to the kind of thing analyzed byWiener (1921) are in Gerlach (1957); unfor-tunately, to obtain a full interval-scale repre-sentation with thresholds involves very com-plicated axioms. This is true to a lesser extentof the axioms for semiordered qualitativeprobability structures given in Section 16.6.3of FM II. The axioms are complicated whenstated strictly in terms of the relation � ofsemiorders.

Probabilistic Theory of Thresholds

For applications in experimental work, it iscertainly the case that the probabilistic the-ory of thresholds is more natural and easier toapply. From various directions, there are ex-tensive developments in this area, many but



not all of which are presented in FM II andIII. We discuss here results that are simple toformulate and relevant to various kinds of ex-perimental work. We begin with the ordinaltheory.

A real-valued function P on A× A is calleda binary probability function if it satisfies both

P(a, b) ≥ 0,

P(a, b) + P(b, a) = 1.

The intended interpretation of P(a, b) is asthe probability of a being chosen over b. Weuse the probability measure P to define twonatural binary relations.

aW b iff P(a, b) ≥ 1

2,

aSb iff P(a, c) ≥ P(b, c), for all c.

In the spirit of semiorders we now definehow the relations W and S are related to vari-ous versions of what is called stochastic tran-sitivity, where stochastic means that the in-dividual instances may not be transitive, butthe probabilities are in some sense transitive.Here are the definitions. Let P be a binaryprobability function on A × A. We define thefollowing for all a, b, c, d in A:

Weak stochastic transitivity: If P(a, b) ≥12 and P(b, c) ≥ 1

2 , then P(a, c) ≥ 12 .

Weak independence: If P(a, c) > P(b, c),then P(a, d) ≥ P(b, d).

Strong stochastic transitivity: If P(a, b) ≥12 and P(b, c) ≥ 1

2 , then P(a, c) ≥max[P(a, b), P(b, c)].

The basic results for these concepts aretaken from Block and Marschak (1960) andFishburn (1973). Let P be a binary probabil-ity function on A × A, and let W and S bedefined as in the previous equations. Then

1. Weak stochastic transitivity holds if W istransitive.

2. Weak independence holds if S is con-nected.

3. Strong stochastic transitivity holds ifW = S. Therefore strong stochastic transi-tivity implies weak independence; the twoare equivalent if P(a, b) �= 1

2 for a �= b.

Random Variable Representations

We turn next to random variable representa-tions for measurement. In the first type, anessentially deterministic theory of measure-ment (e.g., additive conjoint measurement) isassumed in the background. But it is recog-nized that, for various reasons, variability inresponse occurs even in what are apparentlyconstant circumstances. We describe here theapproach developed and used by Falmagne(1976, 1985). Consider the conjoint indiffer-ence (a, p) ∼ (b, q) with a, p, and q givenand b to be determined so that the indiffer-ence holds. Suppose that, in fact, b is a randomvariable which we may denote B(a, p; q). Wesuppose that such random variables are in-dependently distributed. Since realizations ofthe random variables occur in repeated tri-als of a given experiment, we can define theequivalents we started with as holding whenthe value b is the Pth percentile of the dis-tribution of the random variable B(a, p; q).Falmagne’s proposal was to use the median,P = 1

2 , and he proceeded as follows. Let φ1

and φ2 be two numerical representations forthe conjoint measurement in the usual deter-ministic sense. If we suppose that such an ad-ditive representation is approximately correctbut has an additive error, then we have thefollowing representation:

ϕ1[B(a, p; q)] = ϕ1(a) + ϕ2(q)

− ϕ2(p) + ε(a, p; q),

where the εs are random variables. It is ob-vious enough how this equation provides anatural approximation of standard conjointmeasurement. If we strengthen the assump-tions a bit, we get an even more natural the-ory by assuming that the random variable



ε(a, p; q) has its median equal to zero.Using this stronger assumption about all theerrors being distributed with a median of zero,Falmagne summarizes assumptions that mustbe made to have a measurement structure.

Let A and P be two intervals of real num-bers, and let U = {Upq(a) | p, q ∈ P, a ∈ A}be a collection of random variables, each witha uniquely defined median. Then U is a struc-ture for random additive conjoint measure-ment if for all p, q, r in P and a in A, themedians m pq(a) satisfy the following axioms:

(i) They are continuous in all variables p, q,

and a.

(ii) They are strictly increasing in a and p,and strictly decreasing in q .

(iii) They map A into A.

(iv) They satisfy the cancellation rule withrespect to function composition *, i.e.,

(m pq ∗ mqr )(a) = m pr (a),

whenever both sides are defined.

For such random additive conjoint mea-surement structures, Falmagne (1985, p. 273)proved that there exist real-valued continuousstrictly increasing functions φ1 and φ2, de-fined on A and P respectively, such that forany Upq(a) in U ,

ϕ1[Upq(a)]

= ϕ2(p) + ϕ2(q) − ϕ1(a) + εpq(a),

where εpq(a) is a random variable with aunique median equal to zero. Moreover, if ϕ′

1

and ϕ′2 are two other such functions, then

ϕ′1(a) = αϕ1(a) + β

and

ϕ′2(p) = αϕ2(p) + γ,

where α > 0.Statistical tests of these ideas are not a

simple matter but have been studied in or-der to make the applications practical. Major

references are Falmagne (1978); Falmagneand Iverson (1979); Falmagne, Iverson, andMarcovici (1979); and Iverson and Falmagne(1985). Recent important work on probabil-ity models includes Doignon and Regenwetter(1997); Falmagne and Regenwetter (1996);Falmagne, Regenwetter, and Grofman (1997);Marley (1993); Niederée and Heyer (1997);Regenwetter (1997); and Regenwetter andMarley (in press).

Qualitative Moments

Another approach to measuring, in a repre-sentational sense, the distribution of a randomvariable for given psychological phenomenais to assume that we have a qualitative methodfor measuring the moments of the distributionof the random variable. The experimental pro-cedures for measuring such raw moments willvary drastically from one domain of experi-mentation to another. Theoretically, we needonly to assume that we can judge qualita-tive relations of one moment relative to an-other and that we have a standard weak order-ing of these qualitatively measured moments.The full formal discussion of these matters israther intricate. The details can be found inSection 16.8 of FM II.

Qualitative Density Functions

As is familiar in all sorts of elementary prob-ability examples, when a distribution has agiven form, it is often much easier to char-acterize it by a density distribution of a ran-dom variable than by a probability measureon events or by the method of moments asjust mentioned. In the discrete case, thesituation is formally quite simple. Each atom(i.e., each atomic event) in the discrete den-sity has a qualitative probability, and we needjudge only relations between these qualitativeprobabilities. We require of a representing dis-crete density function p on {a1, . . . , an} the



following three properties:

(i) p(ai ) ≥ 0.

(ii)∑n

i=1 p(ai ) = 1.

(iii) p(ai ) ≥ p(a j ) iff ai �∼ a j .

Note that the ai are not objects or stimuli inan experiment, but qualitative atomic events,exhaustive and mutually exclusive. Also notethat in this discrete case it follows thatp(ai ) ≤ 1, whereas in the continuous casethis is not true of densities.

We also need conditional discrete densi-ties. For this purpose we assume that theunderlying probability space X is finite or de-numerable, with probability measures P onthe given family F of events. The relation ofthe density p to the measure P is, for ai anatom of X ,

p(ai ) = P({ai })Then if A is any event such that P(A) > 0,

p(ai | A) = P({ai } | A),

and, of course, p(ai | A) is now a discrete den-sity itself, satisfying (i) through (iii).

Here are two simple, but useful, examplesof this approach. Let X be a finite set. Thenthe uniform density on X is characterized byall atoms being equivalent in the qualitativeordering �∼, that is,

ai ∼ a j .

We may then easily show that the unique den-sity satisfying the equivalence and (i), (ii), and(iii) is

p(ai ) = 1

n,

where n is the number of atoms in X .Among the many possible discrete distri-

butions, we consider just one further exam-ple, which has application in experiments inwhich the model being tested assumes a prob-ability of change of state independent of thetime spent in the current state. In the case ofdiscrete trials, such a memoryless process has

a geometric distribution that can be tested orderived from some simple properties of thediscrete but denumerable set of atomic events{a1, . . . , an, . . .}, on each of which is a posi-tive qualitative probability of the occurrenceof the change of state. The numbering of theatoms intuitively corresponds to the trials ofan experiment. The atoms are ordered in qual-itative probability by the relation �∼. We alsointroduce a restricted conditional probability.If i > j then ai | A j is the conditional eventthat the change of state will occur on trial igiven that it has not occurred on or beforetrial j . (Note that here A j means no changeof state from trial 1 through j .) The qualita-tive probability ordering relation is extendedto include these special conditional events aswell.

The two postulated properties, in additionto (i), (ii), and (iii) given above, are these:

(iv) Order property: ai �∼ a j iff j ≥ i ;

(v) Memoryless property: ai+1 | Ai ∼ a1.

It is easy to prove that (iv) implies a weakordering of �∼. We can then prove that p(an)

has the form

p(an) = c(1 − c)n−1 (0 < c < 1).

Properties (i) through (v) are satisfied, but theyare also satisfied by any other c′, 0 < c′ < 1.For experiments testing only the memorylessproperty, no estimation of c is required. If itis desired to estimate c, the standard estimateis the sample mean m of the trial numbers onwhich the change of state was observed, sincethe mean µ of the density p(an) = c(1−c)n−1

satisfies the following equation:

µ = 1 − c

c.

For a formal characterization of the fullqualitative probability for the algebra ofevents—not just atomic events—in the case ofthe geometric distribution, see Suppes (1987).For the closely related but mathematically



more complicated continuous analogue (i.e.,the exponential distribution), see Suck (1998).

GENERAL FEATURESOF THE AXIOMATIC APPROACH

Background

History

The story of the axiomatic method begins withthe ancient Greeks, probably in the fifth cen-tury B.C. The evidence seems pretty convinc-ing that it developed in response to the earlycrisis in the foundations of geometry men-tioned earlier, namely, the problem of incom-mensurable magnitudes. It is surprising andimportant that the axiomatic method as wethink of it was largely crystallized in Euclid’sElements, whose author flourished and taughtin Alexandria around 300 B.C. From a mod-ern standpoint, Euclid’s schematic approachis flawed, but compared to any other standardto be found anywhere else for over two mil-lennia, it is a remarkable achievement. Thenext great phase of axiomatic developmentoccurred, as already mentioned, in the 19thcentury in connection with the crisis gener-ated in the foundations of geometry itself. Thethird phase was the formalization within logicof the entire language used and the realizationthat results that could not be proved otherwisecan be achieved by such complete logical for-malization. In view of the historical reviewpresented earlier in this article, we will con-centrate on only this third phase in this section.

What Comes before the Axioms

Three main ingredients need to be fixed inan axiomatization before the axioms are for-mulated. First, there must be agreement onthe general framework used. Is it going tobe an informal, set-theoretical framework orone formalized within logic? These two al-ternatives are analyzed in more detail later.

The second ingredient is to fix the primi-tive concepts of the theory being axioma-tized. For example, in almost all theories ofchoice we need an ordering relation as a prim-itive concept, which means, formally, a binaryrelation. We also often need, as mentionedearlier, a binary operation as, for example,in the cases of extensive measurement andaveraging. In any case, whatever the prim-itives may be, they should be stated at thebeginning. The third ingredient, at least asimportant, is clarity and explicitness aboutwhat other theories are being assumed. It is acharacteristic feature of empirical axiomatiza-tions that some additional mathematics is usu-ally assumed, often without explicit notice.This is not true, however, of many qualita-tive axiomatizations of representational mea-surement and often is not true in the founda-tions of geometry. In contrast, many varietiesof probabilistic modeling in psychology doassume some prior mathematics in formulat-ing the axioms. A simple example of thiswas Falmagne’s axioms for random addi-tive conjoint measurement, presented earlier.There, such statistical notions as the medianand such elementary mathematical notionsas that of continuity were assumed withoutfurther explanation or definition.

Another ingredient, less important from aformal standpoint but of considerable impor-tance in practice, are the questions of whethernotions defined in terms of the primitive con-cepts should be introduced when formulatingthe axioms and whether auxiliary mathemati-cal notions are assumed in stating the axioms.The contrasting alternative is to state the ax-ioms strictly in terms of the primitive notions.From the standpoint of logical purity, the lat-ter course seems desirable, but in actual fact itis often awkward and intuitively unappealingto state all of the axioms in terms of the prim-itive concepts only. A completely elementarybut good example of this is the introduction ofa strict ordering and an equivalence relation


General Features of the Axiomatic Approach 33

defined in terms of a weak ordering, a movethat is often used as a way of simplifying andmaking more perspicuous the formulation ofaxioms in choice or preference theory withinpsychology.

Theories with Standard LogicalFormalization

Explicit and formally precise axiomatic ver-sions of theories are those that are formalizedwithin first-order logic. Such a logic can beeasily characterized in an informal way. Thislogic assumes

(i) one kind of variable;

(ii) logical constants, mainly the sententialconnectives such as and;

(iii) a notation for the universal and existen-tial quantifiers; and

(iv) the identity symbol =.

A theory formulated within such a logi-cal framework is called a theory with stan-dard formalization. Ordinarily, three kinds ofnonlogical constants occur in axiomatizing atheory within such a framework: the relationsymbols (also called predicates), the opera-tion symbols, and the individual constants.

The grammatical expressions of the the-ory are divided into terms and formulas, andrecursive definitions of each are given. Thesimplest terms are variables or individual con-stants. New terms are built up by combiningsimpler terms with operation symbols in themanner spelled out recursively in the formu-lation of the language of the theory. Atomicformulas consist of a single predicate and theappropriate number of terms. Compoundformulas are built up from atomic formu-las by means of sentential connectives andquantifiers.

Theories with standard formalization arenot often used in any of the empirical sciences,including psychology. On the other hand, theycan play a useful conceptual role in answering

some empirically important questions, as weillustrate later.

There are practical difficulties in castingordinary scientific theories into the frameworkof first-order logic. The main source of thedifficulty, which has already been mentioned,is that almost all systematic scientific theo-ries assume a certain amount of mathematicsa priori. Inclusion of such mathematics is notpossible in any elegant and reasonable way ina theory beginning only with logic and with noother mathematical assumptions or apparatus.Moreover, a theory that requires for its for-mulation an Archimedean-type axiom, muchneeded in representational theories of mea-surement when the domain of objects consid-ered is infinite, cannot even in principle be for-mulated within first-order logic. We say moreabout this well-known result later. For theseand other reasons, standard axiomatic formu-lations of most mathematical theories, as wellas scientific theories, follows the methodol-ogy to which we now turn.

Theories Definedas Set-Theoretical Predicates

A widely used alternative approach to formu-lating representational theories of measure-ment and other scientific theories is to axiom-atize them within a set-theoretical framework.Moreover, this is close to the practice of muchmathematics. In such an approach, axioma-tizing a theory simply amounts to defining acertain set-theoretical predicate. The axioms,as we ordinarily think of them, are a part ofthe definition—its most important part froma scientific standpoint. Such definitions were(partially) presented earlier in a more or lessformal way (e.g., weak orderings, extensivestructures, and other examples of qualitativecharacterizations of empirical measurementstructures). Note that the concept of isomor-phism, or the closely related notion of homo-morphism, is defined for structures satisfying



some set-theoretical predicate. The languageof set-theoretical predicates is not ordinarilyused except in foundational talk; it is just away of clarifying the status of the axioms.It means that the axioms are given within aframework that assumes set theory as the gen-eral framework for all, or almost all, mathe-matical concepts. It provides a seamless wayof linking systematic scientific theories thatuse various kinds of mathematics with math-ematics itself. An elementary but explicit dis-cussion of the set-theoretical approach to ax-iomatization is found in Suppes (1957/1999,chap. 12).

Formal Results about Axiomatization

We sketch here some of the results that wethink are of significance for quantitative workin experimental psychology. A detailed treat-ment is given in FM III, Chap. 21. We shouldemphasize that all the systematic results westate here hold only for theories formalized infirst-order logic.

Elementary Languages

First, we need to introduce, informally, somegeneral notions to be used in stating the re-sults. We say that a language L of a theoryis elementary if it is formulated in first-orderlogic. This means that, in addition to the ap-paratus of first-order logic, the theory onlycontains nonlogical relation symbols, opera-tion symbols, and individual constants. Intu-itively, a model of such a language L is sim-ply an empirical structure, in the sense alreadydiscussed; in particular, it has a nonempty do-main, a relation corresponding to each primi-tive relation symbol, an operation correspond-ing to each primitive operation symbol, andindividuals in the domain corresponding toeach individual constant.

Using such logical concepts, one major re-sult is that there are infinite weak orders thatcannot be represented by numerical order. A

specific example is the lexicographic order ofpoints in the plane, that is (x, y) �∼ (x ′, y′) ifand only if either x > x ′ or x = x ′ and y ≥ y′.

In examining the kinds of axioms givenearlier (e.g., those for extensive measure-ment), it is clear that some form of anArchimedean axiom is needed to get a numer-ical representation, and such an axiom cannotbe formulated in an elementary language L, apoint to which we return a little later.

A second, but positive, result arises whenthe domains of the measurement structures arefinite. A class of such structures closed un-der isomorphism is called a finitary class ofmeasurement structures. To that end, we needthe concept of a language being recursivelyaxiomatizable; namely, there is an algorithmfor deciding whether a formula of L is an ax-iom of the given theory. It can be shown thatany finitary class of measurement structureswith respect to an elementary language L isaxiomatizable but not necessarily recursivelyaxiomatizable in L.

The importance of this result is in show-ing that the expressive power of elementarylanguages is adequate for finitary classes butnot necessarily for the stating of a set of recur-sive axioms. We come now to another positiveresult guaranteeing that recursive axioms arepossible for a theory. When the relations, op-erations, and constants of an empirical struc-ture are definable in elementary form wheninterpreted as numerical relations, functions,and constants, then the theory is recursivelyaxiomatizable.

Nonaxiomatizability Results

Now we turn to a class of results of direct psy-chological interest. As early as the work ofWiener (1921), the nontransitive equivalencerelation generated by semiorders was defined(see the earlier quotation); namely, if we thinkof a semiorder, then the indistinguishability orindifference relation that complements it willhave the following numerical representation.


General Features of the Axiomatic Approach 35

For two elements a and b that are indistin-guishable or indifferent with respect to thesemiorder, the following equivalence holds:

| f (a) − f (b)| ≤ 1 iff a ∼ b.

Now we have already seen that finitesemiorders have a very simple axiomatiza-tion. Given how close the indistinguishabil-ity relation is to the semiorder itself, it seemsplausible that this relation, too, should have asimple axiomatization. Surprisingly, Roberts(1968, 1969) proved that this is not the case.More precisely, let L be the elementary lan-guage whose only nonlogical symbol is thebinary relational symbol ∼. Then the finitaryclass J of measurement structures for thebinary relation of indistinguishability is notaxiomatizable in L by a universal sentence.

Note that there is a restriction in the result.It states that ∼ is not axiomatizable by a uni-versal sentence. This means that existentialstatements are excluded. The simple axiom-atization of semiorders, given earlier, is sucha universal axiomatization because no quan-tifiers were required. But that is not true ofindistinguishability. A little later, we discussthe more general question of axioms with ex-istential quantifiers for elementary languages.

This result about ∼ is typical of a groupof theorems concerning familiar representa-tions for which it is impossible to axiomatizethe class of finite structures by adjoining auniversal sentence to an elementary languageL. Scott and Suppes (1958) first proved thisto be true for a quaternary relation symbolcorresponding to a difference representation.Titiev (1972) obtained the result for additiveconjoint measurement; he also showed that itholds for the n-dimensional metric structureusing the Euclidean metric; and in 1980 heshowed that it is true for the city-block met-ric when the number of dimensions n ≤ 3. Itis worth mentioning that the proof for n = 3given by Titiev required computer assistanceto examine 21,780 cases, each of which

involved 10 equations and 12 unknowns in arelated set of inequalities. To our knowledge,nothing is known about n > 3.

This last remark is worth emphasizing tobring out a certain point about the results men-tioned here. For any particular case (e.g., anexperiment using a set of 10 stimuli), a con-structive approach, rather than the negativeresults given here, can be found for each par-ticular case. One can simply write down theset of elementary linear inequalities that mustbe satisfied and ask a computer program todecide whether this finite set of inequalitiesin a fixed number of variables has a solu-tion. If the answer is positive, then a numericalrepresentation can be found, and the very re-stricted class of measurement structures builtup around this fixed number of variables andfixed set of inequalities is indeed a measure-ment structure. What the theorems show isthat the general elementary theory of suchinequalities cannot be given in any reason-able axiomatic form. We cannot state for thevarious kinds of cases that are considered anelementary set of axioms that will guaranteea numerical solution for any finite model(i.e., a model with a finite domain) satisfyingthe axioms.

Finally, in this line of development, wemention a theorem requiring more sophisti-cated logical apparatus that was proved by PerLindstrom (stated as Theorem 17, p. 243, FMIII), namely, that even if existential quanti-fiers are permitted, the usual class of finitemeasurement structures for algebraic differ-ence cannot be characterized by a finite set ofelementary axioms.

Archimedean andLeast-Upper-Bound Axioms

We have mentioned more than once thatArchimedean axioms play a special role informulating representational theories of mea-surement when the domain of the empirical



structures is infinite. Recall that theArchimedean axiom for extensive measure-ment of weight or mass asserts that for anyobjects a and b, there exists some integer nsuch that n replicas of object a, written asa(n), exceeds b, that is, a(n) �∼ b. This ax-iom, as well as other versions of it, cannotbe directly formulated in an elementary lan-guage because of the existential quantificationin terms of the natural numbers. In that sense,the fact that an elementary theory cannot in-clude an Archimedean axiom has an immedi-ate proof. Fortunately, however, a good dealmore can be proved: For such elementary the-ories, of the kind we have considered in thischapter, there can be no elementary formulasof the elementary language L that are equiv-alent to an Archimedean axiom. After all, wemight hope that one could simply replace theArchimedean axiom by a conjunction of ele-mentary formulas, but this is not the case. Fora proof, and references to the literature, seeFM III, Section 21.7.

It might still be thought that by avoiding theexplicit introduction of the natural numbers,we might be able to give an elementary formu-lation using one of the other axioms invokedin real analysis. Among these are Dedekind’s(1872/1902) axiom of completeness, Cantor’s(1895) formulation of completeness in termsof Cauchy sequences, and the more standardmodern approach of assuming that a boundednonempty set has a least-upper-bound in termsof the given ordering. We consider only thelast example because its elementary form al-lows us to see easily what the problem is. Toinvoke this concept, we need to be able to talkin our elementary language not only about in-dividuals in the given domain of an empiri-cal structure, but also about sets of these in-dividuals. But the move from individuals tosets of individuals is a mathematically pow-erful one, and it is not permitted in standardformulations of elementary languages. As inthe case of the Archimedean axiom, then,

we have an immediate argument for reject-ing such an axiom. Moreover, as in the case ofthe Archimedean axiom, we can prove that noset of elementary formulas of an elementarylanguage L is equivalent to the least-upper-bound axiom. The proof of this follows nat-urally from the Archimedean axiom, since ina general setting the least-upper-bound axiomimplies an Archimedean axiom.

Proofs of Independence of Axiomsand Primitive Concepts

All the theorems just discussed can be formu-lated only within the framework of elemen-tary languages. Fortunately, important ques-tions that often arise in discussions of axiomsin various scientific domains can be answeredwithin the purely set-theoretic framework anddo not require logical formalization. The firstof these is proving that the axioms are inde-pendent in the sense that none can be deducedfrom the others. The standard method for do-ing this is as follows. For each axiom, a modelis given in which the remaining axioms aresatisfied and the one in question is not sat-isfied. Doing this establishes that the axiomis independent of the others. The argument issimple. If the axiom in question could be de-rived from the remaining axioms, we wouldthen have a violation of the intuitive conceptof logical consequence. An example of lackof independence among axioms given forextensive measurement is the commutativityaxiom, a ◦ b ∼ b ◦ c. It follows from theother axioms with the Archimedean axiomplaying a very important role.

The case of the independence of primitivesymbols requires a method that is a little moresubtle. What we want is an argument that willprove that it is not possible to define one ofthe primitive symbols in terms of the others.Padoa (1902) formulated a principle that canbe applied to such situations. To prove thata given primitive concept is independent of


References 37

the other primitive concepts of a theory, findtwo models of the axioms of the theory suchthat the primitive concept in question is es-sentially different in the two models and theremaining primitive symbols are the same inthe two models.

As a very informal description of a triv-ial example, consider the theory of preferencebased on two primitive relations, one a strictpreference and the other an indifference rela-tion. Assume both are transitive. We want toshow what is obvious—that strict preferencecannot be defined in terms of indifference. Weneed only take a domain of two objects, forexample, the numbers 1 and 2. Then for the in-difference relation we just take identity: 1 = 1and 2 = 2. But in one model the strict prefer-ence relation has 1 preferred to 2, and in thesecond preference model the preference rela-tion has 2 preferred to 1. This shows that strictpreference cannot be defined in terms of in-difference because indifference is the same inboth models whereas preference is different.

CONCLUSIONS

The second half of the 20th century saw anumber of developments in our understandingof numerical measurement. Among these arethe following: (a) examples of fundamentalmeasurement different from extensive struc-tures; (b) an increased understanding of howmeasurement structures interlock to yield sub-stantive theories; (c) a classification of scaletypes for continuous measurement in termsof properties of automorphism groups; (d) ananalysis of invariance principles in limitingthe mathematical forms of various measures;(e) a logical analysis of what sorts of the-ories can and cannot be formulated usingpurely first-order logic without existential orArchimedean statements; and (f) a number ofpsychological applications especially in psy-chophysics and utility theory.

A major incompleteness remains in the so-cially important area of ability and achieve-ment testing. Except for the work of Doignonand Falmange (1999), no representational re-sults of significance exist for understandinghow individuals differ in their grasp of certainconcepts. This is not to deny the extensive de-velopment of statistical models, but only toremark that fundamental axiomatizations arerarely found. This is changing gradually, butas yet it is a small part of representational mea-surement theory.

REFERENCES

Aczel, J., Luce, R. D., & Ng, C. T. (2001). Func-tional equations arising in a theory of rank de-pendence and homogeneous joint receipts. Sub-mitted.

Alper, T. M. (1987). A classification of all order-preserving homeomorphism groups of the realthat satisfy finite uniqueness. Journal of Mathe-matical Psychology, 31, 135–154.

Block, H. D., & Marschak, J. (1960). Random or-derings and stochastic theories of responses. InI. Olkin, S. Ghurye, W. Hoeffding, W. Madow,& H. Mann (Eds.), Contributions to probabil-ity and statistics (pp. 97–132). Stanford, CA:Stanford University Press.

Cameron, P. J. (1989). Groups of order-automorphisms of the rationals with prescribedscale type. Journal of Mathematical Psychology,33, 163–171.

Campbell, N. R. (1920/1957). Physics: The ele-ments. Cambridge: Cambridge University Press.[Reprinted under the title Foundations of sci-ence: The philosophy of theory and experiment,New York: Dover.] (Original work published1920)

Campbell, N. R. (1928). An account of theprinciples of theory and experiment. London:Longmans, Green.

Cantor, G. (1895). Beitrage zur Begrundung dertransfiniten Mengenlehre. Mathematische An-nalen, 46, 481–512.



Cohen, M. A., & Narens, L. (1979). Fundamen-tal unit structures: A theory of ratio scalability.Journal of Mathematical Psychology, 20, 192–232.

Davidson, D., McKinsey, J. C. C., & Suppes, P.(1955). Outlines of a formal theory of value: I.Philosophy of Science, 22, 140–160.

Davidson, D., Suppes, P., & Segal, S. (1957).Decision-making: An experimental approach.Stanford, CA: Stanford University Press.[Reprinted as Midway Reprint, 1977, Chicago:University of Chicago Press.]

Debreu, G. (1960). Topological methods in cardi-nal utility theory. In K. J. Arrow, S. Karlin, &P. Suppes (Eds.), Mathematical methods in thesocial sciences, 1959. Stanford, CA: StanfordUniversity Press (pp. 16–26).

Dedekind, R. (1872/1902). Stetigkeit und Irra-tionale Zahlen. Brunswick, 1872. [Essays onthe theory of numbers. New York: Dover, 1963(reprint of the English translation).]

Doignon, J.-P., & Falmagne, J.-C. (1999). Knowl-edge spaces. Berlin: Springer.

Doignon, J.-P., & Regenwetter, M. (1997).An approval-voting polytope for linear or-ders. Journal of Mathematical Psychology, 41,171–188.

Dzhafarov, E. N., & Colonius, H. (1999). Fechne-rian metrics in unidimensional and multidimen-sional spaces. Psychonomic Bulletin and Review,6, 239–268.

Dzhafarov, E. N., & Colonius, H. (2001). Multidi-mensional Fechnerian scaling: Basics. Journalof Mathematical Psychology, 45, 670–719.

Edwards, W. (Ed.). (1992). Utility theories: Mea-surements and applications. Boston: Kluwer.

Ellemeirer, W., & Faulhammer, G. (2000). Em-pirical evaluation of axioms fundamental toStevens’ ratio-scaling approach: I. Loudnessproduction. Perception & Psychophysics, 62,1505–1511.

Euclid. (1956). Elements, Book 10 (Vol. 3, T. L.Heath, Trans.) New York: Dover.

Falmagne, J.-C. (1976). Random conjoint mea-surement and loudness summation. Psychologi-cal Review, 83, 65–79.

Falmagne, J.-C. (1978). A representation theo-rem for finite random scale systems. Journal ofMathematical Psychology, 18, 52–72.

Falmagne, J.-C. (1985). Elements of psychophysi-cal theory. New York: Oxford University Press.

Falmagne, J.-C., & Iverson, G. (1979). ConjointWeber laws and additivity. Journal of Mathe-matical Psychology, 20, 164–183.

Falmagne, J.-C., Iverson, G., & Marcovici, S.(1979). Binaural “loudness” summation: Prob-abilistic theory and data. Psychological Review,86, 25–43.

Falmagne, J.-C., & Narens, L. (1983). Scales andmeaningfulness of quantitative laws. Synthese,55, 287–325.

Falmagne, J.-C., & Regenwetter, M. (1996). A ran-dom utility model for approval voting. Journalof Mathematical Psychology, 40, 152–159.

Falmagne, J.-C., Regenwetter, M., Grofman, B.(1997). A stochastic model for the evolution ofpreferences. In A. A. J. Marley (Ed.), Choice,decision, and measurement: Essays in honor ofR. Duncan Luce (pp. 111–129). Mahwah, NJ:Erlbaum.

Fechner, G. T. (1966). Elements of psychophysics(H. E. Adler, Trans.). New York: Holt, Rinehart& Winston. (Original work published 1860)

Ferguson, A., Meyers, C. S. (Vice Chairman),Bartlett, R. J. (Secretary), Banister, H., Bartlett,F. C., Brown, W., Campbell, N. R., Craik,K. J. W., Drever, J., Guild, J., Houstoun, R. A.,Irwin, J. O., Kaye, G. W. C., Philpott, S. J. F.,Richardson, L. F., Shaxby, J. H., Smith, T.,Thouless, R. H., & Tucker, W. S. (1940). Quan-titative estimates of sensory events. The ad-vancement of science. Report of the British As-sociation for the Advancement of Science, 2,331–349.

Fishburn, P. C. (1970). Utility theory for decisionmaking. New York: Wiley.

Fishburn, P. C. (1973). Binary choice probabilities:On the varieties of stochastic transitivity. Journalof Mathematical Psychology, 10, 327–352.

Fishburn, P. C. (1988). Nonlinear preference andutility theory. Baltimore, MD: Johns HopkinsPress.


References 39

Fourier, J. (1955). The analytical theory of heat(A. Freeman, Trans.). New York: Stechert.(Original work published 1822)

Gauss, K. F. (1821/1957). Theory of least squares(H. F. Trotter, Trans.). Princeton, NJ: PrincetonUniversity Press.

Gerlach, M. W. (1957). Interval measurementof subjective magnitudes with subliminaldifferences (Tech. Rep. No. 7). Stanford, CA:Stanford University, Behavioral SciencesDivision, Applied Mathematics and StatisticsLaboratory.

Goodman, N. (1951). Structure of appearance.Cambridge: Harvard University Press.

Grassmann, H. G. (1895). Die Wissenschaft derExtensiven Grosse oder die Ausdehnungslehre,republished in Gesammelte Mathematische undPhysikalische Werke. Leipzig. (Original workpublished 1844)

Green, D. M., & Swets, J. A. (1974). Signal detec-tion theory and psychophysics. Huntington, NY:Krieger. (Original work published 1966)

Halphen, E. (1955). La notion de vraisem-blance. Publication de l’Institut de Statistiquede l’Universite de Paris, 4, 41–92.

Heisenberg, W. (1930). Quantum theory. Chicago:University of Chicago Press.

Helmholtz, H. V. (1887). Zahlen und Messenerkenntnis-theoretisch betrachet. Philosophis-che Aufsatze Eduard Zeller gewidmet, Leipzig.Reprinted in Gesammelte Abhandl., Vol. 3,1895, 356–391. [English translation byC. L. Bryan, Counting and Measuring.Princeton, New Jersey: van Nostrand,1930.]

Hilbert, D. (1899). Grundlagen der Geometrie(8th edition, with revisions and supplements byP. Bernays, 1956). Stuttgart: Teubner.

Holder, O. (1901). Die Axiome der Quantitat unddie Lehre vom Mass. Berichte uber die Verhand-lungen der Koniglich Sachsischen Gesellschaftder Wissenschaften zu Leipzig, Mathematische-Physische Klasse, 53, 1–64.

Iverson, G., & Falmagne, J.-C. (1985). Statisti-cal issues in measurement. Mathematical SocialSciences, 10, 131–153.

Iverson, G., & Luce, R. D. (1998). The rep-resentational measurement approach to psy-chophysical and judgment problems. In M. H.Birnbaum (Ed.), Measurement, judgment, anddecision making (pp. 1–79). San Diego:Academic Press.

Klein, F. (1872/1893). A comparative review ofrecent researches in geometry. Bulletin of theNew York Mathematical Society, 2, 215–249.

Krantz, D. H. (1972). A theory of magnitude esti-mation and cross-modality matching. Journal ofMathematical Psychology, 9, 168–199.

Krantz, D. H., Luce, R. D., Suppes, P., & Tversky,A. (1971). Foundations of measurement, Vol. I.New York: Academic Press.

Luce, R. D. (1956). Semiorders and a theory ofutility discrimination. Econometrica, 24, 178–191.

Luce, R. D. (1959). On the possible psychophysicallaws. Psychological Review, 66, 81–95.

Luce, R. D. (1990). “On the possible psychophys-ical laws” revisited: Remarks on cross-modalmatching. Psychological Review, 97, 66–77.

Luce, R. D. (1992). Singular points in generalizedconcatenation structures that otherwise arehomogeneous. Mathematical Social Sciences,24, 79–103.

Luce, R. D. (2000). Utility of gains and losses:Measurement-theoretical and experimental ap-proaches. Mahwah, NJ: Erlbaum. [Errata: seeLuce’s Web page at http://socsci.uci.edu.]

Luce, R. D. (2001). A psychophysical theory ofintensity proportions, joint presentations, andmatches. Psychological Review, in press.

Luce, R. D., & Edwards, W. (1958). The derivationof subjective scales from just-noticeable differ-ences. Psychological Review, 65, 227–237.

Luce, R. D., & Fishburn, P. C. (1991). Rank- andsign-dependent linear utility models for finitefirst-order gambles. Journal of Risk and Uncer-tainty, 4, 29–59.

Luce, R. D., Krantz, D. H., Suppes, P., & Tversky,A. (1990). Foundations of measurement, Vol. III.San Diego: Academic Press.

Luce, R. D., & Tukey, J. (1964). Simultaneous con-joint measurement: A new type of fundamental



measurement. Journal of Mathematical Psy-chology, 1, 1–27.

Macmillan, N. A., & Creelman, C. D. (1991).Detection theory: A user’s guide. Cambridge:Cambridge University Press.

Marley, A. A. J. (1993). Aggregation theorems andthe combination of probabilistic rank orders. InD. E. Critchlow, M. A. Fligner, & J. S. Verducci(Eds.), Probability models and statistical anal-yses for ranking data (pp. 216–240). New York:Springer.

Maxwell, J. C. (1873). A Treatise on electricity andmagnetism. London: Oxford University Press.

Michell, J. (1990). An introduction to the logicof psychological measurement. Hillsdale, NJ:Erlbaum.

Michell, J. (1999). Measurement in psychology:A critical history of a methodological concept.Cambridge: Cambridge University Press.

Moody, E., and Claggett, M. (Eds.). (1952).The medieval science of weights (scientiade ponderibus): Treatises ascribed to Euclid,Archimedes, Thabit ibn Qurra, Jordanus deNemore, and Blasius of Parma. Madison:University of Wisconsin Press.

Narens, L. (1981a). A general theory of ratio scal-ablity with remarks about the measurement-theoretic concept of meaningfulness. Theory andDecision, 13, 1–70.

Narens, L. (1981b). On the scales of measurement.Journal of Mathematical Psychology, 24, 249–275.

Narens, L. (1985). Abstract measurement theory.Cambridge: MIT Press.

Narens, L. (1996). A theory of magnitude estima-tion. Journal of Mathematical Psychology, 40,109–129.

Narens, L. (2001). Theories of meaningfulness.Mahwah, NJ: Erlbaum, in press.

Niederee, R., & Heyer, D. (1997). Generalizedrandom utility models and the representationaltheory of measurement: A conceptual link, InA. A. J. Marley (Ed.), Choice, decision, andmeasurement: Essays in honor of R. DuncanLuce (pp. 153–187). Mahwah, NJ: Erlbaum.

O’Brien, D. (1981). Theories of weight in the an-cient world. Paris: Les Belles Lettres.

Padoa, A. (1902). Un nouveau systeme irre-ductible de postulats pour l’algebre. CompteRendu du deuxieme Congres International deMathematiciens, 249–256.

Pasch, M. (1882). Vorlesungen uber Neuere Ge-ometrie. Leipzig: Springer.

Pfanzagl, J. (1959). A general theory ofmeasurement-applications to utility. NavalResearch Logistics Quarterly, 6, 283–294.

Pfanzagl, J. (1968). Theory of measurement. NewYork: Wiley.

Plateau, J. A. F. (1872). Sur la measure des sensa-tions physiques, et sur loi qui lie l’intensite deces sensation a l’intensite de la cause excitante.Bulletin de l’Academie Royale de Belgique, 33,376–388.

Quiggin, J. (1993). Generalized expected util-ity theory: The rank-dependent model. Boston:Kluwer.

Ramsey, F. P. (1931). Truth and probability. InF. P. Ramsey (Ed.), The foundations of math-ematics and other logical essays (pp. 156–198).New York: Harcourt, Brace.

Regenwetter, M. (1997). Random utility represen-tations of finite n-ary relations. Journal of Math-ematical Psychology, 40, 219–234.

Regenwetter, M., & Marley, A. A. J. (in press).Random relations, random utilities, and randomfunctions. Journal of Mathematical Psychology.

Roberts, F. S. (1968). Representations of indiffer-ence relations. Unpublished doctoral disserta-tion, Stanford University, Stanford, CA.

Roberts, F. S. (1969). Indifference graphs. InF. Harary (Ed.), Proof techniques in graph the-ory (pp. 139–146). New York: Academic Press.

Roberts, F. S., & Luce, R. D. (1968). Axiomaticthermodynamics and extensive measurement.Synthese, 18, 311–326.

Rozeboom, W. E. (1962). The untenability ofLuce’s principle. Psychological Review, 69,542–547. [With a reply by Luce, 548–551, andan additional comment by Rozeboom, 552].

Savage, L. J. (1954). The foundations of statistics.New York: Wiley.

Scott, D., & Suppes, P. (1958). Foundational as-pects of theories of measurement. Journal ofSymbolic Logic, 23, 113–128.


References 41

Shepard, R. N. (1981). Psychophysical relationsand psychophysical scales: On the status of“direct” psychophysical measurement. Journalof Mathematical Psychology, 24, 21–57.

Stevens, S. S. (1951). Mathematics, measurementand psychophysics. In S. S. Stevens (Ed.), Hand-book of experimental psychology (pp. 1–49).New York: Wiley.

Stevens, S. S. (1959). Measurement, psycho-physics and utility. In C. W. Churchman &P. Ratoosh (Eds.), Measurement: Definitionsand theories (pp. 18–64). New York: Wiley.

Stevens, S. S. (1975). Psychophysics: Introductionto its perceptual, neural, and social prospects.New York: Wiley.

Suck, R. (1998). A qualitative characterization ofthe exponential distribution, Journal of Mathe-matical Psychology, 42, 418–431.

Suppes, P. (1999). Introduction to logic. New York:Dover. (Original work published 1957)

Suppes, P. (1980). Limitations of the axiomaticmethod in ancient Greek mathematical sciences.In P. Suppes (Ed.), Models and methods in thephilosophy of science: Selected essays (pp. 25–40). Dordrecht, Netherlands: Kluwer.

Suppes, P. (1987), Propensity representations ofprobability. Erkenntnis, 26, 335–358.

Suppes, P., & Alechina, N. (1994). The definabil-ity of the qualitative independence of events interms of extended indicator functions. Journalof Mathematical Psychology, 38, 366–376.

Suppes, P., Krantz, D. H., Luce, R. D., & Tversky,A. (1990). Foundations of measurement, Vol. II.San Diego: Academic Press.

Thurstone, L. L. (1927a). A law of comparativejudgement. Psychological Review, 34, 273–287.

Thurstone, L. L. (1927b). Psychophysical anal-ysis. American Journal of Psychology, 38,368–389.

Thurstone, L. L. (1927c). Three psychophysicallaws. Psychological Review, 34, 424–432.

Thurstone, L. L. (1959). The measurement ofvalues. Chicago: University of Chicago Press.

Titiev, R. J. (1972). Measurement structures inclasses that are not universally axiomatizable.Journal of Mathematical Psychology, 9, 200–205.

Titiev, R. J. (1980). Computer-assisted resultsabout universal axiomatizability and the three-dimensional city-block metric. Journal of Math-ematical Psychology, 22, 209–217.

Todhunter, I. (1949). A history of the mathemat-ical theory of probability. New York: Chelsea.(Original work published 1865)

von Neumann, J. (1932/1955). Mathematicalfoundations of quantum mechanics (R. T.Byer, Trans.). Princeton: Princeton UniversityPress.

von Neumann, J., & Morgenstern, O. (1947). Thetheory of games and economic behavior. Prince-ton, NJ: Princeton University Press.

Wakker, P. P. (1989). Additive representations ofpreferences: A new foundation of decision anal-ysis. Dordrecht, Netherlands: Kluwer AcademicPublishers.

Wiener, N. (1915). Studies in synthetic logic. Pro-cedings of the Cambridge Philosophical Society,18, 14–28.

Wiener, N. (1921). A new theory of measurement:A study in the logic of mathematics. Proceedingsof the London Mathematical Society, 17, 181–205.


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