Philosophical and Foundational Issues in Measurement Theory

C. Wade Savage, Philip Ehrlich, C. Wade Savage, Philip Ehrlich

• Publisher: Psychology Press
• Year: 2013
• ISBN:
  9781134758739
• Edition: 1
• Topic: Psychologie
• Subtopic: Geschichte & Theorie in der Psychologie

Measurement theory has only recently become recognized as a legitimate, specialized field of inquiry. This text covers a wide range of issues of central concern to contemporary measurement theorists, and a broad range of philosophical perspectives are represented. The formalist, representationalist approach defines measurement as the assignment of numbers to entities and events to represent their properties and relations. It also states that measurement theory is supposed to analyze the concept of a scale of measurement, describe various types of scales and their uses, and formulate the conditions required for the existence of scales of various types. Since this approach dominates contemporary measurement theory, the volume begins with essays by some of its leading architects. In order to allow for diverse points of view, the book also includes articles that attempt to broaden this approach, and several that even criticize the approach.

1	A Brief Introduction to Measurement Theory and to the Essays

C. Wade Savage

University of Minnesota

Philip Ehrlich

Brown University

Until approximately three decades ago, measurement theory was widely assumed to be the avocation of a few physicists, mathematicians, and philosophers of science, and the obsession of a few social scientists who hoped to secure for their fields the authority enjoyed by mathematical physics and chemistry. The recognition of a legitimate, specialized field of inquiry called “measurement theory” is of even more recent origin.

Admittedly theories concerning the nature of quantity date from at least the ancient Greeks, as described in Aristotle’s writings on these subjects in the Categories and the Metaphysics (c. 330 B.C.) and later axiomatized in Euclid’s Elements (c. 300 B.C.). Furthermore, portions of Euclid’s treatise can be regarded as a theory of measurement of spatial extent, in the sense that one line segment, surface, or solid “measures” another by being compared with it. However, this is what we now call synthetic geometry. It does not assign numbers in the abstract, arithmetic sense to nonnumerical continua such as lengths, areas, and regions; therefore, it cannot compare magnitudes within such continua by comparing such numbers. Euclidean geometry compares lengths, areas, and regions by comparing physical, nonnumerical ratios of these magnitudes and in effect uses such ratios in the place of our arithmetic numbers.

The analytic geometry Descartes and Fermat pioneered assumes that numerical measures of length, area, and volume can be assigned to line segments, surfaces, and solids by counting congruent unit objects in a collection that successively approximates the object being measured. It is a theory of measurement in the contemporary sense in which measurement is the assignment of abstract arithmetic numbers to objects. However, it is not a full, explicit theory; it makes the assumption of measurability naively, without attempting to justify it in the manner currently required.

Today, measurement in general is taken to be the assignment of numbers (numerals, say the nominalists) to entities and events to represent their properties and relations. Furthermore, measurement theory is supposed to analyze the concept of a scale of measurement or numerical representation, distinguish various types of scale and describe their uses, and formulate the conditions required for the existence of scales of various types, not just for the case of length and other extensive properties but for measurable properties of all types.

The previous characterization of contemporary measurement theory is heavily influenced by the formalist, representationalist approach to the subject presented in Krantz, Luce, Suppes, and Tversky (1971), hereafter referred to as KLST. As the following essays illustrate, this approach currently dominates the field, serving as a model for most measurement theorists and a target for the rest. To provide an organizing context for the essays and to orient nonspecialist readers, we offer a brief historical survey of contemporary measurement theory.

Contemporary measurement theory can be said to begin with Helmholtz’s Counting and measuring (1887) and Hölder’s Die Axiome der Quantität und die Lehre vom Mass (1901), in which axioms for such (extensive) properties as length and mass were formulated. These works, together with influential treatments by Bertrand Russell (1903) and N. R. Campbell (1928, 1957/1920), created what may be called the conservative conception of measurement. On this conception counting is defined as placing the members of a collection in one-to-one correspondence with a segment of the natural numbers, and direct (extensive) measurement of a property of an object is then defined as counting concatenations of standard objects that approximately equal the object with respect to the property, where the concatenation has, like addition on numbers, such properties as commutativity, associativity, and so on. What most philosophers knew of formal measurement at the close of the first half of the century was gleaned largely from Campbell (1957/1920) and Cohen and Nagel (1934). Psychologists had, by that time, been exposed to a similar treatment by Bergmann and Spence, (1953/1944).

It appears that, on the conservative definition, length, weight, duration, angle, electric charge, and several other physical properties are directly measurable. However, hardness and temperature seem not to be directly measurable. In addition, such allegedly psychological attributes as hue, pitch, taste, and pain intensity seem even more clearly not to be directly measurable. These latter properties do not seem to possess a natural, empirical operation of concatenation that can be used to define their measure in the manner required by the conservative conception. Of course such properties may still be indirectly measurable by measuring a directly measurable, correlated property. Thus, temperature is measured by measuring the length of a column of thermometric fluid in a thermometer. (Henceforth, “measurable” will mean “directly measurable.”) Some of the conservative theorists (Nagel, 1931, for example) distinguished extensive and intensive attributes, and they tentatively conceded that some of these other properties might be intensively measurable. Extensive measurement is accomplished by counting concatenations and is supposed to make such statements as “The length of a is n times greater than the length of b” meaningful; intensive measurement does not proceed by counting concatenations and is supposed to make only such statements as “The temperature of a is greater than the temperature of b” meaningful. However, many conservative theorists did not recognize intensive measurement and claimed that properties without an empirical concatenation are not properly said to be measurable.

Even as the conservative view was being formulated, many psychologists, especially psychophysicists such as Thurstone (1959), were insisting that various psychological properties are measurable; indeed some believed that every property is measurable. In the words of Guilford’s (1954/1936) treatise:

Many psychologists adopt the dictum of Thorndike that “Whatever exists at all exists in some amount” and they also adopt the corollary that whatever exists in some amount can be measured: the quality of handwriting, the appreciation of a sunset, the attitude of an individual toward Communism, or the strength of desire in a hungry rat. (p. 3)

Accordingly, at mid-century, the psychophysicist S. S. Stevens (1946, 1951) formulated what may be called the liberal conception of measurement, according to which measurement is defined simply as the assignment of numbers to things and properties according to rule. (For the distinction between the two conceptions of measurement, see Savage, 1970, chapters 4 and 5, where they are called the “narrow” and “broad” views.) Depending on the rule employed, the numerical assignment will constitute a scale of some type. Among conservatives, philosophical questions about measurement often took the form “Is E measurable?” where E is some property such as hardness, loudness, hue, pain, afterimage area, perceived length, desire, subjective time, welfare, probability, prestige of occupation, value, beauty, and so forth. The liberal theorists argued that this question is fruitless, because everything is held to be capable of measurement of some sort, and substituted for it the question “Of what sort of measurement is E capable?”. They concluded that the task of measurement theory is to classify and describe the types of measurement.

Stevens (1946, 1951) distinguished four main types of scales of measurement. A nominal scale represents only differences among objects (for example, numbers assigned to football players). An ordinal scale represents the order of objects with respect to some property (for example, numbers used to rank restaurants). An interval scale represents intervals of a property (for example, the Centigrade scale of temperature). A ratio scale represents ratios of a property (for example, the inch scale of length). Stevens suggested that scales may also be classified by means of the transformations that leave the scale-form invariant, the “admissible” transformations. If ϕ is the original scale, and ϕ’ is the transformed scale, then the defining transformation for an interval scale is ϕ’ = kϕ + c (for example, ϕ’ is the Fahrenheit scale of temperature; ϕ is the Centigrade scale; k = 1.8; and c = 32); the defining transformation for a ratio scale is ϕ’ = kϕ (for example, ϕ’ is the centimeter scale of length; ϕ is the inch scale; and k = 2.54). On the liberal conception, measurement does not require counting, and it does not require an operation of concatenation on the measured objects. This claim is intuitively obvious for nominal and ordinal scales with a small, finite number of values; for here, numbers can be assigned to the objects one by one, checking each assignment to assure that it represents identity and difference and order. However, Stevens also claimed to have constructed interval and ratio scales of continuous perceptual properties such as perceived loudness, perceived electric shock, and perceived heaviness from the numerical responses of subjects in psychological experiments, without employing any operation of concatenating physical objects, or sensations, or responses.

In the two decades following Stevens’ formulation of his conception of measurement, several logical empiricist philosophers of science provided semiformal treatments of the subject—Hempel (1952), Carnap (1966), and Ellis in his Basic concepts of measurement (1966). In the operationalist-instrumentalist tradition of Mach, Bridgman, and Stevens, Ellis argued that the function of measurement is not to represent independently existing nonnumerical quantities; indeed quantities (except when identified with an ordering relation) are in effect created by the operations that measure them. Consequently one cannot choose between two additive scales of the same quantity that use different operations of concatenation or even between an additive and a nonadditive scale, on the ground that one better represents length than the other. The only rational ground is that one scale leads to simpler numerical laws of length, area, mass, force, and so forth than the other. In so arguing, Ellis adopted a conventionalist view of measurement to some extent. To the extent that he claimed quantities do not exist independently of their measuring operations, his position is antirealist (operationalist). Carnap’s writings on measurement express a more strongly conventionalist, operationalist view than those of Ellis. Hempel’s work is comparatively neutral on the metaphysical issues involved.

Most of the major contributors to formal measurement theory have been mathematicians and psychologists. For most of the century, psychologists had been concerned with the theory of measuring psychological magnitudes, such as subjective brightness and hue, pain and pleasure, and attitudes and preferences. Most of this work was published in psychological journals and labeled “psychometrics” or “psychological scaling,” as if to imply that the issues under discussion concerned only psychological magnitudes. Research of this sort was summarized during the period following Stevens’ formulation in Torgerson’s Theory and methods of scaling (1958). Meanwhile a group of mathematical logicians and mathematical psychologists, building on the results of Cantor, Hölder, Birkhoff, and other mathematicians, began to employ model-theoretic and set-theoretic methods to investigate the conditions and properties of scales of measurement in general, scales from areas as diverse as economics, psychophysics, and physics. This work culminated in 1971 in the landmark Foundations of measurement, Volume I by Krantz et al., a book that effectively defines the field known today as formal measurement theory. Volumes II (1989) and III (1990) have recently appeared and will undoubtedly have a comparably profound impact. The subjec...