The Measurement of Sensation: A Critique of Perceptual Psychophysics [Reprint 2019 ed.] 9780520337039

Citation preview

The Measurement of Sensation

THE MEASUREMENT OF SENSATION A CRITIQUE OF PERCEPTUAL PSYCHOPHYSICS

C. Wade Savage

UNIVERSITY OF CALIFORNIA PRESS BERKELEY

LOS ANGELES 1970

LONDON

UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES, CALIFORNIA UNIVERSITY OF CALIFORNIA PRESS, LTD. LONDON, ENGLAND COPYRIGHT ©

1 9 7 0 BY THE REGENTS OF THE UNIVERSITY OF CALIFORNIA

L I B R A R Y OF CONGRESS CATALOG CARD NUMBER:

69-15941

SBN 5 2 0 - 0 1 5 2 7 - 4 DESIGNED BY DAVE COMSTOCK PRINTED IN THE UNITED STATES OF AMERICA

IN MEMORY OF

Lauren AND

Edith

Acknowledgments My thanks to: —Norman Malcolm, my teacher and friend, for his encouragement throughout the seven-year project that produced this book. —My research assistants, who helped with the details of preparing the book, and whose interest sustained my belief that the project was worthwhile (James Shaw, James Rea, Paul Bridges, Nancy Hollander, Donna Lichtenstein, Charles Whitt, and Victor Morton). —Keith Donnellan, Sydney Shoemaker, and Robert Yost, for their interest in this book's ancestor: my doctoral dissertation for Cornell University. —Gustav Bergmann, who introduced me to psychophysics and to its founder, Gustav Theodor Fechner. —Grace Stimson, my editor at the University of California Press. —Those members of the secretarial staff of the UCLA Department of Philosophy who efficiently and amiably converted my drafts into typed copy. —The University of California, Los Angeles, which supported the book with three grants providing the services of research assistants. —The American Psychological Association, for permission to incorporate into chapter 9 my monograph on ratio scales published in volume 80 of Psychological Monographs. —Diane.

Contents x: THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT x The Problem 2 The Solution 6 Deeper Questions xx 2: SOUNDS AND SENSATIONS Sensations as Perception-Dependent Sensations as Private Sensations as Nonlocatable The Sensationist Theory of Perception Sensations and Measurability

x8 19 25 36 47 54

3: SOUNDS AND SOUND WAVES .. Direct and Indirect Measurement The Measurement of Sound Waves v4re Sounds and Sound Waves Identical? The Question of Measuring Sounds

60 62 74 88 96

4: THE NARROW VIEW OF MEASUREMENT The View Presented The Axioms of Addition The Purpose of Measurement Operations and Measurement

99 xox 108 123 133

5: BROADER VIEWS OF MEASUREMENT Requirements for an Acceptable View Stevens' View Hempel's View An Acceptable View

157 158 163 189 196

6: THE MEASUREMENT OF LOUDNESS Objections from Subjectivity

2x4 2x5

CONTENTS

X

Objections from Relativity Objections Concerning Addition Objections from the Purpose of Measurement Applications and Conclusions 7: THE MEASUREMENT

OF PITCH

Objections to the Absolute Zero Method Objections to Using an Arbitrary Zero Objections to the Halftone as an Interval Objections from Scientific Usefulness Applications and Conclusions 8: THE JND MEASUREMENT

OF SENSATIONS

Fechner's Law Three JND Methods Are Sensation JND's Equal? Sensations and Thresholds Sensations versus Sensitivity

221 228 238 242 246 248 251 257 264 272 283 284 300 316 331 355

9: RATIO SCALES OF PERCEPTUAL MAGNITUDES A Sample Ratio-Scaling Experiment The Introspectionist Interpretation The Behaviorist Interpretation Confusion of the Two Interpretations Perceptual Magnitudes versus Perceptual Abilities

364 367 373 385 408 419

10: PSYCHOLOGICAL AND PHYSICAL DIMENSIONS Quality and Quantity Primary and Secondary Qualities The Asymmetrical Distinction The Symmetrical Distinction Psychological and Physical Observations

428 429 435 442 457 491

11:

PERCEPTUAL PSYCHOPHYSICS

501

The Measurement of Mind Traditional Psychophysics Psychophysics: Old, New, and Radical

501 510 530

BIBLIOGRAPHY

549

INDEX

565

Figures 1. 2. 3. 4. 5. 6. 7. 8.

Auditory sensations, sounds, and sound waves Equal-frequency contours Equal-loudness contours Loudness fractionations of one-half and one-tenth Relation of the jnd to differential sensitivity Length fractionations of one-half and one-fourth Ratio, category, and jnd scales for length Loudness function and cochlear potential

50 84 86 233 356 367 369 401

Tables 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Measurement of a triangle by ruler Li Measurement of a triangle by ruler L2 Relation between rulers Li and Ls Measurement of a triangle by ruler Ls Loudness measurement by the addition method Pitch measurement by the halftone method JND measurement of sensations by the algebraic method JND measurement of sensations by Fechner's method Measurement of the just noticeable stimulus Measurement of the just noticeable difference Measurement by the method of limits Relation of the jnd to differential sensitivity Symmetrical and asymmetrical distinctions Simple interpretation of the symmetrical distinction Psychological dimensions as molar dimensions Psychological dimensions as sensation dimensions An interval scale of psychological weight Exponents in Stevens' power law

129 129 130 130 241 253 303 310 334 335 337 355 444 459 466 468 538 539

Figures 1. 2. 3. 4. 5. 6. 7. 8.

Auditory sensations, sounds, and sound waves Equal-frequency contours Equal-loudness contours Loudness fractionations of one-half and one-tenth Relation of the jnd to differential sensitivity Length fractionations of one-half and one-fourth Ratio, category, and jnd scales for length Loudness function and cochlear potential

50 84 86 233 356 367 369 401

Tables 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Measurement of a triangle by ruler Li Measurement of a triangle by ruler L2 Relation between rulers Li and Ls Measurement of a triangle by ruler Ls Loudness measurement by the addition method Pitch measurement by the halftone method JND measurement of sensations by the algebraic method JND measurement of sensations by Fechner's method Measurement of the just noticeable stimulus Measurement of the just noticeable difference Measurement by the method of limits Relation of the jnd to differential sensitivity Symmetrical and asymmetrical distinctions Simple interpretation of the symmetrical distinction Psychological dimensions as molar dimensions Psychological dimensions as sensation dimensions An interval scale of psychological weight Exponents in Stevens' power law

129 129 130 130 241 253 303 310 334 335 337 355 444 459 466 468 538 539

1 The Problem of Psychophysical Measurement Psychophysics, as traditionally conceived, is the science of relations between mind and body, between psychological phenomena and physical phenomena, between mental entities and their stimuli, or, as is often said, between psychological dimensions and physical dimensions. The task of psychophysics is therefore to discover numerical laws relating psychological dimensions and physical dimensions, laws of the form [i where Y is a psychological dimension expressed in units of psychological measurement, $ is a physical dimension expressed in units of physical measurement, and / is a numerical function discovered by employing the two types of measurement. Perceptual psychophysics, in this conception, is the science of relations between psychological dimensions of perception—such as loudness, brightness, sweetness, and acridity—and physical dimensions of stimuli—such as intensity of sound waves, frequency of light waves, length, weight, and hardness. The prevailing conception of psychophysics, indeed, perhaps the only one in use, is the traditional one. It underlies the "old" psychophysics of Fechner and his adherents, and also the "new" psychophysics of S. S. Stevens and his adherents. This conception has produced more than a century of controversy, much of it revolving around philosophical issues, most of it still unsettled. The aim of this book is to destroy the traditional conception of perceptual psychophysics, and to dissolve the philosophical problems generated by it. The most important of these, and the focus of this book, is the problem of the psychophysical measurement of perception. This chapter provides a brief introduction to the problem and a sketch of the way in which the book attempts to dissolve it.

]v = m,

2

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

THE PROBLEM Psychophysical measurement is the measurement of psychological dimensions in order to discover their numerical relations to physical dimensions. The problem is that psychological dimensions seem not to be measurable. Let us illustrate with the pitch of sounds. A psychophysical law of pitch supposedly relates pitch to some physical dimension, say, frequency of sound waves. It is well known that (roughly speaking) pitch increases as frequency increases. This statement of covariation is a nonnumerical law relating pitch and frequency. A numerical law would have the form, [2] P = f(F), where P is pitch expressed in the units of some system of pitch measurement, F is frequency expressed in the units of some system of frequency measurement, and / is a numerical function whose value is empirically determined by employing the two systems of measurement. Many methods of frequency measurement are available. The method most commonly used employs some type of electronic meter— an oscilloscope, for instance—which is sensitive to and is calibrated for variations in sound-wave frequency. But how is pitch to be measured? There are, it would seem, no meters that are sensitive to pitch (as opposed to frequency). Nor, it seems, can pitch be measured in the w a y w e measure such physical dimensions as length and weight: one cannot lay a measuring rod on sounds, nor place sounds on a balance. Indeed, it can easily seem that apriori arguments show that pitch is immeasurable. Pitch is, apparently, a qualitative dimension. We can say that one pitch is higher, or lower, than another. But can any sense be given to saying that the pitch of one sound is greater, or less, than the pitch of another? If not, pitch is clearly incapable of measurement, since all measurement consists in making numerical determinations of relative magnitude with respect to some dimension. Perhaps we are forced to conclude that pitch, and all other psychological dimensions of perception, are immeasurable, and that numerical laws governing such dimensions are impossible. This conclusion will seem even more inevitable if sounds are taken to be sensations, and pitch is taken to be a dimension of sensations. Pains are paradigm examples of sensations. Pains possess qualitative dimensions: there is burning pain, aching pain, throbbing pain,

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

3

and so on; but they also possess quantitative dimensions: pains may be more or less intense than one another. That the qualitative dimensions of pain are immeasurable does not entail that the quantitative dimensions are immeasurable. Still, it seems impossible to measure the intensity of pain. Measurement consists in assigning numerals to things in order to represent some quantitative dimension in terms of a dimension unit. Hence, if the intensity of pain is measurable, it must be possible to define or specify a unit of intensity. But this does not seem possible. We might specify that the pain, pi, felt by an observer, Oi, at a given time, ti, has an intensity of one unit. Pains, however, are private, which seems to entail that Oa cannot determine whether the pain he now feels has the same intensity as the pain Oi feels. If this is correct, pi can serve at best as a unit of Oi's pains, not as a unit for the pains of 02, O3, and others. Worse still, pi seems unacceptable even as a unit for the pains of Oi. How could Oi determine whether a pain had by him several days later than ti is or is not equal in intensity to pi? Finally, even if pains could be accurately compared by different observers, or by a single observer at different times, it does not seem possible to express the magnitude of pains by comparison with a unit pain. To do so apparently requires some way of adding unit pains, and adding pains seems impossible. The view that loudness, brightness, sweetness, and acridity are, like intensity of pains, dimensions of sensations, is part of what may be called the sensationist theory of perception. The theory holds that perception is a process whereby stimuli in the physical environment produce sensations in the perceiver, which he then uses to perceive (by inference, or interpretation) the environment. Sounds are held to be the sensations thus produced in auditory perception, and loudness and pitch are construed as their most familiar dimensions. Colors are held to be sensations produced in visual perception, and brightness and hue are construed as among their dimensions. Usually, the sensationist theory is accompanied by the view that psychophysical laws of perception are laws relating the sensations produced in perception to their physical stimuli. In this view, numerical psychophysical laws of perception are impossible if, as appears at first sight, sensations are incapable of measurement. It is highly doubtful that normal instances of auditory, visual, and other types of perception involve the production of sensations in the

4

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

perceiver. But even if we reject the sensationist theory and its usual interpretation of psychophysical laws of perception, even if we deny that psychological dimensions are dimensions of sensations, the problem of psychophysical measurement seems to remain. For loudness, pitch, brightness, and hue—even if they are not dimensions of sensations—seem to qualify as psychological dimensions, and whether they can be measured seems doubtful. Existing attempts to solve the problem of psychophysical measurement consist in arguing that psychological dimensions are measurable, but measurable by methods unlike those available for such physical dimensions as length and weight. The solution favored by those with an orientation toward physics consists in establishing that loudness, pitch, and other psychological dimensions can be measured, not directly as can length and weight, but indirectly. In this view, a dimension is indirectly measured by measuring some other dimension with which it is correlated or in terms of which it is defined. Thus, pitch of sounds, which is correlated with frequency of sound waves, can be measured by measuring frequency and assigning the numerals thus obtained to pitch. For instance, sounds with pitches of C below middle C , middle C , and C above middle C are correlated, respectively, with frequencies of 128, 256, and 512 cycles per second. The pitches are indirectly measured by assigning to them the numerals assigned to their corresponding frequencies. The simplest objection to this procedure is that it does not guarantee an assignment of numerals to sounds which accurately represents their relative pitch. Adjacent frequencies in our example stand to each other in the ratio 1 :z, but adjacent pitches do not stand in this ratio, since the intervals between adjacent pitches are octaves and octaves are equal intervals. The solution to the problem of psychophysical measurement favored by psychophysicists consists in establishing that psychological dimensions can be measured, not by indirect procedures parasitic on physical measurement, but by special psychological procedures. One such procedure, suggested by Fechner, employs the just noticeable difference (jnd) as a unit of psychological magnitude. When the difference in loudness between two sounds is so small that an observer can just barely hear it as a difference, it is said to be a just noticeable difference. If such differences are equal, as Fechner and others believe, they can perhaps be used as unit intervals, and numerals can be as-

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

5

signed to sounds so as to represent the number of loudness jnd's they lie above loudness zero (either an arbitrary zero, or the lower limen for loudness). Fechner believed that the use of the jnd method would confirm the logarithmic law, [3] Y = fclog where V is a psychological magnitude expressed in jnd units, $ is a physical magnitude expressed in suitable physical units, and k is a constant that varies according to the department of perception being measured. The simplest objection to this procedure concerns its assumption that jnd's are equal. Given two sounds, one of which is twice as loud as the other, if the softer sound is n jnd's above loudness zero, the second will be less than 2n jnd's above loudness zero, which entails that loudness jnd's are not equal. A recent solution is that offered by S. S. Stevens and his adherents. Stevens proposes that ratio scales of psychological dimensions be erected by requiring observers to make quantitative estimates of the dimensions in question: either fractionation estimates or absolute numerical estimates. In the first of these methods, illustrated with loudness, the observer is asked to say what loudness ratios (1:2, 2:5, etc.) one tone has to another. After estimates for a large number of sound pairs have been obtained, the experimenter arbitrarily designates one tone as the unit tone, and then assigns numerals to various tones which stand to one another in the ratios estimated by the observer. In the second method, the observer is asked to assign numerals to various tones so that the numerals represent the loudness ratios perceived by the observer. Stevens believes that the use of these methods confirms the power law, [4] ¥

=

where ¥ is a psychological magnitude expressed in the units of an estimation scale, $ is a physical magnitude expressed in suitable physical units, and n is an exponent that varies according to the department of perception being measured. The simplest objection to Stevens' methods is that they confuse estimation with measurement. It is one thing to estimate the length ratios in which various rods stand to one another, quite another to measure the length of the rods with a meterstick. Similarly, it is one thing for an observer to estimate loudness ratios, quite another for him or for an experimenter to measure loudness. Stevens may have provided us with a method for conveniently

6

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

recording the quantitative estimates of observers, but he has not provided a method for measuring loudness or any other psychological dimension. THE S O L U T I O N Existing solutions to the problem of psychophysical measurement are unsatisfactory, as the brief sketch above indicates, and as the sequel shows in detail. This book presents a radical solution to the problem. I do not propose a new method for measuring psychological dimensions, nor do I defend any of the methods previously offered. Instead, I argue that the traditional notion of measuring the psychological dimensions of perception is confused, and refers to no realizable goal. M y contention is that the traditional conception of psychophysics, its distinction between psychological and physical dimensions, and, consequently, its construal of the nature of psychophysical measurement are unacceptable. I thus dissolve, rather than solve, the problem of psychophysical measurement. The first stage of my argument is an extended demonstration that loudness, pitch, and similar dimensions are not correctly regarded as psychological dimensions, and that they can be measured by procedures essentially similar to those available for length and weight. The second stage of the argument is a demonstration that existing procedures for measuring psychological dimensions do not measure what the psychophysicists who employ them think they measure. The final stage of the argument is a direct and systematic attack on the backbone of the traditional conception of psychophysics: its distinction between psychological and physical dimensions. First stage (chapters 2-7).—The focal question in these chapters is whether loudness and pitch of sounds—paradigm examples of the so-called psychological dimensions—are measurable. The question is often summarily answered as follows: "Either loudness and pitch are dimensions of sound waves, in which case they are measurable, since sound waves are measurable; or loudness and pitch are dimensions of auditory sensations, in which case they are measurable only indirectly (if at all) via the measurement of the sound waves that produce them." Chapters 2 and 3 show that the question cannot be so easily disposed of. Chapter 2 argues that loudness and pitch are not dimensions of sensations. The argument is that sensations, by definition, are

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

7

perception-dependent, private, and not genuinely locatable, and that sounds possess none of these features. Chapter 3 shows that there are powerful objections to identifying sounds and sound waves. The inclination to identify sounds either with auditory sensations or with sound waves is a refusal to allow sounds to exist in their own right as nonpsychological dimensions on a par with length and weight; and it begs the question of whether sounds, as opposed to auditory sensations and sound waves, are measurable. The question of whether loudness and pitch are measurable cannot be properly addressed without an investigation of the nature of measurement. It is easy to "prove" that loudness and pitch are measurable, or that they are not measurable, simply by adopting a prejudicial definition of measurement. Suppose measurement is narrowly defined as the assignment of numerals to things by the physical addition of dimension units (Campbell's definition.) It then follows that pitch is not measurable, since there is no physical operation of addition for pitch. Suppose, on the other hand, measurement is broadly defined as the assignment of numerals to things by some rule or other (Stevens' definition). It then follows that pitch is measurable, since numerals can be assigned to sounds according to the rule: to the higher pitch assign the higher numeral. Chapters 4 and 5 criticize both unacceptably narrow and unacceptably broad definitions of measurement, and chapter 5 recommends that measurement be defined as the assignment of numerals to things by comparing the things (by addition or some other operation) with units. Using this acceptable definition, chapters 6 and 7 demonstrate that loudness and pitch are measurable. I propose that loudness be measured by a method employing the simultaneous playing of tones as a physical operation of addition, much as weight is measured by physically adding unit weights on a balance. I propose that pitch be measured by a different kind of method, using the halftone as a unit interval of pitch and counting the number of such unit intervals between pitch zero and the tone to be measured. The defense of these two methods consists almost entirely in defeating every objection that can be raised against them. My conclusion is that loudness, pitch, and, by extension, brightness, hue, and similar dimensions are on a par with the physical dimensions of length and weight, and are measurable in the same sense as those dimensions.

8

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

This conclusion throws enormous doubt on the traditional conception of psychophysics. If loudness and pitch are not psychological dimensions, laws relating loudness and pitch to the intensity and frequency of sound waves are not psychophysical laws. What, then, are the psychophysical laws of auditory perception? The traditional psychophysicist seems forced to say that they are laws relating dimensions of auditory sensations to physical dimensions (to dimensions of sounds, or dimensions of sound waves?). But it seems doubtful that normal instances of auditory perception involve the production of auditory sensations. If neither loudness and pitch, nor dimensions of auditory sensations, are the psychological dimensions involved in psychophysical laws of auditory perception, then what are these psychological dimensions? Second stage (chapters 8-9).—The conclusion of the first stage helps to reveal the problematic character of the procedures of psychophysical measurement which traditional psychophysicists have proposed and used. The proper methods for measuring loudness and pitch are those described in chapters 6 and 7. But if this is so, what do existing psychophysical procedures for measuring these dimensions really measure? If the traditional psychophysicist continues to insist that they measure psychological dimensions, he must explain what these psychological dimensions are, and, if he cannot do so, must admit that his methods of psychophysical measurement are, as he conceives them, spurious. That they are spurious is the burden of the second stage of my argument. Chapter 8 describes and criticizes the jnd method of measurement proposed by Fechner, the method that employs the just noticeable difference as a unit of sensation. This method has been subjected to a barrage of criticism during the hundred years since its invention, but none of it has been conclusive. I argue that the method is conceptually confused, that it confuses the quite legitimate notion of a just noticeable difference between stimuli with the illegitimate notion of a just noticeable difference between sensations. The illegitimacy of the notion begins to emerge when we reflect on the fact that the just noticeable difference is by definition the smallest difference perceivable in a range of differences. This definition entails, of course, that some differences are unperceivable. Hence, to apply the notion of a just noticeable difference to sensations is to imply that some sensation differences are

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

9

unperceivable (unintrospectible) by their possessor, which seems absurd. It would seem that the possessor cannot be mistaken about the existence and the properties of his own sensations. The illegitimacy of the notion of a sensation jnd becomes obvious when we note that the precise determination of the jnd requires a method for measuring the entities whose differences are, some of them, just noticeable. W h e n these entities are sensations, no such method seems available. In sum, the jnd method errs by forcing the just noticeable difference, which is a unit or index of sensitivity, into service as a unit of sensation. Chapter 9 describes recent methods of psychophysical measurement proposed principally by S. S. Stevens. These are held to be methods for measuring, not private, introspectible sensations, but "perceptual magnitudes"—perceptual length, perceptual weight, perceptual loudness, and so on. I show that a host of introspectionist assumptions underlie the use of such methods. For example, one method requires observers to make ratio estimates of physical stimuli, and then assigns numerals to the perceptual magnitudes correlated with stimuli so as to reflect estimated ratios. This method assumes that when an observer estimates that stimuli and $2 stand in the ratio 1:2, the corresponding perceptual entities Y i and Y s (what are these?) also stand in the ratio 1:2. The assumption implies that the observer estimates the $>'s on the basis of his perception of the Y ' s , which suggests that Y ' s are private, introspectible sensations. When such introspectionist assumptions are discarded, or purified by substituting behaviorist definitions for them, the notion of a perceptual magnitude and the numerical methods for quantifying such magnitudes lose much of their theoretical significance. Stevens' methods may be useful as a w a y of recording the numerical responses of observers to perceptual stimuli, but it is a mistake to regard them as methods for measuring psychological magnitudes. Third stage (chapters 10-11).—If the measurement of loudness, pitch, and similar dimensions does not qualify as psychophysical measurement, and if the methods of psychophysical measurement proposed and used by psychophysicists are spurious, it seems likely that the traditional conception of psychophysical measurement is confused. The final stage of my argument shows that this is in fact the correct conclusion. Psychophysical measurement is viewed, in the traditional con-

io

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

ception, as the measurement of psychological dimensions in order to discover their relations to correlated physical dimensions. The core of this conception is a distinction between psychological and physical dimensions. Chapter 10 is a systematic attempt to demonstrate that the distinction is completely confused and ultimately illegitimate. The distinction is capable of a large number of interpretations. It can be interpreted asymmetrically, so that it corresponds roughly to one version of Locke's distinction between secondary and primary qualities. O r it can be interpreted symmetrically, so that such "psychological" dimensions as loudness, hue, perceptual length, and perceptual weight each have their "physical" counterparts—intensity of sound waves, frequency of light waves, length, and weight. The symmetrical interpretation has several versions: psychological dimensions may be viewed as molar dimensions, or as dimensions of sensations, or as physiological dimensions, or as perceptual dimensions. Each interpretation and each version is shown to be unacceptable. The suggestion that the distinction is really between types of observation, rather than types of dimension, is also shown to be unacceptable. Thus the distinction collapses and, along with it, the traditional conception of psychophysics and its conception of psychophysical measurement. Chapter 1 1 shows that traditional psychophysics is the result of improper conceptual analysis of the objects to be measured. The psychophysicist wishes to measure perception and other mental entities and processes. But what precisely is he supposed to measure? The traditional answer is that he measures psychological dimensions of perception corresponding to physical stimulus dimensions. In the "old" psychophysics of Fechner, these were held to be dimensions of sensations ; in the " n e w " psychophysics of Stevens, they are held to be perceptual magnitudes. Both ways of viewing perceptual measurement are wrong. Instead of viewing perception as a mental process in a mental world, paralleling a physical world with its physical stimulus processes, the psychophysicist ought to view perception as a collection of perceptual abilities, and it is these abilities he should attempt to measure. Under this conception the problem of psychophysical measurement simply dissolves, since there is no special problem in measuring perceptual abilities. Such measurement is accomplished by measuring stimuli with standard "physical" methods, obtaining the responses of observers to these stimuli, and processing the results by

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

11

some statistical method. No stage in this procedure provides any philosophical problem. Although this book attempts to destroy the traditional conception of psychophysics and its way of viewing the measurement of perception, its aim is by no means entirely negative. Traditional psychophysics locks perception in a conceptual prison, making its measurement apparently impossible, and a true psychological science of perception apparently unattainable. By demolishing the prison, I hope to contribute to the growth and development of the psychological science of perception. DEEPER Q U E S T I O N S Even if we adopt the conception of perceptual psychophysics recommended in this book, some profound questions remain. It would seem that psychophysics as the science of perceptual abilities attempts to determine (a) the limits and (b) the reliability (accuracy) of perception. The remaining questions are: (1) Is psychophysics possible? (2) If so, how is psychophysics possible? That is to say: Is it possible to determine the limits and the reliability of perception? If it is, what conditions must be satisfied in order to make such determinations possible? These questions are not conclusively answered in the book, and indeed they are not usually raised in explicit form. But since they are basic threads in the fabric, the reader must be aware of them in advance if the book is to have maximum value. I will illustrate the two questions with the example of the lower limit—or limen—of hearing. Suppose I wish to discover the softest sound I can hear. I cannot do so if I am forced to rely on my unaided hearing in selecting sounds to present to myself in a psychophysical experiment. To discover that a sound, Si, is the softest I can hear, I must select sounds that are louder than Si and sounds that are softer than Si and then discover that I can hear the louder ones but not the softer ones. But if I have used my unaided hearing to select the softer sounds, then I have heard these sounds, from which it follows that Si is not the softest sound I can hear. The point can be made in a slightly different w a y : In order to discover through the use of my unaided hearing that Si is the softest sound I can hear, I must hear sounds that I cannot hear (namely, those softer than Si), which is of course contradictory.

12

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

There are two possible ways of avoiding the difficulty above. The first is to use the auditory perceptions of some perceiver whose hearing is more acute than mine to select presentation sounds. If I can hear Si and louder sounds, but cannot hear sounds softer than Si, and if the more acute perceiver can hear both the louder and the softer sounds, then I can conclude that Si is the softest tone I can hear. But why should I assume that the other perceiver's hearing is more acute than mine? Why not assume, on the contrary, that when he claims to hear sounds softer than Si he suffers from auditory hallucinations, and that the softer sounds he claims to hear are "in his head"? On this assumption I cannot conclude that Si is the softest sound I can hear, since I cannot assume that sounds softer than I can hear have been produced. Such doubts about the other perceiver's abilities could perhaps be allayed by bringing in additional perceivers. If many perceivers claim to hear sounds softer than Si, and I cannot hear these sounds, then I can perhaps conclude that Si is the softest sound I can hear. Even here a doubt, at least a philosophical doubt, can arise. How can I be certain that the other perceivers, all of whom claim to hear sounds softer than Si, are not suffering from mass hallucination? Or, if this doubt seems insane, how can I be certain that the other perceivers are not responding to some dimension of sounds other than loudness (for example, to the physical impact of the sounds, or sound waves, on some sensitive part of their skin)? The other possible way of avoiding the difficulty of determining my auditory limen is for me to use some method other than my unaided hearing in detecting the presentation sounds. I might, for example, employ some sound-amplifying device, like the old ear trumpet or the newer hearing aids used by persons with hearing loss. (In the same way, we use magnifying glasses to increase the limits of our visual perception of size.) With such a device I can detect sounds softer than those I can hear unaided, then present the sounds to my unaided hearing, and finally discover that I can hear Si and louder sounds but not softer sounds; that is, I discover that Si is the softest sound I can hear unaided. The difficulty in this method is that with the amplifying device I hear sounds that seem no softer than Si and infer—on the ground that the device amplifies—that the sounds being amplified are softer than Si. (In the same way, with a magnifying glass I see a speck that looks no smaller than the tiny speck I see with the unaided eye and in-

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

13

fer—on the ground that the glass magnifies—that the speck under the glass is smaller than the smallest speck I can see with the unaided eye.) And this inference can be challenged. Perhaps the device does not really amplify sounds; perhaps I hear sounds with it which are in reality no softer than Si. If so, I cannot conclude that Si is the softest sound I can hear. The most familiar method of detecting presentation sounds in psychophysical experiments does not rely on hearing, but on what are usually called "operations." These operations are of the instrumental or the noninstrumental variety (usually the former), and they are, more often than not, procedures of measurement. Since the intensity of sound waves is a dimension correlated with the loudness of sounds, it might seem that an intensity meter can be used to detect presentation sounds softer than those I can hear. If the intensity meter detects Si and sounds both louder and softer than Si, whereas I can hear Si and louder sounds but no sounds softer than Si, then perhaps I can conclude that Si is the softest sound I can hear. The difficulty with this method lies in its assumption that the intensity meter detects the same dimension that I hear, namely, loudness. Experiment has shown that loudness of sounds varies, not merely with intensity, but also with frequency, of sound waves. Loudness therefore cannot be identified with intensity. And even if the correlation between the two dimensions were perfect, there would still be a question as to whether the dimensions are identical. The correct view may be that sounds are caused by sound waves, and are not identical with them. If this view is correct, I cannot assume that intensity meters detect sounds, much less sounds softer than those I can hear. Nor do the difficulties end here. Even if I assume that intensity meters detect sounds and their loudness, I can question the instrument's detection ability in the same way that I question the perceptual ability of other perceivers. Why should I assume that the meter is more sensitive to soft sounds than are my own ears? It should not be supposed that the questions of whether and how perceptual psychophysics is possible arise only for the perception of sounds, colors, tastes, and so on—that is, for types of perception whose objects (stimuli) are commonly, although erroneously, assumed to be "psychological" entities. The questions arise as quickly for the perception of weight, length, hardness, and so on—that is, for types of perception whose objects are commonly assumed to be "physical" en-

14

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

tities. I will illustrate this point with the example of a determination of the accuracy of length perception. Suppose there are three short, very thin rods lying on the table before me and that I see Ri as longer than R2 and R2 as longer than Rs. T o determine whether my relative length perceptions are accurate, I must determine the actual lengths of the three rods and then compare actual length with perceived length. Quite obviously I cannot employ the length perceptions whose accuracy is being studied as my tests for actual rod lengths. Such a procedure would be circular and would produce a spurious experiment that could have only one outcome: perceptual accuracy. Two methods might acceptably be used to determine the actual rod lengths. The first of these would employ other perceivers whose length perceptions are more reliable than my own. But a difficulty immediately arises: W h y should I assume that another perceiver or group of perceivers perceives relative rod length more accurately than I do? The second method would employ some "operation" to determine actual rod lengths, an operation of either the instrumental or noninstrumental variety. I might, for example, look at the rods through a magnifying glass. Or I might move my eye within an inch or two of the rods, thus producing a "magnifying" effect without an instrument. But a difficulty arises: W h y should I trust the magnifying glass, and w h y should I assume that looking closely at the rods will more nearly reveal their actual size than looking at them from a greater distance? Another operation is to measure the three rods with a ruler. But if there is a doubt about the accuracy of my visual perception of the rods, there can also be a doubt about the reliability of my measurements, which require looking at the rods in order to align them with the ruler. If I saw the first rod to be longer than the second, and my measurements revealed the first to be shorter than the second, then I could just as easily distrust the measurements as distrust my visual perceptions. Thus far I have discussed difficulties in determining the limits and the reliability of the perception of a particular observer, namely, myself, for the difficulties arise most quickly and in their clearest form by asking how the observer would himself determine the limits and the reliability of his own perception. The difficulties are, in short, ingredients in the philosophical problem of solipsism, which arises most easily

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

15

by adopting the first-person point of view, by considering oneself in his relation to other perceivers and in relation to his perceptual environment. Now it may be supposed that the difficulties will vanish if we abandon the first-person point of view, and bear in mind the dictum that in a psychophysical experiment a perceiver (O) cannot be his own experimenter (E). This suggestion is ineffectual. To begin with, there is no good reason to think that an O cannot be his own E, as long as he does not confuse the two roles. More important, E is in the same position with respect to other perceivers and to his environment as O. E's determinations of stimuli to be presented to O in psychophysical experiments are acceptable only if these determinations are reliable. But to discover that they are reliable would require performing psychophysical experiments in which E became the O, and then the old difficulties would again arise. We would require some method, either perceptual or operational, by means of which to select the stimuli to be presented to E (now the O) in determining his perceptual limens and perceptual accuracy. And whether such a method is possible is doubtful for the reasons described earlier. The above arguments are incomplete and inconclusive. Nonetheless, they do provide an indication of the theoretical and logical difficulties that must be solved if the psychophysical enterprise of determining the limits and the reliability of perception is to be possible. That they show such an enterprise to be impossible is highly questionable, not merely because the arguments are inconclusive, but also because the difficulties they present can be totally undercut by reconstruing the goals and nature of perceptual psychophysics. It can be maintained that the notion of a perceptual limit and the notion of perceptual reliability have no place in psychophysics. On this conception, psychophysics does not attempt to determine the limits or the reliability of any perceiver's perception. Rather, it attempts to determine (a) the relation between aided and unaided perceptions of the same kind (aided and unaided hearing, for example); (b) the relations between perceptions of one kind and perceptions of a different kind (between auditory and visual perceptions, for example); (c) the relations between the perceptions of one observer and the perceptions of other observers; and (d) the relations between perceptions of various kinds and the observations made by means of instruments or other

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

operations. On this conception, there is no need to suppose that different perceptions or observations are more acute or reliable than others. There is, therefore, no need to employ the notion of a perceptual limit or the notion of perceptual reliability. That this attenuated conception is required for a science of psychophysics to be possible is suggested in some parts of this book (particularly toward the end of chapter 10), but it is not established. I raise the issue here, not to promise a solution, but rather to call attention to aspects of later discussions which might otherwise go unnoticed. The above discussion suggests that a rather large philosophical question lies beneath the surface of this book. Psychophysics, at its deepest level, is or pretends to be the science that determines the limits and the reliability of perception. It is, at this level, analogous to epistemology, which is the study of the limits and the reliability of knowledge. One major difference is that psychophysics attempts to be scientific, to rest its results on observation and experiment, whereas epistemology is, in the opinion of most philosophers, apriori and nonempirical. As with psychophysics, there is a serious question as to whether epistemology is possible. In order to fix the limits of our knowledge, we must possess some method of determining what facts there are which might or might not be known, and must then discover which of these we can know and which we cannot know. But this method apparently requires that we know facts we cannot know (those beyond the epistemological limit), which is, of course, contradictory. In order to determine whether our knowledge is reliable—that is, whether the facts are as we think we know them—we must again have some method of determining what the facts are, and must then discover whether we know them as they are. But if we have such a method, then we know the facts as they are, and cannot know them as they are not. It thus seems that epistemology is impossible, at least where its goal is taken to be the determination of the limits and the reliability of knowledge. Perhaps, then, epistemology is possible only if reconstrued as the study, not of limits and reliability, but of the relations between the knowledge possessed by one knower or group of knowers and the knowledge possessed by other knowers. The questions of whether and how perceptual psychophysics is possible are intimately connected with the questions of whether and how epistemology is possible. Perceptual psychophysics attempts to dis-

THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

17

cover the relation between the perceiver and the objects perceived; epistemology (which is virtually indistinguishable from general psychophysics) attempts to discover the relations between the knower and the objects known; and there is, in both instances, a serious question as to whether the relations can be discovered. I believe that the deep motivation of this book lies in the connection between psychophysics and epistemology, and in the question of their possibility. Such motivation is not altogether surprising in view of the fact that Gustav Theodor Fechner was not only a physicist, mathematician, and psychologist, but also a philosopher in the post-Kantian scientific tradition. Fechner's psychophysics was always intended by him to be a part of his metaphysics. But his metaphysical aims were from the beginning ridiculed or ignored, and he is remembered today as the brilliant founder of psychophysics and the mad architect of a strange philosophy. I know almost nothing of his philosophy, and do not wish to defend what I do know of it. I do, however, share Fechner's conviction that psychophysics has philosophical as well as scientific importance.

2 Sounds and Sensations Sounds are those entities described in ordinary speech as loud, faint, high-pitched, deep-throated, piercing, grating, staccato, dulcet, and so on. They are commonly said to have certain "dimensions": loudness, pitch, and, according to some technicians, volume and density. Dimensions of sounds are widely held to be psychological dimensions, and thus to be candidates for psychological, or psychophysical, measurement. But are sounds capable of being measured? If so, is their measurement essentially different from the measurement of physical entities? On one interpretation, psychological dimensions are regarded as dimensions of sensations. Now if loudness and pitch are dimensions of sensations it would appear that they cannot be measured, at least not in the same way as length and weight. Pains and afterimages are paradigm examples of sensations. Obviously, we cannot lay a ruler on a pain, or place an afterimage on a balance. So if pains and afterimages can be measured at all, it must be by methods quite unlike those employed for such physical dimensions as length and weight. Perhaps the difficulty goes even deeper; perhaps pains are simply incapable of being measured. Is it possible to find a zero for pain, to select a unit, and then to employ these in a method that is capable of intersubjective, or even intrasubjective, confirmation? At first sight it does not seem so. If sounds are sensations, perhaps they too are incapable of being measured. We are not yet ready to face the question as to whether sensations can be measured, but it is not too early to argue that its answer is irrelevant to the question as to whether loudness and pitch can be measured. In this chapter I show that sounds are not sensations and that the alleged impossibility of measuring sensations is no argument for the impossibility of measuring sounds. In the process of making these points it also becomes clear that there is no reason to regard loudness and pitch as "psychological" dimensions, on any interpretation of that

SOUNDS AND SENSATIONS

19

term. Sounds are not, like rods and coffee cups, physical objects; but they are, as is shown later, physical entities. Hence, any law relating loudness or pitch to string length, sound-wave intensity, or any other physical dimension must be regarded as a physical, not as a psychophysical, law. It is, in a way, obvious that sounds are not sensations. Sounds differ in important ways from rods, coffee cups, books, and other socalled physical objects. But this difference is not sufficient to classify sounds as sensations. The class of perceivable entities is much broader than the brief list above suggests. That list contains only "moderatesized specimens of dry goods," 1 to use a phrase of J. L. Austin's. We must also include among the entities we perceive flashing lights, rainbows, fires, clouds, hazes, lakes, reflections in water, facial expressions, gestures, movements, waves, explosions, sandstorms, and winds. Using this expanded list as a basis for comparison, it seems obvious that sounds are, like the items on the list, physical entities and not sensations. Because it goes against an old and unfortunately respectable tradition, a full-scale argument for the view that sounds are not sensations must attempt to define the concept of a sensation, to devise criteria for sensations, and to show that sounds do not satisfy that definition or those criteria. The argument is that sensations, in contrast with physical entities, are (1) perception-dependent, (2) private, and (3) nonlocatable; and that sounds are none of these. SENSATIONS AS PERCEPTION-DEPENDENT One important difference between sensations and physical entities might be expressed as follows: sensations cannot exist unperceived, physical entities can exist unperceived. In simpler terms, sensations are, and physical entities are not, perception-dependent. Of necessity, if O has a pain he feels the pain; he cannot have a pain he does not feel. But he may have a bug on his neck which neither he nor any other perceiver feels or perceives in any other way. Again, O cannot have an afterimage he does not see. But he may have a smudge on his sleeve which neither he nor any other perceiver sees or perceives in is w o Ä O 3B m B0)ïC 2 ^ C b < >0 S 13 u .s ^ ° dS^ a; u> ¿3uoSi> .t! J3¡um ä« È Ö h 0JT3Ï ^H ?3 O H '-0 . 3 fi iJ-TgS ÍUl TM "O Ul U1 , = , and + . W e must be able to specify, within the measurable dimension, a relation, G, which, like the relation, > , is 4 From various parts of Campbell, "Symposium: Measurement and Its Importance for Philosophy," pp. 122-127.

THE NARROW VIEW OF MEASUREMENT

IO5

transitive and asymmetrical; and a relation, E, which, like the relation, = , is transitive and symmetrical; and an operation, @ , which, like the operation, + , is summative, commutative, associative, and summative for equals. Bergmann expresses this view concisely. "The essence of measurement is that some arithmetical relations among the numbers assigned correspond, by virtue of a shared logical structure, to descriptive relations among the things to which they are assigned. The measurement we prefer to others is so constructed that a maximum number of arithmetical relations has such descriptive correlates or, as one also says, empirical meaning."5 Axioms (1) through (8) are the logical isomorphs of (i') through (8'): (1') through (8') are the axioms of mathematical equality and inequality, and mathematical addition; (l) through (8) are the axioms of physical equality and inequality, and physical addition. In the classical view, only a dimension that is described by such physical isomorphs is measurable. Statements (i') through (8') are arithmetical statements. Consequently, they are (in my view of arithmetic) necessary, not contingent, statements. What is the logical status of (l) through (8)? The orthodox view is that these axioms, and any others that might be proposed as axioms of measurement, are contingent statements. Referring to most of these axioms, interpreted in terms of weight and the operation of balancing, Campbell says: Now these statements concern experimental facts; they assert that, in certain circumstances, we shall observe something. The statements may be true or false; and, as with all statements of experimental fact, experiment only can determine whether they are true or false. If they are true, there will be a certain similarity between the arithmetical process of addition and the arithmetical relation of equality on the one hand and the physical process of addition and the physical relation of equality on the other; if they are false, there will not be this similarity. 8

Pap argues that this view is not generally true. The relation "longer than" is, he contends, asymmetrical and transitive by definition, from 8 Gustav Bergmann, "The Logic of Measurement," State University of Iowa Studies in Engineering (1956), p. 28. * Norman Robert Campbell, Physics: The Elements (Cambridge, Eng., 1920), pp. 279-280. See Bergmann, "The Logic of Measurement," for a more formal statement of this view.

i o6

THE NARROW VIEW OF MEASUREMENT

which it follows that axioms (i) and (2) are necessarily true for length. But the matter is, he says, different for the relation "heavier than." Whether we take "heavier than" to mean "feels heavier than" or to mean "overbalances," "it remains a question of empirical fact whether the relation is transitive."7 The problem here is a deep one, and we must content ourselves with some tentative suggestions concerning it. If G is taken to mean "heavier than" and E to mean "equal in weight to," then (1) through (4) are necessarily true. In general, if we construe these as axioms about quantitative relations—such as "heavier than," "longer than," "warmer than," "harder than," and so on—then they are necessary statements. But if C is taken to mean "feels heavier than" and E to mean "feels equal in weight to," or if C is taken to mean "overbalances" and E to mean "balances," then (a) through (4) are contingently true. How objects feel when lifted, or what they do to a balance, are tests for their relative weight. In general, if (1) through (4) are construed as statements about tests for quantitative relations, they are contingent statements. Now many writers, under the influence of the slogan that "measurement is operational," construe the axioms of measurement as statements about tests for quantitative relations. But, as we shall discover in a later section, this slogan constitutes an exceedingly poor argument. It appears then that (1) through (4) can be construed either as necessary or as contingent statements. What of (5) through (8)—the axioms of addition? Are they necessary or contingent statements? Consider axiom (5), "If (X E Z) then (X @ Y) G Z , " as it applies to length. There seem to be two ways of interpreting this statement. We may take it to mean (a) "If X is equal in length to Z, then X and Y together are longer than Z." Construed in this way, (5) seems to be necessarily true. For if X and Y are together equal to Z or less than Z, then X must be shorter than Z. The word "together" in (a) does not refer, at least not specifically, to any physical operation of addition, such as endwise juxtaposition. Nor do "equals" and "longer than" refer, at least not specifically, to any empirical tests for equality and inequality. But we can interpret (5) so that such operations and tests are referred to. We then obtain, (b) "If X and Z align at both ends, then Z will not align at both ends with X and Y in 7

Pap, An Introduction

to the Philosophy

of Science,

p. 128.

THE NARROW VIEW OF MEASUREMENT

IO7

endwise juxtaposition." This statement is contingently true. A world is clearly possible in which the following sequence of events is standard. X and Z are placed side by side and found to align at both ends with each other. Then X and Y are juxtaposed end to end and placed side by side with Z, and found to align at both ends (of the juxtaposition) with Z. In this world endwise juxtaposition causes objects to shrink. In such a world, although (a) will still be true, (b) will be false. In summary, (1) through (8) may be given a particular or an abstract interpretation. When interpreted in the particular manner they refer, in their application to length, to some physical operation of addition, such as endwise juxtaposition, and to tests for determining length equality and inequality. Thus interpreted they are contingent statements. When interpreted in the abstract manner, (1) through (8) refer to length addition, though not to any operation of addition; and to the relations of length equality and inequality, though not to any tests for these relations. Thus interpreted they are necessary statements. If (1) through (8) are to serve as axioms of measurement, as conditions for measurable dimensions, they must be interpreted in the particular, contingent manner. In a world where objects shrink as a result of endwise juxtaposition, axiom (5) will be true in the abstract sense. But measurement by means of endwise juxtaposition will be impossible in this world. In a world where an object aligns at both ends with a second, the second aligns at both ends with a third, and the third fails to align at both ends with the first, axiom (3) will still be true in the abstract sense. But a procedure for assigning numerals to rods which employs the alignment test for equality will be impossible. It follows that the question of whether a given dimension is measurable is a contingent question. Whether a dimension is measurable depends on whether it satisfies axioms (a) through (8) in their particular sense. And whether the dimension satisfies these axioms in that sense is a contingent question. Let us apply these remarks to the abbreviated definition of measurement given at the beginning of this chapter, which states that measurement is the assignment of numerals to objects within a dimension by means of a physical operation of addition of dimension units. Whether a dimension is measurable by this definition depends on whether a physical operation of addition for the dimension can be performed. And whether a physical operation of addition for the dimension can be performed is a contingent question.

lo8

THE NARROW VIEW OF MEASUREMENT

Whether length, pitch, brightness, and sourness are measurable cannot be settled by apriori methods. To learn whether a dimension is measurable we must ask whether there are tests for determining equality and inequality of objects within the dimension, and whether there is a physical operation of addition for objects within the dimension. These questions can be answered only by empirical examination. Such examination reveals that many dimensions commonly believed immeasurable—brightness and loudness, for instance—are in fact measurable on the classical definition. THE AXIOMS OF ADDITION Axioms (5) through (8) are the axioms of physical addition which, in the classical view of measurement, must be satisfied by any measurable dimension. In this section I challenge the classical view by attempting to prove that neither these nor any other axioms of addition must be satisfied in a system of measurement. AXIOMS ( 6 ) , ( 7 ) , AND ( 8 )

That axioms (6) and (7) are not required can be shown by describing two simple systems for measuring the weight of cubes of a single size. Both systems make use of a simple balance consisting of a flat, even lever placed at its midpoint on a fulcrum. When no objects are placed on it, the lever remains in a horizontal position. In both systems, two objects are said to be equal in weight when, placed on opposite extreme ends of the lever, they balance each other (i.e., the lever remains in a horizontal position). At this point the descriptions of the systems diverge. In system one, the operation of addition is defined as placing two or more objects on one extreme end of the lever, one on top of the other. Let us call this vertical addition. An indefinite number of equal unit objects are discovered and labeled Wi, W2, Ws, and so on. The labeling is for purposes of reference only: no order of addition of unit objects is prescribed. Measurement of a given object is accomplished by balancing that object against the required group of unit objects added vertically. A count of the objects in this group provides the numeral to be assigned to the object being measured. In system two, the operation of addition is differently defined. X is said to be added to Y when Y is placed on one extreme end of the lever (let us call this first position) and X is placed adjacent to Y be-

THE NARROW VIEW OF MEASUREMENT

IO9

tween Y and the lever's midpoint (let us call this second position). X is said to be added to Y and Z—X @ (Y @ Z)—when Z is placed in first position, Y in second position, and X in third position. Let us refer to these operations as horizontal addition. Unit objects are chosen in the following manner. Wi (a unit object in system one) is placed on one extreme end of the lever and left there. Then an object is discovered which, when placed on the opposite end in first position, balances W1. This object is labeled Ui. Next we discover an object with the following property: when Ui is removed from the lever and the object is placed in second position, it balances Wi. This object is labeled U2. Then an object with the following property is discovered: when U2 is removed from the lever and the object is placed in third position, it balances Wi. We label this object Us. In the manner thus indicated, an indefinite number of unit objects are selected and labeled. The measurement of a given object is accomplished by placing it on one extreme end of the lever and horizontally adding unit objects on the other end until a balance is obtained. A count of unit objects in the requisite collection supplies the numeral to be assigned to the object being measured. Unlike the first system, system two prescribes the order of addition of unit objects. The rule is: Ui is placed in first position, U2 in second position, Us in third position, and so on. It can be deduced from the theory of levers that axioms (6) and (7) are satisfied by the operation of vertical addition but are not satisfied by that of horizontal addition. This theory tells us that the moment of an object resting on a lever (i.e., the tendency of that object to cause rotation about the fulcrum) is the product of the object's weight and its distance from the fulcrum. Hence, the total moment of X @ Y is equal to the total moment of Y @ X, where @ designates the operation of vertical addition, for X and Y are at the same distance from the fulcrum, regardless of which is placed on top of the other. Now suppose X and Y are objects of unequal weight. Let m be the moment of X in second position, and n the moment of Y in second position. Let Am be the increase in the moment which results from moving X from second to first position, and let An be the increase in the moment which results from moving Y from second to first position. Then the total moment of X and Y, when X is in second position and Y is in first position, is m + n + An, and the total moment of X and Y , when X is in first position and Y is in second position, is n -f- m + Am.

no

THE NARROW VIEW OF MEASUREMENT

Since X and Y are of unequal weight. Am and An are unequal. Therefore, the total moment of X and Y in one position on the arm does not equal their total moment when their positions are reversed. Hence, X @ Y does not always equal (balance) Y @ X, where @ designates the operation of horizontal addition. That is to say, axiom (6) is not satisfied by the operation of horizontal addition. This fact does not, however, disqualify the horizontal system as measurement. The units of this system were chosen so that its measurements of single objects would agree with those of the vertical system. The vertical addition of a given number of W's will balance the same number of U's added horizontally, since the moment of W i in first position equals the moment of Ui in first position, and the moment of Ui in first position equals the moment of U2 in second position, and the moment of U2 in second position equals the moment of Us in third position, and so on. There will not be agreement in measurements of complex objects, since the vertical system will, and the horizontal will not, assign to (X @ Y) and (Y @ X) the same numeral. But this fact seems insufficient to disqualify the horizontal system as measurement. By redefining physical addition we can create a horizontal system of measurement which satisfies neither axiom (6) nor axiom (7). Suppose addition is defined so that an object's position on the lever depends inversely on how deeply nested the object's name is in the addition formula. Thus, "X @ Y " means "X in first position and Y in second," " Y @ X " means " Y in first position and X in second," " X @ (Y @ Z ) " means "X in first position, Y in second, and Z in third," "(X @ Y) @ Z " means " Z in first position, X in second, and Y in third," and so on. It can be deduced from the theory of levers that, under this definition of addition, axioms (6) and (7) are false. With its new operation of addition, the horizontal method still qualifies as measurement. To falsify axiom (8) the test for equality must also be changed. If "X E X' " means "X in first position on one end balances X' in second position on the other end," and (X @ Y) E (X' @ Y')" means "X and Y in first and second position respectively on one end balance X' and Y ' in second and third position respectively on the other end," then not only axiom (8) but also axioms (3) and (4) will be false. The resulting horizontal system qualifies, nonetheless, as measurement.

THE NARROW VIEW OF MEASUREMENT

111

The general point is that operations of addition and tests for equality and inequality can be chosen in such a way that systems of measurement employing them fail to satisfy some of, if not all, the classical axioms of measurement. THE AXIOM OF ADDITIVITY

Proponents of the narrow view of measurement frequently maintain that a dimension has not been measured unless numerals have been assigned to objects within that dimension in such a way that the following statement is true: (9) N(X) + N(Y) = N(X @ Y) (additivity) where N means "the number assigned to" and @ designates some physical operation of addition.8 Some writers express this requirement by saying that the numerals assigned must form an "additive scale" for the process to be one of measurement. Axiom (9) is therefore referred to as the axiom of additivity. According to one view, an operation of addition will satisfy axiom (9) only if it satisfies axioms (5), (6), (7), and (8); that is, axiom (9) entails the other four axioms. So it may be argued, from the essentiality of (9), that the other four are essential to a process of measurement. There are, obviously, two possible lines of reply. First, we may ask whether axiom (9) does entail the other four. It is clear that (5) is entailed. Suppose the operation of juxtaposition never results in a juxtaposition longer than either of the objects juxtaposed. Then no numeral can be assigned, through the use of this operation, to objects X and Y ; and no numeral can be assigned to the juxtaposition of X and Y. Precisely speaking, unless (5) is true (9) will be meaningless, that is, neither true nor false. We will not attempt to determine whether (9) entails (6), (7), and (8); however, we may note, first, that there is no obvious reason for thinking that these are entailed, and, second, that if either (6) or (7) is entailed, then we have established that (9) is not essential to a system of measurement, since we have shown that (6) and (7) are not essential. 8 Bergmann and Spence, " T h e Logic of Psychophysical Measurement/' p. 1 0 7 ; Pap, An Introduction to the Philosophy of Science, p. 130; Suppes, " A Set of Independent A x i o m s for Extensive Quantities/' p. 166 (theorem 9). Pap seems to regard (9) as an axiom, and Suppes regards it as a theorem; Bergmann and Spence do not address the question.

112

THE NARROW VIEW OF MEASUREMENT

Second, we may directly challenge the assertion that (9) is essential to a system of measurement. Reflection on the two systems of weight measurement described earlier will show that the assertion is false. Suppose, using the first system (vertical addition and W's as units), X is found to have a weight of 3 W's and Y a weight of 4 W's. Then the vertical addition of X and Y will balance 7 W's, thus satisfying axiom (9). We saw that a given number of W's added vertically will balance the same number of U's added horizontally. Therefore, using the second system (horizontal addition and U's as units), X will be found to have a weight of 3 U's and Y a weight of 4 U's. Furthermore, and for the same reason, the vertical addition of X and Y will balance the horizontal addition of 7 U's (in the prescribed order, of course). But, since the total moment of X and Y added vertically is greater than the total moment of X and Y added horizontally, X and Y added horizontally will not balance 7 U's added horizontally. It follows that axiom (9) is not satisfied in the second system of weight measurement. Later an additional reason will be offered for maintaining that (9) is inessential. A possible explanation for the inclination to regard (9) as essential will buttress our criticism. Most of the familiar systems of measurement enable us to determine the total magnitude—total weight, for instance—of a group of objects by calculation, which is commonly supposed to be one of the purposes of measurement. Thus Guild says: . . . to m a k e a connection [between phenomenal structure and number] w e m u s t artificially associate a phenomenal criterion with numerical equality and a phenomenal operation with numerical addition. W h e n w e h a v e done this, but not before, w e can predict b y

arithmetical

calculation those phenomenal relations w h i c h involve only the stipulated practical criteria of equality and addition.*

Stevens, although he is an important opponent of the classical view, describes one function of measurement in a similar way: . . . it is desirable to assign numbers in each scale w h i c h not only denote the order within the scale (for which the letters of the alphabet w o u l d serve well enough), but also designate the relative magnitudes of the p h e n o m e n a to w h i c h the scale is applied. W h e n this is done, the 9 A . Ferguson et al, "Quantitative Estimates of Sensory Events," Report of the 106th Annual Meeting, British Association for the Advancement of Science (1938), pp. 297-298.

THE NARROW VIEW OF MEASUREMENT

113

scale numbers can be manipulated in accordance with arithmetical laws in order to determine additional relationships such as the sum of two magnitudes, the relative separation of two pairs of magnitudes, etc.10 Such appraisals of the purpose of measurement may lead to the argument that (9) must be satisfied in a system of measurement because only then can the total magnitude of a group of objects be determined by calculation. This argument is mistaken. Suppose a candymaker sells his product by weight, which is determined by a procedure employing a simple balance and an operation of vertical addition. Suppose also that the price of all candies sold is the same. When a customer wishes to purchase amounts of several different kinds of candy, the total cost of the candy can be determined in either of two ways. On the one hand, all the candy selected by the customer may be balanced together against the requisite number of unit objects. If the candy equals 30 W's, and if the price is 500 per 10 W's, the customer's bill is $1.50. On the other hand, each kind of candy may be separately weighed and the resulting three figures added to give the total cost. Thus, if the customer has selected 9 W's of chocolate, 1 3 W's of taffy, and 8 W's of divinity, the total weight is 30 W's and the customer's bill is $1.50. The second method requires a calculation, and may be described in this way. By means of a balance, numbers are assigned to several different objects (or groups of objects). An arithmetical operation—the addition of these numbers —is then performed to predict the result of weighing the objects collectively, that is, to predict the collective weight of the objects. The two methods will not have identical results (i.e., the second will not correctly predict the result of the first) unless the physical operation of addition involved satisfies axiom (9). Now suppose that the two methods do not have identical results. Does this imply that the first is not a system of measurement? Surely not. If we employ in the two methods the operation of horizontal rather than vertical addition, axiom (9) is not satisfied, and the second method will not correctly predict the results of the first. But, as we have already seen, a procedure employing horizontal addition can as legitimately be regarded as one of measurement as can a procedure employing vertical ad10 S. S. Stevens, "A Scale for the Measurement of a Psychological Magnitude: Loudness," Psychological Review, 43 (1936), 405.

114

THE NARROW VIEW OF MEASUREMENT

dition. When the second method (involving calculation) agrees with the first (not involving calculation), it may be preferred to the first because of its greater convenience. But if the second method does not agree with the first, it does not follow that the first is not one of measurement. There is yet another reason for the prevalent view that (9) must be satisfied by any system of measurement. Stated in terms of weight and the operation of placing objects in the same pan of a balance, the argument goes as follows. It is not enough that a combination of unit weights is merely heavier than any of the component weights. The combination of two equal unit weights must be twice as heavy as either component, the combination of three unit weights three times as heavy as either component, and so on. A procedure that does not enable us to say that an object is twice, or three times, or one-fourth as heavy as another does not qualify as weight measurement. This argument is an application of axiom (9) to the addition of equal unit objects. It involves a confusion which it is important to reveal. In terms of the operation of vertical addition described earlier, that X is twice as heavy as Y can be determined in either of two ways. First, if Y is equal to (balances) Z, and if Y and Z together equal (balance) X, then X is twice as heavy as Y. Second, if the number of unit weights in a collection that equals (balances) X is twice the number in a collection that equals (balances) Y , then X is twice as heavy as Y . Consider now the special case where X is a collection of two unit weights, Y and Z. X must be twice as heavy as Y , since the two objects cannot fail to satisfy either of the tests above. X is equal to the collection consisting of Y and Z, since, by hypothesis, X is this collection. And the number of unit weights in a collection equaling X is twice the number in a collection equaling Y , since, by hypothesis, X is a collection of two unit weights one of which is Y. Hence, the addition of two equal unit objects of weight cannot fail to be twice as heavy as either of the components, so long as the operation of addition satisfies axiom (5). Generally, if the addition of equal unit objects satisfies axiom (5), it must also satisfy axiom (9). Axiom (9) is not an additional requirement to be met. Failure to understand this point probably rests upon the failure to distinguish between what we may call numerical estimates, on the one hand, and measurements, on the other. When X and Y are hefted

THE NARROW VIEW OF MEASUREMENT

«5

individually, X may feel twice as heavy as Y even though X is not twice as heavy as Y. For a person to say, simply on the basis of having hefted two objects, that one is twice as heavy as the other is for him to make a numerical estimate of their relative weight. But it is only an estimate, and needs to be verified either by adding Y and some equal object, Z , and balancing the collection against X; or by measuring X and Y. If there were no such procedures for verifying statements about relative weight which are based on the way objects feel, it would not be proper to speak of these as "estimates" of relative weight. To speak of an estimate is to imply that there is some way of checking or verifying the estimate. The estimate may be accurate or inaccurate, and there must be some method, other than another estimate, for deciding which. We assess estimates on the basis of how closely they predict the results of their verifying procedures—measurements, for instance. But we cannot criticize the measurement for failing to conform to the estimate. There is no requirement that the results of measuring the weight of two objects must conform to numerical estimates of their weight. This is true even if a large number of such estimates made by different persons are in agreement. The claim that the addition of equal unit objects must satisfy axiom (9), in addition to axiom (5), may rest on a failure to understand the distinction between numerical estimates of magnitude and measurements of magnitude. That is to say, those who insist on axiom (9) in this context may do so because they feel, in some vague way, that an operation of addition can be rejected or accepted on the basis of how closely it verifies numerical estimates of a given dimension. But this is, as has been pointed out, to make an estimate logically prior to its verification. A X I O M ( 5 ) A N D ABSOLUTE ZERO

It may seem obvious that, to be measurable, a dimension must satisfy at least axiom (5). If not even this axiom of addition is true, it would appear that there is no operation of addition for the dimension. And without an operation of addition, how can the dimension be measured? A rod can be measured by juxtaposing equal unit rods until a juxtaposition equal in length to the rod has been obtained. But if juxtaposition of rods never results in a combination longer than any of the rods in the juxtaposition, it is impossible to measure, by the operation of juxtaposition, any rod longer than the unit rods employed.

ii6

THE NARROW VIEW OF MEASUREMENT

T o put the point in a somewhat whimsical way, any rod longer than a unit rod will be found to be infinitely long. Axiom (5), then, apparently must be true of any operation of addition employed in measurement. The claim that a measurable dimension must satisfy axiom (5) is intimately related to the claim that a measurable dimension must possess an "absolute zero." To show that the former claim is false, I show that the latter is false. I demonstrate that if there is an operation of addition for a given dimension, and if that operation satisfies axiom (5), then the dimension possesses an absolute zero. I then argue that some measurable dimensions do not possess an absolute zero, from which it follows that some measurable dimensions do not satisfy axiom (5). The proponents of the narrow view of measurement do not explicitly claim that a measurable dimension must possess an absolute zero, which is surprising in view of the relation between axiom (5) and the notion of absolute zero. But Stevens, who does not hold the narrow view of measurement, says: "Once a ratio scale is erected, its numerical values can be transformed (as from inches to feet) only by multiplying each value by a constant. A n absolute zero is always implied, even though the zero value on some scales (e.g., absolute temperature) may never be produced." 11 Many proponents of the narrow view would insist that only a ratio scale, in Stevens' classification, is a scale of measurement. Such theorists are therefore committed to saying, if Stevens is correct in his analysis of a ratio scale, that measurable dimensions must possess an absolute zero. Campbell (the leading exponent of the narrow view) seems to say in one passage that some measurable dimensions do not possess an absolute zero, 12 but his intent in the passage is not completely clear. Unfortunately, neither the concept nor the requirement of an absolute zero is clearly and explicitly treated in the literature on measurement. It may seem that, if measurable dimensions must possess an absolute zero, then pitch, hue, temperature, and certain other dimensions are not measurable. The example of temperature is especially useful. Both the centigrade and the Fahrenheit scales of temperature have what is usually called an "arbitrary" or "conventional" zero. A certain temperature is "arbitrarily" chosen as the zero temperature. O n the 1 1 S. S. Stevens, "Mathematics, Measurement, and Psychophysics," in Handbook of Experimental Psychology, ed. S. S. Stevens (New York, 1951), p. 28. 1 2 Campbell, Physics: The Elements, pp. 320-521.

THE NARROW VIEW OF MEASUREMENT

117

centigrade scale it is the temperature of freezing water; on the Fahrenheit scale it is 32 degrees below that of freezing water. Hence, when we say that one object is 5 degrees centigrade and another 1 0 degrees centigrade, we do not imply that the latter is twice as warm as the former. When we say that one object is 1 0 degrees Fahrenheit and another 30 degrees Fahrenheit, we do not imply that the former is one-third as warm as the latter. Either of these scales can be converted to an "absolute" scale (the so-called Kelvin scale) which does contain an "absolute zero." It is universally conceded that the centigrade, Fahrenheit, and Kelvin scales enable us to measure temperature dependency, not independently. Regardless of which scale is used, temperature is measured by means of some other dimension (volume of mercury, for instance). Temperature does not possess an absolute zero; rather, an absolute zero is assigned to this dimension in a procedure of dependent measurement. One view is that a dimension can be independently measured only if it possesses an absolute zero, and that, consequently, temperature cannot be independently measured. Length can be independently measured and, in this view, possesses an absolute zero. The first question regarding this view is: What is meant by "absolute zero"? The concept of zero is in some respects a baffling one. It seems intimately related to the concept of nothing. This relation might be stated in the following way. If there is nothing on the table, then there are no objects on the table; and if there are no objects on the table, then there are zero objects on the table. Even if the preceding statement is in itself unobjectionable, it by no means makes the concept of zero clear. A person—for instance, a child totally unfamiliar with mathematics—may know how to use the phrases "nothing on the table" and "no objects on the table" without being able to use the term "zero." Because o is a numeral, it is governed by certain rules for the manipulation of numerals. Some of these rules are: (a) a + o = a (from which it follows that a — a = o and a — o = a); (b) if a > b, then a — b > o; (c) if a < b, then a — b < o. Until one understands such rules he does not understand the concept of zero. The statement of these and other numerical rules governing zero does not however explain the concept of zero in its application to such dimensions as length and weight. It does not tell us what it means to say that a dimension possesses an absolute zero; nor does it tell us what the phrase "object of zero length" means, or whether it means anything at all. M y pro-

n8

THE NARROW VIEW OF MEASUREMENT

cedure in addressing these questions is an oblique one. I begin by describing a system of measurement for a dimension that seems not to have an absolute zero. The normal inclination is probably to regard weight as a paradigm case of a dimension possessing an absolute zero. Whether this view is correct, however, seems to depend on how the operation of placing an object on a balance is described. Let us describe it in a somewhat unusual way. To place an object in the pan of a balance, place it in and attach it to the pan. If Y is a balloon filled with just the right mixture of helium and air, and if X balances (equals) X', then X and Y placed in the same pan (added) will balance X'. Formally stated, (X @ Y ) E X', which contradicts axiom of addition (5). If Y is a balloon filled entirely with helium, and if X balances X', then X' will overbalance X and Y. Formally stated, X' G (X @ Y), which also contradicts axiom of addition (5). The first balloon—the one filled with a mixture of helium and air—is an object of zero weight. The second balloon—the one filled entirely with helium—is an object of negative weight. Note, however, that "object of zero length" and "object of negative length" are to all appearances meaningless. It may be objected that this contrast between weight and length is specious, the result of considering weighing under the nonideal conditions of air rather than in the ideal conditions of a perfect vacuum. In a perfect vacuum, it may be said, neither a balloon nor any other object will remain suspended or rise, but will exert some force downward on the pan of a well-constructed balance. T w o points may be made in reply. First, the statement that no object will remain suspended or rise in a vacuum is contingent. It is logically possible for an object to remain suspended or to rise (i.e., move outward from the surface of the earth) in a vacuum. Hence, there is no contradiction in the notion of an object of no weight (a weightless object) in a vacuum. The second point is taken from one of Campbell's footnotes: "Usually when we are speaking of weighing, and speaking accurately, we mean the weight in vacuo; but the weight of a body in air of a given density is quite as definite a magnitude as its weight in vacuo. A n d of course what we measure is always the weight in air." 13 With the introduction of objects of zero weight and objects of 13

Campbell, Physics:

The Elements,

p. 31911.

THE NARROW VIEW OF MEASUREMENT

119

negative weight, a question arises: Can a system of measurement be devised for these as well as for objects of positive weight? Certain preparatory definitions must be stated. We will use a balance constructed so as to make it possible to attach objects to the pans: a balance, say, with large, deep pans with lids that can be fastened. Placing an object in a pan consists in placing it in the pan, closing the lid, and fastening the lid. "Equal to," "greater than," and "added to" are given the usual definitions. E means "balances," G means "overbalances," and @ means "placed in the same pan with." "Object of positive weight" is defined as follows: If (X @ Y) G X', then Y is an object of positive weight (a positive object). "Object of zero weight" is defined as follows: If (X @ Y) E X', then Y is an object of zero weight (a zero object). "Object of negative weight" is defined as follows: If X' G (X @ Y), then Y is an object of negative weight (a negative object). It can be shown that these definitions of positive, zero, and negative objects are legitimate. Let us adopt the convention of designating positive objects by capital letters with a bar underneath, zero objects by capitals enclosed in parentheses, and negative objects by capitals with a bar on top. Given these conventions and the above definitions, the statements on the left side of the list below become true by definition. (a + b) > a (X @ Y) G X' a < [a + (_fe)] X' G (X @ ?) a+ o= a K @ (Y)]EX' That the statements on the left are the analogues of the mathematical rules on the right is evidence that my definitions are legitimate. A simple system for the measurement of positive and negative weight begins with the selection of an object of zero weight, which is labeled (A). Select next a positive object, labeling it U- Then select an indefinite number of objects which balance U, labeling each of these U. Now select an object that, when placed in the same pan with one of the U's, balances (A), and label it 0 (a negative object). Then select an indefinite number of objects which equal 0, labeling each of them U. To measure any given positive object, 2£/ place in one pan and in the other pan the number of positive unit objects (U's) required to balance X. This number is the weight of X in U's. To measure any given negative object place X in one pan and in the other pan the number of negative unit objects (U's) required to

120

THE NARROW VIEW OF MEASUREMENT

balance X. This number is the weight of X in U's. Notice that the choice of the unit of negative weight ensures that the addition of a positive and a negative object, when these are the same number of units in weight, will equal an object of zero weight. This result corresponds to the arithmetical rule: a + (—a) = o, or, a—a = o. We must anticipate the protest that the above is not a procedure for the measurement of weight. It may be said that I have described a procedure for measuring two intersecting dimensions, buoyancy in air and weight in air, that what I have called "negative weight" is really buoyancy in air. The point of this objection is that I have not described a single dimension having no absolute zero, but rather two dimensions—buoyancy and weight—both of which have an absolute zero, and intersect at this zero. The objection is not well taken. We may admit, for the sake of argument, that the dimension, for which a procedure of measurement has just been described, is not properly called "weight." But it does not follow that it is really two intersecting dimensions instead of one. The symbols G, E, and @ have been consistently defined so as to apply to all objects within the measured dimension. For instance, G is defined in such a way that X G Y if and only if X overbalances Y, and this definition applies whether X and Y are objects of positive weight or objects of negative weight. Perhaps some term other than "weight" should be used to refer to this single dimension—"bweight," let us say. We may cheerfully concede this point, for it can still be argued that there are some dimensions, bweight, for instance, which do not have an absolute zero and are nevertheless independently measurable. The foregoing analysis indicates that the following precise meanings should be given to the phrases, "object of zero magnitude" and "dimension having an absolute zero." An object, Y, is one of zero magnitude if (X @ Y) £ X' (where @ and E are defined for the dimension in question). A dimension has an absolute zero if (X @ Y) G X' (where @ and G are defined for the dimension in question). Hence, a dimension having an absolute zero can contain no objects of zero magnitude. Length is a dimension that contains no objects of zero magnitude and has an absolute zero, since (X @ Y) G X' (where @ designates the operation of juxtaposing objects and G means "greater in length than"). Weight, or bweight, on the other hand, contains objects of zero magnitude and does not have an absolute zero, since

THE N A R R O W VIEW OF MEASUREMENT

121

there are objects (e.g., a balloon with a certain helium and air mixture) such that (X @ Y) E X' (where @ and E are defined for weight in the now familiar manner). A n important result of the above definitions of "object of zero magnitude" and "dimension having an absolute zero/' and of the analysis that led to these definitions, is that axiom of addition (5)— (X @ Y) G X'—need not be satisfied by a measurable dimension. For (X @ Y) G X' is true of a dimension having no absolute zero only when Y is a positive object. And some dimensions—weight, for instance—are measurable even though they do not possess an absolute zero. To put the point another way, the physical operation of addition employed in a system of measurement need not satisfy axiom (5). The operation of placing two objects in the pan of a balance, closing the lid, and fastening it is a physical operation of addition for weight (or "bweight," if you will). But this operation does not satisfy the axiom, (X @ Y) G X', since if Y is a negative object the statement is not true. It may be supposed that axiom (5) can be restated so as to apply to this operation and can therefore be retained as a condition for a measurable dimension. But how is this to be done? On the one hand, axiom (5) might be restated as three alternatives: either (X @ Y) G X' or (X @ Y) E X' or X' G (X @ Y). The operation in question does satisfy this complex axiom. However, not only is this not axiom (5); it is inconsistent with axiom (5). O n the other hand, the conventions suggested earlier for designating positive, zero, and negative objects might be adopted: capital letters with a bar underneath for positive objects, capitals enclosed in parentheses for zero objects, and capitals with a bar on top for negative objects. Axiom (5) would then become: (X @ Y) G X'. Unfortunately this statement is true by definition (as we saw earlier), and is necessarily satisfied by every operation of combination. In other words, it is no longer an axiom of addition, in the usual sense of this term. It is no longer a contingent statement which may or may not be true of a given operation of combination. Another important result of our analysis is that the common practice of employing "arbitrary (conventional) zero" as an antonym for "absolute zero" is a misleading use of these terms. Weight (or "bweight," if you prefer) does not have an absolute zero; but neither does it have an arbitrary zero. The word "arbitrary" is one of a list of

122

THE NARROW VIEW OF MEASUREMENT

appallingly vague terms too often employed by scientists and philosophers of science (others in the list are "relative," "subjective," "operation"). It is probably wisest to avoid using the term altogether. Let us assume for the present, however, that it is correct to say of the centigrade and Fahrenheit scales of temperature that each has an arbitrary zero. This statement means, presumably, that the use of thermometers to measure heat would not have been prevented by the selection of zeros other than those that were in fact selected. All that is required is that various experimenters agree on a zero (or zeros). In this sense, the zero in the system of weight measurement described above is not arbitrary. To construct a system for the measurement of positive and negative weight, "object of zero weight" must be defined as follows: If (X @ Y) E X', then Y is an object of zero weight. T o give "object of zero weight" the definition provided for "object of negative weight" or for "object of positive weight" would make the selection of acceptable units of positive and negative weight impossible. The point is that the proper antonym of "absolute zero" is nonabsolute zero," and the proper antonym of "arbitrary zero" is "nonarbitrary zero." A s the terms are defined above, to say either that a dimension has an absolute zero or that it has a nonabsolute zero implies that there is an operation of addition for the dimension. If the operation of addition satisfies axiom (5), the dimension has an absolute zero; if not, it has a nonabsolute zero. If there is no operation of addition for the dimension, it is meaningless to say either that it has an absolute zero or that it has a nonabsolute zero. Length has an absolute and a nonarbitrary zero. Weight (or bweight) has a nonabsolute and a nonarbitrary zero. Temperature, insofar as the use of the centigrade and Fahrenheit scales is concerned, has an arbitrary zero. But it is meaningless to say either that temperature has an absolute zero or that it has a nonabsolute zero, since (so far as I know) there is no operation of addition for temperature. A t this point the theorist who is inclined to maintain that a dimension, to be measurable, must have an absolute zero may restate his view as follows. To be independently measurable a dimension must, in the terms given herein, possess a nonarbitrary zero, and this is really what is meant by the assertion that it must possess an absolute zero. W e can agree that both length and weight are independently measurable since, although the former has an absolute zero and the

THE NARROW VIEW OF MEASUREMENT

123

latter a nonabsolute zero, neither zero is arbitrary in the sense described herein. But unless there is a physical operation of addition for a given dimension, the choice of a zero in that dimension will be arbitrary, and the dimension incapable of independent measurement. The examination of this rejoinder is deferred to chapter 7, where I examine the possibility of independently measuring pitch, a dimension for which there is apparently no physical operation of addition. In that chapter, as well as in the one concerning loudness measurement, some other possible meanings of the terms "zero" and "absolute zero" are explored. I also consider the suggestion that the zeros or absolute zeros for loudness and pitch are liminal tones—that is, the softest and the lowest tones that can be heard, or else tones just below the limen. It has been established that axioms (5) through (9) can be false for measurable dimensions. This effectively destroys the classical view of measurement, and yet seems to leave the core of the view untouched. That core is the belief that measurement involves a physical operation of addition. And, contrary to first appearances, it is just possible that there exist physical operations of addition, or perhaps we should say combination, that satisfy none of the axioms of addition thus far discussed. In what follows I argue that no operation of combination is required in a system of measurement. THE PURPOSE OF MEASUREMENT Particular analyses of measurement frequently seem to depend, at least in part, upon how the proponents view the purpose, or principal use, of measurement. A theorist, observing that such-and-such is the purpose of measurement, stipulates that a system of numeral assignment qualifies as one of measurement only if it serves that purpose. Then he attempts to discover what features a system of numeral assignment must possess in order to serve the purpose in question. Some writers apparently believe that such an argument can be used to show that a system of measurement necessarily employs a physical operation of addition. In the present section I attempt to disprove this belief. One use of measurement is that it permits us to indicate, in convenient fashion, the rank or order of objects within a given dimension. If a group of objects have been weighed on a balance, we can tell, simply by reading off the numerals assigned to each, which are

124

THE NARROW VIEW OF MEASUREMENT

heavier or lighter than others. That this use is, however, nothing more than a convenience of measurement, and not its principal use or purpose, is obvious from the fact that the order of objects can be indicated by assigning letters of the alphabet—a procedure that clearly is not measurement. There would seem to be two leading candidates for the principal use of measurement. (A) The principal use of measurement is that it permits the application of arithmetical statements to some dimension; that is, it provides an interpretation of arithmetical statements. A person who takes this view may go on to stipulate that a system of numeral assignment qualifies as one of measurement only if it permits the application of arithmetical statements to some

dimension.

Campbell provides a clear statement of this view: If a property is to be measurable it must be such that (1) two objects which are the same in respect of that property as some third object are the same as each other; (2) by adding objects successively we must be able to make a standard series one member of which will be the same in respect of the property as any other object we want to measure; (3) equals added to equals produce equal sums. In order to make a property measurable we must find some method of judging equality and of adding objects such that these rules are true. . . . It is because these rules are true that measurement [of such properties as length] is useful and possible; it is these rules that make the measurable properties so similar to numbers, that it is possible and useful to represent them by numerals the primary purpose of which is to represent numbers. It is because of them that it is possible to find one, and only one numeral, which will fitly represent each property; and it is because of them, that these numerals, when they are found, tell us something useful about the properties. One such use arises in the combination of bodies possessing the properties. We may want to know how the property varies when bodies possessing it are added in the way characteristic of measurement. When we have assigned numerals to represent the property we shall know that the body with the property 2 added to that with the property 3 will have the same property as that with the property 5, or as the combination of bodies with properties 4 and 1 . This is not the place to examine exactly how these conclusions are shown to be universally valid; but they are valid only because the three rules are true."

Measurement may be said to have a second use. (B) The principal use of measurement is that it permits the formulation and veri14

Norman Robert Campbell, What Is Science? (London, 1921), pp. 1 1 7 - 1 1 8 .

THE NARROW VIEW OF MEASUREMENT

fication of numerical scientific laws. A person taking this view may go on to stipulate that a system of numeral assignment qualifies one of measurement only if it permits the formulation and

as

verifica-

tion of numerical scientific laws. Again, Campbell gives expression to the view: For why is the process of measurement of such vital importance; why are we so concerned to assign numerals to represent properties. . . . The true answer to our question is seen by remembering . . . that the terms between which laws express relationships are themselves based on laws and represent collections of other terms related by laws. When we measure a property, either by the fundamental process or by the derived process, the numeral which we assign to represent it is assigned as the result of experimental laws; the assignment implies laws. [Campbell is here referring to the axioms of measurement.] And therefore, in accordance with our principle, we should expect to find that other laws could be discovered relating the numerals so assigned to each other or to something else; while if we assigned numerals arbitrarily without reference to laws and implying no laws, then we should not find other laws involving these numerals. This expectation is abundantly fulfilled, and nowhere is there a clearer example of the fact that the terms involved in laws themselves imply laws. When we can measure a property truly, as we can volume (by the fundamental process) or density (by the derived process) then we are always able to find laws in which these properties are involved; we find, e.g., the law that volume is proportional to weight or that density determines, in a certain precise fashion, the sinking or floating of bodies. But when we cannot measure it truly, then we do not find a law. An example is provided by the property "hardness." 1 8 It may seem that a system of measurement, if it is to serve either of the uses mentioned above, must employ a physical operation of addition. T o be more precise, it may seem that if either of the italicized statements under (A) and (B) is true, every system of measurement necessarily employs an operation of addition, for it may seem that a system of numeral assignment can serve the uses mentioned in the italicized statements only if it employs an operation of addition. I propose to show that this view is false. THE FIRST USE

The statement " 3 +

4 =

7 " is a simple arithmetical statement.

A system of weight measurement employing a balance and unit 18

Campbell, What Is Science? pp. 132-133.

126

THE NARROW VIEW OF MEASUREMENT

objects makes it possible to apply this statement arid others like it to the dimension of weight. It provides an interpretation of such statements in terms of the weight of objects. With the selection of a group of unit objects, " 3 " comes to designate any collection of three unit objects, or any object (or collection of objects) that balances such a collection; " 4 " comes to designate any collection of four unit objects, or any object (or collection of objects) that balances such a collection; and so on for every numeral. Given this interpretation, any object (or collection of objects) designated by " 3 " together with any object (or collection of objects) designated by " 4 " will, when placed in a single pan of a balance, balance any object (or collection of objects) designated by " 7 . " Obviously this interpretation is useful. For instance, with suitably chosen units the interpretation enables a candymaker not only to sell candy by weight, but also to keep a record of the amount of candy sold in individual transactions and to total these amounts at the end of the day or the week. Let us grant, for the sake of argument, that a system of numeral assignment qualifies as one of measurement only if it permits the application of arithmetical statements to some dimension. It does not follow that the operation of addition employed in a system of measurement must satisfy either axiom (6), (7), or (9). To apply " 3 + 4 = 7 " to the dimension of weight, an interpretation must be provided, not only for " 3 , " " 4 , " and " 7 , " but also for " + " and " = . " But it is not necessary to interpret " A r " in terms of the same operation of addition employed in the system of numeral assignment. This point can be illustrated by means of the two systems of weight measurement described earlier. We may employ an operation of horizontal addition in assigning numerals to the weight of objects, but interpret the in arithmetical statements such as " 3 + 4 — 7 " in terms of an operation of vertical addition. If this is done, an object to which " 3 " has been assigned (by horizontal addition of units) added vertically to an object to which " 4 " has been assigned (by horizontal addition of units) will equal an object to which " 7 " has been assigned (by horizontal addition of units), despite the fact that horizontal addition satisfies neither axiom (6), (7), nor (9). Even more striking is the following. Let us grant, for the sake of argument, that a system of numeral assignment qualifies as one of measurement only if it permits the application of arithmetical state-

THE NARROW VIEW OF MEASUREMENT

12 7

ments to some dimension. It does not follow that a system of measurement must employ an operation of addition of some kind. Suppose numerals are assigned to length in the following way. A group of standard objects is selected by visual estimation. To X is assigned the numeral " 1 , " to Y (which looks twice as long as X) the numeral "2," to Z (which looks three times as long as X) the numeral " 3 , " and so on. It is conceivable that numerals have been assigned in such a way that an object to which " 3 " has been assigned and an object to which " 4 " has been assigned will, when juxtaposed, align at both ends with an object to which " 7 " has been assigned. If they do, the application of the arithmetical statement " 3 + 4 = 7 " to length will have been made possible by a procedure for assigning numerals which does not employ an operation of addition, that is, without the addition (juxtaposition) of unit objects. I do not maintain that the system described above is one of measurement. Indeed, in chapter 5 I present reasons for saying that it is not. My only point here is that a system of numeral assignment which employs no operation of addition may provide an interpretation for arithmetical statements. Someone will be quick to reply that in the system of numeral assignment described above the application of arithmetical statements to length is achieved fortuitously, and that there is no assurance that a procedure of numeral assignment will permit the application of arithmetical statements to length unless it employs an operation of addition of unit objects. But, first, my point still stands. If we grant that a system of numeral assignment qualifies as measurement only if it permits the application of arithmetical statements to some dimension, it does not follow that it employs an operation of addition of unit objects. Second, the objector asserts that our system of numeral assignment fortuitously provides an interpretation for arithmetical statements because he assumes that systems of numeral assignment which employ an operation of addition achieve this result in a nonfortuitous manner. He assumes, for example, that the assigning of numerals to rods by juxtaposing unit rods provides an interpretation for arithmetical statements in a nonfortuitous manner. But is this assumption correct? What, precisely, is the difference between a "fortuitous" and a "nonfortuitous" result? It seems clear that we could not know, in advance of experience with rods, that a rod of 3 unit lengths and one of 4 unit lengths will, when juxtaposed, align at both ends

128

THE NARROW VIEW OF MEASUREMENT

with a rod of 7 unit lengths. Is this result, then, fortuitous? It will perhaps be said that we can confidently predict, on the basis of our experience with them, that systems of numeral assignment employing an operation of addition will provide an interpretation of arithmetical statements, and that, in this sense, the result obtained through the use of such systems is nonfortuitous. But, in this sense, systems of numeral assignment which do not employ an operation of addition may provide an interpretation for arithmetical statements in a nonfortuitous manner, including perhaps even the system in which numerals are assigned to rods on the basis of visual estimation of their length. Can we not predict, on the basis of our experience with adult human beings, that if a normal person by visual estimation assigns " 3 " to one object, " 4 " to another, and " 7 " to a third, then the first two will align at both ends (at least approximately) with the third? THE SECOND USE

The use of measurement in the sciences may suggest that the principal use of measurement is to permit the formulation and verification of numerical scientific laws (henceforth referred to simply as numerical laws) between or within dimensions. It is difficult to find simple and illuminating examples to illustrate this use. The one employed here, although possibly artificial, should not prejudice any of the conclusions reached. Suppose we wish to discover the relation holding between a side, s, and the hypotenuse, h, of any isosceles right triangle. It would be possible, without employing a system of measurement, to ascertain that h is always longer than s, and that the relation between h and s remains approximately constant from triangle to triangle. But to ascertain the precise character of this relation, a system of numeral assignment is required. One system that might be employed for this purpose makes use of a ruler, Li, constructed in the following manner. A unit object, which is short in relation to the sides of the triangles to be measured, is chosen. Marks are made at equal intervals on Li by laying the unit object in successive adjacent positions along its length; these marks are labeled 1, 2, 3, and so on. To measure a side of a triangle, Li is aligned at the " 1 " end with one end of the side, and the numeral lying nearest the other end of the side is noted. The data in table 1, which might be collected through this use of Li, show that the hypotenuse of each

THE NARROW VIEW OF MEASUREMENT

12Ç

triangle examined is 1.4 times as long as the associated side. To express the relation in a formula, [1] h = 1.4s. A second ruler, Ls, is constructed by making and labeling marks TABLE 1 TRIANGLE

LENGTH OF SIDE

LENGTH OF HYPOTENUSE

a b c d e f

5 10 15 20 25 3°

14 21 28 35 42

7

arbitrarily along its length. Suppose that, by striking coincidence, Li and L2 are so related that when they are aligned at the " 1 " ends each number on L2 is the square of the corresponding number on Li. If the same six triangles above are measured by means of L2, the data in table TABLE 2 TRIANGLE

a b c d e

f

LENGTH OF SIDE

LENGTH OF HYPOTENUSE

25

49 196 441 784 1,225 1,764

IOO

225 400

625

goo

2 are obtained. Using these data, the relation between the hypotenuse and the side of each triangle is [2] V h - i . 4 V 7 . (It is, of course, not necessary to examine the data in table 2 to arrive at this formula, since it can be deduced from the numerical relation between Li and L2.) A third ruler, Ls, is constructed by making and labeling marks arbitrarily along its length, observing only the rule that the farther the mark from the " 1 " end of Ls the higher the numeral assigned to that mark. Suppose that the relation between Li and Ls, when they are aligned at the " 1 " ends, is as shown in table 3. If the same six triangles

1

3°

THE NARROW VIEW OF MEASUREMENT

are measured by means of La, the data in table 4 are obtained. Examination reveals that no single formula expresses the data in table 4. Measurement of the triangles by means of La would, therefore, suggest that there is no numerical law relating the sides and hypotenuses of isosceles right triangles. (Again, if we know the relation between Li and Ls, we can discover that there is no such numerical law without actually measuring the triangles by means of La.) TABLE 3 Ll

L3

Li

L3

5 7 10 14 15 20

10 20 70 210 220 800

21 25 28 30 35 42

880 1,400 1,600 2,000 4,200 6,000

TABLE 4 TRIANGLE

a b c d e f

LENGTH OF SIDE IO

70 220 800 1,400 2,000

LENGTH OF HYPOTENUSE

20 210 880 1,600 4,200 6,000

It is possible to argue that the hypotenuse is found, with La, to be a function of the side; that the function is described by table 4; indeed, that the table is the function. But even if this argument is accepted, there is still a difference between the function given in table 4 and the functions in tables 1 and 2. And it is functions of the type in tables 1 and 2 which are preferred in science. An obvious reason for this preference is that only functions of this type can be used as rules for predicting the results of further measurements of the dimensions in question. The above systems for measuring triangles show that measurement can have its second principal use without employing any phys-

THE NARROW VIEW OF MEASUREMENT

131

ical operation of addition. Let us grant, for the sake of argument, that a system of numeral assignment qualifies as one of measurement only if it permits the formulation and verification of numerical laws. It does not follow that a system of measurement must employ a physical operation of addition, as is made clear by comparing the three systems of numeral assignment just described. Although L2 was not constructed by means of any operation of addition, using this ruler results in the foundation and verification of numerical law [2]. It may be objected that this result is achieved fortuitously, whereas a system of numeral assignment employing an operation of addition assures the possibility of formulating and verifying numerical laws. But, first, my point still stands. Fortuitously or not, the system employing L2 achieves this result, and L2 was not constructed by means of any operation of addition. Second, the meaning of the term "fortuitous" poses a problem. W e cannot know, prior to experience, that systems of numeral assignment employing an operation of addition do permit the formulation and verification of numerical laws. Is it therefore true that such systems achieve this result fortuitously? Perhaps it will be said that we can confidently predict, on the basis of our experience with them, that systems of numeral assignment employing an operation of addition permit the formulation and verification of numerical laws, and that, in this sense, such systems achieve the desired result nonfortuitously. But, in this sense, systems of numeral assignment which employ no operation of addition may also produce the desired result nonfortuitously. For a possible example consider the procedure, described earlier, in which numerals are assigned to standard rods by visual estimation of length. Can we not predict, on the basis of our experience with adult human beings, that this system of numeral assignment will permit the formulation and verification of numerical laws that are more or less precise? Thus far we have discussed the usefulness of measurement only at what might be called the experimental level. Let us say that a system of numeral assignment is experimentally useful if it permits the formulation and verification of numerical laws. Let us say that a system of numeral assignment is theoretically useful if the numerical statements to which it leads can be incorporated in some fruitful scientific theory. A theory may be regarded as an axiom system. A scientific theory is an axiom system in which some of the axioms, and any theorems de-

THE NARROW VIEW OF MEASUREMENT

duced from them, are contingent statements. 16 A fruitful

scientific the-

ory is one in which a significant portion of the theorems have been verified as true. T o incorporate a statement into a theory is to show that it can be deduced from the axioms of that theory. Under these definitions, Li and L2 are both experimentally useful; Li is, and L2 is not, theoretically useful. T h e statement [1] to which w e are led through the use of Li can be deduced from the axioms of Euclidean geometry. But [2], the statement to which w e are led through the use of L2, cannot be deduced from the axioms of any theory (or, at least, has not been so deduced). It may be said that w e ought not to regard as measurement any system of numeral assignment which is not theoretically useful. A n d it may be supposed that to be theoretically useful a system of numeral assignment must employ an operation of addition, and that, as a consequence, every system of measurement must employ an operation of addition. But this supposition is erroneous. It is true that, among the examples given, the theoretically useful systems of numeral assignment employ an operation of addition of unit objects and that the theoretically useless systems do not. A n d it m a y be that this is true in general of theoretically useful systems of numeral assignment. If true, however, it is true as a matter of fact. There is no apriori reason f o r saying that theoretically useful systems of numeral assignment must employ an operation of addition. W e k n o w that the system that makes use of Li is theoretically as w e l l as experimentally useful. Since w e also k n o w the numerical relation between Li and L2, w e can deduce that L2 is experimentally usef u l and theoretically useless. But if w e do not k n o w the relation between Li and L2, or if w e do not k n o w whether Li is experimentally or theoretically useful, w e cannot say whether L2 is experimentally or theoretically useful without putting it to actual use and appraising the results. N o w w e may find ourselves in precisely this position w i t h regard to a n e w l y proposed system of measurement: w e may have n o other system of measurement with which to compare it. In that event the experimental and theoretical usefulness or uselessness of the proposed system can be determined only b y putting it to actual use. If a system proposed under these conditions does not employ an operation 1 8 See R. B. Braithwaite, Scientific Explanation (New Y o r k , 1953), p. 12, and J. H . Woodger, The Technique of Theory Construction, International Encyclopedia of Unified Science, Vol. II, no. 5 (Chicago, 1939), p. 66.

THE NARROW VIEW OF MEASUREMENT

*33

of addition, we may perhaps have reason, from the history of scientific progress, to be suspicious of it. But such suspicions do not amount to an apriori reason for saying that the proposed system will prove experimentally, or theoretically, useless. Before concluding this section I must explain the qualification made at its beginning—that the example of discovering the numerical relation between a side and the hypotenuse of an isosceles right triangle is possibly artificial. The reason for this qualification lies in the dispute as to whether Euclidean geometry is a scientific theory, that is, as to whether its axioms and theorems are contingent statements.17 This dispute need not be settled here. If some of its axioms and theorems are contingent statements, Euclidean geometry is an example of a fruitful scientific theory. On the other hand, if none of its axioms and theorems are contingent statements, Euclidean geometry can still serve as a useful model or analogy for a scientific theory. It can still be employed to show what is meant by a theory and to show what it is to incorporate a statement into a theory. OPERATIONS AND MEASUREMENT It is commonly said that "measurement is operational." This statement is as much a slogan as a philosophical remark, and, like most slogans, is suggestive, vague, and ambiguous. The present section is concerned with discovering what its meaning might be, and whether it can be given a true interpretation. There are two points at which operations might be said to enter into measurement. Of the eight axioms of measurement listed earlier, (5) through (8) contain the symbol which designates a physical operation of addition for the dimension in question. All the axioms contain either G or E which designate quantitative relations for the dimension in question. It is commonly supposed that we must specify tests or operations for determining these relations. In these two respects measurement may be said to be "operational." OPERATIONS OF ADDITION

The availability of operations of addition.—It is commonly supposed that length and weight can be measured, whereas loudness, pitch, 17

See Carl G. Hempel, "Geometry and Empirical Science/' in Readings in Philosophical Analysis, ed. H. Feigl and W. Sellers (New York, 1949), pp. 244-249, for the orthodox, positivist view.

*34

THE NARROW VIEW OF MEASUREMENT

brightness, hue, and sourness cannot, because there is a physical operation of addition for length and weight but not for the other dimensions. Two important points must be made: first, this supposition is false; second, it is contingently false. Two lights placed next to each other and made to shine simultaneously are together brighter than either alone. Few writers on measurement seem to have realized that this operation of brightness addition makes it possible to measure brightness by methods like those used for length and weight. 18 Loudness provides another clear example. Two tones produced simultaneously are together louder than either produced alone. In chapter 6 a system of numeral assignment for loudness which employs this operation of loudness addition is described and defended. It is a system that qualifies as measurement under the classical definition. The second point above is that statements asserting the existence or nonexistence of an operation of addition for a given dimension are contingent. There are undoubtedly many persons who would not know prior to being told that two tones produced simultaneously are together louder than either produced alone. The italicized statement is not self-evidently true, nor does there seem to be any way of deducing it from a definition of loudness. In general, the question of whether a given operation satisfies axiom (5) must be settled by experiment and not by apriori methods. This may seem untrue with regard to the operation of juxtaposition. Two juxtaposed objects are in fact longer than either alone, yet a world in which this is not true is conceivable. It would, of course, be an extremely bizarre world, one in which, as we would probably say, objects shrink when juxtaposed. Of course we might stipulate that an operation does not qualify as one of juxtaposition unless the two (or more) objects involved in the operation are, at its completion, longer than either alone; even so, the question of whether an operation so defined can be performed will remain. And it is a question of fact, not one that can be settled apriori. Similar considerations obtain for weight. Under normal conditions two objects placed together on one arm of a balance exert more downward force than either alone. This will not be true, however, if balance and objects are freed from the gravitational attraction of the earth and other bodies. Of course we might arbitrarily specify that an operation does not 18

Nagel, "On the Logic of Measurement," p. 23, is an exception.

THE NARROW VIEW OF MEASUREMENT

135

qualify as one of placing two or more objects on a balance unless the combined objects exert more force than either alone; even so, the contingent question of whether such an operation can be performed will remain. I have argued that the question of whether a dimension permits an operation of addition is contingent. The question of whether there is an operation of addition for loudness, pitch, hue, brightness, and sweetness is not one that can be settled by apriori methods. There are, in fact, operations of addition for loudness and brightness. Perhaps there also are operations that will qualify as physical addition for other dimensions, operations that have not even been thought of, much less tried. The question is further complicated by unclarity, in a crucial area, as to what qualifies as an operation of addition. It might be agreed that, by definition, an operation of addition has not been performed on several objects within a dimension unless the objects added are together greater in respect to the dimension than either alone. But consider the dimension of sweetness and the following operation, Q, for that dimension. Q consists in mixing two or more liquids and boiling the mixture for a period of time proportional to the volume of mixture. If this operation is properly specified, the sweetness of the mixture at the end of the operation is greater than that of either liquid before mixing. Ought we to call Q an operation of addition? The normal inclination would probably be to say that we should not. But why not? In what crucial respect does Q differ from those operations that are properly said to be operations of addition? Bergmann would object to calling Q an operation of addition for the dimension of sweetness, as can be inferred from his discussion of temperature. He considers a proposal that "Ti @ T2" be defined as the sum of the temperatures of two objects that have been brought into contact and have established thermic equilibrium. He argues that the symbol @ denotes an operation of addition only if Ti @ T2 is uniquely determined by Ti and T2, and that unique determination can be assured only by specifying the weights and physical composition of Ti and T2. He then asserts that even if this condition were fulfilled, "the operation would still not be one within the dimension since, in order to secure the definiteness of [Ti @ T2] we had to draw upon extraneous factors, namely, the dimension of weight and chemical com-

i36

THE NARROW VIEW OF MEASUREMENT

position." 19 Bergmann would presumably apply the same form of objection to my suggested operation of addition for sweetness. He would say that such a sum of sweetnesses can be uniquely determined only by drawing upon the extraneous factors of volume and time. This objection is inadequate, because in one sense the unique determination of any physical sum draws upon "extraneous factors." In juxtaposing rods in a measurement, we must not use rods that react with one another, or rods made of ice or rubber. The composition of the rods is, according to Bergmann, an extraneous factor. (He anticipates this criticism in a footnote to the passage above, but his reply seems totally inadequate.) Perhaps the objection that really underlies Bergmann's complaints is that the specification of an operation of addition for sweetness, Q, requires a reference to the measurement of time and the measurement of the volume of liquids because these "extraneous" dimensions must be measured in order to know whether Q has been performed. Under this requirement, a system of measurement employing Q is one of dependent measurement, and therefore cannot be used to show that sweetness is independently measurable by a procedure employing an operation of sweetness addition. A first point of reply is that many procedures for measuring length and weight require the measurement of "extraneous" dimensions. For instance, the most precise weighings are performed under certain conditions of temperature, pressure, and humidity which can be determined only by measuring these dimensions. But the more important point is that the operation of sweetness addition can be redescribed so as to eliminate all reference to the measurement of extraneous dimensions. The redescribed operation, Q', consists in mixing two or more amounts of sweet liquids and boiling down the mixture until its volume is equal to the amount of the sweetest liquid in the mixture. Since boiling evaporates water in the mixture while leaving the sweetening agent in solution, the resulting solution will contain more sweetening agent than was in the sweetest of the liquids mixed. The resulting liquid will, consequently, be sweeter than the sweetest of the sweet liquids mixed, and a fortiori sweeter than any of the liquids mixed. Note that Q' can be performed without measuring the volume of any liquid; we need only mark the container at the surface of the sweet18

Bergmann, "The Logic of Measurement," pp. 29-30.

THE NARROW VIEW OF MEASUREMENT

137

est liquid before mixing the liquids, and such marking is not a procedure of numeral assignment. What reason is there to deny that Q ' can be employed in a procedure of assigning numerals to sweetness which qualifies as measurement? Addition and the concept of measurement.—It has been argued in earlier sections that the operation of addition employed in measurement need not satisfy axiom (5), (6), (7), (8), or (9), and that there are no very good reasons for asserting that a system of measurement must employ an operation of addition of any kind. T o this latter it may be replied that measurement is, by definition, a procedure of numeral assignment employing an operation of addition. Such a reply will appeal either to a technical use or to the ordinary use of the term "measurement." The writings of Campbell and his followers produce the impression that in technical scientific contexts "measurement" means a system employing an operation of addition. Other writers, however, use the term in a broader sense. For instance, Stevens and Hempel maintain that certain scales for "psychological magnitudes" are scales of measurement, even though they employ no operation of addition. So, in point of linguistic fact, the term "measurement" is not restricted by all technical writers to a procedure employing an operation of addition. It may be said, however, that, apart from facts about technical use, there are reasons for preferring Campbell's use to Stevens' use: (a) in the past scientists have generally used the term "measurement" in the manner prescribed by Campbell; and (b) procedures that would, in this use, be called "measurement" have been shown to be the most fruitful in the history of science. To verify (a) would require a survey of the history of science and is obviously beyond the scope of this study. Even if (a) is true, however, it must be pointed out that Campbell's use of the term "measurement" seems especially appropriate for (and was developed in connection with) physics, Campbell himself being a physicist. There is considerable contemporary controversy as to whether Campbell's use is appropriate for psychology, where many procedures of measurement do not employ an operation of addition. N o w the history of experimental psychology is considerably shorter than the history of experimental physics. The attempt to develop scales of psychological measurement is usually regarded as having begun with Weber and Fechner, who worked and wrote a mere hundred years ago.

i38

THE NARROW VIEW OF MEASUREMENT

Therefore, it would not be surprising to find that Campbell's use of "measurement" has a longer history than Stevens' use. Related considerations apply to assertion (b). It may be true that the procedures of psychological measurement which employ no operation of addition have not yet proved their fruitfulness, but there has been relatively little time since their development to test them. More important, some of these procedures have, it seems, proved fruitful. Stevens and others have employed such procedures to discover what they consider to be numerical laws relating pitch and frequency, and loudness and intensity. If the claim that measurement is by definition a procedure employing an operation of addition cannot be supported by appealing to the technical use of the term, will an appeal to ordinary use be more successful? The answer is that reflection on ordinary use firmly establishes precisely the opposite conclusion. In two of the four methods for measuring length described in chapter 3, the fundamental operation is the juxtaposition of unit objects; in the other two the fundamental operation is the laying of a unit object in successive adjacent positions along a rod. We would without hesitation call any of these methods a procedure for the measurement of length, but only two of them employ an operation of addition of unit objects. We can legitimately speak of the addition of unit objects only when an operation of combination is performed on two or more of the objects. Since laying a single unit object in successive adjacent positions along a rod is not an operation of combination performed on two or more unit objects, it is not properly called an operation of addition. A less obvious illustration is available for the dimension of weight. Let us say that a single cube, U, selected as the unit object of weight, is in zero position when placed on the arm of a balance directly over the fulcrum, in first position when in the adjacent space, in second position when in the next adjacent space, and so on. The higher the position the greater the moment of force of U and, hence, the heavier the object on the other arm that can be balanced by U. Weighing a given object, X, consists in placing X on one end of an arm of the balance and placing U on the other arm in the position required to achieve a balance. The number of U's position gives the numeral to be assigned to X; no operation of combination is employed. A proponent of the narrow view of measurement may object

THE NARROW VIEW OF MEASUREMENT

139

that the point above rests on an improper conception of physical addition. He may insist that laying a single unit object along a rod in successive adjacent positions can properly be called an operation of addition. (He may argue, for example, that spaces, rather than rods, are added.) This objection, however, may very well subvert the view it is designed to support, the view that measurement necessarily involves a physical operation of addition. Proponents of the narrow view usually assert that such dimensions as pitch and hue are incapable of independent measurement, since they do not permit a physical operation of addition. But if the notion of physical addition is so broadly construed that some operations, which are not operations of combination of two or more objects, qualify as physical addition, then it can be argued that pitch permits a physical operation of addition. The system of measurement for pitch described in chapter 7 involves counting off equal pitch intervals. Under the broad interpretation of the notion of physical addition, it can be argued that counting off equal pitch intervals is, like laying a unit object along another rod in successive adjacent positions, a physical operation of addition. Addition and subjectivity.—Undoubtedly one reason for thinking that measurement necessarily involves a physical operation of addition is the assumption that a procedure of numeral assignment is "subjective" unless based on such an operation. Numerals can be assigned to objects in a variety of ways. For instance, an observer, O , may visually estimate the relative lengths of a group of rods, and then attempt to assign numerals to them so that the ratio between their lengths is the same as the ratio of the numerals assigned. This procedure is, one is inclined to say, subjective. O may fail to match length ratios and number ratios, or he may fail to be consistent in his numeral assignments. Contrast this method with one in which O uses equal unit rods, comparing juxtapositions of these with the rods to be measured, in order to make his numeral assignments. This procedure is, one is inclined to say, objective. Its use will insure that length ratios match number ratios, and that numeral assignments are consistent. One difficulty in this argument lies in the incredible obscurity of the terms "subjective" and "objective." But let us pass this by for two other objections that are more easily made. First, procedures of numeral assignment based on an operation of addition are not automatically objective or consistent. Every measurement requires a mea-

140

THE NARROW VIEW OF MEASUREMENT

surer, and the measurer may use the operation of addition well or badly. Although he compares rods to be measured with juxtapositions of unit rods, he may, if careless, assign numerals to the rods in an inconsistent manner. Second, a procedure of numeral assignment, even if not based on an operation of addition, may still be objective. In all probability, most observers who assign numerals to rods by visual estimation assign them inconsistently and in such a way that length ratios are not accurately represented. But we may discover an observer whose assignments are never defective in these ways. Furthermore, we may discover other procedures of numeral assignment which, although they employ no operation of addition, are just as objective as those that do. Just such a procedure for pitch is described and defended in chapter 7. TESTS FOR EQUALITY AND INEQUALITY

Every procedure of measurement requires a method for determining the relative magnitude of objects and collections of objects within the relevant dimension. If the procedure employs an operation of addition and a number of equal unit objects, some method is necessary for selecting the unit objects and for comparing collections of these with objects being measured. Even if the procedure employs a single unit object and no operation of addition, some method is necessary to determine the magnitude of the object being measured relative to the magnitude of the unit of measurement. For length the method thus employed is usually alignment; for weight it is usually balancing objects (and collections of objects) against each other. Some writers seem to believe that relative magnitude cannot be determined in a procedure of measurement by direct perceptual comparison of the objects within the dimension. They are, as we shall see, mistaken. Consider the following pairs of determinations. (1) We determine whether X is heavier than Y (a) by lifting the objects or (b) by placing them on opposite arms of a balance. (2) We determine whether X is longer than Y (a) by simply looking at and comparing the objects or (b) by aligning them. (3) We determine whether X is harder than Y (a) by feeling the objects or (b) by attempting to scratch one with the other. (4) We determine whether X is warmer than Y (a) by feeling the objects or (b) by bringing them into contact with a mercury ther-

THE NARROW VIEW OF MEASUREMENT

141

mometer. Some writers apparently believe that determinations of equality and inequality in a procedure of measurement must be of the b-type. One gets this impression from those writers who assert that G and E (the two relational terms in the eight axioms of measurement) must be "operationally defined." Reese provides an example. He says that "in order to establish an ordinal scale it is necessary a) to define > , < and = by means of a set of operations used in establishing these relations, and b) it is necessary to demonstrate experimentally that the relation > and < defined by this set of operations is asymmetrical and transitive; and that the relation expressed by the symbol = , defined by this same set of operations, is symmetrical and transitive."20 Many writers give operational definitions for G and E. Thus, in connection with length, E is usually defined as "is congruent with" or "aligns at both ends with." In connection with weight, E is usually defined as "balances" and G as "overbalances." In connection with hardness, G is usually defined as "scratches."21 Presumably, then, the «-determinations would not be regarded as "operations" by many theorists. Whether or not this interpretation is correct, it is important to expose the error in the view that determinations of equality and inequality in a procedure of measurement must be of the b-type, because some objections to the procedures of measurement presented in chapters 6 and 7 are based on that view. What is the difference between the a- and the b-determinations? A few negative points must be made at the outset. First, we cannot say that the fa-determinations involve instruments, whereas the adeterminations do not. Procedures (1 b) and (4b) do involve instruments—a balance and a thermometer, respectively—but (2 b) and (3 b) do not. The latter procedures use only the objects about which the determinations are being made. Second, we cannot say that the «-determinations are, and the b-determinations are not, determinations made by means of sense perception, for determinations of both types involve 20 T. W. Reese, "The Application of the Theory of Physical Measurement to the Measurement of Psychological Magnitudes," Psychological Monographs, 55 (1943), 46. 21 Carl G. Hempel, fundamentals of Concept Formation in Empirical Science, International Encyclopedia of Unified Science, Vol. II, no. 7 (Chicago, 1952), pp. 60-61.

142

THE NARROW VIEW OF MEASUREMENT

sense perception. In (ib) an observer must touch or look at the balance to determine whether either arm sinks; in (2b) an observer must touch or look at the objects being aligned. Fuller and more formal descriptions of both types of determination will be useful. (1 a) If, when X and Y are looked at in good light from a position where they are neither difficult to see nor obstructed, X looks longer than Y, then X is longer than Y.

(ib) If, when X and Y are placed side by side and aligned at one end (either by sight or by touch), the opposite end of X extends past the corresponding end of Y , then X is longer than Y.

(2a) If, when X and Y are separately lifted with the same hand, X feels heavier than Y , then X is heavier than Y.

(2 b) If, when X and Y are placed on opposite arms of a balance, the X-arm sinks, then X is heavier than Y.

(3a) If, when X and Y are successively pressed with the same finger—one that has normal feeling in it (e.g., is not anesthetized)—X feels harder than Y, then X is harder than Y.

(jb) If, when an attempt is made to scratch X with Y and Y with X, X scratches Y but Y does not scratch X, then X is harder than Y.

(4a) If, when X and Y are successively touched with the same hand, X feels hotter than Y, then X is hotter (has a higher temperature) than Y.

(4b) If, when X and Y are successively brought into contact with the same mercury thermometer, the mercury level for X is higher than that for Y, then X is hotter (has a higher temperature) than Y.

Logical differences.—What, if anything, do the a-definitions have in common? In the italicized phrases, the verbs "looks" in (ia) and "feels" in (2a), (3a), and (4a) are used to state that objects appear to some particular sense to have a certain relation or property. It is essential to understand that these terms imply from their context the use of some one sense, not two or more senses. With this understanding, we may generalize as follows. In the «-determinations we deter-

THE NARROW VIEW OF MEASUREMENT

143

mine whether two objects have a certain relation by discovering whether they appear to some particular sense to have that relation. In the ^-determinations a different procedure is employed. In (2b), for instance, we determine whether X is heavier than Y by discovering whether the X-arm of the balance sinks. In order to do so, we must discover whether the X-arm of the balance appears to some particular sense to sink. Nevertheless, this discovery is to be distinguished from the discovery that X appears to some particular sense to be heavier than Y. Differently stated, in the a-definitions "greater than" is defined in terms of how objects appear to some particular sense, whereas the b-definitions are of a different character. ("X looks longer than Y , " "X feels heavier than Y , " etc., are what some philosophers have called reports of "immediate" or "direct" perception.) The above statement of the differences between a- and bdeterminations helps to explain the temptation to regard the former as subjective and the latter as objective. In saying that X feels heavier than Y we sometimes express uncertainty as to whether X is heavier than Y. If we are uncertain whether X is heavier than Y , we ought not to conclude that X is heavier than Y. Such a conclusion would indeed seem to be subjective. In a different and quite acceptable sense, however, the word "feels" does not express the speaker's uncertainty. A person can say of a rough surface that it feels rough in order to contrast the way it feels with the way it looks, but he does not thereby necessarily imply that he is uncertain as to whether the surface is rough. In this second sense of "feels," there is in general no subjectivity in concluding that X is heavier than Y from the fact that X feels heavier than Y. It is this second sense that is involved in the italicized statement in the description of (2a) above. It may be objected, however, that (2a) is a subjective determination in the sense that only the person making the determination can judge whether one object feels heavier to him than another. In determination (2 b), on the other hand, the observer tries to discover which object overbalances the other, and a judgment on this matter is one that any normal person can make. Therefore, (2a) involves a subjective, (2b) an objective, judgment. This view is mistaken. For the unclear terms "subjective" and "objective" let us substitute the precise terms "incorrigible" and "corrigible," as defined in chapter 2: an incorrigible statement may be corrected by an auditor only for an error

144

THE NARROW VIEW OF MEASUREMENT

of use; a corrigible statement may be corrected for errors in addition to those of use, notably errors of perception. First-person, presenttense afterimage and pain statements are excellent examples of incorrigible statements. If O i describes an afterimage presently seen by himself as "elliptical and circular," O2 can challenge the description on the ground that it is self-contradictory, thus correcting an error of use. If O i then changes his description to "elliptical and red," a correction by O2 would be meaningless. If O2 should say to O i , "That's incorrect; you must be too far away from the afterimage to see it clearly," his statement would be meaningless. Consider the following example. O i says, after having lifted two objects, " X feels heavier than Y . " O2 then lifts X and Y and says, " N o , X feels lighter than Y ; there must be something wrong with your arms." O2 is here correcting, not an error of use, but an error of perception. Whether or not his explanation of the error is correct, the meaningfulness of his statement shows that "X feels heavier than Y " is a corrigible statement. In a parallel example O i says, after having looked at the objects, "X looks longer than Y . " O2 then looks at the objects and says, " N o , X looks equal to Y ; you must be too far away from the objects to see them clearly." The meaningfulness of Oa's statement shows that Oi's statement is corrigible. After being challenged by O2, O i may go on to say, "Well, X feels heavier than Y to me," or "Well, X looks longer than Y to me." These statements may be incorrigible. If so, they must be distinguished from O i ' s former statements, which were corrigible. The description of (2«) directs us to conclude that X is heavier than Y if X feels heavier than Y . "X feels heavier than Y " must be understood in this description as the type of corrigible statement discussed above. A n observer may make an error in perception in finding that X feels heavier than Y . If he exercises maximum caution he will compare his judgments as to how the objects feel with those of other persons, and compare his present judgments with his past judgments regarding the objects. The tendency to regard determinations of the «-type as subjective is probably connected with the tendency to think of them as estimates. It is frequently said that an observer who looks at two rods and judges their relative length is milking a "direct perceptual estimate," or that the data obtained from such observers are data on "estimated

THE NARROW VIEW OF MEASUREMENT

145

length." From this point of view, «-determinations are estimates of what the corresponding ^-determinations will reveal concerning the relative magnitude of objects. For instance, "X feels heavier than Y " is an estimate of relative weight—as determined by a balance—of X and Y ; and "X looks longer than Y " is an estimate of the relative lengths—as determined by the alignment method—of X and Y ; and so on for the other pairs of determinations. In this view the adeterminations will indeed seem "subjective." An estimate is a kind of guess, and by definition a guess, even an educated guess, is made by a person who does not have impeccable grounds for the statement describing the guess. Guesses are by definition inconclusive, and in fact frequently wrong; in short, one may be inclined to say, they are subjective. The view that «-determinations are estimates of what b-determinations will reveal is objectionable in two ways. First, the term "estimate" is employed in an imprecise, if not an incorrect, way. W e frequently make estimates regarding the relative weight of objects by lifting them, but when we do so we are estimating either that one object is twice, three times, or half as heavy as another, or that a certain object weighs a certain number of pounds or grams. To speak of estimating that X is heavier than Y is, at best, to use the term "estimate" in an imprecise manner. For, to be precise, an estimate is by definition made in numerical terms; thus an estimate presupposes some system of measurement for the dimension with regard to which the estimate is made. If O estimates that X is 10 grams in weight, he is predicting or inferring (on the basis, for instance, of having lifted the objects) that the use of a balance and a group of standard weights will reveal that X has a weight of 10 grams. When O estimates that X is half as heavy as Y, either he means that X and an object of equal weight together will be found equal to Y in weight, or he means that if X and Y are measured X will be found to have a weight of n grams and Y a weight of 2ti grams. That is to say, O's estimate concerning the relation in weight between X and Y presupposes a system of measurement, either one in which X serves as the unit or one that employs some other unit. Since the a-determinations are not determinations of relative magnitude in numerical terms, and since they do not imply any system of measurement for the relevant dimensions, they are not, strictly

146

THE NARROW VIEW OF MEASUREMENT

speaking, estimates of what the fa-determinations will reveal concerning relative magnitude. Although important, this point is mainly verbal. It is possible to concede the point, abandon the term "estimate," and go on to insist that the a-determinations are predictions of or inferences as to what ^-determinations will reveal regarding relative magnitude. The argument that this latter view is equally incorrect constitutes my second objection. Using (2a) as my example, the view I wish to refute is this: In concluding that X is heavier than Y , from the fact that X feels heavier than Y , O is predicting or inferring that if X and Y are properly placed on the lever of a properly constructed balance, the X-arm will sink. If this is true, then O must always withdraw the conclusion if the X-arm does not sink when a weighing is made. But since O need not always withdraw the conclusion when the X-arm does not sink, the view in question is false. The italics indicate the critical premise in my argument. An illustration will make its truth apparent. Suppose that X is a cubic inch of lead and Y is a cubic inch of sponge rubber. When a normal person lifts the two objects he will find X to be undeniably heavier than Y. That is to say, X will feel heavier than Y , and the conclusion that X is heavier than Y will immediately be drawn. If X and Y are then properly placed on a properly constructed balance and the Y-arm sinks, O is not required to withdraw his previous conclusion that X is heavier than Y. He may conclude instead that the balance is defective. The situation just described is bizarre. To make it more so, let us suppose that in all pairs the object found by all persons to be heavier by lifting is found by all persons to be lighter by using a balance. In this event we should probably not know what to say. We might decide to regard the use of a balance as the method for determining "weight," and lifting as a method for determining a characteristic under some other name. Or, on the contrary, we might decide to regard lifting as the method for determining "weight," and the use of a balance as a method for determining some other characteristic. The point is that a decision would have to be made at that time; it cannot now be argued that we would be required, in those bizarre circumstances, to take the indications of a balance as the criteria for weight. The relation between the a- and ^-determinations in examples (1)

THE NARROW VIEW OF MEASUREMENT

147

and (2) may be compared with that between shape determinations b y sight and shape determinations by touch. In general, the shape of objects can be determined by either sight or touch: a normal person can tell that a ball is round in a dark room when it is within reach, and he can tell that it is round in a lighted room when it is not within reach. In the first instance he is not inferring that it would look round, nor in the second instance is he inferring that it would feel round. The two instances illustrate two independent methods for determining that an object is round. 22 O n e method may, however, be superior to the other. It is extremely difficult to distinguish, by touch alone, between a large table whose edges are very slightly curved and one whose edges are perfectly straight. But the problem does not exist when sight is employed. This and other similar instances suggest that sight is the superior method for determining shape. If true, however, this does not imply that the way an object looks is the criterion for its shape, or that we can only infer its shape from the way it feels. If, to a normal person, an object feels round and looks square, he is not necessarily forced to conclude that it is square, and that his sense of touch has deceptively led him to draw an incorrect inference. In so bizarre a situation, he might not know what to conclude. But no argument can now be offered that he ought then to conclude that the object is square. Empirical differences.—There is little question that many writers regard ¿-determinations as superior to «-determinations. It may be maintained that they are superior in that (i) they lead to a higher degree of observer agreement about the relative magnitude of objects in various dimensions; (ii) the certainty with which the determinations of relative magnitude can be made is greater; and (iii) a greater degree of sensitivity in the determinations is possible. I wish to show that, contrary to what may be assumed, the question of superiority in any of these three respects is an empirical one, a question of fact. i) Observer agreement.—Owing to the requirement that experiments be reproducible, intraobserver agreement and interobserver agreement are desiderata in science. Therefore, if the ¿-determinations produce a higher degree of intra- or interobserver agreement than the «-determinations, the former are more acceptable than the latter. Con22 Berkeley would not agree. See his New Theory of Vision, pars. 44, 46, 49, 55, 59, in The Works of George Berkeley, I, ed. A. A. Luce (London, 1948), 187 ff.

148

THE NARROW VIEW OF MEASUREMENT

centrating on determinations of weight, let us expand the description of (x b). A simple balance consists of a lever (of uniform width and breadth) and a fulcrum. To determine whether X is heavier than Y , support the lever by means of the fulcrum so that the lever remains freely in a horizontal position, and then place X and Y on opposite ends of the lever. If the lever remains in a horizontal position, X and Y are equal in weight. If it does not, the object resting on the end of the lever which sinks is the heavier. Will determinations of relative weight exhibit a higher degree of observer agreement when made by means of a simple balance than when made by the method of lifting? T o this factual question an apriori answer cannot be given. The outcome of weight determinations made by a simple balance depends on the care exercised in following the weighing rules stated above and on the conditions of the weighing. If, in determining the relative weight of X and Y , O i places X and Y at different distances from the fulcrum, whereas O2 places them at equal distances, their results probably will not agree. W e cannot expect agreement if some of the persons making the determinations do not adhere strictly to the weighing rules. Whether they do so or not is a factual question. Furthermore, the conditions of weighing may vary from one weighing to the next. If, for example, a piece of iron is being weighed, and during a first, but not during a second, weighing a magnet is underneath the balance, the results of the weighings will probably not agree. W e cannot reasonably expect agreement if crucial conditions are not kept constant from one weighing to the next. Whether or not this requirement will be satisfied is a question of fact. Parallel considerations obtain for relative weight determinations made by the method of lifting. Let us elaborate (1a) as follows. (The description here applies only to the determination of the relative weight of heavy objects.) To determine the relative weight of X and Y , first place X in a satchel with a handle, grasp the handle with one hand, and then stand erect. Next place Y in the satchel and lift it following the same procedure. Then judge the relative weight of X and Y. W e cannot expect observer agreement if some of the persons making the determination fail to observe the rules—for example, by lifting X with one hand and Y with two hands. Furthermore, if conditions are not kept constant from one determination to the next (the example of magnetic interference may again be used) we cannot rea-

THE NARROW VIEW OF MEASUREMENT

149

sonably expect agreement. Whether or not these requirements are satisfied is a question of fact. It is conceivable, then, that the method of lifting will produce greater intra- or interobserver agreement than the use of a balance in determinations of relative weight. Conceivable, yes, but unlikely in the light of our past experience with balances. It is also conceivable that one method of lifting, such as the one described above, will produce greater observer agreement than another, such as the method of lifting X with one hand and Y with the other. We have considered some examples of factors that may produce lack of agreement in determinations of relative weight. The method that eliminates the larger number of these factors is the one that produces the greater degree of agreement. The degree to which these disturbing factors are eliminated depends on the level of scientific knowledge and the extent to which the rules governing a determination take account of this knowledge. For example, if X and Y are placed at equal distances from the fulcrum of a balance during one weighing and at unequal distances during another, the results of the two weighings will probably not agree. A comprehensive scientific theory (the theory of levers) is available to explain this lack of agreement. Consequently, the rules governing the use of a balance require that X and Y be placed at equal distances from the fulcrum. Now consider methods of lifting. Perhaps the results of a relative weight determination in which X and Y are lifted with different hands will not agree with the results of lifting X and Y successively with the same hand. There is, unfortunately, no scientific theory that can predict agreement or disagreement between the two methods. Consequently, it is difficult to devise rules for a method of lifting which will ensure a high degree of observer agreement. Since the same difficulty does not obtain for rules governing the use of a balance, the balance method for relative weight determination has an advantage over the lifting method. But we must be clear about the nature of the advantage. It is possible that in the future a physiological theory of lifting will be developed, a theory enabling us to devise rules insuring as much observer agreement for the method of lifting as presently results from the balance method. This possibility makes it clear that the superiority of the balance method, in regard to observer agreement, is due to the present state of empirical science, and not to any

THE NARROW VIEW OF MEASUREMENT

inherent features of balances or human arms and hands. That the balance method is superior in regard to observer agreement is merely a matter of fact, since we can, by anticipating certain developments in science, imagine a situation in which the lifting method is equally if not more advantageous in this respect. ii) Certainty of determination.—In lifting objects we are frequently in doubt as to whether they are equal or different in weight. Often we feel required to say, "They may be equal, they may not be equal; I can't tell," especially when the objects in question are of unequal size and similar weight (e.g., a feather pillow and a book). The uncertainty can be reduced to some degree by, for example, placing the object in a standard satchel and lifting it by means of a handle. But even when a highly sophisticated method is employed, areas of uncertainty will probably remain. Balances (e.g., the simple one described earlier) are, of course, neither certain nor uncertain about the weights of objects placed on them. The question is, rather, whether the use of a given balance will enable a person to say categorically, in a significantly large number of cases, that objects are or are not equal in weight. Even a person skilled in using balances may be uncertain as to whether the objects have been placed at equal distances from the fulcrum, or uncertain as to whether the balance is sensitive enough to record a weight difference in objects of the kind being weighed. For these reasons it is entirely conceivable that an observer may (at least in certain instances) feel required to check the results of a balance determination by the method of lifting the objects. The following objection may be made. Although we may be uncertain as to whether weighings made on a balance are correct, still the instrument indicates (correctly or incorrectly) either that the objects placed on it are equal in weight or that they are unequal in weight, with no third possibility admitted. With the lifting method, on the other hand, we may find either that the objects are equal, or that they are unequal, or that their relative weight is indeterminate. This objection cannot stand as stated. As weighed on the simple balance described earlier, X is heavier than Y if the end of the lever supporting X sinks from its original position. But there may sometimes be doubt as to whether that end has or has not sunk. A slight sinkage may be extremely difficult to detect, so that occasionally an observer may feel

THE NARROW VIEW OF MEASUREMENT

required to say, "It may have sunk, it may not have sunk; I simply can't tell." He could then not say either that X is heavier than Y or that it is not, for he would find the relative weight indeterminate. There may be, then, an area of uncertainty in balance determinations, just as there may be (and is) an area of uncertainty in lifting determinations. Nonetheless, the use of a method other than lifting may reduce the area of uncertainty. For example, suppose that some relative weight determinations made by lifting are certain and some are uncertain. Suppose further that all the certain determinations are corroborated by using a simple balance. If the balance gives a clear indication of relative weight in some determinations that are uncertain with the lifting method, the balance can be used to reduce the area of uncertainty, providing it is properly constructed and skillfully used. No apriori judgment can be made regarding the usefulness of the balance in reducing the area of uncertainty. Furthermore, there is no reason to assume that only an instrument can have this advantage. As noted earlier, a sophisticated lifting method for relative weight determination may produce fewer uncertain results than an unsophisticated balance method. Whether it does or not is a question of fact which cannot be settled by apriori methods. iii) Sensitivity.—It may be said that the b-methods are more sensitive than the «-methods, that is, that methods of the fa-type can detect finer differences than methods of the «-type. Suppose ten stones are selected arbitrarily and labeled A, B, . . . J. Suppose further that most observers, employing the lifting method of relative weight determination, rank the stones in the following ascending order of weight, (a) A

F

5 D E G I J,

H with B and C ranked equal in weight and F, G, and H ranked equal in weight. Suppose also that most observers, employing a simple balance, rank the ten stones in the following ascending order of weight, c

(b) A B C D E £ H I J, with only F and G ranked equal in weight. The fact that more differences in weight are indicated by the balance method is not, in itself, a sufficient reason for regarding it as

152

THE NARROW VIEW OF MEASUREMENT

the more sensitive. Suppose that by using the lifting method most observers rank the stones as in (a). Now consider a new method for relative weight determination. The observer places his hand palm upward on a table, and places first one stone and then another in his palm —the heavier the stone, the more pressure on the hand. Suppose most observers using this method rank the stones as in (b). There is no inclination to regard the palm method as the more sensitive, even though it produces order (b). In short, simple comparison of ranking orders does not tell which of two methods of ranking is the more sensitive. To say that one method is more sensitive is to imply that it produces more nearly correct results than the other. And a mere comparison of (a) and (b) cannot tell which set of results is the more nearly correct. What, then, would be an adequate reason for regarding the balance method as more sensitive than the lifting method? We can, by lifting, easily detect a difference in weight between a large bag of sand and a small bag of sand. By lifting, however, we cannot detect a difference in weight between a large bag of sand and the same bag with one grain added. Nevertheless, there is a difference, even if it cannot be detected, for grains of sand are not weightless; otherwise the bag, which is a collection of grains of sand, would be weightless. Consequently, if a balance indicates an increase in weight in the bag after the grain has been added, we are entitled to regard the balance method as more sensitive than the lifting method (on condition that weighings with the balance agree in general with lifting determinations). It is important to note that a given balance method may be less sensitive than a particular method of lifting. It may be less sensitive, for example, when the lever is heavy in relation to the objects being weighed, or when the part of the fulcrum which makes contact with the lever is not sharp. It is also noteworthy that one method of lifting may be more sensitive than another. For example, lifting objects in a standard satchel may be a more sensitive (or less sensitive) method than that of lifting objects in the hands. Whether one method is more sensitive than another is a question that cannot be settled apriori. Possible tests for equality and inequality.—To return to the original question of this section: Must equality and inequality of objects be determined by methods of the b-type in a procedure of measurement? Must G and £ (the relational terms in the axioms of measure-

THE NARROW VIEW OF MEASUREMENT

153

ment) be given definitions of the b-type? Consider the following system of weight measurement: (A) A unit object is selected. A number of other objects, each of which feels equal in weight to the unit object, are selected by lifting. Measurement of a given object, X, is accomplished by lifting X and by lifting collections of unit objects, until a collection that feels equal in weight to X is discovered. The weight of X is obtained by counting the number of unit objects in the requisite collection. Now consider a parallel system: (B) A unit object is selected. A number of other objects, each equal in weight to the unit object, are selected by means of a balance. Measurement of a given object, X, consists in discovering a collection of unit objects which balances X. The number of objects in this collection is the weight of X in terms of the selected unit. The analysis is in the preceding section shows there is no good reason for refusing to call procedure (A) one of measurement, although determinations of equality and inequality are made by a method (lifting) of the a-type, and although C and E are defined by definitions of the a-type. " X feels heavier than Y " is not subjective in the sense that it does not (or need not) express the observer's uncertainty; it is not incorrigible; and it is neither an estimate nor a prediction of the results of a (possibly superior) balance determination of relative weight. In these respects it compares favorably with any of the observation statements that report the data of the sciences. It may be claimed that the determination of relative magnitude by means of a balance is superior to determination by means of lifting —superior in observer agreement, certainty of determination, sensitivity, or in all three—and that, therefore, procedure (B) is superior to procedure (A). And this claim may very well be true. There is a strong presumption that at least those methods employing balances of the kind found in scientific laboratories (chain balances, for instance) are superior in each of these respects. But it is important to understand that if a claim for the superiority of procedure (B) is true, it is true as a matter of fact. And it is absolutely essential to be clear about what does and does not follow from the truth of this claim. It does not follow that (A) is not properly regarded as a procedure of measurement; it

154

THE NARROW VIEW OF MEASUREMENT

follows only that it is a less acceptable procedure of measurement than some other. An example will make the point clear. The ordinary chemistry laboratory usually has several types of balance: some that are calibrated to the nearest thousandth of a gram, some that are calibrated only to the nearest tenth of a gram. That a procedure employing the latter type of balance is less sensitive than a procedure employing the former does not entail that the former procedure is not properly regarded as one of measurement. THE CONCEPT OF AN OPERATION

One of the most popular, and at the same time most confused, beliefs about measurement in particular and science in general is that it is "operational." On the basis of this belief, scientific procedure is contrasted with the procedures used in common life to make quantitative and other determinations, which latter are said to be "subjective" and "imprecise." Scientific procedure, by virtue of its operational character, is thought to be "objective" and "precise." But what is an "operation"? What are operations to be contrasted with? That is to say, what are nonoperations? And how does the use of operations ensure that a procedure will be objective and precise? Many theorists insist that scientific terms be "operationally defined." What is an "operational definition"? What kind of definition is it being contrasted with? And in what way are operational definitions advantageous? It is probably not an accident that these questions have never been adequately answered, that the concept of an operation has never been properly analyzed. For the term "operation" has a curious, but nonetheless important, use. Some scientists use it in a sort of incantation to exorcise the demon of "subjectivity." Some experimenters believe that if they can specify the "operations" they employ, in measurement and elsewhere, then their experiments are automatically unobjectionable. Some theorists believe that if their theoretical terms can be "operationally defined," their theory is automatically precise and fruitful. It takes only a moment's reflection to see that these beliefs are false. Operations and operational definitions can be self-contradictory (e.g., "If a liquid turns litmus red it is acid; if it turns litmus red it is not acid") or vacuous (e.g., " I f a liquid turns litmus red it is either acid or not acid"). More realistically, operations can be specified, and operational definitions given, in vague and imprecise ways. More

THE NARROW VIEW OF MEASUREMENT

155

important still, operations and operational definitions may be unprofitable. An experiment may be "operational" and still be irrelevant to any scientific theory, or be of no importance to anyone but the experimenter. An operational definition of a theoretical term is usually useless if it divests the term of any conceptual connections with all other theoretical terms. These completely obvious remarks do not constitute anything like a full analysis of the concept of an operation. It is beyond the scope of this book to expose all the superstitions surrounding this concept, since only those operations involved in measurement are under examination. Furthermore, it cannot be assumed that the concept contains nothing but superstition. What lies in the minds of most writers, when they use the term "operation," is some procedure employing an instrument. And it is obvious that science, as we know it today, is almost wholly based on the use of instruments. Even here there is unclarity, for the concept of an instrument is obscure and stands in need of analysis. Moreover, the concept of an operation seems to be broader than that of an instrument. Some operational procedures involve instruments and some do not. There would seem to be at least two senses of the technical term "operation." In the first sense, operations are to be contrasted with concepts. Concepts are given empirical content by specifying operations, that is to say, by providing operational definitions for them. The concept of adding rods is given empirical content by means of the operation of endwise juxtaposition, and may possibly be given content by means of some other operation on rods. The concept of weight equality is given empirical content by means of the operation of balancing objects, and can also be given content by means of the operation of lifting them. In this sense, both the «-determinations and the bdeterminations discussed earlier are operations. In any correct view of measurement, measurement necessarily requires operations in this sense, since it consists in applying certain quantitative and arithmetical concepts to objects within some dimension. It is not possible to apply these concepts unless they can be given empirical content. And supplying this empirical content is equivalent to specifying operations, in the present sense, for the concepts. This point, however, provides no argument for the classical view of measurement. It does not specify the quantitative and arithmetical con-

i56

THE NARROW VIEW OF MEASUREMENT

cepts that must be applied in measurement, nor does it tell us that the concept of addition is one of these. In the second sense, operations are to be contrasted with perceptual determinations. More precisely, operational procedures of verification—the ^-determinations discussed earlier—are to be contrasted with nonoperational procedures of verification—the «-determinations. Thus, aligning rods (a noninstrumental operation) and balancing objects (an instrumental operation) are operational procedures for verifying statements about length and weight equality and inequality. Simply looking at rods and simply lifting weights are nonoperational procedures for verifying the same statements. In this sense of "operation," there is nothing with which to contrast such operations of addition as juxtaposing rods and placing objects on a single pan of a balance. In no correct view of measurement does measurement necessarily require operations in this sense. It is possible to describe and use a system of measurement, even one involving the addition of units, which employs nonoperational procedures for determining equality and inequality. Such a system may be less advantageous than one employing operational procedures, but it is nonetheless measurement. Many writers who hold the classical view of measurement state or imply that G and E in the axioms of measurement must be operationally defined. In the present sense of the term "operation," this claim is false. One last point. The two senses of "operation" distinguished above are connected in an important but deceptive way. Although operations are contrasted in the one sense with concepts, and in the other sense with perceptual determinations, an operation is in either sense a method for pinning a concept down to empirical reality. This connection probably explains why the ambiguity in the term usually goes unnoticed. And it explains why it is easy to pass from the correct view that measurement is operational in the first sense, to the incorrect view that measurement is operational in the second sense.

5 Broader Views of Measurement The preceding chapter criticizes the classical view of measurement, which holds that measurement is the assignment of numerals to objects within a dimension by means of the physical addition of unit objects. It is called the narrow view because, by comparison with the views examined in this chapter, it prescribes the most limited application of the term "measurement." Although the classical definition is unacceptably narrow, other definitions have been proposed which are just as unacceptable because they are too broad. In this chapter some of these unacceptably broad definitions are examined and criticized, and then a definition of measurement which seems to satisfy the criterion of acceptability stated below is suggested. The first view I criticize is usually associated with S. S. Stevens, and it has gained wide acceptance. Stevens defines measurement as any rule-governed procedure for assigning numerals to things; different rules for assigning numerals are said to create scales of measurement, and scales are distinguished in terms of the transformations that leave the scales invariant. I argue that Stevens' definition is implicitly circular, and that it begs such questions as: Is loudness measurable? I argue also that the definition is at once unacceptably narrow and unacceptably broad. The second view I criticize is proposed by Hempel, who defines measurement as any procedure for univocally assigning numerals to things. I argue that this definition is unacceptably broad even on Hempel's own terms. I propose that measurement can be acceptably defined as a procedure for assigning numerals to things by empirically comparing them with a unit. This definition sets the stage for asking whether loudness and pitch—and, by extension, similar dimensions—are capable of measurement, and, later, for asking whether dimensions of sensations are capable of measurement.

i58

BROADER VIEWS OF MEASUREMENT

REQUIREMENTS FOR AN ACCEPTABLE VIEW An acceptable definition of measurement is one that is as broad as it can be without doing undue violence to either the ordinary meaning or the technical meaning of the term. The definition should be as broad as possible (consistent with ordinary and technical meaning) in order to encompass the question as to whether loudness and pitch are measurable. It seems likely that, if measurable, these dimensions (especially pitch) cannot be measured by procedures exactly like those normally employed for length and weight. It might therefore be easy to "prove" that loudness and pitch are incapable of measurement by devising a certain kind of definition—one that is as narrow as it can be without implying that length and weight are incapable of measurement. For instance, the definition might specify that any procedure of measurement must employ an instrument. Length and weight qualify (a ruler for the former, a balance for the latter), but loudness and pitch apparently do not. To conclude that the latter dimensions are not measurable on the basis of this definition would be illegitimate. Length can be measured by juxtaposing equal unit rods and comparing the juxtaposition with the object to be measured. This procedure employs no instrument. A broad definition is needed, then, but it should not do undue violence to either the technical or the ordinary meaning of the term. I readily admit that mistakes may be made in a zealous attempt to adhere to ordinary use. Some of the facts about the ordinary use of the term "measurement" are of no importance for this study. For example, in ordinary contexts we frequently speak of measurement in which the use of numerals is not implied. When one stretches and marks a string between two door facings, he is said to have measured the width of the door (perhaps to determine whether a piece of furniture can be carried through it). To take another example, the term "measurement" is rarely applied in nontechnical contexts to weight and time; it is applied chiefly to length and derivative dimensions (e.g., area, volume). We speak of weighing objects but not of measuring their weight, of timing a footrace but not of measuring its time. It might be said that the second example presents "inessential linguistic facts." But how are the "essential" facts to be distinguished from the "inessential" facts? It might be said that the first example illustrates a "loose" sense of the term

BROADER VIEWS OF MEASUREMENT

159

"measurement," whereas in the "strict" sense measurement is a certain type of procedure for numeral assignment. But what justification is there for regarding one rather than the other as the "loose" sense? The linguistic facts cited above and the problems they raise (if they do raise any) are of no importance to this study. The significance of the question—Are loudness and pitch measurable?—cannot be fully understood apart from its scientific context; and within this context it is of no importance that in ordinary speech "measurement" is sometimes applied to procedures that do not involve the use of numerals, or that ordinarily we do not speak of "measuring" weight and time. The significance of the question derives in large part from the fact that the procedures the scientist calls "measurement" are designed to enable (and usually do enable) him to discover numerical laws relating the dimensions measured. A procedure that does not involve the use of numerals obviously cannot have this result. Hence, we desire a definition that qualifies only those procedures that involve the use of numerals as procedures of measurement. Whether such a definition excludes a few procedures ordinarily called measurement is not important. Procedures of numeral assignment are available for length (and derivative dimensions), weight, and time, and they have made possible the discovery of a host of numerical laws relating these dimensions. In view of this fact, and in view of the similarity among the various procedures, it would be foolish to insist, on the basis of ordinary use, that weighing objects and timing the fall of bodies are not procedures of measurement. The reader may wonder then, why we do not concern ourselves exclusively with the technical meaning of "measurement." A central question of this essay is whether the so-called psychological dimensions are measurable. The interest of the question derives in large part from an apparent dissimilarity between such dimensions as loudness and pitch, on the one hand, and length and weight on the other; from the fact that loudness and pitch do not seem to be measurable in the ordinary sense in which length and weight are measurable. An opponent may assert that loudness and pitch are incapable of measurement because "You can't lay a meterstick on tones" or "Pitch is not additive." Suppose, in an attempt to defeat his view, we first define measurement as a procedure of numeral assignment which observes the following rule: To different objects or magnitudes assign different

16o

BROADER VIEWS OF MEASUREMENT

numerals, to the same object or magnitude assign the same numeral (thus, the assignment of numerals to houses in a street qualifies as measurement). We then point out that numerals can be assigned to pitch and loudness so as to satisfy this rule, and that these two dimensions are therefore measurable. The opponent will not be and should not be satisfied. His reply will be something like the following. "I said that loudness and pitch are not measurable. I did not deny that numerals can be assigned to these dimensions so as to satisfy the rule: To different magnitudes assign different numerals, to the same magnitude assign the same numeral. This procedure of numeral assignment is not properly called one of measurement. When I say that loudness and pitch cannot be measured, I mean that they cannot be measured in the sense in which length and weight can be measured. You must show me procedures of numeral assignment for loudness and pitch comparable to those available for length and weight, which are normally and quite properly called measurement." This reply shows that the importance of the question—Are loudness and pitch capable of measurement?— depends on the ordinary meaning of the term "measurement." That is to say, in the context of the question as it normally arises, "measurement" is used in its ordinary sense. To define the term in a manner that does undue violence to its ordinary meaning, and then to argue that loudness and pitch are capable of measurement, leaves an opponent with the impression that he has been cheated, that his question has not been answered. This is precisely the impression Stevens leaves us with when he defines measurement as any rule-governed procedure for assigning numerals to things so as to represent facts and conventions about them, and then asserts that the troublesome question—what is measurement?— reduces to the manageable question: What are the rules for various procedures of numeral assignment? We are left with the feeling that Stevens has not addressed the question of whether loudness, pitch, and so on, are measurable, that he has instead sidestepped it.1 As a question of philosophical interest, then, the question—Are loudness and pitch capable of measurement?—presupposes, in at least a rough way, the ordinary meaning of the term "measurement." This 1 S. S. Stevens, "Mathematics, Measurement, and Psychophysics," in Handbook of Experimental Psychology, ed. S. S. Stevens (New Y o r k , 1951), p. 29. See the next section of this chapter.

BROADER VIEWS OF MEASUREMENT

l6l

consideration, among others, shows that we need pay no attention to a protest that the question is a scientific one and that we ought to concern ourselves exclusively with the technical meaning of "measurement." What the measurement of a given measurable dimension will reveal (e.g., by what numerical laws it is related to other dimensions) is exclusively a scientific question. But whether a given dimension is measurable is as much a philosophical question as a scientific one. Psychologists occasionally claim to have discovered and developed unusual procedures for measuring such so-called psychological dimensions as loudness and pitch. Fechner's procedure, which employs the just noticeable difference as a unit of magnitude, is an example (see chap. 8). Although it is not the philosopher's task to devise and employ such procedures, it is within his province to ask whether they are properly called procedures of measurement for the relevant dimensions. Scientists are not automatically immune, even with their professional training and experience, from conceptual confusion. And it is entirely conceivable that a physicist or a psychologist will announce (perhaps as a startling discovery) that he has found a procedure of measurement for loudness and pitch, when the procedure cannot properly be subsumed under that concept. In this connection a warning by Campbell, himself a physicist, is instructive: [A philosopher] will suppose that the logical analysis of measurement is familiar to every physicist who actually measures, and he will not expect me to say anything that is not to be found in every competent textbook. He is reminded therefore that most physicists have a horror of logic and regard an accusation that their doings conform to logical principles as a personal insult. The most distinguished physicists, when they attempt logical analysis, are apt to gibber; and probably more nonsense is talked about measurement than any other part of physics. When an international congress meets to discharge the dull but necessary duty of fixing the conventions of measurement (which duty it performs admirably), a flood of incomprehensible verbiage about "units and dimensions" is let loose, which leaves everyone even more muddled than they were before. The only conclusion that could be drawn from "competent" textbooks is that there are no principles of measurement. 2

My criterion for an acceptable definition of measurement is admittedly a difficult one to employ, partly because of vagueness in the * Norman Robert Campbell, "Symposium: Measurement and Its Importance for Philosophy," Proceedings of the Aristotelian Society, Supplement, 17 (1938), 121.

162

BROADER VIEWS OF MEASUREMENT

phrases "undue violence" and "ordinary meaning." Certain definitions clearly do undue violence to the ordinary meaning of "measurement"; an example is "Measurement consists in the assignment of numerals to things or properties" (too broad). But it is not always so easy to apply the criterion. The phrase "ordinary meaning" has been so troublesome that many philosophers find it useless, partly because of its inherent vagueness. But vagueness does not necessarily make a term useless. Occasionally a political scientist, a political analyst, or a pollster refers to the opinions or attitudes of, and the information possessed by, the "man in the street." The intent of the phrase is clear: it is to distinguish the class of professional politicians, diplomats, tax experts, economists, and so on, from the class of nonprofessionals. Of course, "man in the street" is a vague phrase; nonprofessionals usually have some of the attitudes or opinions of, and the information possessed by, the professionals, and vice versa. But it does not follow that the phrase is meaningless, or even useless. The foregoing provides an analogy for understanding the phrase "ordinary meaning" as it is used here. The technical meaning of "measurement" is determined by its use among scientists: physicists, psychologists, and the like. Its ordinary meaning is determined by its use among nonscientists. As in the analogy, "ordinary meaning" is a vague phrase, since some scientists use the term "measurement" much as nonscientists do, and vice versa. It should be apparent by the end of this book, however, that the notion of ordinary meaning is useful. It is important (especially for those oriented toward science) to note that the terms "ordinary meaning" and "technical meaning" are, as we use them, antonyms, neither of which can be understood apart from the other, each of which is as vague or as precise as the other. A related point is that establishing the technical meaning is sometimes as difficult as, or more difficult than, establishing the ordinary meaning of a term. "Measurement," for example, is defined and used by physicists, psychologists, and philosophers of science sometimes in strikingly different, sometimes in subtly different, ways. Whoever expects to find a single, wellestablished definition current among specialists in all these fields will be disappointed. Disputes as to the possibility of measuring sensations (sensory events) have been sufficiently acute within the past three or four decades that one learned society—the British Association for the Advancement of Science—appointed a special committee of phys-

BROADER VIEWS OF MEASUREMENT

163

icists and psychologists to study the matter. This committee devoted most of its time to what may fairly be called a discussion of the meaning of the term "measurement". Its reports revealed striking disagreement among its members, who failed to reach a mutually acceptable conclusion.3 STEVENS' VIEW It is easy to obtain the impression that controversy over the nature of measurement is verbal in character, and that different views of measurement are little more than the proponents' preferences, grounded in their special areas of scientific competence, to use the term in a certain way. Having formed this impression, one may be inclined to define measurement so as to avoid the allegedly verbal disputes. This motivation partly underlies the theory of measurement provided by S. S. Stevens. THE VIEW PRESENTED

Stevens begins his theoretical treatment of measurement with a discussion of the question as to whether it is possible to measure sensations, that is, sensory events. He notes that, after seven years of deliberation on this question, the members of a special committee of the British Association for the Advancement of Science "came out by that same door as they went in," one faction stoutly maintaining that sensory events are capable of measurement, another faction taking the opposite position, primarily on the ground that no "meaning can be given to the concept of addition as applied to sensation." Stevens then says: It is plain from this and from other statements by the committee that the real issue is the meaning of measurement. This, to be sure, is a semantic issue, but one susceptible of orderly discussion. Perhaps agreement can better be achieved if we recognize that measurement exists in a variety of forms and that scales of measurement fall into certain definite classes. These classes are determined both by the empirical operations invoked in the process of "measuring" and by the formal (mathematical) properties of the scales. 4 8 A. Ferguson et al., "Quantitative Estimates of Sensory Events," The Advancement of Science, 1 (1939-40), 331-349. 4 S. S. Stevens, "On the Theory of Scales of Measurement," Science, 103 (1946), 677.

164

BROADER VIEWS OF MEASUREMENT

Stevens believes that futile verbal disputes can be avoided, and that the problems that generate them can be resolved by adopting a liberal definition of measurement: . . . the most liberal and useful definition of measurement is . . . the assignment of numerals to things so as to represent facts and conventions about them. . . . The problem of what is and what is not measurement then reduces to the simple question: What are the rules, if any, under which numerals are assigned? If we can point to a consistent set of rules, we are obviously concerned with measurement of some sort, and we can then proceed to the more interesting question as to what kind of measurement is it.5

This is a proposal that measurement be defined as broadly as possible: Definition 2. Measurement is the assignment of numerals to objects or events within a dimension according to some consistent rule or set of rules. Since a rule of numeral assignment creates what Stevens calls a "scale," he develops his proposal by distinguishing four different types of scales. Brief descriptions of each of these, in the manner of his presentation, follow.® 1) Nominal scale.—This type of scale is used to identify objects or classes of objects. For example, numerals are assigned to the houses on a street, and to the players on a football team. The rule of assignment here is simply: Assign numerals in such a way that there is a oneone relation between objects (houses, players, etc.) and numerals. For the purposes served by a nominal scale, symbols possessing a conventional order (e.g., numerals or letters of the alphabet) need not be employed. 2) Ordinal scale.—Scales of this type indicate the order or rank of objects within a dimension. Mohs's scale for the hardness of minerals is an example. The rule for assigning numerals here is: To the greater (lesser) degree assign the higher (lower) numeral. Within these limits any assignment is possible. Instead of the numerals i , 2, 3, . . . 10, Mohs could have used 1, 10,100, . . . 1,000,000,000 to represent different degrees of hardness. The only difference is that the second 5 Stevens, "On the Theory of Scales of Measurement," p. 680. * What follows is primarily a summary of Stevens, "Mathematics, Measurement, and Psychophysics," pp. 25-29, and Stevens, "On the Theory of Scales of Measurement," pp. 678-680.

BROADER VIEWS OF MEASUREMENT

16 5

series would have been more cumbersome. These different possibilities for numeral assignment show that the intervals between successive points on an ordinal scale cannot be regarded as either equal or unequal, and that it is improper to regard a mineral of hardness 1 0 as twice as hard as one of hardness 5. 3) Interval scale.—As we have just seen, in one sense the assignment of numerals creating an ordinal scale is arbitrary. The rule that creates an interval scale does not permit, in this sense, the arbitrary assignment of numerals. The Fahrenheit and centigrade scales of temperature serve as examples. (It is important to keep in mind that these are generally considered scales of indirect, rather than direct, measurement.) Since such scales have an "arbitrary" zero point, no meaning can be given to saying that objects so scaled are, in terms of the scale values, twice, half, or three times as large as others. On the centigrade scale the zero point is the temperature of freezing water; on the Fahrenheit it is 32 degrees below the temperature of freezing water. If judgments of relative temperature were predicated on the centigrade scale, the boiling point of rubidium (700 degrees) would be seven times that of water (100 degrees). If such judgments were predicated on the Fahrenheit scale, the boiling point of rubidium (1,292 degrees) would be approximately six times that of water (212 degrees). Such ratio judgments are as arbitrary as the choice of zero which produces them. 4) Ratio scale.—The numerals of a ratio scale represent order, the equality of intervals, and the equality of ratios. The centimeter scale of length illustrates these features. A length of 2 centimeters is greater than a length of 1 centimeter (order); the length between the 1 - and the 2-centimeter points is equal to that between the 2- and the 3-centimeter points (equality of intervals); and a length of 1 centimeter stands to a length of 2 centimeters as a length of 2 centimeters stands to a length of 4 centimeters, that is, in the ratio of 1 : 2 (equality of ratios). In a ratio scale an absolute zero is implied; otherwise the scale numerals would not represent equality of ratios. This tidy classification is not so free from difficulty as it might seem. For example, there is some uncertainty as to whether an interval scale should be regarded as a scale of equal intervals. In one place Stevens contrasts interval and ordinal scales in this way: "Whenever, in addition to knowing how to order a set of items, we are able to devise a rule telling us when to assign the numeral adjacent to the one

166

BROADER VIEWS OF MEASUREMENT

previously assigned, we possess the makings of what we might call an intensive [earlier term for "interval"] scale." 7 He then goes on to point out that " a rule for assigning adjacent numerals does not necessarily create a unit of measurement." Thus the rule of assigning adjacent numerals to degrees (of loudness, brightness, etc.) which are just noticeably different from one another does not necessarily create a unit of measurement, since just noticeable differences may not be equal. In later essays Stevens adopts a different position, listing "the determination of equality of intervals or differences" as the "basic empirical operation" for an interval scale.8 In his latest discussion of scales of measurement he takes the same position, although not with complete confidence, when he attacks the suggestion that there is a sort of interval scale intermediate between the ordinal scale and the equalinterval scale.9 Bergmann and Spence maintain that the question of equality of intervals in scales of this type is, in one sense, meaningless. Both Fahrenheit and centigrade scales of temperature have advantages over ordinal scales of this dimension, but, according to these authors, the advantages have not always been correctly described. Their view is that we prefer those scales whose "expression [is] . . . intuitive and (or) . . . mathematically simple" and which "play or which we expect to play a prominent role in our ever changing theoretical structure." "It is," they claim, "an inaccurate and misleading way of speaking when such choice of scales is described as an attempt to equalize the unit distances of the scale." 10 The difficulty here is related to, if not identical with, the problem of whether Stevens' fourfold classification of scales is exhaustive. In one essay he considers the possibility of a fifth type of scale, one based on "the three empirical operations: determination of equality, determination of greater or less, and determination of equal ratios," but not on any "empirical operation for determining equal intervals." He calls 7 S. S. Stevens and J. Volkmann, "The Relation of Pitch to Frequency: A Revised Scale/' American Journal of Psychology, 53 (1940), 530; see also p. 331. 8 Stevens, "On the Theory of Scales of Measurement/' p. 678, table 1 ; Stevens, "Mathematics, Measurement, and Psychophysics," p. 25, table 6. See also J. P. Guilford, Psychometric Methods (rev. ed.; New York, 1954), p. 14. ® S. S. Stevens, "Measurement, Psychophysics, and Utility," in Measurement: Definitions and Theories, ed. C. West Churchman and Philburn Ratoosh (New York, 1959), pp. 35-36. 10 Gustav Bergmann and K. W. Spence, "The Logic of Psychophysical Measurement," in Readings in the Philosophy of Science, ed. H. Feigl and May Brodbeck (New York, 1953), pp. 1 0 9 - 1 1 0 .

BROADER VIEWS OF MEASUREMENT

this a logarithmic interval scale, since the mathematical transformation that leaves the scale invariant involves raising each number of the scale to some power. Stevens concedes that such scales have so far proved useless to science, but he holds out the possibility that they may in future find application in certain areas of psychophysics. 11 A fourfold, or even a fivefold, classification of scales is probably not exhaustive. It may be possible to distinguish a very large number, perhaps even an indefinitely large number, of scale types. Since this possibility is not directly relevant to the criticism later brought against Stevens, there is no need to examine it further. A s for the question of whether an interval scale is a scale of equal intervals, it is clearly possible to specify a rule of numeral assignment which creates a scale of equal intervals, as is shown in the discussion below. Since it is this type of scale that Stevens seems to have in mind, and since the possibility of another type of interval scale does not affect later assessment of Stevens' view, this question is not pursued further. Stevens does not explicitly formulate the rules of numeral assignment for the various scales (except for the nominal). Instead, he lists in a chart the "basic [empirical] operations needed to create a given scale," as follows. "Nominal scale—determination of equality, ordinal scale—determination of greater or less, interval scale—determination of equality of intervals, ratio scale—determination of equality of ratios." 12 This list suggests the following formulations. [1] To equal objects assign the same number (nominal scale). [2] To the greater (smaller) object assign the greater (smaller) number (ordinal scale). [3] T o objects separated by equal intervals assign numbers separated by equal intervals (interval scale). [4] T o objects that stand to one another in equal ratios assign numbers that stand to one another in equal ratios (ratio scale). N o one of these rules is logically implied by any of the others; that is, it is logically possible to satisfy any one of the rules without satisfying Stevens, " M e a s u r e m e n t , P s y c h o p h y s i c s , and U t i l i t y , " pp. 3 1 - 3 4 . Stevens, " M a t h e m a t i c s , M e a s u r e m e n t , and P s y c h o p h y s i c s , " p. 25; see a l s o Stevens, " O n the T h e o r y of Scales of M e a s u r e m e n t / ' p. 678, table 1. Patrick Suppes, " M e a s u r e m e n t , Empirical M e a n i n g f u l n e s s , and T h r e e - V a l u e d L o g i c , " in Measurement: Definitions and Theories, ed. C h u r c h m a n and R a t o o s h , p . 1 3 1 , adopts Stevens' device of distinguishing t y p e s of measurement in terms o f the transformations appropriate to each type. 11

12

168

BROADER VIEWS OF MEASUREMENT

any other. Consequently, it is possible to imagine a scale that employs only one of the rules. Stevens, however, intends the list of rules to be cumulative. He defines the four scales in such a w a y that if a scale employs a given rule on the list then it also employs every rule listed above it. This requirement might be expressed by saying that higher scales include lower ones, but not vice versa. The ratio scale thus includes all scales. Most ratio scales assign a unit number to some standard object, whose magnitude is then taken to be the unit for the scale. In the centimeter scale of length this standard object is the standard meter in Paris, and the number assigned to it is 1 (when the unit is the meter) or 100 (when the unit is the centimeter) or 1,000 (when the unit is the millimeter). Length, as we say, has a zero, but the centimeter scale does not assign the number o to the length of any object. Most interval scales assign both the number o and a unit number to some objects among those scaled. In the centigrade scale for temperature o is assigned to freezing water and 100 to boiling water. To take account of scales of these familiar types, two more rules must be added to the four above. [5] To some object assign the number o (rule for zero). [6] To some object assign some number greater than o (rule for a unit). The centimeter scale of length employs each of the six rules except [5]; the centigrade scale of temperature, each of the six except [4]. Leaving aside the question of whether the above list of rules is exhaustive, Stevens' definition of measurement can now be stated precisely, as follows. Measurement is the assignment of numerals to objects or events according to some one or more of rules [1] through [4], with or without the additional employment of rules [5] and [6]. D i f ferent types of measurement are distinguished on the basis of the rules employed: the highest type of rule employed determines the type of measurement. These rules create (or, perhaps we should say, are) scales, so we can also distinguish types of scales. A scale is classified on the basis of the highest type of rule involved. For example, a scale created by rule [3] but not rule [4] is an interval scale. Measurement can be alternatively defined, therefore, as the construction of a scale of some type for objects or events. Stevens says that each of his scales "is best characterized by its

BROADER VIEWS OF MEASUREMENT

169

range of invariance—by the kinds of transformations that leave the 'structure' of the scale undistorted."18 He lists the transformations that leave each of the four types invariant. Nominal scale: any one-one substitution of numerals is legitimate. This transformation implies that the numerals of such a scale do not represent even the order of the objects to which they are assigned. Ordinal scale: any one-one substitution of numerals which leaves the original order unchanged is legitimate. This transformation implies that the intervals of the scale may not be regarded as equal. Interval scale: to each number may be added a constant. This transformation implies that the scale zero is conventional and that the numbers of the scale do not establish ratios among the objects to which they are assigned. Ratio scale: each number may be multiplied by a constant. The transformations in this list are cumulative in the reverse order listed; that is, a transformation that leaves a given type of scale invariant also leaves a scale of a lower type invariant. For example, multiplication by a constant leaves a scale of any of the four types invariant. Stevens' discussion of the concept of invariance14 is not helpful. As one example of the use of the concept, he points out that projective geometry is in large part the study of invariant relations. If a figure is drawn on a rubber sheet and the sheet is then stretched, the order of the intersections in the figure is left invariant, although areas, lengths, and angles are not. As Stevens sees it, the notion of invariance has extremely broad application. Invariant relations are discovered and studied by every science, for every science is interested in discovering laws that are true "in all sorts of reference systems," true "under a wide assortment of conditions." Newton's second law of motion and the law of conservation of energy are given as examples, but they throw little light on the notion of invariance as it applies to scales of measurement. Indeed, the "invariance" exhibited by physical laws, such as the second law of motion, under certain transformations seems positively disanalogous to the invariance exhibited by certain scales under certain transformations. The former transformations are changes in the circumstances and reference systems under which the law is tested, whereas the latter transformations are changes (substitutions) of the numerals that constitute the scale of measurement. is Stevens, "Mathematics, Measurement, and Psychophysics," p. 23. 14 Ibid., pp. 1 9 - 2 1 .

BROADER VIEWS OF MEASUREMENT

T o say that a given scale, S, defined b y rule R, remains invariant under a certain transformation, T , should be taken to mean that a n y assignment of numbers to objects which satisfies R can be transformed into another assignment to those objects which satisfies R b y substituting new numbers for those in the original assignment according to the rule of T . T o illustrate, a ratio scale, defined by rule [4], remains invariant under the rule of multiplication b y a constant. T h u s , if the assignment of 2, 5, 7, n , respectively, to X, Y , Z , U , respectively, satisfies rule [4], then the assignment of 4 , 1 0 , 1 4 , 22, respectively, to X, Y , Z , U, respectively, also satisfies rule [4], since the second assignment is obtained b y substituting n e w numbers for those in the original assignment b y the rule of multiplication b y 2. This precise definition of scale invariance reveals one respect in w h i c h Stevens' treatment of that concept is confused. W h e t h e r a scale is invariant under a given transformation is a mathematical question and not, as Stevens suggests, an empirical one. That a ratio scale remains invariant under multiplication b y a constant can be mathematically deduced from rule [4] which defines the scale. It is also important to note that the invariance transformations for a given scale depend solely on the rule that defines the scale. Simply to identify a group of objects together with the numerals that have been assigned to them provides insufficient information to determine the invariance transformations for that scale; w e must k n o w , in addition, the rule b y which those numerals have been assigned. Invariance is, then, a mathematical property of rules for numeral assignment. T o say that such a rule is invariant under transformation T is to say that if one assignment satisfies the rule then so will another obtained from the first b y means of T . If invariance is a property of rules of numeral assignment, h o w are w e to assess Stevens' practice of calling scales invariant? B y " s c a l e " m a y be meant (a) a particular assignment of numerals to objects, (b) a rule for assigning numerals to objects, or (c) a particular assignment together with the rule of assignment. T h e term is used in sense (a) w h e n the assignment of numerals to minerals proposed b y M o h s — 1 to talc, 2 to g y p s u m , 3 to calcite, and so o n — i s called a scale. In this sense, it is clear that a scale cannot be said to be invariant, since this sense contains no reference to the rules of numeral assignment which determine invariance transformations. T o use the term " s c a l e " in sense (b) would be unnatural, because it would make it impossible for us to say that

BROADER VIEWS OF MEASUREMENT

171

different scales can be produced by employing a single rule for assigning numerals. We want to be able to say that a different scale from Mohs's—for instance, one that assigns 1 to talc, 1 0 to gypsum, 100 to calcite, and so on—can be produced by using rule [2], the same rule used by Mohs. The proper sense for "scale," therefore, is (c). In this sense, Mohs's scale and the alternative assignment are different scales obtained by the same rule. And, in this sense, we can speak of scale invariance. For although invariance is primarily a property of rules of numeral assignment, it is secondarily a property of the particular assignments produced by such rules. The above clarifications of the concept of invariance and of the concept of a scale are partly designed to prevent an important misunderstanding of Stevens' view. It is possible to obtain the impression that Stevens believes scales can be classified merely by reference to the transformations that leave them invariant, without any reference to the rules for numeral assignment which create the scale. Thus Stevens might be interpreted as holding that if an experimenter has assigned numerals to loudness, or brightness, or "psychological weight," or "psychological length," then obviously he has in some way measured the dimension in question, and all we have to do to learn the type of measurement involved is to see whether the assignment of numerals remains invariant under this transformation or that, without any reference to the rules used in making the assignment. There is no very good reason to interpret Stevens in this way, but, in any event, it is important to point out the error in such a view. Suppose a dispute arises as to whether the assignment of numerals to temperature by means of a centigrade thermometer is an interval scale or a ratio scale. Without referring to any rules used to make the assignment, one disputant argues that scale numbers can only be multiplied by a constant and that the scale is therefore a ratio scale, and the other disputant argues that to scale numbers a constant can be added and that the scale is therefore an interval scale. Thus conducted, the dispute is meaningless and cannot be settled. To settle it, the rules for assigning numerals to temperature must be formulated, which is a rather difficult task, as we will see later. Once this is done, the question of whether those rules remain invariant under a given transformation can be settled mathematically and without difficulty. Similarly, to decide whether a "psychological" dimension, such as loudness or

172

BROADER VIEWS OF MEASUREMENT

"psychological weight," has been measured, we must discover the rules, if any, which have been used in assigning numerals to the dimension. If no consistent rule has been employed, the dimension has not been measured in any way. If consistent rules have been employed, they must be formulated, which may be a difficult task. Then and only then can the question of which transformations leave the scale invariant, and thus the question of what type of measurement scale has been constructed, be settled. OBJECTIONS TO STEVENS' VIEW

First Objection.—Stevens' definition of measurement begs and trivializes such questions as: Is loudness measurable? Is brightness measurable? Stevens believes that these questions tend to become verbal and, consequently, insignificant. He suggests, as a way of avoiding this result, that various types of measurement be distinguished on the basis of rules for numeral assignment, and that the questions be reformulated as follows: What type of measurement is available for loudness, and for brightness? This suggestion subtly begs the question of whether loudness and brightness are measurable. The definition of measurement which lies behind the suggestion is Definition 2, according to which measurement is simply the assignment of numerals to things according to rule. Stevens' sole defense of this definition is that it resolves disputes as to whether such dimensions as loudness and brightness are measurable. The disputes are resolved because on Definition 2 loudness and brightness are obviously measurable. It is obvious that we can assign numerals to tones and to colors by means of the rule used to assign them to football players (the rule for a nominal scale). It is equally obvious that we can assign numerals to loudness and brightness by means of the rule used to assign them to hardness in Mohs's procedure (the rule for an ordinal scale). In brief, when asked whether such dimensions as loudness and brightness are measurable, Stevens replies that they are, on Definition 2, and that the definition should be accepted because it resolves the question in favor of saying that the dimensions are measurable. This is surely to beg the question. Stevens' suggestion implies that an apparently interesting and significant philosophical question is in reality uninteresting and insignificant. It is obvious that numerals can be assigned to loudness and

BROADER VIEWS OF MEASUREMENT

173

brightness by the rules for a nominal or an ordinal scale. It follows from Definition 2 that loudness and brightness are obviously measurable. Since, in truth, it is far from obvious that these dimensions are measurable, Definition 2 is unacceptable. Interesting and significant philosophical questions cannot be resolved by verbal maneuvers like that employed by Stevens. (This objection should not be taken as denying that some philosophically interesting questions contain linguistic confusions and evaporate when the confusions are laid bare. It seems clear, however, that the question of whether loudness and brightness are measurable is not of this type.) Stevens would probably reply that his definition does not beg or trivialize any question, but merely requires reformulation of the questions at issue. Anyone who asks whether loudness and brightness are measurable wants to know whether they are measurable in the way that length and weight are measurable. Now numerals can be assigned to length and weight so as to create ratio scales for these dimensions. Consequently, the question about loudness and brightness becomes: Can numerals be assigned to loudness and brightness so as to create ratio scales for these dimensions? And this reformulation is much more precise and manageable than the original question. This reply is cogent only if Definition 2, the definition presupposed by the reformulation, is acceptable. The remaining objections show that it is not. Second Objection.—Stevens' definition of measurement is, at certain levels of its applications, implicitly circular. Stevens defines measurement as the assignment of numerals to things according to rules. Now a rule that cannot be applied is, in reality, not a rule, and no one can be said to have understood a rule unless he knows how to apply it. The application of rules [1] and [2] listed earlier is relatively unproblematic. We can, for example, assign numerals to objects to represent their hardness by using the scratch test: if one object scratches another the first is harder than the second and may be assigned the larger number; if neither object scratches the other they are equally hard and are to be assigned the same number. But the application of rules [3] and [4] presents special problems. Suppose we attempt to assign numerals to the length of rods by applying rule [4]. According to this rule, we are to assign to rods numbers that stand in equal ratios only if the rods stand in those ratios. But how will we determine whether the rods stand in those ratios? It would seem

174

BROADER VIEWS OF MEASUREMENT

that the only acceptable method is to measure the length of the rods, by using a meterstick, for example. If this is the only acceptable method, rule [4] can be applied only by employing a procedure of measurement, and, consequently, Stevens' definition of measurement is circular. The circularity consists of the following. Stevens tells us that measurement consists of the assignment of numerals to things according to rules. To understand any one of these rules we must understand its application. But to understand the application of rule [4] we must already know what measurement consists of, since measurement is required to apply the rule. To make the circularity even more obvious, Stevens' definition of the type of measurement involved in the creation of a ratio scale must read, in its complete form, as follows. Measurement of a dimension through the creation of a ratio scale consists in assigning numerals to objects within the dimension according to the rule: To objects that a procedure of measurement has shown to stand in equal ratios, assign numerals that stand in equal ratios. Since the term defined occurs in the definiens, the definition is circular. Its circularity renders the definition useless in answering such questions as: Are loudness and brightness measurable? To know whether these dimensions can be measured through the creation of ratio scales, we must know whether numerals can be assigned to objects within the dimensions according to the rule that creates a ratio scale. But in order to know this latter we must know whether the dimensions are measurable (since measurement is required in applying the rule), which is simply a reassertion of the original question. The same objection can be made against Stevens' definition of interval-scale measurement. Again, how would we apply rule [3] which creates such a scale? There is reason to think that the only w a y of determining whether objects in a given dimension are separated by equal intervals is by measuring them. The interval between X and Y is equal to that between U and V if the interval between the numbers assigned to X and Y in a procedure of measurement equals the interval between the numbers assigned to U and V in a procedure of measurement. But if this is the only way of determining equality of intervals, the definition of the type of measurement involved in creating an interval scale is, like that involved in creating a ratio scale, circular. One is just as useless as the other in addressing such questions as: Are loudness and brightness measurable?

BROADER VIEWS OF MEASUREMENT

175

The only way of escaping this objection is to provide an alternative method for determining equality of intervals, perhaps the method of "direct perception." Illustrated in terms of temperature, the suggestion is that we can determine whether the temperature interval between X and Y equals that between U and V by touching the four objects: if the interval between the first two feels equal to the interval between the last two, the intervals are equal and numbers separated by equal intervals can be assigned. But is this method acceptable? It is far from clear that an experimenter ought to conclude that intervals are equal from the fact that they feel equal. Such a conclusion ought, at best, to be tentative. Obviously the method of touching can qualify as a method for estimating the equality of temperature intervals, but it does not seem to qualify as a method for determining their equality. Even if it does so qualify, it would be regarded by all scientists as exceedingly "subjective," and unacceptable in comparison with the method that employs a thermometer. It may be said in reply to my objection that it is possible to charge Stevens with circularity only because of imprecision in his statement of the rules that create various types of scale, and that the rules should be generally stated as follows, where T is the positive result of some empirical test or empirical identification performed on the objects in question, and N means "the number assigned to": [1'] [2'] [3'] [4'] [5'] [6']

If If If If If If

Ti (X,Y) then N(X) = N ( Y ) T2 (X,Y) then N(X) > N ( Y ) Ts ( X X U / V , ) then N(X) — N(Y) = N(U) — N(V) T4 (X,Y,U,V) then N(X)/N(Y) = N(U)/N(V) Ts (X) then N(X) = o Ta (X) then N(X) = 1

T o illustrate, if we use the scratch test, [2'] becomes, "If X scratches Y , then the number assigned to X is greater than the number assigned to Y . " If we use the thermometer test, [3'] becomes, "If the difference in length between the columns of mercury for X and Y equals the difference in length between the columns for U and V , then the number assigned to X minus the number assigned to Y equals the number assigned to U minus the number assigned to V . " Thus stated, the rules do not mention the equality of intervals, or equality of ratios, within the dimension to which numerals are assigned; consequently, the ob-

176

BROADER VIEWS OF MEASUREMENT

jection of circularity in a definition of measurement in terms of these rules cannot be made. For example, rule [3'], although it might be used to assign numerals to temperature, does not mention equality of temperature intervals; hence, no procedure of measurement for determining temperature intervals is presupposed. This reply does not succeed. As an initial indication that it does not, consider the instance of rule [3'] stated above, which incorporates the thermometer test. In order to apply this rule, it is necessary to measure the lengths of four columns of mercury. A procedure of measurement is therefore required to apply the rule, though not a procedure for measuring temperature. It follows that if we define the measurement of temperature as the assignment of numerals by means of rule [3*] with the thermometer test, the definition is still circular, although in a slightly different way. Such measurement can be construed only as dependent measurement, the definition of which, as chapter 2 has shown, is circular. Dependent measurement is defined as a procedure for assigning numerals to things which contains a subprocedure of measurement. Stevens' definition of measurement may serve, at certain levels, as a definition of dependent measurement, and if we thus interpret it, its circularity is unobjectionable. But it will not then serve as an acceptable, noncircular definition of independent measurement. Such a definition is required here, for we wish to ask whether such dimensions as loudness and brightness are independently measurable. Perhaps the example of the thermometer test is prejudicial to my case. Is it possible to specify tests for assigning numerals in the manner of rules [3'] and [4'] without employing some procedure of measurement? It does not seem so. An experimenter can, of course, specify any test he pleases for assigning numerals to objects, and he may specify that a group of rules such as [1'] through [6'] be used in such assignments. He may even stipulate that a particular group of these rules creates, by definition, a dimension to which numerals are assigned. But if he wishes to specify a test for assigning numerals to a preexisting dimension—such as temperature, loudness, brightness, or weight— his selection of tests cannot be arbitrary. If he wants the numerals to represent equality of temperature intervals, or equality of weight ratios, his test is acceptable only if it does in fact assign numerals that (i) represent the dimension in question, and (ii) represent the equality in question. For most if not all dimensions, a procedure of measurement is required to insure that conditions (i) and (ii) are simultaneously met.

BROADER VIEWS OF MEASUREMENT

x 77

An example will illustrate the point. The ease with which a mineral can be drilled into is, apparently, some function of its hardness. If the same drill is applied with the same force for the same duration to minerals X and Y, and if the drill penetrates more deeply into Y than into X, then X is harder than Y. In view of this fact, it may seem possible to use the operation of drilling as a test for hardness ratios in rule [4']. Let us abbreviate "the length of the hole made in X by a drill, D, operating with force, F, for the duration, T " to "the drilled length for X." The rule can then be stated as follows. [4^1] If the drilled length for X minus the drilled length for Y equals the drilled length for U minus the drilled length for V, then N(X)/N(Y) = N(U)/(V). Obviously, this test can be applied only by measuring the drilled lengths for the various minerals. Consequently, the assignment of numerals by means of rule [4'h] can qualify at best as dependent measurement of hardness. But leave this earlier point aside. The point here is that we cannot know whether the equality of differences in drilled length is indicative of equality of hardness ratios without measuring hardness. Equality of drilled-length differences may indicate equality of hardness ratios, equality of hardness intervals, or neither. W e cannot know which it indicates without (a) measuring the hardness of X, Y, U, and V, (b) measuring drilled lengths for these minerals, and (c) comparing the results. (We might be able to deduce from physical theory that equality of drilled-length differences corresponds to equality of hardness ratios. Still, the verification of the theory requires the measurement of length, force, duration, and other dimensions.) What I have argued thus far is that Stevens' definition of measurement is implicitly circular, at least as regards its application to such nonadditive dimensions as temperature and hardness. But what of dimensions like length and weight, which permit of a physical operation of addition? Can legitimate tests employing these operations be devised for rules [3'] and [4*] ? At first sight, it seems so. The following statement describes such a test for weight, carried out on a balance. [3'w] If Y placed together in the same pan with Z balances X, and U placed together in the same pan with Z balances Y, then N(X) — N(Y) = N(Y) — N(U). The test in this rule requires no further procedure of measurement, nor

178

BROADER VIEWS OF MEASUREMENT

is a procedure of measurement required to determine whether the test is a proper test for equality of weight intervals (as opposed to equality of weight ratios). But the question is whether the test is not itself

a

procedure of measurement. Since the rule for assigning zero is not employed, if objects X, Y , and U satisfy rule [3'w], w e can assign to them the sequence — 1 , o, 1 , or the sequence o, 1 , 2, or the sequence 1 , 2, 3, or the sequence 2, 3, 4, and so on. In spite of this arbitrary feature of the assignment, one is inclined to say that to assign any sequence is to measure weight. This inclination derives f r o m the fact that the test in [3'w] seems to be a procedure of measurement in itself. T h e test can be alternatively described in the f o l l o w i n g w a y . T o X is assigned some numeral; to Y , w h i c h equals the addition of X and Z , is assigned the next highest numeral; to U, w h i c h equals the addition of Y and Z , is assigned the next highest numeral. This description shows that Z is used as a unit and that numerals are assigned b y physically adding this unit and other objects. Thus, the procedure qualifies as measurement on the narrow, classical definition discussed in chapter 3. W h a t the above example establishes concerning rule [3'] can also be established concerning rule [4']: In these rules acceptable tests that employ a physical operation of addition either are or involve a procedure of measurement. Consequently, Stevens' definition of measurement in terms of such rules is circular even in its application to additive dimensions such as length and weight. Third Objection.—Stevens'

definition of measurement is unac-

ceptably narrow, because it fails to q u a l i f y familiar procedures for measuring length and weight as measurement. It will be useful in introducing this objection to show h o w the scaling of temperature w i t h a thermometer fits Stevens' definition. A mercury thermometer is a sealed capillary tube with a reservoir containing mercury at one end. Heat in substances with which the tube is brought in contact causes the mercury to rise in the capillary: the hotter the substance the longer the mercury column.

Abbreviating

" t h e length of mercury column obtained w h e n a thermometer is brought in contact with X " to "the length of mercury for X , " the following rules govern the assignment of numerals to temperature on the centigrade scale. [l't] If the mercury length for X equals the mercury length for Y , then N ( X ) = N ( Y ) .

BROADER VIEWS OF MEASUREMENT

179

[2't] If the mercury length for X is greater than the mercury length for Y , then N(X) > N(Y). [3't] If the mercury length for X minus the mercury length for Y equals the mercury length for U minus the mercury length for V , then N(X) — N ( Y ) = N(U) — N(V). [4't] If the temperature of X is or equals the temperature of freezing water, then N(X) = o. [5't] If the temperature of X is or equals the temperature of boiling water, then N(X) = 100. In practice, none of the above rules is used in assigning centigrade scale numerals to temperature. Instead, the following procedure is employed, (a) A thermometer is calibrated as follows: (i) o is written on the tube at the end of the mercury column for freezing water, (ii) 100 is written on the tube at the end of the mercury column for boiling water, (iii) the tube length between o and 100 is divided into one hundred equal segments and labeled 1 , 2, 3 , . . . 99, (iv) the remainder of the tube is divided into segments the size of those in (iii) and labeled — 1 , — 2 , — 3 , and so on, and 101,102,103, and so on. (b) Numerals are assigned to objects by bringing the thermometer in contact with the object and reading off the numeral at the end of the mercury column thus obtained. None of the five bracketed rules for temperature are actually employed in procedure (a)-(b), but it seems clear that they are implicit in the procedure. The bracketed rules can easily be restated as rules for calibrating a thermometer in the manner described under (a), and an assignment of numerals by means of a thermometer thus calibrated will satisfy the bracketed rules as originally stated. Consider now the four paradigmatic methods for measuring length described in the first part of chapter 3. Each of these explicitly employs Stevens' rule [6^, the rule for a unit. None employs, even implicitly, rule [5^, the rule for assigning o; nor are they required to employ this rule to qualify as measurement on Stevens' definition. But since each of them creates a ratio scale for length, they must employ, implicitly if not explicitly, rules [1'] through [4*]. There is an initial inclination to think that each of the four methods of length measurement employs the following instances of Stevens' four rules. [l'l] If X and Y align at both ends, then N(X) = N(Y). [2'1] If X aligns at one end with Y and overextends Y at the other end, thenN(X) > N ( Y ) .

i8o

BROADER VIEWS OF MEASUREMENT

[3I] If the juxtaposition of Y and Z aligns at both ends with X, and the juxtaposition of V and W aligns at both ends with U, and Z aligns at both ends with W , then N ( X ) - N ( Y ) = N ( U ) - N ( V ) . [4I] If a juxtaposition of n X's aligns at both ends with a juxtaposition of m Y's, and a juxtaposition of n U's aligns at both ends with a juxtaposition of m V's, then N(X)/N(Y) = N(U)/N(V). Examination shows that this inclination is in error. None of the paradigmatic methods of length measurement satisfies every one of these four rules, and one of them employs none of the four rules. This conclusion, together with the observation that the length methods do not employ other instances of Stevens' four rules, constitutes the current objection to Stevens' definition of measurement. Consider the method of repeatedly applying a unit object to measure length. This method consists in (a) selecting a single rod as the unit rod, (b) laying this rod in successive adjacent positions along the entire length of the rod being measured, (c) counting the number of such placements, and (d) assigning the numeral thus obtained to the rod being measured. It is obvious that this method does not explicitly employ rules [l'l] through [4!]. Furthermore, since the method nowhere involves operations of aligning rods or juxtaposing rods, it would seem that the method does not implicitly employ any of the four rules. In reply it may be said that the method of repeatedly applying a unit implicitly employs the four rules in a subtler sense than any yet entertained, in the sense that if the method did not assign numerals in such a way as to satisfy the four rules, then it would have to be abandoned. But this point is simply false. A world is possible in which the operations of aligning and juxtaposing rods affect the length of the rods. In such a world we might continue to use the method of repeatedly applying a unit, which employs neither operation, to measure length. Or, again, a world is possible in which rods repel one another and thus cannot be aligned or juxtaposed. In such a world we might still use the method of repeatedly applying a unit, even though it would be impossible to determine whether the four rules were satisfied. In what sense, then, are rules [ i l ] through [4!] implicitly employed in the method of repeatedly applying a unit? And if these instances of Stevens' rules are not employed, what instances are?

BROADER VIEWS OF MEASUREMENT

181

Consider now the method of adding unit objects to measure length. This method consists in (a) selecting a unit rod, (b) discovering other rods that align at both ends with the unit rod, (c) juxtaposing unit rods until a juxtaposition is obtained which aligns at both ends with the rod being measured, (if) counting the number of unit rods in the required juxtaposition, and (e) assigning the numeral thus obtained to the rod being measured. Since this method uses operations of aligning rods and juxtaposing rods, it appears easier to argue that the method employs rules [l'l] through [4I]. But the appearance is deceptive. The operations of aligning rods and juxtaposing rods are separately employed in the method of adding units, but they are never combined into the complex tests described in [3'!] and [4'1]. Contrary to first appearance, rule [2!] is not employed in the method. If a juxtaposition of four unit rods, X, aligns at one end with Y and falls short of Y at the other end, we do not assign 4 to X and some larger number to Y. Rather, we continue juxtaposing unit rods until the juxtaposition aligns at both ends with Y. Rule [1'] is employed in steps (c) through (c), but only in a trivial sense. If a juxtaposition of four unit rods, X, aligns at both ends with Y, then 4 is assigned to Y. But 4 is assigned to X only in the trivial sense that X contains four unit objects. It is only in step (b) that rule [x'l] is employed nontrivially. Here the method calls for aligning the unit object at both ends with some other rod and assigning to the other the numeral assigned to the one, namely, the numeral 1. Even here the assignment is to unit rods only, and not to those being measured. Since the tests in rules [Yl], [3!], and [4'1] are not used in the method of adding units, we must conclude that these rules are not employed either explicitly or implicitly in the method. In reply it may be said that the three rules are implicit in another sense, in the sense that if the method did not assign numerals in such a way as to satisfy the rules, we would be forced to abandon the method. But why should we be forced to do this? Is it because the method would then fail to qualify as measurement? To say this is to beg the question at issue, the question of whether a procedure of measurement must satisfy rules [l'l] through [4!]. Is it because the method would then be useless? Perhaps the method of adding units is more useful if it satisfies rules [l'l] through [4'!] than if it does not. But if so, it does not follow that it is useless. Is it because the method would then be inconsistent? If rules (a) through (e) do not assign numerals in such a way as to satisfy rules [l'l] through

182

BROADER VIEWS OF MEASUREMENT

[4'1], then the one set of rules is inconsistent with the other set. But it does not follow that either set is internally inconsistent. It may be said that the method of adding units implicitly employs the four rules in a different and stronger sense, in the sense that an assignment of numerals to rods which satisfies rules (a)-(e)

neces-

sarily satisfies rules [l'l] through [4I]. Now it may in fact be true that an assignment that satisfies the first set of rules satisfies the second; that is, it may be true in the present world. But it is possible to imagine a world in which it is not true. Suppose the rods, X, Y, Z, and so on, measured in method (a)—(e) affect the length of one another during alignment, but that the unit rods, Ui, U2, Us, and so on, of the method neither affect the length of one another nor the length of X, Y, Z, and so on, during alignment. Then the following may occur: the numeral 4 is assigned to X and also to Y by method (a)-(e);

but when X and Y

are aligned, X becomes longer than Y. In that event, the numeral assignment will not satisfy rule [2'1] even though it does satisfy rules (fl)-(e). Still, we could legitimately retain method (a)-(e) to measure length. If the method of adding units does not employ rules [l'l] through [4!] either explicitly or implicitly, then what instances of Stevens' four rules does it employ? What tests does this method use or imply that might be specified as the tests in Stevens' four rules? It is illegitimate to answer by saying that the method of adding units is itself the test. Method (fl)-(c) is indeed a test for assigning to rods the same number, different numbers, numbers in equal intervals, and numbers in equal ratios. But, obviously, (a)-(e) does not employ tests for assigning numbers in these ways; rather, it is such a test. To say that the method of adding units is a method for assigning numerals to rods in accordance with Stevens' four rules, and then to specify (a)-(e)

as the test in each

of these rules is circular: it begs the question of whether (fl)-(e) is a method for assigning numerals in accordance with Stevens' four rules. Furthermore, to specify a procedure of measurement as the test in any of Stevens' four rules circularizes his definition of measurement, as noted in the discussion of the Second Objection. Stevens defines measurement as the assignment of numerals to things by means of some or all of the four rules. To go on to specify some procedure of measurement as the test in these rules is to say that measurement is the assign-

BROADER VIEWS OF MEASUREMENT

183

ment of numerals to things by means of a procedure of measurement. Parallel considerations apply to the method of repeated application of a unit, and to the other paradigm examples of length measurement. We may conclude that neither the method of repeated application of a unit nor the method of adding units can be regarded as the assignment of numerals to rods by means of rules [1'] through [4']. Consequently, these methods do not qualify as measurement in Stevens' definition. Since they are paradigm examples of measuring length, Stevens' definition is unacceptable. It is important to note that this conclusion is perfectly consistent with the admission that a procedure of measurement assigns numerals to things within a dimension in such a way as to represent dimension equalities, differences, equalities of intervals, and equalities of ratios. If the method of repeated application of a unit, or the method of adding units, is successfully used to assign numerals to the length of rods, then Stevens' four rules, as they were originally stated in [1] through [4], will be satisfied. That is to say, the following will then be true. [il] If X is equal in length to Y , then N(X) = N(Y). [2I] If X is longer than Y , then N(X) > N(Y). [3I] If X and Y, and U and V, are separated by equal intervals of length, then N(X) - N(Y) = N(U) — N(V). [4I] If the ratio between the lengths of X and Y is equal to the ratio between the lengths of U and V , the N(X)/N(Y) = N(U)/N(V). It may seem that the above admission is nothing less than a concession that Stevens' definition of measurement is, after all, acceptable. T w o points will show that this is not so. In the first place, [il] through [4I] are instances of Stevens' four rules as they were originally stated, and must be distinguished from rules [l'l] through [4'!]. Unlike the latter, they are abstract, and contain no reference to tests, such as aligning rods and juxtaposing rods. Hence, there is no contradiction between my earlier assertion that a method of numeral assignment such as (a)-(e) need not satisfy rules [l'l] through [4'!] and my present assertion that a method of length measurement such as (a)-(e) will, if successfully employed, satisfy rules [il] through [4I]. (For a discussion of the difference between the abstract and the particular interpretation of rules or axioms of measurement, see the first part of chapter 4.)

184

BROADER VIEWS OF MEASUREMENT

In the second place, it is one thing to say (i) that statements [ll] through [4I] will be true of rods to which numerals have been assigned by a successfully employed procedure of measurement, and quite another thing to say (2) that a procedure of length measurement employs the rules embodied in [ll] through [4I]. Statement (1) is a description of the result of measuring rods, and as such it is true. Statement (2) is a description of the method of measuring rods. As such it is incomplete, since it does not specify the tests that may legitimately be employed to determine length equalities and differences among rods. And as such it is false, since whatever tests are thus specified are either not involved in paradigmatic methods of length measurement, or are, circularly, the method of length measurement itself. Statement (x) may be incorporated in an acceptable definition of measurement; (2) cannot. It is likely that Stevens' definition of measurement gains its initial plausibility from a failure to distinguish between (a) and (2). This explanation of the initial plausibility of Stevens' view enables us to locate his error with some precision. Having correctly ascertained that [1] through [4] describe the result of successfully employing a procedure of measurement, but having failed to distinguish the result from the method of assigning numerals in such a procedure, Stevens leaps to the conclusion that any method that has the result of making statements [1] through [4] true is a method of measurement. That this conclusion is mistaken is easily seen by noting that the random assignment of numerals to objects in a dimension is not measurement, even if by accident it succeeds in making [1] through [4] true for that dimension. The method is crucial: some methods of numeral assignment are, others are not, methods of measurement. Stevens does not tell us which are and which are not. Only an acceptable definition of measurement can tell us this. Fourth Objection.—Stevens' definition of measurement is unacceptably broad, because it includes procedures of numeral assignment which are not properly regarded as measurement. Stevens says that measurement is the assignment of numerals to things according to rules [1] through [4]. But it is clear that his general definition does not restrict measurement to the use of just these four rules, since he says (as quoted earlier): "If we can point to a consistent set of rules, we are obviously concerned with measurement of some sort." Any consistent rule or set of rules can, according to Stevens'

BROADER VIEWS OF MEASUREMENT

185

definition, be employed in a procedure of measurement. It is this feature of his definition which accounts for its "liberality" and makes it useful in avoiding "verbal disputes" as to what measurement does and does not consist in. Gut the definition is excessively liberal. In addition to the four rules mentioned by Stevens, we can formulate an indefinitely large number of consistent rules for assigning numerals to things, many of which Stevens himself would surely be reluctant to dignify with the term "measurement." T w o examples will illustrate the point. (1) Numerals are assigned to twenty football players in the following way. The numerals 1 through 20 are printed on twenty slips of paper. The players are arranged in a circle, and the slips in a concentric circle inside them. A t the center of the circles is placed a pointer capable of being spun. First we spin the pointer and note which player it comes to rest on. Then we spin the pointer again and note which numeral it comes to rest on. To that player is assigned that numeral. This procedure may or may not result in the assignment of different numerals to different players. (2) Twenty objects of different hardness are ranked in ascending order of hardness. Beginning with the least hard object and the numeral 1, numerals are consecutively assigned to the series of objects, except that every fifth object in the series is assigned the same numeral as the preceding object. Thus, 1 is assigned to the least hard object, 2 to the next hardest, 3 to the next, 4 to the next, 4 to the next, and so on. This procedure results at certain points in the assignment of the same numeral to different hardnesses. Surely no one, not even Stevens, would call procedures (1) and (2) procedures of measurement. Note that Stevens cannot reply by saying that the rules in (1) and (2) are inconsistent. The latter does, and the former may, result in the assignment of the same numeral to different degrees of different objects, but this is not to say that the rules are inconsistent. Stevens might reply that (1) and (2) are procedures of measurement, since they are types of numeral assignment according to a consistent rule. This reply would make it clear that his definition is impossibly broad. O r he could say that the rules in (1) and (2) are not rules that create scales of measurement. This line is, of course, the correct one to take, but to adopt it is to abandon Stevens' very liberal definition of measurement, according to which any consistent rule of numeral assignment defines a procedure of measurement.

i86

BROADER VIEWS OF MEASUREMENT

It is possible to show that Stevens' definition is too broad without instancing rules so unusual as those given in (i) and (2). It is too broad simply because it includes the so-called nominal scale in the class of scales of measurement. According to Stevens, football players have been measured by means of a nominal scale when numerals are assigned to them according to the rule: to different players assign different numerals, to the same player assign the same numeral. Stevens takes note of the objection that nominal scales are not scales of measurement, and implies that it is unimportant: The nominal scale is a primitive form, and quite naturally there are many who will urge that it is absurd to attribute to this process of assigning numerals the dignity implied by the term measurement. Certainly there can be no quarrel with this objection, for the naming of things is an arbitrary business. However we christen it, the use of numerals as names for classes is an example of the "assignment of numerals according to rule." 15

But the objection is not unimportant, since, if it is sound, Stevens' definition is too broad. One hesitates to call a nominal scale a "scale of measurement" because its values do not represent the relative magnitude of the objects scaled. To assign numerals to football players is simply to give them convenient names, and surely measuring is to be distinguished from naming. Surely one cannot be said to have measured a dimension unless the numerals assigned indicate whether objects in the dimension are greater or less than one another. Stevens' statement, "the oft-debated question whether the process of classification underlying the nominal scale constitutes measurement is one of those semantic issues that depends upon taste,"16 is simply not true, as my first objection to his definition of measurement shows. His definition trivializes and begs such questions as whether loudness and brightness are measurable, and is thus useless in addressing the questions. Whether a definition has this important use is neither a "semantic issue" nor a matter of "taste." The above objection forces us to abandon Stevens' definition, or to narrow it. It can be narrowed only by restricting the rules of numeral assignment which may legitimately be employed in measurement. Sup18 Stevens, "Mathematics, Measurement, and Psychophysics," p. 26; see also Stevens, "On the Theory of Scales of Measurement/' p. 679. 16 Stevens, "Measurement, Psychophysics, and Utility," p. 25.

BROADER VIEWS OF MEASUREMENT

187

pose we say, then, that measurement is the assignment of numerals to things according to one or more of rules [1] through [4]. This definition excludes the procedure described in (2), since at certain points that procedure assigns the same numeral to different hardnesses, thus violating rule [2]. The definition also excludes the procedure described in (1), and any procedure that creates a nominal scale. Rule [1] calls for the assignment of the same numeral to equal objects, and for the assignment of the higher numeral to the greater object. These two rules are not observed in the nominal scaling of football players, since the relative magnitude of the players (height, weight, age, and so on) is not taken into consideration. Furthermore, the same numeral is never assigned to different players, contrary to rule [2]. When narrowed as above, Stevens' definition of measurement is clearly less objectionable than in its original, completely liberal form. But it still seems too broad, since it includes ordinal scales—those created by rules [1] and [2]—in the class of scales of measurement. Mohs's scale of hardness of minerals is an ordinal scale. Ten minerals, from least hard to hardest, are selected by the operation of scratching. If mineral X scratches Y, X is harder than Y ; if X is scratched by Y, Y is harder than X; if X and Y do not scratch each other, they are equally hard. The numerals 1 through 10 are then assigned to the ten minerals in ascending order of hardness, and numerals are assigned to other minerals by comparing them with the standard ten. How could a claim that Mohs's scale of hardness is not properly regarded as one of measurement be supported? There would seem to be two main lines of argument. First, it might be argued that Mohs's scale does not serve the principal use (or purpose) of measurement and therefore does not merit the term. Second, it might be argued, either from technical or from ordinary use, that the scale is by definition not a scale of measurement. Serious difficulties confront the first line of argument. There are two major candidates for the principal use of measurement: (1) measurement is a system of numeral assignment which permits the application or arithmetical statements to a dimension; (2) measurement is a system of numeral assignment which makes possible the formulation and verification of numerical laws. (See chapter 4 for a discussion of these two uses.) We may admit that either (1) or (2) is the principal use of measurement without committing ourselves to the view that no

188

BROADER VIEWS OF MEASUREMENT

system of numeral assignment can qualify as measurement unless it possesses one or the other of these uses. But even if this point is waived, another difficulty remains. There is no apriori reason for maintaining either that Mohs's scale of hardness does not permit the application of arithmetical statements to hardness, or that the scale does not make possible the formulation and verification of numerical laws pertaining to hardness. In point of fact, the scale has not as yet found either of these uses. It is impossible, however, to predict with certainty that it will never come to have either use, and difficult to argue, therefore, that it cannot properly be regarded as a scale of measurement. The second line of argument, that such scales as Mohs's are not scales of measurement, proceeds from technical or from ordinary use. The argument from technical use does not succeed, since many writers use the term "measurement" in technical contexts to refer to the construction and use of such scales as Mohs's. The argument from ordinary use is the viable one. When "measurement" is applied to the construction and use of such scales as Mohs's, one feels that a familiar term has been misused. Whether this feeling can be justified is considered in the sequel. Hempel refuses to include Mohs's scale in the class of scales of measurement on the ground that it does not, as every procedure of measurement must, provide for the univocal assignment of numerals to the dimension in question. The next section of this chapter shows that Hempel's argument does not succeed. The correct basis for excluding ordinal scales becomes clear in the final section of the chapter, where it is argued that measurement is the assignment of numerals to things in a dimension by comparing them with a dimension unit. Mohs's scale makes no use of a unit of hardness, and it is for this reason that to call it a scale of measurement is a gross violation of the ordinary use of the term. Stevens' definition of measurement is unacceptably broad, and must be abandoned. But it may still be retained as a definition of scaling. There is no harm in defining scaling as the assignment of numerals to things according to rules, so long as it is recognized that not all scaling is measuring. Such a definition does seem to describe, with considerable accuracy, the use recently made of the term "scaling" in psychology and social science. If there is a "semantic issue" here, it surrounds the use of the term "scaling." Should we say that the use of any con-

BROADER VIEWS OF MEASUREMENT

sistent rule or set of rules—including, for example, the rule used to assign identifying numerals to football players—creates a scale? How this question is answered may very well, in Stevens' phrase, "depend upon taste." But in any event, scaling must be distinguished from measuring. Some scales, such as the centimeter scale of length, are scales of measurement; other scales, such as a nominal scale of football players, are not scales of measurement. The error in Stevens' theory of "measurement" can perhaps be explained as follows. Stevens has constructed what may very well be an adequate theory of scaling, but has confused scaling with measuring. The confusion is a natural one, because measuring is a kind of scaling: like all scaling, measuring is the assignment of numerals to things according to rules. Having noted this similarity, Stevens has expanded the use of the term "measurement" to cover scaling of all kinds. This expansion nicely serves one of his principal aims, which is to show that psychological magnitudes are measurable. But the expansion is illegitimate. Even if psychological magnitudes can be scaled, it does not follow that they can be measured. HEMPEL'S VIEW Hempel believes that ordinal scales such as Mohs's scale of hardness are not properly regarded as scales of measurement. Presumably, then, he would agree with my charge that Stevens' definition of measurement is excessively broad. Such scales as Mohs's can, of course, be excluded by adopting the classical definition of measurement discussed in chapter 4. But Hempel believes, as I do, that this definition is excessively narrow. He attempts to formulate a new, liberal definition of measurement which will avoid both extremes. THE VIEW PRESENTED

In characterizing what he calls "rank orders . . . of a nonmetrical character," Hempel makes three observations17 which are here summarized and illustrated by reference to Mohs's scale of hardness. To form this scale, ten standard minerals from least hard to hardest are selected and ranked by the operation of scratching. If one mineral 17 Carl G. Hempel, fundamentals of Concept Formation in Empirical Science, International Encyclopedia of Unified Science, Vol. II, no. 7 (Chicago, 1952), pp. 61-62.

19"

BROADER VIEWS OF MEASUREMENT

scratches another, the one is considered harder than the other; if neither mineral scratches the other, they are considered equally hard. Numerals are then assigned to the ten minerals in ascending order of hardness: 1—talc 2—gypsum 3—calcite 4—fluorite 5—apatite

6—orthoclase 7—quartz 8—topaz 9—corundum 10—diamond

Now for Hempel's observations. (1) More than one sequence of numerals can be assigned to the objects being ranked (see the description of an ordinal scale in the preceding section for this point). (2) The numeral assigned to a given object depends on the other objects being ranked. If we were ranking all the minerals except talc, we would assign 1 to gypsum, 2 to calcite, and so on. Furthermore, if, after the ten minerals are ranked and numbered as above, we wish to include an eleventh that falls between two of the others, a new assignment of numerals is necessary. If the new mineral, X, is harder than apatite and less hard than orthoclase, we must assign 6 to X, 7 to orthoclase, and so on. (3) Such reassignment may be avoided by assigning a fractional numeral, say 5.5, to X. But this method is imperfect, since any mineral harder than apatite and less hard than orthoclase will receive the numeral 5.5. Minerals of different hardness may thus receive the same numeral. Hempel apparently thinks that features (i), (2), and (3) can be summarized by saying that the assignment of numerals prescribed by Mohs's scale is an equivocal assignment, whereas measurement consists in a univocal assignment of numerals. This conclusion leads him to a definition of measurement, restated here with less precision than he uses and in the terminology of the present study. Definition 3. Measurement of a dimension consists in specifying criteria that assign to each object in the dimension exactly one real number so as to satisfy the following conditions: (a) X E Y if and only if N(X) = N(Y), and (b) X G Y if and only if N(X) > N(Y), where E means "equal to," C means "greater than," and N means "the number of." 18 18

Hempel, Fundamentals of Concept Formation, pp. 62-63.

BROADER VIEWS OF MEASUREMENT

191

On its face Definition 3 seems indistinguishable from Stevens' definition of an ordinal scale, which would be unacceptable to Hempel. He would argue, however, that the two definitions are significantly different in that his own calls for criteria that assign exactly one real number to each object in the dimension and, therefore, does not include ordinal scales such as Mohs's. If this interpretation were correct, it could be argued that Definition 3 has the advantage of being neither too broad for ordinary use nor too narrow for technical use. On the one hand, it would reflect our ordinary disinclination to call ordinal scales such as Mohs's scales of measurement; on the other hand, it would permit us to include as measurement procedures that do not employ any operation of addition. Hempel shows how certain procedures, weighing for instance, which rely on an operation of addition qualify as measurement in his definition. He then points out that "measurement based on stipulations of the kind discussed so far [the addition of units], while probably the only type of nonderivative [independent] measurement used in physics, is not the only type that is conceivable. Psychology, for example, has developed procedures of a quite different character for the fundamental [independent] measurement of various characteristics."19 Hempel describes Stevens' method for constructing a "scale of pitch," noting that, although "it nowhere relies on any mode of combination," it enables us "to assign, in a reasonably univocal fashion, numerical pitch values." His conclusion is that, since the procedure "presupposes no other scale of measurement (theoretically not even that of frequency), it has to be qualified as fundamental [independent] measurement."20 But Hempel's interpretation is not correct, for Definition 3 does include ordinal scales. Perhaps, for the three reasons given by Hempel, Mohs's scale does not prescribe univocal assignment of numerals. It is, however, possible to describe a procedure that, although essentially similar to Mohs's, does prescribe univocal assignment. A large number of minerals (say, fifty or sixty thousand) are collected from all areas where minerals are known to exist. Subclasses of this large class are then formed so that the members of each subclass are equal in hardness. Next, a representative member is chosen at random from each subclass. These representatives are ranked in respect to their hardness, 19 20

H e m p e l , Fundamentals Ibid., p. 69.

of Concept Formation, p. 68.

BROADER VIEWS OF MEASUREMENT

from least hard to hardest. Finally, numerals are assigned to the representatives according to the following rule: To the least hard assign x, to the next hardest assign 2, and so on, until every representative has been assigned one and only one numeral. If, as is likely, a representative of every existing degree of hardness is obtained, there will be no need to assign the same fractional numeral to minerals of different hardness. By selecting representatives from an adequate sample of minerals, we may produce a hardness scale that assigns numerals to hardness univocally. It is quite true, as Hempel would point out, that the order of hardness of the mineral representatives could have been indicated by a different sequence of numerals from the one prescribed. But, in fact, one particular sequence was prescribed; and if the rule is followed, the result will probably be a univocal assignment of numerals to hardness. (Notice in this connection that we can prescribe a particular sequence of numerals with the expectation that it will lead to the formulation and verification of numerical laws concerning hardness.) It is also true, as Hempel again points out, that the numeral assigned to a given object depends upon the other objects to which numerals are assigned. The numeral assignment for a group of a thousand objects will not be the same as that for a group of fifteen hundred. But the fact still remains that the method described above will in all probability provide for the univocal assignment of numerals to hardness. OBJECTION TO HEMPEL's VIEW

My objection to Hempel's definition of measurement can now be stated. Let us call the above method for assigning numerals to hardness the S-method, and let us suppose that it does assign numerals to hardness univocally. The S-method qualifies as measurement on Hempel's definition, but it creates only an ordinal scale of hardness, essentially of the same type as Mohs's and differing from it only in its greater "density." The numerals the S-method assigns do not represent equal hardness ratios or even equal hardness intervals, and surely a method that does not give this significance to numerals is not properly regarded as measurement. There are two possible lines of reply to this objection. x) It may be said in reply that the univocal assignment of the S-method depends on a happy accident in the selection of representatives of every existing degree of hardness. Since the method does not

BROADER VIEWS OF MEASUREMENT

193

insure that such a selection will be made, it does not insure that numerals are univocally assigned, and therefore cannot properly be regarded as a method for univocally assigning numerals to hardness. This reply is unsuccessful. Whether the S-method produces a univocal assignment of numerals is, indeed, a question of fact. But, as Hempel himself seems to say at one point, 21 the same is true of every method of numeral assignment, even those that rely on an operation of addition of unit objects. A numeral may be assigned to the weight of an object by balancing it against a group of unit objects. Whether this method will or will not assign different numerals to objects of different weight depends upon a host of factors: the sensitivity of the balance, its position at the time of use, the atmospheric and magnetic conditions surrounding it at the time of use, the durability of the unit objects, the w a y the latter are placed on the balance, and so on. Of course, one may reduce the probability that such factors will produce an equivocal assignment through an extremely careful and exhaustive description of the method to be employed (e.g., level the balance before weighing, use the balance only if the relative humidity is below 70 percent, cluster the unit objects in the center of the pan). But no description of this kind, however lengthy and careful, logically entails that the method will produce a univocal assignment of numerals. A t most we can only say that a given method, if followed, makes it highly probable that univocal assignment will result. 2) The second reply to my objection is that the possibility remains that new minerals will be formed or created whose hardness lies between two adjacent representatives in the S-scale. Since the S-method contains no rule to cover this possibility, it does not insure that numerals will be univocally assigned, and cannot therefore properly be regarded as a method for univocally assigning numerals to hardness. This reply is also unsuccessful. For, in the first place, so long as no new intermediate minerals are formed or created, the method remains one of univocal assignment. Second, numerals can be univocally assigned to new intermediate minerals by extending the S-method to the assignment of fractional numerals. Suppose, for example, a number of new minerals have been created whose hardness lies between those already scaled at 10 and 1 1 . W e collect a sample of these new minerals, 21

Hempel, Fundamentals

of Concept

Formation,

p. 68.

194

BROADER VIEWS OF MEASUREMENT

divide the sample into classes whose members are equal in hardness, select representatives of these classes, and then rank the representatives from least hard to hardest. Finally, the numerals 10.1, 10.2,10.3, and so on, are assigned to these representatives. If, when the numeral 10.9 has been reached, the group of representatives has not been exhausted, we continue numbering with 10.91,10.92,10.93, so on. If the S-method thus extended is not one of univocal assignment, then neither is any method that has ever been used or described, including methods that employ an operation of addition of unit objects. In measuring the weight of objects by addition of unit objects (U's) on a balance, we may encounter an object, X, which outweighs a given collection of unit objects, say, one of ten members, but is outweighed by the next largest collection, one of eleven members. What numeral shall be assigned to X—10.5? If this rule is adopted, any object lying between 10 and 1 1 will receive the numeral 10.5, and the assignment will not be univocal. O f course, we may, in order to measure X, select a group of ten lighter unit objects (u's) which together balance one of the original units. Then perhaps X will balance 10 U's plus some number of u's. If, for example, X balances 10 U's together with 5 u's, the weight of X is 10.5 U. But if X is found to be heavier than 10 U's together with 5 u's, and lighter than 10 U's together with 6 u's, a group of ten objects, o's, must be selected which together balance one u, and these o's must be employed in the manner of the u's. It is impossible to predict how far this process of fractionating units will have to be carried before a collection of units and fractional units can be obtained which balances X. Therefore, the method just described—which carries the process of fractionation through only two stages—does not logically insure univocal assignment of numerals to all the objects that may be weighed. W e may, of course, try to select unit objects small enough to provide univocal assignment without the use of fractional numerals. But at best we can only say that a given method, if followed, makes it highly probable that univocal assignment will be achieved. In this respect, the method for assigning numerals to weight is on a par with the S-method for assigning numerals to hardness. If it is said that the description of levels of unit-weight fractionation indicates how to proceed, through deeper and deeper levels, until a balance for X is obtained, we may reply that our description of levels of fractionation for the hardness scale indicates how to proceed,

BROADER VIEWS OF MEASUREMENT

195

through successive levels, until an intermediate mineral can be located fractionally within the scale. In both methods it is possible, if the fractionation is carried far enough, to assign one and only one numeral to each object within the dimension in question. In this sense both methods provide for univocal assignment of numerals. If there is a sense in which one method does not so provide, then, in that sense, neither does the other. Hempel shows that he is partly aware of these difficulties. In the midst of describing a procedure that employs an operation of addition on a balance for assigning numerals to weight, he says: This account of the fundamental measurement of mass is necessarily schematized with a view to exhibiting the basic logical structure of the process. W e have to disregard such considerations as that the equilibrium of a balance carrying a load in each pan may not be disturbed by placing into one of the pans an additional object which is relatively light but whose mass is ascertainable by fundamental measurement. This means that fundamental measurement does not assign exactly one number to every object in D1. 2 2

That the italicized qualification is relegated to a footnote, and that it is thought necessary only as the result of a "schematized" account, show that Hempel has not fully understood the problem raised here, and does not realize that the qualification is tantamount to an admission that his definition of measurement as univocal assignment of numerals is unacceptable even to himself. For it does not, with this qualification, exclude ordinal scales, such as that for hardness described earlier. Indeed, to make this qualification is nothing more nor less than to give up his definition of measurement as univocal assignment of numerals. Hempel's definition, then, is unacceptable. I have described a procedure—the S-method—for assigning numerals univocally to hardness, which creates only an ordinal scale of this dimension, a scale essentially similar to Mohs's scale of hardness. The numerals of the S-scale, like those of Mohs's, do not represent equality of hardness intervals or equality of hardness ratios; consequently, neither scale is properly regarded as a scale of measurement. The reply to the objection that the S-method is not really one of univocal assignment is that there is equal justification for saying that weighing objects on a balance is not a procedure of univocal assignment. If the latter procedure is one 22

Hempel, Fundamentals of Concept Formation, p. 86 and n. 79. Italics added.

196

BROADER VIEWS OF MEASUREMENT

of univocal assignment, so is the former. If the latter is not one of univocal assignment, Definition 3 is clearly unacceptable, since weighing objects on a balance is a paradigm example of measurement. A N ACCEPTABLE VIEW It would appear that at the core of the concept of measurement, whether used in ordinary or technical contexts, are the concept of a unit and the concept of comparison. Dimensions have been measured, or metricized, as one sometimes says, when the magnitude of objects within the dimension can be expressed in terms of a dimension unit or units. The selection of a unit enables us to assign numerals to objects within the dimension in such a way that the mathematical significance (or, at least, part of it) of numbers is transferred to the dimension objects. T o put the point metaphorically, dimension units serve as numbers, which can be counted and manipulated in others ways that may (or may not) correspond to mathematical manipulations. The other part of the core is the concept of comparing one thing with another. When we measure the objects in a dimension we compare them with some standard or standards within the dimension which have been previously settled upon. That the standards have been established by convention is irrelevant. For if we all accept the same conventions, the same standards, we will be able to describe the magnitude of dimension objects in terms of those standards. This type of description may prove extremely useful in a number of respects. It is clear that the comparison involved in measurement must be of a specific type. It is possible to imagine laying a particular rod in successive adjacent positions along the edge of a particular table, and thus to predict (i.e., estimate) how many rod lengths will equal the length of the table edge. This process, though it might be called "comparing," is not the sort required for measurement. Measurement requires real, not imaginary, processes of comparing, for otherwise it would not be possible to distinguish measuring from estimating. Again, it is possible to compare the length of a particular table expressed in inches with its length expressed in centimeters. If the length in inches is 10, the length in centimeters is 25.4 (1 inch = 2.54 cm). The comparison that produces this result is the mathematical process of converting values of one scale into values of another. Measurement, in contrast, requires a nonmathematical process of comparing dimension

BROADER VIEWS OF MEASUREMENT

197

objects with a standard. There is no reason to suppose, however, that the process of comparing must involve the physical addition of dimension objects. There may be (and are, as is shown in a later chapter) processes of comparison suitable for measurement which do not involve the physical addition of objects. Let us characterize the process of comparing required for measurement as an "empirical" process, to distinguish it from imaginary processes, mathematical processes, and any others that need to be excluded. Measurement may then be defined as follows: Definition 4. Measurement is a procedure of assigning numerals to objects within a dimension by means of an empirical process of comparing these objects with a dimension unit or units. This definition satisfies the criterion of acceptability laid down earlier: it is as broad as it can be without doing undue violence to the ordinary meaning or to the technical meaning of the term "measurement." To assist in making this and other points, the descriptions of the four methods of length measurement given in chapter 3 are here reproduced. Method 1. An object is selected and labeled U; it is the unit object of length. Next an indefinite number of objects, each equal in length to U, are discovered and labeled U. To measure the length of a given object, X, the U's are juxtaposed until a juxtaposition is obtained which is equal in length to X. The number of U's in the juxtaposition is then determined by counting, thus giving the length of X in number of U's. Let us call this the method of addition of unit objects. Method 2. A unit object is selected and labeled U. To measure a given object, X, one end of U is aligned with one end of X, and a mark is made on X at the other end of U. Then one end of U is aligned with this mark and a second mark made on X at the other end of U. This procedure is continued until the end of X has been reached. The number of placements required to reach the end of X is its length in number of U's. Let us call this the method of repeated application of a unit object. Method 3. A unit object is selected and labeled U. A group of standard objects is then selected by the following procedure. An object,

198

BROADER VIEWS OF MEASUREMENT

equal in length to U, is discovered and labeled Ui. U and Ui are juxtaposed, and a third object, equal in length to the juxtaposition, is discovered and labeled U2. U and U2 are juxtaposed, and another object, equal in length to the juxtaposition, is discovered and labeled Us* This procedure is continued as long as necessary to measure all the objects to be measured. Measurement of a given object, X, is accomplished by determining which of the standard objects it equals. The subscript numeral of the standard object gives the length of X in number of U's. Let us call this the method of comparison with standard objects. Method 4. A unit object is selected and labeled U. Then a rod L, considerably longer than U, is selected. One end of U is aligned with one end of L, and a mark is made on L at the other end of U. The numeral 1 is placed beside this mark. One end of U is aligned with this mark and a second mark is made on L at the other end of U. The numeral 2 is placed beside this mark. This process is continued through the entire length of L. T o measure a given object, X, the " 1 " end of L is aligned with one end of X. The numeral beside the mark on L lying nearest the other end of X designates the length of X in number of U's. Let us call this the method of using a ruler. A study of these four methods shows that Definition 4 is broader than the classical Definition 1 , which defines measurement as the assignment of numerals to objects within a dimension by means of a physical operation of addition of unit objects. Only Methods 1 and 3 employ an operation of addition, and thus only they qualify as measurement under Definition 1. But all four methods use a process of comparing objects with a unit object or objects, and therefore all four qualify as measurement under Definition 4. Chapter 6 presents a procedure for assigning numerals to pitch which does not employ an operation of adding tones. This procedure qualifies as measurement under Definition 4, but not under Definition 1. In spite of its relative broadness, Definition 4 does not seem to do undue violence to the ordinary meaning of measurement; in this respect it is superior to any of those previously considered. When we are told that a rod is 6 centimeters in diameter, or that a test tube contains a sample of 3 grams, or that an automobile has a rz-volt battery, or that a certain sound has a loudness of 60 decibels, or that a patient

BROADER VIEWS OF MEASUREMENT

199

has received 400 roentgens of radiation, or that a certain liquid contains 100 gram calories of heat, or that the sound wave produced by the A string of a violin is a 44o-cycle-per-second wave, w e assume that a procedure of measurement is available for the dimension in question and that these statements can be verified by employing that procedure. W e make this assumption because the magnitude of the relevant dimension has been described in terms of number of dimension units. O n the other hand, when we are told that a certain mineral has a hardness of 6 on the Mohs scale, or that the brightness of a certain star is 4 (in terms of the 6-point scale of stellar magnitude used by early astronomers), or that wind has a force of 1 2 (hurricane force on the Beaufort scale), we hesitate to say that a procedure of measurement is available for the relevant dimension. Definition 4 explains our hesitation. Numerals have indeed been assigned to objects in the relevant dimension, but they do not indicate number of dimension units. Mohs's scale of hardness enables us to say that talc has a hardness of 1 , gypsum a hardness of 2, and so on, but these numerals do not indicate number of hardness units. N o unit of hardness is defined or used in the construction of this scale. It is worth noting that the distinction between a "quantitative" scale and a "qualitative" scale is compelling even at the ordinary level. The centimeter scale of length would be regarded as "quantitative," Mohs's scale of hardness as "qualitative." Definition 4 explains this difference. It is more difficult to decide whether Definition 4 does undue violence to the technical meaning of "measurement," not because there are strong indications that it does, but because it is extremely difficult, as we have seen, to obtain a consensus among scientists as to what the term ought to mean. A t the one extreme is Definition 1 (proposed by Campbell and his adherents); at the other extreme is Definition 2 (proposed by Stevens and his adherents). One point in favor of Definition 4 is that it seems to lie somewhere between these two extremes. Another is that Stevens, although the unbearably broad Definition 2 is his proposal, in practice employs Definition 4. In his scales for the measurement of "subjective magnitudes" he attempts to define units for the various psychological dimensions in question. In this respect he is followed and preceded by numerous other psychologists. These scientists, although they reject Definition 1 as too narrow, proceed as though Definitions 2 and 3 are too broad. In practice, when they at-

200

BROADER VIEWS OF MEASUREMENT

tempt to formulate procedures of measurement for psychological dimensions, they attempt to specify units of measurement for the dimensions in question. Thus the mak has been proposed as the unit of "subjective length" (see chap. 9), the temp as the unit of "subjective time," the veg as the unit of "subjective weight," the bril as the unit of "subjective brightness," the sone as the unit of loudness, and the mel as the unit of pitch.23 In view of these considerations it seems reasonable to suggest that Definition 4 does not do undue violence to the technical meaning of the term "measurement." A s stated. Definition 4 does not distinguish between independent and dependent measurement. The following definition of their distinction is based on the analysis in chapter 3: Definition 4a. Dependent measurement is a procedure of assigning numerals to objects within a dimension by means of an empirical process of comparing these objects with a unit or units, which includes at least one subprocedure of measurement. Independent measurement is a similar procedure except that it includes no subprocedures of measurement. There is, however, an alternative definition that may, at least for certain purposes, prove just as acceptable: Definition 4b. Dependent measurement is a procedure of assigning numerals to objects within a dimension by means of an empirical process of comparing these objects with a unit or units that are either identified with or defined in terms of units for some other dimension. Independent measurement is a similar procedure except that its units are neither identified with nor defined in terms of units for some other dimension. That Definitions 4a and 4b are not equivalent can be seen b y considering the four methods of length measurement just described. O f these, only the method of addition of unit objects and the method of repeated application of a unit object qualify as independent measurement under Definition 4a, since each of the other two methods involves one of the former two as subprocedures. Under Definition 4b, however, 2 8 S . S. Stevens and E. H. Galanter, "Ratio Scales and Category Scales for a Dozen Perceptual Continua," Journal of Experimental Psychology, 54 (1957), 377411.

BROADER VIEWS OF MEASUREMENT

201

all four methods qualify as independent measurement, since in none of them is the unit of length identified with or defined in terms of a unit for some dimension other than length. Thus, the definition of dependent measurement in 4« is narrower than that in 4b. This might be construed as an argument in favor of the latter, since there is a strong inclination to say that each of the four methods of length measurement should qualify as independent (direct) measurement. UNITS

Definition 4 is rather sketchy as it stands. It defines measurement as a procedure for expressing the magnitude of objects in number of units, but it does not say what a unit is, nor what the requirements of units are. As a first step toward filling out the definition, we need to distinguish among units, unit objects, and unit intervals. Let us begin with the distinction between units and unit objects. In Method 2 for measuring length—the method of repeated application of a single unit object—the object, U, is the unit object of length: its length is the unit of length. Method 1—the method of addition of unit objects—makes use of a number of U's, all equal in length. The length of any of these U's is the unit of length. As a first approximation at a definition, we may say that the unit in a system of measurement is the magnitude of the unit object or objects in that system. Special terms are usually devised to refer to units: for example, "inch," "meter," "gram," and the like. The unit of length in the metric system is the meter, and the meter is the length of a rod kept under special conditions in a vault near Paris (the "standard meter"). The distinction between unit intervals and unit objects is more difficult. This distinction is introduced because in chapter 7 I describe a procedure for the measurement of pitch in which a unit interval— the half tone—replaces the unit object. In methods for measuring length, a "unit interval" could only be the difference in length between an object equal in length to the unit object and one twice the length of the unit object, or the difference in length between an object twice the length of the unit object and one three times the length of the unit object, and so on. That is, the unit interval here apparently is defined in terms of the unit object. Method 4 for measuring length, helps to clarify the point. A ruler is constructed by laying a unit object in successive adjacent positions along a rod, and marking it and numbering

202

BROADER VIEWS OF MEASUREMENT

the marks. The distance between any two adjacent marks is the unit interval of length, and it is defined by the unit object that is laid along the rod. The classical view of measurement requires that a unit interval be defined by a unit object. This requirement is attacked in chapter 7, which defends a system of pitch measurement employing the unit interval of a half tone, a unit not defined by any unit tone. In an earlier paragraph I suggested the following definition of "unit": The unit in a given system of measurement is the magnitude of the unit object in that system. But if, as chapter 7 proves, a system of measurement can employ a unit interval not defined by any unit object, the definition of "unit" must be revised as follows: The unit in a system of measurement is the magnitude of the unit object or unit interval in that system. In the present section I attempt to state the salient requirements of units of measurement. Since chapter 7 contains an extensive discussion of the notion of a unit interval, the major concentration here is on the notion of a unit object. It is reasonable to assume that we cannot require of unit intervals what we do not require of unit objects (although the converse, if true, is not obviously so). The size of the unit.—It is often said that the unit in a system of measurement is "arbitrary" or "conventional." In some senses of these vague terms the assertion is true; in other senses it is false. If we wish to measure the length of golf clubs by the method of repeated application of a unit object, we do not select as the unit object one that is equal in length to the longest club to be measured. If a unit this size were used, the system would assign the numeral 1 to every club measured, and would consequently be worthless. Perhaps it could not even be regarded as a system of measurement. If, on the other hand, we wish to measure the length of telephone poles, a unit the length of a golf club could serve the purpose quite well (depending on the degree of precision required). The choice of a unit is determined by the magnitude of the objects it will be used to measure and is, in that respect, not arbitrary. But in most instances there is an area of choice within which the unit can easily be one magnitude as another; in this respect the choice is arbitrary. The precision obtainable in a given system of measurement is partly a function of the magnitude of its unit: the shorter the unit object, the more precise the measurements. Suppose that in one system

BROADER VIEWS OF MEASUREMENT

203

for measuring telephone poles the unit is 1 / 1 0 0 the length of the longest pole, and that in another system the unit is 1/25 the length of the longest pole. In the first system, different numerals are assigned to the longest pole and to one that is 99/100 the length of the longest, but in the second system, the same numeral is assigned to the two poles and the system is therefore less precise. Precision is, in general, a desirable feature in a system of measurement. But it is no more than this. A system does not cease to be a system of measurement simply because it is relatively imprecise. Note, further, that the notion of "absolute precision" in a system of measurement is without meaning. The 1 / 1 0 0 unit mentioned above yields more precise results than the 1/25 unit, and a 1/200 unit would yield more precise results still. But we cannot, by decreasing the magnitude of the unit, finally devise a system that yields "absolutely precise" measurements. A final point worth noting is that increasing precision can bring with it certain disadvantages. As the length of the unit object of length is decreased, the assignment of numerals to telephone poles becomes more precise. But at the same time it becomes increasingly harder to use the unit object, to lay it correctly in successive adjacent positions along the telephone pole. If the unit object is so short that it cannot be detected by the senses, or by available instruments, it cannot be used at all. (For a more detailed discussion of precision, see chapter 6). The unit in a system of measurement is commonly said to be "conventional," which in one sense is plainly true. The choice of units is based, not on the ground that a certain magnitude is the "correct" magnitude for a unit, but on the ground that a certain magnitude best serves a certain group of human purposes. Several magnitudes may serve these purposes equally well. If so, we are at liberty to select any one of them which suits our fancy, but we are not at liberty to change the unit without warning, or to use one that most investigators do not or will not use. Measurement is useful only to the extent that we are able to duplicate our own measurements and compare them with those of other investigators. Duplication and comparison, however, are not possible if different investigators use different units and the unit used by one investigator cannot be converted into the unit used by others (as the units of the gram-centimeter scale can be converted into those of the pound-inch scale). For this reason units are adopted and fixed

204

BROADER VIEWS OF MEASUREMENT

by international bodies of scientists, and changes are made only by official decision. These adoptions and decisions are "conventions" which the scientific community agrees to follow. The foregoing remarks, so obvious as to be almost trivial, serve as a reminder of the requirements that are, for good reason, imposed on the units employed in measuring length, weight, and other familiar physical dimensions. The reminder is necessary because there is a tendency, when discussing the possibility of measuring the so-called psychological dimensions, to impose more stringent or different requirements on the units proposed for measuring them. In chapters 6 and 7 systems of measurement are proposed for loudness and pitch. It is unfair to demand of the units in these systems that they satisfy requirements in addition to or more stringent than those used to measure length, weight, and time. Stability of the unit of measurement.—Most of the remaining requirements normally imposed on units by theorists of measurement can be reduced to these two: (i) the unit in a system of measurement must not change from one measurement to another; and (2) the unit in a system of measurement must not change during a single measurement. 1) T w o reasons may be offered in support of the requirement that the unit remain unchanged from one measurement to another. First, unless the requirement is satisfied it is impossible to duplicate the measurement of a given object, and thus impossible to confirm the measurement. Suppose we measure the length of object X by the method of repeated application of a unit object (Method 2) on two separate occasions. Suppose also that the length of the unit object changes between the two occasions, but that the length of X remains unchanged. Then lack of agreement between the two measurements of X is not a disconfirmation of the earlier measurement. Suppose, on the other hand, that the length of the unit rod changes from one measurement to the next, but that the length of X also changes so as to cause the two measurements to agree. Agreement between the two measurements of X is not, under these circumstances, a confirmation of the earlier measurement. A second reason for the requirement is that unless it is satisfied it is impossible to compare the results of measuring different objects,

BROADER VIEWS OF MEASUREMENT

205

for example, a floor and a carpet to cover it. If the unit of measurement changes between the time the floor is measured and the time the carpet is measured, the carpet will not fit the floor (unless, of course, either the floor or the carpet changes in size after being measured). If two different objects, when measured, are found to be the same number of units in length, the two objects can be expected, when aligned, to equal each other. If one object is found to be twice the length of the other, it is expected that two placements of the shorter in adjacent positions are required to equal the length of the longer. These expectations will not be fulfilled if the unit of measurement changes from one measurement to the next (so long as the objects being measured do not change in length). For these reasons we employ units in our systems of measurement that we believe to be stable. In devising a system of length measurement we do not select a unit object made of ice or rubber, or one composed of a substance that reacts easily with other substances. In devising a system of weight measurement we do not select unit objects that chip easily, or objects made of magnetized iron, or those that absorb moisture from the atmosphere. Furthermore, we mark the unit object in such a way as to be able to identify it easily on any occasion of its use, since the use of a different unit object may result in a change in the unit magnitude. Of course, we may use any object equal in magnitude to the designated unit object as easily as the unit object itself. We may wish to make a number of copies of the unit object so that different investigators may easily carry on a joint enterprise. The unit object is then comparable to the standard meter in Paris of which all metersticks are supposed to be copies. In spite of these precautions the unit of measurement may change from one measurement to the next. How can such changes be discovered? For most systems of measurement in actual use, we are rarely able to make this discovery by means of the unaided senses. If a meterstick should one day look no longer than a pencil, we would be justified in concluding that it had changed in length (unless, of course, we knew that our eyes were defective, or that the meterstick was seen through a distorting medium, etc.). If a weight marked " 1 0 grams" should one day feel light as a feather, we could conclude that its weight had changed (unless, of course, it had carried out of the earth's gravitational

206

BROADER VIEWS OF MEASUREMENT

field, or the observer had some muscular disease, etc.). In carefully devised systems of measurement, however, gross changes of this kind do not occur. The chain balance found in most chemistry laboratories is accurate to 1/10,000 of a gram. Some of the unit weights used with these balances are so light (1/50 gm, for example) that a minute loss of metal changes its weight and affects the results of weighing. Changes of this kind, which occur when the weights are handled carelessly, cannot be detected by the unaided senses. Whether such a change has occurred is determined in practice by comparing the questionable set of weights with a new set on the balance. In the scientific laboratory and in daily life, we insure that changes in units do not occur by using instruments of measurement (metersticks, graduated cylinders, sets of weights) manufactured by a reliable firm under controlled conditions. But this precaution is irrelevant if the instruments are manufactured and the unit objects are chosen by ourselves, as they are in the four simple methods of length measurement described earlier. Furthermore, the question as to whether the unit of measurement has changed can arise for the models used by manufacturers to make copies for widespread use. It can arise, even more basically, for internationally accepted standards stored in vaults tinder special conditions, like the one near Paris. At this basic level, whether the unit of measurement has undergone some perceptually undetectable change can be determined only by reference to the reasons that lead us to adopt stable units. If Method 2 for length—repeated application of a unit object—is properly employed to measure object X on two occasions, but with differing results, and if X has not changed in length from one occasion to the other, we may conclude that the unit object in Method 2 has changed in length. If objects X and Y are found to be the same number of units in length by using Method 2, and yet X and Y do not equal each other in length, then, if X and Y have not changed in length, we may conclude that the unit in Method 2 has changed in length between the measurement of X and the measurement of Y. That is to say, our inability to duplicate the result of measuring the same object, and our inability to compare the results of measuring different objects are, at this basic level, tests for detecting instability in the unit of measurement. Is the requirement that the unit in a system of measurement remain unchanged from one measurement to the next a logical or an em-

BROADER VIEWS OF MEASUREMENT

207

pirical requirement? To call it a logical requirement is to say that it is part of the definition of measurement, that no system in which the unit is unstable qualifies as a system of measurement. To call it an empirical requirement is to say that it is a desideratum of measurement, but not a part of the definition. A system that fails to satisfy an empirical requirement is undesirable or relatively useless, but it does not thereby fail to qualify as measurement. Whether the requirement of stable units is logical or empirical is not an easy question, for at least two reasons. First, systems with a unit that changes from one measurement to the next permit the duplication of measurements of a single object and the comparison of the results of measuring different objects if the objects being measured also undergo changes directly proportional to any change in the unit. This statement makes it appear to be an empirical fact that a system with an unstable unit does not permit duplication and the comparison of results and, consequently, seems to imply that the desirable features of duplicability and comparability can be achieved by systems that do not possess a stable unit. But matters are not so simple. For if, as the hypothesis states, units and measured objects always underwent directly proportional changes, so that duplication and comparison of results were always possible, the so-called changes would not be detectable. They would be changes without a change, which are, perhaps, not really changes. The second area of unclarity concerns the status of the features of duplicability and comparability of results. Are they merely desirable features of a system of measurement, or features that a system must possess in order to qualify as a system of measurement? If they are merely desirable features, then any requirement designed to insure their presence, such as the requirement of stable units, is only an empirical requirement of a system of measurement. The following tentative assessment is made with these unclarities in mind. Measurement may be regarded as a kind of test, a test for the amount of a property possessed by an object. Now a test must allow for the duplication and comparison of its results. If each liquid that is capable of changing the color of litmus sometimes turned it red, sometimes blue, in a quite unpredictable manner, litmus could not be used in a test for acidity and alkalinity. Similarly, a procedure for assigning numerals to objects which has unpredictable, inconsistent results, cannot be used as a test for the amount of a property of the

208

BROADER VIEWS OF MEASUREMENT

object. If the unit in a system of numeral assignment is unstable, the system will lead to unpredictable, inconsistent results, unless the objects to which numerals are assigned change in direct proportion to changes in the unit. But the hypothesis of such concomitant changes either is so unlikely that it merits no consideration, or else it is meaningless (since it hypothesizes a change without a change, which is not a change). Consequently, the requirement that the unit in a system of measurement remain relatively stable from one measurement to another may be regarded as a logical requirement. I say that the unit must be relatively stable because certain kinds of change in the unit clearly would not disqualify the system as one of measurement. Suppose, after a system has been used for several years to make a large number of measurements, the unit changes, and that we continue to use the system to measure, among others, some of the objects measured before the change in the unit. By comparing measurements of the same object, we could then discover precisely how the unit had changed and could translate the results of the earlier measurements into those of the later. If, for instance, later measurements of object X assign it n units, and earlier measurements assigned it in units, the results of each of the later measurements must be multiplied by 2. The possibility of making this translation depends on a relatively stable unit, one that remains unchanged throughout a substantial number of measurements. If the unit should change between each measurement and the next, it would not be possible to translate the results of one measurement into those of another. 2) At first sight, the requirement of stability of the unit within a single measurement seems to be a logical requirement, owing to the following considerations. By definition, a system of measurement for a given dimension enables us to express the magnitude of any object within the dimension in terms of dimension units. Thus, after measuring a rod, X, by the method of repeated application of a unit object, U, we can say that the rod is 10 U's in length. But suppose U undergoes changes in length while being laid in successive adjacent positions along X. What, then, can be the meaning of the assertion that "X is 10 U's in length"? The quoted statement refers to a single unit and presupposes that it is everywhere the same. If the unit object changes within a single measurement, this presupposition is falsified and the quoted statement is meaningless. A system of length measurement

BROADER VIEWS OF MEASUREMENT

209

must, by definition, render statements like the one quoted meaningful, and no system can impart meaning to such statements if the unit changes within a single measurement. The above reasoning does not withstand examination. The statement " X is 10 U's in length" means simply that if unit object U is laid in successive adjacent positions along the length of X, ten such positionings are required to reach the end of X. Hence, the quoted statement can be true even though the length of U changes as it is laid along X. To suppose that the quoted statement means something else is to divorce it from the empirical procedures that give it meaning. It is to misconstrue it as a pseudomathematical, abstract statement without empirical content. The most viable defense for the requirement that the unit remain unchanged within a single measurement is the one offered for its predecessor, namely, that otherwise the system of measurement will not permit the duplication of the results of measuring a single object or the comparison of the results of measuring different objects. If, in the method of repeated application of a unit object, the unit object, U, shrinks slightly with each successive placement along the length of X, and if it always shrinks in the same way for a given placement, then several different measurements of X will duplicate one another. Furthermore, comparison of the results of measuring X and Y will be possible. If X and Y are found to be the same number of units in length, X and Y will align at both ends with each other. If, on the other hand, U shrinks with successive placements along the length of X, but in different ways during different measurements, then different measurements of X will not duplicate one another, and comparison of the results of measuring different objects will not be possible. These considerations show that it is a requirement of any system of measurement either that the unit remain unchanged during a single measurement or that it change only in a manner that is consistent from one measurement to the next. Even if the unit object changes within a single measurement in a manner that is consistent from one measurement to the next, its use does not permit certain sorts of comparison. If, with a shrinking unit object, X is found to be n units in length and Y is found to be zn units, then X requires less than two adjacent placements to equal the length of Y (unless, of course, X and Y , while being positioned, also shrink in such a way as to offset the shrinkage in the unit object).

210

BROADER VIEWS OF MEASUREMENT

Another reason for the requirement that the unit remain unchanged within a single measurement is that otherwise arithmetical statements cannot be interpreted in terms of and cannot be applied to physical entities. The axiom of additivity, (9) N(X) + N(Y) = N(X @ Y), says that, for a given system of measurement, the number it assigns to X added (mathematically) to the number it assigns to Y must equal the number it assigns to the (physical) addition of X and Y. If the axiom is not satisfied by a system of measurement, that system cannot be used to apply such arithmetical statements as "3 + 3 = 6" to rods, weights, and other physical objects. If the unit in a system of measurement changes within a single measurement, even in a manner that is consistent from one measurement to the next, axiom (9) will not be satisfied. Suppose U, the unit object in the method of repeated application of a unit, shrinks with each successive placement along the rod being measured. Suppose we measure rod X with U and find it to be 3 U's in length, and that we measure Y with U and find it also to be 3 U's in length. If we now juxtapose X and Y and measure the juxtaposition with U, we will discover their combined length to be less than 6 U's, thus falsifying (9). Of course, if X and Y shrink on juxtaposition in such a way as to offset the shrinkage in U, we will find their combined length to be 6 U's, and (9) will remain true. Is the requirement that the unit remain unchanged within a single measurement a logical, or an empirical, requirement of a system of measurement? At first sight it seems to be empirical. If the requirement is not met, the system does not enable us to apply arithmetical statements to the objects measured. Although it is clearly desirable for a system to have this use, a system without it may nevertheless qualify as one of measurement. Consider, for example, a system that has the other principal use of measurement—a system that enables us to discover numerical laws relating the dimensions measured. Furthermore, it may be possible to describe a system of measurement—that is, a procedure of assigning numerals to things by empirically comparing them with a unit—which does not possess either of the two principal uses of measurement. Other considerations suggest that the requirement of a stable unit within a single measurement is not merely an empirical, but a logical, requirement. A system that does not meet this requirement does not,

BROADER VIEWS OF MEASUREMENT

211

as we have seen, permit various comparisons of the results of measurement, comparisons the possibility of which seems essential to a system properly regarded as one of measurement. In any event, there is a requirement very like requirement (2) which a system must meet if it is to permit any duplication and any comparison of the results of using the system. The requirement is that either the unit must not change within a single measurement or it must change in a manner that is consistent from one measurement to the next. On balance, it would appear that requirements (1) and (2) are both sufficiently critical to be regarded as logical requirements of a system of measurement. A system must, in order to qualify as one of measurement, employ a unit that remains unchanged from one measurement to the next and unchanged within a single measurement. The equality of units.—We are now in a position to discuss the requirement usually expressed in the statement that "the units in a system of measurement must be equal." In one sense this statement is plainly false. In the metric system of length measurement there are several units: the meter, the centimeter (hundredth of a meter), the millimeter (thousandth of a meter), the micron (millionth of a meter). These other units are fractions of the meter, fractional units. Now the assertion that units must be equal is not intended to deny the possibility of fractional units. But what is meant by the claim that units must be equal in systems that have no fractional units, such as the four methods of length measurement described earlier? In Method 2—the method of repeated application of a unit—a single object, U, is designated and used as the unit object. The length of this object is the unit of length. It makes no sense to say that the unit objects in this system must be equal, for this implies that there is more than one unit object. Similarly, it makes no sense to say that the units in this system must be equal, for this implies that there is more than one unit magnitude. All that can be meant by saying that the units in this system must be equal is that the length of the unit object must not change, either from one measurement to the next or within a single measurement. Therefore the reason for saying that units must be equal is just as strong or just as weak as the reason for saying that the unit must not change. In Method 1—the method of addition of unit objects—a unit object, U, is first selected. Next a number of additional objects equal

212

BROADER VIEWS OF MEASUREMENT

in length to U are discovered. All these equally long unit objects are used to form juxtapositions with which to compare the object being measured. Let us call the first object selected the primary unit object, and the objects equal to it secondary unit objects. Since there is only one primary unit object, it makes no sense to ask whether the primary unit objects are equal. But we can ask whether the unit objects, both primary and secondary, are equal. Consequently, to say that the units in Method i are equal must mean (a) that the unit objects, primary and secondary, are equal, and (b) that the unit magnitude remains unchanged from one measurement to the next and within a single measurement. The preceding section contains a discussion of (b) as a requirement of a system of measurement. Is (a) a requirement of a system of measurement? It is, by definition, a requirement of Method 1 , the description of which stipulates that secondary unit objects be equal in length to the primary unit object. The interesting question, however, is whether every method that, like Method i , involves the addition of unit objects must contain such a stipulation. If a number of unit objects are unequal in length, and there is no rule for juxtaposing them, then in one measurement X may be found equal to a juxtaposition of 5 U's, whereas in another measurement X may be found equal to a juxtaposition of 6 U's. The duplicability of results is therefore not insured by this method. If, on the other hand, a rule for adding unit objects is given, so that they must be juxtaposed in a certain order, then different measurements of X will assign to it the same numeral, in spite of the inequality of unit objects, and duplication of results is possible. Even if a rule for the addition of unequal unit objects is given, however, the system does not permit certain kinds of comparisons. If the system assigns 2 to X and 4 to Y , two adjacent placements of X will not equal Y. Also, the system does not permit the application of arithmetical statements to the objects measured, since it does not satisfy such axioms as (9), the axiom of additivity. Perhaps a system that does not permit the application of arithmetical statements is not thereby disqualified as a system of measurement, but one that does not permit the comparison of results does seem not to qualify. Let us summarize and try to reach a conclusion. With respect to systems of measurement which employ only one unit object (such as Method 2), the requirement that "units must be equal" is simply the

BROADER VIEWS OF MEASUREMENT

213

requirement that the unit must be stable, must remain unchanged from one measurement to another and within a single measurement. Since we have tentatively concluded that the requirement of a stable unit is a logical requirement of a system of measurement, we must also tentatively conclude that the requirement of equal units is a logical requirement for systems employing only one unit object. With respect to systems of measurement employing more than one unit object (such as Method 1), the requirement that units must be equal is the dual requirement that the unit be stable and that the unit objects be equal. Here again, our tentative conclusion that the requirement of unit stability is a logical requirement of a system of measurement forces us to the tentative conclusion that the requirement of equal units is also a logical requirement. This assessment is reinforced by the fact that if the unit objects are not equal, certain kinds of comparisons of the results of measurement are impossible. A system that does not permit such comparisons does not seem to qualify as one of measurement.

6 The Measurement of Loudness In this chapter a system of independent measurement for the loudness of tones is described and defended. I say "tones" rather than "sounds" because a distinction is usually made between noises and other kinds of sounds. Noises have no clearly identifiable pitch. A noise is either so brief in duration that it is extremely difficult to say what its loudness and pitch are, or its loudness and pitch change so rapidly and widely that it seems to have neither loudness nor pitch. (Perhaps the proper thing to say is simply that a noise has neither loudness nor pitch.) Tones are different; most persons are able to say, by listening to two tones, whether they are or are not of the same loudness or pitch. Tones are, we may say, "musical." The system of measurement described herein applies to tones; noises are not considered. The discussion in this chapter is based on the working assumption that loudness is not identical with any dimension or dimensions of sound waves, such as intensity and frequency. The relation between loudness and dimensions of sound waves is taken to be one of correlation (perhaps causation), not one of identity. (The basis for these assumptions is provided in chapter 3.) The procedure followed in the chapter consists in describing a system of numeral assignment for loudness which qualifies as independent measurement, by either Definition 1 or Definition 4, and defending it against various objections. It is a fact, verifiable by anyone who can hear and has a reasonably good ear, that two tones sounded or played simultaneously are louder than either one sounded or played alone. That is, there is a physical operation of addition for loudness of tones. A physical operation of addition is an operation of combination which, when performed on objects in a dimension, results in a combination greater in respect to the dimension than any of the components. Since playing tones simul-

THE MEASUREMENT OF LOUDNESS

215

taneously satisfies this definition, it qualifies as a physical operation of loudness addition. It would therefore seem possible to devise a system of measurement for loudness which employs the addition of unit tones. First, a relatively soft tone, U, is designated as the unit tone. Then an indefinite number of additional tones (U's), each of which is equal in loudness to the first, are discovered. Measurement of the loudness of a given tone, X , is then accomplished as follows. Different numbers of unit tones are played simultaneously until a collection of tones is obtained whose combined loudness is equal to that of X. A count of the number of tones in the required collection provides the numeral to be assigned to X. (This method should be compared with Method 1 for the measurement of length—the method of addition of unit objects —described in chapters 3 and 6.) A more elegant and efficient system may be built on the one above by selecting a group of standard tones. A tone, Ti, equal in loudness to the unit tone, U , is discovered. Then a tone, T2, equal in loudness to the combined loudness of U and Ti played simultaneously, is discovered, and then a tone, T3, equal in loudness to the combined loudness of U and T2 played simultaneously. This process is continued for an indefinite number of T's. Measurement of a given tone, X, is accomplished by ascertaining which of the T ' s X most nearly equals in loudness. The subscript numeral of this standard tone is assigned to X. (This method should be compared with Method 3 for the measurement of length—the method of comparison with standard objects—described in chapters 3 and 6.) OBJECTIONS FROM SUBJECTIVITY Objection One.—One

objection to the method above is that there

is no adequate w a y of identifying unit tones, and that, therefore, there is no assurance (a) that the unit tones in the procedure remain equal and unchanged during a single measurement, or (b) that the unit tones remain equal and unchanged from one measurement to the next. A s shown in chapter 5, if (a) or (b) is false, the procedure is useless, since it cannot be used to confirm earlier measurements and the results of using it at different times are incomparable. There are ways of identifying unit objects of length and weight: writing " U " on them, painting them all one color, and so on. But tones are events and cannot be labeled or

216

THE MEASUREMENT OF LOUDNESS

painted. Only memory can be relied upon, and memory of tone loudness is unreliable when the tones are separated by any considerable length of time. My recollection that the tone played yesterday has the same loudness as the one I hear today is not to be trusted. Furthermore, another person's recollection of the relative loudness of the two tones may not agree with mine. How could such disagreement be resolved? Surely some criterion other than the purely subjective one of memory is required for identity of unit tones; but no other is available. This objection is not sound. Admittedly, one cannot label an event as one labels a physical object, but there is a way of identifying tones which is just as good for measurement as labeling physical objects. Tones may be identified through their causes. For example, if a tone is produced on Monday by dropping a steel ball on a tuning fork from a certain height, we may assume that a tone produced on Tuesday by dropping the same steel ball on the same tuning fork from the same height will have the same loudness as the tone produced on Monday (unless, of course, the characteristics of the fork have changed, or the earth's gravitational field has been altered, etc.). Again, the loudness of a vibrating string is a function of the amplitude of its vibration. Hence, if the same string stretched with the same tension is vibrated on two different occasions, the loudness of the two tones produced will be the same. It may be said that these causal criteria for the identity of unit tones are inadequate, that the same string stretched with the same tension between the same two points may produce on Tuesday a tone having a different loudness from the tone produced on Monday. But the same point can be made against the criteria employed for the identity of unit objects of length and weight. A unit rod labeled U is used on Monday and again on Tuesday. Although the rod may have changed in length in the interval between its two uses, unit objects of length are chosen so as to prevent such changes, in accordance with what is known about the nature of various substances. We do not employ a rubber object, or one made of ice, as the unit object of length because we know that these materials are subject to sudden alterations in length. Similarly, we can produce unit tones in such a way as to prevent changes in loudness from one playing to the next, and how we decide to produce them will depend on what we know about tone production. We will not produce unit tones by means of

THE MEASUREMENT OF LOUDNESS

217

strings whose tension may suddenly and unpredictably relax or increase, for we know that such strings may very well produce tones of different loudness on different occasions. A causal criterion is therefore available for the identity of unit tones which is just as acceptable as the criteria employed for the identity of unit objects of weight and length. Objections in principle against employing the former criteria apply equally well to the latter criteria. The above considerations show that there is no greater necessity for relying on memory of relative magnitude in the identification of unit tones than there is in the identification of unit rods. The label criterion is available for unit rods, the causal criterion for unit tones. Since Objection One regards memory of relative magnitude as a "subjective" criterion, it is important to point out that both types of identification rely, in two noteworthy respects, on memory of relative magnitude. First, the use of the label criterion for rods, as well as the use of the causal criterion for tones, can be justified only through the use of memory of relative magnitude. We are justified in assuming that the same tuning fork struck with the same force on widely separate occasions will produce tones of the same loudness, because we have observed that tones of the same loudness are produced when the occasions are close together in time. These observations require the use of memory of relative loudness. The same thing is true of the justification of the label criterion. We may assume that the rod labeled U has the same length on Monday as it does on Tuesday, because we have observed that U does not undergo changes in length from one moment to the next. These observations require the use of memory of relative length. Second, in identifying either unit rods or unit tones, the memory criterion may be used, in certain circumstances, to overrule any other that may be employed. If we remember at 12:02 that a rod was much shorter at 1 2 : 0 1 , we conclude that it does not have the same length as it did previously, even though it may be labeled U on both occasions. If at 12:02 we remember that a tone played at 12:01 was much less loud, we conclude that the two tones do not have the same loudness, even though they are produced by the same tuning fork struck with the same force. In the identification of unit tones we rely on memory of relative magnitude in the two respects above. If such reliance vitiates any procedure of numeral assignment employing unit tones—because of

218

THE MEASUREMENT OF LOUDNESS

the alleged unreliability of memory of relative magnitude—then the measurement of length is no less impossible than the measurement of loudness, for in the identification of unit rods we also rely on memory of relative magnitude in the two respects above. The correct view, it would seem, is that memory of relative magnitude is not completely unreliable. It is reliable when (a) the objects being compared are perceived on occasions close together in time, and (b) the difference in magnitude between the objects being compared is relatively large and, therefore, easily recognizable. Now, in regard to the unit objects employed in a procedure of measurement, conditions (a) and (b) are rarely satisfied. In the first place, these objects are usually employed in measurements made on occasions widely separated in time. One's recollection that the unit objects have or have not changed in magnitude over a long period of time is, on the whole, not to be trusted. Second, in applying a procedure of measurement we wish to avoid even very slight changes in the magnitude of the unit objects. Memory of relative magnitude cannot be trusted in the detection of such changes. Since conditions (a) and (b) are rarely satisfied in regard to unit objects of measurement, a criterion for identifying these objects other than memory of relative magnitude is needed. Such a criterion is available, as we have seen, for loudness as well as for length. A final point should be made in replying to Objection One. As noted in chapter 5, in some instances it may be extremely difficult to say whether changes have or have not occurred in the unit objects of a system of length or weight measurement. The question may then have to be settled on the basis of whether different measurements of the same object have the same result. Because a measurement of rod X at time ti shows X to be 10 U's in length and a measurement of X at time t2 shows X to be 1 1 U's in length, we may conclude that the unit objects of length have changed in length from one measurement to the next. Obviously the same test is available for the proposed system of loudness measurement. Here, then, is my general conclusion: whatever criteria or tests for the identity of or change in unit rods may be proposed, similar and equally acceptable criteria or tests can be proposed for the identity of or change in unit tones. Objection Two.—A second possible objection to the proposed system of loudness measurement is that no acceptable method has been or can be specified for determining whether two tones (or collections

THE MEASUREMENT OF LOUDNESS

219

of tones) played in immediate succession are equal in loudness. Some way of determining whether they are is presupposed in the system at two points. First, the selection of unit tones requires selecting a primary unit tone and then discovering an indefinite number of secondary unit tones equal in loudness to the primary one. Second, to measure a given tone, T, various numbers of unit tones are simultaneously played until a collection equal in loudness to T is discovered. The test for equality is listening to tones (or collections of tones) played in immediate succession. If two tones (or collections of tones) sound equal in loudness, they are taken to be equal in loudness. Objection Two says that the determination of relative loudness through listening, a method relying solely on sense perception, is unacceptable. Anyone pressing this objection would probably say that the method is "subjective." Before dealing with Objection Two it is necessary to show in what respect it differs from Objection One, with which it may easily be confused. According to Objection One, there is no acceptable criterion for the identity of unit tones; that is, there is no acceptable method for determining whether tones not played in immediate succession are equal in loudness. Memory of relative loudness is deemed unreliable for this purpose, and no other criterion is seen to be available. The point of Objection Two is that there is no acceptable method for determining whether tones played in immediate succession (e.g., half a second apart) are equal in loudness. Here no question of the reliability of memory can arise, not because one's memory of the magnitude of a tone played half a second ago is reliable, but because it is incorrect to say that one remembers a tone played half a second ago. We do not say, of two tones played half a second apart, "The tone I hear now has the same loudness as the one I heard previously," but rather, "The two tones have the same loudness." Two tones played in immediate succession are both in the "specious present." Objection One deals with the question of criteria for the identity of unit tones; Objection Two, with the question of determinations of relative magnitude for tones played in immediate succession. That these are different questions will become clearer by reflecting on procedures of length measurement. The question of how we determine that the unit rod used today has the same length as the one used yesterday is not to be confused with the question of how we

220

THE MEASUREMENT OF LOUDNESS

determine that two rods presently before us have the same length. Similarly, the question of how we determine that the unit tone employed today has the same loudness as the unit tone employed yesterday must be distinguished from the question of how we determine that two tones played in immediate succession have the same loudness. Objection Two is concerned with the latter question, and it maintains that the proposed system of loudness measurement provides no satisfactory answer. An acceptable method for determining whether two rods presently before us are equal in length consists in aligning them. But, the contention is, the proposed system of loudness measurement does not provide, and cannot provide, an acceptable method for determining whether two tones played in immediate succession are equal in loudness. The analysis in the last part of chapter 4 provides a basis for arguing that this objection is unsound. There it was established that procedures of measurement may employ methods for determining relative magnitude which, to use the phrase of the present objection, rely solely on sense perception. Characterized more precisely, the type of method in question has the following feature: From the fact that X and Y appear to some particular sense to be equal (unequal) with respect to a given dimension, it is concluded that X and Y are equal (unequal) with respect to that dimension. Any method fitting this description was called an «-method or an «-determination. It was established that «-methods are not, in one crucial sense, subjective; and that, although they may or may not be subjective in a second sense, the fact that they are does not entail that they cannot be employed in procedures of measurement. Let us refer to subjectivity in the first sense as logical subjectivity, and to subjectivity in the second sense as empirical subjectivity. (The analysis in chapter 4, where logical and empirical differences between a- and b-methods are explored, explains this choice of terminology.) That a-methods are not logically subjective may be illustrated with the example of loudness. If, because tones Ti and Tz sound equal in loudness, we conclude that Ti and T2 are equal in loudness, we are employing an «-method for determining relative loudness. That the italicized statement is corrigible can be established by the same line of argument used in chapter 4 to show that "X feels heavier than Y " is corrigible. Furthermore, in concluding that Ti and T2 are equal in

THE MEASUREMENT OF LOUDNESS

221

loudness because they sound equal in loudness, we are neither estimating nor predicting that Ti and T2 would be found equal in loudness by some method other than the method of simply listening to the two tones. Again, this point can be established by the same line of argument used in chapter 4 to show that "X feels heavier than Y " is neither an estimate nor a prediction of the results of some method other than lifting. On the other hand, the «-method for determining relative loudness may be empirically subjective; that is, it may provide a relatively low degree of sensitivity, observer agreement, or certainty of determination. But the fact that it has these disadvantages does not entail that the procedure of numeral assignment in which it is employed cannot properly be regarded as one of measurement. At most, what follows is that the procedure of measurement is relatively useless. Finally, it should be noted that the «-method may not be empirically subjective. Whether it is or is not is a factual question, one that cannot be settled without putting the method to use; perhaps the method will provide a relatively high degree of observer agreement, certainty of determination, or sensitivity. These points may be emphasized by referring to the unusual system of weight measurement described in chapter 4. A unit object of weight is selected and labeled U. Then an indefinite number of objects equal in weight to U are discovered by the method of lifting and are also labeled U. Two objects that feel equally heavy when lifted at the same time, one in each hand, are taken to be of equal weight. The weight of a given object, X, is measured by lifting X in one hand and collections of U's in the other, until a collection equal in weight to X is discovered. The fact that this system of weight measurement employs an a-method for determining relative weight—a method that relies solely on sense perception and is, perhaps, empirically subjective—does not entail that it cannot properly be regarded as one of measurement. Neither does this conclusion follow from the fact that a system of loudness measurement employs an «-method for determining the relative loudness of tones. OBJECTIONS FROM RELATIVITY Objection Three.—A critic who compares my procedure for measuring loudness with methods for weight measurement employing

222

THE MEASUREMENT OF LOUDNESS

highly sensitive balances, or even methods for measuring length by means of a ruler, may say the former is too imprecise (or inexact) to be regarded as one of measurement. In order to assess this objection, clear meanings must be given to the terms "precise" and "imprecise." The most suitable definition would seem to be one in terms of univocal assignment of numerals. The assignment of numerals to objects within a dimension is univocal when the following conditions are satisfied: (a) if X E Y then N(X) - N(Y), (b) if X G Y then N(X) > N(Y), where E means "equals," C means "greater than," and = and > are the conventional arithmetic symbols. Whether the correct employment of a system of measurement does or does not result in a univocal assignment of numerals depends on (i) the sensitivity of the method employed for determining relative magnitude, and (ii) the magnitude of the unit relative to the magnitudes of the objects to be measured. The role of (i) is obvious. If a given balance does not detect differences between the weights of a cubic centimeter each of salt, sand, iron, and lead, then a procedure of measurement employing this balance will not assign numerals univocally to the several samples. If the cc of lead is chosen as the unit object, the numeral i will be assigned to the other three samples, since the sample of lead will be balanced by any of the other samples. If the cc of salt is chosen as the unit object, again the numeral i will be assigned to the other three samples, since the sample of salt will balance any of the other three samples. In general, if a system of measurement is never to result in an equivocal assignment of numerals to a given set of objects, the system must employ a method for determining relative magnitude which is capable of detecting every difference, no matter how small, between objects in the set. Now let us examine the role of (ii). Using a balance more sensitive than the one just described, let us select as the unit an object that weighs 1 gram by the conventional system. Next let us measure three objects, X, Y , and Z, which actually weigh 1.8 grams, 2.0 grams, and 2.2 grams, respectively. The resulting assignment of numerals will be equivocal, X, Y , and Z each receiving the numeral 2. Suppose, however, using the same balance and unit, we measure three different objects, U, V, and W, respectively 2.0 grams, 3.2 grams, and 4.4 grams in weight. U will receive the numeral 2, V the numeral 3, and W the numeral 4, making the assignment univocal. Or, by selecting a smaller

THE MEASUREMENT OF LOUDNESS

223

unit, say one that is 0.2 grams in weight, numerals may be assigned univocally to X, Y , and Z as well as to U, V , and W. These six objects will then receive, in the order listed, the numerals 9, 1 0 , 1 1 , 1 0 , 1 6 , 22. In general, for any group of objects, a univocal assignment of numerals to those objects requires the selection of a unit whose magnitude is equal to or less than the smallest difference between objects in the group. The notions of complete and relative precision, as applied to systems of measurement, can now be defined. (1) Of two systems of measurement for the same dimension, Mi and M2, Mi is the more precise if its correct employment would result less frequently in equivocal numeral assignments to objects within the dimension than would the correct employment of M2. (2) A system of measurement is completely precise if its correct employment would never result in an equivocal assignment of numerals to objects within the dimension. These definitions suffer (as any definitions of the same terms must) from an ambiguity in the phrase "objects within the dimension." The phrase may refer to any object in the dimension which has existed, now exists, or will exist; let us call any object of this type an actual object. On the other hand, "objects within the dimension" may refer to any conceivable object in the dimension; let us call any object of this type a possible object. Precision may be defined in terms of actual objects or in terms of possible objects. Under the first option, "actual objects" must be substituted for "objects" in the two definitions above, thus producing definitions ( l A ) and (2A). Under the second option, "possible objects" is substituted for "objects," producing definitions (iB) and (2B). If complete precision is defined as in (2B), no system of measurement can be completely precise. As noted earlier, if a system of numeral assignment is never to result in equivocal assignments, the unit in the system must be no larger than the smallest difference between objects to which numerals may be assigned. Now it is not possible to select a unit meeting this requirement when numerals may be assigned to any possible object. Given a unit of any specific magnitude, however small, there will always be possible objects the difference between which is smaller than the unit. One might suppose that a completely precise system of length measurement could be devised by selecting an extremely small unit of length, say the micron (a

224

THE MEASUREMENT OF LOUDNESS

millionth of a meter). But this system will assign numerals equivocally to objects whose lengths are 1/2, 2/3, 3/5, 1 - 1 / 2 , 1-2/3, and 1-3/5 microns. T o the first three will be assigned the numeral 1 , to the latter three the numeral 2. A system of completely precise measurement, in the sense of (2B), would require an infinitely small unit. But an infinitely small unit would have no magnitude; it would, in other words, not be a unit. Hence a completely precise system of measurement, in the sense of (2B), is logically impossible. This point has the following significance in the context of this chapter: Since, by definition (2B), no system of measurement can be completely precise, it is no objection against my system of loudness measurement that it is not, in this sense, completely precise. If, on the other hand, we adopt (2A) as our definition of complete precision, it is logically possible for a system of measurement to be completely precise. It is, however, highly unlikely that any such system will ever be devised because it would require (a) a method for determining relative magnitude which is capable of detecting every difference, however small, between actual objects in the dimension, and (b) a unit that is no larger than the smallest difference between actual objects in the dimension. And it is highly unlikely that a system of measurement meeting these requirements will be devised. In any event, it is no more difficult to devise a system of loudness measurement which meets these requirements than it is to devise similar systems for length or weight. That the method of listening to tones played in immediate succession is incapable of detecting every difference between actual tones is of no consequence, because a similar point applies to every existing procedure of length or weight measurement. In measuring length with a meterstick whose smallest unit is the millimeter, the method for determining relative length is to align the object being measured with the graduations on the ruler by means of the naked eye. It is impossible to detect differences in length of diameter between smoke particles by this method, but if an electron microscope is employed, it is possible to detect such differences. Even the electron microscope, however, is incapable of detecting differences between some particles smaller than smoke particles. T o summarize, complete precision in measurement is either logically impossible or highly unlikely, depending on which definition, (2B) or (2A), is adopted. Hence, regardless of the definition adopted, it

THE MEASUREMENT OF LOUDNESS

is illegitimate to maintain, on the ground that it is not completely precise, that the proposed system for assigning numerals to loudness is not measurement. This contention is equally illegitimate when supported by the argument that the system of numeral assignment is relatively imprecise. It is obvious that the relative imprecision of a procedure of numeral assignment is never a reason for refusing to qualify it as measurement. In establishing this point the difference between definitions (lA) and (iB) is immaterial. The measurement of length with a meterstick is less precise than its measurement with an electron microscope; that is, the former procedure would result more frequently in equivocal assignments of numerals to objects of length than would the latter. But it clearly does not follow that the former procedure is not one of measurement. Measurement by means of automobile scales would probably result more frequently in equivocal assignments of numerals to weight than would measurement by means of an electron microscope to diameter. But it clearly does not follow that the former procedure is not one of measurement. The above argument has interesting implications regarding Hempel's definition of measurement discussed—and rejected—in chapter 5. According to Hempel, measurement of a dimension consists in specifying criteria that assign to each object in the dimension exactly one real number, in such a manner that the following conditions are satisfied: ( f l ) X E Y if and only if N(X) = N(Y), and (b) X G Y if and only if N(X) > N(Y). More briefly, measurement consists in the univocal assignment of numerals to the objects within a dimension. On Hempel's definition, measurement must be precise: either relatively precise or completely precise, depending on how the definition is interpreted. But, as we have just seen, precision is not one of the defining features of a system of measurement: a system of measurement may be either precise or imprecise. Hempel's definition is therefore unacceptable. Objection Four.—It may be argued that the dimension of loudness of tones has no absolute zero and that, consequently, no procedure of numeral assignment for loudness can qualify as independent measurement. Those who maintain that independently measurable dimensions must have an absolute zero may do so for either or both of the following two reasons. First, a system of independent measurement for a given dimension must permit the relative magnitude of objects within the dimen-

226

THE MEASUREMENT OF LOUDNESS

sion to be expressed in ratios; a system of numeral assignment for a dimension that has no absolute zero cannot have this use. The dimension of temperature provides an illustration. We can assign numerals to objects in this dimension by employing either a centigrade or a Fahrenheit thermometer. But, because the zero on both scales is arbitrary, the numerals thus assigned do not represent temperature ratios. That X is found to be 2 degrees and Y to be 4 degrees centigrade does not imply that Y is twice as warm as X. Such statements are meaningless when the dimension does not possess an absolute zero. Second, a system of numeral assignment qualifies as independent measurement only if it permits the formulation of numerical laws concerning the dimension in question; a system of numeral assignment for a dimension that has no absolute zero cannot have this use. The system of numeral assignment which employs a centigrade thermometer and an arbitrary zero assigns positive numerals to some objects—those above the freezing point of water—and negative numerals to other objects—those below the freezing point of water. No single numerical law can describe the relation between temperature and some other dimension, such as pressure, when some temperature values are negative and some are positive. Two numerical laws would be required: one for the positive temperature values and one for the negative temperature values. There is a twofold reply to Objection Four. (1) In the only clear sense of absolute zero relevant in the present context, loudness does have an absolute zero. (2) Even if loudness had no absolute zero, it would not follow that it is incapable of independent measurement: the above attempts to show that it does follow are unsound. At this stage only (1) is argued; the argument for (2) is deferred to chapter 7, where the dimension of pitch, which does not possess an absolute zero, is considered. In chapter 4 a clear definition of the phrase "absolute zero" was formulated. It specifies that a dimension possesses an absolute zero if and only if the combination of any two objects in the dimension is greater than either of the component objects. Formally stated, a dimension possesses an absolute zero if and only if all its objects satisfy the axiom, (X @ Y) G X', where @ and C are defined in terms of the dimension in question, and X equals X'. On this definition, length possesses an absolute zero, since the juxtaposition of any two rods is

THE MEASUREMENT OF LOUDNESS

227

longer than either rod alone. Weight may or may not possess an absolute zero, depending on how the operation of weight addition is defined. If it is defined as the placing of two or more objects in the same pan of a conventional balance, weight possesses an absolute zero. But if it is defined as the placing of two or more objects in a balance pan with a lid and closing the lid, weight has a nonabsolute zero because not all objects in the dimension satisfy the axiom, (X @ Y) G X'. If Y is an object of no weight or an object of negative weight (a heliumfilled balloon, for instance) and X is an object of positive weight, it is not true that (X @ Y) G X'. In the sense defined above, loudness, like length, possesses an absolute zero. The addition (simultaneous playing) of two tones produces a combined tone that is louder than either of the component tones. That is to say, (X @ Y) G X' is satisfied for all tones, where @ means "played simultaneously with" and G means "is louder than." If someone wishes to argue that loudness does not possess an absolute zero, in a sense other than the one clearly defined above, it is up to him to provide the alternative definition. Having provided it, he must then be able to show that loudness, unlike length, does not possess an absolute zero in his sense. I contend that it is not possible to do so. Possibly the inclination to argue that loudness does not possess an absolute zero rests on a confusion of two notions: the notion of a tone having a loudness just less than the least audible loudness, and the notion of a tone of zero loudness. This confusion might lead to the argument that, since there are tones of zero loudness (i.e., tones having a loudness just less than that of the least audible loudness), loudness cannot be said to possess an absolute zero. For if we add (play simultaneously) a tone of positive loudness and one of zero loudness, the combined tone will not be greater than either of the components. That is, the axiom, (X @ Y) G X', is not satisfied if X is a tone of positive loudness and Y is a tone of zero loudness. If this axiom is not satisfied by the objects in a dimension, the dimension does not possess an absolute zero. This argument errs in its assumption that a tone having a loudness just less than the least audible loudness is a tone of zero loudness. Such a tone is, rather, a tone of positive loudness, but of so small a degree that it cannot be heard. Similarly, a rod having a length just less than the least visible length is not a rod of zero length; it is

228

THE MEASUREMENT OF LOUDNESS

a rod of positive length, so short that it cannot be seen. According to the clear definition developed in chapter 4, Y is a tone of zero loudness only if (X @ Y) £ X', where @ means "played simultaneously with" and E means "equals." But there are no such tones. The confusion of the notion of a tone having a loudness just less than the least audible loudness with the notion of a tone of zero loudness may lead in another way to the conclusion that loudness does not possess an absolute zero. Because the least audible loudness for one subject may not be the same as the least audible loudness for another subject, the loudness of a tone just below that of the least audible loudness may vary, depending on the subject. And, if zero loudness is just less than the least audible loudness, zero loudness may also vary, depending on the subject. But which subject should be used in establishing zero loudness? The choice is entirely arbitrary, and zero loudness is therefore an arbitrary zero. Hence, loudness does not possess an absolute zero. Rather, a zero must be assigned to loudness, and such assignment is arbitrary. This argument, whatever other failings it may have, errs in its assumption that a tone having a loudness just less than the least audible loudness is a tone of zero loudness. It is indeed true that the least audible loudness varies according to the auditory capacities of the subject, just as it is true that the least visible length varies according to the visual capacities of the subject. But neither fact means that the dimension in question does not possess an absolute zero. OBJECTIONS CONCERNING ADDITION Objection Five.—This objection appeals to certain apparently established facts concerning the simultaneous playing of tones. A passage from Stevens and Davis contains a useful summary: . . . two tones of equal loudness, which are sufficiently separated in frequency as not to stimulate overlapping areas on the basilar membrane, yield, when presented together, a loudness twice as great as either one alone. . . . If the tones introduced into the same ear are too near in frequency, they stimulate overlapping areas of the basilar membrane, whereupon some degree of masking may occur and may interfere with the summation of the two loudnesses. Strikingly different, however, is the effect when the two tones are led to each ear separately. In this case, summation occurs, but only when the frequencies are close together. . . . When the two tones are of different

THE MEASUREMENT OF LOUDNESS

229

frequency, and are led to the two ears separately, loudness does not sum. It appears that, in order for loudness to sum arithmetically in one ear, the tones must be far apart in frequency—for it to sum in two ears separately, the tones must be identical in frequency. Of course if the tones in one ear are identical in frequency and phase, their intensities must sum.1 In this chapter I am assuming a distinction between loudness and pitch (dimensions of sounds or tones), and intensity and frequency (dimensions of sound waves). The passage above does not seem to observe this distinction, since it speaks of the "intensity of a tone" and the "frequency of a tone." These locutions are, nonetheless, convenient, and I sometimes use them myself in the ensuing discussion. The meaning of the terms "sum" and "sum arithmetically," as used in the passage above, are far from clear. An analysis of their unclarity is undertaken in the reply to Objection Six. For present purposes it is sufficient to note that the loudness of a combination tone depends on (a) the relative frequency of the tones, and (b) whether the component tones are introduced into the same or different ears. Let us consider only instances of component tones introduced into a single ear. Imagine a group of tones, Ti, T2, Ts, and so on, which are equal in loudness and have widely different frequencies. Imagine now another group of tones, T'i, T'2, T'3, and so on, each of which is equal in loudness to any tone in the first group and which have slightly different frequencies. It can be inferred from the passage in Stevens and Davis that a given number of tones in the first group played simultaneously will not be equal in loudness to the same number of tones in the second group played simultaneously; the tones in the first group will sum arithmetically with respect to loudness, whereas those in the second group will not. Hence, a combination of tones in the first group will be louder than a combination of an equal number of tones in the second group. It might be supposed that therein lies an objection to my system of loudness measurement, which assigns a numeral to any given tone, T, by playing equal unit tones simultaneously until a collection has been obtained which is equal in loudness to T. A system of measurement employing the tones of the first group above as unit tones 1 S. S. Stevens and Hallowell Davis, Hearing: Its Psychology and Physiology (New York, 1938), pp. 1 1 5 - 1 1 6 .

230

THE MEASUREMENT OF LOUDNESS

will not produce the same results as a system that employs the tones of the second group. In general, to one and the same tone, T , the first system of measurement assigns a higher numeral than does the second. How are we to decide which group of unit tones to employ? The objection cannot stand, for if consistently applied it would produce the absurd conclusion that no dimension can be measured. A system of length measurement which juxtaposes equal unit objects made of soft rubber will produce different results from those produced by a system that employs unit objects made of brass. For example, if we assign numerals to the height of a wall by vertically juxtaposing rubber units objects, the weight of the upper objects will compress the lower ones, and the length of any juxtaposition will be shorter than a juxtaposition of brass unit objects of the same length and number. Obviously, this fact does not entail that a system of measurement employing the juxtaposition of equal unit objects is unavailable for length. All we must do is specify that unit objects must be of a certain type (e.g., that they must be rigid). Similarly, I can specify that unit tones in my system of loudness measurement must be of a certain type (e.g., that they must be widely different in frequency). (A second alternative is not discussed here. Perhaps there is no objection to employing unit rods made of soft rubber, so long as the conditions for their use are carefully specified for all types of measuring situations. Perhaps, similarly, there is no objection to employing unit tones that have similar frequencies.) The other fact about playing tones simultaneously, which may suggest that my system of loudness measurement is spurious, is that combinations of equally loud tones differ in loudness, depending on whether the component tones are introduced into the same ear or into different ears of the person making the measurement. Let us consider only a system in which the unit tones are far apart in frequency. A combination of component tones introduced simultaneously into a single ear is louder than a combination of the same component tones introduced into separate ears. Nevertheless, it does not follow that loudness cannot be measured by a system that employs the simultaneous playing of equal unit tones, for we can simply specify that the tones are to be introduced into a single ear of the person making the measurement. The point can be illustrated by an analogy. In the unusual system

THE MEASUREMENT OF LOUDNESS

231

of weight measurement described at the end of chapter 4, an indefinite number of equal unit objects are discovered by the method of lifting and are labeled U. Two objects that feel equally heavy, when lifted at the same time one in each hand, are taken to be of equal weight. To measure the weight of a given object, X, the observer lifts it in one hand and collections of U's in the other until he discovers a collection that is equal in weight to X. A count of the U's in the collection supplies the numeral to be assigned to X. Now suppose this procedure is altered in one respect: the collection of U's is lifted with both hands instead of a single hand, then X is lifted also with both hands. Although the weight of an object measured by the new system might be entirely different from the weight when measured by the old, it does not follow that one of the procedures is not measurement. Careful description of the procedure to be used and faithful adherence to the procedure in practice may qualify either procedure as measurement. Objection Six.—The discussion of Objection Five reveals that the loudness of a combination tone depends on (a) whether the component tones have the same or different frequencies, and (b) whether the component tones are introduced into the same ear or into different ears of the observer. M y system of loudness measurement is not adequately described, then, until the relative frequency of the unit tones in the system, as well as the manner of listening to combinations of these, is specified. Let us stipulate that the equally loud unit tones in the system are similar in frequency, and that these tones are to be introduced into a single ear of the person making the measurement. Objection Six maintains that the unit tones in my system of numeral assignment do not sum arithmetically with respect to loudness, and that consequently the system does not qualify as one of measurement. When two equal unit rods are juxtaposed the resulting juxtaposition is twice as long as either component; when three equal unit rods are juxtaposed the resulting juxtaposition is three times as long as either component; and so on. Two equally heavy objects, when placed in one pan of a balance, will balance an object in the other pan which is twice as heavy as either of the first two; three equally heavy objects, when placed in one pan of a balance, will balance an object in the other pan which is three times as heavy as any one of the first three; and so on. But two equally loud unit tones are not, when played

232

THE MEASUREMENT OF LOUDNESS

simultaneously, twice as loud as either component; three equally loud unit tones are not three times as loud as either component; and so on. This objection might be supported in any of several ways, but each supporting argument can be defeated. The first argument may be stated as follows: (a) Two equally loud tones of similar frequencies do not sum arithmetically with respect to intensity when the component tones are introduced simultaneously into a single ear (i.e., the intensity of the combination tone is not twice as great as the intensity of either of the component tones); hence, (b) two equally loud tones of similar frequencies do not sum arithmetically with respect to loudness when the component tones are introduced simultaneously into a single ear (i.e., the loudness of the combination tone is not twice as great as the loudness of either of the component tones). This argument succeeds only (1) if intensity and loudness are identical, or (2) if there is a precise empirical correlation between intensity and loudness. But (1) is not obviously true, as has been shown in chapter 3. Indeed, it is so far from being obviously true that we may proceed on the working assumption that it is false. The indications of intensity meters cannot, we assume, be used as the criteria for loudness. Once this point is acknowledged, it becomes impossible to show that (2) is true without begging the question of argument (a)-(b). For if the unit tones in my system sum arithmetically with respect to loudness, there is no precise empirical correlation between loudness and intensity, since the unit tones do not in fact sum arithmetically with respect to intensity. And if the unit tones do not sum arithmetically with respect to loudness, there may be a precise empirical correlation between loudness and intensity. The point is that in order to know whether there is such a correlation, we must first know whether loudness does or does not sum arithmetically, and that is just the question at issue in argument (a)-(b). In brief, since the indications of intensity meters cannot be used as the criteria for loudness, we cannot argue from the facts of intensity summation to the facts of loudness summation. The question then becomes: What are the criteria for loudness? How can we know whether the unit tones in my system sum arithmetically with respect to loudness? This question is answered in the sequel. The second argument in support of the assertion that loudness does not sum arithmetically appeals to certain experimental facts concerning what is usually called "loudness estimation." The graph in

233

THE MEASUREMENT OF LOUDNESS

figure 4, adapted f r o m one devised b y Stevens and D a v i s , is a synthesis of data obtained b y several experimenters. In the experimental procedure described in detail b e l o w , subjects made w h a t are usually called "direct estimates" of fractional relations of various types b e t w e e n t w o tones sounded successively. T h e subject is presented w i t h a standard tone, T . , w h o s e inten-

/

trt

_J Ld CO

O

UJ

80

o -

60

u. 0

>• 40

H (/)

z 20 u

A

20

-

60 40 I N T E N S I T Y OF

80

—

-

— - -

100

120

IN DECIBELS

Fig. 4. Results of loudness fractionations, which are the basis of one possible argument against the tone-addition method of loudness measurement. The double unbroken line shows the intensities of two tones when one (T c ) is estimated to be one-half as loud as the other (T,). The double broken line represents "accurate" estimates of one-half and is provided for comparison. The single unbroken line shows the intensities of two tones when one is estimated to be one-tenth as loud as the other. The single broken line represents "accurate" estimates of one-tenth and is provided for comparison. Adapted from S. S. Stevens and Hallowell Davis, Hearing: Its Psychology and Physiology (New York, 1938), p. 113. sity has been determined beforehand. H e is then asked to adjust the intensity of a comparison tone, Tc, until he estimates it to be one-half as loud as the standard. T h e intensity of the comparison tone is then determined. T h i s procedure is repeated f o r several standard tones of different intensities. T h e results are represented b y the double u n -

2

34

THE MEASUREMENT OF LOUDNESS

broken line in figure 4, which gives the relative intensities of pairs of tones when one of the pair, Tc, is estimated to be one-half as loud as the other, T,. Thus, a tone of 80 decibels is estimated to be one-half as loud as a tone of 90 decibels. It is easily seen that "one-half" estimates of relative loudness do not correspond to relative intensities of one-half. This nonexistent correspondence has been represented on the graph, for purposes of comparison, by the broken double line. In a second, similar experiment the subject is asked to adjust the intensity of one tone until he estimates it to be one-tenth as loud as the standard. The results are represented by the unbroken single line in figure 4, which gives the relative intensities of pairs of tones when one of the pair, T c , is estimated to be one-tenth as loud as the other, T a . Thus, a tone of 20 decibels is estimated to be one-tenth as loud as a tone of 40 decibels. "One-tenth" estimates of relative loudness do not, therefore, correspond to relative intensities of one-tenth. This nonexistent correspondence has been represented on the graph, for comparison purposes, by the broken single line. The second argument for the assertion that the loudness of the unit tones in my system of measurement does not sum arithmetically is, then, simply this: Relative loudness does not correspond to estimated relative loudness, as figure 4 shows. For example, a tone of 20 decibels (db) is estimated to be one-tenth as loud as a tone of 40 db (as shown by the unbroken single line). If the relative loudness of these two tones did correspond to estimated relative loudness, the first tone would have been estimated to be one-half the loudness of the second. The reply to this argument is twofold. First, the argument confuses loudness with intensity. The numerals along the ordinate and the abscissa of figure 4 signify decibels, which are units of intensity, not units of loudness. The figure shows that a tone with an intensity of 20 db is estimated to be one-tenth as loud as a tone with an intensity of 40 db, not that a tone with a loudness of 20 loudness units is estimated to be one-tenth as loud as a tone with a loudness of 40 loudness units. Hence, figure 4 shows that relative intensity does not correspond to estimated relative loudness. It does not show that relative loudness does not correspond to estimated relative loudness. Figure 4 would show that relative loudness does not correspond to estimated relative loudness only if there were a linear relation between tone intensity, expressed in db, and tone loudness, expressed

THE MEASUREMENT OF LOUDNESS

235

in loudness units. If there were such a relation, loudness would equal intensity multiplied by a constant: L = cl. The numbers along the ordinate and the abscissa of figure 4 could then be multiplied by c to produce a graph showing the relation between loudness, expressed in loudness units, and estimated relative loudness. And it would show that the two do not correspond. But, as the discussion of Objection Eight shows (see below), there is no linear relation between tone intensity, expressed in decibels (db), and loudness, expressed in loudness units. The relation is instead logarithmic. There is, however, a linear relation between tone intensity, expressed in dynes per square centimeter (dynes cm2), and loudness, expressed in loudness units. Since the relation between db and dynes cm2 is known, the graph in figure 4 can be renumbered in terms of the latter unit. Then the figure will show that relative intensity, expressed in dynes cm2, corresponds roughly to estimated relative loudness. And, since loudness, expressed in loudness units, is a linear function of intensity, expressed in dynes cm2, it can be inferred that relative loudness, expressed in loudness units, corresponds roughly to estimated relative loudness. The second part of the reply to the argument is as follows. Even if it could be shown that relative loudness, expressed in the loudness units of my system, does not correspond to estimated relative loudness, it would not follow that the system is not one of measurement. Experiment has shown that relative weight (as determined by a balance) does not correspond to estimated relative weight, and that relative length (as determined by a meterstick) does not correspond to estimated relative length.2 But these facts clearly do not entail that assigning numerals to weight by means of a balance, and assigning numerals to length by means of a meterstick, are not procedures of measurement. There is a clear distinction between the measurement of relative magnitude and the estimation of relative magnitude. The distinction is easily observed in connection with length or weight. It is much more easily overlooked in connection with loudness, however, partly because of the inclination to regard loudness as a dimension of sensation, as a "subjective dimension." We can easily distinguish between "subjective weight" (estimated weight) and "objective weight" (measured 2 S. S. Stevens and E. H. Galanter, "Ratio Scales and Category Scales for a Dozen Perceptual Continua/' Journal of Experimental Psychology, 54 (1957)» 37 8 3®4

ff.

236

THE MEASUREMENT OF LOUDNESS

weight). But what is meant by "objective loudness" as opposed to "subjective loudness"? It may be said that all one can mean by "objective loudness" is the intensity of sound waves. If so, it will readily be admitted that there is a distinction between the measurement of loudness (i.e., the measurement of intensity) and the estimation of loudness. It is my contention, however, that a distinction between "objective loudness" and "subjective loudness" is in order, even on the assumption that "objective loudness" is not the intensity of sound waves (see chaps. 2 and 3). I further contend that "objective loudness," construed in this way, is capable of measurement (as opposed to estimation) by means of the procedure defended in this chapter. Hence, the loudness of tones could be measured by my system in order to determine the relation between relative loudness and estimated relative loudness. This experiment would be comparable to experiments in which length is measured by the juxtaposition of equal unit rods in order to determine the relation between relative length and estimated relative length. The third argument for the assertion that loudness does not sum arithmetically can be stated with greatest economy in these terms: Playing tones simultaneously violates the axiom of additivity, (9) N(Ti) + N(T 2 ) = N(Ti @ T2), where Ti and T2 designate tones, @ means "played simultaneously with," N means "the number of," and = and + are the conventional arithmetical symbols. If playing tones simultaneously does violate this axiom, results like the following will be obtained by employing the operation in a system of loudness measurement. Tone Ti is found to equal 3 unit tones, tone T2 is found to equal 4 unit tones; but Ti and T2 played simultaneously are found to equal 6 unit tones. Similarly, the operation of juxtaposing rods would violate the axiom of additivity if a system of numeral assignment employing this operation assigned to rod X the numeral 3, to rod Y the numeral 4, and to the juxtaposition of X and Y the numeral 6. This argument fails for two reasons. First, there is no reason whatever to believe that the operation of playing tones simultaneously violates axiom (9). Whether it does or not can be determined only by putting the system of measurement employing this operation to use, as is true of any system employing a physical operation of addition. Second, even if playing tones simultaneously does violate axiom (9),

THE MEASUREMENT OF LOUDNESS

237

it does not follow that the loudness of unit tones in my system of loudness measurement does not sum arithmetically. T o make this point, part of the argument in chapter 4 must be restated and elaborated. Suppose we wish to determine whether two equally loud tones, Ti and T2, are, when played simultaneously, twice as loud as either component. If no system of loudness measurement is available, w e cannot make this determination. But so long as a system of loudness measurement is available, and Ti and T2 are not unit tones in that system, we can easily determine whether T i and T2 played together are twice as loud as either played alone. Using the available system, a number is assigned to Ti and one to T2 (it will be the same number, since Ti and T2 are equally loud). Then T i and T2 are played simultaneously and a number is assigned to the combination tone. If the two numbers assigned to the component tones are, when added, equal to the number assigned to the combination tone—that is to say, if N(Ti) + N(J%) = N(Ti @ T2)—then the combination tone is twice as loud as either component. If my system of loudness measurement is the only one available, and if Ti and T2 are unit tones in the system, the combination of Tx and T2 must be twice as great as either component, for, by definition, Ti is 1 unit of loudness, T2 is 1 unit of loudness, and Ti and T2 played simultaneously are 2 units. Hence, it is true by definition that N(Ti) + N(T 2 ) = N(Ti @ T2), where Ti and T2 are unit tones in the only available system of loudness measurement. If a second and more precise system of loudness measurement is available—that is, one that has unit tones of lesser loudness—the unit tones in the first system do not necessarily sum arithmetically, for they can now be measured by means of the more precise system. But the unit tones in the most precise system of loudness measurement available necessarily sum arithmetically. To summarize my treatment of Objection Six, there is no evidence available to show that the loudness of tones does not sum arithmetically, even if the tones being added are similar in frequency and are introduced into one ear only. Furthermore, even if experiment would show that the loudness of some tones fails to sum arithmetically, it could never show that the loudness of all tones fails to sum arithmetically, for the loudness of the unit tones in the most precise system of loudness measurement sums arithmetically by definition. Stevens and

238

THE MEASUREMENT OF LOUDNESS

Davis, in the passage quoted under Objection Five, maintain that tones do not sum arithmetically when they are similar in frequency and are introduced into a single ear. These authors may be basing their contention on the facts of loudness estimation. If so, the reply is that the fact that two equal tones are estimated to be less than twice as loud as either component does not entail that they are less than twice as loud. Or perhaps Stevens and Davis are basing their contention on the facts of intensity summation. Perhaps they even identify loudness with intensity. If so, the reply is that, although tones do not sum arithmetically with respect to intensity under all conditions, it does not follow that tones do not sum arithmetically with respect to loudness, for loudness and intensity cannot be identified. In this book I am defending only a system of loudness measurement, which employs the operation of playing tones simultaneously. And there is no good reason to suppose that the loudness of tones does not sum arithmetically for this operation of addition. OBJECTIONS FROM THE PURPOSE OF MEASUREMENT Objection Seven.—This objection maintains that my system for assigning numerals to loudness does not permit the application of arithmetical statements—such as "3 + 4 = 7"—to loudness, and therefore does not qualify as measurement. With the selection of a group of unit tones, the numeral 3 comes to designate any collection of three unit tones, or any tone (or collection of tones) equal in loudness to such a collection; the numeral 4 comes to designate any collection of four unit tones, or any tone (or collection of tones) equal in loudness to such a collection; and so on for every numeral. If this interpretation permits the application of "3 + 4 = 7 " to loudness, then any tone (or collection of tones) designated by 3 and any tone (or collection of tones) designated by 4 will, when played simultaneously, be equal in loudness to any tone (or collection of tones) designated by 7. The present objection maintains that such statements as the one in italics are false. This question cannot be debated on apriori grounds. It is a question of fact, and the answer can be obtained only by putting my system of loudness measurement to use. So far as I know, there are no experimental facts to indicate that my system does not permit the application of arithmetical statements to loudness. The eight axioms

THE MEASUREMENT OF LOUDNESS

239

of measurement discussed in chapter 4 must, according to many theorists, be true in any system of measurement. Although I contend that a system of measurement need not satisfy all these axioms, there is a presumption that any system that does satisfy them will permit the application of arithmetical statements to the dimension to be measured. If my system of loudness measurement satisfies each of these axioms, perhaps we can safely conclude that it does permit the application of arithmetical statements to loudness. It is noteworthy, then, that there do not seem to be any experimental facts indicating that all or even some of these axioms are not satisfied by the system proposed in this chapter. Some of the axioms of measurement cannot even be stated for my system of loudness measurement, and therefore cannot possibly be found to be false. In axioms of addition (6) (X @ Y ) E (Y @ X), and (7) [X @ (Y @ Z)] E [(X @ Y) @ Z], let us interpret E to mean "balances" and @ to mean "placed together on the same arm of a balance with." More specifically, let X @ Y designate the operation of placing Y on one extreme end of the balance arm with X next to it, and let Y @ X designate the operation of placing X on one extreme end of the balance arm with Y next to it (the operation of horizontal addition described in chapter 4). On this interpretation, (6) and (7) are contingent statements which require verification by experimentation with balances and objects placed on the balances. If, however, we interpret E to mean "is equal in loudness to" and @ to mean "played simultaneously with," then (6) says that X and Y played simultaneously are equal in loudness to Y and X played simultaneously. This statement would be meaningful only if playing X and Y simultaneously and playing Y and X simultaneously were different operations, in the way in which X @ Y and Y @ X are different operations when @ is interpreted to mean the operation of horizontal addition. But they are not different operations, and (6) is meaningless. It is, therefore, not a contingent statement, not a statement that can be falsified or found to be true by playing tones simultaneously. A similar analysis applies to (7). Objection Eight.—It may be argued that my system of loudness measurement does not make possible the formulation and verification of numerical laws relating loudness and other dimensions, and that in consequence it does not qualify as measurement. Again, this is not

240

THE MEASUREMENT OF LOUDNESS

an apriori question, but a question of fact, which can be settled conclusively only by putting the system to use. There are indications, however, that use of the system would lead to the formulation of an empirical law relating loudness of tones to intensity of sound waves. If two sound waves are identical in frequency and phase, their intensities sum arithmetically when the waves are added (made to impinge simultaneously on some measuring instrument). If the waves are not identical in frequency and phase, something approximating arithmetic summation takes place. Suppose we select as unit tones in the system of loudness measurement tones produced by sound waves identical in frequency and phase. Now, by definition, the unit tones when played simultaneously sum arithmetically with respect to loudness. The loudness of (Ti @ T2 @ . . . T»), where the T's are equal unit tones, is by definition n times the loudness of any tone in the collection. Since the unit tones are chosen so as to be equal in loudness, the sound-wave intensities producing the tones are equal. If, as we have supposed, these sound waves are identical in frequency and phase, the intensity of (Wi @ W2 @ . . . W»), where the W's are sound waves of equal intensity, is n times that of the intensity of any sound wave in the collection. It follows that the relation between loudness, measured in units of my system, is a linear function of intensity, measured in appropriate units; and that there is a numerical law relating loudness and intensity. Let us make some precise suggestions as to what the law is. In most experiments on sound perception, the intensity of sound waves is taken to be sound-wave pressure, which is expressed in dynes per square centimeter. The greatest pressure that will produce an audible tone, without impairing hearing, is about 200,000,000 dynes cm*, a trillion times the pressure at the lowest threshold. The range of hearing is, therefore, fantastically broad. For convenience in representing this wide range, units of sound pressure are usually converted into decibels,8 which express the ratio between two sound-wave pressures (intensities). The logarithm of the ratio of the greater intensity to the lesser intensity is the intensity of the greater in bels; a decibel is one-tenth of a bel. In using decibels to express intensity, the lesser of the two intensities in the ratio is always taken as .0002 dyne cm2, * See Stevens and D a v i s , Hearing,

pp. 29-31, for an explanation of this unit.

241

THE MEASUREMENT OF LOUDNESS

the intensity at the threshold of hearing. Hence, the decibel can be defined as follows: Db =

10

log

i

.0002

,

where Db is the number of decibels and I is the intensity of the greater sound-wave intensity expressed in dynes cm2. Table 5 relates the intensity of sound waves, expressed first in dynes cm2 and then in decibels, to the loudness of associated tones, expressed in the units of my tone-addition method. Unit tones in the system have been selected in such a way that they are produced by sound waves whose intensity is .0020 dyne cm2, or 1 0 decibels, thus putting them well within the audible range. The table shows that the following laws should obtain: [ 1 ] Ip =

.0020 La,

and log (La + 1 ) , where IP is sound-wave intensity in dynes cm2, La is loudness in loudness addition units, and la is sound-wave intensity in decibels. These, then, are the laws we may expect to confirm by using the tone-addition method of loudness measurement recommended in this chapter. If [2] la =

10

TABLE 5 INTENSITY IN

INTENSITY IN

LOUDNESS IN TONE-

DYNE CM2

DECIBELS

ADDITION UNITS

200,000,000

120

100,000,000,000

20,000,000

HO

10,000,000,000

2,000,000

IOO

1,000,000,000

200,000

90

100,000,000

20,000

80

10,000,000

2,000

70

1,000,000

200

60

100,000

20

50

10,000

2

40

1,000

.2000

30

IOO

.0200

20

IO

.0020

IO

1

.0002

O

O

242

THE MEASUREMENT OF LOUDNESS

the method should falsify [1] and [2], there may still be other laws that its use would confirm; and this possibility is sufficient to argue that the method is not intrinsically unsuitable for the discovery of numerical laws governing loudness. Objection Seven assumes that no system of numerical assignment which does not permit the application of arithmetical statements to some dimension qualifies as measurement. Objection Eight assumes that no system of numeral assignment which does not make possible the formulation and verification of numerical laws qualifies as measurement. Both assumptions can be challenged, although the foregoing discussion makes the challenge unnecessary. We may admit that no system of numeral assignment which has either of the uses would be useful, without being forced to admit that such a system would fail to qualify as measurement. Or, less drastically, we may admit that a system of numeral assignment must have at least one of the two uses, without conceding that it must have both, to qualify as measurement. APPLICATIONS A N D CONCLUSIONS A central question in this book is whether psychological dimensions, of which loudness is often taken to be an example, are measurable. To many, the question seems a simple one. After all, we know perfectly well what measurement consists in. By means of a yardstick the length and width of a floor may be measured to discover what size rug it will require. By means of a balance the precipitate of a chemical reaction is measured (weighed) to discover how much iron there is in a sample of a certain ore. Can the loudness of a tone be ascertained by such procedures as these? Is it not clear that we cannot lay a yardstick on a tone to gauge its loudness? Is it not obvious that tones cannot be added—as weights are added in the pan of a balance—in order to compare the loudness of a given tone with the combined loudness of unit tones? This it-is-a-simple-question attitude leads straight to the conclusion that loudness cannot be measured. It should now be clear that the question is not so simple. In the first place, there are different ways of measuring length: some do and some do not employ yardsticks. When even length can be measured without a yardstick, it is quite unfair to demand that loudness be measured with a yardstick. In asking whether loudness and other

THE MEASUREMENT OF LOUDNESS

243

so-called psychological dimensions are measurable, we must operate with some definition of measurement, and not with simpleminded paradigms. As shown in this chapter, loudness is measurable even on the narrow, classical definition of measurement rejected in chapter 4. Second, the question of whether a given dimension admits of a physical operation of addition cannot be settled in an apriori manner. We must look and see. When we do so, as in the present chapter, we find such an operation for loudness. The literature on measurement abounds with simplistic arguments attempting to show that psychological dimensions are incapable of measurement. The following passage is from William James. T o introspection, our feeling of pink is surely not a portion of our feeling of scarlet; nor does the light of an electric arc seem to contain that of a tallow-candle in itself. Compound things contain parts; and one such thing may have twice or three times as many parts as another. But when we take a simple sensible quality like light or sound, and say that there is now twice or thrice as much of it present as there was a moment ago, although we seem to mean the same things as if we were talking of compound objects, we really mean something different. W e mean that if we were to arrange the various possible degrees of the quality in a scale of serial increase, the distance, interval, or difference between the stronger and the weaker specimen before us would seem about as great as that between the weaker one and the beginning of the scale. It is these RELATIONS, these DISTANCES, which we are measuring and not the composition of the qualities themselves, as Fechner thinks. 4

The foregoing discussion makes it clear that James's remarks are little better than prejudices. My method for measuring loudness employs the operation of playing tones simultaneously as an operation of loudness addition. This operation enables us to say that a compound, (Ti @ T2), of two equal tones is twice as loud as its components. This statement does not mean that the loudness interval between Ti and (Ti @ T2) seems equal to the interval between Ti and the beginning of the scale. It means the same thing as the statement that two equal rods are, when juxtaposed, twice as long as either component. My system of loudness measurement is exactly comparable to a system of length measurement which employs equal unit rods 4

The Principles of Psychology (New York, 1950), I, 546.

244

THE MEASUREMENT OF LOUDNESS

and the operation of endwise juxtaposition. It legitimizes the same quantitative statements about loudness which are commonly made about length. In measuring loudness by the operation of playing tones simultaneously, we are no more measuring distances and relations than when we measure length by the operation of endwise juxtaposition of rods. It is likely that James's prejudice rests on a tendency to think of sounds and colors as sensations. He speaks at one point of "our feeling" (i.e., our sensation) of pink. There is a strong inclination to say of sensations and feelings that they cannot be added, that one (compound) sensation cannot contain others as components. But even if this inclination is correct, it does not follow that sounds and colors cannot be added; sounds and colors are not sensations, as has been shown in chapter 2. Brightness, hue, loudness, and pitch are not dimensions of sensations. In this chapter I have attempted to show that one so-called psychological dimension—loudness—is measurable in the narrow, classical sense, that is, by a method employing a physical operation of addition. The argument can, of course, be extended to any of the so-called psychological dimensions which admits of a physical operation of addition. For example, two lights, when placed together or made to shine on the same reflecting surface, are brighter than either alone. This operation of brightness addition may be called the operation of contiguous exposure. Brightness can thus be measured exactly as loudness can be measured. First a number of equally bright lights are selected as units. The test for equality here is visual comparison of the lights; that it is a legitimate test can be argued as in the reply to Objections One and Two. A given object is measured by contiguously exposing a sufficient number of unit lights to make the collection equal in brightness to the light being measured. That this operation of brightness addition is legitimate can be seen from considerations like those advanced in reply to Objections Three, Four, and Five. There is just as much reason to think that this method of brightness measurement will prove useful as there is to think that my method of loudness measurement will prove useful. It is not clear how many of the so-called psychological dimensions can be measured by an addition method, because it is not clear which of them do and which do not admit of a physical operation of

THE MEASUREMENT OF LOUDNESS

245

addition, a difficulty due partly to unclarity in the concept of physical addition. It was asked in chapter 4 whether boiling liquids together can qualify as a physical operation of addition for sweetness of liquids. If so, sweetness can be measured by an addition method. But for some of the so-called psychological dimensions it does not seem possible even to imagine an operation of addition. Hue and pitch are the chief examples. The question whether these dimensions can be measured by some procedure that does not employ an operation of addition is considered in chapter 7.

7 The Measurement of Pitch Is pitch capable of independent measurement? According to the definition developed in chapter 5, a procedure of independent measurement for pitch must assign numerals to tones so as to express the pitch of tones in number of pitch units, units that are neither identified with nor defined in terms of other units. Pitch can perhaps, be dependency measured; one simple method depends on the fact that, when tones are produced by vibrating strings, pitch is a function of the length of the string. Since length can be measured (in centimeters, say), we can express each pitch in terms of the numeral assigned to the length of the string producing that pitch. Because this procedure either identifies pitch units with or defines them in terms of length units, it is dependent measurement. Another, less simple procedure depends on the fact that pitch is a function of the frequency of the sound wave associated with the tone, and can therefore be expressed in terms of the numerals that designate corresponding frequencies. Again, because this procedure either identifies pitch units with or defines them in terms of other units (i.e., units of frequency), it is dependent measurement. The question here is whether pitch is independently measurable, measurable by procedures significantly similar to those available for length and weight, measurable by a procedure that neither identifies pitch units with nor defines them in terms of other units. Some would say that pitch obviously cannot be measured in this way, an incautious assertion that possibly rests on an unsuccessful attempt to imagine a method of measuring pitch which exactly resembles measuring length by means of a ruler, or by means of successive applications of a unit object. A theorist who tries to imagine "laying a ruler on sound" and realizes that the process makes no sense may conclude that sound is not independently measurable. The conclusion is illegitimate, since there is no good reason for assuming that methods of independent measurement for different dimensions must be similar in every detail.

THE MEASUREMENT OF PITCH

247

One cannot "lay a ruler on weight" either, but it does not follow that weight is not independently measurable. Such incautious, intuitive answers to my question must be rejected. If I can describe a method for the independent measurement of the pitch of tones so that every objection to the method can be overcome, my question will have been affirmatively answered. Musicians employ a standard vocabulary to describe the interval in pitch between two tones. There are "halftones," "whole tones," "minor thirds," "perfect fifths," "octaves," and so on. Among musicians there is virtually unanimous agreement concerning the equality or inequality of given pairs of intervals, and the "size" of given intervals. Moreover, a few persons are able to say, simply by listening to a tone, whether it has this pitch or that, whether it is middle C, F below middle C, and so on. These persons possess what is sometimes called "perfect pitch." These facts suggest the following methods of pitch measurement. Method A.—The halftone is selected as the unit interval of pitch. Middle C is chosen as the reference or "zero" pitch. To measure the pitch of a tone, T, which is above middle C, play middle C; then play a tone that is a halftone higher in pitch than middle C; then a tone that is a halftone higher in pitch than the preceding tone; and so on until a tone that has the same pitch as T has been played. Then count the number of halftones between middle C and the last tone played to obtain the numeral to be assigned to T. A similar method is used to measure tones lower than middle C. For these, move down instead of up the pitch scale. A simple way of performing the procedure just described is to use a piano, an instrument so tuned that the tones it produces are separated by intervals of a halftone. The piano serves as a sort of "ruler" for pitch. Through its use we can measure the pitch of a given tone, T, by discovering which key must be depressed to produce a tone of the same pitch as T. This key will be a certain number of keys either above or below middle C. The determination of this number by counting furnishes the numeral to be assigned to T. Thus, a given tone, Ti, may be 6 halftones above middle C (+6); another tone, T2, 3 halftones below middle C (—3); another tone, Ts, 12 halftones below middle C (—12); and so on. Although Method A enables us to assign numerals to tones, and

248

THE MEASUREMENT OF PITCH

to express intervals between tones numerically, it has certain limitations. The method enables us to say that the intervals between Ti and T2 and between T2 and Ts have a magnitude of 9, and that they are therefore equal. It does not, however, enable us to say that the pitch of Ti is some fraction of the pitch of T2, or that the pitch of T2 is some fraction of the pitch of Ts. The reason we cannot make statements of the latter kind is, apparently, that they presuppose a system of measurement which employs an "absolute" zero for pitch, not the "arbitrary" zero stipulated above. Let me try to describe a method that does not suffer from this limitation. Method B.—The halftone is again selected as the unit interval of pitch, but the lowest pitch that can be heard is chosen as the zero pitch. To measure a given tone, T, we first play the lowest tone that can be heard; then we play a tone a halftone higher in pitch than the lowest pitch that can be heard; then a tone a halftone higher in pitch than the preceding tone; and so on until a tone that has the same pitch as T has been played. By counting the number of halftones between the lowest tone that can be heard and the last tone played, we obtain the numeral to be assigned to T. In both Methods A and B we designate a zero pitch and describe a unit that is neither identified with nor defined in terms of other units. It seems, therefore, that these are two methods by which pitch may be measured independently. (In Stevens' terminology, Method A creates an interval pitch scale, and Method B creates a ratio pitch scale. The latter seems to have all the properties of the centimeter scale of length.) Before accepting this conclusion, however, a number of objections must be considered. The first three are directed against Method B. OBJECTIONS TO THE ABSOLUTE ZERO METHOD Objection One.—One of the most obvious objections is that the allegedly "absolute" zero point in Method B may not be the same for all persons performing measurements. Suppose that an instrument capable of producing sound waves at any frequency produces tones that are increasingly lower in pitch. If at one point Oi says that he can no longer hear anything, whereas Oj maintains that he hears a tone lower than the preceding one (which Oi could also hear), then the lowest tone that can be heard by O2 is lower than the lowest tone that can be heard by Oi. It should follow, then, that O2, will judge a given

THE MEASUREMENT OF PITCH

249

tone, which both he and Oi can hear, to be more halftones above zero than will Oi, and that the measurements of Oi and Os will not agree. There is an easy reply. First, this difficulty can be overcome simply by recognizing the possibility described in the objection. If Oi wishes to confirm measurements taken by O2, he must first ascertain whether they hear the same lowest pitch. If they do not, the measurements of 0 1 and O2 neither confirm or disconfirm each other. Second, within certain limits adjustments can be made for observers who hear different lowest pitches. If Oi's lowest tone, Ti, is lower than ( V s lowest tone, T2, O2 can ask Oi how many halftones separate Ti and T2 and can add this figure to each of his own measurements. Oa's measurements should then agree with Oi's, other things being equal. Of course, O2 cannot check Oi's determination of the number of halftones separating Ti and T2, and in this sense Ox is unable to confirm or disconfirm Oi's measurements. This limitation does not appear to be critical. If, after the adjustment described above, Oi and 02 generally agree in their measurements, confirmation of one by the other is admissible. Objection Two.—The lowest audible tone may be different in pitch at different times for the same observer. The limits of hearing vary among different observers (e.g., the young can hear tones of higher pitch than can the aged); and the limits of hearing may vary at different times for the same observer. If the lowest pitch O can hear on Monday is different from the lowest pitch he can hear on Tuesday, measurements of a given tone made on Monday will not agree with measurements of the same tone made on Tuesday. In general, the numerals assigned on Monday will not have the same significance as those assigned on Tuesday, and the measurements will be incomparable. It cannot be said in reply that the lowest audible pitch on Monday is necessarily the same as the lowest audible pitch on Tuesday, as any one of a number of examples shows. Suppose O's hearing is normal on Monday and that he hears tones several octaves lower in pitch than middle C, but that on Tuesday he suffers an ear injury making him unable to hear any tone lower than middle C. It would be absurd to contend that the middle C that O hears on Tuesday is equal in pitch to the lowest tone he hears on Monday. Under some conditions O can on Tuesday adjust Monday's measurements to bring the two into agreement. On Tuesday he can repro-

250

THE MEASUREMENT OF PITCH

duce Ti, the lowest tone audible to him on Monday, in order to compare it with T2, the lowest tone audible to him on Tuesday. There are three possible outcomes, (a) If Ti is higher in pitch than T2, O will be able to hear Ti and to discover how many halftones higher than T2 it is. If he adds this number to his Monday measurements, they should agree with Tuesday's. (b) If Ti is equal in pitch to Ti, O will be able to hear Ti and to discover its equality to T2. He will then know that no adjustment of Monday's measurements is necessary, (c) If Ti is lower in pitch than T2, O will be unable to hear Ti, since T2 is the lowest tone audible to him on Tuesday, the day of the comparison. Consequently, O will not be able to discover how many halftones lower Ti is them T2, and will not be able to adjust the measurements so as to bring them into agreement. If, of course, another observer has heard Ti on Monday, and if Ti is the lowest tone audible to that observer on Tuesday, he will be able to compare Ti and T2 even in case (c). In the absence of such an observer, there is a limitation on the use of the lowest audible tone as the zero tone for pitch. The limitation can be overcome by designating a relatively high tone, Ti, produced by a certain frequency, F*, as the tone having zero pitch. Measurements predicated on this zero will be immune to the limitations thus far discussed so long as T t is not lower in pitch than the lowest audible tone for most of the persons making pitch measurements. But such a procedure of measurement does not employ the lowest audible tone as a zero tone. It is a procedure, not of type B, but of type A, which employs an "arbitrary" or "conventional" zero. Objection Three.—Even if the two preceding objections are discounted, a third seems to be fatal: The difficulty in determining the lowest audible pitch makes it impractical for use as a zero point. Stevens and Davis state the objection in technical terms: The lower limit for pitch is difficult to determine with precision for two reasons. First, there is the problem of distinguishing between a very low frequency which is heard as a tone and one which is heard simply as a series of distinguishable pulsations. Second, the ear itself introduces so much distortion (the production of aural or "subjective" harmonics) at these low frequencies that the task of distinguishing between the perception of the fundamental tone and the hearing of higher harmonics becomes difficult. 1 1

S. S. Stevens and Hallowell Davis, Hearing: Its Psychology and Physiology

(New York, 1958), p. 69.

THE MEASUREMENT OF PITCH

251

Stated less technically, the objection is that the area toward the bottom of the perceivable pitch scale is nebulous, as is instantly clear to anyone who has listened to the pedal tones of an organ. It is difficult to say whether one low pedal tone is lower in pitch than another, and whether the lowest pedal tones are vibrations that are felt rather than heard. The difficulty created by this nebulous area is obvious. If every measurement of the pitch of a given tone, T, requires us to begin at the lowest audible pitch and count upward by halftones until we arrive at the pitch of T, most measurements will be either impossible or hopelessly out of agreement with others. A count of halftones in the area just above the lowest audible pitch is virtually impossible. To accomplish it at all requires a great deal of guesswork. As a consequence, measurements of T made by Oi and O2 can be expected to disagree. The three objections above, all directed toward a system of measurement employing the lowest audible pitch as an absolute zero, can be avoided by dropping the requirement of an absolute zero and adopting instead a system that employs an arbitrary zero. (In chapter 4 objections to this use of "absolute zero" and "arbitrary zero" are offered.) Let us therefore adopt Method A, again employing the halftone as the unit, but taking as the zero point on the pitch scale the tone designated "middle C " by musicians. In some respects, such a scale is comparable to that employed in measuring temperature. Some point along the scale of degree is selected as the zero point: on the centigrade scale it is the freezing point of water; on the Fahrenheit scale it is below freezing. (Notice that, although the pitch scale is comparable to either of the temperature scales in possessing an arbitrary zero, the pitch scale provides for independent measurement, whereas the temperature scales at best provide for dependent measurement.) OBJECTIONS T O USING A N ARBITRARY ZERO Objection Four.—One objection to Method A is that we cannot know that what we call middle C today is a tone of the same pitch as the one we called middle C yesterday or last month; that is, we cannot know whether the same pitch is being employed as the zero pitch from one measurement to the next. (The same objection is raised against

THE MEASUREMENT OF FITCH

the system of loudness measurement defended in chapter 6.) There are two lines of reply. 1) Among musicians a distinction is frequently made between persons who have perfect pitch and persons who do not. The person with perfect pitch can correctly say whether the pitch he hears today is the same pitch he heard in the past. If he is familiar with the system of musical notation he can say, for example, whether the pitch of a tone presently being played is or is not middle C. Persons having this ability exhibit wide agreement in identifying the tones they hear. Hence, if some pitch, say middle C , is chosen as the zero point in a system of measurement, there should be wide agreement in pitch measurements among such persons. If, however, we rely on what might be called "pitch memory" in identifying the zero point, considerable disagreement may be expected among persons who do not possess perfect pitch. Therefore a system that employs middle C as the zero point would be useful only in an elite group—a group composed of persons who have perfect pitch. 2) Fortunately, reliance on pitch memory is not required, for there is in use another criterion for the same pitch. In common practice we take it that two tones have the same pitch if they have the same cause. Thus, if a tone heard yesterday and one heard today are both produced by the same tuning fork, and if the tuning fork has undergone no change from one tone to the next, the tones are equal in pitch. The tone produced by a certain standard tuning fork may be designated "middle C . " (Such a standard fork is employed by piano tuners.) Therefore, a person need not have perfect pitch to measure tones by the system of measurement under consideration here. He needs only the ability to identify the unit interval of a halftone and the use of a tuning fork that produces the zero tone of middle C. He can then determine that a given tone is so many units (halftones) above or below zero (middle C). His measurements may be expected to agree with those of other observers who can identify halftones and who have access to the standard tuning fork. (For a defense of the causal criterion of equality for dimensions of tones, see chapter 6, Objection One.) Objection Five.—Another objection to Method A is that it will not lead to the discovery of numerical laws relating pitch to other di-

253

THE MEASUREMENT OF PITCH

mensions. Suppose we attempt to formulate a numerical law relating pitch and frequency, using the middle C system of pitch measurement. We investigate a number of pitches, determining the frequency in cycles per second of the sound waves producing each pitch and the magnitude of each pitch in halftones above or below middle C. Suppose the data thus acquired are those given in table 6. (These are not the actual data that an experimenter would produce, as the discussion of Objection Twelve shows, but this point is irrelevant here.) These data TABLE 6 PITCH IN HALFTONES

CORRESPONDING FREQUENCY IN CPS

+4 +3 +2 +1 0 —1 —2 -3 -4

272 268 264 260 256 252 248 244 240

show that, at least in one part of the pitch scale, equal increases in pitch are accompanied by equal increases in frequency. The numericeli law would be [x]F = cP, where F is frequency, P is pitch, and c is a constant. But obviously, if pitch must be expressed in halftones above or below zero, in positive or negative numerals, this formula cannot be used to state the relation between pitch and frequency. It can be so used only if pitch, as well as frequency, is expressed in positive numerals alone, numerals assigned by a system of measurement that gives pitch an absolute zero. The reply to this objection is obvious. The relation between pitch and frequency indicated by the hypothetical data in table 6 can be represented either (1) by devising a different type of formula, or (2) by transforming the middle C pitch scale into an "absolute" scale, a scale that has an absolute zero.

THE MEASUREMENT OF PITCH

254

1 ) A n increase of 1 halftone in pitch is accompanied by an increase of 4 cps in frequency, and the zero on the pitch scale corresponds to a frequency of 256 cps above the zero on the frequency scale. Thus any given frequency can be determined by multiplying the corresponding pitch value by 4 and then adding 256. The resulting formula, [2] F = 4P + 256, states the relation between pitch and frequency when pitch is expressed in number of halftones above (a positive numeral) or below (a negative numeral) the arbitrary zero of middle C. In order to discover this formula it is necessary to know what pitch increase is accompanied by what frequency increase, and where zero is on the pitch scale in relation to zero on the frequency scale. 2) The relation between pitch and frequency indicated by the hypothetical data of table 6 can be stated in a formula of the form, F = cP, by transforming the middle C pitch scale into an absolute scale. Since the zero of the middle C scale is 256 cps above the zero of the frequency scale, and since i-unit pitch increases are accompanied by 4-unit frequency increases, 256 divided by 4 gives the number that must be added to every value of the middle C pitch scale in order to transform it into an absolute scale. Thus, + 1 will become 65, o will become 64, — 1 will become 63, —64 will become o, and so on. In terms of the new scale, the relation between pitch and frequency on the basis of the hypothetical data can be stated in the formula,

[3] F = 4 PT o discover this formula it is necessary to know, as it is to discover formula [2] for the middle C scale, what pitch increase is accompanied by what frequency increase, and where the arbitrary zero is on the pitch scale in relation to zero on the frequency scale. Formulas [2] and [3] differ in presupposing different types of pitch scale, but they state the same relation between pitch and frequency. Transformation of the middle C pitch scale into an absolute scale may, in some respects, be compared with the transformation of the centigrade scale of temperature into the absolute Kelvin scale. The arbitrary zero point on the centigrade scale is the temperature at which water freezes. (This provides an easy means of identifying the zero point, much as a standard tuning fork provides a way of identifying the arbitrary zero pitch of middle C.) The Kelvin scale enables us to

THE MEASUREMENT OF PITCH

255

discover such laws as Charles's law which, in its nonnumerical formulation, states that equal increases in the temperature of a gas are accompanied by equal increases in its volume, so long as the pressure of the gas is held constant. The centigrade scale, however, cannot be used to verify the numerical formulation of Charles's law, [ 4 ] T = CV, where T is temperature, V is volume, and c is a constant. To provide a scale that can be used to verify this law, the values on the centigrade scale must be transformed into values on the absolute or Kelvin scale. Absolute zero (i.e., zero on the absolute scale) has been set at approximately —273 degrees centigrade, so to convert a centigrade value into an absolute value one must add 273 to the former. Thus, o degrees centigrade is 273 degrees Kelvin, —273 degrees centigrade is o degrees Kelvin, and so on. If gases are measured on the absolute scale, they will approximately satisfy [4]. Objection Six.—The middle C pitch scale is not, in Stevens' terminology, a ratio scale; it does not enable us to say that one tone is half or twice or three times as high in pitch as another, since it employs an arbitrary and not an absolute zero. Suppose we measure three pitches, Pi, P2, and P3, and find that Pi is three halftones below middle C (i.e., —3), P2 is three halftones above middle C (i.e., + 3 ) , and P3 is six halftones above middle C (i.e., + 6 ) . In what ratios do these tones stand to one another? Is P3 twice, three times, one and three-fourths times as high in pitch as Pi? If we use middle C as the zero pitch, it would seem that P3 is twice as high in pitch as P2. But how do P3 and P2 stand to Pi? It is impossible to say. Pi, since it lies below zero, cannot be related to pitches above zero by a ratio; to make such a ratio possible a zero point below Pi must be chosen. To compare pitches below this new zero point with those above it, another zero point must be chosen. And to compare pitches below this new zero point with those above it, another zero point must be chosen. And so on indefinitely. It is true that the middle C scale cannot be used to establish the ratios in which any pitch, however low, stands to another. The limitation may be overcome, however, in the manner suggested in the answer to Objection Five. If we can determine how many halftones below middle C the lowest audible pitch lies, this number can be added to each of the values on the middle C pitch scale to convert the latter to a ratio

256

THE MEASUREMENT OF PITCH

scale, one that can be used to establish tlie ratio between any two pitches. If 40 is chosen as the transformation number, the zero on the new pitch scale will be roughly the lowest audible pitch for the average human perceiver. That is, for the average human perceiver there will be no pitches lower than the zero pitch on the new scale; therefore, the new scale can be used to establish the ratio between any two pitches. Pi will be 37 units above zero, P2 will be 43 units above zero, and Ps will be 46 units above zero. Pa is, therefore, 46/37 as high in pitch as P2. (No claim is made here that the selection of a r.«o point 40 halftones below middle C will produce a scale permitting the formulation and verification of numerical laws relating pitch to other dimensions, such as frequency.) It may be argued that the method just described for establishing pitch ratios is a piece of chicanery, and that, however zero points are manipulated, to say that one tone is twice or (especially) 46/37 as high in pitch as another is meaningless. To this argument I reply that the system of measurement which employs a zero point 40 halftones below middle C gives sense to saying that one tone is 46/37 as high as another. If no system of measurement existed for length, it would be meaningless to say that one rod is 46/37 as long as another, just as it seems meaningless to a person unfamiliar with my system of pitch measurement to say that one tone is 46/37 as high as another. A system of measurement can give sense to saying that one object in a dimension is twice or one-half or three times as great as another, even in a dimension in which such a statement at present seems meaningless. Furthermore, the character of intuitive judgments or estimates regarding the ratios of entities in a dimension is largely irrelevant to the measurement of those entities. For example, suppose that the selection of a zero point 40 halftones below middle C creates a scale that assigns to Pi the number 20 and to P2 the number 40. Suppose, however, that most persons would refuse to say, simply on the basis of having listened to Pi and P2 played successively, that P2 is twice as high as Pi. In that event, the scale in question is not useful in predicting the pitch estimates of listeners. But it does not follow that P2 is not twice as high in pitch as Pi. Nor does it follow that the scale would be useless in the formulation and verification of numerical laws relating pitch to other dimensions. And it does not follow that the scale is not properly

257

THE MEASUREMENT OF PITCH

regarded as one of measurement. (See Objection Six in chapter 6 for more on this point.) ^ OBJECTIONS TO THE HALFTONE AS AN INTERVAL Objection Seven.—Presumably an interval of a halftone is equal to any other interval of a halftone, whether the two are or are not in the same part of the pitch scale. Otherwise a halftone would not be a valid candidate for a unit interval of pitch. It may be said that statements about the equality of dimension intervals are, if meaningful, reducible to statements about the equality and addition of dimension objects, and that, since there is no operation of addition for pitch, statements about the equality of pitch intervals are meaningless.2 Consider the dimension of length. Suppose that three rods of unequal lengths are aligned at one end according to the following diagram: X Y Z When we say that the length interval between X and Y is equal to the length interval between Y and Z, we mean that there are two rods, U and V, such that if X added to (juxtaposed with) U equals (aligns at both ends with) Y and Y added to V equals Z, then U equals V. Thus statements about the equality of length intervals are reducible to statements about the equality and addition of rods. Objection Seven may be formally stated as follows: (a) If there is no operation of addition of objects within a given dimension, no meaning can be given to statements about the equality of intervals within that dimension; since (b) there is no operation of addition for the pitch of tones, (c) no meaning can be given to statements about the equality of pitch intervals. A perfectly acceptable way of meeting this objection is simply to deny the truth of its conclusion. If the conclusion is false, then, since the argument is valid, one of its premises is false. The conclusion seems false for the following reasons. Many persons, musicians at least, do make such statements as "These are the same intervals," which is to 2

See A. Ferguson et al, "Quantitative Estimates of Sensory Events," Report of the 108th Annual Meeting, British Association for the Advancement of Science (1938), p. 512, for an objection of this kind.

258

THE MEASUREMENT OF PITCH

say that the intervals are equal. It is absurd to suggest that all such statements, concerning whose truth or falsity there is wide agreement, are meaningless. If the conclusion (c) is false, which of its two premises is false? Not (b), surely, because clearly there is no operation of addition for pitch. The offending premise is (a). Premise (a) rests on the claim that statements about the equality of dimension intervals are, if meaningful, reducible to statements about the equality and addition of dimension objects. This claim, like (a) itself, is false, even for dimensions such as length and weight, which do admit of an operation of addition. Suppose a person looking at the three rods diagramed above says, "The interval between X and Y is equal to the interval between Y and Z." Now it is certainly relevant to the truth of his claim to ask whether, if X added to U equals Y and Y added to V equals Z, U equals V. If the italicized statement is false, the person's claim is usually taken to be false; if true, his claim is usually taken to be true. Suppose, however, that all persons assert that the intervals X - Y and Y-Z are equal, but that the italicized statement is false. In this event the italicized statement might very well be discounted. If such discrepancies became the rule rather than the exception, we would probably not know what to conclude. But it would not be necessary to conclude that immediate visual judgments of length intervals are unreliable. We might conclude instead that rods shrink when aligned. At this point an antagonist may take the line that argument (a)~ (c) is poorly stated, and that with revision it will emerge as sound. The revised argument is as follows: (a') If there is no operation of addition of objects within a dimension, statements about the equality of intervals within that dimension are subjective; since (b') there is no operation of addition for the pitch of tones, (c') statements about the equality of pitch intervals are subjective. The reply to this argument has the same form as the reply to its predecessor. I deny the truth of the conclusion and infer that, since the argument is valid, one of its premises—undoubtedly the first—is false. The term "subjective," although widely used by scientists and some philosophers of science, is virtually useless. What does it mean in the above argument? Suppose it means "in frequent conflict." Among competent observers (musicians, for example) there is virtually unanimous agreement in judgments of the kind in question. Disagreement

THE MEASUREMENT OF PITCH

259

among incompetent observers (those, for instance, who have a poor ear for pitch) may be discounted. On the other hand, if "subjective" means "not verifiable by means of an operation of tone addition," I cheerfully concede the argument, for it is then trivial. It remains to be shown that immediate judgments of pitch-interval equality are inferior to determinations of pitch-interval equality made by means of an operation of tone addition. It remains also to be shown that statements about the equality of pitch intervals are reducible to statements about the equality and addition of tones. Specifically, it remains to be shown that procedures that rely on immediate judgments of pitch-interval equality cannot legitimately be regarded as procedures for the independent measurement of pitch. Even with these arguments my antagonist will undoubtedly be dissatisfied. He may respond as follows. Imagine a system of length or weight measurement analogous to Method A or Method B for pitch, a system employing equal unit intervals of length or weight and providing a scale for the measurement of the dimensions. Would not such a procedure be absurd? Let us describe a procedure for the measurement of length analogous to Method A for pitch. First, a rod is chosen as the rod of zero length. Then other rods, some longer and some shorter than the zero rod, are chosen so as to be separated by equal length intervals. The following diagram pictures a group of standard rods selected in this way. 4 3 2 1 o —1 — 2 —3 —4 A given rod is measured by aligning it with the standard rod to which it is most nearly equal and assigning to it the numeral on that standard rod. This method for measuring length seems absurd in contrast with the other familiar methods for measuring length, such as those that juxtapose equal unit rods or lay a single unit rod in successive adjacent

z6o

THE MEASUREMENT OF PITCH

positions along the rod to be measured. The proposed method is so absurd that it ought not to be regarded as measurement at all. There are several points of reply to this argument. First, there is no alternative to the method for measuring pitch, as there is to the method for measuring length; hence the charge of absurdity cannot be made against it. Second, where is the "absurdity" in the unfamiliar, interval system of length measurement? Perhaps the familiar systems of length measurement would be easier to devise and employ; perhaps the scale resulting from the familiar systems would yield more precise measurements; possibly the familiar systems would, and the interval system would not, lead to the discovery of numerical laws. But if there are these advantages, do they not obtain as a matter of fact? I must confess that I can envisage no apriori proof that advantages of the kind mentioned must lie on the side of the familiar systems. If there is no such proof, the absurdity of using the interval system of length measurement is not a logical absurdity, but simply the absurdity of employing the least advantageous system of measurement. It follows that there is no reason for refusing to regard the interval system as a system of measurement. That one system of numeral assignment for a dimension is in fact less advantageous than another does not entail that it is not a system of measurement. Objection Eight.—Methods A and B both use a pitch interval called the halftone as a unit of pitch. Although there is wide agreement among musicians regarding the equality or inequality of this and other pitch intervals, there are some persons who cannot distinguish pitch intervals at all. Not only are they unable to say whether an interval is a halftone or a major third, they are also unable to determine whether two intervals are equal or unequal. Some of the persons who have never learned to make such distinctions can be taught to do so, but others, misleadingly described as "tone-deaf," cannot. The current objection is that measurement by means of either Method A or Method B can be accomplished only by persons who have a good ear for pitch and by persons who have been trained to identify pitch intervals. On the other hand, the measurement of length by means of a ruler is not restricted to an elite group; anyone can use a ruler to mark off equal intervals of length. More important, the fact that pitch intervals can be identified only by an elite group suggests that these intervals do not exist in the tones themselves, but are merely the artifacts of musical acculturation.

THE MEASUREMENT OF PITCH

261

Part of the reply to this objection is obvious. There is nothing surprising or objectionable in the fact that training may be required before a person can employ my system of pitch measurement, because training may also be required to enable a person to measure length or weight. There is no reason to think that a person who has never attempted to measure length will automatically be able to employ a ruler in determining the length of a floor. For one thing, he must understand what it is to lay the ruler along the floor in adjacent but nonoverlapping positions. Children frequently cannot perform this task in a consistently successful manner. The "tone-deaf" person may be compared with the person who, for one reason or another, cannot learn to lay a ruler along a floor in the proper manner. Neither person will be able to perform measurements in the dimension in question by means of the proposed system. This reply may not seem to go to the heart of the objection, for the following reason. Inability to identify and compare pitch intervals is on a par, not with inability to use a ruler, but with inability to identify and compare lengths. An observer who is unable to pick out from a collection of rods those that are of approximately equal length has abnormal vision. Length equalities are similarities really existing in rods, and inability to perceive these similiarities indicates abnormal vision. But equalities among pitch intervals are not real similarities in tones; they are merely the artifacts of musical training and acculturation, and inability to perceive them indicates merely a poor ear for pitch, not abnormal hearing. But why should we not say that inability to pick out equally long rods indicates merely a "poor eye for length"? It is true, of course, that fewer people are unable to compare lengths than are unable to compare pitch intervals; but training in length comparison is given children from a very early age, whereas many children never receive training in pitchinterval comparison. For those who have been trained and have become skillful, pitch-interval comparisons are as easy as length comparisons. The difference between length similarities and pitch similarities is not that the former are real and the latter artifacts, but rather that the former are more important and more familiar to us. Objection Nine.—A useful system of measurement must have a unit that does not vary from one measurement to another. For example, an elastic band, or an object that undergoes substantial alterations in

262

THE MEASUREMENT OF PITCH

length with changes in temperature, cannot be the unit object of length. It may be said that the halftone is "elastic": what is identified as a halftone in a given part of the scale at one time may not be the same interval identified as a halftone in the same part of the scale at another time, so that the unit of pitch may vary without the observer's realizing it. The reply to this objection is that the possibility of variation in the unit is no more serious for pitch measurement than for length measurement. First, a variation in the unit of pitch can manifest itself in the same way as a variation in a unit of length. If a certain object is found to be ten units in length at one time and eleven units in length at another, either a mistake has been made in one of the measurements, or the object measured has changed in length, or the unit of measurement has altered. Which hypothesis is adopted depends upon the details of the particular case. Similarly, if a tone is determined to be ten halftones above middle C at one time and eleven at another, either a mistake has been made in one of the measurements, or the two tones are not really of the same pitch, or intervals of different size have been regarded as equal (as halftones). Again, the details of the particular case determine the correct hypothesis. Second, the same-cause-same-pitch criterion can be used, at least to some extent, to determine whether equal intervals in the same part of the scale are being designated as halftones in different measurements. Suppose that during one measurement the interval between tones Ti and T2 is said to be a halftone, and that during a second measurement the interval between Ts and T4 is said to be a halftone. If the sound waves, Wi and W3, producing Ti and T3 are the same in all respects (equal in frequency and intensity), and if the sound waves, W2 and W4, producing T2 and T4 are the same in all respects, then the intervals between Ti and T2 and between Ts and T4 are equal. If Wi and W3 are the same in all respects, but W2 and W4 are not the same in all respects, the intervals between Ti and T2 and between Ts and T4 are not equal. Objection Ten.—Objection Nine suggests that halftone intervals in the same part of the pitch scale may not remain equal from one measurement to the next. Objection Ten holds that halftone intervals in different parts of the scale may not in fact be equal. Suppose a musician identifies the interval between two tones played in the base register

THE MEASUREMENT OF PITCH

263

as a halftone, and then identifies the interval between two tones played in the treble register also as a halftone. Perhaps he is mistaken; perhaps the intervals are not really equal. Even more strikingly, perhaps the interval called a halftone by musicians progressively widens as one ascends the musical scale, and perhaps this variation is the same for all competent observers. If the variation is the same for all observers, and the same from one measurement to the next, it could not be shown to exist by comparing the results of different measurements. The hypothesis just stated does not seem meaningful. How can it be verified? The only possible way, given complete observer agreement, requires reliance on some criterion for interval equality other than hearing. Can the criterion be the readings of a frequency meter? The interval between B below middle C and middle C is a halftone; so also is the interval between B below C below middle C and C below middle C, although it is lower in the musical scale than the first. The frequencies that produce these tones are, respectively, 240 cps, 256 cps, 120 cps, and 128 cps. The frequency interval between the first two tones is 16 cps; that between the last two tones is 8 cps. Since the frequency intervals are unequal, the use of frequency readings as the criterion would establish that the pitch intervals between the two pairs of tones are unequal. This conclusion is illegitimate; the pitch intervals between the two pairs of tones are equal, as every musician would insist. T o use the readings of a frequency meter as a criterion for pitch-interval equality or inequality is to identify pitch and frequency, and this identification is entirely questionable. M y conclusion is that intervals in different parts of the scale are equal if all or most competent observers call them equal. The hypothesis that they are nevertheless unequal makes sense only if a criterion other than hearing is employed for pitch-interval equality, only if "unequal" refers to intervals in a dimension other than pitch, such as frequency. Unequal frequency intervals, however, do not necessarily imply unequal pitch intervals, for frequency and pitch are apparently causally related, not identical. It may be replied that pitch has two aspects, a "molecular" aspect and a "molar" aspect, and that pitch intervals may be unequal in the former aspect and yet equal in the latter. Whether this dual-aspect theory is acceptable is a difficult question, but let us grant its acceptability for the sake of argument. The system of pitch measurement herein described is a system of measurement for pitch

264

THE MEASUREMENT OF PITCH

in its molar aspect. Therefore, if all or most competent observers call intervals in different parts of the scale equal, the intervals are equal in their molar aspect. Whether they are equal in their molecular aspect is irrelevant for my purposes. Notice also that if my system measures pitch in its molar aspect, the system of measurement which employs a meterstick measures length in its molar aspect. If a dual-aspect theory applies to pitch, it also applies to dimensions such as length and weight. And the systems of length and weight measurement usually taken to be paradigm examples of measurement of a dimension must be regarded as systems for the measurement of molar aspects of dimensions. (See chapter xo for a full treatment of this point.) OBJECTIONS FROM SCIENTIFIC USEFULNESS Objection Eleven.—Is the halftone method of measuring pitch of any use in the discovery of numerical laws relating pitch and other dimensions? Or is it rather the useless invention of a philosopher who concerns himself only with the analysis of such concepts as that of measurement, or, worse yet, only with the meaning of words? To the hardheaded scientist, who wishes to measure pitch in order to discover precise relationships between it and other dimensions, who wishes to explain and predict the behavior of tones, any success thus far in defending the halftone method against objections may seem a pyrrhic victory, or perhaps a verbal one. Of what importance is it that we may call a method for assigning numerals to pitch "measurement," if the method is useless? One possible reason for thinking that the halftone method is scientifically useless is that it seems imprecise. No smaller unit than the halftone is feasible in a system of pitch measurement, since the halftone is almost the smallest interval that even musicians can clearly identify. The interval between B below middle C and middle C is one halftone. But the frequency that produces the first tone is 240 cps, and the frequency producing the second is 256 cps, a frequency interval of 16 cps. The halftone is thus seen to be a relatively gross unit. How, using it, can we expect fruitful results, such as the discovery of a law relating pitch and frequency? The reply is twofold. First, the halftone system of measurement may have uses other than that of discovering a numerical law relating pitch and frequency. Even if the halftone system does not lead to the discovery of such a

THE MEASUREMENT OF PITCH

265

law, it still might lead to the discovery of numerical laws relating pitch and the length of vibrating strings, or to numerical laws relating pitch and other dimensions of sound, such as loudness or volume. (Volume is recognized by many psychologists as being a dimension of sound distinct from either loudness or pitch.) When it is said that measurements made by means of a certain system are "too imprecise," we must ask, "Too imprecise for what?" For some experimental purposes weight measurements made on a balance accurate to the nearest tenth of a gram are adequate, whereas other purposes require weighings accurate to the nearest thousandth of a gram. What is required depends upon the particular line of investigation being pursued. Second, as is shown in the reply to Objection Twelve, a system of independent pitch measurement employing a unit even larger than the halftone (the octave) does in fact lead to the formulation of a numerical law relating pitch and frequency. Charges of imprecision against a system of measurement, when made apriori and without reference to the experimental and theoretical usefulness of the system, are dangerous. The proof of the pudding is in the eating. Objection Twelve.—The reply to Objection Five shows that the use of an arbitrary zero in the halftone method of pitch measurement does not make it impossible to use that method to discover numerical laws relating pitch and other dimensions. But no evidence was provided that use of the method does in fact lead to such laws. It may be said that the halftone method is scientifically useless because it has not led and will not lead to numerical laws governing pitch. I will show that this claim is false by deducing certain numerical laws relating pitch and frequency from what is already known about the relation between these two dimensions. As the reply to Objection Five shows, there is no apriori reason that the middle C pitch scale cannot be converted, by adding to each of its values a suitable transformation number, into an absolute scale that will permit the formulation and verification of numerical laws relating pitch and other dimensions. The analogy between the centigrade scale of temperature and the middle C scale of pitch helps to make this point clear. The Kelvin or absolute scale of temperature was established in the following way. The first step was the discovery of Charles's law, which states simply that equal increases in the temperature of a gas

266

THE MEASUREMENT OF PITCH

under constant pressure are accompanied b y equal increases in the volume of the gas. It w a s then discovered that w h e n the temperature of a given volume, V i , of gas under constant pressure is lowered f r o m o° centigrade to — 1 ° centigrade, its volume is V i — 1 / 2 7 3 V i . From this fact together with Charles's l a w it was inferred that the volume of a gas cooled to — 2 7 3 ° centigrade is n e g l i g i b l e — f o r all practical purposes, zero. Consequently, — 2 7 3 0 centigrade w a s designated absolute zero w i t h respect to temperature. (It is impossible w i t h present techniques to cool a gas to this temperature; all gases liquefy, that is, become nongases, before this point is reached.) T h u s a n e w scale of temperature, called the Kelvin scale, was created, in w h i c h each degree Kelvin corresponds to a single degree centigrade and in w h i c h o° Kelvin equals — 2 7 3 ° centigrade. T h e new scale makes it possible to express Charles's law numerically, as [4] T = c V , where V is volume, c is a constant, and T is temperature in degrees Kelvin. Just as the use of the centigrade scale of temperature (which employs an arbitrary zero) led to the discovery of an absolute scale of temperature and thus to the formulation of a numerical law relating gas temperature and volume, so the use of a pitch scale employing an arbitrary zero will lead to the discovery of an absolute scale of pitch and thus to the formulation of a numerical law relating pitch and frequency. T o prove this assertion, I shall first discuss a system of pitch measurement which employs the octave

(instead of the half-

tone) as the unit of pitch, and middle C (256 cps) as the arbitrary zero. It has been k n o w n since the Pythagoreans that the sources for t w o tones separated b y an octave stand in the relation of 2:1. T h i s is true for frequency as well as for string length. Middle C is produced b y a sound w a v e of 256 cps; C below middle C , b y a frequency of 128 cps; the C below that, b y a frequency of 64 cps; and so on. In this p o w e r series, each number equals 2 raised to some power. T h e absolute zero for pitch, then, should correspond to a frequency of 1 cps, since 2 or any other number raised to the zero power equals 1 (20 = 28 =

1). Since

256, absolute zero on the middle C octave pitch scale is — 8

octaves. Therefore, to convert the middle C octave pitch scale to an absolute octave scale, the number 8 must be added to each value of the former. T h u s , o (middle C ) becomes 8, 1 (C above middle C) becomes 9, and so on.

THE MEASUREMENT OF PITCH

267

In terms of the values of this absolute pitch scale, a numerical law relating pitch and frequency can be formulated. It is [ 5 ] F = 2p, where F is the frequency in cycles per second and P is the pitch in units of the absolute octave scale. For example, middle C is 8 units above zero on the absolute scale, and is produced by a frequency of 256 cps. When F = 256 and P = 8, equation [5] is satisfied. In two important respects [5], the frequency-pitch law just formulated, is analogous to [4], Charles's law numerically formulated. (1) Just as gases liquefy at extremely low temperatures, so extremely low sounds cannot be heard. As a matter of fact, gases cannot attain a temperature of —273 0 centigrade; as a matter of fact, sounds cannot attain a pitch of 8 octaves below middle C. To speak of either gases or sounds of these low magnitudes is to employ a useful theoretical fiction, useful because it makes possible the construction of absolute scales which lead to the formulation of numerical laws. Strictly speaking, however, the laws thus formulated do not apply for extremely low temperatures or extremely low pitches. (2) Neither [4] nor [5] is a completely accurate description of the relations between the dimensions referred to. Charles's law is a better description of the behavior of some gases than it is of the behavior of others; and it applies more accurately to gases to under certain (constant) pressures than to gases under other pressures. Equation [5] describes with a fair degree of accuracy the relation between pitch in its midrange and frequency. Departures from the law can be observed in the upper and lower pitch ranges. The physicist says that the gas laws are completely accurate descriptions of "ideal gases," but an "ideal gas" is defined by the physicist as one that satisfies a certain set of gas laws. Therefore his statement is a tautology. It is, however, an interesting tautology within his theoretical structure. In the same way we might say that [5] describes the behavior of "ideal sounds," defining "ideal sounds" as those that satisfy a specified set of laws, such as [5]. The tautology might (or might not) prove to be theoretically interesting. When two tones are separated by the interval of an octave their frequencies stand to each other in the ratio of 2 : 1 , in whatever part of the pitch scale they occur. The fact that octaves, in whatever part of the pitch scale they occur, are equal intervals suggests that the

268

THE MEASUREMENT OF PITCH

frequencies producing tones separated by equal pitch intervals stand to each other in equal ratios. It suggests, for example, that the frequencies producing an interval of a halftone in one part of the scale will stand to each other in the same ratio as the frequencies producing an interval of a halftone in another part of the scale, and that in both instances the value of the higher frequency is the value of the lower multiplied by some number. Now a halftone is one-twelfth of an octave: if one counts twelve halftones up the pitch scale, the first and last tones will be one octave apart. Let P0, Pi, P2,... P12 be the thirteen tones separated by halftone intervals, and let F0, Fi, F 2 . . . F12 be the thirteen frequencies that produce the tones. We wish to discover a number, n, such that if frequency F0 is multiplied by n, and the result by n again, and so on for twelve such multiplications, frequency F12 (the result of the twelfth multiplication) will be twice F0. The number that satisfies this requirement is 1.059463—the twelfth root of 2. The discovery of this number permits the formulation of a numerical law relating pitch and frequency when pitch is expressed in terms of a halftone scale of measurement. Again, the pitch corresponding to a frequency of 1 cps is the zero point. The absolute zero for pitch is then 96 halftones below middle C (8 octaves below, 12 halftones to the octave, 8 x 12 = 96). The numerical law is [6] F = I.O59463p, where F is the frequency in cycles per second and P is the pitch in terms of the absolute halftone scale just described. To illustrate the law, consider a tone that has the pitch of middle C. On the absolute scale it has a pitch of 96 unit halftones (i.e., P = 96). The frequency producing this tone is 256 cps (i.e., F = 256). These values satisfy equation [6]. (The computation, most easily performed by using logarithms, is not given here.) Laws [5] and [6], relating pitch and frequency, have the same form, [7] F = c* They differ in that the value of the constant, c, is different. But they are, in an important sense, the same law—the same law expressed in terms of different units of pitch. Suppose the constant, c, in Charles's law of gases, [4] T = cV, is n, where temperature, T, is expressed in degrees Kelvin and volume, V, in cubic inches. There are 10.648 cubic centimeters to the cubic inch. Hence, we can express [4] differently by tak-

THE MEASUREMENT OF PITCH

269

ing c — nl 10.648 and expressing volume in terms of cubic centimeters. In the same way [6] is a different way of expressing [5]. When pitch is expressed in octaves, c = 2. When pitch is expressed in halftones, then, (i.e., since there are twelve halftones to the octave, c becomes 1.059463). The remarks immediately above prepare us for a further objection against regarding either the octave or the halftone system of numeral assignment as a system of pitch measurement. In physics the choice of smaller units in systems of length, weight, and temperature measurement leads to the formulation of more accurate numerical laws relating the dimensions in question (this rule is a matter of fact). For example, the use of a centimeter scale of length leads to a more accurate formulation of Charles's law than does the use of an inch scale of length. The laws relating pitch and frequency, however, do not fulfill a comparable expectation. A s a matter of fact, [6] is a less accurate law than [5]. In the midrange of pitch, the frequency producing a tone one octave higher than another is always twice as great as the frequency producing the latter tone, but the frequency producing a tone one halftone higher than another is not always 1.059463 times as great as the frequency producing the latter tone. The two systems of pitch measurement are, therefore, different in character from the familiar systems of length, weight, and temperature measurement. This difference may seem to suggest that the systems here described are not really systems for the measurement of pitch, or, at least, that they are not scientifically useful systems for pitch measurement. This suggestion is not correct. Suppose a system of length measurement, Li, is replaced by another system, L2, employing a smaller unit. Suppose further that the laws discovered through the use of La are less accurate than those discovered through the use of Li. W e would then conclude either (a) that the laws discovered through the use of Li are false, that we have not, as we first thought, discovered a law relating the dimensions in question; or (b) that the laws discovered through the use of Li and L2 are only approximate, that additional variables have not been taken into consideration, and that when they are a more exact law which includes these variables can be formulated. W e would not and need not conclude that Li and La are not systems of measurement for length, or that Li and La are not scientifically useful. Similarly, the fact that laws discovered through the use of the halftone system

270

THE MEASUREMENT OF PITCH

of pitch measurement are less accurate than those discovered through the use of the octave system does not mean that one or both systems are not systems of measurement, or that they are not scientifically useful. Objection

Thirteen.—To

obtain numerical law [5] it was neces-

sary to convert the octave scale employing middle C as an arbitrary zero into an absolute scale where absolute zero is 8 octaves below middle C . That is, to convert the middle C scale into an absolute scale, — 8 was added to each value of the former. (The method of obtaining the conversion number, — 8 , is explained above.) Objection Thirteen maintains that the selection of this conversion number was possible only because w e knew which frequencies to associate with pitches of C , only because frequency had previously been measured. Since the measurement of pitch b y means of the absolute octave scale depends on the measurement of frequency, it is a procedure, not of independent, but of dependent measurement. The reply to this objection must be made in several stages. T h e two definitions of dependent and independent measurement presented in chapter 5 are here reproduced for convenience. Definition

4a. Dependent measurement is a procedure of assigning

numerals to objects within a dimension by means of tin empirical process of comparing these objects with a unit or units, which includes at least one subprocedure of measurement. Independent measurement is a similar procedure except that it includes no subprocedures of measurement. Definition

4b. Dependent measurement is a procedure of assigning

numerals to objects within a dimension by means of an empirical process of comparing these objects with a unit or units that are either identified with or defined in terms of units for some other dimension. Independent measurement is a similar procedure except that its units are neither identified with nor defined in terms of units for some other dimension. According to Definition 4a, the measurement of pitch b y means of the absolute octave scale does not qualify as independent measurement if, as Objection Thirteen asserts, construction of the scale requires the measurement of frequency. For then the measurement of pitch consists in (a) measuring associated frequencies, (b) using the

THE MEASUREMENT OF PITCH

271

numerals thus obtained to arrive at the conversion number —8, (c) assigning numerals to pitch by means of the middle C pitch scale, and (d) converting the values thus obtained to those on an absolute scale by adding —8 to each. Since (a) is in itself a procedure of measurement, the complex procedure (a)-(b)—(c)-(d) includes a subprocedure of measurement and is, by Definition 4a, dependent measurement. There are two points of reply. First, why must we accept the above analysis of the absolute octave procedure? Let us say, rather, that the measurement of pitch by means of this procedure consists in (i) assigning numerals to pitch by means of the middle C pitch scale, (ii) obtaining a conversion number, and (iii) adding this number to each of the numbers assigned in (i). How the conversion number is obtained is irrelevant: perhaps it is arrived at by intuition or guessing, or perhaps in some other way. The procedure (i)-(ii)-(iii) is a description of the absolute octave system of measurement, and, as described, it contains no subprocedures of measurement. It qualifies, therefore, as independent measurement under Definition 4a. Second, we need not accept Definition 4a. Definition 4 b has, as noted in chapter 5, certain advantages over its competitor. O n Definition 4b, the measurement of pitch by means of the absolute octave scale clearly does qualify as independent measurement, for the unit in this system—the octave— is neither identified with nor defined in terms of units of frequency or any others. To clarify this last point it will be helpful to contrast the absolute octave pitch scale with the Kelvin scale of temperature. In the latter the values of the centigrade scale are converted into those of an absolute scale by the addition of —273. In this respect the absolute octave pitch scale compares with the Kelvin scale. But the unit of the Kelvin scale, unlike the unit of the pitch scale, is defined in terms of a unit for another dimension (i.e., length). A centigrade thermometer is constructed in the following manner. A n uncalibrated mercury tube is placed in freezing water, and the level reached by the mercury under these conditions determines the zero-degree point. The tube is then placed in boiling water, and the level reached by the mercury under these conditions determines the xoo-degree point. Then 100 equal intervals are marked off between the zero- and the 100-degree points b y some system of length measurement. Finally the scale is extended below the zero-degree point and above the 100-degree point in the

272

THE MEASUREMENT OF PITCH

same length intervals. Each such interval represents a degree—the unit of temperature. It is plain that the unit of temperature is defined in terms of the unit of length in the system of measurement used to mark off the intervals on the mercury tube. Let P* be the point to which the mercury rises or falls when the thermometer is brought into contact with X, Py the point reached when the thermometer is brought into contact with Y, Pt the point reached by contact with freezing water, and Pb the point reached by contact with boiling water. Let P x - P r represent the distance (length) between points P * and P 7 , and P r - P b the distance (length) between points Pr and Pb, expressed in number of units of some system of length measurement. The unit of temperature may then be defined as follows. A degree is the interval in temperature between any two objects, X and Y, which satisfy the following condition: Pr — Px —

Pb

— = xoo. Py

Since P r - P b and P*-P y must be expressed in terms of the unit in some system of length measurement, the unit of temperature is defined in terms of the unit of another dimension, both in the centigrade scale and in the Kelvin scale based on it. Both scales are, therefore, scales for the dependent measurement of temperature under Definition 4b. But the octave in the system of pitch measurement is neither identified with nor defined in terms of a unit for some other dimension. The scales that employ this unit are, therefore, scales of independent measurement under Definition 4b. Parallel points apply to the halftone system of measurement: it too is a system of independent measurement under both Definition 4a and Definition 4b. APPLICATIONS AND CONCLUSIONS The method for assigning numerals to pitch proposed in this chapter qualifies as measurement under the definition recommended in chapter 5: Measurement is the assignment of numerals to objects within a dimension by means of an empirical process of comparing these objects with a dimension unit or units. The unit in my method of pitch measurement is the halftone, and the empirical process of comparing these halftones with the one to be measured is the process

THE MEASUREMENT OF PITCH

273

of counting off adjacent halftones from zero (absolute or arbitrary). Does this method qualify as measurement by the classical definition discussed in chapter 4? Measurement is there defined as the assignment of numerals to objects within a dimension by means of a physical operation of adding dimension units. Whether my method qualifies under this definition is not a crucial issue, since the definition has been rejected as unacceptably narrow. But if it does qualify, there is an even stronger argument for saying that pitch is capable of measurement. To say that measurement involves an operation of adding dimension units leaves open the question of whether the units must be unit objects or unit intervals. If the addition must be of unit objects, as in the familiar systems of length and weight measurement, pitch is not measurable according to the classical definition. There is no operation of adding tones with respect to pitch, no operation of adding pitches. If the addition may be of unit intervals, a different assessment is in order. M y method of measuring pitch employs the unit interval of a halftone, and there is an inclination to say that halftones are added in the method. It would appear that the classical definition has been intended by its proponents to require the addition of unit objects. But there seems to be no compelling reason that it should not be more broadly interpreted so as to permit the addition of unit intervals. Its characteristic feature—the requirement that a physical operation of addition is essential to measurement—would still be preserved. Is it correct to say that the halftone method of pitch measurement employs a physical operation of adding intervals? The operation employed in the method is the successive playing of adjacent halftones. Suppose the zero pitch in the system is To and that we wish to measure Tn, a tone above zero. We do so by playing To and Ti, then Ti and T2, then T2 and T3, and so on, each pair played being separated by an interval of a halftone, until we reach a tone equal to or nearly equal to T n . Is this operation one of adding intervals? It is, of course, not an operation like that of juxtaposing unit rods, or like that of placing unit weights in the pan of a balance. It is more like the operation of laying a single unit rod in successive adjacent positions along the length of another, but it is unlike that operation in some respects. It is not to be expected, however, that adding intervals will be just the same as adding objects.

274

THE MEASUREMENT OF PITCH

These remarks are merely suggestive. How, precisely, can the operation of pitch addition be described? One point seems clear. Two tone intervals cannot be added unless they have one tone in common. Thus, T1-T2 and T2-T3 can be added, but T1-T2 and Ts-T< cannot. Now, what do we do when we add two tone intervals? The only possible answer is that we play the noncommon tones in the two intervals. To add T1-T2 and T2-T3 we play Ti and T3; that is, we play Ti-Ts. This description is unsettling, for it seems to imply that the results of an operation of tone-interval addition are known in advance, and that the operation is therefore not an empirical, physical one. In one sense, of course, we know the outcome of any operation of addition in advance. We know in advance that if we juxtapose rods Ri and R2 the result will be the juxtaposition (Ri @ R2). Similarly, we know in advance that if we add pitch intervals T1-T2 and T 2 - T 3 the result will be the addition T1-T3. But there is no rod, Rn, of which we can know in advance that it will be equal in length to the juxtaposition of Rx and R2. In contrast, there is one interval, T1-T3, of which we can know in advance that the addition of T1-T2 and T 2 - T 3 will equal it. Further evidence that the so-called operation of pitchinterval addition is not an empirical one can be obtained by interpreting the axioms of addition in chapter 4 in terms of the operation. Thus interpreted, axioms (6) and (7) seem to be logically, not empirically, true. This baffling question need not be pursued further. It is sufficient to point out that even if there is, properly speaking, no such thing as the physical addition of pitch intervals, still the system of assigning numerals to pitch satisfies the definition of measurement recommended in this book. Pitch can be measured by an operation of counting off equal pitch intervals from some stipulated zero, whether or not the operation is called "addition of pitch intervals." This method is possible because pitch is interoallic. That is to say, certain pitch intervals can be confidently identified and reidentified by a person with an ear for that sort of thing. Such a person can tell whether two intervals in different parts of the pitch scale are the same or different, whether both are perfect fifths, or whether one is a halftone or a minor third. His report can be confirmed (or disconfirmed) by other persons with an ear for intervals.

THE MEASUREMENT OF PITCH

275

No dimensions other than pitch seem to be intervallic. No names exist for intervals of hue, or intervals of acridity, or intervals of sourness. And, so far as I know, there are no observers who can confidently say whether hue intervals or sourness intervals in different parts of their respective scales are the same or different, and expect to have other observers confirm or disconfirm their reports. For this reason no method for measuring hue or sourness, like that provided for pitch in this chapter, can be described. Does it follow that these other dimensions cannot be measured? The question is a difficult one, for its answer depends on whether these other dimensions are by nature intervallic. Is it in the nature of hue that hue intervals cannot be identified and reidentified, or given names? Or is it in the nature of human observers? Or, again, is it an accident of history? Perhaps if hue played a role in our aesthetic experience similar to that played by pitch, hue would be intervallic. Perhaps if a "music" of hue had been developed, it would for its "musicians" be intervallic. To consider another possibility, perhaps the nonintervallic character of hue is a function of the discriminatory abilities of human beings. Perhaps other possible perceivers would respond to hue intervals in the same way musicians respond to pitch intervals, so that for those perceivers hue would be intervallic. With so many possibilities left open, it would appear that we cannot with any confidence pronounce hue, acridity, sourness—and other dimensions traditionally classified as "qualitative"—immeasurable. They are, at most, immeasurable only if they are nonintervallic by nature. And even this conclusion seems rash. For as long as it is uncertain whether there are physical operations of addition for hue, acridity, and sourness, it remains uncertain whether they are immeasurable. The difficulties here are that (a) there may be operations of addition for some or all of these dimensions which no one has ever performed or has even thought of, and (b) the concept of a physical operation of addition needs further analysis. Should we say that pouring two sour liquids into the same beaker and boiling the mixture until it is more sour than either ingredient is an operation of addition for sourness? Should we say that mixing two red colors and treating them chemically so that the mixture becomes redder than either ingredient is an operation of addition for hue? I provide no answer to these

276

THE MEASUREMENT OF PITCH

questions. A n answer must be provided, however, before w e can conclude that the traditionally immeasurable dimensions are really so. THE MEASURABILITY OF PSYCHOLOGICAL DIMENSIONS

O n e of my central contentions in this book is that some of the dimensions traditionally, but wrongly, regarded as "psychological" are measurable. I have described methods of measurement for loudness and pitch; I have suggested the possibility of similar methods for other dimensions of the same type. Several points must be kept in mind if the full significance of this conclusion is to be felt. A brief statement of these points will also serve as a partial summary of my argument. For simplicity, only loudness and pitch are discussed. x) W h e n w e say that loudness and pitch are measurable, b y "loudness" and " p i t c h " w e neither mean nor refer to any dimension or dimensions of sound waves. Failure to understand this point could result in a charge of triviality: " W h e n you say that loudness and pitch are measurable, you simply mean that intensity and frequency are measurable. But no one would dispute that statement. Y o u still have not shown that heard loudness and heard pitch—that

is, loudness and

pitch in the ordinary, nontechnical sense of these terms—are measurable." It is quite clear that in the ordinary, nontechnical sense of the terms, "loudness" does not mean "intensity," and " p i t c h " does not mean "frequency." But it is in just this sense that I assert loudness and pitch to be measurable. I do mean "heard loudness" and "heard pitch." This point should be perfectly clear, given my methodological assumption that neither loudness nor pitch may be identified with any dimension or dimensions of sound waves. That "loudness" and " p i t c h " do not mean "intensity" and " f r e q u e n c y " does not entail that loudness and pitch are not identical with intensity and frequency. (Similarly, that " c l o u d " does not mean "collection of dust particles" does not entail that clouds are not collections of dust particles.) But if, as I have consistently assumed throughout m y argument, loudness and pitch are not identical with intensity and frequency, then "loudness" and " p i t c h " cannot mean "intensity" and "frequency." (Similarly, if clouds are not collections of dust particles, " c l o u d " cannot mean "collection of dust particles.") 2) I have argued that loudness and pitch are independently

mea-

THE MEASUREMENT OF PITCH

2 77

surable. Hardly anyone would deny that these dimensions can be dependently measured, via measurement of the intensity and frequency of correlated sound waves. That they are capable of independent measurement is, however, contrary to widespread belief. I have described and defended procedures of measurement for both loudness and pitch which contain no subprocedures of measurement. Or, to put the matter differently, I have advanced procedures for expressing loudness and pitch in units neither identified with nor defined in terms of units for other dimensions. In this way I have shown that loudness and pitch are measurable by procedures essentially similar to those for length and weight. There is a tendency to think that certain dimensions are basic or fundamental, from the point of view of science, in that all other dimensions can be measured only b y means of them, in that the measurement of all other dimensions depends on their measurement. Length and weight (and, at times, number and duration) are frequently mentioned as the chief examples of these fundamental dimensions. But I have shown that, insofar as measurement is concerned, loudness and pitch are no less "fundamental" than length and weight, since their measurement does not depend on the measurement of other dimensions. 3) M y conclusion that loudness and pitch are independently measurable has not been established at the expense of adopting an unacceptable definition of measurement. It is a simple matter to "prove" that loudness and pitch are measurable if wide latitude is allowed in defining the term "measurement." Such proofs have no philosophical significance. If measurement is defined simply as the assignment by rule of numerals to objects, obviously loudness and pitch are measurable because numerals can be assigned by rule to tones. But this definition is unacceptably broad. It counts as measurement the numbering of players on a football team. Again, if measurement is defined as the assignment of numerals to objects so as to represent their rank within a given dimension, obviously loudness and pitch can be measured. Tones can be ranked with respect to loudness or pitch, and numerals can be assigned to them so as to represent their rank. But, again, the definition is unacceptably broad. To prove that loudness and pitch are measurable by adopting such definitions has no philosophical significance. Such proofs do not show

278

THE MEASUREMENT OF PITCH

that there are procedures for assigning numerals to loudness and pitch which are essentially similar to those available for length and w e i g h t — the paradigms of measurable dimensions. There is clearly a difference between measuring length with a ruler or weighing objects on a balance, and assigning numerals to minerals by means of Mohs's scale of hardness or numbering football players. W e must first devise a definition of measurement which includes the former procedures and excludes the latter, and then show that in this sense loudness and pitch are measurable, if we are to produce a conclusion of philosophical significance. That is precisely what I have done. I proposed a definition of measurement which, broad as it is, unduly violates neither technical nor ordinary use. Measurement, I said, is the assignment of numerals to objects within a dimension so as to express their magnitudes in terms of dimension units. I then argued that loudness and pitch are measurable in this philosophically interesting sense. PSYCHOPHYSICAL LAWS OF AUDITORY PERCEPTION

M y conclusion and the analysis that led to it have an important bearing on the nature of psychophysical laws of hearing. Loudness and pitch are usually regarded as psychological or subjective dimensions, and are contrasted with the physical dimensions of length and weight. More specifically, they are usually regarded as dimensions of sensation. In chapter 2 I demonstrate that this view is erroneous by showing that sounds are, like physical objects possessing length and weight, public, perception-independent, and genuinely locatable. That loudness and pitch are not dimensions of sensations is confirmed by my conclusion that they are, like length and weight, capable of independent measurement. Perhaps hearing always involves having auditory sensations. But, if so, auditory sensations are not to be confused with sounds, and loudness and pitch are not to be confused with "sensation loudness" and "sensation pitch" (dimensions of auditory sensations). From the foregoing it is clear that laws relating loudness and pitch to physical dimensions are physical, not psychophysical, laws. This conclusion places me at variance with a large number of philosophers and with the majority of psychologists, especially those who share Fechner's conception of psychophysics. This conception, which was dominant during the past century and strongly persists in the

THE MEASUREMENT OF PITCH

279

present, may be briefly characterized as follows. A physical law is one relating a physical dimension, $1, to another physical dimension, $2. It may be a nonnumerical law or a numerical law. If the former, it states simply that varies, either directly or inversely, with $2. If the latter, the law states that = /(2), where $1 and $2 are expressed in units of measurement appropriate to each. A psychophysical law is, as the name suggests, one relating a psychological dimension, Y i , to a physical dimension, $1. Again, it may be a nonnumerical law or a numerical law. If the former, it states that varies, directly or inversely, with $1. If the latter, the law states that = f($i), where Y i and i>i are expressed in units of measurement appropriate to each. One of the critical questions for this conception is whether it is possible to formulate numerical psychophysical laws. The problem has been that it does not seem possible to measure psychological dimensions independently, as is desirable (perhaps, even essential) if we are to establish numerical laws relating psychological dimensions to physical dimensions. Because loudness and pitch have been regarded as psychological dimensions, this problem has been thought to apply to them as much as to pain intensity, afterimage size, or any other clearly psychological dimension. I have shown that there is no problem about the measurement of loudness and pitch, that they are, like length and weight, capable of independent measurement, and that it is perfectly possible to formulate and verify numerical laws relating them to physical dimensions. But in the process I have also shown that these laws will be, not psychophysical, but physical laws, since loudness and pitch are physical rather than psychological dimensions. To cite several examples of such laws, there can be (a) laws relating loudness to pitch, (b) laws relating loudness to string tension, (c) laws relating pitch to string length, (d) laws relating loudness to intensity of sound waves, (e) laws relating pitch to frequency of sound waves. Laws of types (d) and (e) are possible only on the assumption that loudness and pitch cannot be identified with intensity and frequency. O n the opposite assumption, loudness has to be identified with a certain function of frequency and intensity (as density is identified with a certain function of weight and volume). With this identification, laws of types (b) and (c) can alternatively be described as, respectively, (b') laws relating intensity and frequency to string ten-

28o

THE MEASUREMENT OF PITCH

sion, and (c') laws relating frequency and intensity to string length. The important point is that, whether or not loudness and pitch are identified with dimensions of sound waves, laws relating loudness and pitch to dimensions of sound waves, or to other physical dimensions, are physical laws. This conclusion creates a grave problem for the traditional psychophysicist. What are the psychophysical laws of auditory perception, if not those relating loudness and pitch to physical dimensions? Auditory perception has always seemed a safe haven for the traditional conception of psychophysics. Everyone knows what sounds are and that their dimensions are loudness and pitch. All normal observers are able to make perceptual discriminations within these two dimensions. When, therefore, the traditional psychophysicist says that psychophysical laws of audition are those relating loudness and pitch to, say, intensity and frequency, he illustrates his conception in what appears to be a clear and incontrovertible manner. Sounds are enough like sensations for the unwary to confuse the two. At the same time, dimensions of sounds are enough like length and weight to suggest that they too ought to be measurable. Consequently, superficial reflection on auditory perception can easily lead to the belief that a science of the relations between psychological and physical dimensions is possible. My conclusion shows that the belief thus founded is confused. Loudness and pitch are, indeed, measurable; but they are not psychological dimensions. Their measurement will produce physical laws, not psychophysical laws, not laws of mind and body. The most natural response to this argument is to say that the psychological dimensions involved in psychophysical laws of auditory perception are dimensions of auditory sensations. Auditory sensations are perception-dependent, private, and nonlocatable, and thus clearly qualify, where sounds and sound waves do not, as psychological dimensions. Consequently, laws relating these sensations to their physical stimuli seem to qualify as psychophysical laws of auditory perception. If auditory sensations are measurable, then numerical laws relating them to their physical stimuli are possible, and the aim of traditional psychophysics is apparently attainable. Numerous thinkers in many of the sciences have held that sensations are incapable of measurement. The traditional psychophysicists believe otherwise. My methods for measuring loudness and pitch do

THE MEASUREMENT OF PITCH

281

not settle this dispute, since these dimensions are not dimensions of sensations. Nonetheless, they have important similarities to dimensions of sensations, and can serve to some extent as useful analogies. Sounds are like sensations in that they are processes and not objects. And, although they are public and perception-independent, they are perhaps less so (if the notion of degree makes any sense here) than inkwells and coffee cups, the standard examples of physical objects. Finally, although sounds possess genuine location, they possess it in a different sense from inkwells and coffee cups. (See chapter 2 for these points.) A discussion of the measurement of sounds can serve, then, as a useful preliminary to a discussion of the measurement of sensations. Perhaps the difficulties in measuring sensations are like the difficulties in measuring sounds. Perhaps the difficulties are merely apparent for the one, as we have found them to be for the other. Perhaps, to go one step further, sensations can be measured by methods like those described for measuring sounds. There seem to be powerful objections to this last suggestion. Loudness, as we have seen, can be measured by a method that employs the operation of playing tones simultaneously as an operation of addition, a method precisely comparable to methods used for length and weight. But sound sensations are apparently incapable of being added. Pitch, as we have seen, can be measured by employing the recognizable interval of a halftone as a unit, but weight sensations and sound sensations do not seem to be intervallic. Even if they were, there would be other difficulties. Because sensations are private and perceptiondependent, it does not seem possible to select either units or zeros for them. Suppose we select Y i as the zero sensation and Y i - Y j as the unit interval of sensation. These sensations necessarily belong to a particular observer, Oi. If the method of measurement in which they figure is to apply to other observers, such as O2, we must be able to say which of Oz's sensations are equal to V i , and which of CVs sensation intervals are equal to that between Y i and ^2. How is this possible, in view of the privacy of sensations? Perhaps the sensation used by O2 as a zero sensation is unequal to that used by Oi. Perhaps the sensation intervals used by O2 as units are unequal to those used by Oi. If so, the results of measurements made by Oi and O2 are incomparable. Even more disturbing, perhaps the sensation used by Oi as a zero sensation varies from one measurement to the next, or the unit interval of sensation

282

THE MEASUREMENT OF PITCH

u s e d b y h i m v a r i e s w i t h i n a g i v e n m e a s u r e m e n t . In that e v e n t , d i f f e r e n t m e a s u r e m e n t s m a d e s o l e l y b y O i are i n c o m p a r a b l e . T h e s e o b j e c t i o n s to t h e possibility o f m e a s u r i n g s e n s a t i o n s , c o m p e l l i n g as t h e y s e e m , n e e d a g r e a t deal o f critical s c r u t i n y . T h e y m a y o r m a y n o t b e s o u n d . N o n e t h e l e s s , t h e y (or similar o b j e c t i o n s ) h a v e c o n v i n c e d m a n y t h i n k e r s t h a t if s e n s a t i o n s are m e a s u r a b l e , t h e y are m e a s u r a b l e o n l y b y m e t h o d s that are r a d i c a l l y d i f f e r e n t f r o m t h o s e u s e d to m e a s u r e p h y s i c a l d i m e n s i o n s s u c h as l e n g t h , w e i g h t , a n d , w e m u s t n o w include, l o u d n e s s a n d pitch. T h e s e p h y s i c a l d i m e n s i o n s are measured b y physical methods. Psychological dimensions, dimensions o f s e n s a t i o n s , m u s t b e m e a s u r e d b y p s y c h o l o g i c a l p r o c e d u r e s that h a v e their o w n special features. In e x p e r i m e n t a l p s y c h o l o g y t h e first o f t h e s e p s y c h o l o g i c a l p r o c e d u r e s to b e r e c o m m e n d e d e m p l o y e d the just able difference

(jnd) as a s e n s a t i o n unit a n d the absolute

notice-

threshold

of

s e n s a t i o n as a sensation zero. T h e n e x t c h a p t e r is d e v o t e d to an e x a m i n a t i o n of this m e t h o d of s e n s a t i o n m e a s u r e m e n t . T h e chief a p p e a l o f t h e j n d m e t h o d is its apparent e l i m i n a t i o n of at least s o m e of the p r o b l e m s o f sensation m e a s u r e m e n t s k e t c h e d a b o v e . Let us select as the z e r o f o r O i ' s w e i g h t s e n s a t i o n s the s m a l l e s t s e n s a tion c a p a b l e o f b e i n g p r o d u c e d in h i m ; a n d let us select as a u n i t f o r his sensations the smallest s e n s a t i o n d i f f e r e n c e capable o f b e i n g p r o d u c e d in h i m . Let us select a similar z e r o a n d a similar u n i t f o r

Oi's

w e i g h t sensations. W i t h t h e s e choices, it a p p e a r s that the z e r o s e n s a t i o n f o r O i m u s t equal the z e r o s e n s a t i o n f o r O2, a n d that t h e s e n s a t i o n u n i t f o r O i m u s t equal the s e n s a t i o n unit f o r O2. F u r t h e r m o r e , it s e e m s that s e n s a t i o n units f o r a single o b s e r v e r m u s t r e m a i n equal b o t h f r o m o n e m e a s u r e m e n t to the n e x t a n d w i t h i n a single m e a s u r e m e n t .

These

p o i n t s are m a d e o n l y to e x p l a i n the a p p e a l o f the jnd m e t h o d o f s e n s a t i o n m e a s u r e m e n t , n o t to j u s t i f y it. For, as is s h o w n later in c h a p t e r 8, t h e jnd m e t h o d s e e m s t o c o n t a i n the a d v a n t a g e s a b o v e o n l y b e c a u s e o f c o n f u s i o n in the c o n c e p t s o f a just n o t i c e a b l e d i f f e r e n c e a n d of an a b s o l u t e threshold.

8 The JND Measurement of Sensations Traditional psychophysics incorporates a sensationist theory of perception, according to which a perceiver invariably has a sensation or sensations whenever he perceives something. On this theory, if O sees something, he has a visual sensation; if he feels something he has a tactual sensation; if he hears something he has an auditory sensation; and so on. Psychophysical laws of perception are held to be laws relating sensations to their physical causes, or stimuli. If these laws are to be numerical, the sensations involved in perception must be measurable. In this chapter several closely related methods for measuring sensations are examined, each of which employs just noticeable differences (jnd's) between sensations as sensation units. So far my principal example has been auditory perception, but its complexity makes it an undesirable example for the purposes of this chapter. Auditory sensations must be distinguished both from sounds and from sound waves, since they are, unlike sounds and sound waves, perception-dependent, private, and nonlocatable. If, as I have assumed, sounds cannot be identified with sound waves, difficult questions arise. Are the stimuli for auditory perception sounds, or are they sound waves? Are psychophysical laws of perception those relating auditory sensations and sounds, or those relating auditory sensations and sound waves? These questions apparently do not arise for the perception of length and the perception of weight (among other examples). Here there seems to be only one candidate for the stimulus, namely, the length seen and the weight felt, and only one candidate for the psychophysical law of perception, namely, one relating length sensations to length and one relating weight sensations to weight. Because of its apparent simplicity, and because it is one of the

284

JND MEASUREMENT OF SENSATIONS

most common subjects of investigation by early psychophysicists, weight perception serves as the example in this chapter. FECHNER'S LAW Major interest in jnd methods of sensation measurement dates from the publication of Fechner's Elemente der Psychophysik in i860. Fechner conceived of psychophysics as the science of the relation between mind and body, a science that attempts to discover numerical laws relating psychological and physical dimensions. In the area of perception, such laws relate sensations to their physical stimuli: sound sensations to sound waves, color sensations to light waves, weight sensations to weights, length sensations to lengths, and so on. In the Elemente, Fechner proposed a general law of the relation between sensations and stimuli. In deriving this law from a law previously discovered by Weber, he employed an assumption usually described as the assumption that all jnd's are equal. Consequently, Fechner was widely regarded as having proposed a method of sensation measurement which employs the just noticeable difference as a unit. A brief description of these developments follows. WEBER'S LAW

Over the period 1829-1849, the German psychologist Ernst Heinrich Weber collected evidence for a law that may be informally stated as follows: For every sense department, the difference between one stimulus and another that is just noticeably different is a constant fraction of the first. Let d $ be the amount by which a given stimulus, must be changed (decreased or increased) in order to produce a second stimulus just noticeably different than the first. Weber's law may then be stated as

where c is a constant whose value depends on the sense department in question. The fraction d $ / $ is usually referred to as the Weber fraction; [1] tells us that the fraction is constant within a given sense.1 Weber's law can be partially confirmed by ordinary observation. 1 See E. B. Titchener, Experimental Psychology (New York, 1905), II, pt. 2: Instructor's Manual, pp. xiii-xx, for a history of Weber's discoveries.

JND MEASUREMENT OF SENSATIONS

285

If a candle is lighted in a room where a single candle is already burning (with no other source of light), the increase in the light within the room is easily noticed. But if a candle is lighted in a room where fifty others are already burning, the increase is barely (if at all) noticed. Again, the difference in length between a rod of 1 inch and a rod of 2 inches is easily noticed (let the rods be held in separate hands). But the difference in length between a rod of 50 inches and one of 5 1 inches is barely noticed (if noticed at all). Ordinary observation thus shows that the increase in magnitude which can be noticed—the just noticeable increase—is proportional to the magnitude increased. Such simple observations only partially confirm Weber's law, however, because they do not show that the just noticeable increase is a constant proportion of the magnitude increased. To confirm that the proportion is constant, as Weber claimed, we must be able to measure the magnitude in question. Weber offered evidence for [1] by experimenting only with weight perception and length perception, and by citing previous experiments with pitch perception, but he inferred that the law held generally, for all types of perception. An illustration of the law for weight perception follows. Experiment has shown that the normal observer is able by lifting to tell the difference between a weight of 53 grams and one of 54 grams, but is not able to distinguish a weight of 53 grams from a weight less than 54 grams. Again, the normal observer can distinguish a weight of 106 gm from a weight of 108 gm, but not a weight of 106 gm from one less than 108 gm. (This statement of the facts in question, although imprecise, serves the immediate purpose.) The just noticeable (greater) difference for a weight of 53 gm is 1 gm; the just noticeable difference for a weight of 106 gm is 2 gm. Although the jnd's are thus different, depending on their location in the weight scale, they are the same proportion of their limiting stimuli. A jnd of 1 gm is 1 / 5 3 of 53 gm; a jnd of 2 gm is 1 / 5 3 of 106 gm. The value of c in equation [1] is, therefore, 1 / 5 3 for weight. (Weber found the value of c to be 1/30, but later experiment has corrected his figure.) Weber's experiments were limited to just two sense departments, and were, by present standards, carelessly performed. Extensive and careful experimentation during the intervening years has served to bring his law under severe attack. Some psychologists simply pro-

286

JND MEASUREMENT OF SENSATIONS

nounce the law false and conclude that it is of no value to experimental psychology. A mellower view is also widely held, as the sequel shows. FECHNER'S LAW

Gustav Theodor Fechner (1801-1887), a German physicist, psychologist, and philosopher, is by all accounts the founder of the science of psychophysics. Some scholars even regard him as virtually the founder of experimental psychology. According to Titchener, "If Weber laid the foundation stone of experimental psychology, Gustav Theodore Fechner . . . may be said to have planned, and in a large measure to have erected, a whole building."2 For us, Fechner's chief importance lies in the fact that he believed it possible to express mental phenomena in quantitative terms, and was the first to propose a way of doing so. More specifically, Fechner wanted to discover numerical laws relating psychological and physical dimensions, laws of the form Y = /(, is today known as Fechner's law. CRITICISM OF WEBER's LAW

Fechner's law, [7], is derived from Weber's law, [1]. Indeed, the latter is the sole empirical basis for the former. Taking [a] to be experimentally established, Fechner mathematically deduced [7] from it, together with certain assumptions (which he did not verify and which may be unverifiable) and stipulations. Fechner's edifice is thus constructed primarily on Weber's law. If that law can be shown to be false, the edifice will collapse. And Weber's law does seem to be false. Woodworth and Schlosberg describe the latest experimental results: "Perhaps the best single way to summarize the results is to say that Weber's fraction is fairly constant for the middle ranges of stimulus values, but that it increases rapidly at the extremes, near the RL [lower limen] and TL [upper limen]."5 If the Weber fraction is not constant, formula [1] is false. The proper reply is suggested by Woodworth and Schlosberg: A

vast

amount of argument has been devoted to the question of

whether or not W e b e r ' s l a w holds. If w e stop w o r r y i n g about whether or not it is a universal l a w of judgement, the answer is simple. It holds as a rough empirical generalization in the mid-ranges of most senses — b o t h f o r intensity and for quality, although it w a s originally p r o 5 Robert S. Woodworth and Harold Schlosberg, Experimental Psychology York, 1954), p. 223.

(New

JND MEASUREMENT OF SENSATIONS

289

posed for intensity only. By and large, these mid-ranges are the working area of the senses. . . . Weber's law furnishes us with a valuable description of the discriminating power of the important sensory ranges. As such, it is extremely valuable.6 To expand on this suggestion, [1] is an approximation, or a rough generalization. Science is filled with laws of this sort, which, rough as they are, nonetheless prove useful. The gas laws of physics, for example, do not hold for gases at extremely low temperatures, but they prove useful and are retained because they hold throughout a vast range of temperatures. Such laws, which the experimental data approximately fit, are called "ideal" laws. Since Weber's law holds throughout a large range—the midrange—of most stimulus continua, it too may be retained and regarded as an "ideal" law. 7 A s Woodworth and Schlosberg point out, attempts have been made to supplant Weber's law by another more accurate one. In 1892 Fullerton and Cattell proposed the square root law, which says that the just noticeable difference increases in direct proportion to the square root of the stimulus: [8] d $ = c But this law fits the experimental data no better than [1]. Later, in 1932, Guilford suggested a general law of which both [1] and [8] are instances: [9] dS> = c ) from a law of the form $ = /($)? In other words, is it legitimate to deduce a genuine psychophysical law (one containing both sensation variables and stimulus variables) from a law containing only physical (stimulus) variables? This question can profitably be investigated by studying the derivation of Fechner's law from Weber's law, whether the latter is or is not acceptable. CRITICISM OF FECHNER's LAW

Fechner's derivation of his law has been subjected to examination and criticism that are perhaps without parallel in the history of psychology. In the half century following publication of the Elemente, virtually every psychologist felt compelled to take a position on his law. Scores of theorists devised what seem to be all conceivable objections, and Fechner and his supporters responded in numerous books and articles. Titchener, in the introduction to his Experimental Psychology, Vol. II, pt. 2, has written the nearest thing to a definitive history of this controversy. In the sequel I review and comment on what seem to me the salient objections to Fechner's law. M y treatment is brief and sometimes sketchy, since my main purpose here is to introduce questions that the remainder of the chapter is designed to answer. The final assessment of Fechner's proposal and of the objections to it comes later. What did Fechner do?—Unfortunately, there is more than one way of construing Fechner's accomplishment. The standard interpretation is that in deriving [7] Fechner measured sensations, or, at least, suggested a method for measuring sensations. The unorthodox interpretation is that Fechner hypothesized that a method of sensation measurement (which he himself did not provide) would reveal [7] to be the empirical law relating sensations to their stimuli. These two interpretations are discussed in turn. 1 ) The measurement interpretation. Virtually all commentators say that in deriving his law Fechner measured sensation, a characterization that at first sight seems inaccurate. In this book measurement is defined as a procedure of assigning numerals to objects within a dimension by means of an empirical process of comparing these objects with a dimension unit or units. Fechner does not seem to have provided any procedure of assigning numerals to sensations. Nevertheless, he

JND MEASUREMENT OF SENSATIONS

291

would himself agree with the standard interpretation. Of [2], the first of the formulas deduced from Weber's law, Fechner says: "The fundamental formula does not presuppose the measurement of sensation, nor does it establish any; it simply expresses the relation holding between small relative stimulus increments and sensation increments."® But of [3], the next formula obtained, he says: "In the measurement formula one has a general dependent relation between the size of the fundamental stimulus and the size of the corresponding sensation and not one which is valid only for the cases of equal sensations. This permits the amount of sensation to be calculated from a relative amount of the fundamental stimulus and thus we have a measurement of sensation."10 Why did Fechner believe he had achieved measurement in [3] but not in [2] ? The reason would seem to be as follows. Suppose that by measuring a given stimulus, $ 1 , we obtain the numeral that represents its magnitude. Even if we know the value of the constant, c, in formula [2], we cannot use this formula to obtain the numeral to be assigned to the sensation, Y i , produced by $1. If we know the values of the constants in [3], however, we can use this formula to obtain the sensation numeral. Consequently, [3] does, but [2] does not, constitute a rule for assigning numerals to sensations, a rule that is required in a procedure of sensation measurement. Perhaps it was for this reason that Fechner called [3] the "measurement formula." Another requirement must, on my definition, be met by a procedure of sensation measurement. The numerals assigned to sensations must represent the magnitude of sensations in terms of sensation units. Although Fechner does not specify any unit of sensation, most commentators believe that such a unit is implied by his derivation. This implication is elicited from premise (i) in the proof of formula [2], which states that just noticeable increases (differences) in the stimulus correspond to equal increases (differences) in sensation. This premise seems to imply that sensation differences corresponding to just noticeable differences between stimuli are being employed as sensation units. The adoption of these units fixes the value of the constant, c, in formula •Fechner, Elemente der Psychophysik, II, 10; translation from Wayne Dennis, Readings in the History of Psychology (New York, 1948), p. 209. 10 Fechner, Elemente der Psychophysik, II, 16-17; translation from Dennis, Readings, p. 213.

292

JND MEASUREMENT OF SENSATIONS

[3], and in the formulas derived from [3]. Any one of these formulas thus constitutes a rule for assigning to sensations numerals that represent the magnitude of the sensations in terms of sensation units, and, consequently, constitutes a procedure of sensation measurement. Because Fechner did not explicitly propose the sensation jnd as a unit, it is difficult to judge whether the measurement interpretation of his derivation is correct. A further difficulty is that another, equally plausible, interpretation is available. 2) The hypothesis interpretation. At first glance, Fechner's derivation reveals no method of sensation measurement. Instead, Fechner seems to have deduced, from Weber's law and certain assumptions, an empirical numerical law relating sensations to their stimuli. According to the hypothesis interpretation, Fechner's problem should be described as follows. Since Fechner neither possessed nor was able to offer a method for measuring sensations, it was impossible for him to discover experimentally any numerical law relating sensations to stimuli. So he did the next best thing: he turned from measurement to hypothesis, and inferred that the law relating sensations and stimuli would be discovered by measurement (when and if it became available) to be [7]. His law is, in brief, a hypothesis about the results of sensation measurement, and as such it is uncertain. It can be conclusively verified only by a method of sensation measurement, which Fechner himself could not offer. A simple analogy will make this interpretation clearer. Suppose we possess (as we do) a method for measuring the volume of water. Suppose (contrary to fact) we do not possess a method for measuring weight. We could not, then, discover experimentally a numerical law relating the volume and the weight of water. We could, nonetheless, hypothesize such a law. For example, we might first assume that equal differences in volume correspond to equal differences in weight. From this assumption we could deduce mathematically that [10] W = c V, where W is weight, V is volume in cubic centimeters, and c is an undetermined constant. We would not say that in deducing [10] we had measured weight, or had provided a method for measuring weight. Rather, we would say we had formulated a hypothesis as to what numerical law the measurement of weight (if it could be accomplished)

JND MEASUREMENT OF SENSATIONS

293

would discover. It is possible to give [10] a more precise form by stipulating a unit of weight. If we stipulate that a volume of 1 cc has a weight of 1 U (unit of weight), then the value of the constant, c, in [10] is 1 , and the formula becomes [ 1 1 ] W = 1 V. Formula [ 1 1 ] has the same status as [10]. Both are hypotheses about the results of measuring the weight and volume of water. But [10] is a hypothesis about the results of any system of weight measurement, whereas [11] is a hypothesis about the results of a system of weight measurement in which the unit is stipulated as the weight of 1 cc of water. In the absence of a method of weight measurement, both [10] and [ 1 1 ] would remain uncertain. They could be conclusively verified only by employing some method of weight measurement (like the one we do in fact have). Fechner's law [7], is analogous to [10]. In deducing it he does not measure sensations, but instead offers a hypothesis as to what sensation measurement would discover. This fact remains unchanged even if we stipulate as units the sensation differences corresponding to just noticeable differences between stimuli. This stipulation enables us to compute the value of the constant, c, in [7], but it does not convert that law into a method of sensation measurement. The important disanalogy is that we do of course possess a method of weight measurement by means of which to verify [10], whereas we apparently possess no method of sensation measurement. Consequently, Fechner's law remains unverified and uncertain. In the possibility of the two interpretations presented above, an objection to Fechner's derivation can be found. It is not entirely clear what Fechner intended his derivation to accomplish, and even less clear what it did in fact accomplish. Commentators, who generally subscribe to the standard interpretation, seem to be unaware that another interpretation is possible, and at times they seem to vacillate uneasily between the two. We will discover that unclarity about what Fechner accomplished introduces unclarity into some of the traditional objections to his derivation. When the two interpretations are not distinguished, there is a tendency to shift unconsciously back and forth between them. Thus critics of Fechner may offer an objection to one interpretation which properly applies to the other. And defenders of

JND MEASUREMENT OF SENSATIONS

2Ç4

Fechner may answer an objection to one interpretation with a reply that properly applies to an objection to the other interpretation. Debate containing this sort of confusion is bound to be endless. The objection to the equality of jnd's.11—The

most popular objec-

tion to Fechner's law has a l w a y s been and still is that " j u s t noticeable differences (jnd's) are not equal," or, at least, that w e have no reason to suppose that they are equal. Stated in this fashion the objection is quite imprecise. Does it refer to stimulus jnd's or to sensation jnd's? It follows f r o m Weber's law, even if it is only approximately correct, that just noticeable differences between stimuli are unequal. This law tells us that the just noticeable stimulus difference, dS*, is a constant fraction of the stimulus,

which means that the just noticeable

stimulus difference increases as the stimulus increases. Fechner could not have said consistently that stimulus jnd's are equal, since he accepted Weber's law and derived his own law f r o m it. Fechner's derivation assumes the equality, not of stimulus jnd's, but the equality of sensation jnd's; more precisely, it assumes the equality of sensation differences corresponding

to just

noticeable

differences

between

stimuli. This assumption is stated in premise (i) of Fechner's derivation as reconstructed above. Whether the assumption is true is crucial, for if it is false, Fechner's fundamental formula, [2] d\F = c

is false.

N o w w h y should we suppose that sensation jnd's are equal? W h y not suppose, as Brentano suggested, 1 2 that Weber 7 s law applies to sensations as well as to stimuli? W h y not suppose, that is to say, that f o r any two sensations produced b y just noticeably different stimuli, the difference between the one sensation and the other is a constant fraction of the one? O n this assumption, instead of [2], the following formula is true: r

, d¥ _

d$ $ •

If this formula is treated as a differential equation and then integrated 11 See Titchener, Experimental Psychology, II, pt. 2, pp. lxviii-lxxxix, for a history of this objection. 12 See S. S. Stevens, ' T o Honor Fechner and Repeal His Law," Science, 1 3 3 (1961), 80, for the reference.

JND MEASUREMENT OF SENSATIONS

295

the result will be, not Fechner's logarithmic law [7], but Stevens' power law: [13] V = k The above criticism implies that Fechner's law is a hypothesis, derived from the assumption that sensation jnd's are equal, concerning the numerical relation between sensations and their stimuli. If, on the other hand, Fechner is interpreted as having suggested a method for measuring sensations, it is still necessary to assume that sensation jnd's are equal. Measurement is a procedure for assigning numerals to objects by comparing them with a unit or units. As shown in chapter 5, these units must be equal for the procedure to qualify as measurement. If Fechner did provide a method of sensation measurement, the units in the method are sensations jnd's, and these must be equal. It might be argued that sensation jnd's are equal on the grounds that (a) the corresponding stimulus jnd's are equally noticeable, that (b) the corresponding stimulus jnd's are psychologically equal, or that (c) sensation jnd's are the smallest possible sensation differences. Each of these arguments is considered in the sequel and found to be objectionable. Quite apart from the objections, it is clear that each of the three arguments is highly inferential and inconclusive. It seems that the only conclusive way of determining whether sensations jnd's are equal is by measuring sensations. If we wish to know whether the difference in weight between two objects equals the difference in weight between two other objects, we must either compare these differences directly or else measure the four objects and compute their differences. Similarly, if we wish to know whether two sensation jnd's in different parts of the sensation scale are equal, we must either compare the jnd's directly or measure the sensations involved and compute their differences. Since direct comparison of jnd's in different parts of the sensation scale seems impossible, we must measure the sensations. But what procedure of sensation measurement is available? If Fechner's derivation contains a method for measuring sensations employing sensation jnd's as units, we cannot use his method to confirm that sensation jnd's are equal. Such confirmation would be circular, comparable to using a meterstick to confirm that its own centimeter markings are equally spaced. But if the jnd method cannot be used, what other method is available? Stevens suggests measuring

296

JND MEASUREMENT OF SENSATIONS

sensations by using the observer's ratio judgments of stimuli to assign numerals to the sensations produced in him by the stimuli. Chapter 9 establishes that this method is suspect, and that by altering certain of Stevens' assumptions, which are themselves unverifiable, his method can be used to confirm Fechner's law and, therefore, the equality of sensation jnd's. It appears, then, that no method of sensation measurement has been suggested which could confirm or disconfirm Fechner's assumption that sensation jnd's are equal. Of course it does not follow that the assumption is unverifiable. There may be methods of sensation measurement which no psychophysicist has used or even entertained. If such a method is discovered, it may (or may not) show that Fechner's law and his assumption that sensation jnd's are equal are both true. At present, however, both the law and its assumption remain what they were when Fechner advanced them, unconfirmed hypotheses. The quantity objection.1S—Many critics have charged that Fechner's derivation assumes sensations to be measurable and, in so doing, begs the very question Fechner was trying to answer. Most of these critics go on to argue that since sensations cannot be added they cannot be measured. Here are some examples of this line of argument: "Our feeling of pink is surely not a portion of our feeling of scarlet; nor does the light of an electric arc seem to contain that of a tallow candle within itself" (James); "This sensation of 'gray' is not two or three of that other sensation of ' g r a y ' " (Kulpe); " A blue surface is something other than a green, but the latter has in itself, apart from the memory of colors, nothing of the doubleness or the threefoldness of the green" (Ebbinghaus); "Whatever it is called, a pain exactly ten times as strong as another does not admit of such absolute statement" (Von Kries).14 The point of these remarks is that since we cannot add sensations, we cannot say of any sensation that it is composed of others; consequently we cannot say that it is one, two, or three times another. Numerical comparisons of this kind, and therefore measurement, are impossible for sensations. But the objection is unsound. It may be true that to introspection no sensation appears to be composed of smaller 13 See Titchener, Experimental Psychology, II, pt. 2, pp. xlviii-lxviii, for a history of this objection. 14 Quoted by Edwin G. Boring, "The Stimulus Error," American Journal of Psychology, 32 (1921), 453-454-

JND MEASUREMENT OF SENSATIONS

297

sensations, but it does not follow that no sensation is composite, or that sensations cannot be added. Two rods can be so cleverly juxtaposed as not to appear visually to be composite; it does not follow that they are not composite. Furthermore, some rods neither appear to be nor are composite; it does not follow that rods cannot be juxtaposed (added). What reason, then, is there to say that sensations cannot be added? The objection fails even if sensations cannot be added, for it assumes that only dimensions that permit an operation of adding dimension objects can be measured. Measurement is the process of assigning numerals to objects by means of an empirical operation of comparing the objects with a unit or units. This operation may or may not be one of addition. Chapter 7 shows that even though there is no operation of adding pitches pitch can be measured; numerals can be assigned to pitch in such a way that one pitch can be said to be one, two, three times as great as another. Now there may be some argument to show that no empirical operation for comparing sensations with a unit is possible. And there may be some argument to show that, even if such an operation is possible, sensation units cannot be meaningfully defined. But until these arguments are presented, it is reasonable to assume that sensations are measurable. If Fechner proposed a method for measuring sensations, it is his critics who are guilty of begging the question at issue. To propose a method of sensation measurement is obviously not to assume that sensations are measurable, but rathr to attempt a proof that sensations are measurable. If the proposed method is acceptable, sensations are measurable. What Fechner's critics must show is that the method is unacceptable. If, on the other hand, Fechner did not propose a method for measuring sensations, but offered a hypothesis concerning the numerical relation between sensations and their stimuli, the quantity objection has some force. A numerical law relating two dimensions is meaningful only if it is possible (in principle if not in practice) to measure the two dimensions. Hence, if Fechner's law is meaningful, it must be possible to measure sensations. In this sense, Fechner certainly assumed that sensations are measurable. But no objection flows from this fact unless it can be shown that it is impossible to measure sensations. Can this be shown? This chapter argues that the jnd method for measuring

298

JND MEASUREMENT OF SENSATIONS

sensations is suspect. Chapter 9 argues that the methods of sensation measurement recommended by Stevens are also suspect. These conclusions strongly suggest that sensations are immeasurable, since the jnd method and those recommended by Stevens are the only methods thus far proposed or entertained by psychophysicists. Nevertheless, some method of sensation measurement may yet be discovered which has never been entertained, much less recommended, by any psychophysicist. Until it is shown that the nature of sensations and the nature of measurement make such a discovery impossible, we cannot conclude that Fechner's law is meaningless. The objection of indirectness.—Commentators, both critical and sympathetic, agree that Fechner's method is indirect. Their assessment is certainly true in some sense. But the term "indirect" is highly ambiguous, and it is difficult to know what sense a given commentator has in mind. Furthermore, the sense of the term, and the significance of its application, depend on how Fechner's derivation is interpreted. If Fechner offered a hypothesis concerning the numerical relation between sensations and their stimuli, what he did could perhaps be called "indirect" by comparison with what he might have done. He might have devised a method for measuring sensations and, by using this method, discovered a law of sensations. But Fechner did not intend, on the present interpretation, to do the more "direct" thing. To criticize him for not doing something else is not to criticize him for what he in fact did. Fechner's hypothesis that [7] is true could perhaps be called "indirect" in virtue of resting on premises and assumptions. It rests on the premise of Weber's law and on the assumption that sensation jnd's are equal. But to record this fact by calling his hypothesis "indirect" is to imply that the hypothesis would have been more acceptable had it been based on no premises and on no assumptions. That is, it would have been more acceptable had Fechner arrived at it through unsupported intuition, thought it up out of a clear sky, or had it revealed to him in a dream, all of which is clearly absurd. If we interpret Fechner as having suggested a method for measuring sensations which employs sensation jnd's as units, then the charge of indirectness seems well taken. An indirect (dependent) procedure of measurement is one that contains some subprocedure of measurement, or one whose unit is defined in terms of some other unit. A direct (independent) procedure of measurement is one whose unit is

JND MEASUREMENT OF SENSATIONS

299

not defined in terms of any other unit, and one that does not contain any subprocedure of measurement. The procedure suggested in Fechner's derivation is clearly indirect, since it contains the subprocedure of measuring stimuli in order to verify Weber's law. Furthermore, the unit in his procedure, the sensation jnd, seems to be defined in terms of the stimulus unit. The sensation jnd is defined as the difference between sensations corresponding to a just noticeable difference between stimuli, and the just noticeable difference between stimuli is expressed as a ratio between stimulus magnitudes (for example, the ratio for weight is 54:53). Since this ratio is expressed in number of stimulus units, it would seem to follow that sensation jnd's are defined in terms of stimulus units. That a method of measurement is indirect may or may not constitute an objection to the method. Some indirect methods for measuring a dimension imply a direct method for measuring the dimension, and are therefore inconclusive. Some indirect methods for a dimension, Di, define their units in terms of the unit for some other dimension, D2, and cannot be used to discover empirical laws relating the two dimensions. For example, suppose we define the weight sensation unit as the sensation difference corresponding to a weight difference of 1 gm. It then follows by definition that the relation between weight sensations and weights is linear. But it is incorrect to call this relation a "law" of weight sensations and weights, since it is trivially true, true by definition. Does Fechner's indirect method for measuring sensations possess either of the disadvantages mentioned above? And if it does, are direct methods employing sensation jnd's as units possible? Neither of these questions can be settled without an attempt to describe various jnd methods of sensation measurement in detail. This attempt is made in the sequel. The foregoing review provides a fair idea of the scope, the difficulty, and, above all, the inconclusiveness of the controversy generated by Fechner's law. James, one of Fechner's most hostile critics, describes the controversy in acid terms: "Fechner's book [the Elemente] was the starting point of a new department of literature, which it would be perhaps impossible to match for the qualities of thoroughness and subtlety, but of which, in the humble opinion of the present

300

JND MEASUREMENT OF SENSATIONS

writer, the proper psychological outcome is just nothing."11 Fechner, in contrast, was encouraged by the controversy: "The tower of Babel was never finished because the workers could not reach an understanding on how they should build it; my psychological edifice will stand because the workers will never agree on how to tear it down." 18 Fechner's prophecy has come true. Despite extensive criticism, his law has never been clearly invalidated. Of the numerous attacks on it, none has been conclusive. Recently, Stevens and his followers have advanced and argued extensively for a psychophysical power law, as opposed to Fechner's logarithmic law. As chapter 9 shows, Stevens' arguments in favor of his law are inconclusive, and certain of his assumptions, which are empirically univerifiable, can be so altered as to produce Fechner's law from the same empirical data. The remainder of this chapter is an attempt to tear down Fechner's edifice, not primarily by advancing some of the usual objections, but rather by showing that the just noticeable difference is a unit of sensitivity, and cannot be pressed into service as a unit of sensation. THREE JND METHODS The preceding discussion of Fechner's law shows that there are two ways of interpreting his derivation: as a hypothesis about the law relating sensations to their stimuli, or as a method for measuring sensations which employs sensation jnd's as units. The latter interpretation, which is examined in this section, is the more important of the two, since if sensation measurement is impossible, numerical laws relating sensations and stimuli are impossible (meaningless). To determine whether jnd measurement of sensations is possible, we must attempt to describe the method (or methods) in complete detail. Only then can we decide whether the method is free from objection, whether it is, as claimed, a method for measuring sensations, and whether it can be used to verify Fechner's law. In supplying this description I take weight sensations as my example. THE THREE METHODS

There are, it turns out, three possible methods of sensation measurement employing sensation jnd's as units. Two of these methods 15

William James, The Principles of Psychology (New York, 1950), I, 554. G. T. Fechner, In Sachen der Psychophysik (Leipzig, 1877), p. 215, quoted by S. S. Stevens, "On the Psychophysical Law," Psychological Review, 64 (1957), 153. 18

JND MEASUREMENT OF SENSATIONS

301

are independent, one is dependent. Only the latter was suggested by Fechner. All three must be described in detail and carefully distinguished from one another if any progress is to be made in assessing them. Method A.—Weight sensations of steadily decreasing magnitude are produced in O by having him lift progressively lighter objects until he reports that he no longer has a weight sensation. This point is designated the zero point. (We might say that at this point O has a zero sensation, were it not for the implication that O can have sensations he does not perceive, or introspect.) Then O is presented with the same series of weights, but in reverse order, until he reports a weight sensation. To this weight sensation is assigned the numeral 1 . Next, weights above this point are presented to O in ascending order until he reports a weight sensation just noticeably greater than the preceding one. To this weight sensation is assigned the numeral 2. This process is continued until a weight sensation is produced in O which he finds equal to the weight sensation being measured. The numeral assigned to this equal weight sensation, which represents the number of sensation jnd's above zero required to reach it, is the numeral to be assigned to the sensation being measured. It would be extremely inconvenient to repeat the process described above each time a weight sensation is to be measured. The inconvenience can be avoided by employing a single set of weights under standard conditions of lifting, for then it can be assumed that O will obtain the same weight sensation each time he lifts a given weight. Hence, we can discover and number a set of weights, 3>o, $ 1 , $ 2 , . . . such that i>o produces no sensations in O; i>i produces Y i , the least sensation; $2 produces ^2, the sensation just noticeably greater than Y i ; and so on. A given sensation, ^m, produced by is then measured by having O select the $ from the standard set which produces a sensation equal to ^m (i.e., by having O find the $ that feels equal in weight to ^m). The numeral assigned to this $ is the numeral to be assigned to ^m. Method B.—This method employs a conventional sensation zero, rather than the natural zero of Method A. Let us select as the zero sensation, Yo, the sensation produced in O by lifting a weight of 100 grams under standard conditions. A weight sensation heavier than Yo is then measured as follows. Weights increasingly heavier then 100 grams are

JND MEASUREMENT OF SENSATIONS

3o, $ + 1 , $+2, . . . $ „ , which produce sensations separated by jnd intervals both above and below sensation zero. The numeral assigned to a given weight is assigned also to the correlated weight sensation. A given sensation can then be measured by having O select the $ that produces an equal sensation. The numeral on this $ is the numeral to be assigned to the sensation being measured. Method C.—This method, in contrast with Methods A and B, requires the measurement of the weights used to produce weight sensations. By presenting O with a number of objects that have been weighed on a balance, we discover that when one of his weight sensations is just noticeably greater than another, the weight producing the one sensation stands in the ratio 54:53 to the weight producing the other sensation. That is, we discover that the following version of Weber's law is true: [14]

= 54/53,

where is a weight that produces a sensation just noticeably greater than the sensation produced by weight Let us stipulate that the sensation produced by a weight of 100 grams is the zero sensation, To. By employing equation [14] we can construct a table that will enable us to assign numerals to weight sensations, a table indicating the magnitude of the sensation in terms of the number of sensation jnd's above or below the zero sensation. Equation [14] tells us that a sensation,

303

JND MEASUREMENT OF SENSATIONS

T + i , just noticeably greater than Yo will be produced by a weight of 54/53X100 = 101.900 gm; that a sensation, Y+2, just noticeably greater than Y + i will be produced by a weight of 54/53 X 101.900 = 103.836 gm; and so on. Also, [14] tells us that a sensation, T - i , just noticeably less than ^o will be produced by a weight of 53/54 X 100 = 98.150 gm; that a sensation, Y-2, just noticeably less than Y - i will be produced by a weight of 53/54 X 98.150 = 96.383 gm; and so on. Table 7 contains values surrounding the zero sensation computed in the manner above. By means of an expanded table 7, any weight sensation can be measured simply by determining the weight of the object producing the sensation, looking up this weight in the left-hand column of the table, and assigning the corresponding numeral in the righthand column to the sensation being measured. TABLE 7 WEIGHT

NUMBER ASSIGNED TO

IN GRAMS

SENSATION

109.867 107.818 105.808 103.836 101.900 100.000 98.150 96.383 94.648 92.944 91.271

+5 +4 +3 +2 +1 0 —1 —2 -3 -4 -5

The relation between weights and weight sensations in table 7 is capable of more economical expression in a formula,

where Y is the magnitude of the weight sensation expressed in number of jnd steps above sensation zero (a positive number) or below sensation zero (a negative number), $ is the weight in grams of the

JND MEASUREMENT OF SENSATIONS

stimulus producing the sensation, and $o is the weight in grams of the stimulus producing the zero sensation. Since ] 0 g 54/53^

12 3-45/

and 4>o = ioo, [15] becomes 1 0 0 )[16] = 123.45 0°g ^ ~ This equation has the same status as table 7: it is a rule for assigning numerals to weight sensations on the basis of the weight of the producing stimuli. If 1 gm, rather than 100 gm, is selected as the weight of the stimulus producing a zero sensation, then, since log 1 = 0, [16] becomes [17] V = 123.45 log which is an instance of Fechner's law [7].17 Method C closely corresponds to the method Fechner may have been suggesting. Like his, it is an indirect (dependent) method, since one of its subprocedures is the measurement of weights in order to discover Weber's law. Unlike Fechner's procedure, the law relating sensations to their stimuli is deduced from Weber's law in Method C by a simple algebraic process, rather than by differentiation and integration. There are two advantages in the simpler deduction. First, it does not imply that sensations are a continuous function of the stimuli producing them, does not imply, in other words, that every difference between perceived stimuli corresponds to a difference between sensations. Second, it more clearly exhibits the status of Fechner's law. Equation [16] is a rule for assigning numerals to weight sensations. It does not result from measuring weight sensations, but is, rather, a precondition for their indirect measurement. To be more precise, equation [16] is not discovered by a procedure of (a) measuring stimuli, (b) directly measuring the sensations produced by these stimuli, and (c) finding an equation to fit the results. If [16] cannot be discovered by a procedure of type (a)-(b)-(c), the suspicion will remain that it is not a genuine empirical law, and is true merely in virtue of stipulations in a method of indirect measurement.

The possibility of Methods A and B apparently removes this suspicion. Both are direct (independent) methods for measuring weight 17

James, The Principles of Psychology, 1,538-539; Robert S. Woodworth, Experimental Psychology (New York, 1938), pp. 435-437.

JND MEASUREMENT OF SENSATIONS

sensations; neither contains any subprocedure of measurement for the weight of stimuli or for any other dimension. If they are acceptable methods, and if they can be used in a procedure of type (a)-(b)-(c) to verify Fechner's law, then this law is a genuine empirical law relating sensations and stimuli. There is every reason to believe that Method B will verify law [16], since, like Method C, it employs sensation jnd's as units and stipulates that the zero sensation is produced by a weight of 100 gm. Method A, on the other hand, employs a different zero and cannot be expected to yield immediately the same numerical law. But we can easily convert [16] into a law that we can expect Method A to yield simply by replacing the zero value of 100 by the value of the weight producing a zero sensation in Method A. If this value is 0.02 gm, then [18] Y = 123.45 (log

— log 0.02)

is a law we can reasonably expect to obtain by using Method A. The points above can be clarified by describing methods for measuring length analogous to those for measuring weight sensations. Suppose we wish to measure the length of uniform rods of a standard diameter. Method A': A unit rod, U, is selected and laid in successive adjacent positions along the rod to be measured from one end to the other. A count of the placements required gives the numeral to be assigned to the rod being measured. Method B': The length of a rod, Ro, is stipulated to be the "zero length." (Suppose Ro weighs 100 gm.) To measure a rod, R, shorter than Ro, the two rods are aligned at one end and, beginning at the other end of Ro, U is laid along its length until the corresponding end of R is reached. A count of the placements of U supplies the negative numeral to be assigned to R. If R is longer than Ro, the two rods are aligned at one end and, beginning at the other end of Ro, U is laid along the length of R until its corresponding end is reached. A count of the placements supplies the positive numeral to be assigned to R. Method C': We discover, by using a balance, that when the length interval between any two rods is the length of U, the longer rod is 2 gm heavier than the shorter rod. Stipulating that the length of a 100-gm rod is the "zero length," we assign + 1 to a rod of 102 gm, + 2 to a rod of 104 gm, and so on; and we assign —1 to a rod of 98 gm, —2 to a rod of 96 gm, and so

306

JND MEASUREMENT OF SENSATIONS

on. The rule of numeral assignment here can be expressed in the formula [19] L = W - Wo, where L is length, W is weight, and Wo is the weight of the rod of zero length. Method C' is, unlike Methods A' and B', a method of indirect (dependent) measurement, since it contains the subprocedure of measuring the weight of rods. Equation [19] is one way of expressing the rule of numeral assignment in Method C'. This equation is not discovered by a procedure of (a) measuring weight, (b) directly measuring length, and (c) finding an equation to fit the results. Consequently, there can be doubt as to whether the equation is a genuine empirical law or whether it is true merely by stipulation. The possibility of Method B', which is a method of direct (independent) measurement, removes these doubts. Since this method employs the same length unit and the same length zero as Method C', its use will verify equation [19]. Analogously, since Method B is a method of direct sensation measurement, and since its use will verify law [16], that law is a genuine empirical law, if Method B is acceptable. Method A' strains the analogy, since it neither stipulates nor discovers a "zero length." If, however, we pretend that the zero length in this method is the length of some rod of negligible weight, say one of 0.00001 gm, the method will also verify law [19]. The important general disanalogy is that, whereas A', B', and C' are clearly acceptable methods of length measurement, A, B, and C are, as the sequel shows, entirely problematic as methods of sensation measurement. Objection to the three methods.—The unit in Methods A, B, and C is said to be the just noticeable difference between sensations. But the concept of a just noticeable difference, which is clearly applicable to physical stimuli, such as rods and weights, seems inapplicable to sensations. Two stimuli and $2, are just noticeably different for an observer, O, if and only if there are no intermediate stimuli (stimuli greater than $ 1 but less than $2) which he would perceive to be different from both and $2. Restated for sensations, this definition becomes: Sensations Y i and ^ 2 are just noticeably different for Oi if and only if there are no intermediate sensations

JND MEASUREMENT OF SENSATIONS

(sensations greater than Y i but less than T j ) which he would perceive to be different from both Y i and Y*. Thus defined, sensations jnd's cannot be determined unless there is some way of discovering intermediate sensations. And if sensations jnd's cannot be determined, the concept of a just noticeable difference does not apply to sensations. How can intermediate sensations be discovered? It is clear that Oi's perceptions (introspections) of his own sensations cannot be employed for the purpose. If Y i and are just noticeably different for Oi, then by definition he does not perceive any of his sensations to be greater than V i but less than Y2. Nor can ( V s perceptions of Oi's sensations be employed, for there are no such perceptions: Oi's sensations are private to him and cannot be perceived by another observer. The point can be made another way. If Oi honestly reports that none of his sensations lie between ^Fi and Y2, O2 must conclude that in fact none of Oi's sensations lie between Y i and In other words, Oi's sensation reports are incorrigible, and O2 cannot know that Oi has intermediate sensations when Oi himself fails to know it. Not only does it seem impossible to discover intermediate sensations; their very existence is problematic. Sensations and ^ 2 are just noticeably different if and only if there are no intermediate sensations that O would perceive to be different from both Y i and This definition implies that there can be differences between O's sensations which he fails to perceive; it implies that there can be a sensation, Y*, greater than T i and less than which O does not perceive to be greater than and less than V2. The definition implies, in short, that O can make perceptual (introspective) mistakes about his sensations. But this implication seems unintelligible. Observers can make perceptual mistakes about the properties of and relations between physical entities. But one important difference between physical entities and sensations is that comparable mistakes about sensations seem impossible. What could explain O's perceptual mistakes about his sensations? How would such mistakes be conceived? Presumably, O perceives unequal weights as equal because the weights produce equal sensations in him. Shall we say, similarly, that O perceives unequal weight sensations as equal because they produce unequal sensations in him? Such an explanation is obvious nonsense. If, as the above argument suggests, an observer cannot fail to perceive

308

JND MEASUREMENT OF SENSATIONS

differences between his sensations, then the notion of a just noticeable difference between sensations is unintelligible, and the concept of just noticeability cannot be applied to sensations. The implication that O can fail to perceive some of his own sensation differences is comparable to one created by applying the concept of the absolute threshold (limen) to sensations. To say that a physical stimulus is liminal for O means that O is unable to perceive any stimulus of lesser magnitude. If this definition is applied to sensations, to say that a sensation is liminal for O means that O is unable to perceive any sensation of lesser magnitude, which implies that O can have sensations he fails to perceive. If unperceived sensations are an impossibility, if sensations are necessarily perceived by their owner, the implication that O can have sensations he does not perceive is unintelligible, and the concept of a limen cannot be applied to sensations. THE THREE METHODS RECONSTRUED

If the concept of just noticeability does not apply to sensations, Methods A, B, and C are unacceptable as described. In order to rescue the methods from this objection, their unit must be redefined. It must be defined, not as (a) the just noticeable difference between sensations, but as (b) the difference between sensations corresponding to a just noticeable difference between stimuli. The difference between these two definitions cannot be understood without carefully distinguishing the notion of a just noticeable difference between sensations from that of a just noticeable difference between stimuli. The first notion is unacceptable, and so is definition (a) which employs it. But definition (b) employs the second notion, and is entirely acceptable. Methods A, B, and C can be reconstrued as employing the unit defined in (b), thus avoiding the objection that the concept of just noticeable differences between stimuli. This reconstrual is mainly a matter of redescribing the methods, so that they speak of sensation differences corresponding to just noticeable differences between stimuli and to just noticeably greater stimuli, rather than just noticeable differences between sensations and just noticeably greater sensations. In addition, law [14], j/o), which is TABLE 8

1

2

3

STIMULUS

STIMULUS IDENTIFIED

SENSATION

MAGNITUDE

SOLELY BY LABEL

MAGNITUDE

105.808 103.836 101.900 100.000 98.150 96.383 94.648

C

b a zero —a -b —c

3

2 1 O

—1 —2 -3

INSTRUCTIONS FOR METHOD B

1 ) Stipulate that the zero sensation is one produced by a certain stimulus, and label this stimulus "zero" (suppose it is a stimulus of 100 gm). 2) Discover just noticeably different stimuli above and below the zero stimulus, labeling them + a , + b , and so on, above zero, and - a , - b , and so on, below zero. Column 2 in table 8 is a partial list of such labels. 3) Taking the sensation unit to be a difference between sensations corresponding to a just noticeable difference between stimuli, correlate numerals representing sensation magnitude with stimuli separated by jnd intervals. Column 3 is a partial list of these numerals. 4) Use columns 2 and 3 to assign 4') Measure the stimuli separated numerals to the sensation proby jnd intervals labeled in duced by any stimulus, Ascolumn 2. Column 1 contains sign the numeral in column 3 in part the results of these lying opposite the stimulus in measurements. column 2 which feels most nearly equal to $>«. 5') Infer from columns 1 and 3 5) Use columns 1 and 3 to assign Fechner's law, W = k (log numerals to the sensation pro$ - log $ 0 ) . duced by any stimulus, 3>x. OR Assign the numeral in column 3 lying opposite the value ob- 5 " ) Infer Weber's law, = c, from columns 1 and 3. tained by measuring i>«.

JND MEASUREMENT OF SENSATIONS

3«

simply equation [16] with two elements replaced by constants. The instructions refer to table 8. In one sense, entities have been measured if a scale for the entities—a rule for assigning numerals to them—has been devised. In this sense, step (i)-(2)-(3) constitute an independent method for measuring sensations, since they produce the scale represented in part by columns 2 and 3. In another sense, entities have not been measured until numerals have been assigned to them by using a scale (rule of numeral assignment). In this sense, steps (a)—(2)—(3)—(4) also constitute an independent method for measuring sensations. In the first sense—in which to measure is to devise a scale—steps (a)—(2)—(3)—(4') constitute a dependent method for measuring sensations. The scale devised in these steps is shown in part in columns 1 and 3. The method is dependent because it presupposes the measurement of stimuli to obtain the values in column 1. In the second sense—in which to measure is to use a scale—steps (a)—(2)—(3)—(4')—(5) also constitute a dependent method for measuring sensations. Steps (a)—(2)—(3)—(4')—(5') constitute a method for deriving Fechner's law relating sensations to their stimuli. Only steps (2) and (4') in this method are empirical, experimental procedures. Step (1) is merely a stipulation. Step (3) consists in applying a definition to previously obtained experimental results. It is a mathematical operation, comparable to transforming the values obtained for a group of weights, expressed in grams, into values expressed in ounces. Step (5') is also mathematical: it requires finding, by intuition or calculation, a formula that will satisfy columns 1 and 3. Steps (i)-(2)-(3)-(4')- (5") constitute a method for deriving Weber's law governing stimulus jnd's. Fechner's law and Weber's law are, therefore, derivable at the same stage in Method B. The former law is not, as it is in Method C, based on the latter law. Instructions for Method A would read the same as those for Method B, with two exceptions. First, step (1) would not be a stipulation, but would consist in discovering the just not noticeable stimulus to serve as the zero stimulus. Second, step (2) would consist in discovering just noticeably different weights above, but not below, the zero stimulus. Method A thus contains the same independent and dependent procedures of sensation measurement, and the same procedure for discovering Weber's law and Fechner's law, as Method B. Let us now devise instructions for Method C.

312

J N D MEASUREMENT OF S E N S A T I O N S

INSTRUCTIONS FOR METHOD C

1) Discover Weber's law = c) governing just noticeable differences between stimuli by (a) selecting pairs of stimuli that are just noticeably different and (b) measuring them. 2) Stipulate that the zero sensation is one produced by a stimulus of a certain magnitude (e.g., 100 gm). 3) Infer from Weber's law the values of stimuli above and below the zero stimulus separated by jnd intervals. A partial list of these values is given in column 1 in table 8. 4) Taking the sensation unit to be a difference between sensations corresponding to a just noticeable difference between stimuli, correlate numerals representing sensation magnitude to the stimulus values given in column 1. Column 3 is a partial list of these numerals. 5) Use columns 1 and 3 to assign 5') Infer Fechner's law, W = k numerals to the sensation pro(log $ - log $0), from colduced by any stimulus, Asumns 1 and 3. sign the numeral in column 3 lying opposite the value in column 1 which most nearly equals the value obtained by measuring In one sense—in which to measure is to devise a scale—steps (i)-(2)-(3)-(4) constitute a dependent method for measuring sensations. The scale consists of columns 1 and 3. The method is dependent because it presupposes the measurement of stimuli. In the other sense —in which to measure is to use a scale—steps (a)—(2)—(3)—(4)—(5) also constitute a dependent method for measuring sensations. Steps (i)-(2)-(3)-(4)-(5') constitute a method for deriving Fechner's law relating sensations to their stimuli. Only step (1) in this method is an empirical, experimental procedure. Step (2) is merely a stipulation. Step (3) is a mathematical operation, consisting in multiplying stimulus values by the fraction given in Weber's law. Step (4) is also a mathematical operation consisting in applying a definition to previously obtained results. Since the discovery of Weber's law is the first step in this method, Fechner's law is based on Weber's. The two lists of instructions help us to see that Fechner's law can be obtained in either of two ways. Obtained at step (C5') it is an artifact of a method of dependently measuring (scaling) sensations. Here the law is simply a more convenient way of expressing columns

J N D MEASUREMENT OF SENSATIONS

313

x and 3 in table 8. It is a rule for assigning numerals to sensations when the corresponding values of their stimuli are known. It is a scale of sensations based on a stimulus scale. Obtained at step (B5'), Fechner's law is apparently a genuine empirical law, discovered by (a) measuring stimuli, (b) independently measuring the sensations produced by the stimuli, and (c) finding an equation to fit the results. I say "apparently" because Fechner's law is a genuine empirical law only if the independent method of sensation measurement used to discover it—in steps (x)—(2)—(3)—is an acceptable method. The rest of this chapter is concerned with showing that all jnd methods of sensation measurement are unacceptable. The lists of instructions show (i) that both methods of obtaining Fechner's law by means of jnd measurement of sensations are based on essentially the same experimental data. Steps (B2) and (B4') are the only experimental steps in Method B. These steps correspond, almost exactly, to steps (Cia) and Cib), which are the only experimental steps in Method C. That is, the experimental data on which both derivations are based are obtained by discovering and measuring just noticeably different stimuli. A closely related point is (ii) that the experimental data that lead to Fechner's law lead also to Weber's and vice versa. It appears that the empirical content of Fechner's law is exhausted by the experimental data obtained in jnd methods of sensation measurement. For the law to have any additional empirical content, there would have to be acceptable methods for measuring sensations in addition to jnd methods. But apparently there are no other acceptable methods of sensation measurement. We will see in chapter 9 that although Stevens disputes this claim, the methods of sensation measurement he offers are suspect. From the unavailability of alternative methods of sensation measurement, and from (i) and (ii), it follows (iii) that the empirical content of Fechner's law is precisely the data obtained in measuring just noticeable differences between stimuli. This fact is of the highest importance in assessing the traditional, Fechnerian conception of psychophysics, for it entails that the empirical content of Fechner's law and Weber's law is the same. Consequently, it is completely unclear in what sense the two laws can be regarded as different. Both laws are based on the measurement of just noticeable differences between stimuli. Every such measurement that

314

JND MEASUREMENT OF SENSATIONS

tends to confirm, or disconfirm, the one tends to confirm, or disconfirm, the other. And there are, apparently, no other measurements or experiments that tend to confirm or disconfirm either. What reason, then, is there to regard the laws as distinct? Fechner's law, as it is usually written, has the form, Y = It seems to be a psychophysical law in the traditional sense, a law relating sensations to their stimuli. Weber's law, on the other hand, has the form, $ = f($). It contains only physical, stimulus variables, and therefore cannot be regarded as a psychophysical law in the traditional sense. But if Weber's law and Fechner's law are indistinguishable, it seems legitimate to conclude that the latter is not, any more than the former, a law relating sensations to their stimuli, appearances to the contrary notwithstanding. A similar conclusion can be reached in a somewhat different manner. As we have seen, steps (i)-(2)-(3) under B constitute a jnd method for measuring sensations which is, in my sense of the term, independent, or direct. This method does not presuppose any system of measurement, nor are its units defined in terms of the units for any other system of measurement. It is, however, indirect in a different but nonetheless important sense. Like the other jnd methods of sensation measurement, it defines the unit difference between sensations in terms of a stimulus difference. This definition was adopted to avoid the objectionable application of the concept of just noticeability to sensations. It does have the desired effect. But, by the same token, it has the effect of removing sensations from the experimental picture. Sensations are manipulated only indirectly, through their stimuli, in the reconstrued jnd methods. Only indirectly are they compared with one another, and with sensation units. To see this point, consider each of the steps in the independent method under B. Step (Bi) consists in selecting a stimulus, thus indirectly specifying a zero sensation. Perception of the sensation is not required. Step (B2) consists only in discovering and labeling stimuli. Sensations are not involved, either directly or indirectly. Step (B3) consists in correlating numerals with stimuli, thus indirectly correlating them with sensations. The numerals are intended by the correlator to represent sensation magnitudes. But no sensations need be or are perceived. Numerals are correlated with sensations in thought, not in actual empirical fact. Step (B4) is described as "assigning numerals to sensations." But what it consists in, in actual empirical fact, is perceiv-

JND MEASUREMENT OF SENSATIONS

315

ing stimuli and comparing them with other stimuli. The "assignment of a numeral to a sensation" is an operation of thought. Sensations need not be perceived, and no numerals are, in actual empirical fact, assigned to them. These observations make it clear that sensations, as measured by reconstrued A, B, and C, need not be viewed as perceivable (introspectable), empirically real entities. They may be regarded as theoretical entities, or theoretical fictions, as these devices are sometimes called. Indeed, an even stronger point can be made. Methods A, B, and C could proceed quite as well as they do even if there were no perceivable, empirically real sensations produced in the discriminal perception of stimuli. The assumption that such sensations do not exist requires only simple changes in the description of the methods. These changes consist in labeling column 1 of table 8 "Measured value of the stimulus" and column 3 "Number of just noticeable (stimulus) differences of the stimulus from (stimulus) zero"; in substituting "stimulus as discriminated" for "sensation" in the instructions, and in removing phrases suggesting the "production of sensations"; and, finally, restating Fechner's law to read = k (log - log $ m o), where is the number of just noticeable differences of the stimulus from zero, and is the measured value of the stimulus. Thus construed, Fechner's law is not a psychophysical law in the traditional sense. Since it contains only physical variables, it cannot be said to relate psychological and physical dimensions. The possibility of construing the law in this way, without disturbing the experimental processes of measurement on which it is based, suggests that Fechner's law is not really a law relating sensations to stimuli, appearances to the contrary notwithstanding. The foregoing arguments are sufficient to cast serious doubt on A, B, and C as methods of sensation measurement, and a further consideration supports this doubt. Why should it occur to anyone to use sensation differences corresponding to just noticeable stimulus differences as units of sensation measurement? Why select these sensation differences over any others? One possible answer calls attention to the difficulty of identifying sensation differences. Sensations, unlike physical objects, are private and transitory. It is, consequently, difficult to recognize or identify a given sensation difference as the same one observed previously. The just noticeable difference between stimuli is,

3

X6

JND MEASUREMENT OF SENSATIONS

however, identifiable; consequently, the sensation difference corresponding to it is identifiable. This answer is weak, since there are other stimulus differences even more easy to identify than jnd's. The difference of a gram is an example. Why not use the sensation difference corresponding to the stimulus difference of a gram as the sensation unit? The only possible answer is that we are not assured that such sensation differences are equal, whereas we are assured that the sensations differences corresponding to just noticeable stimulus differences are equal. This alleged advantage is challenged in what follows. ARE SENSATION JND'S EQUAL? It is argued in chapter 5 that the units in an acceptable procedure of measurement must be equal. Do the units in Methods A, B, and C satisfy this requirement? On one interpretation, the units of these methods are just noticeable differences between sensations. I have argued that this interpretation is unacceptable on the ground that the concept of just noticeability does not apply to sensations. On the other interpretation, the units of the three methods are sensation differences corresponding to just noticeable differences between stimuli. The difference between and the relative adequacy of these two interpretations are irrelevant to the question of whether their units are equal. Every just noticeable sensation difference (if there are such) is identical with some sensation difference corresponding to a just noticeable stimulus difference. And every sensation difference corresponding to a just noticeable stimulus difference is identical with some just noticeable sensation difference. Hence, if sensation differences corresponding to just noticeable stimulus differences are equal (unequal), then just noticeable sensation differences are equal (unequal). And, if just noticeable sensation differences are equal (unequal), then sensation differences corresponding to just noticeable stimulus differences are equal (unequal). In view of the relation between these two units, in this section I refer to both as sensations jnd's. ARGUMENTS FOR THE EQUALITY OF SENSATION JND's

Several theoretical arguments have been advanced in support of the contention that sensation jnd's are equal. Some of these have been

JND MEASUREMENT OF SENSATIONS

317

explicitly used by psychologists; others have been merely suggested. None has been adequately examined. The argument from equal noticeability.—One familiar argument is as follows: Since just noticeable differences between stimuli are equally noticeable (i.e., are noticed equally often), the corresponding differences between sensations must be equal. T o illustrate the argument, let $1, $2, $3, and $4 be weights of 53, 54, 106, and 108 grams, respectively; and let V i , V2, Y3, and ^4 be, respectively, the sensations produced by lifting these weights. In accordance with familiar practice, let us define a just noticeable stimulus difference as one noticed, or perceived, 50 percent of the time. N o w suppose that O perceives a difference between $1 and $2, and also between and i n 50 percent of a given number of trials. The difference between i>i and i>2 and between $3 and are, then, equally noticeable, and they are, according to my definition, just noticeable differences. The argument from equal noticeability concludes that the differences between Y i and W2 and between ^3 and ^4 must be equal. N o w w h y should anyone suppose that equally noticeable stimulus differences correspond to equal sensation differences? The answer (which, to my knowledge, has never been explicitly stated) will probably be the following. O perceives differences between stimuli by perceiving differences between the sensations produced by these stimuli. When O perceives Y i and V2 as equal he judges and $2 to be equal. When O perceives 'Pi and Y 2 as unequal he judges 3>i and to be unequal. N o w suppose that the difference between and $2 and the difference between $3 and $4 are perceived equally often. If the difference between Y i and is greater than the difference between and ^"4, there are more sensations possible between Y i and ^2 than between ^3 and Y4. Thus, there is more likelihood that and $2 will produce sensations that O perceives as different than that $3 and $4 will produce sensations that O perceives as different. Consequently, there is more likelihood that and $2 will be perceived as different than that $3 and $4 will be perceived as different. But this conclusion is contrary to the hypothesis that the difference between and $2 is perceived equally as often as the difference between $3 and $4. Therefore, the difference between Y i and ^2 is not greater than the difference between ^3 and

318

JND MEASUREMENT OF SENSATIONS

By the same reasoning it can be shown that the difference between T 3 and Y4 is not greater than the difference between Y i and Y j . Hence, these differences must be equal. The argument above is objectionable, to begin with, because it assumes that the greater the difference between two sensations, the more noticeable (more likely to be noticed) it is. It assumes that if the difference between and T2 is greater than the difference between Y a and Y 4 , 0 will be more likely to notice the former. This assumption is entirely questionable. It is as questionable as the corresponding assumption for stimuli is false. If $ 1 , $2, $3, and $4 are weights of 53, 54, 106, and 108 gm, respectively, the difference between i>i and $2 is equally as noticeable as the difference between $3 and $>4. Since the latter difference is greater than the former, however, it is false that the greater the difference between stimuli the more noticeable it is. Why should not the same principle apply to weight sensations? Perhaps it is equally false that the greater the difference between sensations the more noticeable it is. The point here can be stated even more dramatically. Weber's law says that equally noticeable stimulus differences (just noticeable differences) increase in size as one ascends the weight scale. Why should Weber's law not apply to weight sensations? Perhaps equally noticeable sensation differences increase in size as one ascends the sensation scale. In conclusion, there is no good reason to suppose that equally noticeable stimulus differences correspond to equal sensation differences. There may (or may not) be some reason to suppose that equally noticeable stimulus differences correspond to equally noticeable sensation differences. But the two italicized statements have entirely different meanings, and the latter does not entail the former. The argument in question is objectionable, in the second place, because it assumes that O notices differences between weights by noticing differences between the sensations they produce in him. This assumption seems to generate an infinite regress. If O notices the difference between and $2 by noticing the difference between the corresponding sensations, Y i and Y2, and if he notices the difference between and $2 50 percent of the time, it would seem that he must notice the difference between Y i and ^ 2 50 percent of the time. But why does he notice the latter difference 50 percent of the time? To answer this question it seems we must posit some further entities, T j

JND MEASUREMENT OF SENSATIONS

319

and 1*2, corresponding to and and suppose that O notices a difference between these entities 50 percent of the time. And so on ad infinitum. Not only does this way of explaining how O notices stimulus differences lead to an infinite regress, but the nature of the entities posited, like Yi and Y2, is obscure. Are they second-order sensations? If not, then what are they? Confronted by difficulties of this nature, we ought to consider the possibility that O notices weight differences directly, in an unmeditated manner. If he does notice weight differences directly, any weight sensations he may have should be regarded as mere epiphenomena in the process of weight perception, not as an essential or operative part of it. The argument from psychological equality.—A second argument for the assumption that just noticeable differences between stimuli correspond to equal differences between sensations is usually stated as follows. Since just noticeable differences between stimuli are psychologically (subjectively) equal, the differences between sensations corresponding to them must be equal. The chief difficulty in this argument lies in its use of the phrase "psychologically equal." The phrase might be defined as follows: psychologically equals i>3-i>4" means " ^ i - T ^ equals ^ 3 - ^ 4 . " This definition makes it trivially true (true by definition) that if just noticeable differences between stimuli are psychologically equal, their corresponding sensation differences are equal. It thus begs the question of the argument under consideration, and cannot be legitimately employed in making that argument. What must be meant by the statement that "the difference between and $2 is for O psychologically equal to the difference between $3 and $4" is that " O perceives or estimates that the difference between $ 1 and $2 is equal to the difference between $3 and $4." But it is doubtful that just noticeable differences between stimuli are equal in this interesting sense of the phrase. First, the available evidence suggests that observers do not perceive just noticeable stimulus differences to be equal. There are (so far as I know) no experiments in which observers have been asked to estimate the relative sizes of individual stimulus jnd's. But observers have been asked to estimate the relative sizes of larger stimulus intervals in numerous experiments. These experiments reveal that, for a certain class of continua (the socalled prothetic continua, such as length, weight, loudness, and bright-

320

JND MEASUREMENT OF SENSATIONS

ness), psychologically equal stimulus intervals do not contain equal numbers of stimulus jnd's; that when one stimulus interval is perceived to be n times as great as another the former does not contain n times as many jnd's as the latter.18 Second, the intervals between and $2 and between $3 and if just noticeable, are not genuine perceptual intervals. They are intervals that O sometimes perceives and sometimes does not perceive (perceives 50 percent of the time, by one definition). So it is misleading to say that O perceives these intervals, and even more misleading to say that he perceives them to be equal, or unequal. It would seem that the only way to avoid the objection above is to shift from saying that stimulus jnd's are psychologically equal to saying that they are psychologically similar, and to use this similarity as a reason for thinking that the corresponding sensation differences are equal. Similarity of two stimulus differences is, however, not a good reason to suppose that the corresponding sensation differences are equal. Consider an analogy. The pitch intervals between the tones C and F# and between C and C# are similar in that both are dissonant. But that kind of similarity provides no basis for thinking that the two intervals are equal; the first interval (an augmented fourth) is much larger than the second (a halftone). Nor does it provide a basis for thinking that the frequency intervals corresponding to the pitch intervals are equal. The first (an interval of 107 cycles per second) is much greater than the second (an interval of 6 cycles per second). The most that can be concluded from the fact that two stimulus intervals are similar is that the corresponding sensation intervals are similar. This conclusion, of course, does not establish that just noticeable stimulus differences correspond to equal sensation differences. Even the conclusion itself is questionable, for it seems that all we can mean by saying that two just noticeable stimulus differences are similar is that they are just noticeable differences (what else could be meant?). From similarity in this trivial sense similarity of sensation differences cannot be inferred. That two stimulus differences are just noticeable does not seem to be an adequate reason for supposing that the corresponding sensation differences are just noticeable. The argument from minimal differences.—The last argument for 18

Stevens, "On the Psychophysical Law/' pp. 172-173.

JND MEASUREMENT OF SENSATIONS

321

the assumption that just noticeable stimulus differences correspond to equal sensation differences is rather complex. Briefly, it claims that the sensation differences that correspond to just noticeable stimulus differences are the smallest sensation differences possible and therefore must be equal. In stating this argument, I refer to a difference between a sensation, Y , and any other sensation as a " Y difference." A) If one sensation difference, Y1-Y2, is greater than a second, Y3-Y4, there is a Y i difference that is smaller than Y 1 - Y 2 and equal to Y 3 - Y 4 ; but, B) If Y 1 - T 2 is a sensation difference corresponding to a just noticeable stimulus difference, there are no smaller Y i differences; hence, C) No sensation difference that corresponds to a just noticeable stimulus difference can be greater than another; similarly, D) If one sensation difference, Y1-T2, is smaller than a second, ^ 3 - ^ 4 , there is a Ya difference that is smaller than Y a - Y * and equal to Y i - Y a ; but again, E) If is a sensation difference corresponding to a just noticeable stimulus difference, there are no smaller Y s differences; hence, F) No sensation difference that corresponds to a just noticeable stimulus difference can be smaller than another; (C) and (F) entail that G) No sensation difference that corresponds to a just noticeable stimulus difference can be unequal to another; therefore, H) All sensation differences that correspond to just noticeable stimulus differences are equal. One group of objections to this argument is directed against premises (B) and (E), which assert that there are no sensation differences smaller than those that correspond to just noticeable stimulus differences. This assertion, which is not obviously true, probably rests on an argument like the following. Let Y1-Y2 be for O the sensation difference corresponding to the just noticeable stimulus difference, Any smaller Y i difference must be unperceivable by O, since otherwise he would, on perceiving these smaller sensation differences, perceive smaller $1 differences than But there are no differences

JND MEASUREMENT OF SENSATIONS

between O's sensations which he fails to perceive; sensations are entities about which the observer cannot make such mistakes. Therefore, there are for O no Y i differences smaller than Y i - T i . This argument is unpersuasive for two main reasons. First, it employs a crude and unacceptable notion of the just noticeable difference. It assumes that, if i>i-3>2 is a just noticeable difference for O, he can perceive no smaller i>i difference. But it is now generally recognized that the just noticeable difference is not a sharp one. O may perceive a difference between and $2 50 percent of the time, and a slightly smaller difference 40 percent of the time, and a still smaller one 25 percent of the time, and so on. Now, we must decide which of these differences to regard as just noticeable for O, and the decision will be in large part arbitrary. If, as is often done, we select the difference perceived 50 percent of the time as just noticeable, we say that i»i-4>2 is a just noticeable difference. It does not follow, however that every smaller $ 1 difference is unperceivable by O. Second, the argument assumes there is a contradiction in the supposition that some sensations and sensation differences are unperceived and unperceivable by the possessor. But this assumption is not shared by all psychologists, and even some philosophers have denied it. The history of philosophical and psychological debate over this point suggests that it is extremely difficult to resolve, if not unresolvable. Moving to another objection, premises (B) and (E) assert, it will be recalled, that there are no sensation differences smaller than those that correspond to just noticeable stimulus differences. This assertion may be challenged on the ground that the smallest sensation differences in one observer may not be the smallest in a different observer. Let and ^'2 be sensations of observer O', and V ' i and V ' 2 sensations of observer O" which, respectively, equal those of O'. The difference between V i and ^ ' 2 then equals the difference between and ^"2. Now suppose that V 1 - V 2 corresponds to a stimulus difference that is just noticeable for O', and that this fact entails that V 1 - V 2 is the smallest Y i difference for O'. Even so, it does not follow that W V ' 2 is the smallest Y i difference for O " , for V ' i - T ' 2 may correspond to a stimulus difference that is more than just noticeable for O". This possibility seems to show that (B) and (E) are false. Of course, it may be replied that (B) and (E) refer to minimal sensation differences relative to a single observer, but the

JND MEASUREMENT OF SENSATIONS

3*3

reply seems ad hoc and unacceptable. If sensations in a single observer can be compared with respect to their equality or inequality, then surely sensation differences in different observers can be similarly compared. If they can, there is no reason to suppose that the smallest sensation difference for a given observer is the smallest for every observer. In short, there is no reason to suppose that the smallest sensation difference for a given observer is the smallest sensation difference simpliciter. The final objection to argument (A) through (H) concerns, not the truth of any of its premises, but the validity of its conclusion. In particular, the step from (G) to (H) is dubious. Let us grant, for the sake of argument, that no sensation difference corresponding to a just noticeable stimulus difference can be unequal to another sensation difference corresponding to a just noticeable stimulus difference. Still, it does not follow that all such sensation differences are equal. If, as argument (A) through (H) suggests, sensation differences that correspond to just noticeable stimulus differences are the smallest possible sensation differences, then perhaps the concepts of equality and inequality do not apply to them. Perhaps it is meaningless to say of differences that are the smallest possible either that they are equal or that they are unequal. If so, it is true, in a sense, that no minimal sensation difference can be unequal to any other minimal sensation difference. But, in the same sense, it is false that all such sensation differences are equal. EVIDENCE BEARING ON THE EQUALITY OF SENSATION JND'S

Each of the above three arguments for the equality of sensation jnd's is theoretical, and, in a sense, inconclusive, since none of them advances empirical evidence bearing directly on the equality or inequality of sensation jnd's. It would appear that such evidence can be obtained only by directly comparing sensation jnd's or by examining the results of measuring sensations with a method employing sensation jnd's as unit intervals. In this section, I examine evidence of this direct empirical kind. Psychologists use the term "equality" in several senses. When they say that sensation jnd's are equal they sometimes mean that sensation jnd's are "psychologically" or "subjectively" equal, sometimes they mean that sensation jnd's are equal relative to some other system

324

JND MEASUREMENT OF SENSATIONS

of sensation measurement, and sometimes they mean that sensation jnd's are absolutely equal. These three senses must be carefully distinguished and treated separately. Psychological equality.—To say that two weight intervals, $1— $2 and $>2- ( i > 3, are psychologically equal means that the intervals appear (feel) equal to some observer or observers. Similarly, to say that two sensation intervals, ^ l - f i and are psychologically equal must mean that the intervals appear equal to some observer or observers. If the two intervals are jnd intervals, no sense can be given to the assertion that they appear equal or unequal. It is tempting to argue this point by saying that jnd intervals are so small, or are of such uncertain magnitude, that the observer cannot make a judgment on their relative magnitude. But the real reason is that jnd intervals are not genuine perceptual intervals, since, by definition, they sometimes appear equal and sometimes appear unequal. This point becomes clear in the sequel, when the concept of a just noticeable difference is made precise. Perhaps it cannot be said that sensation jnd's individually appear equal or unequal, but this fact does not entail that they cannot be said collectively to appear equal or unequal. To say that halftones of pitch individually appear equal is to say that if individual halftones are compared, they will appear equal. To say that halftones collectively appear equal is to say that if pitch intervals containing equal numbers of halftones are compared they will appear equal. Similarly, to say that sensation jnd's individually appear equal is to say that if individual sensation jnd's are compared they will appear equal. And to say that sensation jnd's collectively appear equal is to say that if sensation intervals containing equal numbers of sensation jnd's are compared they will appear equal. No experiments have been performed which would enable us to determine whether sensation jnd's individually or collectively appear equal or unequal. Observers have been asked, in many psychophysical experiments, to say how weights and weight intervals appear to them. But no observer has been asked to say how weight sensations and weight-sensation intervals appear to him. And there may be a good reason for not asking this question. It is highly doubtful that sensations can meaningfully be said to appear this way or that. If such assertions were meaningful, it ought to make sense to say that O's sensations

JND MEASUREMENT OF SENSATIONS

325

appear to him to stand in relation R when they do not stand in relation R. But such a statement seems to violate the concept of a sensation. Sensations, it would seem, appear as they are by definition. Apparently no meaning is attached to the notion of a mistake on O's part in perceiving (introspecting) his own sensations or their relations. For this reason, it can be tentatively concluded that the notion of psychological equality or inequality of sensation intervals (including just noticeable sensation intervals) has no meaning. This conclusion will not be accepted unless careful distinctions are drawn between stimulus-interval equality and sensation-interval equality, and between psychological equality and absolute equality. Stimulus intervals, such as those between weights, can meaningfully be said to appear equal or unequal, even if sensation intervals cannot. Some writers have suggested that from the fact that stimulus intervals appear equal (are psychologically equal) we can infer that the corresponding sensation intervals are equal (absolutely equal). This inference was attacked earlier in discussing one argument for the equality of sensation jnd's. But the point here is that it is one thing to infer absolute sensation-interval equality from apparent stimulus-interval equality, and quite a different thing to infer psychological sensationinterval equality from apparent stimulus-interval equality. The latter inference is objectionable simply on the ground that the concept of psychological equality does not apply to sensations or sensation intervals, so that even if the former inference is acceptable, the second seems not to be. The question of whether sensation jnd intervals are psychologically equal is ultimately irrelevant to the question of whether a jnd method for measuring sensations is possible. The fact is that weight intervals of one gram feel unequal to most observers. From this fact it does not follow that weight intervals of one gram are unequal. Furthermore, if weight intervals of one gram felt equal to most observers, it would not follow that these intervals were equal. In general, from the psychological equality or inequality of units, whether stimulus units or sensation units, nothing follows as regards the absolute equality of those units or their acceptability as units in a system of measurement. Relative equality.—The only methods of sensation measurement thus far discussed have been jnd methods. Chapter 9 examines the most interesting alternative method thus far proposed by psychophysicists

326

JND MEASUREMENT OF SENSATIONS

—the ratio scaling method developed by Stevens and others. Working on the assumption that the magnitude of an observer's sensation corresponds to his estimate of stimulus magnitude, this method stipulates a sensation unit—the veg for weight sensations, the mak for length sensations, and so on—and assigns numerals to sensations which represent their magnitude in terms of the unit. Proponents of this method claim to have discovered that, for a large class of sensation continua (the socalled prothetic continua), sensations are a power function of the stimuli producing them. Weight sensations are among those governed by the power law. That is, [20] Y t = where is the magnitude of the weight sensation in vegs, $ is the weight of the stimulus in grams, and the value of n is 1.45. 1 9 Converting [20] to logarithms, we obtain [21] log = n log A s we saw in the preceding part, weight sensations are, when measured in sensation jnd's, a logarithmic function of weight stimuli; that is, [22] V) = k log 9. where Y j is the magnitude of the weight sensation in jnd's. By combining [21] and [22], we obtain U3] t , = 4 i o g Y T . Thus, weight sensations measured in jnd's are a logarithmic, not a linear, function of weight sensations measured in vegs. It follows that sensation jnd's are unequal relative to the veg method of sensation measurement. This result, however, provides no objection to the jnd method of sensation measurement. It is equally true that vegs are unequal relative to the jnd method of sensation measurement, but no one will count this as an objection to the veg method. If we opt for the veg method of sensation measurement, we shall have to say that sensation jnd's are unequal. If we opt for the jnd method of sensation measurement, we shall have to say that vegs are unequal. These observations provide no assistance at all in deciding which method to adopt. Stevens claims that the units of his ratio scaling method are, unlike those of the jnd method, subjectively (psychologically) equal, and 19

Stevens, "On the Psychophysical Law," pp. 1 6 2 , 1 6 6 .

JND MEASUREMENT OF SENSATIONS

3*7

he uses this claim to argue for the superiority of his method. 10 The claim is unclear, since it fails to distinguish between stimulus equality and sensation equality. If Stevens means that sensation intervals containing equal numbers of vegs appear equal, then his claim is open to the objection that no meaning is attached to the assertion that sensations or sensation intervals appear this way or that. If, on the other hand, Stevens means that stimulus intervals corresponding to sensation intervals containing equal numbers of vegs appear equal, then his claim is meaningful and probably true. On this interpretation, however, the claim provides no basis for asserting the superiority of ratio scaling methods. That two stimulus intervals appear equal does not entail that the corresponding sensation intervals are apparently (psychologically, subjectively) equal, or that they are absolutely equal. An analogy may help to make the point clear. That two weight intervals between liquids appear equal does not entail that the corresponding volume intervals are apparently equal, or that they are absolutely equal. From the psychological equality or inequality of weight intervals nothing follows as to the psychological or absolute equality or inequality of volume intervals. In a similar way, from the psychological equality or inequality of stimulus intervals nothing follows as to the psychological or absolute equality or inequality of sensation intervals. Absolute equality.—Whether sensations jnd's are psychologically equal and whether they are relatively equal are not critical questions, since their answers provide no reason for adopting or abandoning any method of sensation measurement. If, however, we find that sensation jnd's are absolutely unequal, then we have a powerful argument against the use of any jnd method. The most significant criterion for the absolute equality of sensation units is satisfaction of the axiom of additivity. Stated for the addition of sensations, the axiom is [24] + NpF 2 ) = N ( T , @ Y 2 ) , where N is the number assigned to a given sensation or collection of sensations by the method of measurement in question (here a jnd method). This criterion is applicable only if there is an operation of addition for sensations. Is there such an operation? Two weights lifted at the same time feel heavier than either lifted alone. It would seem to follow, 20

Stevens, "On the Psychophysical Law," pp. 1 5 4 , 1 7 2 .

328

JND MEASUREMENT OF SENSATIONS

on the assumption that weights produce sensations when lifted, that the weight sensations produced by lifting two weights simultaneously are together greater than the sensation produced by lifting either weight alone. If so, there is an operation of adding weight sensations, the operation of producing sensations simultaneously by lifting weights simultaneously, and the axiom of additivity does have meaning when applied, as in [24], to weight sensations. This conclusion is contrary to widespread belief. James and Stumpf, and numerous contemporary theorists, would argue that the sensation produced by lifting Y i is not part of the sensation produced by lifting Y i and T 2 simultaneously, and that there is no composite sensation (^Fi @ T2). This argument is specious. It rests on the fact that an observer lifting two weights simultaneously does not introspectively distinguish two weight sensations. But neither does he, by feeling, distinguish the two weights if he lifts them in a single cup; and yet it can be said that he lifts and feels two weights. A balance does not distinguish two weights when they are placed in one of its pans; yet we can say that the two objects are being weighed. We should not refuse to say, then, that O has a composite sensation when he lifts two weights, if the only reason for refusing is that he does not introspect the sensation as composite. Since it seems that the axiom of additivity can be interpreted in terms of the operation of producing sensations simultaneously, let us ask whether, thus interpreted, the axiom is true. Let Y i and ^ 2 be the sensations produced by lifting, respectively, weights and $2. Let (Vi @ ^2) be the composite weight sensation produced by lifting @ $2), that is, by lifting and $2 together in one hand. If and $2 have the same weight, then the number assigned to @ $2) by weighing it will be twice the number assigned to either or $2. Now suppose that ($1 @ $2) weighs 200 gm and and $2 weigh 100 gm each. Experiment has shown that equal weights feel (approximately) equal to normal observers. And we can infer that when weights feel equal to an observer, the weight sensations produced in him are equal. Hence, in normal observers, to which we will confine our attention, ^Fi and are equal. The jnd method will therefore assign the same number to these two sensations. If the method is to satisfy axiom [24], it must assign to ( ^ i @ ^2) a number twice that assigned to either T i or ^2. But this condition is not fulfilled. Fechner's law tells us that weight, expressed in grams, is a logarithmic function of the

JND MEASUREMENT OF SENSATIONS

3*9

magnitude of weight sensations, expressed in jnd units. That is, weight differences corresponding to sensations jnd's grow larger as one ascends the scale. The number of sensation jnd's between o and 100 gm, then, is larger than the number between 100 and 200 gm. Hence, the jnd method assigns a number to ( Y i @ Y2) which is less than twice the number it assigns to either Y i or ^"2. Therefore, axiom [24] is false for the jnd method of weight sensation measurement, and it follows that weight sensation jnd's are absolutely unequal. Surprisingly, the above argument has, to my knowledge, never been used or even stated. One part of the explanation is the widespread view that sensations do not permit of an operation of addition. Although this view may be correct for some sensations, there is no good reason to think it correct for sensations produced by stimuli that themselves permit of an operation of (physical) addition, such as weight sensations and length sensations. Another part of the explanation is unclarity concerning or inadequate investigation of the concept of equality. Criteria for absolute equality are rarely proposed and discussed, but it seems unlikely that such discussion will produce a better criterion than the one employed in the above argument, that is, satisfiability of the axiom of additivity. In reply to my discussion of jnd equality, it may be said that sensation jnd's are equal by definition, or by stipulation, and that arguments and evidence for and against the equality or inequality of sensation jnd's are irrelevant. This reply is unacceptable. One cannot merely stipulate that sensation jnd's are equal. This stipulation is no more legitimate than the stipulation that sensation differences corresponding to stimulus differences of 1 gm are equal, or that sensation differences corresponding to aesthetically pleasing stimulus differences are equal, and so on. If such stipulations, or definitions, were legitimate, we could stipulate virtually any sensation differences we pleased as unit differences, even those that are patently unequal. It may be argued in reply that the only scientifically useful test for equality of units of measurement is whether the method employing them leads to the discovery of scientific laws. 21 Since the use of methods employing sensations jnd's leads to the discovery of Fechner's 21 Gustav Bergmann and K. W. Spence take a similar position in "The Logic of Psychophysical Measurement," in Readings in the Philosophy of Science, ed. H. Feigl and May Brodbeck (New York, 1953), pp. 1 0 9 - 1 1 0 .

33°

JND MEASUREMENT OF SENSATIONS

law, which relates sensations to their stimuli, sensation jnd's can be pronounced equal. This reply may have the practical value of encouraging experimental activity, but it cannot justify that activity. For usefulness in discovering scientific laws is unacceptable as a test for the equality of units in a system of measurement. The test is unacceptable, in the first place, because it conflicts with the clearly acceptable test of satisfiability of the additivity axiom. The conflict was demonstrated earlier in the proof that a jnd method of measurement for weight sensations, although it leads to the discovery of Fechner's law, does not satisfy the axiom of additivity. In the second place, the test of usefulness in discovering scientific laws leads to inconsistency. Applying the test produces the conclusion that sensation differences corresponding to stimulus differences of i gm (let us call them gram sensation differences) are equal, since by using a system of measurement employing these sensation differences as units we obtain the following law: V = — C, where Y is the sensation expressed in gram sensation differences, $ is the stimulus expressed in grams, and C is the value of the stimulus selected as producing the zero sensation. Applying the same test produces the conclusion that sensation differences corresponding to just noticeable stimulus differences (sensation jnd's) are equal, since by using a system of measurement employing these sensation differences as units we obtain Fechner's law: W = k log 3», where Y is the sensation expressed in sensation jnd's. But gram sensation differences and sensation jnd's cannot both be equal. Weber's law says that just noticeable stimulus differences, measured in grams, increase as one ascends the weight scale. Consequently, weight sensation differences in different parts of the scale containing equal numbers of gram sensation differences contain unequal numbers of sensation jnd's. It follows that gram sensation differences and sensation jnd's cannot both be equal. The treatment in this part of the chapter has shown that there are no good reasons for supposing that sensation jnd's are equal, and that there is one very persuasive reason for thinking that sensation jnd's are unequal. This point is, in itself, sufficient to disqualify any method employing sensation jnd's as units of measurement. But, although important, the point does not yet reveal what is fundamentally wrong with jnd methods of measurement. The fundamental objection to any such method is not that its units are unequal, but rather that

JND MEASUREMENT OF SENSATIONS

33*

the introduction of sensation jnd's as units of sensation measurement rests on a confusion between stimulus and sensation: between stimulus differences and sensation differences, between stimulus jnd's and sensation jnd's, between stimulus units and sensation units. To present this objection requires considerable discussion of the concept of a threshold. SENSATIONS AND THRESHOLDS Certain objections to jnd methods of sensation measurement can be understood only after detailed examination of the concept of a threshold. These jnd methods employ the sensation jnd as a unit of sensation, and some of them stipulate the zero sensation in terms of the just noticeable stimulus. Reference to just noticeable entities and to just noticeable differences is an imprecise way of expressing absolute and relative thresholds. The concept of a threshold was originally intended to apply to 6timuli. An absolute threshold is specified as a stimulus or stimulus value below which perception fails. A relative (differential) threshold is specified as a stimulus difference below which perception fails. Can the concept be extended to sensations? In what follows I argue that it cannot. The argument depends heavily on distinctions between stimuli and the sensations they produce, between stimulus thresholds and sensation thresholds, and between the various methods for determining each. These distinctions are badly blurred in most of the relevant literature. Although not all jnd methods presuppose sensation thresholds, they all presuppose that sensations play a crucial role in the perceptions that define stimulus thresholds. This presupposition needs examination. If, in defining and explaining thresholds, we find it unnecessary to posit or mention sensations, most of the motivation for proposing jnd methods of sensation measurement will have been destroyed. THRESHOLDS AND THEIR MEASUREMENT

One of the familiar facts of perception is that the ability to perceive stimuli is limited in humans and animals. In most stimulus dimensions there are some items that can be perceived (noticed) and some that cannot. Particles with a diameter of 1 mm can be seen by human observers with the unaided eye, but particles with a diameter

JND MEASUREMENT OF SENSATIONS

332

of .001 mm cannot. Stimuli may be roughly divided, then, into two categories—the perceivable and the unperceivable—with respect to a given magnitude. The line below represents such a division with respect to weight. o

A

Z

The left end of the line represents the zero point for weight. The right end of the line has no terminal point, since objects of any weight, however large, are theoretically possible. Objects lying between points o and A are too light to be detected by lifting. Objects lying to the right of point Z are too heavy to be lifted, and consequently too heavy to be judged, by lifting, with respect to weight. The weight of objects lying between A and Z can be perceived by lifting. Points A and Z represent what are usually called absolute thresholds, or absolute limens: A the lower and Z the upper. An equally obvious fact of perception is that the ability to distinguish stimuli is limited in animals and humans. Most persons can easily see the difference between a length of 1 cm and a length of .5 cm, but no person can with the unaided eye see the difference between lengths of .001 cm and .002 cm. Differences may be roughly divided into two groups—those that can and those that cannot be perceived— with respect to a given magnitude. In making this division it is necessary to bear in mind that a given difference may be perceivable in one part of the magnitude in question and unperceivable in another part. Human perceivers can perceive the difference between a weight of 53 gm and one of 54 gm, but not the difference between a weight of 106 gm and one of 1 0 7 gm. The diagram below represents a division of weight differences into the perceivable and the unperceivable for several selected points along the weight continuum. o

AB

CD

EF

G H

Z

A and Z again represent the lower and upper absolute thresholds. Line segments A B , C D , EF, and G H represent the point of division between differences that can and those that cannot be felt by lifting, at the points A , C, E, and G on the weight continuum. Any point, X, between A and B defines a difference, A - X , which cannot be felt; any point, Y , to the right of B defines a difference, A - Y , which can be felt. Any

333

JND MEASUREMENT OF SENSATIONS

point, X, between C and D defines a difference, C - X , which cannot be felt; any point, Y , to the right of D defines a difference, C - Y , which can be felt. And so on for EF and G H . A complete diagram would mark off similar line segments from every point on the line, but it is of course impossible to draw. The line segments represent what are usually called differential thresholds, or differential limens. (Some psychologists have denied that there are unperceivable stimuli and unperceivable stimulus differences, and thus have maintained that, in one sense, there are no thresholds. 22 ) The concept of an absolute threshold and that of a differential threshold have been presented thus far in rather vague, pictorial form. W e turn now to an examination of some of the attempts to make these concepts precise. The method of just noticeability.—The

earliest attempt to un-

derstand the concept of a threshold was in terms of just perceivability, or least perceivability, as it is sometimes called. The absolute threshold was held to be determined by the just noticeable entity, and differential thresholds were held to be determined by just noticeable differences between entities. A rough definition of the concept of just noticeability in its absolute sense is: X is for O the just noticeable entity within a given dimension if and only if (a) X is perceivable by O , and (b) O is unable to perceive entities within the dimension which are smaller than X. A n d now for the relative sense: X and Y are for O just noticeably different within a given dimension if and only if (a) X and Y are different, (b) O can perceive that X and Y are different, and (c) no intermediate entity—an entity lying between X and Y — c a n be perceived by O as intermediate. The following methods for determining the just noticeability of stimuli and differences between stimuli are suggested b y these definitions. 1 ) The just noticeable

stimulus.

Suppose the just noticeable

weight is approximately o.i gm. E, the experimenter, presents to O , the observer, several weights above and below o . i gm, either in ascending or in descending order. O is asked to say, upon lifting an object, 22 For an example see L. L. Thurstone, The Measurement of Values (Chicago, 1959), p. 55. For a recent discussion see J. A . Swets, ' I s There a Sensory Threshold?" Science, 1 3 4 (1961), 165-177.

334

JND MEASUREMENT OF SENSATIONS

whether he feels its weight or not by answering " Y e s " or " N o . " Table 9 gives the results of such a hypothetical experiment. On the basis of TABLE

9

WEIGHT OF STIMULUS STIMULUS

X Y Z u V

IN GM

•25 .20 .10 .07

.04

O ' S RESPONSE

Yes Yes Yes No No

this experiment, the just noticeable stimulus may be expressed either numerically or nonnumerically. Expressed nonnumerically, it is object Z ; expressed numerically, it is .10 gm, the weight of object Z. (An alternative numerical expression takes a point halfway between Z and U, i.e., .085 gm, as the just noticeable stimulus.) An obvious defect in this method of determining the absolute threshold is its imprecision or, perhaps we should say, its inconclusiveness. The experiment fails to show whether there are any weights between Z and U which O can perceive. If E had included a weight of .09 gm among the presentation stimuli, O might have perceived that weight, thus producing a different just noticeable stimulus. Or he might have perceived a weight of .08 gm, and so on. This inconclusiveness can be removed only by using other methods for determining the absolute threshold. 2) The just noticeable difference. A just noticeable difference is always relative to a given stimulus, which is called the standard. To see how such differences are determined, consider an experiment in which a weight of 1 5 gm is selected as the standard. Although either the just noticeably greater difference or the just noticeably smaller difference may be determined, only a method for the former is described here. E presents to O, in descending order, let us say, a number of comparison weights that differ from one another by amounts considerably less than the jnd to be determined. In each trial O is required to lift a comparison weight and then the standard, and to say whether the first is heavier than or equal to the second. Table 1 0 gives the results of a hypothetical experiment in which X, Y , Z, U, V , and W are comparison weights, and S is the standard.

JND MEASUREMENT OF SENSATIONS

335

TABLE 10 STIMULUS

WEIGHT OF STIMULUS

O ' s RESPONSE

X

15.6

Heavier

Y

Heavier

Z

15-5 15.4

Heavier

V

15-3 15.2

Equal

15.1

Equal

u w

s

Equal

15.0

As with the just noticeable stimulus, there is a numerical and a nonnumerical way of expressing the just noticeable difference discovered in this experiment. Expressed nonnumerically, the jnd is the difference between S and Z (or between any two objects that feel equal in weight to S and Z, respectively); expressed numerically, it is .4 gm, or the difference in weight, measured in grams, between S and Z. (Alternatively, the weight halfway between Z and U may be substracted from the weight of the standard, to obtain a jnd of .35 gm.) This method suffers from the same defect as does the method for determining the just noticeable stimulus. It is imprecise and inconclusive, for it fails to show whether there are any weights between Z and U which the observer would perceive to be heavier than S. If the experimenter had included among the comparison objects a weight of 15.38 gm, for example, O might have perceived that one to be heavier than S, thus producing a different jnd. The above methods for determining thresholds are no longer employed by psychologists. One of their defects, as already noted, is imprecision or inconclusiveness. A related defect lies in the assumption that there is a sharp breaking point between what an observer can notice and what he cannot notice. This criticism is nicely stated by Fullerton and Cattell, who say that psychologists who employ the method of just noticeable differences . . . adopt the curious supposition that stimuli seem exactly alike so long as the difference is less than a certain amount, whereas, w h e n the difference is made greater than this amount, it becomes suddenly apparent. T h i s is b y no means the case. T h e clearness w i t h w h i c h a difference is distinguished varies gradually f r o m complete doubt

to

complete certainty. T h e variation is continuous, and no point can be

336

JND MEASUREMENT OF SENSATIONS

taken and called "just noticeable difference," and kept constant f o r different observers, or even f o r the same observer at different times. 2 3

What these authors mean can be illustrated by removing the restrictions placed on O's responses in the jnd experiment described earlier. If E allowed O to make any response he wished, not just "heavier" and "equal," O's responses would probably indicate an area of considerable vagueness in his perceptions of equality and inequality. He might respond as follows (reading down the third column in table 1 0 ) : "Heavier"; "Heavier"; "Heavier, I think"; "Equal, I think"; "Equal"; "Equal." If such responses were obtained, how would the just noticeable difference be computed? What comparison weight should be selected from which to subtract the weight of the standard? We cannot, from the concept of the just noticeable difference, deduce any answer to this question. We might take the breaking point between responses of "Heavier" and the uncertain responses, and the breaking point between responses of "equal" and the uncertain responses, average these two values, and subtract the weight of the standard from the average. But this solution would be a step toward a different and superior method for determining the threshold, the method of limits described below. A final criticism of the method of just noticeability is that it takes no account of the familiar fact that an observer's responses to stimuli depend on a number of factors in addition to the individual stimuli themselves. O's response to a given stimulus depends on the instructions given him by the experimenter, on his psychological state (fatigue, expectation, etc.), on the manner in which the stimuli are presented to him, and so on. For instance, if E had presented the comparison weights of table 1 0 in ascending rather than in descending order, O might have judged the weight of 15.3 gm to be heavier than the standard, rather than equal to it. If E had presented the comparison weights a second time, even in descending order, different responses might have been obtained because of fatigue, or because of expectations formed in the first presentation. The method of limits.24—This method, also called the method of 28 G. S. Fullerton and J. M. Cattell, On the Perception of Small Differences (Philadelphia, 1892), pp. 1 0 - 1 1 . 24 For an extended treatment of this method see J. P. Guilford, Psychometric Methods (rev. ed.; New York, 1954), chap. 5. For a brief description see Woodworth and Schlosberg, Experimental Psychology, pp. 196-198.

337

JND MEASUREMENT OF SENSATIONS

minimal changes, is used in determining the relative threshold, or differential limen (DL), as it is more commonly known. For the standard in a hypothetical experiment, a weight of 15.0 gm is selected. E then makes a rough determination of the DL, perhaps by using the method of just noticeability. Next he selects and weighs a group of comparison weights to be presented to O in both ascending and descending series. In each trial, O is required to say whether the comparison weight is heavier than the standard, equal to the standard, or lighter than the standard. (These responses are symbolized by a plus sign, an equals sign, and a minus sign, respectively.) Table 1 1 gives the result of such a hypothetical experiment. TABLE 1 1 COMPARISON WEIGHT

16.( 15 15 15 15 15 15 15 15 15 15 (S) 14 14 14 14 M 14 14 14 14 14 T. Ti

+ + + + + +

1545 14.45

+ + +

15-55 14-65

+ + + + +

+ + +

+ +

+ + + +

15-35 14-65

15-35 14-35

15-35 14-65

*5-*5 14-55

338

JND MEASUREMENT OF SENSATIONS

The comparison weights are presented to O six times, in alternating descending and ascending series. The columns headed by a downward-pointing arrow show responses for a descending series; those headed by an upward-pointing arrow show responses for an ascending series. Each column has two breaking points: the point where O shifts from a plus to an equals response (or vice versa), and the point where he shifts from an equals to a minus response (or vice versa). The upper point is designated as T u and the lower point as Ti. The values for both points are entered at the bottom of each column in table 1 1 . The DL is computed as follows. The average Tu is 15.38, and the average Ti is 14.55. Ti subtracted from T u yields what is called the interval of uncertainty (IU). In the hypothetical experiment IU = T u — Ti = .83. Since the DL is taken to be half the interval of uncertainty, in our hypothetical experiment DL = IU/2 = .83/2 = -415. The use of the method of limits to determine the absolute threshold, or absolute limen (RL), is similar to its use above. Again E presents O with weights in ascending and descending series, but with single weights rather than pairs. For each presentation O is required to say either that he feels the weight, or that he is uncertain whether he feels it, or that he does not feel it. (These responses are symbolized by a plus sign, a question mark, and a minus sign, respectively.) The experiment will produce a table like table 1 1 , with a question mark in place of the equals sign. Again, there will be two breaking points, Tu and Ti, for each series, and again the two are averaged. But now the computation differs. The RL for weight is the average of the average Tu 1.1

-T-

dt

and the average Ti: RL =

Av. T u +1 Av. Ti

•

The method of limits is clearly superior to the method of just noticeability. Since it consists in averaging breaking points in O's pattern of responses, it takes into account the fact that the threshold is not a sharp point; moreover, there is no imprecision or inconclusiveness in the determination of the threshold. Furthermore, it recognizes the bias that may result from the manner in which stimuli are presented to O. The breaking point in O's pattern of responses may be influenced by the manner in which weights are presented to him. Since, for example, the point may be higher for an ascending series than for a descending series, both types of series are presented to him and the results are averaged. O's responses may also be biased by the length

JND MEASUREMENT OF SENSATIONS

339

of the series presented to him. For instance, if all series were descending and equal in length in a DL experiment, O might come to expect equality between the comparison and the standard weight after, say, the first eight comparison weights had been presented. He might then begin reporting his expectations rather than his perceptions of weight equality. To avoid the bias of such expectations, the length of the series presented to O is varied. In the method of limits the concept of a threshold becomes exclusively a numerical concept. In the methods for determining the just noticeable stimulus and just noticeable differences between stimuli, thresholds may be expressed either numerically or nonnumerically. For example, we may say either that the relative threshold is .4 gm, or that it is the difference between stimuli S and Z. In the method of limits the latter expression is meaningless. In the hypothetical experiment leading to table 1 1 , the DL for a standard of 15.0 gm is .4x5. But we cannot say that the DL thus obtained is the difference between S (which weighs 15.0 gm) and another stimulus, Z (which weighs 1 5 . 4 1 5 gm). To say this would suggest that O always perceives this difference, whereas he does not, according to table 1 1 . And there is no reason that the DL should be the difference between a weight of 15.0 and 1 5 . 4 1 5 gm rather than the difference between a weight of 14.9 and 1 5 . 3 1 5 gm: in either instance the difference is .415. The DL, as determined by the method of limits, is not a difference between stimuli. Rather, it is a numerical value, computed in the manner described, which is associated with the value of a specified standard stimulus. If relative thresholds should not be regarded, in the method of limits, as differences between stimuli, it is an even more serious error to regard them as perceptual differences. The difference between two tones separated by an interval of a halftone, or by an interval of a minor third, is a genuine perceptual difference. A normal observer can perceive such differences and, if he has an ear for music, recognize them for what they are. When it occurs to a theorist that relative thresholds might be used as unit intervals for sounds or sensations, he is in all probability thinking of relative thresholds as genuine perceptual differences, like halftones, which can, as shown in chapter 7, be used as unit intervals. This view is encouraged by equating relative thresholds with just noticeable differences, which are perceptual differences by definition. But when the notion of a threshold is made precise, as it

34°

JND MEASUREMENT OF SENSATIONS

is in the method of limits, relative thresholds can no longer be regarded as perceptual differences. In the hypothetical experiment from which table 1 1 was obtained, the DL was found to be .415. But O does not always perceive the difference between the standard stimulus of 15.0 gm and stimuli of 1 5 . 4 1 5 gm. Sometimes he perceives such differences; sometimes he does not. Consequently, the DL is not a genuine perceptual difference. For this reason threshold intervals are quite unlike such units of measurement as unit rods, or halftones. This fact raises serious doubts about the propriety of employing relative thresholds as units of measurement. It would seem that one logical requirement of units is that they be detectable by an instrument, or by the observer who will employ the units to measure things. But relative thresholds—when the notion of a relative threshold is precisely understood—are by definition detectable by the observer only part of the time. Other methods.—There are several precise methods for determining thresholds in addition to the method of limits described above. One of these is called the method of average error,25 and is commonly used to determine the relative threshold. O is required, in several trials, to select a comparison stimulus that appears to him to equal the standard. The results are then processed by one of several statistical procedures. For instance, the standard deviation (SD) of O's choices may be computed and used as the value of the relative threshold. Another method, called the method of constant stimuli,26 is commonly used to determine both absolute and relative thresholds. To determine the RL, E selects several stimuli above and below the threshold and presents them to O several times in random order. At each presentation O is asked to say whether he does or does not perceive the stimulus. The physical magnitude of the stimulus perceived 50 percent of the time is the RL, or absolute threshold. If none of the stimuli is perceived 50 percent of the time (which usually happens), E must use one of several statistical procedures to calculate the value of the stimulus that would be so perceived. These highly technical procedures are not examined here. My comments on the method of limits apply also to these other methods for determining thresholds. With their use the concept of a threshold becomes precise, sophisticated, and numerical. Thresholds 25 26

See Guilford, Psychometric Methods, chap. 4. Ibid., chap. 6.

JND MEASUREMENT OF SENSATIONS

341

cannot be regarded as differences between stimuli, much less as perceptual differences between stimuli. Consequently, there is doubt that they can serve as units of measurement. ARE THERE SENSATION THRESHOLDS?

Thus far we have examined the concept of a threshold in its application to stimuli, which are of course physical entities. Does the concept apply also to those entities called sensations? Are there sensation thresholds? Could we devise experiments for determining sensation thresholds? The answer to these questions depends in part on how the concept of a threshold is understood and on the method for determining the threshold. If the method of just noticeability is employed, thresholds may be expressed either numerically or nonnumerically. Let us begin with the latter. Expressed nonnumerically, the just noticeable weight sensation for O would be a sensation such that O perceives it and fails to perceive every sensation of lesser magnitude. To discover this sensation, E would have to conduct an experiment like the one that produced table 9, except that sensations instead of stimuli are presented to O. E would present O with a descending series of sensations, Y i , Y2, Y s , Y4, asking him whether he does or does not perceive them. If O perceives Y i , and Y s , but fails to perceive V t and Y s , then is the just noticeable sensation. This experiment presupposes (1) that O has sensations that he is unable to perceive whose existence and magnitude E can nevertheless detect. A just noticeable difference between O's sensations would be a difference such that O perceives it and cannot perceive any smaller difference. To discover such a difference, E would have to conduct an experiment like the one that produced table 1 0 , except that sensations instead of stimuli are presented to O. E would select a standard sensation, and a group of greater comparison sensations, Y i , V2, Y s , Y4, ^ 5 , Ye, to present to O along with the standard. In each trial O is asked to say whether the comparison is greater than or equal to the standard. If Y i , and are perceived by O to be greater than and ^4, ^5, and are perceived to equal V s - Y , is a just noticeable difference for O. This experiment presupposes (2) that there are differences between O's sensations which he is unable to perceive whose existence and magnitude E can nevertheless detect.

34*

JND MEASUREMENT OF SENSATIONS

Presuppositions (a) and (2) are highly problematic. Everyone is prepared to admit that O can be presented with weights that he fails to perceive, or weights whose differences he fails to perceive. But it seems virtually a contradiction to say that O can have a weight sensation that he does not perceive, or that he may fail to perceive differences between his weight sensations. Sensations, unlike stimuli, seem to be just the kind of entities about which such mistakes cannot be made. Pains are often taken to be paradigm examples of sensations. Can O have a pain in his tooth and fail to feel it? Can he have a pain in one tooth more intense than a pain in another tooth, and yet feel the pains to be equal? Surely not, we are inclined to say, if O is paying attention to his somatic field. But perhaps pains are not typical in this respect. They are by definition unpleasant; by definition the possessor must react negatively to them (otherwise they would not be called pains). And perhaps part of this reaction is the possessor's noticing or feeling them. Afterimages, on the other hand, can be effectively neutral (neither pleasant nor unpleasant). Can O have an afterimage that he does not see? Surely not, if he is paying attention to the contents of his visual field. Can O have two afterimages that differ in size, color, or shape, and fail to see the difference? Here our inclinations are not so clear: we do not know quite how to answer. The above treatment of the problem of the misperception of sensation is inconclusive, and it must be left that way. Enough has been said to show at least that presuppositions (1) and (2) are highly problematic. There are still other difficulties. How can E know that O has a sensation if O fails to perceive and report it? How can E know that O's sensations are different if O fails to perceive and report a difference? These questions present no difficulty when raised about stimuli. E can detect weights and weight differences as well as O, and better than O if he uses a balance, because weights are public entities. Sensations, however, are private, perceivable only by their possessor. Consequently, the normal tests for sensations and sensation differences are the honest perceptual reports of the possessor. If O honestly reports that he has a sensation, he does have one; if he reports that he does not have a sensation, he does not have one. If O honestly reports that his sensations are different, they are different; if he reports that they are not different, they are not different. These tests cannot be employed to detect sensations and sensation differences that O does not

343

JND MEASUREMENT OF SENSATIONS

perceive. But what other tests are available? Since E cannot himself perceive O's sensations, it seems that he must rely on O ' s perceptual reports for his knowledge of O ' s sensations. The only way to get around this difficulty is to argue that E can infer O's sensations and sensation differences from the stimuli to which O is exposed. The five weight sensations (listed in descending order of m a g n i t u d e ) — ^ 2 , ¥3, ¥4, Y s — u s e d in the experiment to determine O's just noticeable weight sensation are produced in O by having him lift weights (listed in descending order of magnitude) $2, i>3, i»s. Although O is unable to perceive and Y s , we can infer that he has these sensations from the fact that he lifts Su and $5. Since lifting the heavier weights produces sensations (as we know from O's perceptual reports), it seems reasonable to suppose that lifting the lighter weights produces lesser sensations. The stimulus continuum, we can infer, is paralleled by a sensation continuum, even though some of the sensations on this continuum are below the limen for O. From the experiment in which a just noticeable sensation difference for O was discovered, we can infer that, since lifting $8 and produces sensations ^ s and T3, whose difference O can perceive, lifting i>8 and also produces different sensations, Y s and ^4, even though O is unable to perceive the difference. This argument is unconvincing. Even if sensations do lie on a continuum, there is no reason to suppose that it parallels the weight continuum, that is, that the zero points of the two continua lie opposite each other. Weights and weight sensations may be related in a manner represented by the following diagram: Weight sensation Weight

°

° >

There is simply no reason to suppose that lifting an object of any weight whatsoever produces a sensation in O. Some objects are so light that lifting them does not activate any of the nerves in the lifting arm or hand, neither the pressure nerves nor the kinesthetic nerves. If nerves are not activated, sensations are not produced. Furthermore, there seems no compelling reason to suppose that sensations lie on a continuum. Even physics recognizes discontinuous magnitudes, such as the energy of electron shifts in atoms. Sensation magnitude seems to be another likely candidate. It is therefore rash to suppose that be-

344

JND MEASUREMENT OF SENSATIONS

tween every pair of O's sensations there is another possible sensation that would be produced in him by lifting an intermediate weight. It appears, then, that the only acceptable way of determining whether different lifted weights uniformly produce different weight sensations in O is by using the test of his honest perceptual reports. This test shows, of course, that sensation magnitudes are discontinuous (otherwise there would be no relative sensation thresholds). The foregoing difficulties relate to the presuppositions of experiments to determine sensation thresholds which express these thresholds in nonnumerical terms. Experiments that express sensation thresholds in numerical terms also presuppose (i) and (2), and thus inherit all the same difficulties. In addition, they presuppose (3) that sensations are capable of being measured. When the method of just noticeability is used to determine thresholds, the thresholds may be expressed numerically or nonnumerically. When precise and acceptable methods are employed, however, the threshold becomes a numerical value, computed from the numerical data of the experiment. An experiment using the method of limits to determine relative sensation thresholds must produce a table of data like table 1 1 , with magnitudes of weight sensations entered in the left column. These magnitudes must be entered as numerical values, which presupposes that E possesses some method for assigning numerals to O's sensations, some method of sensation measurement. What method of sensation measurement can E employ? Are weight sensations capable of measurement? Methods A, B, and C, described earlier, employ just noticeable differences between sensations as units. Since all these methods are questionable, it is questionable whether sensation thresholds can be expressed numerically. In any event, it is perfectly clear that Methods A, B, and C cannot be used to measure sensations in an experiment that attempts to determine the relative sensation threshold, because the sensation units of these methods are just noticeable differences between sensations. These differences are of approximately the same magnitude as relative sensation thresholds determined by the method of limits; and they are of the same magnitude, since they are the same differences, as relative thresholds determined by the method of just noticeability. Consequently, the units of A , B, and C are not small enough to express sensation differences smaller than relative sensation thresholds. But

JND MEASUREMENT OF SENSATIONS

345

these differences must be expressible if we are to conduct experiments to discover which of O's sensation differences he can and which he cannot perceive. The same point applies to reconstrued A , B, and C, since their unit differences between sensations are of approximately the same (if not the same) magnitude as those in the original methods. If the concept of a threshold is understood precisely (i.e., numerically), it is circular to employ sensation thresholds in an attempt to solve the problem of sensation measurement. Methods A, B, and C employ nonnumerical sensation thresholds as units, and one of them designates the threshold sensation nonnumerically as a zero sensation. Precisely understood, however, sensation thresholds are numerical values, and their determination requires a procedure of sensation measurement. Hence, if we are to devise a method of sensation measurement which employs precise sensation thresholds as units, we must already possess a procedure of sensation measurement. Consequently, we cannot solve the problem of how to measure sensations by employing precise sensation thresholds as units, for that "solution" entails that the problem is already solved, which is circular. Experiments for determining sensation thresholds expressed nonnumerically presuppose (1) and (2). Experiments for determining sensation thresholds expressed numerically presuppose (1), (2), and (3). Each of these presuppositions, however, is highly problematic. For this reason it would seem that the concept of a threshold does not apply to sensations. In any event, it is obvious that psychologists really do not mean to apply it in this way, because no experiment has ever been performed to determine the absolute or relative sensation thresholds of observers. Experiments employing the methods of just noticeability, limits, average error, and constant stimuli are in fact used to discover stimulus thresholds, not sensation thresholds. The fact has not been apparent to some psychologists, owing to their tendency to confuse the stimulus with the sensation it produces. This confusion is effectively corrected by doing what is done above: first describing the use of the various methods to determine stimulus thresholds, and then describing their use to determine sensation thresholds. In this way it becomes apparent that the threshold experiments psychologists perform are intended to discover stimulus thresholds. It is to stimuli that the concept of a threshold is meant to apply and ought to be applied. If this analysis is correct, methods of sensation measurement

346

JND MEASUREMENT OF SENSATIONS

which employ relative sensation thresholds as units (i.e.. Methods A , B, and C) are unacceptable for the most fundamental of reasons: They assume, falsely, that the concept of a threshold applies to sensations. If it does not so apply, it is meaningless to speak of or try to determine sensation thresholds, and meaningless to debate whether they are unequal or otherwise unacceptable. The same criticism does not apply to Methods A, B, and C when they are reconstrued, since they then speak, not of sensation thresholds, but of sensations that correspond to sensation thresholds. Since the reconstrued methods do not presuppose either (i), (2), or (3), at least not necessarily, the objections to these presuppositions cannot be leveled against them. The fundamental objection to the reconstruals—and it applies also to the originals—is to their implicit assumption that weight sensations must be posited in order to explain the phenomena of thresholds. The next section exhibits the problematic nature of this assumption. THE SENSATIONIST EXPLANATION OF THRESHOLDS

I have argued that the concept of a threshold is inapplicable to sensations. What connection, then, do sensations have with thresholds? The answer will be that without sensations, thresholds cannot be explained. It is a fact that observers do not perceive the weight of every object they lift; some objects are too light to be felt. It is also a fact that observers do not feel every difference in weight between objects they lift; some weight differences are too small to be felt. These facts seem to need explanation. Why are observers unable to feel every weight and every weight difference? The answer may be given that some weights are just too light to be felt, and some weight differences are too small to be felt. But this is no answer at all; it merely restates the fact that there are thresholds. It seems that the only way to answer the question is to advance the theory that O perceives weights and weight differences by introspecting his weight sensations and inferring weights and weight differences from them. This theory may be called the sensationist explanation of thresholds. The sensationist explanation has two versions. One of them holds that sensations are a continuous function of their stimuli, that every stimulus produces a sensation, and that different stimuli produce different sensations. The other version holds that sensations are not a continuous function of their stimuli, that some stimuli (those below

JND MEASUREMENT OF SENSATIONS

347

the threshold) do not produce sensations, and that different stimuli sometimes (when their difference is below the threshold) produce the same sensation. Sensations as a continuous function.—Consider first the absolute threshold. O perceives $ 1 , $2, and but fails to perceive the smaller stimuli $4 and $5. These stimuli produce in O sensations Y i , Y 2 , Y s , and ^ s , on the assumption that sensations are a continuous function of stimuli. (Both sensations and stimuli are listed in descending order of magnitude.) Given this assumption, the explanation of the threshold may be completed in either of two ways. We hypothesize (1) that O perceives (notices) Y i , ^ 2 , and and infers a stimulus for each of them, but fails to perceive (notice) Y4 and T s , because they are so small, and infers the absence of stimuli for them. Or we may suppose (2) that O perceives (notices) each of the sensations produced in him but, because and are so small, infers the absence of stimuli for them. Explanation (1) is unacceptable because it is regressive. It holds that O perceives some of his sensations and fails to perceive others. But why should this be so? The theoretical situation here is the same as it is for the perception of stimuli. Why does O perceive $2, and $3 but fail to perceive $4 and $5? It is no answer to reply that $4 and are too small to be perceived, because this reply merely restates the fact that O does not perceive some (the smaller) of the stimuli to which he is exposed. The only genuine explanation seems to require a reference to O's sensations. Similarly, it is no explanation to say that O fails to perceive and Y 5 because some sensations are too small to be perceived, since this reply merely restates the fact to be explained. A genuine explanation of O's inability to perceive sensations must have the same form as a genuine explanation of his inability to perceive stimuli. Consequently, if the explanation of O's inability to perceive $4 and $5 is that he fails to perceive the associated perceptual entities, W* and Vs, the explanation of his inability to perceive ^4 and must be that he fails to perceive some associated perceptual entities, and V s , and the explanation of his inability to perceive V 4 and V s must be that he fails to perceive some associated perceptual entities, V U and and so on ad infinitum. In this way, the current explanation of the absolute threshold leads to an infinite regress, and is therefore unacceptable. To be ac-

348

JND MEASUREMENT OF SENSATIONS

ceptable, explanations must end somewhere. Furthermore, the perceptual entities required at the third and higher levels of the explanatory scheme are unintelligible. What are W* and T's? Are they sensations produced by sensations? What are T"« and Sensations produced by sensations produced by sensations? Surely there are no such sensations. But then what are these strange perceptual entities? The final objection to explanation (i) is directed against its assertion that O has sensations he fails to perceive (notice). The difficulties in this assertion, presented in the preceding section, will be reviewed briefly. An examination of paradigm instances of sensations, such as pains, seems to reveal that sensations are by nature things about which their possessor can make no perceptual mistakes. Furthermore, if O can have sensations he does not perceive, there seems to be no adequate test for the presence of sensations in him. The test cannot be O's perceptual reports, and other tests seem unacceptable. Explanation (2) is not regressive and does not posit any unperceived sensations. It holds that O perceives all the sensations produced in him, even those whose stimuli are below the threshold, but infers that the latter, because of their small size, are not produced by stimuli (an inference that is of course in error). The major objection here is that no good reason is given, or can be given, for this strange inference on O's part. O often infers a stimulus from a small sensation : he does so for sensations whose stimuli are just above the threshold. Why then does he infer no stimulus for other small sensations (those whose stimuli are below the threshold)? Has he learned that sensations below a certain magnitude are not caused by stimuli? Clearly not, since the lesson to be learned here is the opposite one. If in fact lifting a weight, however light it may be, produces a weight sensation in O, he will learn this fact from experience. When he feels a weight sensation, he will see an object in his hand. If he doubts that the object has weight, he can place it on a balance to check. Thus he should learn to infer a weight each time he has a weight sensation. O's inference of no stimulus from very small sensations is therefore inexplicable. The same objections may be made to the continuous-function explanation when it is applied to relative thresholds. Suppose O perceives the stimulus differences, and í>i-$4, but fails to perceive < , the smaller differences, and $i- I 2. On the basic assumption of the continuous-function explanation, these stimulus differences are

JND MEASUREMENT OF SENSATIONS

349

associated with sensation differences, Y1-Y4, Yi-Ys, Yi-Yj. (In ascending order of magnitude, the stimuli are $ 1 , $2, $3, $4, the sensations are Y i , T2, Y 3 , ^4, Y V ) The explanation can be completed in either of two ways. We can hypothesize (1') that O perceives the first two sensation differences and infers that the corresponding stimulus pairs are unequal, but fails to perceive the last two sensation differences because they are so small, and infers that the corresponding stimulus pairs are equal. Or we may suppose (2') that O perceives each of the four sensation differences, but infers that the last two, because they are so small, correspond to stimulus pairs that are equal. These two hypotheses are open to the same objections brought against their counterparts—(1) and (2)—which explained absolute thresholds. Hypothesis (1') is open to the objection that it is regressive. O's failure to perceive the last two sensation differences must be explained in the same way as his failure to perceive the corresponding stimulus differences is explained. The hypothesis is also objectionable because it posits unperceived sensation differences. It would seem that O cannot make mistakes about the equality or the inequality of his sensations. Hypothesis (2') is objectionable because it neither offers nor can offer any reason why O infers stimulus equality from small sensation differences. Sensations as a discontinuous function.—This explanation asserts that if O perceives a stimulus it produces a sensation in him, and that if he fails to perceive a stimulus it fails to produce a sensation in him. If O perceives a difference (inequality) between two stimuli, those stimuli produce different (unequal) sensations in him; if he fails to perceive a difference between two stimuli, those stimuli produce equal sensations in him. Since O fails to perceive stimuli and stimulus differences below the threshold, it follows that not every stimulus produces a sensation in him, and that not every stimulus difference is associated with a sensation difference. That is, sensations are not a continuous function of stimuli. Let us begin with the relative threshold. Suppose O perceives the stimulus differences, and but fails to perceive the smaller differences, i>i-i>3 and (In ascending order of magnitude, the stimuli are $ 1 , $2, £>3, $4, and $5.) On the theory that sensations are a discontinuous function of stimuli, this threshold is explained as follows. The sensation pairs produced by these stimulus

35°

JND MEASUREMENT OF SENSATIONS

pairs axe related in the following w a y : ^ i C Y s , V i K W * , T i = Y 3 , = V2. If the sensations produced by a stimulus pair are unequal, O infers that the stimuli are unequal; if the sensations are equal, O infers that the stimuli are equal. This explanation is immune to the objections brought against its predecessors. It is not regressive, does not hold that O's perception of his sensations can be mistaken, and does not impute any strange or inexplicable inferences to O. What, then, if anything, is wrong with it? A first objection is that the explanation is internally inconsistent. Imagine the following typical situation. O is unable to perceive stimulus differences i-$>2 and p are symbols for one physical dimension, weight, but for two methods of measuring it: the one method employing the jnd as the unit, the other employing the gram. The jnd is, of course, not a unit of weight measurement, but rather an interval or quantity defined by differential responses to weight. The least misleading way of writing Fechner's law, on the suggested interpretation, is N = k log §>, where N means "the number of

JND MEASUREMENT OF SENSATIONS

363

stimulus jnd's $ lies above the zero stimulus" and $ means "the stimulus expressed in number of appropriate units." This statement of the law does not suggest that a sensation dimension is related to a physical dimension, nor that sensations have been measured. Nor does it suggest that one physical dimension is related to another: it does not even presuppose the dubious distinction between psychological and physical dimensions. Therefore, it does not imply that a gap between mind and body has been spanned by a numerical law. I do not know whether Fechner's law, as interpreted above, will be of any use to psychophysicists. It is at least clear that this interpretation removes most of its traditional significance. Since the law no longer spans the mind-body gap, it loses its former metaphysical significance. Since the law no longer relates psychological and physical dimensions, it loses much of its former theoretical significance. For example, a central question in contemporary psychophysics is whether Stevens' power law or Fechner's logarithmic law is the true psychophysical law. This question presupposes that both laws relate psychological and physical dimensions (otherwise the laws would not compete). Consequently, the question simply dissolves when Fechner's law is correctly interpreted. In the next chapter I argue that the question also dissolves under a correct interpretation of Stevens' law, since his law does not relate psychological and physical dimensions either.

9 Ratio Scales of Psychophysics at present is generally regarded as the attempt to measure or scale psychological magnitudes. Very roughly we may distinguish jnd (confusion, discriminability) scales, partition (category, equisection) scales, and ratio (magnitude) scales of psychological magnitudes. The " n e w " psychophysics, for which S. S. Stevens is largely responsible, holds that the ratio scale is most desirable, for the following reasons. It is superior to a jnd scale because it is constructed by a "direct" method, 1 and because jnd's are not "subjectively equal" on so-called prothetic continua.2 It is superior to a partition scale because it enables us to say not only that one psychological entity is greater than another, but also how much greater,8 and because it contains the partition scale.4 * Most of the material in this chapter originally appeared as a monograph, "Introspectionist and Behaviorist Interpretations of Ratio Scales of Perceptual Magnitudes," Psychological Monographs, Bo (1966), 1-32. (Copyright 1966 by the American Psychological Association, and reproduced by permission.) Some additions have been made at the end of the section entitled "The Behaviorist Interpretation," and also at the end of the chapter. A reply by Stevens accompanied the monograph, and the additions are partly in response to his comments. 1 S. S. Stevens, "Measurement and Man," Science, 127 (1958), 387. 2 S . S. Stevens, "A Scale for the Measurement of a Psychological Magnitude: Loudness," Psychological Review, 43 (1936), 411-416; S. S. Stevens, "Biological Transducers," Convention Record of the Institute of Radio Engineers (1954), pp. 30-31; S. S. Stevens, "On the Psychophysical Law," Psychological Review, 64 (1957), 154,172; S. S. Stevens, "The Psychophysics of Sensory Function," American Scientist, 46 (i960), 227; S. S. Stevens, "Ratio Scales, Partition Scales, and Confusion Scales," in Psychological Scaling: Theory and Applications, ed. Harold Gulliksen and Samuel Messick (New York, i960), pp. 57-59; Ira J. Hirsh, The Measurement of Hearing (New York, 1952), pp. 10-11. 8 Stevens, "A Scale for the Measurement of . . . Loudness," pp. 406-407; S. S. Stevens, "The Quantification of Sensation," Daedalus, 88 (1959), 6 1 1 ; S. S. Stevens, "On the New Psychophysics," Scandinavian Journal of Psychology, 1 (i960), 28; Stevens, "Psychophysics of Sensory Function," pp. 228-230. * Stevens, "Ratio Scales, Partition Scales, and Confusion Scales," pp. 35-54.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

365

Furthermore, ratio-scaling procedures have led to the discovery of a psychophysical law of great generality and theoretical power* Stated as a first approximation, the law is

[1] W

=

where V is the psychological magnitude in psychological units, $ is the stimulus magnitude in physical units, and n varies according to the sense modality in question. Formula [1] is a power law and contrasts with Fechner's logarithmic law, [ 2 ] Y = fcl0g*, where Y is the psychological magnitude measured in jnd units, $ is the stimulus magnitude in physical units, and k varies according to the sense modality in question. The new psychophysics claims that [2] is invalid, because the method for obtaining it is indirect and employs jnd's as psychological units. On the other hand, [1] is said to be based on direct methods and consequently to employ psychological units that accurately represent psychological magnitudes.6 Modern psychophysicists usually maintain or imply that they have cast off the dualist metaphysics and the introspectionist methodology that frustrated their predecessors. According to Galanter, The name psychophysics derives from the classical question about the relation between the physical environment and the mind. Today, modern psychophysicists are not professionally concerned with this philosophical issue of the mind-body relation, but rather with the constraints that are placed upon the behavior of a person in his judgments, actions, and so on, by the sea of physical energies that surround him. 7

And Hirsh writes: The influence of behaviorism in American psychology is easily seen in modern psychophysics. We no longer look for relations between stimuli and sensations but rather relations between stimuli and re5

Stevens, "On the Psychophysical Law," p. 162; S. S. Stevens, "Problems and Methods of Psychophysics," Psychological Bulletin, 55 (1958), 192-194; Stevens, "On the New Psychophysics," pp. 28-29; Stevens, "Psychophysics of Sensory Function," pp. 235 ff.; S. S. Stevens, ' T o Honor Fechner and Repeal His Law," Science, 133 (1961), 84; S. S. Stevens, "The Surprising Simplicity of Sensory Metrics," American Psychologist, 17 (1962), 30-32. 8 For reviews of recent developments in psychophysical scaling see Gosta Ekman and Lennart Sjoberg, "Scaling," Annual Review of Psychology, 16 (1965), 451-474; and Joseph L. Zinnes, "Scaling," Annual Review of Psychology, 20 (1969), 447-4787 E. H. Galanter, "Contemporary Psychophysics," in New Directions in Psychology, ed. Roger Brown (New York, 1962), pp. 92-93.

366

RATIO SCALES OF PERCEPTUAL MAGNITUDES

sponses. We can observe responses, the elements of behavior, and measure them, whereas the private sensation, which remains as untouchable as it was in Fechner's day, does not concern us. We do not ask whether or not a man hears a tone. We seek only to find whether or not he responds in a specified way to a tone. We can have measurement, then, on both sides of the psychophysical relation. The "psycho" part refers merely to behavior.8 Such declarations of philosophical enlightenment seem premature. Many of the new psychophysicists officially subscribe to a behaviorist and operationist philosophy, but in their experimental and theoretical work employ introspectionist assumptions, or lapse into an introspectionist view of the nature of psychophysical measurement. They assert that theirs is an attempt to measure behavioral responses, and yet they employ methods whose rationale seems to be that of enabling the experimenter to quantify those private sensations formerly regarded as directly inaccessible. This indictment could be completely substantiated only by considering a large number of scaling methods and the work of a large number of psychophysicists. The analysis here concentrates on ratio-scaling methods and on the work of S. S. Stevens, who is the principal architect of the new psychophysics. The principal feature of the analysis in this chapter is a distinction between introspectionist and behaviorist interpretations of perceptual magnitudes, the psychological magnitudes involved in perception. The two interpretations are sketched in an early section of the chapter, and then presented in detail and criticized in succeeding sections. In the final section the unfortunate consequences of the almost universal failure to distinguish the two interpretations are illustrated and discussed. The analysis indicates that the concept of psychological magnitude is illegitimate, and that the attempt to scale such magnitudes ought to be abandoned. To abandon this attempt is to abandon not only the orientation of Stevens' new psychophysics, but also that of the "old," which Fechner founded. Even if this sweeping conclusion cannot be sustained, the analysis at least shows that neither ratio, partition, nor jnd scales can be properly assessed and compared without distinguishing the introspectionist from the behaviorist view of psychophysical measurement. 8

Hirsh, Measurement

of Hearing, pp. 1 5 - 1 6 .

RATIO SCALES OF PERCEPTUAL MAGNITUDES

367

A SAMPLE RATIO-SCALING EXPERIMENT Ratio scales can be constructed either by numerical estimation methods—magnitude production and magnitude estimation—or by fractionation methods—ratio production and ratio estimation.9 The analysis in this section is based on a sample experiment in which a scale for psychological length is constructed by the method of ratio estimation, but my main conclusions apply equally well to ratio scales constructed by any appropriate method. Although the experiment presented here is imaginary and stylized, its main features have been extracted from actual experiments, if not for length then for similar continua. With only trivial adjustments, my analysis can be applied to any actual ratio-scaling experiment. The sample experiment is labeled M, for "mak scale." The experiment has two parts. In each trial in the first part, the

LENGTH OF STANDARD IN CM

Fig. 6. A plot of data from two hypothetical length-fractionation experiments. The upper solid line represents O's visual estimates of one-half. The lower solid line represents O's visual estimates of one-fourth. The dashed lines are derived by procedures and for purposes explained in a later section. 9

For an exhaustive classification and description of scaling methods see Stevens, "Problems and Methods of Psychophysics," pp. 177-196.

368

RATIO SCALES OF PERCEPTUAL MAGNITUDES

experimenter, E, presents the observer, O, with a standard rod, and asks him to select a comparison rod, $c, that looks one-fourth as long. In each tried in the second part, O is asked to select comparison rods that look one-half as long as the standard. The data thus obtained are represented by the solid lines in figure 6. In both fractionations O consistently underestimates the comparison rod (or, perhaps we should say, overestimates the standard). The equation for the lower line is [3]

= -33

and for the upper line, [4] 1>c = .58 The numerals along each axis of figure 6 represent physical rather than psychological magnitudes. To measure the psychological magnitudes involved in O's perception, a unit of psychological length must be chosen, and the data on which figure 6 is based must be restated in terms of that unit. The choice of a unit of measurement is based primarily on convenience. Let us stipulate, then, that the psychological length associated with a stimulus rod of 100 cm is 100 psychological units, and let us call the unit thus defined the mak.10 This choice of unit determines the first point (marked A in fig. 7) in a plot of psychological length against physical length. Figure 6 shows that O would estimate a 58-cm rod to be half as long as a 100-cm rod. Hence the psychological magnitude associated with the former must be half that associated with the latter. Since the latter is 100 maks, the former must be 50 maks (the point labeled B in fig. 7). Again according to figure 6, O would estimate a 33.6-cm rod to be half as long as a 58-cm rod. Hence, the psychological magnitude associated with the former must be 25 maks, half that associated with the latter (point C in fig. 7). Points D and E, and any further points desired, are obtained by the same method, and a curve is fitted visually to these points. The equation for the curve is [5] Y = S>116, where Y is the psychological magnitude expressed in maks and $ is the physical magnitude expressed in hundreds of centimeters, curve can also be constructed in a similar manner from line [3], by 1 0 For the origins of both the unit and its name see E. P. Reese, T. W. Reese, J. Volkmann, and H. H. Corbin, eds., Psychophysical Research Summary Report, NAVEXOS P-1104, S D C 1 3 1 - 1 - 5 (1953), p. 41.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

369

assuming that when O says one rod looks one-fourth as long as a second the psychological magnitudes associated with the two rods stand in the ratio 1 ¡4. Points a, b and c (fig. 7), which precisely correspond to points A, C, and E of the preceding construction, are thus obtained. The fact that the two psychophysical curves coincide presumably shows that O's one-half and one-fourth estimates of length are made on the same psychological continuum. The other two curves in figure 7 are concave downward. The lower one is constructed from data obtained in an imaginary categoryscaling experiment, in which O is asked to assign numerals from 1 to 1 1 to rod lengths so that the difference between a rod to which 1 is assigned and a rod to which 2 is assigned is the same as the difference between a rod to which 2 is assigned and a rod to which 3 is assigned, and so on. The category scale is represented on the inside left-hand

0

20 40 60 80 100 PHYSICAL LENGTH IN CENTIMETERS

Fig. 7. Three psychological curves for length estimates. Points A-E on the ratio curve are obtained from the upper solid line in figure 1 , points a-c from the lower solid line in figure 1. The category curve represents the data of an imaginary category scaling experiment. The jnd curve represents the data of an imaginary jnd experiment. The relations among the three curves are discussed in later sections.

37°

RATIO SCALES OF PERCEPTUAL MAGNITUDES

ordinate. The upper curve in figure 7 is constructed from data obtained in an imaginary jnd experiment; the number of jnd's is represented on the right-hand ordinate. Although all three curves are based on imaginary data, they are nevertheless typical of the results of actual experiments dealing with length, area, finger span, and so on. 11 It is obvious that any interpretation of experiment M must contain (a) a distinction between psychological ("subjective," "apparent") magnitudes and physical ("objective," "real") magnitudes}2 The physical entities $ 1 , $2, $3, and so on, of which physical length is a magnitude, are associated with psychological entities, Y i , Y2, W3, and so on, of which psychological length is a magnitude. The latter are involved in O's perception of physical entities. (Identity of subscript indicates association of physical entity with psychological entity.) Magnitudes may be classified as intensive or extensive. If psychological length is an intensive magnitude, any can meaningfully be said to be greater than, equal to, or less than another. Consequently, we may ask whether psychological length varies with physical length and, if so, whether it varies inversely or directly. If psychological length is an extensive magnitude, any given V can meaningfully be said to be half as great, twice as great, ten times as great, and so on, as another. Consequently, we may sensibly attempt to measure psychological length in the fullest sense: that is, erect a ratio scale for psychological length and determine precisely how, in terms of its scale and the centimeter scale, it varies with physical length. Experiment M makes such an attempt, and psychophysical law [5] is the result. Any interpretation of experiment M must commit the procedure therein employed to (P) a principle of correspondence between relative psychological length and O's estimates of relative physical length. In the experiment, (a) E measures the physical length of rods to be presented to O; (b) O estimates the relative physical length of these rods; (c) E represents the data thus obtained in figure 6; and (d) E scales psychological length by constructing figure 7 from figure 6. Steps 1 1 See Reese, Reese, Volkmann, and Corbin, Psychophysical Research Summary Report, chaps, v, vi; 5. S. Stevens and E. H. Galanter, "Ratio Scales and Category Scales for a Dozen Perceptual Continua," Journal of Experimental Psychology, 54 (1957), 377-411; S. S. Stevens and G. Stone, "Finger Span: Ratio Scale, Category Scale, and JND Scale," Journal of Experimental Psychology, 57 (1959), 91-95. 1 2 See J. P. Guilford, Psychometric Methods (rev. ed.; New York, 1954), p. 21.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

371

(a), (b), and (c) are straightforward enough, but (d) is puzzling. How, from the data of physical measurement and physical estimates, is a psychological magnitude scaled? Such scaling will seem to be a conjuring trick unless certain of its assumptions are brought to light. Chief among these is the principle of correspondence, the general statement of which is: If O estimates that $2, $3, and so on, stand in the physical relation R, then Y i , ^2, ¥3, and so on, stand in the psychological relation R. The instance of this principle used in constructing a jnd scale is (A): If O estimates that $ 1 and $2 are just noticeably different and that $2 and are just noticeably different, then the interval between T i and is equal to that between and Y3. The instance used in constructing a partition scale is (B): If O estimates that the interval between and equals that between and $3, then the interval between and equals that between W* and Vs. The instance used in constructing a ratio scale is (C): If O estimates that and stand in the same ratio as $2 and $3, then Y i and ^ 2 stand in the same ratio as ^2 and Vs. The more specific instance used in constructing the ratio scale in experiment M is (Ci): If O estimates that and $2 stand in the ratio m:n, then Y i and Y2 stand in the ratio m:n.ia The method of numerical estimation also employs a principle of correspondence intimately related to (C). The principle is (D): If O assigns to $ 1 , $2, $3, and so on, the numerals m, n, o, and so on, respectively, then Y i = k(m/n) Y2, V2 = k(n/o) Vs, and so on, where either k = 1 or k ¥= 1. If we make the standard assumption that k = 1, the principle becomes (Di): If O assigns to $ 1 , $2, $3, and so on, the numerals m, n, o, and so on, respectively then Y i , V2, Y3, and so on, stand in the ratios m :n, n :o, and so on. The introspectionist interpretation of experiment M holds that (Vi) T ' s and their magnitudes are privately observable and that the principle of correspondence embodies a theory of O's perceptual mechanism. This interpretation preserves a simple, attractive, and familiar view of the nature of psychophysics. Psychological entities are held to occupy a private realm distinct from the public realm occupied by ls

For examples of the explicit use of (Ci) see R. S. Harper and S. S. Stevens, " A Psychological Scale of Weight and a Formula for Its Derivation," American Journal of Psychology, 61 (1948), 345; and T. W. Reese, "The Application of the Theory of Physical Measurement to the Measurement of Psychological Magnitudes," Psychological Monographs, 55 (1943), 22-23.

372

RATIO SCALES OF PERCEPTUAL MAGNITUDES

physical entities. Nonetheless, psychological magnitudes are as "empirically real" as and no less fundamental than physical magnitudes. Consequently, it is a legitimate scientific enterprise to try to measure V magnitudes, much as we measure $ magnitudes (e.g., with a meterstick), and to attempt to discover the mathematical relation between Y magnitudes and $ magnitudes, just as we attempt to discover the mathematical relation between two $ magnitudes. Psychophysics is thus "an exact theory of the functionally dependent relations of body and soul,"14 the attempt to extend the best procedures of physical science into the psychological realm. Another advantage is that the introspectionist interpretation leaves no doubt that M is a psychological experiment. How do the physical measurements and physical estimates obtained in M become transformed into a scale of psychological length? The introspectionist answers that O perceives psychological entities by some inner sense and uses these inner estimates to make outer estimates. Although O reports only the latter, E can make use of the former by hypothesizing a certain perceptual mechanism inO. This hypothesis, however, produces one of the disadvantages in the introspectionist interpretation. The introspectionist assumes that O visually estimates the length of rods, «fc's, by introspecting the magnitude of psychological entities, Y's. But it is doubtful that any perceptual mechanism of this sort is at work in O when he estimates rod length. Putting the difficulty in a different way, principle of correspondence (Ci) is based on premises describing O's perceptual mechanism which are entirely problematic. These premises are stated and thoroughly examined in the sequel. A related disadvantage is that the introspectionist interpretation does not seem to fit the phenomenological facts of experiment M. O observes rods and estimates their relative length, but there is no phenomenological evidence that he observes private, psychological entities and estimates their magnitude. A further disadvantage is that Y's are held to be private, hidden from the view of everyone but O. E must therefore rely on O's unconfirmed "V estimates in constructing a ratio scale of psychological magnitude. Psychological magnitudes thus seem to be placed outside the pale of "objective" scientific investigation and measurement. 14

G. T. Fechner, Elements of Psychophysics,

1966), 1,7.

trans. Helmut E. Adler (New York,

RATIO SCALES OF PERCEPTUAL MAGNITUDES

373

The behaviorist interpretation holds that (Y2) Y ' s and their magnitudes are nonobservable and that the principle of correspondence is true by definition. This interpretation is consistent with the phenomenological fact that experiment M seems to require observation of physical entities alone. A further advantage is that psychological entities are no longer located in some private realm, directly accessible only to O. Now they can be regarded as generally available to scientists, like any other magnitude capable of metricization and scientific treatment. A price must be paid, however, for these advantages. The distinction between psychological and physical magnitudes loses its sharpness and obviousness. It is now seen as unwise to think of experiment M as an attempt to discover the relation between mind and body, since the dualism of parallel realms implicit in such a characterization has been brought into question. The dualist view implies that psychological entities are, although private, as "empirically real" as physical entities. But the behaviorist holds that they are theoretical constructs or "fictions," conceptual inventions of the experimenter devised for some anticipated scientific use. The principle of correspondence, (Ci), is the tool for building these constructs. It is a stipulative definition, not a description of some mechanism underlying O's perception of physical length. And the psychophysical law, [5], made possible by these conceptual maneuvers must be regarded, not as an explanation, but as a description of O's length perceptions. The T ' s thus become a useful but dispensable fagon de parler. THE INTROSPECTIONIST INTERPRETATION THE NATURE OF PSYCHOLOGICAL ENTITIES

One difficulty for the introspectionist interpretation arises from its assertion that O observes private entities of which psychological length is a magnitude. O sees rods, walls, and the experimenter; he feels his chair and other physical objects; he hears the sound of E's voice; and so on. This description of what O observes does not mention any private, psychological entities. If O does observe such entities, how does he do it: by seeing them, smelling them, "intuiting" them? This difficulty cannot be assessed until the notion of a psychological entity is given more content. Several suggestions deserve consideration.

374

RATIO SCALES OF PERCEPTUAL MAGNITUDES

The first suggestion is that Y ' s are visual sensations: private mental events or processes that occur within O during rod perception. Thus briefly stated, the suggestion remains obscure, for no instances of visual sensation and no reason to believe they exist have as yet been given. The obscurity can be removed by offering visual afterimages as paradigm examples of visual sensations. Then, to say of O that he has a visual sensation is to say that he has a visual afterimage, or something like a visual afterimage. A clear meaning is thus given to the assertion that visual sensations are private. O's afterimages are logically private, since it is a logical, not merely an empirical, fact that his afterimages can be perceived only by him. Furthermore, the mode of perception now becomes clear: visual afterimages and whatever is like them are seen. A second suggestion distinguishes between psychological and physical magnitudes, but not between psychological and physical entities. The psychological length of rods is their length as perceived by O ; their physical length is their length as measured by E. And Y i and refer, not to different entities, but to different aspects of the same entity. As it stands, this suggestion is unclear, since we still do not know what "length as perceived by O " means. One way of removing the unclarity is to suggest that perceived length is the lengthin-his-visual-field of rods seen by O. (Other ways of removing the unclarity lead into the behaviorist interpretation.) This suggestion preserves the logical privacy of psychological magnitudes, since it is a logical fact that only O can perceive the length-in-his-own-visual-field of a rod. It does not preserve the privacy (or the separateness) of psychological entities. In addition, the suggestion implies that psychological length is perceived by sight. A third suggestion holds that Y magnitudes are magnitudes of physiological processes occurring within O during rod perception. These processes may be specified as retinal processes (length or area of retinal stimulation, retinal electrical potential, etc.); as optic-nerve processes (frequency of nerve impulses, number of activated fibers, etc.); or as brain processes (area of stimulation in the occipital lobes, electrical potential in the lobes, etc.). It is not so strange as it may seem to classify this suggestion as introspectionist, for, like the other two, it contends that Y magnitudes are privately observed by O. Unlike the others, it does not make clear by what faculty Y ' s are perceived. Are optic-nerve impulses felt? Are retinal processes seen? Furthermore,

RATIO SCALES OF PERCEPTUAL MAGNITUDES

375

physiological processes, unlike visual sensations, are contingently rather than logically private. Although it may in fact be true that only O perceives the retinal processes occurring to him, these processes can be observed by other perceivers, either now or in the future, by means of suitable instruments. Each of the three suggestions above must be rejected for the same two reasons. First, there is no phenomenological evidence whatsoever that O observes, during experiment M, any of the magnitudes suggested. In regard to the first suggestion, no afterimages are induced in O ; he sees nothing like an afterimage. The complainant who says that an afterimage is a poor paradigm for a visual sensation must produce a better one, on pain of leaving the notion of a visual sensation obscure. In defense of the second suggestion, it may be said that any rod seen by O must have a length-in-his-visual-field. Even if we concede this less than clear contention, still there is no reason to believe that O is aware of every—or even of tiny—rod's length-in-his-visual-field. He makes no reports of and seems to pay no attention to such magnitudes. From the objector who complains that length-in-the-visual-field is a poor paradigm for the notion of apparent length, we must demand a better one, else the notion of apparent length will remain obscure. As for the third suggestion, one way of emphasizing that O perceives no retinal, nerve, or brain processes is to point out that an observer who had never heard of retinas, optic nerves, or brains could function quite as well in experiment M as a professional physiologist. The general criticism in this paragraph can be reinforced by contrasting the kind of instructions actually given in M ("Select the rod that is half as long as the one I now hold up") with the instructions that would be required in any of the suggestions mentioned (e.g., "Select the rod that has a size in your visual field half that of the rod I now hold up"). The new instructions would produce entirely different experiments. Second, experiment M cannot be construed as a procedure for measuring the Y ' s mentioned in any of the three suggestions. The standard procedures for measuring the size of an afterimage, or the size-in-the-visual-field of a rod, require O to view the image or the rod against a screen at a fixed distance and to indicate the area on this screen occluded by the image or the rod. E can then express the size of the afterimage, or the size-in-O's-visual-field of the rod, in terms of the size in centimeters or inches of the occluded area. No screen is

376

RATIO SCALES OF PERCEPTUAL MAGNITUDES

used in experiment M and no determinations of occluded area are made. It is even more obvious that M is not a procedure for measuring physiological processes. The area of retinal stimulation is measured by applying an electroretinoscope, or, less directly, by using computations based on the construction of the eye and the laws of optics. Instrumental and computational methods are also employed in measuring nerve impulses, and brain processes. None of these methods is found in M. The foregoing analysis shows that the concept of a psychological entity is unclear and scientifically unacceptable. When we try to understand the concept by producing possible instances—afterimages, things in the visual field, physiological processes—the two objections just presented become applicable. These objections are so conclusive, and in a way so obvious, as to make it seem that psychological entities could not be any of the things suggested. Thus the concept dissolves like a mist exposed to the light of day. And it is a mist, a vague penumbra, in the thinking of the introspectionist. He has some vague notion of psychological entities—"sensations," he usually calls them—located in a private mental realm within the perceiver. But they are not afterimages, or things in the visual field, or physiological processes, all of which are measurable. Even if this obscurity in the concept is overlooked, sensations still seem to be beyond the reach of scientific investigation and measurement. O's sensations cannot be measured by E because of their privacy. But they do not seem capable of measurement even by O, since they do not seem to be the kind of entities to which the operations required in measurement can be applied. In chapter 5 measurement is defined as a procedure for assigning numerals to objects within a dimension by an empirical operation of comparing the objects with a unit or units. For physical length the operation of comparison is laying a ruler alongside the object; for physical weight it is placing the object on a balance. What operation of comparison with a unit can O apply to his own sensations? Faced with this problem, the introspectionist may concede that the mak scale does not qualify as sensation measurement under the above definition of that term; but he may insist that the definition is too narrow. Measurement, he may say, is any assignment of numerals to a class of objects which represents the magnitude ratios of the objects. This definition entails that it is the result and not the manner of

RATIO SCALES OF PERCEPTUAL MAGNITUDES

377

numeral assignment which is important. Numerals may be assigned by means of an operation of comparing objects with a unit, or in a quite different manner, as in experiment M . A s long as the result is an assignment that represents magnitude ratios, measurement can be said to have taken place. The content of this reply is suggested by Stevens' discussions of the nature of measurement. A s noted in chapter 5, Stevens maintains that measurement is "the assignment of numerals to aspects of objects or events according to rule." O n the basis of four different rules for numeral assignment he distinguishes four types of scales: nominal, ordinal, interval, and ratio scales. But the rules mentioned describe only the result of numeral assignment. The rule for an interval scale is: Assign numerals so as to represent equal intervals. The rule for a ratio scale is: Assign numerals so as to represent equal ratios. Stevens denies that a physical operation of addition is required to create even a ratio scale. 15 A n d he says that Fechner's mistake was in believing that "measurement must be reducible to counting [the constituents of sensation]." 16 The suggestion is that any method that produces a ratio scale (or any of the other types) is properly regarded as measurement, which is just to say that it is the result and not the manner of a procedure of numeral assignment which makes it one of measurement. Since Stevens officially subscribes to a behaviorist position, he would probably deny vehemently that his theory of measurement suggests an introspectionist interpretation of psychological scaling. Nevertheless, see the extensive discussion of his writings in a later section. Whether the introspectionist reply can or cannot be found in Stevens, it is important to formulate it and to see that it is unacceptable, unacceptable because it confuses sensation estimation with sensation measurement, and illegitimately substitutes the one for the other. This distinction, discussed in earlier chapters, is presented here simply by means of an analogy. Suppose that O sees from a distance the shadows cast by rods on a wall, but that E can neither see nor apply his meterstick to the shadows. Wishing nevertheless to measure them, and believing that O is able to make accurate estimates of shadow length, 1 5 S. S. Stevens, "Measurement, Psychophysics, and Utility," in Measurement: Definitions and Theories, ed. C. West Churchman and Philburn Ratoosh (New York, 1959), p. 24; S. S. Stevens, "Mathematics, Measurement, and Psychophysics," in Handbook of Experimental Psychology, ed. S. S. Stevens (New York, 1951), pp. 25-29. 1 6 Stevens, "Quantification of Sensation," pp. 614-615.

37»

RATIO SCALES OF PERCEPTUAL MAGNITUDES

E uses him in a two-part experiment. In each trial in the first part he asks O to locate the rod, $1, that casts a shadow, ^ i , one-fourth as long as the shadow, T i , cast by rod $2. In each trial in the second part he asks O to designate the rod that casts a shadow one-half as long as the shadow cast by a given rod. With these data in hand, E constructs first a figure like figure 6 and then one like figure 7, and announces with elation that he has measured the inaccessible shadows. It is clear, however, that shadows have not been measured. The shadow observer determines shadow length, not by measurement, but by direct estimation. The shadow experimenter determines shadow length by relying on the observer's estimates, that is, by indirect estimation. Similarly, the sensation observer determines the magnitude of his sensations, not by measurement, but by direct estimation. And the sensation experimenter determines the magnitude of O's sensations by relying on the latter's sensation estimates, that is, by indirect estimation. To call any of these procedures measurement is to violate a distinction that must be preserved by any acceptable definition of the term, the distinction between estimates and measurements. THE PRINCIPLE OF CORRESPONDENCE

According to the introspectionist interpretation, the principle of correspondence is derived from certain hypotheses about the nature of O's perceptual mechanism. The derivation for the principle employed in obtaining line [5] from line [4] may be illustrated as follows: (Ca) If O (indirectly) estimates that and $2 stand in the ratio 1. :2, then O (directly) estimates that Y i and stand in the ratio 1 :z; (Cb) if O (directly) estimates that T i and ^2 stand in the ratio 1:2, then Y i and Ya stand in the ratio 1 :2; (Ci) hence, if O (indirectly) estimates that and $2 stand in the ratio 1:2, then and stand in the ratio 1:2. Informally, (Ca) says that O's estimates of psychological ratios are the same as his estimates of associated physical ratios; (Cb) asserts that O's estimates of psychological ratios are accurate; and (Ci) concludes that psychological ratios correspond to estimated physical ratios. To understand this argument and the perceptual mechanism it describes, the reader may find it helpful to employ the shadow analogy presented earlier, that is, to think of *F's as shadows observed only by O and $'s as rods that cast the shadows. Premise (Cb).—This

premise is often assumed, but rarely justi-

RATIO SCALES OF PERCEPTUAL MAGNITUDES

379

fied.17 How can we determine whether O's psychological estimates are accurate? It would seem that the method must be analogous to that for determining whether estimates of physical ratios are accurate. If we wish to know whether O's estimate that a given rod is half as long as another is accurate, we simply measure the physical lengths of the two rods and then compare measured physical length with estimated physical length. If the ratio as determined by measurement is the same as the ratio as estimated by O then O's estimate is accurate; otherwise it is inaccurate. The rods must be measured by a procedure that does not depend on O's estimates of rod lengths, for otherwise the accuracy test will be circular and will not be a genuine test. By analogy, a test of accuracy for O's estimates of psychological length seems to require a procedure for measuring psychological length. After the procedure has been applied, the measured psychological length must be compared with the estimated psychological length. If the test is to be noncircular, the procedure for measuring psychological length must be independent of O's estimates of psychological length. The mak-scale procedure is thus precluded, since it relies on O's psychological estimates. If, as it appears, no other procedure is available, (Cb) is unverifiable. Let us recapitulate and broaden the difficulty. Experiment M is put forward by its introspectionist proponents as a method for measuring psychological length. The experiment rests on the assumption that O's estimates of psychological length are accurate. This assumption cannot be verified, however, without a method for measuring psychological length which is prior to and independent of M. Hence, to recommend M as a method for measuring psychological length begs the question of whether psychological length is measurable. Since physiological processes are only contingently, and not necessarily, private, the above objection does not apply unmodified to the suggestion that O estimates physiological processes in M. When O's estimates of psychological length are construed, for instance, as estimates of nerve-impulse frequency, we can test their accuracy in 17 Stevens, " A Scale for the Measurement of . . . Loudness/' p. 408; S. S. Stevens, "The Direct Estimation of Sensory Magnitudes—Loudness," American Journal of Psychology, 69 (1956), 2-3,18, 23; J. P. Guilford and H. F. Dingman, "A Validation Study of Ratio-Judgement Methods," American Journal of Psychology, 67 (1954), 395; Stevens, "Mathematics, Measurement, and Psychophysics," pp. 40-41; Stevens, "Biological Transducers," p. 30.

380

RATIO SCALES OF PERCEPTUAL MAGNITUDES

a noncircular manner by connecting an oscilloscope to electrodes placed on the nerve in question. Even so, related difficulties arise. First, O's (putative) nerve-impulse estimates have not in fact been tested for accuracy. Therefore the assumption of their accuracy is, although not unverifiable, unverified. Second, in testing O's (putative) nerve-impulse estimates for accuracy—which must be done before the mak scale can be accepted—we measure with instruments the very magnitude that M is supposed to scale. M is therefore superfluous, since in justifying it we accomplish its purpose. Third, M cannot be construed as a method for measuring, as opposed to estimating, nerve-impulse frequency. The only possible justification for employing M to quantify a physiological magnitude is that the magnitude is presently inaccessible to existing intruments and physiological techniques, and that we may scale the magnitude by means of O's inner observations of it until improved instrumentation and technique make reliance on such a second-best method unnecessary. But this justification clearly implies that experiment M provides us only with a method for estimating a physiological magnitude (allegedly) observed by O, and that measurement of such magnitudes is accomplished by instruments and physiological techniques. The importance of the distinction between estimation and measurement has already been sufficiently emphasized. Premise (Ca).—This premise says that O's estimates of psychological ratios are the same as his estimates of the associated physical ratios. But what is its basis? Remember that O provides E with estimates only of physical magnitudes in experiment M. He does not say, " Y i is half as great as but rather, "The length of this rod ($1) is half as great as the length of that rod ($2)." Why assume that when O makes this latter report Y i and ^ 2 stand in the ratio 1 : 2 ? Perhaps he has learned to say " $ 1 is half as great as $ 2 " when T i and ^ 2 stand in the ratio 1:3. 1 8 If so, and if his Y estimates are accurate, the mak scale in figure 7 does not correctly represent ^ ratios. For, on this assumption, point B should be placed at 33.3 on the y-axis, point C at 1 1 . 1 , and point D at 3.7, thus producing a different scale of psychological magnitude, as well as a different psychophysical law. We must know precisely how O's internal estimates relate to his 18 W. R. Gamer ("A Technique and a Scale for Loudness Measurement," Journal of the Acoustical Society of America, 26 [1954], 74) is the only experimenter known to me who seems to consider such possibilities.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

381

external estimates in order to obtain the correct curve in figure 7. But no one possesses this knowledge at present, and there is no clear w a y of obtaining it. The introspectionist holds that Y magnitudes are private to the observer. How, then, can E discover which V estimates serve as a basis for O ' s $ estimates of one-half? It seems that the only conclusive w a y is to ask O : " W h e n you estimate that $ 1 and $2 stand in the ratio 1:2, what estimate do you make of the relative magnitude of Y i and Y 2 ? " T o see that the meaning of this question is not clear w e need only phrase it in accordance with any of the suggestions regarding the nature of psychological entities mentioned earlier: " W h e n you estimate that a given rod is half as long as another, what estimate do you make of your afterimages (sizes in your visual field, areas of stimulation on your retina)?" Sophisticated observers, as well as naive ones, would be at a complete loss in the face of such questions. It is important to realize that neither intraobserver agreement nor interobserver agreement can be used to determine the truth or falsity of ( C i ) . There is an inclination to suppose that if several observers estimate that and $2 stand in the ratio m:n, then and stand in the same ratio. There is an even stronger inclination to suppose that if a single observer estimates on several occasions that and i>2 stand in the ratio m-.n, then Y i and ^ 2 stand in the same ratio. But how are these assumptions to be justified? It is possible for several observers to estimate that and $2 stand in the ratio m:n when Y i and ^"2 stand in the ratio m :n for one observer, n :o for another, p :r for a third, and so on. It is possible for a single observer to estimate that and $2 stand in the ratio m:n when ^Fi and ^2 stand in the ratio m:rt on one occasion, n:o on another, p:r on a third, and so on. These possibilities can be ruled out only by comparing estimated physical ratios with actual psychological ratios. Such comparison is possible only if there is some way of determining psychological ratios which is independent of the observer's estimates. N o method of this type is available. Nor is one possible if psychological entities are logically private. For if they are, then any method for determining their actual magnitude must rely on the observer's estimates of their magnitude, and is therefore viciously circular. It will be said that misleading estimates like those imagined above occur only when the observer attends to the stimulus rather than to the sensation, that is, when he commits the "stimulus error"; and that

382

RATIO SCALES OF PERCEPTUAL MAGNITUDES

they can be avoided by giving observers instructions to attend only to the sensation. But how can E be certain that O has followed these instructions? Again, it is impossible without an adequate, noncircular method for comparing psychological magnitudes with estimated physical magnitudes. Whatever the instructions to the observer, intraobserver agreement and interobserver agreement provide no better grounds for thinking that O's estimates accurately reflect psychological length ratios than they do for thinking that his estimates accurately reflect physical length ratios. Consequently, statements like the following are erroneous: "In scaling experiments we are forced to assume the uncontrolled ability of the subject accurately to report his sensations. . . . the reproducibility of the data upon repetition of [the] experiment lends some support to this assumption."19 THE PURPOSE OF RATIO SCALING

What, on the introspectionist interpretation, is the purpose of constructing the mak scale of psychological length? To some the answer may seem obvious. Many, if not all, empirical scientific laws are statements relating two or more variables that are defined independently of one another. And laws that supply the mathematical relation are obviously more useful than those that do not. For instance, [6] V = JcT is the physical law relating the volume and the temperature of a gas where pressure is held constant. No further justification is needed for measuring pressure and temperature than that it makes the formulation and verification of [6] possible. And the attempt to discover [6] needs no justification. Similarly, psychophysical law [5] relates two variables, except that one of them is psychological. Law [5] tells us how psychological length varies with physical length when the experimenter's instructions to the observer, as well as certain other elements of the perceptual situation, are held constant. Measuring physical length by a meterstick, and measuring psychological length by the mak scale, are adequately justified by pointing out that they make the discovery of [5] possible. The above argument is dubious, first of all, in its contention that 19

Galanter, "Contemporary Psychophysics," p. 142.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

383

the attempt to discover correlations between two or more variables needs no justification. Surely some correlations are more useful than others, and surely some are completely useless. But let this pass. The more important criticism for our purpose is that psychological length does not seem to be a magnitude like pressure or temperature. Pressure and temperature are observable, "empirically real" magnitudes, whereas psychological length does not seem to be so. In addition, pressure is defined independently of temperature, and vice versa; but psychological length does not seem to be definable independently of physical length. These points become clear in the discussion of the behaviorist interpretation to follow. A second introspectionist justification for measuring psychological length points out that (Ca)-(Cb)-(Ci) is a theory of O's perceptual mechanism, a theory designed to explain how O visually estimates length; and that psychophysical law [5], which is the mathematical completion of this theory, depends on the mak scale. The argument for this theory of O's perceptual mechanism is that failure to accept it leaves unexplained O's ability to make reliable estimates of physical length. This argument is far from conclusive. In the first place, it is doubtful, as I have argued, that (Ca) and (Cb) are verifiable. If they are not, the theory in which they figure is not scientifically respectable. Second, alternative theories of the same type with equal explanatory power can be obtained by replacing (Cb) with some other assumption, an assumption that physical estimates of 1 : 2 are based on psychological estimates of 1 or 1 ¡4, or 2 ¡3, and so on. Third, it has yet to be shown that any of the alternative theories just indicated is required to explain O's ability to make estimates of relative physical length. Why must we suppose that O bases his estimates of physical ratios on estimates of psychological ratios? Why is the following explanation not sufficient? O's eyes are normal, and he has learned how rods look when they stand in the ratio 1 : 2 ; hence, when he looks at rods he can say with some accuracy whether or not they stand in that ratio. That this explanation implies no perceptual mechanism like that in (Ca)-(Cb)-(Ci) is not enough to reject it. It may be well to include a warning against attempting an epistemological justification of ratio scaling. The essential contention of the introspectionist theory of O's perceptual mechanism is that O bases

384

RATIO SCALES OF PERCEPTUAL MAGNITUDES

his (always indirect) estimates of physical entities on his (always direct) estimates of psychological entities. This kind of perceptual theory is often subsumed under or associated with the philosophical, epistemological theory that knowers base their (always indirect) knowledge of the external environment on their (always direct) knowledge of private, internal processes, these last being caused results or at least reflections of the environment. This epistemological theory is clearly dualist—since it posits parallel psychological and physical realms— and introspectionist—since O is believed to know the internal realm by some process of inner perception, that is, by "introspection." The theory seems both to establish our knowledge of the external world and to explain how we obtain it, and it has, as a consequence, attracted psychologists and philosophers for centuries. But there are problems, the most important of which is an instance of the skeptical problem of solipsism. If O directly perceives only internal entities, how can he know that his inferences concerning the existence and character (physical length, for instance) of external entities are accurate? Some theorists may suppose that this problem is solved by the discovery of such laws as [5], that if O learns from E the precise relation between his internal entities (Y's) and external entities ($'s), he can make reliable inferences from the one to the other. But the skeptical problem that confronts O also confronts E. E also directly perceives only internal entities; consequently, his inferences concerning the physical length of the experimental rods are equally problematic. Since E must determine the physical length of the experimental rods in order to establish psychophysical law [5], and since E's physical length inferences are just as problematic as O's, O cannot rely on the validity of [5] to support his own inferences concerning physical length. The blind cannot be led by the blind.20 It is no reply to say that E measures the rods—with a meterstick, say—and thus insures the accuracy of his determination of their length. E's measurements are accurate only if he correctly infers the physical length of meterstick segments from their psychological length. And how can he know these inferences are correct? If, as it appears, the solipsist problem arises in an introspectionist 20

This difficulty is related to the "psychologist's circle," discussed by Edwin G. Boring, "The Psychologist's Circle," Psychological Review, 38 (1931), 1 7 7 - 1 8 2 ; Gustav Bergmann and K. W. Spence, "The Logic of Psychophysical Measurement," Psychological Review, 5 1 (1944), 2-5; and others.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

385

but not in a behaviorist interpretation of psychophysical measurement, then this is clearly an argument in favor of the latter. THE BEHAVIORIST INTERPRETATION THE PRINCIPLE OF CORRESPONDENCE

It is often said that scales like the one constructed in experiment M are "scales of observer response." This statement is either imprecise or false, for it suggests that psychological magnitudes are literally magnitudes of O's responses: the loudness of his verbal reports, or their pitch, or the time required to make them, or some other magnitude. Obviously none of these suggestions is acceptable. There is no reason to think that the loudness, pitch, or duration of O's verbal reports has any interesting or systematic relation either to the length of the rods or to estimates of their length. The behaviorist must not identify psychological length with some magnitude of O's responses; rather, he must define psychological length in terms of O's responses in some useful manner. Many of Stevens' remarks suggest the erroneous identification; for example, ". . . brightness . . . [is] the name for a response of a human organism to an external configuration of the environment."21 Since every magnitude, psychological or otherwise, is defined by the relational terms "greater than," "equal to", and "less than," we must, in defining psychological length, provide a rule for the application of at least these three terms. Such a rule is contained in the following definition. Definition (i). If O estimates that is (physically) greater than (equal to, less than) $2, then Y i is (psychologically) greater than (equal to, less than) Ya. This statement defines psychological length as an intensive magnitude, but fails to give it all the features of the extensive magnitude scaled in experiment M. To generate an extensive magnitude a definition of psychological ratios is also required. Definition (ii). If O estimates that and $2 stand in the (physical) ratio n-.m, then Y i and ^ 2 stand in the (psychological) ratio n-.m. 21

Stevens, "Measurement, Psychophysics, and Utility," p. 52.

386

RATIO SCALES OF PERCEPTUAL MAGNITUDES

Definitions (i) and (ii) can be used, together with an arbitrarily chosen unit of psychological length, to construct the ratio scale of psychological length in figure 7. Definition (ii) is, of course, the principle of correspondence encountered earlier. By making the principle a definition, however, the behaviorist denies that it requires a supporting argument such as (Ca)-(Cb)-(Ci), a theory of O's perceptual mechanism. I have argued that (Ca) is dubious and apparently unverifiable, that (Cb) is dubious and potentially circular, and that there are alternative theories to the one embodied in (Ca)-(Cb)-(Ci). The behaviorist is unaffected by these objections, since he subscribes neither to (Ca) and (Cb), nor to any theory of O's perceptual mechanism which posits internal estimates as the basis for external estimates. He simply obtains O's numerical estimates of rod length, and then constructs the mak scale as a means of expressing mathematically the relation between those estimates and actual rod length. Although immune to some objections, the behaviorist interpretation of the principle of correspondence may be open to others. If the principle is merely a definition, why should we accept it? The reply cannot be that O's estimates of $ ratios are based on his estimates of Y ratios and that his estimates of Y ratios are accurate. To make this reply would be to fall back on the introspectionist view. Nor can the behaviorist maintain that (ii) is a natural definition, a definition of ordinary terms, like " A vixen is a female fox." Psychological length is a technical notion which the behaviorist psychophysicist introduces for extraordinary purposes. Definition (ii) must, therefore, be regarded as a stipulative definition, which can be supported only by appealing to its scientific usefulness. Whether it is in fact useful is discussed in the sequel. Definitions (i) and (ii) are examples of what are sometimes called "operational definitions" of a psychological magnitude. Most behaviorist psychophysicists call for such definitions, assure the reader that they can be provided, and then fail to provide them.22 In failing to provide explicit statements of the definitions, the psychophysicist risks 22 S. S. Stevens, "The Operational Definition of Psychological Concepts," Psychological Review, 42 (1935), 517-527, illustrates this malpractice. Gunnar Goude, On Fundamental Measurement in Psychology (Stockholm, 1962), pp. 28-29, m a y be an exception.

387

RATIO SCALES OF PERCEPTUAL MAGNITUDES

overlooking several troublesome but extremely important questions concerning the behaviorist interpretation. One such question is: Are there as many types of psychological magnitude as there are types of estimate? Are psychological magnitudes constructed from different types of estimate incomparable? Suppose O estimates in a halving experiment for length that [7] - - V2 í>2 and $2 = V2 $3, and in an experiment requiring length estimates of one-third, he says that

[8] $1 = 'A

í>3.

Together with Definition (ii), [7] entails that [9] = 1/2 W2 and W2 = Vi Together with certain elementary rules of algebra, [9] entails that [10] T , = 1/4 T 3 . On the other hand, [8] together with Definition (ii) entails that [11] V 1 = H Ts. Apparently [10] and [ 1 1 ] are inconsistent; consequently, the estimates in [7] and [8] apparently conflict. The behaviorist may try to remove the inconsistency by maintaining (a) that Definition (ii) is not applicable to all O's fractionations, (b) that the elementary laws of algebra do not apply to psychological magnitudes, (c) that O is mistaken in some of his estimates of psychological length, or (d) that [10] and [xi] describe two different psychological magnitudes. Solution (a) is unacceptable, since it undermines the definition that makes the construction of ratio scales of psychological length possible. Solution (b) is also unacceptable, since it implies that the numerals assigned to psychological entities in experiments like M do not represent ratios. Solution (c) is completely out of the question, since it carries the implication that psychological entities are observable, and thus plunges us back into an introspectionist framework. Only solution (d) remains. The estimates on which figure 6 is based do not lead to the kind of inconsistency illustrated above; but there is no assurance that all or even most actual experiments will be like M in this respect. It is always possible—indeed, it is likely—that O will make conflicting estimates. What the behaviorist is forced to say about conflicting estimates shows that, even in experiments that contain no conflicts, different fractional estimates create different psychological magnitudes.

388

RATIO SCALES OF PERCEPTUAL MAGNITUDES

If they do, then it is wrong to suppose that points A-E and points a-c in figure 7 lie on the same psychological continuum. The two sets of points do lie on the same line, but this fact is simply the accidental result of particular estimates made by a particular observer. The same observer at another time, or another observer, may easily produce conflicting estimates. Then the two sets of points will not even lie on the same line. When we compare ratio estimates with interval estimates the "operationist" features of Definition (ii) become even more obvious. Suppose O halves three rods as in [7], and then estimates length intervals for the same rods by saying that [12] $ 3 — $2 = $2 — $ 1 . Together with a principle of correspondence for interval estimates, [12] entails that [13] = - Yl. Apparently [9] and [13] are inconsistent; consequently, the estimates in [7] and [12] apparently conflict. The behaviorist may try to remove the inconsistency with solutions analogous to (a), (b), or (c), but the solutions will be unacceptable for similar reasons. The conflict can be acceptably removed only by maintaining that [9] and [ 1 3 ] describe different psychological magnitudes, different psychological lengths. In brief, if we hold that principle of correspondence (Ci) and other similar principles are stipulative definitions, we seem forced to admit that there are as many types of psychological length as there are types of estimate of physical length, that there is no single continuum called "psychological length." If this is so, then no scale of psychological length can conflict with any other. A ratio scale constructed from m/n estimates cannot conflict with a ratio scale constructed from n/o or p/r estimates. Hence, Campbell is mistaken in his criticism of ratioscale measurement. He argues that if a loudness scale is constructed from estimates of one-half, then "x sone will not be estimated as a tenth of IOX sone." What then, he wonders, is the advantage of a figure like figure 7 over one like figure 6? " W h y do not psychologists accept the natural and obvious conclusion that subjective measurements of loudness in numerical terms (like those of length or weight or brightness) are mutually inconsistent and cannot be the basis of measurement?" 23 The behaviorist ought to reply that one-half scales and one23 A. Ferguson et al., "Quantitative Estimates of Sensory Events," of Science, 1 (1939-40) 338.

Advancement

RATIO SCALES OF PERCEPTUAL MAGNITUDES

389

tenth scales are scales of different psychological loudnesses, and cannot be inconsistent. More obviously, a ratio scale constructed from fractionation estimates cannot conflict with a partition scale constructed, say, from equisection estimates; and neither of these can conflict with a jnd scale. It may be supposed that since none of the three curves in figure 7 coincides with any of the others, only one can represent a "valid" scale of psychological length. But this supposition is true only if the ratio scale, Sr, the partition scale, S P , and the jnd scale, Si, are all scales of one and the same psychological magnitude, Va. According to the behaviorist, Sr, S P , and Sj are scales of different psychological magnitudes, ^b, and Vc, respectively. Hence, the three scales do not compete with one another, and the fact that they do not coincide raises no question about their validity. Discussing the lack of correspondence between ratio and partition scales, Stevens says that "observers are so constituted that they are unable to partition a prothetic continuum without a systematic bias." 24 He tries to explain this fact by suggesting that the observer's sensitivity is not uniform on the scale, being greater in the lower ranges. To say that partition estimates exhibit a systematic bias implies that the partition scale and the scale with which it is being compared are scales of a single psychological continuum. More precisely, the implication is that SP and Sr are scales of ^ a , that Sp and Sr do not coincide, and that Sr is the "true" scale. The behaviorist reaction is that S„ and Sr are scales of different psychological magnitudes and therefore do not compete. Since the one is as "true" a scale as the other, the estimates producing the one are no more biased than the estimates producing the other. Stevens sometimes takes a different tack. At one point he writes that the question of scale validity is " a matter of opinion," that a "judgement about validity always reduces ultimately to a value judgement," and that "In the long run . . . it is the scientific community that will decide the issue." 25 There is no issue for the behaviorist, since different types of psychological scale are scales of different psychological magnitudes and do not compete for validity. At other points Stevens adopts this "operationist" point of view. "Since the three kinds 24

Stevens, "Ratio Scales, Partition Scales, and Confusion Scales," p. 52. S. S. Stevens, "On the Validity of the Loudness Scale," Journal of the Acoustical Society of America, 3 1 (1959), 996. 25

39°

RATIO SCALES OF PERCEPTUAL MAGNITUDES

of scales are nonlinearly related on prothetic continua, it seems clear that they must measure different things. Each is probably a valid scale of something." 28 Speaking of the three different types of scale for subjective finger span, he says: "Obviously, three different aspects of finger span are being measured by these three functions. Although a certain amount of argument has revolved around the question of which of these functions is the 'true' scale, it should be apparent that all three are true scales of something or other." 27 It is impossible to locate Stevens' view precisely, since he constantly shifts back and forth between a behaviorist and some other way of treating the question of scale validity. THE NATURE OF PSYCHOLOGICAL ENTITIES

According to the behaviorist, the V s defined in (i) and (ii) are not observed by E, O, or anyone else. This statement makes the interpretation consistent with the phenomenological facts of experiment M, which are as follows. O sees rods and walls, feels chairs and tables, hears voices, and so on, but he does not observe, through some mysterious faculty of perception, psychological entities of which psychological length is a magnitude. O observes by sight the experimental rods, and he estimates their relative physical length. E measures the physical length of the rods, records O's estimates, and constructs a psychological magnitude in accordance with Definitions (i) and (ii). Neither O nor E observes psychological entities or magnitudes. This feature of the view has an important bearing on the question of observer accuracy. Psychophysicists often say that experiments like M presuppose the ability of the observer to make accurate estimates of psychological magnitudes. Stevens says that in fractionation procedures "we make an assumption that calls for scrutiny. We postulate, among other things, that the subject knows what a given numerical ratio is and that he can make a valid judgement of the numerical relation between two values of a psychological attribute." 28 Such a statement can be understood only in the context of an introspectionist interpretation of scaling experiments. As a previous section has shown, the introspectionist bases principle of correspondence (Cx) on the assumption, (Cb), that O's estimates of psychological magnitude are 26 27 28

Stevens, "On the Validity of the Loudness Scale," p. 998. Stevens and Stone, "Finger Span," p. 94. Stevens, "Mathematics, Measurement, and Psychophysics," pp. 40-41.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

391

accurate. But the behaviorist maintains that O does not make estimates of psychological magnitudes, that O's estimates are of physical magnitudes like length, weight, and the like. Now it is clear that experiment M does not presuppose O's ability to make accurate estimates of physical length. Indeed, O's length estimates are inaccurate, which is typical of experiments of this kind. And E often gives O explicit instructions to provide naive estimates, to make no attempt at being accurate. For example, Garner tells his subjects: "Remember to try to assign the numbers according to how loud the tones appear to you. We are interested in how loud tones seem to be to you, not in some kind of 'accuracy.' " 29 In sum, in the behaviorist view, O makes no estimates of psychological magnitudes, and the estimates he does make—estimates of physical magnitudes—are not required or presupposed to be accurate. There are also important implications for a related question, that of the so-called stimulus error. As the term was originally introduced by Titchener, committing the stimulus error is (SEi) "confusing sensations with their stimuli" or "read[ing] the character of the stimuli... into the 'sensations.' " 3 0 As Boring, who reviewed the history of the notion, put it, "We commit the stimulus-error if we base our psychological reports upon objects rather than upon the mental material itself, or if, in the psychophysical experiment, we make judgements of the stimulus and not judgements of sensation."31 The term has been used, however, in a different, or at least an extended, sense, in which committing the stimulus error is (SE2) confusing one stimulus with another. Stevens, in criticizing the so-called physical correlate theory, characterizes it as holding that "all quantitative estimates of sensory magnitude are really based on some form of 'stimulus error.' ' /S2 The theory maintains, for example, that typical observers, knowing little or nothing about the magnitudes of sound waves, base their estimates of loudness on the (estimated) distance of the sound source.83 Now, to 29

Quoted by Stevens, "Direct Estimation of . . . Loudness," p. 17. E. B. Titchener, Experimental Psychology (New York, 1905), II, pt. 1 : Student's Manual, p. xxvi. 31 Edwin G. Boring, "The Stimulus Error," American Journal of Psychology, 32 (1921), 451. 32 Stevens, "On the Validity of the Loudness Scale," pp. 1002-1003. 33 R. M. Warren, " A Basis for Judgements of Sensory Intensity," American Journal of Psychology, 7 1 (1958), 675-687; R. M. Warren, E. A. Sersen, and E. B. Pores, " A Basis for Loudness Judgements," American Journal of Psychology, 7 1 (1958), 700-709. 30

392

RATIO SCALES OF PERCEPTUAL MAGNITUDES

commit this error is not to confuse sensations with their stimuli, but rather to confuse one stimulus with another. Thus we have a different meaning of the term "stimulus error." The behaviorist may with perfect consistency attack the physical correlate theory. He may regard it as an error to confuse one stimulus with another, and he may attempt to show by experiment that observers do not commit the error. (He may also, again with perfect consistency, attempt to show that observers do commit the error when estimating loudness, brightness, etc. But what, in sense (SE2), would the stimulus error be for length, weight, etc.?) The behaviorist may, therefore, attack the stimulus error in sense (SE2), but he may not attack it in sense (SEx). In his view, typical observers do not commit any error when they make estimates of the stimulus in psychophysical experiments. That is precisely what they are asked by E to do. Indeed, the behaviorist cannot even admit the possibility of stimulus error in sense (SEi). To commit the error in that sense is to be aware of the stimulus, to be aware of the sensation caused, and to "read" the former into the latter. But the behaviorist holds that observers are not aware of any sensations caused by rods, weights, and the like, in psychophysical experiments dealing with these stimuli. The behaviorist experimenter may want his observers to estimate length or weight "as they see it" or "as they feel it," and he may give them instructions to that effect. Nonetheless, these will be instructions to estimate the stimulus. One decided advantage in the behaviorist interpretation is its removal of private entities from the conceptual structure of experiment M. By definition, a public entity can be observed by any normal observer, a private entity by only one observer. Hence, the distinction between public and private entities applies only to observable entities, like rods and afterimages, and not to the theoretical T ' s of the behaviorist interpretation. It follows that we must not say that Y ' s are public. When we recognize the parallel mistake in saying that they are private, however, there is no longer any inclination to regard them as directly inaccessible to everyone but O, or to think of O as an intermediary between the investigating scientist and a realm of private data. The dualism between public and private data entirely collapses, and, as a result, several disturbing theoretical and philosophical problems simply vanish. Psychological entities are indeed unobservable; nevertheless, since they are defined in terms of publicly observable rod

RATIO SCALES OF PERCEPTUAL MAGNITUDES

393

presentations and observer reports, there is no problem about the general availability of psychological data or the "objectivity" of results based on such data. According to the behaviorist, Y ' s are, to use a current phrase, theoretical constructs;34 that is, they are theoretical entities constructed by E by means of Definitions (i) and (ii) for scientific use. An analogy will help to explain their nature. O is asked in a two-part experiment to estimate the dollar value of automobiles. In each trial of the first part, E presents O with a standard auto and asks him to select a comparison auto one-fourth as expensive. In each trial of the second part O is asked to select a comparison auto that is one-half as expensive as the standard. A plot of the data thus obtained produces lines identical to those in figure 6, except that numerals along the axes represent auto values in thousands of dollars. E now invokes a principle of correspondence, which says: If O estimates that and stand in the ratio 1 :z, then Y i and stand in the ratio 1 :z, where $>'s represent the real value and T ' s the estimated value of automobiles. E stipulates that one unit of estimated value equals one thousand units (dollars) of real value, and, using the principle of correspondence above, constructs a scale of estimated value, plotting estimated value against actual value by the same method used to obtain figure 7. It is clearly a mistake to attempt to identify the V s in this analogy with some group of privately observable entities, of which estimated value is a magnitude. The attempt would lead to theories like the following. When O estimates the value of a $5,800 automobile he has a thought, Y i ; when he estimates the value of a $10,000 automobile he has another thought, T2. Since is (psychologically) half as great as O estimates that the values of the two automobiles stand in the ratio 1 : 2 . This theory is specious, first, because no meaning can be given to the assertion that one thought is less great or greater than another, and, second, because it is an obvious fact that O makes estimates, not about his thoughts, but about automobiles. Similarly, it is an introspectionist mistake to try to identify the ^F's of the mak scale with privately observable sensations. Such identification commits us to the unclear and possibly meaningless view that sensations can be greater 34 This view is vaguely suggested by Garner, " A Technique and a Scale for Loudness Measurement," pp. 86-87, and incompletely by Stevens, "On the New Psychophysics/' p. 27.

394

RATIO SCALES OF PERCEPTUAL MAGNITUDES

or less than one another; and it implies, falsely, that O makes estimates of sensations. It is also a mistake to ask for examples of the entities of which estimated value is a magnitude. Estimated value cannot be construed as a magnitude of O's utterances (their loudness, duration, etc.), nor of automobiles, nor of any other observable group of entities. Analogously, it is a mistake to ask for examples of the entities of which psychological length is a magnitude. Behind this request lies the introspectionist belief that Y's can be ostensively defined and are observable entities, examples of which can literally be pointed to by someone. One completely understands the nature of psychological entities and their magnitudes when he understands Definitions (i) and (ii). To ask for examples, after having carefully studied the definitions, reveals misunderstanding. The above considerations tempt us to say that in the behaviorist view psychological entities are fictions and do not exist. Although true in one sense of "exist" (the sense in which observables exist), to say that the psychophysicist is trying to measure something that does not exist is paradoxical and misleading. What we should say, instead, is that the concept of psychological magnitude is the product of a façon de parler. Consider again the automobile analogy. When we say, (a) "The estimated value of the one automobile is half the estimated value of the other," we do not wish to create the impression that automobiles have two kinds of value, estimated value and actual value, in the way that they have ballast value on a ship as well as financial value on the market. Statement (a) is just another way of saying, (b) "O estimates that the value of the one automobile is half the value of the other." Statement (a) means nothing more than what is meant by (b), and (b) does not imply that estimated and actual value are two kinds of value possessed by automobiles. Only money value is involved, although an economist or someone else may be interested in O's estimates of money value. (Other examples: "alleged" age and "confirmed" age are not two kinds of age; "probable" area and "determined" area are not two kinds of area; "apparent" cause and "actual" cause are not two kinds of cause.) Analogously, when we say, (c) "The estimated length of the one rod is half the estimated length of the other," we do not imply that estimated length and actual length are two kinds of length possessed by rods. Statement (c) is just another way of saying, (d) "O

RATIO SCALES OF PERCEPTUAL MAGNITUDES

395

estimates that the length of the one rod is half the length of the other." And (d) carries no implication of two kinds of length. Rods possess one kind of length: physical length, if you will. But the psychophysicist is interested in the estimates that observers make the physical length of rods. E may express facts about such estimates by speaking of "estimated length" as opposed to "actual (real) length." Such talk, however, is merely a façon de parler which E finds it convenient to use in describing the behavior of his observers. THE PURPOSE OF RATIO SCALING

As noted in an earlier section, the behaviorist can argue for the acceptance of principle of correspondence (Ci) only on grounds of scientific usefulness. The reason for accepting (Ci) is that it makes the construction of the mak scale possible. But what reason is there for constructing the scale? Well, its construction makes the formulation and verification of psychophysical law [5] possible. But what reason is there for formulating and verifying [5]? The behaviorist cannot answer, in the manner of the introspectionist, that [5] is a law relating two variables, Y and defined independently of each other, and that the discovery of such laws needs no justification, for Y is defined in (i) and (ii) in terms of O's $ estimates. Nor can the behaviorist plead that [5] explains how O makes reliable estimates of physical length, because [5] could provide such an explanation only in conjunction with a theory of O's perceptual mechanism like (Ca)-(Cb)-(Ci), whereas the behaviorist does not link [5] to any such theory. Consequently, he seems reduced to saying that [5] does not tell us how O estimates physical length, but, rather, that it tells us what O estimates physical length to be. Law [5] is therefore not an explanation, but rather a mathematical description of O's physical length estimates. And the reason for accepting (Ci) is that it makes this mathematical description possible. But the reason seems insufficient, since there are other ways of mathematically describing O's length estimates, for example, in terms of laws [3], [4], and others like them. Whether [5] is a descriptive or an explanatory law cannot be settled without discussing the nature of psychological explanation. This is a large topic which we cannot and need not enter more than a little way here. The behaviorist can consistently adopt either of two views of the nature of psychological explanation. In the first view, (PEi), a

396

RATIO SCALES OF PERCEPTUAL MAGNITUDES

psychological explanation consists in discovering the mechanisms, mental or physiological, on which the behavior of organisms depends. It is clear that a good behaviorist can attempt to discover the physiological mechanisms on which perceptual behavior depends. Perhaps he can even attempt to discover the "mental" mechanisms of perception, so long as the mental entities involved in such explanations are operationally defined in terms of observer responses, and there is no suggestion that observers introspect these mental entities. But the point here is that [5] seems to contain no reference to the mental or physiological mechanisms on which O's length estimates depend. It seems, rather, to be nothing more than a mathematical description (among others available) of O's perceptual responses to length. If this is correct, then the behaviorist cannot recommend the adoption of [5] by claiming that it is, in sense (PEi), an explanatory law. In a second view, (PE2), psychological explanation consists in discovering laws that record and predict the behavior of organisms. In this sense, [5] is clearly an explanatory law, since it not only records O's length estimates, but also enables us to predict new behavior, to predict ratio estimates not made by O in experiment M, and perhaps even to predict (by making certain assumptions) interval estimates that observers would make in category-scaling experiments. A law with predictive power of this sort clearly qualifies as an explanatory law of perception. And it may be misleading to call it a descriptive law, since this designation may be taken to imply that it has no predictive power. On the other hand, if the implication is avoided, [5] may be called a descriptive law to contrast it with laws that are explanatory in sense (PEi). Law [5] does, then, provide an explanation of O's perception in sense (PE2). This fact, however, does not in itself seem to constitute a sufficient reason for adopting [5]. Apparently we can substitute for [5] a bundle of laws which have the same reportive and predictive use. Equation [3] reports and predicts O's one-fourth estimates; equation [4] reports and predicts O's one-half estimates; and other laws of the same type report and predict other fractionation estimates. None of these laws presupposes the concept of psychological length. It is obviously useful to construct the lines and discover the equations of figure 6. But what additional purpose is served by constructing the lines and formulating the equations of figure 7, a construction that presupposes

397

RATIO SCALES OF PERCEPTUAL MAGNITUDES

the concept of psychological magnitude? More generally, the question is this: Is there any legitimate purpose in constructing ratio scales of psychological magnitude which cannot be equally well achieved without the concept of psychological magnitude? If the answer is negative, the concept may and should (for reasons to be given later) be dispensed with. In what follows I attack four arguments that attempt to show that the concept is indispensable. First argument.—One argument insists that without the concept of psychological magnitude we would not be able to formulate the general psychophysical laws that have been advanced in the past few decades. Equations [3] and [4], and similar equations for other estimation ratios, can be obtained from [5], and [5] can in turn be obtained from them. This can be accomplished by using the equation [14] a =

b\

or, logarithmically, [15 ]n = log a/log b, where a is the fraction (one-fourth, one-half, etc.) O is required to estimate, b = $c/s, and n is the exponent in an equation like [5]. Thus it is possible to perform the useful operation of predicting the results of new fractionation experiments.85 In view of the algebraic relations above, [5] may be considered a kind of general law, of which [3], [4], and certain other fractionation equations are, in a sense, instances. It may be supposed that such general laws must take the form of [5], that is, that they must relate some psychological magnitude to a stimulus magnitude. This claim appears to be false. Given [14], we can write, r

, .

[16] b =

n

t—

and can then substitute into the equation, [17]

= b

to obtain:

= ^V^Q,.

Where n has the value determined in experiment M, [17] becomes [18] 4>c = M

Gosta Ekman, "Two Generalized Ratio Scaling Methods," Journal of Psy-

chology, 45 (1958), 288.

398

RATIO SCALES OF PERCEPTUAL MAGNITUDES

Equations [3] and [14], and all the other fractionation equations obtainable from [5] by the method described in the preceding paragraph, can also be obtained from [18] by the same method. And from every fractionation equation from which we can obtain [5] by the algebraic method described in the preceding paragraph, we can also obtain [18] by the same method. Thus [18] is as general, and as useful in prediction, as is [5]; and yet it does not employ the concept of psychological length. Since the above argument assumes that all fractionation equations are linear, its conclusion is not a general one. The general claim is as follows. Given any equation, Y = F($), from which we can obtain and which can be obtained from any of the fractionation equations, = M®»),... = by the method above, there = G($0/ from which we can obtain and which is another equation, can be obtained from any of the same fractionation equations by the same method. If this claim is true, the most general psychophysical laws can be formulated without employing the concept of psychological magnitude. The G equation is entirely different in type from the F equation, since it contains only physical variables. But it is, nonetheless, a psychophysical equation, an equation that can be used to describe and predict O's ratio estimates of physical length. Second argument.—This argument maintains that without the concept of psychological length we would be unable to make the comparisons made in figure 7 between ratio scales, partition scales, and jnd scales. Thus stated, the argument is trivial and question begging. If we do not construct a scale, Sr, of psychological length, we cannot compare it with scales SP and Sj, for there is nothing with which to make the comparison. The question is: W h y should we construct any scales of psychological length? There may indeed be good reasons for constructing psychological scales of some kind, but is there any good reason for constructing scales of psychological magnitude? W h y even introduce the concept of psychological magnitude? The answer will be that without the concept we cannot compare ratio estimates with interval estimates, or compare either of these with jnd estimates. This argument is unconvincing, since there seem to be other ways of comparing the various types of estimate. Let us illustrate an alternative method by comparing jnd estimates with ratio estimates of onehalf. The upper line in figure 7 is the result of plotting number of jnd's against rod length. By using this line we can plot in figure 6 the length

RATIO SCALES OF PERCEPTUAL MAGNITUDES

399

of the stimulus associated with n jnd's (ordinate) against the length of the stimulus associated with 2n jnd's (abscissa). For example, figure 7 shows that the stimuli associated with 20 and 40 jnd's are 5 and 1 1 cm, respectively. Accordingly, we place a point at 5 on the y-axis and a point at 1 1 on the x-axis of figure 6. Arbitrarily selected points obtained in this manner are connected by the lower of the two dashed lines in figure 6. That the dashed line does not coincide with line [4] presumably shows that O does not estimate length ratios of one-half on the basis of length jnd's. (If the lines coincided, would this show that O does estimate length ratios of one-half on the basis of length jnd's?) Similarly, we can compare jnd estimates with ratio estimates of one-fourth by plotting in figure 6 the length of the stimulus associated with n jnd's against the length of the stimulus associated with 4rt jnd's. We can use this same method to compare partition estimates with ratio estimates, so long as the partition interval is relatively small and the smallest stimulus estimated is near physical zero. The numerals on the inside left-hand ordinate of figure 7 can be taken to represent merely the number of apparently equal intervals associated with a given stimulus, just as the numerals on the outside right-hand ordinate represent the number of jnd intervals associated with a given stimulus. Then we may plot in figure 6 the length of the stimulus associated with n apparently equal intervals, and thus obtain the upper of the two dashed lines. That this dashed line does not coincide with line [4] presumably shows that O does not estimate length ratios of one-half on the basis of equal-appearing intervals. It is important to understand that in this method of comparison, the ordinate numerals of the category scale and the jnd scale represent, not the size of intervals along a psychological continuum, but merely the number of intervals associated with various stimuli. The method makes the harmless and probably trivial assumption that jnd's are similar in some respect, and that apparently equal intervals are similar in some respect; but it does not assume that jnd intervals, or apparently equal intervals, are equal on a psychological continuum. (Similarly, to count off the number of octaves in the musical scale assumes that octaves are similar in some respect, but not that they are equal on a psychological continuum.) If the ordinate numerals represent equal intervals on a psychological continuum, the following principle of correspondence is presupposed: If O estimates that — $2 = $>2 —

4oo

RATIO SCALES OF PERCEPTUAL MAGNITUDES

then Y i — T j = T2 — Ys. But the method presupposes no such principle. We can speak of the number of similar intervals without employing the concept of a psychological entity or a psychological magnitude. The method thus makes it possible to compare O's jnd estimates and partition estimates with ratio estimates without measuring, scaling, representing, mentioning, or presupposing a psychological magnitude called psychological length. I am not suggesting that it is the only method with this feature. Avoiding the concept of psychological magnitude seems to depend only on the ingenuity of the theorist or experimenter in representing the data of psychological experiments. Third argument.—This argument contends that without the concept of psychological magnitude we would not be able to compare psychophysical functions with psychophysiological functions. Figure 8, taken from a study by Stevens and Davis, 86 shows that when cochlear potential (multiplied by a constant) and psychological loudness are plotted against sound-wave intensity, the curves obtained have a high degree of correspondence. (That the loudness function in this figure is no longer accepted is irrelevant to my argument.) Where Y = fi(3>) is the psychophysical function and 8 = is the physiological function (0 designates physiological magnitude), fi is linearly related to fa. Consequently, Stevens and Davis suggest that "as a first approximation, the form of the loudness function is imposed by the behavior of the cochlear mechanism"; but they point out that "identification of the loudness function with the recorded potential must [in view of divergences] be made with reservations."37 Such interesting suggestions could not be made without employing the concept of psychological magnitude, according to the current argument. Thus stated, this argument begs the question. Let Y and 8 be a psychological and a physiological magnitude, respectively. It is true that we cannot compare a psychophysical function, Y = fi(i>), with a physiological function, 8 = fi($), without employing the concept of psychological magnitude. But this observation is trivially true, since, by hypothesis, one of the functions to be compared contains W as a variable. If we are to avoid begging the question, we must ask whether the results of psychophysical experiments can be compared with the 3 8 S. S. Stevens and Hallowell Davis, "Psychophysiological A c o u s t i c s : Pitch and Loudness," Journal of the Acoustical Society of America, 8 (1936), 5. 37 Ibid., p. 6.

401

RATIO SCALES OF PERCEPTUAL MAGNITUDES

results of physiological experiments without employing the concept of psychological magnitude, whether such comparisons can be made w i t h out comparing

functions with 8 - $ functions. T h e answer is, ap-

parently, that it is possible. Let 8c be the physiological process (cochlear potential, area of

DB ABOVE THRESHOLD Fig. 8. The loudness function and the size of the cochlear potential. The solid curve represents the loudness function obtained by Churcher from tonefractionation experiments. The circles represent the averaged results of measurements of the size of cochlear potentials at the round windows of six guinea pigs as a function of intensity of the stimulus. Taken from S. S. Stevens and Hallowell Davis, "Psychophysiological Acoustics: Pitch and Loudness," Journal of the Acoustical Society of America, 8 (1936), 5. retinal stimulation, etc.) associated with the physical

comparison

stimulus, c =

fs($*),

In w e can

402

RATIO SCALES OF PERCEPTUAL MAGNITUDES

also formulate and verify associated functions of the form 8 C = f curve that has been discovered. If agreement is worse, it may no longer be possible to argue that the 8-3> curve describes the physiological mechanism underlying O's ratio estimates. The solid line in figure 8 was derived by making use of an assumption like (Ci) for loudness frequency, and also the assumption 88

Ekman and Sjoberg may have some point like this in mind when they say ("Scaling," p. 452): "According to a strictly behavioristic view of perception . . . the psychological scale is an arbitrary and possibly trivial transformation of response data."

RATIO SCALES OF PERCEPTUAL MAGNITUDES

that a tone introduced into one ear sounds half as loud as the same tone introduced into both ears. If these assumptions are replaced by others, it is no longer possible to argue that "the form of the loudness function is imposed by the behavior of the cochlear mechanism." Assuming still other principles of correspondence, we can make experiment M verify a psychophysical log law instead of a psychophysical power law. Each of the alternative principles of correspondence so far considered assumes that equal physical ratios are accompanied by equal psychological ratios. That is, each is an instance of principle (C): If O estimates that $i/S>2 = $2/$3, then W1/W2 = ^ 2 / ^ 3 . But why should we make this assumption? Why not assume instead that equal physical ratios are accompanied by equal psychological differences? This would be to assume that if O estimates that i > i/i ) 2 = $2/^8, then — ^ 2 = T2 — ^3. On the basis of this assumption experiment M establishes that the psychophysical law for length is an instance of [2], which is of course Fechner's law. (This point is made by Treisman. The only weakness in his argument is his assumption that Y is a neural effect in O, which leaves him open to a reply by Stevens, who objects that ratio-scaling experiments need not posit intervening neural variables. Treisman's argument can be restated without the fatal assumption, along the lines suggested above.39) The point above applies to experiments employing the technique of numerical estimation as well as to those employing the technique of fractionation. Many theorists have distinguished between "direct" and "indirect" psychological measurement. Ekman calls Stevens' technique "direct scaling of subjective variables, because the essential steps of the scaling procedure are implied in the experimental situation." Thus, when O assigns 100 to the loudness of a standard tone and 62 to the loudness of a comparison tone, the "two scale values are, by definition, on a ratio scale: the ratio 62/xoo should, according to the instructions, be equal to the subjective ratio of the second loudness to the loudness of the standard." Ekman regards Thurstone's methods, and would also regard Fechner's, as "indirect methods, since they are based on a set of assumptions intervening between the experimental data and the 89

Michael Treisman, "Sensory Scaling and the Psychophysical Law," Quarterly Journal of Experimental Psychology, 16 (1964), 1 1 - 1 7 ; Michael Treisman, "What Do Sensory Scales Measure?" Quarterly Journal of Experimental Psychology, 1 6 (1964), 387-388; S. S. Stevens, "Concerning the Psychophysical Power Law," Quarterly Journal of Experimental Psychology, 16 (1964), 383-384.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

final scale."40 Employing this distinction, one might wish to argue that fractionation methods are indirect because they depend on intervening assumptions, whereas numerical estimation methods are free of such assumptions and are therefore direct. If this were true, then different psychophysical laws could not be derived from numerical estimation experiments, since there would be no intervening assumptions, no principles of correspondence, to manipulate. But it is not true. The method of numerical estimation assumes principle of correspondence (Di): If O assigns to stimuli $ 1 , $2, i s , and so on, the numerals m, n, o, and so on, respectively, then TFi, Y2, Ys, and so on, stand in the ratios m:n, n:o, and so on. Ekman tacitly agrees when he says that scale values in a numerical estimation experiment are, "by definition, on a ratio scale." The "definition" in question is just the principle of correspondence stated above. Other definitions, other principles of correspondence, can as easily be adopted. For instance, if O assigns to stimuli $ 1 , $2, $3, and so on, the numerals m, n, 0, and so on, the difference between Y i and ^ 2 is m — n, the difference between ^ 2 and Ya is n — 0, and so on. Again, this assumption leads to Fechner's logarithmic law rather than to Stevens' power law. The method of numerical estimation, therefore, is not in Ekman's sense direct. (Indeed, no psychophysical method invented is direct in his sense, and probably none could be invented.) Hence, data obtained by employing the method are as subject to arbitrary manipulation as those obtained by any other method. When the concept of psychological length is employed, comparisons between different ratio estimates, between different psychophysical and physiological functions depend entirely on the principles of correspondence employed. But how do we know which principles to use? The choice seems arbitrary, and, consequently, so do the results of the various comparisons. This arbitrariness is not present when we use methods of comparison which do not employ the concept of psychological magnitude. Therefore, it seems not merely possible, but also advisable, to discard the concept. Fourth argument.—Stevens and his associates have conducted a number of experiments in which O's are required to make what are called "cross-modality comparisons" or "matches," that is, to make 40 Gosta Ekman, "Some Aspects of Psychophysical Research," in Sensory Communication, ed. W. A. Rosenblith (New York, 1961), pp. 35,43.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

405

stimuli in one perceptual modality psychologically equivalent to stimuli in some other perceptual modality.41 In one of these experiments, O is asked to squeeze a hand dynamometer in such a way that the force exerted is psychologically equivalent to the magnitude of some physical stimulus, such as weight, being presented to O. (Described introspectionistically, the experiment is one in which O is asked to squeeze the dynamometer so that the "sensation of strain" 42 produced by squeezing is equal to the "sensation of weight" produced by lifting.) These experiments have revealed that for prothetic stimulus magnitudes, the force on the dynamometer is a power function of the magnitude of the matched stimulus. More important, this power function has been found to bear a certain relation to the psychophysical power functions (like [5]) governing the perception of the various stimulus magnitudes. This relation is illustrated below. Fractionation experiments (like experiment M) and numerical estimation experiments have shown that the physical force produced in a hand dynamometer is a power function of the psychological force (the "sensation of strain") produced by squeezing the dynamometer, and that the exponent in the function is 1.7. That is, the following law has been discovered, [19] ^ r = where is psychological force expressed in suitable psychological units, and i f is physical force expressed in suitable physical units. Experiment has also shown that the power function for lifted weights is [20] Ww = S»,1 where V . is psychological weight (the "sensation of weight") expressed in suitable psychological units, and is physical weight expressed in suitable physical units. Since the cross-modality experiment requires the observer to make psychological force and psychological weight equivalent, we can assume that in the experiment [21] = From [19], [20], and [21] it can be deduced that the relation between 41 Stevens, "The Surprising Simplicity of Sensory Metrics"; S. S. Stevens, J. C. Stevens, and J. D. Mack, "Growth of Sensation on Seven Continua as Measured by Force of Handgrip," Journal of Experimental Psychology, 59 (1960), 60-67; S. S. Stevens, "Matching Functions between Loudness and Ten Other Continua," Perception and Psychophysics (1966). 42 Stevens uses this phrase in "The Surprising Simplicity of Sensory Metrics," P- 33-

4o6

RATIO SCALES OF PERCEPTUAL MAGNITUDES

psychological force and psychological weight in a cross-modality experiment is [22] 4», 17 = f w 1 4 5 , or [23] i»f = Stevens has then been led to the following general hypothesis: "If two continua are governed by power functions, Y i = i>im, and TP2 = $ 2 " , . . . cross modality matches, in which Y i is equated to Y2, will determine an equal-sensation function of the form, i>i = $ 2 n / m ." 4 3 That this hypothesis has found considerable experimental support seems to provide additional evidence for Stevens' claim that [5] is the general psychophysical law. "Confronted with this richly interconnected evidence, some of us find it difficult to escape the belief that there exists a general principle of psychophysics—a principle that governs to a good approximation throughout all the prothetic perceptual domain. Psychophysics, we venture to suggest, has found itself a law." 44 It may be argued that psychological magnitudes within different perceptual modalities cannot be compared with one another, and that the results of these comparisons cannot be deduced from previous psychophysical laws, without employing the concept of psychological magnitude, and that the concept is therefore indispensable. This argument is question begging as it stands, since it trivially asserts that comparisons between and laws concerning psychological magnitudes presuppose the concept of psychological magnitude. The interesting, nontrivial question is whether responses to stimuli of different modalities can be compared with one another, and whether the results of such comparisons can be deduced from previous laws, without employing the concept of psychological magnitude. The answer to this question is, apparently, affirmative. In the cross-modality experiment described above, O is not asked to compare "psychological force" with "psychological weight." Rather, he is asked to make the force on the dynamometer equivalent to the weight of the object lifted. He is asked to compare, or match, physical force and physical weight. Consequently, so far as the design of the experiment is concerned (description of materials, instructions to observers, etc.), no concept of psychological magnitude is presupposed. 43 44

Stevens, Stevens, and Mack, "Growth of Sensation on Seven Continua/' p. 60. Stevens, "The Surprising Simplicity of Sensory Metrics," pp. 33-34.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

Furthermore, the results of the experiment can be deduced from laws of the type suggested in the reply to the first objection above. The counterpart of [19] is [24] $ f c =

1.7

Va.ii,,,

where i>fC is the force estimated by O to be the fraction, at, of the force ^tn. The counterpart of [20] is [25] 3>wc =

— •

where is the weight estimated by O to be the fraction, a„, of 3>WB. The assumption of the cross-modality experiment is that O matches the force on the dynamometer (as he perceives it) to the weight lifted (as he perceives it). Thus, if he perceives the ratio between two weights to be 1 :z, his two corresponding squeezes on the dynamometer will be in a ratio he perceives to be 1 :2. That is to say, [26] at = aW/ which corresponds to equation [21]. From [24], [25], and [26], none of which contains any reference to psychological magnitudes, we can deduce [22] and [23] (the deduction is not given here). That is, the relation between stimuli in different perceptual modalities can be deduced from laws that do not presuppose the concept of psychological magnitude. The concept thus appears to be dispensable. Although incidental to the main point above, it is important to note that the discovery of the general law of which [23] is an instance is not, as Stevens thinks, good evidence that [5] is the general psychophysical law. As noted earlier, it is possible to assume, as the principle of correspondence for fractionation and numerical estimation experiments, that equal physical (stimulus) ratios correspond to equal psychological differences. Using this principle, such experiments establish, not that [19] and [20] are true, but that [27] Y , = 1.7 log where is psychological force expressed in suitable psychological units, and $>f is physical force expressed in suitable physical units; and that [28] Tw = 1.45 log where ^"w is psychological weight expressed in suitable psychological units, and i»w is physical weight expressed in suitable physical units. These two equations are, of course, instances of Fechner's logarithmic

4o8

RATIO SCALES OF PERCEPTUAL MAGNITUDES

law [2]. Cross-modality equation [23] can be deduced from [27], [28], and [21]; consequently, the discovery of such equations no more tends to confirm Stevens' law than it does Fechner's law. Here again is an example of the difficulty in deciding whether Stevens' law or Fechner's law is the correct one. One way of avoiding this apparently insoluble difficulty is simply to abandon the concept of psychological magnitude, thus abandoning both Stevens' law and Fechner's law, and any other law that is genuinely of the form. To do so will not affect the discovery or verification of cross-modality equations like [23]. These equations are of the form and contain no reference to psychological magnitudes. Stevens and his associates have discovered that, for prothetic continua, observers equate stimuli within different perceptual modalities according to the general law, [29] = where is the stimulus within one perceptual modality (weight perceptions, for instance) and 3>b the stimulus within some other perceptual modality (length perception, for instance). They have also discovered that t = n/m, where m is the exponent in the power law governing O's perception of and n is the exponent in the power law governing O's perception of It may seem that the discovery of [29] could not have been made without previously discovering laws of the form, but this is not so. Stevens did deduce [29] from laws of the Y - i form, but [29] could also have been directly obtained from cross-modality comparison experiments, and it has been directly verified in that manner. Equation [29] will survive as an experimentally verifiable law even if we abandon laws of the form. CONFUSION OF THE TWO INTERPRETATIONS FAILURE TO DISTINGUISH THE INTERPRETATIONS

If the introspectionist and behaviorist interpretations of ratio scales are explicitly and systematically distinguished anywhere in the literature, I am not aware of it. Psychophysicists in general do not observe the distinction in presenting and discussing the results of their experiments. There are two major symptoms of this deficiency—unclarity as to what the O's in psychophysical experiments estimate, and unclarity as to what the E's in psychophysical experiments measure. It is especially disconcerting, and especially significant, to discover such

RATIO SCALES OF PERCEPTUAL MAGNITUDES

409

unclarities in the work of S. S. Stevens, the major architect

and

methodologist of the so-called new psychophysics. If Stevens is unclear, then so is a vast part of contemporary psychophysics. His writings are therefore given most attention here. First symptom.—On

the behaviorist interpretation the O in an

experiment like M is said to provide quantitative estimates of

that

is, of physical entities or stimuli, like rods, weights, and so on. O is not said to estimate or even to be aware of any Y ' s or Y-magnitudes. O n the introspectionist interpretation, O is thought to make direct quantitative estimates of psychological entities, such as sensations—or at least estimates of psychological magnitudes—in order to make indirect estimates of physical or stimulus magnitudes. Stevens constantly shifts from one of these positions to the other. He mentions all the following as estimated by O : (a) "the standard stimulus and a set of variable stimuli"; (b) "the stimuli that arouse two sensory magnitudes"; (c) "[an attribute of] the stimuli"; (d) "the apparent magnitude [of the stimulus] as he perceives it"; (e) "the apparent magnitude [of the stimuli]"; (/) "the apparent strength [of the stimuli]"; (g) "the apparent strength or intensity of his subjective impressions"; (h) "some aspect of his experience"; (i) "subjective events" or "sensations"; (/) "sensations"; (fc) "the magnitude of a given sensation"; (I) "the relative magnitudes of . . . sensation"; (m) "attributes of sensation"; (n) "the apparent intensity of sensations aroused"; (o) "the apparent strengths of the sensations produced." 45 In (a) through (c) O is said to estimate stimuli or stimulus magnitudes, and so a behaviorist interpretation is implied. In (i) through (o) O is said to estimate psychological entities or magnitudes, and so an introspectionist interpretation is implied. Phrases (d) through (h) are sufficiently ambiguous to be consistent with either interpretation. A t times conflicting interpretations are implied in the same article, or even on 45 (a) "Direct Estimation of . . . Loudness/' p. 25; (b) ibid., p. 1 ; (c) "The Scaling of Subjective Roughness and Smoothness" (with Judith Harris), Journal of Experimental Psychology, 64 (1962), 489; (d) "Ratio Scales, Partition Scales, and Confusion Scales," p. 54; (e) "Problems and Methods of Psychophysics," p. 193; (f) "Psychophysics of Sensory Function," p. 239; (g) ibid., p. 232; (h) "Direct Estimation of . . . Loudness," p. 18; (f) "On the Psychophysical Law," p. 163; (/) "Biological Transducers," p. 30; (Jc) "Growth of Sensation on Seven Continua" (with J. C. Stevens and J. D. Mack), p. 64; (J) "Biological Transducers," p. 30; (m) " A Scale for the Measurement of . . . Loudness," p. 406; (n) "Growth of Sensation on Seven Continua," p. 60; (o) "Psychophysics of Sensory Function," p. 238.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

the same page! Moreover, the vacillation between interpretations has continued for a period of three decades, up to the present. Stevens might reply that the vacillation is only apparent, that it is the result of using convenient locutions. In describing fractionation experiments, he writes: . . . a pair of stimuli are given, and the subject estimates the numerical value of their apparent ratio. (More properly stated, he estimates the numerical ratio between the two magnitudes of an attribute of the sensation aroused by the two stimuli, but for the sake of brevity we say simply that he estimates the apparent ratio of the stimuli.)*6

But this explanation commits Stevens to an introspectionist view of ratio-scaling methods. And it is difficult to believe that he wishes to be so committed. Galanter, in discussing the category scaling of subjective length, provides a clear and instructive example of the same sort of vacillation within a single paragraph: The failure of the subject to recognize the repeated presentations of the stimuli is not relevant in this category scaling experiment; rather

it is his judgements about the relative magnitudes of the stimuli that are sought. The experimenter can never decide whether the subject is right or wrong in a scaling experiment; there is therefore no natural way to introduce an outcome structure into this kind of experiment. In scaling experiments we are forced to assume the uncontrolled ability of the subject to accurately report his sensations. As we shall see, the reproducibility of the data upon repetition of this experiment lends some support to this assumption.47

The italicized phrases show that Galanter describes the scaling experiment in conflicting ways: first O is said to estimate stimuli, and then he is said to estimate sensations. If E cannot decide whether O's estimates are right or wrong, the estimates cannot be of stimuli, for E can decide whether estimates of stimuli are right or wrong. We seem forced, therefore, to take Galanter to mean that O's estimates are of his sensations, since the accuracy of sensation estimates cannot be decided by E. But Galanter would thereby be adopting an introspectionist interpretation, an interpretation he would presumably disavow. What he should say is that O estimates stimuli and that the accuracy of his 48 47

Stevens, "Mathematics, Measurement, and Psychophysics," p. 40. Galanter, "Contemporary Psychophysics," p. 142. Italics added.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

411

estimates can be decided by E, but that E does not attempt to elidt accurate estimates from O. The remark that E must assume the accuracy of O's estimates is a sure sign that the writer either is an introspectionist or is confused. Warren and Warren seem to have noticed the vacillation illustrated above: The New Psychophysics also makes a fundamental distinction between quantitative judgments of subjective (psychological) magnitudes and estimates of physical magnitudes (Stevens, 1958). Thus, in order to construct the veg scale of subjective heaviness, Ss' judgments are considered distinct from estimates of physical weight. However, in the experiment of Harper & Stevens (1948), Ss were instructed to select that weight "which feels half as heavy as the standard." It is difficult to see how this phrasing differs from instructions to choose an object which seems to be half physical weight. Yet this distinction is essential for Stevens' psychological continua.48 A s these writers correctly point out, Stevens and his co-workers sometimes instruct their observers to estimate physical magnitude, but at other times speak as if judgments of psychological magnitude are being required. But Warren and Warren are wrong in implying that the new psychophysicists possess an official distinction between judgments of psychological magnitudes and estimates of physical magnitudes, and that the former are theoretically required of observers in psychophysical experiments. The distinction seems never to have occurred to most of the psychophysicists in question, and they are far from clear as to what they are or should be requiring from their observers. Second symptom.—Failure to observe the distinction between the introspectionist and behaviorist interpretations is also revealed in unclarity as to what experiments like M attempt to measure. In the introspectionist interpretation, measurement in such experiments consists in assigning numerals to a psychological magnitude privately observed and quantitatively estimated by O. On a behaviorist interpretation, it consists in quantitatively defining a psychological magnitude in terms of O's quantitative observations of a class of physical stimuli. Again selecting Stevens to illustrate the vacillation between and unclarity regarding these two positions, we find him writing in 1936 that 48

R. M. Warren and R. P. Warren, " A Critique of Stevens' 'New Psychophysics/ " Perceptual and Motor Skills, 16 (1963), 804-805.

412

RATIO SCALES OF PERCEPTUAL MAGNITUDES . . . scale numbers [should bear] . . . a reasonable relationship to t h e experience of the observer. T h u s , the scale w o u l d be satisfactory if the magnitude of the attribute of sensation to w h i c h the number 1 0 is assigned should appear to b e half as great to the experiencing i n dividual as that to w h i c h the number 2 0 is given, and twice as great a s the magnitude to w h i c h the number 5 is g i v e n . . . . the subjective j u d g e ments (responses) of the observer must provide the ultimate test of the validity of the n u m b e r s o n the scale as representative of degrees of loudness [in general, sensation]. T h e utilization of the observer's discriminations in this w a y presupposes, of course, that he is capable of m a k i n g valid judgements of the numerical ratio of one impression to another. 4 9

This passage contains one of the strongest suggestions in the literature of an introspectionist view of ratio scales. Stevens says that the scale is satisfactory if "sensations appear . . . to the experiencing individual" to have those ratios indicated by the numerals assigned to them by the experimenter, and that the ultimate test for this result is the "subjective judgements . . . of the observer." To speak in this way is to imply that the entities being measured are private sensations, inner events of which the observer—but not the experimenter—is aware through a faculty of inner perception (introspection). In addition, the last sentence quoted contains one of the most explicit uses of the assumption that O's sensation estimates are accurate, an assumption that is the second premise in the introspectionist argument, (C«)-(Cb)-(Ci), discussed earlier. It is surprising to hear these suggestions from a writer who is known for his attempt to apply the philosophy of operationism to psychology.50 It is even more remarkable to find these suggestions in the very same pages where Stevens says: " . . . in case of sensations what we want is a scale for the measurement of some aspect of the response of a living organism to a certain class of stimuli"; " . . . a subjective scale is a scale of response"; and ". . . loudness is a name which we give to a certain class of discriminatory responses." These statements suggest a behaviorist program of defining private sensations, a program laid out in greater detail the year before. Since sensation cannot refer to a n y private or inner aspect of c o n sciousness w h i c h does not s h o w itself in an overt manner, it m u s t 49

Stevens, " A Scale for the Measurement of . . . Loudness," pp. 406-408. S. S. Stevens, "Psychology and the Science of Science," in Psychological Theory, ed. M . Marx (New York, 1 9 5 1 ) , pp. 2 1 - 5 4 . 50

RATIO SCALES OF PERCEPTUAL MAGNITUDES

4*3

exhibit itself to an experimenter as a differential reaction on the part of an organism.... Thus, the sensation red is a term used to denote an "objective" process or event which is public and which is observable by any competent investigator. . . . In the same way that sensation denotes a class of reactions which satisfy certain criteria, attribute of sensation denotes a sub-class which satisfies more restricted criteria.51 This passage is puzzling. Stevens says that an experimenter may concern himself with a "private aspect of consciousness" so long as it "exhibits itself . . . as a differential reaction." Is not this to cling to the private and immeasurable entity while trying to rid oneself of it? It is one thing to say that sensations that are defined as differential reactions can be measured; it is quite another to say that private sensations that are exhibited as differential reactions can be measured. To say the latter suggests that it is the private sensation we really ought to measure, but since we cannot do that, we can only do second best and measure a public substitute, that is, the differential reaction. The pure behaviorist does not cling to private sensations, or try to measure them in terms of substitutes. He holds that private entities, if they exist, are of absolutely no interest or importance to psychological science. His position is that certain theoretical constructs may be defined in terms of quantitative observer estimates, and that the constructs may then be said to have been "measured." He says we may call these constructs "sensations" if we choose, but they are not to be thought of as substitutes for or representatives of some sort of private entities. The constructs must stand on their own feet, in virtue of their scientific usefulness, and not in virtue of going proxy for private entities which the scientist unfortunately cannot observe. The passages analyzed above are taken from writings early in Stevens' career. It is not unreasonable to expect that in the intervening thirty years he would have clarified his position on the interpretation of ratio scaling. And we do indeed find some evidence that he has. In 1958 he says that since the "non-operational aspects of sensation" are "inaccessible," the "operational stance is indispensable to scientific sense and meaning" in psychology. It follows, he thinks, that "verifiable statements about sensation become statements about responses." 52 In a 1959 article we read that "immediate experience," with its privacy, 51 82

Stevens, "Operational Definition of Psychological Concepts," p. 524. Stevens, "Measurement and Man," p. 386.

414

RATIO SCALES OF PERCEPTUAL MAGNITUDES

is not the object of the science of sensation. Sensation is, like temperature, "a construct, a conception built upon the objective operations of stimulation and reaction. We study the responses of organisms, not some nonphysical mental stuff that by definition defies objective test."5® In X960 Stevens cheerfully concedes that we cannot measure the "strength of a sensation" in the "inner, private, subjective" sense; but he insists that there is another sense of "sensation strength" which makes it possible for us to ask "sensible objective questions about the input-output relations of sensory transducers."54 These passages clearly suggest a behaviorist interpretation of psychophysical measurement. But it is one thing to suggest a position, another to describe it in detail, to lay it out for critical scrutiny and examine its implications. It is, therefore, quite impossible to say what position Stevens would take on many of the points discussed in this chapter. Furthermore, we can still find significant evidence of an introspectionist view of the object of psychophysical measurement in Stevens' later writings. In addition to the passages quoted in earlier sections, one more is worth mentioning: Sensations do not come with numbers written on them, and when we try to assess the ratio between a pair of them we find ourselves up against a difficult task of appraisal. It is no wonder then that subtle constraints and biases can influence the result. This is another way of saying that the outcome is a function of the method—as it always is in science. What we want, of course, is an unbiased method, one that on the average lets O make an estimate that is neither too high nor too low. Since we do not know in advance what his estimate should be, we can apply no independent criterion of validity.55 Stevens is here saying that in a ratio-scaling experiment O estimates sensations and E attempts to assign the correct numerals to these sensations. He implies that both tasks are difficult. They are difficult because the sensations have not been previously quantified, which makes it difficult for O, and because the sensations are private to O, which makes it difficult for E. Stevens' lamenting the lack of an "independent criterion of validity" can be understood only by assuming that he regards the scale whose validity is in question as a scale of a private magnitude! 88

Stevens, "The Quantification of Sensation," p. 612. Stevens, "Psychophysics of Sensory Function," p. 226. 85 Stevens, "Direct Estimation of . . . Loudness," pp. 24-25.

84

RATIO SCALES OF PERCEPTUAL MAGNITUDES

415

Hirsh provides another example of confusion about the object of psychophysical measurement. He begins by noting the difficulty of measuring private entities: The end product of our several sensory systems is the sensation— auditory, visual, tactual, olfactory, or gustatory. Each of us knows what a sensation is, because each of us has sensations It is difficult, however, for each of us to know about another person's sensations, because we cannot get inside his world of experience very easily. You and I may both say "red" when we see a particular object; but you cannot be sure that my impression of red is exactly the same as yours. . . . we cannot observe the sensations of others, and we can only measure what we can observe.56 He then suggests a way out of the difficulty: Since we cannot observe the sensation that exists in another individual's world of experience, it would seem indeed that we cannot measure sensation. On the other hand, we can twist the meaning slightly and define the sensation in terms of events that we can measure. When a man says, "I see red," we cannot measure the redness of his visual sensation, nor even be sure that he has one, but we can observe his verbal behavior—"I see red." The phenomena of audition may be studied in the same way. We cannot measure auditory sensations that are private, but we can measure sensations that are defined in terms of behavior or observable responses.57 These passages clearly contain a behaviorist-like interpretation of psychophysical measurement. But, in the first place, they are extremely sketchy and vague. How, precisely, are sensations to be defined? In terms of responses to physical stimuli? Or in terms of responses to private sensations? If the latter, the suggestion is still introspectionist. And how does Hirsh understand "definition"? Does he regard the statements that define sensation as stipulative definitions whose only defense is their usefulness? If not, then his suggestion may still contain introspectionist elements. In the second place, Hirsh seems, even more clearly than Stevens, to cling to the private entity while trying to expel it. He laments our inability to "get inside" another individual's world of experience, as if that is what the psychologist really wants to do. He says that the end products of our sensory systems are private sensations, that we s8 87

Hirsh, Measurement Ibid., pp. 5-6.

of Hearing, pp. 4-5.

4i6

RATIO SCALES OF PERCEPTUAL MAGNITUDES

cannot measure these, and that we must settle for measuring something closely related, something defined in terms of behavior or observable responses. The behaviorist does not attempt to measure private entities in terms of public substitutes. His view is that the end products of our sensory systems should be regarded either as physiological processes or as behavioral responses, not as private sensations. The measurement of physiological processes is accomplished, not by procedures like those in experiment M, but by tapping the organism with instruments, like oscilloscopes. As for behavioral responses, they may be measured in either of two senses. On the one hand, certain attributes of responses— duration, loudness, and so on—can be measured in the ordinary physical sense by using clocks, meters, and so on. On the other hand, certain theoretical constructs can be defined in terms of quantitative behavioral responses, and the entities thus defined can then be said to have been measured. These entities, however, should not be thought of as substitutes for the private entities of which Hirsh speaks. THE IMPORTANCE OF DISTINGUISHING THE INTERPRETATIONS

The preceding section was designed to indicate that the distinction between the introspectionist and behaviorist interpretations of ratio scales has more than academic interest. In spite of its importance, there is almost universal failure even to suggest the distinction, much less explicitly to draw it. And there is a universal tendency, found even among those who mention the distinction, to run the two interpretations together. The explanation for this confusion may be as follows. The most natural and attractive way of viewing ratio scaling in particular and psychophysical measurement in general is to see such measurement as the attempt to quantify privately observable, empirically real magnitudes. But the consequences of this view are that psychological magnitudes cannot be measured or psychophysical laws verified in an acceptable scientific manner. The philosophically sensitive psychophysicist recognizes these consequences, and seeks to avoid them by adopting a behaviorist interpretation of psychophysical measurement. He concedes that although private psychological magnitudes are incapable of scientific treatment, psychological magnitudes as defined in terms of observer responses can be measured and laws regarding them can be verified.

RATIO SCALES OF PERCEPTUAL MAGNITUDES

417

It is difficult, however, to discard the introspectionist interpretation completely, and for more than one reason. First, this interpretation is the more natural and the more attractive of the two, both because it has a longer history and because it is, for some deeper reason, intellectually more satisfying. Second, the psychophysical experimenter tends to be impatient with philosophical and methodological considerations, and to be unwilling to lay the behaviorist definitions out in detail and to examine their implications with care. More fundamental than either of these reasons, however, is the logical similarity between the theoretical fictions of the behaviorist interpretation and the empirically real, observable psychological magnitudes of the introspectionist. Although psychological length as defined in (i) and (ii) is merely a shadow of its introspectively observable counterpart, it is still a magnitude. Apparently one can still speak of Y's as being greater than, equal to, or less than one another. And although the principle of correspondence is, in the behaviorist view, merely a stipulative definition, still it receives the same formulation as the introspectionist principle. These similarities produce a strong tendency to slip from the behaviorist interpretation back into the introspectionist, or, what is virtually the same error, to treat psychological length as defined in (i) and (ii) as a substitute, or proxy, for the privately observable, scientifically unmanageable psychological magnitude posited by the introspectionist. This tendency manifests itself in a number of ways, one of the most important of which is to think of psychological length as a single magnitude that observers may estimate in a number of different ways. Thus O's one-half and one-fourth estimates in experiment M are thought of as different fractional estimates of entities that lie along a single continuum. And jnd, interval, and ratio scales are thought of as different scales of a single magnitude, so that if none of them agrees with either of the others, only one of them can be "valid." Failure to distinguish the introspectionist from the behaviorist interpretation produces vacillation between the two, and serves to conceal the difficulties in both. Psychological length is, on the introspectionist interpretation, an "empirically real" but scientifically unmanageable magnitude; on the behaviorist interpretation, it is a scientifically manageable but "fictional" magnitude. Running the two interpretations together creates the delusion that psychological length is both

418

RATIO SCALES OF PERCEPTUAL MAGNITUDES

"empirically real" and scientifically manageable. On the introspectionist interpretation, psychophysical law [5] is a description of O's perceptual mechanism, but it is unverifiable because only O can observe Y ' s . On the behaviorist interpretation, [5] is verifiable in the standard scientific manner, but it is not an explanation of how O perceives physical length. By amalgamating the two interpretations, [5] seems to emerge both as publicly verifiable and as an explanation of O's perceptual behavior. The amalgam is, of course, unstable, since it incorporates contradictory elements. Only when the two interpretations are carefully distinguished, only when each is set out in detail as has been done in earlier sections of this chapter, does it become possible to assess ratio-scaling procedures. It seems clear that the mak scale is illegitimate if interpreted introspectionistically. O observes no private psychological entities; and even if he did, the mak scale would provide us, not with measurements, but only with estimates of those entities. Furthermore, psychophysical law [5] is unverifiable on this interpretation, both because it rests on principle of correspondence (Ci), which in turn rests on unverifiable assumptions (Ca) and (Cb), and because O's (putative) quantitative estimates of psychological length are unconfirmable ("unverified" must replace "unverifiable," and "unconfirmed" must replace "unconfirmable" where O is said to estimate a physiological magnitude). If we adopt a behaviorist interpretation, these positive objections no longer apply. Rather, the objection becomes the negative one that the scaling of psychological length is unnecessary, because the concept of psychological length is unnecessary. In an earlier section it is argued that any legitimate purpose in constructing a ratio scale of psychological length can be equally well achieved without employing the concept of psychological magnitude. If this is so, then the concept is dispensable and may be abandoned. And there are reasons for thinking that it should be abandoned. One of these is simply Occam's maxim: Do not multiply entities beyond necessity. Put in more contemporary fashion: Do not clutter up the theoretical system with unnecessary constructs. If this reason seems insufficient, we may point to the confusion caused by the constant temptation, described earlier, for the behaviorist to slip back into an unacceptable, introspectionist way of thinking about psychological magnitudes. Other reasons emerge from

RATIO SCALES OF PERCEPTUAL MAGNITUDES

419

consideration of alternatives to systems that include the concept of psychological magnitude. PERCEPTUAL MAGNITUDES VERSUS PERCEPTUAL ABILITIES It may seem that to abandon the concept of perceptual magnitude is to abandon perceptual psychophysics. On the contrary, it is to adopt a conception of perceptual psychophysics which differs both from the "old" psychophysics of Fechner and from the "new" psychophysics of Stevens. Whatever their differences, both theorists view perceptual psychophysics as the attempt to measure perceptual magnitudes in order to discover mathematical laws relating these magnitudes to stimulus magnitudes. It is clearly unorthodox, therefore, to recommend doing away with the concept of perceptual magnitude. But it is one thing to embrace an unorthodox view of the nature of perceptual psychophysics, and another to abandon the science. The most promising unorthodox view would replace the concept of a perceptual magnitude with that of a perceptual ability. Perceptual psychophysics then becomes the experimental discipline that describes, measures, predicts, and explains the perceptual abilities of organisms. Let us examine the consequences of radically applying this view of the science to experiment M. Experiment M is really an attempt to determine O's ability to make fractional estimates of rod length by sight. O's ability to make such estimates may be perfect, or it may be less them perfect in varying degrees. These degrees of ability may be represented in psychophysical functions of the form, = b If O were able to quarter lengths perfectly, the value of b would be .25. If he were able to halve lengths perfectly, the value would be .50. To the extent that b falls below these values, O overestimates the comparison rod (underestimates the standard). To the extent that b exceeds these values, O underestimates the comparison rod (overestimates the standard). Thus the value of b provides a measure of O's ability to make fractional estimates of length. This measure can be obtained merely by constructing the functions of figure 6. The W-Q functions of figure 7 are not only unnecessary; it is difficult to see immediately what connection they have with perceptual ability. The discovery of functions enables us to compare O's abilities to make fractional estimates of one-fourth with his ability

420

RATIO SCALES OF PERCEPTUAL MAGNITUDES

to make fractional estimates of one-half, one-third, two-thirds, and so on. O underestimates the comparison rod in the first part of experiment M by .34 - .25/.25, or 36 percent. He underestimates the comparison rod in the second part by .58 — .50/.50, or 1 6 percent. Thus we can say, not only that O's ability to fourth lengths is not as great as his ability to halve them, but also how much greater one ability is than the other. What this difference in abilities shows is not of concern here, but one might wish to consider the hypothesis that other things than length are among O's cues, or that he does not comprehend fractions perfectly, or even that different physiological mechanisms are employed in the two fractionations. As noted in an earlier section, the functions also make it possible for us to compare O's ability to make ratio estimates with his ability to make interval estimates of length. Again, as noted above, by comparing psychophysical functions with their associated psychophysiological 6 - 8 functions, we can make inferences regarding the physiological mechanisms underlying O's ability to make ratio estimates. There is no philosophical problem of psychological measurement for the science of perceptual abilities. In the orthodox view, psychological measurement is regarded as the measurement of psychological magnitudes. There is a question, however, as to whether such measurement is possible. If psychological magnitudes are private, as the introspectionist believes, then they are, it would seem, incapable of measurement. If we define psychological magnitudes in terms of observer responses, as the behaviorist does, then they appear capable of measurement. But this solution seems to allow the prize to slip through our fingers. What we really wanted to measure was an empirically real, although private, dimension of mind; but all we managed to measure was an anemic substitute, a theoretical construct or fiction. We really wanted to discover the relation between a psychological magnitude correlated with but defined independently of a physical magnitude; but instead we were forced to define the psychological magnitude in terms of (observer responses to) physical magnitude. These difficulties are completely circumvented if psychological measurement is construed as the measurement of perceptual abilities. When we abolish psychological magnitudes, such as psychological length, psychological weight, and so on, there can be no question of how we measure them. Viewed as an attempt to determine ability to

RATIO SCALES OF PERCEPTUAL MAGNITUDES

421

fractionate length, experiment M consists in (a) measuring the lengths of the rods to be presented, (b) eliciting quantitative estimates of rod length from O, and (c), computing the functions of figure 6. Nowhere in these three stages do we find any attempt to measure psychological magnitudes. Speaking loosely, we may say that (a), (b), and (c) constitute a complex procedure for measuring O's ability to fractionate physical length. But strictly speaking, the only measurement that occurs in M is the measurement of physical length in stage (a). And since there is no problem in measuring physical length, there is no problem of measurement when M is regarded as an attempt to determine one of O's perceptual abilities. In the new psychophysics the old philosophical doubts about the possibility of psychological measurement reappear with a somewhat different emphasis, that is, as doubts about the possibility of determining which of the various types of psychological scale is "valid." In the attempt to scale perceptual magnitudes such as psychological length, a problem arises over the validity of competing scales. Shall we employ a jnd, an interval, or a ratio scale? On the introspectionist interpretation this problem is unavoidable and insoluble. Each scale is constructed from what are taken to be O's estimates of private psychological entities. Since there is no way of confirming such estimates, there is no way of determining whether the scales built from them accurately represent the magnitude of the private entities. On the behaviorist interpretation, there still seems to be a problem. Psychological magnitudes are defined in terms of O's publicly observable responses to physical stimuli, so that the privacy problem is solved. But do O's jnd responses, his interval responses, and his ratio responses to length define three different psychological magnitude, three different psychological lengths? It seems odd to say that they do, since they are all responses to the same physical magnitude. On the other hand, if a single psychological magnitude is involved, scales constructed from the different responses compete with one another and we must choose among them. And how shall we do that? If we take ourselves to be scaling perceptual abilities, these difficulties do not arise. There is no philosophical problem concerning the validity of psychological scales for the science of perceptual abilities. A scale constructed from the partition estimates of an observer at best provides a measure only of his ability to make ratio estimates.

422

RATIO SCALES OF PERCEPTUAL MAGNITUDES

T o deny this would be like asserting that a scale of a person's muscular ability to lift rods provides a measure of his ability to discriminate rods, or that a scale of his reading ability provides a measure of his mathematical ability. Of course it m a y be necessary to choose between different measures of perceptual ability. For example, after discovering the

functions of figure 6, w e must decide whether O ' s ability to

fractionate lengths is best represented by the difference between the value f o r a perfect estimate (.25 f o r quartering, .50 f o r halving) and the obtained value of b, or b y the ratio between these values. But this decision can be made on the basis of scientific convenience and usefulness. It involves no philosophical problems. T h e foregoing is a radical application to experiment M of the conception of psychophysics as the science of perceptual abilities. It recommends replacing Stevens' l a w , [ 1 ]

=

b y a law or laws h a v -

ing an altogether different form. A less radical application also seems possible, since Stevens' law can apparently be interpreted in a manner that does not presuppose the concept of psychological magnitude, and thus allows it to be viewed as a law of perceptual ability. In equation [5], let V mean " t h e numeral assigned b y O to and let $> mean " t h e stimulus expressed in appropriate physical u n i t s . " Interpreted in this manner, [5] does not refer to any psychological magnitude, such as "psychological l e n g t h " or "psychological w e i g h t . " It is, therefore, quite misleading to use Y , the usual symbol f o r a p s y chological magnitude, in stating the law. It would be less misleading to state it as [5'] i>e = $ m n , where b y O to

and

means " t h e numeral assigned

means " t h e stimulus expressed in appropriate

physical units." Alternatively, 3>e could be taken to mean merically estimated b y O " and

could be taken to mean

as nuas mea-

sured by E . " This w a y of stating the law makes it completely clear that only physical magnitudes are involved in the experimental and theoretical framework of such experiments as M , and that no special methods f o r measuring psychological magnitudes are presupposed. The only objection to stating Stevens' law as [5'] is that it m a y erroneously suggest that the law relates two types of physical dimension: f o r example, ') speak of a dimension called perceptual length and contrast it with physical length. These dimensions are, in the interpretation presently under discussion, examples of psychological and physical dimensions, respectively. This interpretation can be diagramed in a table identical in all respects with table 16, except that the first column reads, "Perceptual loudness, Perceptual pitch, Perceptual brightness," and so on. The interpretation has at least one advantage over its predecessor. Perceptual dimensions (if there are such) are involved in every instance of perception, at least in every instance of perception of relative magnitude. And, since perceptual dimensions are clearly psychological, laws relating these dimensions to their physical counterparts are clearly psychophysical laws of perception. But in other respects the current interpretation produces special difficulty. Can perceptual dimensions be measured? If so, how? And what are they dimensions of? Let us take the latter question first. When we speak of the perceptual length of a rod, to what are we attributing perceptual length? Not, surely, to the rod. For we would thus be suggesting that the rod possesses length of two different kinds—perceptual length and physical length—which seems absurd. The rod possesses weight, hardness, color, length, and so on, but it does not possess something called "perceptual weight," or "perceptual hardness," or "perceptual color," or "perceptual length." Perhaps we can say that perceptual length is a dimension of the sensations produced in O when he perceives rods. If we do, we seem to convert the current interpretation of psychological dimensions into one discussed earlier, thus failing to produce a different interpretation and inheriting all the difficulties of the old one (more on this in a moment). Perhaps we should say, then, that perceptual length

484

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

is a dimension of the observer's perceptions, or percepts. This answer, natural as it may seem, is the product of deep confusion. Considerable preparation is needed to bring the confusion to the surface. We often use expressions similar to "perceptual length" to describe opinions about unperceived objects. For example, suppose (c) O infers that Ri is longer than R2 by looking, not at the rods, but at their shadows, and that the rods are actually the same length. We can express this fact by saying that (c') the inferred length of Ri is greater than the inferred length of R2, although the real length of Ri equals the real length of Rs. Or, again, suppose (d) O infers that Ri is 6 inches long and Ra is 7 inches long, when actually both are 6 inches long. We can express this fact by saying that (d') the inferred length of Ri is 6 inches and the inferred length of R2 is 7 inches, although the real length of each is 6 inches. Statements (c') and (if)—about inferred length— have an important similarity to statements (a') and (b')—about perceived length. We might describe this similarity by saying that the one set of examples deals with directly perceived length, and the other set with indirectly perceived length. Or we might say that both deal with apparent length, the one with perceived length and the other with inferred length. With apparent length as the generic concept, we can discover numerous instances of it. Suppose (e) O uses a ruler to measure the two rods and finds one to be longer than the other, when in fact the rods are equal in length. We can express this fact by saying that (e') the measured length of one rod is greater than the measured length of the other, although the rods have the same length. Or suppose (f) O simply guesses that one rod is longer than the other, when they are equally long. We can express this fact by saying that ( f ) the guessed length of one rod is greater than the guessed length of the other, although their actual lengths are the same. Such examples can be multiplied indefinitely, each example containing some instance of the concept of apparent length to describe commonplace facts that are not in the least problematic. Problems arise when we ask what apparent length is the length of. Consider example (d'). What is the inferred length the length of? The rods about which the inferences are made? The inferences themselves? The thoughts expressed by the inferences? The reports of the inferences? Each of these answers is absurd. To say that inferred length is

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

485

the length of the rods is to imply that the rods are just as long as they are inferred to be, which rules out the possibility of error. It makes no sense to say that inferences, thoughts, or reports have length, unless we mean the time (or space) it takes to make the inference, have the thought, or complete the report. Length in this sense is not inferred length, since the inference that a length is great can be made in a small period of time, can express a fleeting thought, and can be made in a short report. Attempts to say what inferred length is the length of are contortions produced by asking an illegitimate question. The question is meaningless, and so is any answer to it. Inferred length, guessed length, or perceived length—in general, apparent length—is not the length of anything. This becomes clear when we examine the relation between the primed and unprimed statements in the examples above. The primed statement has the same meaning as its unprimed counterpart. In general, " T h e length of Ri appears greater than the length of R2, but the length of Ri is equal to the length of R2," means the same as "The apparent length of Ri is greater than the apparent length of R2, but the real length of Ri is equal to the real length of R2." Since the first statement does not presuppose entities that apparent length is the length of, neither does the second. Let us illustrate this point for perceptual length. Statement (a)—"O perceives Ri to be longer than R2, but Ri is the same length as R2"—means the same as statement (a')—"The perceptual length of Ri is greater than the perceptual length of R2, but the real physical length of Ri is equal to the real physical length of R2." Since (a) does not presuppose entities that possess perceptual length, neither does (a'). It is quite difficult to keep this logical point in mind, and some explanation for the difficulty is needed. The inclination to posit an entity that apparent length is the length of is the result of an illusion of grammar. Statement (a') speaks of "perceived length." Since "length" is a term designating a property, and since properties must be properties of something, we look for something that perceived length is the length of. T o resist this illusion we must remind ourselves that (a') means the same thing as (a). But we forget that it does. Under the spell of the illusion, we try to find entities to which we can attribute apparent length. Failing to find them, we invent some, often giving them special names. "Percepts" are said to possess perceptual length. "Inferents" might be said to possess in-

486

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

ferred length. In general, "appearances" are said to possess apparent length. The entities referred to by these terms have no real existence: they are neither physical entities, nor physiological entities, nor even psychological or mental entities. They are grammatical fictions, entities invented to meet the needs of an illusion of grammar. One concept of sensation arises in just this way. Instead of using the term "percept" to describe the grammatical fictions demanded by statement (a'), we might use the term "sensation." If we use the term in this way we must say that sensations are always involved in perception. Suppose " O ' s sensation of Ri is greater than his sensation of Rs" is just another way of saying that " O perceives (senses) Ri to be greater than Ra." Suppose, comparably, that " O has a sensation of R i " is just another way of saying that " O perceives (senses) Ri." Then of course it is true that sensations are always involved in perception. Sensations, in this sense, are necessarily involved in perception, since to speak of sensations is just another way of describing perceptions. This use of the term cannot be condemned as illegitimate, but it must be carefully distinguished from the use made of the same term in a previous interpretation of the distinction between psychological and physical dimensions. In that interpretation "sensation" refers, not to a grammatical fiction, but to introspective, private entities of the same type as pains and afterimages, which are produced by the stimuli of perception. In the current sense, sensations are not introspected, nor produced by anything: they are grammatical fictions. Confusion between these two senses probably explains why psychologists and philosophers have so often thought that introspective sensations are always involved in perception. In the grammatical sense, it is trivially true that sensations are always involved in perception. It is easy to confuse the grammatical with the introspective sense, and thus be led to the nontrivial belief that introspective sensations are always involved in perception. It may be supposed that these grammatical fictions are a species of theoretical fictions, or theoretical constructs, or theoretical entities, as they are alternatively labeled by philosophers of science. The electron is often cited as an example of a theoretical entity. Scientists posit such entities in order to explain and predict observable phenomena. By positing electrons we can explain the traces in a cloud chamber, and predict similar phenomena. By incorporating the electron into a theory

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

487

of the atom we can explain why a given substance reacts with another, and predict other reactions. But it is difficult to see what explanatory or predictive function could be served by "percepts." That O's percept of Ri is greater than his percept of Rt does not explain why he perceives Ri to be longer than R2, since the two italicized statements mean the same thing; that is, they are just different ways of saying the same thing. And, for the same reason, the predictive value of the first statement is no greater than that of the second. Suppose O perceives Ri to be longer than R2, R2 to be longer than Ra, and Ra to be longer than R4. Knowing only this, we can predict nothing about O's perceptual responses to length. But if we know in addition that R2 is longer than Ri, R3 is longer than R2, and R4 is longer than R3, then we may hazard the prediction that, within limits at least, O perceives the longer rod to be the longer. This prediction is made without positing any entities called "O's length percepts." The prediction is not facilitated, enlarged, or made more precise by translating statements about the lengths perceived by O into statements about O's length percepts. The same point applies to the prediction of O's numerical responses to length. If O perceives Ri to be 6 inches long and Ra to be 7 inches long, then O's percept of Ri has a perceptual magnitude of 6 and his percept of R2 has a perceptual magnitude of 7. Since the two italicized statements have the same meaning, the predictive value of the latter is no greater than of the former. Suppose O perceives rods Ri, R2, Ra, and R4 to be 6, 7, 10, and 1 3 inches long, respectively. Knowing only this, we can predict nothing about O's numerical perceptual responses to length. But if, in addition, we know that the rods are in fact 6, 7, 10, and 13 inches in length, respectively, then we can hazard a prediction that O will perceive rods to have the numerical length they in fact do have. This prediction neither requires nor is enhanced by positing "length percepts" to which numerals are assigned. Consequently, length percepts have no special explanatory or predictive value. It is, therefore, wrong to compare grammatical fictions like "percepts" with theoretical fictions, or theoretical entities. It would seem more appropriate to compare them with the entities posited in certain games we play with children. The child says, "I met no one on the playground today," eliciting the reply, "And how was Mr. No One?" This reply is made possible by an illusion of grammar. The illusion is much less compelling than the one that produces "percepts" and

488

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

other "appearances." For normal adults it dissolves instantly into whimsy. For of course there is no Mr. No One, and the child's remark was not intended to imply that there is. It would appear that we should say also that there are no "percepts" or "appearances," and that talk about perceptual and other apparent magnitudes is not intended to imply that there are. But the illusion that there are is more compelling here. It is treated by psychologists, not in the whimsical manner it deserves, but quite seriously. If there are no entities to which perceptual magnitudes can be attributed, these magnitudes are strange creatures indeed, quite tinlike the magnitudes of length and weight we attribute to physical objects, even unlike the magnitudes of loudness and pitch we attribute to sounds. There is a serious question as to whether we can even call them "magnitudes." At first sight perceptual magnitudes seem to have the logical properties of magnitudes like length, weight, and loudness. We can compare lengths, weights, and loudnesses; similarly, we compare length percepts, weight percepts, and loudness percepts. But note. We can meaningfully say that Ri is longer than R2, but not that a percept of Ri is longer than a percept of R2. What we must say is that the one percept is greater than the other. Now what is meant here by "greater"? The only possible answer is that it means "perceptually greater," which is totally unhelpful. When O's percept of the weight of Ri is greater than his percept of the weight of R2, is it greater in the same sense or respect in which his percept of the length of Ri is greater than his percept of the length of R2? We do not know how to begin to answer this question. It seems to make no sense. Even more baffling questions arise when we ask what meaning numerals have when they are used to describe percepts. " O perceives Ri to be 6 inches long and R2 to be 7 inches long" becomes, when transformed into percept language, "O's percept of Ri has a magnitude of 6 and his percept of R2 has a magnitude of 7." What do these numerals mean? They cannot mean that O's percepts are 6 and 7 inches long, respectively, for it makes no sense to attribute length to percepts or to express their magnitude in the units employed to express length. Do the numerals signify units? If so, units of what? The only possible reply is that they signify "perceptual units" or units of perceptual magnitude, which is totally unhelpful. Worse than that, it seems to have no

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

489

meaning. For when we try to say how these units are chosen, nothing comes to mind. The fact of the matter is that units of perceptual magnitude are not chosen. The numerals assigned to percepts are just those numerals assigned to the rods with which they are associated. Insofar as they signify units, they signify units of length. They become part of the descriptions of percepts only because percept language is an alternative way of describing numerical perceptions of objects that are measured in units of length. These remarks indicate that the notion of "measuring percepts" is a curious one. It is clear that the measurement of percepts is not to be compared with the measurement of magnitudes like length, weight, loudness, and brightness. For "perceptual magnitudes" are not genuine magnitudes. They are grammatical constructions possessing only some of the logical features of genuine magnitudes like length and loudness. They are analogous to genuine magnitudes, but the analogy is a slim one indeed. We speak of the "relative magnitude of percepts" and of their "measurement" only by slender analogy. Percepts do not literally possess magnitude, and we do not literally measure them. The so-called measurement of perceptual magnitudes thus seems insufficiently nourishing to satisfy the appetite of perceptual psychophysics. What relevance can the measurement of a grammatical construction like perceptual length have to length perception? How can it explain length perception? The proper answer would seem to be that it cannot. Perceptual magnitudes seem to have only a descriptive function. By employing the notion of perceptual length, we can describe the length perceptions of an observer in an alternative manner, but we do not thereby explain his length perceptions. And it is not even clear why we should prefer the alternative description. Instead of saying, "O's percept of Ri is greater than his percept of R2," why not say that O perceives Ri to be longer than R2? Why not dispense altogether with the concept of perceptual magnitude? That the concept is dispensable was shown in chapters 8 and 9, where it was argued that both jnd methods and Stevens' methods of psychological measurement can be interpreted as methods for measuring perceptual abilities, rather than perceptual (or sensation) magnitudes. One final point about construing psychological dimensions as perceptual dimensions is worth making. As noted earlier, perceptual

49°

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

magnitudes are a species of apparent magnitudes. Perceptual length, inferred length, guessed length, and measured length are all species of apparent length. There is no reason, other than to divide the scientific labor, for perceptual psychophysics to concern itself exclusively with those apparent magnitudes we have called perceptual. There is no reason that the science should not also investigate observer responses that are mediated by instruments. For example, a perfectly legitimate psychophysical experiment would employ an observer who is required to measure the frequency of sound waves with an oscilloscope. The stimuli of the experiment are sound waves. The responses are the sound-wave reports made by O through his use of the oscilloscope. The stimuli must of course be independently measured by an experimenter, E, who may also use an oscilloscope. And E's sound-wave observations must be more reliable than those of O, either because his oscilloscope is better, or because he is more experienced in its use, or because his condition is normal, and so on. (For more on this topic, see the next section of this chapter.) If O finds the intensity of sound wave Wi to be greater than that of sound wave W2, then the "measured intensity" of Wi is greater than the "measured intensity" of W2. If O finds the intensity of Wi to be 40 decibels and the intensity of W2 to be 45 decibels, then the "measured intensity" of Wi is 40 and the "measured intensity" of W2 is 45. Measured length is, like perceptual length, a species of apparent length. Like perceptual length, it depends on perception, although it also depends on the use of an instrument or operation of measurement. It is just as legitimate (or illegitimate) for a psychophysicist to try to "measure" measured length, or inferred length, or guessed length, as it is for him to try to "measure" perceptual length. The point above illuminates the oddity in regarding perceptual and other apparent magnitudes as magnitudes. The measured length of a rod—its length as measured by some observer—is neither a magnitude of the rod, nor a magnitude of some internal process in the observer, nor a magnitude of the observer's responses (their duration, for instance). It is best to cease regarding it as a magnitude, and to speak only of O's responses in his attempts to measure rods. In general, it is best to abandon the notion of an apparent magnitude, and to speak only of observer responses to stimuli. And to abandon the notion is

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

491

to abandon the last surviving interpretation of the distinction between psychological and physical dimensions. PSYCHOLOGICAL AND PHYSICAL OBSERVATIONS Although each later interpretation of the symmetrical distinction between psychological and physical dimensions seems an improvement over its predecessor, none seems entirely satisfactory. Perhaps the reason is that a distinction between types of dimension is ill-conceived from the start, and that we should distinguish instead between psychological and physical observations. There is surely a difference between weight observations made by lifting ("psychological") and weight observations made by means of a balance ("physical"), between sound observations made by listening ("psychological") and sound observations made by means of an oscilloscope ("physical"), and so on. Perhaps, then, the troublesome distinction between psychological and physical dimensions is simply a misleading form of the legitimate distinction between psychological and physical observations. In any event, the latter distinction seems adequate for psychophysics and seems, at first sight at least, to present no special problem. Given any dimension—loudness, hue, sweetness, weight, length, hardness—we can separate the psychological observations of that dimension from the physical observations. Physical scientists are presumably interested only in the latter; psychologists are interested in both, and in their relation to each other. Stevens and Davis seem to have something like this distinction in mind in a passage at the beginning of their study of hearing. The word dimension is used here to mean any of what are commonly called the "physical" aspects of sound, such as frequency, energy, velocity and phase. These are ways in which sounds may vary; or they are scales in terms of which sounds may be measured. The "physical" aspects are commonly distinguished from the "subjective" or "psychological" aspects of sound, and it is well for our purposes to ascertain upon what operations or concrete procedures such a distinction rests. The operations involved in the measurement, and hence in the definition, of the energy of sound consist of noting the effect of the sound-wave on some other physical system such as a microphone with its associated amplifier and output meter. On the other hand, the operations involved in the determination of the loudness of a sound consist of the direct procedure of noting the effect of the sound on

492

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

the living organism. The difference between the two procedures lies in the fact that in measuring the "physical" aspect an observer makes a judgement about a scale-reading, whereas in determining the "subjective" aspect an observer makes a judgement directly about the soundwave itself as it affects his sense-organ. There is perhaps no reason for considering the one type of observational judgement as any more basic or "physical" than the other, except for the fact that the observations of physics—those which we call pointer-readings— constitute the class of human reactions which show the greatest uniformity among individuals and which have therefore been made basic in the exact sciences. The direct observation of the aspects of a stimulus, without the aid of instruments, is made with much less precision. Consequently we can say that the general problem of the psychology of hearing is that of observing the aspects of sound, as it affects the organism directly, and comparing the results with observations of the aspects of sound made with the help of instruments. 18

The passage is ambiguous. At many points the authors distinguish the psychological from the physical "aspects" of a sound. The term "aspects" is vague, and it may be taken as synonymous with "dimensions." At other points the authors focus more on a distinction between psychological and physical observations or "judgements." Perhaps the passage illustrates the way in which the distinction between psychological and physical dimensions can be transformed into the superior distinction between psychological and physical observations. As we shall discover, the latter is not wholly acceptable either. Whatever its special advantages may be, the distinction between psychological and physical observations is not without difficulties, some of them already familiar to us. What, for instance, are the criteria for the distinction? The most obvious answer is that the criteria here parallel those for the distinction between psychological and physical dimensions. It may be that (A') psychological observations are perceptual, whereas physical observations are operational; and that (B') psychological observations are not intersubjectively confirmable, whereas physical observations are intersubjectively confirmable. Let us scrutinize these two criteria. First of all, the categories "perceptual observation" and "operational observation" are not mutually exclusive, because all observations of whatever type involve perception of some sort. This division 18 S. S. Stevens and Hallowell Davis, Hearing: Its Psychology and Physiology (New York, 1938), pp. 3-4.

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

493

of categories is, therefore, not so sharp as it seems. Nevertheless, we can distinguish between observations involving perception but no operations, and those involving perception and operations; and, for want of better terms, we may label the former "perceptual" and the latter "operational." But why should we regard the perceptual observations as psychological, and the operational observations as physical? If this issue is more than terminological, if the terms are meant to suggest that perceptual observations are the special subject matter of the psychologist and operational observations the special property of the physical scientist, then there is an objection. Psychologists frequently study operational observations, and there is nothing peculiar in their doing so, as the following historical example demonstrates. The first studies of reaction time arose because astronomers noticed that their observations of stellar transits were not always in agreement. Astronomers A and B, using either adjacent telescopes or the same telescope with two eyepieces, did not always agree in recording the time at which a given star passed the cross hair of the telescope. It was soon discovered that a difference in the reaction times of A and B caused the discrepancy. Although the star passed the two cross hairs at the same instant, A required more time to press the key to his clock than did B. Psychologists became interested in the phenomenon, and with the invention of the chronoscope (an instrument that electrically records the stimulus and the response to it) they began to measure these and other reaction times with great precision.17 This example shows that psychologists may and do investigate operational observations: the observation of stars with a telescope is operational, and yet it is material for psychological study. The branch of psychology now called "human engineering" deals in large part with the abilities of human perceivers (airplane pilots, electrical engineers, etc.) to make certain operational observations (radar observations, oscilloscopic observations, etc.) with the help of instruments. Criterion (B'), which says that physical observations are, and psychological observations are not, intersubjectively confirmable is even more objectionable than the first criterion. Nearly all the observations mentioned in this chapter are intersubjectively confirmable. Obviously this is true of observations of sound-wave frequency and of 17 Edwin G. Boring, "The Beginning and Growth of Measurement in Psychology," in Quantification, ed. Harry Woolf (New York, 1961), pp. 1 1 5 - 1 1 6 .

494

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

light-wave intensity made by appropriate meters, of observations of weight made by means of a balance, and of observations of length made with a meterstick. But it is also true of weight observations made by lifting, and of length observations made by simple visual comparison. After Oi lifts objects X and Y and reports that X is (feels) heavier than Y , O2 can lift the same two objects and make a report about their relative weight. His report will either agree or disagree with Oi's report. Agreement provides confirmation; disagreement, disconfirmation. There is no apriori reason to suppose that there will be less agreement among observations of this kind than among light-wave observations or weight observations made with a balance. It is less obvious but just as true that perceptual observations of brightness, pitch, loudness, and hue are intersubjectively confirmable. After Oi looks at objects X and Y and reports that X is brighter than Y , O2 can look at the same two objects and make a report on their relative brightness which will either agree or disagree with O's report. Again, there is no apriori reason to anticipate less agreement among observations of this kind than among observations of any other kind. The point above is not meant to deny that there is a significant difference between the psychologist's treatment of observations and the physical scientist's treatment of them. In a chemical laboratory, if one technician's weighing of the precipitate of a certain chemical reaction cannot be confirmed by another technician, one or both weight observations are discarded. But in the psychological laboratory, this need not be so. A psychologist may wish to examine differences between the weight observations of two subjects and explain them. In this instance the one observation is as much a part of the experimenter's data as the other. To put the point in a sharper but less sophisticated form, the physical scientist requires that the observations composing his data be correct; the psychologist, on the other hand, may include incorrect observations among his data. Perhaps, then, no distinction between psychological and physical observations can be drawn. But we can and should distinguish between psychological and physical treatments of observations. The distinction between psychological and physical observations is closely related to another distinction often stressed by philosophers of psychology: the distinction between the experimenter's observations and the observer's observations in a psychophysical experiment. In

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

495

general, in a psychophysical experiment an experimenter, E, presents an observer, O, or a group of observers with a stimulus or a group of stimuli. The O is asked to respond to these stimuli in a manner specified by E. A description of the stimuli (the types of stimuli, the conditions and manner in which they were presented, etc.) and the responses of the O's (what they said and did, the instructions and motivations given them, their ages, training, mental and physical condition, etc.) constitute the data obtained in the experiment. The following is a crude but paradigmatic example of a psychophysical experiment. Five rods of different lengths are selected by E with a meterstick. O is required to arrange the five rods in ascending order of length by simply looking at them. (The purpose of the experiment is not of concern here.) E's observations of the stimuli are operational, whereas O's observations of the stimuli are perceptual. In this respect experiment is typical of the large majority of psychophysical experiments actually performed. The view has therefore arisen that in any proper psychophysical experiment experimenter observations must be operational and observer observations must be perceptual. I have already argued that this view is erroneous, but the point is worth laboring. The following experiments, though hypothetical, are legitimate psychophysical experiments. 1 ) E weighs five objects on a chain balance to the nearest thousandth of a gram. He then requires an inebriated O to weigh the same five objects on the same chain balance to the nearest thousandth of a gram. 2) E weighs five objects on a chain balance to the nearest thousandth of a gram. He then requires an O in normal condition to arrange the objects in ascending order of weight by lifting them. 3) E arranges five objects in ascending order of weight by lifting them. He then requires an inebriated O to arrange the objects in ascending order of weight by lifting them. In (1) the stimulus observations of both E and O are operational. In (2) E's stimulus observations are operational and O's are perceptual. In (3) the stimulus observations of both E and O are perceptual. Thus, with respect to their operational or perceptual character, the observations of the experimenter and those of the observer need not be of different types. What, then, is the essential difference between them?

496

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

Is there any reason for drawing the distinction? Clearly there is. In the experiments above E's observations of the stimuli are more reliable than O's. If E and O are interchanged in the experiments above, each of the three new experiments will be illegitimate, since O ' s stimulus observations will in each instance be more reliable than E's. Consequently, we may lay down the following requirement for a legitimate psychophysical experiment. Requirement 1. The observations employed in selecting the stimuli for a psychophysical experiment must be more reliable than the stimulus observations required of the observer. It is sometimes said that the experimenter may not serve as his own observer, or subject. In one sense this is true; in another sense it is not. Let us begin with the latter. Insofar as E responds to the stimuli of the experiment in the manner required of his O's, he becomes just another O , and his responses may no more be used in selecting the stimuli than may those of any other O. It is logically possible, however, and sometimes psychologically possible, for E to select the stimuli, say by weighing them, and then to become an O for his own experiment. If the O ' s are required to respond without the benefit of any previous acquaintance with the stimuli (such as having weighed them on a balance), then to become his own O , E must be able to respond as if he also had had no previous acquaintance with the stimuli. This may be extremely difficult for him to do; indeed, it may for him be psychologically impossible. But for another experimenter it may not be. O n the other hand, in another sense the experiment logically cannot be his own observer. In the paradigm experiment described earlier O was required to arrange five rods in ascending order of length by looking at them. O ' s observations cannot be employed to select the stimuli for the experiment. Suppose small letters have been printed on the rods for purposes of identification, and that O arranges them in the order c, d, a, b, e. Consider the consequences if E should take this arrangement to represent the true relative lengths of the objects. In the first place, it is impossible for O to make a mistake in his arrangement of rods. In the second place, one outcome of the experiment is determined in advance: O will arrange the rods in their "true" order. N o

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

497

legitimate psychophysical experiment can have these features. Consequently, we can lay down the following requirement: Requirement 2. The observations required of the observer in a psychophysical experiment cannot be employed in selecting the stimuli to which the observer is required to respond. Another way of stating this requirement is to say that the observer's observations cannot be criteria for the stimuli to which he is required to respond. Or, even more simply, the observer's observations cannot define the stimuli of the experiment. It is logically possible for the E in a psychophysical experiment to make observations used in selecting the stimuli, and then to switch roles and make the observations of these stimuli which the experiment is designed to study. What is logically impossible is that one and the same group of observations should be both observations to select stimuli and observations to be studied. Therefore it is misleading to describe a psychophysical experiment as a comparison between an experimenter's observations of stimuli and an observer's observations of stimuli. To be precise, the comparison is between observations not being studied and observations being studied. Requirements 1 and 2 make it clear that the distinction between types of observation presupposed by a psychophysical experiment is not between experimenter's observations and observer's observations, but rather between observations that define (are used to select) the stimulus and observations that define (constitute) the response. It is important to understand that no observations are by nature of one kind rather than of the other, as is revealed by the three hypothetical experiments described earlier. The observations defining the stimulus in one experiment may be the observations defining the response in another. Observations defining the stimulus, or the response, may be perceptual or operational; they may have a high degree or a low degree of intersubjective verifiability. The only requirements are the two mentioned earlier: the same observations cannot define both stimulus and response, and the observations defining the response must be the less reliable. It should be noted that this latter requirement does not imply that observer observations cannot be accurate. They may or may not be. But to be the less reliable, the probability of their accuracy must

498

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

be lower than the probability of accuracy of the stimulus-defining observations. The question of how we determine that one observation is more accurate than another is a deep one, and it is not explored here. It leads immediately into the morass called "the psychologist's circle": Can it be known that the experimenter's observations (those defining the stimulus) are more accurate than the observer's observations (those defining the response)? The distinction between observations that define the stimulus and those that define the response is, for perceptual psychophysics, quite simply the distinction between stimulus and response. Perhaps to most readers it will have been clear in advance that the distinction between stimulus and response is presupposed by perceptual psychophysics. What may not have been clear is the relation between this familiar contemporary distinction and the traditional distinction between psychological and physical dimensions. This chapter began with primitive versions of the distinction. Through criticism and analysis, one version was replaced by another, each being less objectionable than its predecessor. But no version was found to be completely acceptable. Not until the distinction was utterly transformed into one between response-defining observations and stimulus-defining observations did we find a completely acceptable distinction. The suggestion of this analysis is that the traditional distinction between psychological and physical dimensions reduces, for perceptual psychophysics, to the contemporary distinction between response and stimulus. The analysis suggests that perceptual psychophysics neither requires nor can tolerate distinction between psychological and physical dimensions. What perceptual psychophysics studies is not relations between dimensions of different types, but relations between perceptual responses and perceived stimuli. Consequently, it is misleading to describe psychophysics as the science of mind and body, because this description suggests that the science spans a gulf between two different worlds: the private mental world and the public physical world. When the distinction between response and stimulus is made the fundamental one, there is no suggestion of different worlds. The response, like the stimulus, is publicly observable and capable of scientific treatment. There is no gulf between response and stimulus. "Psychophysics is the science of relations between stimulus and

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

499

response" is a useful slogan. But, like all slogans, it can be used to gloss over difficult questions, and also to describe positions that are mutually incompatible. Contemporary psychophysicists sometimes use the slogan to dissolve the traditional problem of psychophysical measurement, as well as the larger problem of the relation between mind and body. Stevens, among others, sometimes suggests the following argument. The older psychophysics tried and failed to measure private sensations. If we substitute response for sensation, we make the object of psychophysical measurement public and measurable. Numerical psychophysical laws relating sensation and stimulus may be impossible, but such laws relating response and stimulus are not.18 (See chapter 9 for a full treatment of this argument.) I take a different view. To substitute response for sensation is a step in the right direction, but it is not enough. To conceive psychophysics as the science of relations between stimulus and response, and then to say responses must be measured in order to produce numerical psychophysical laws, is to bring along many of the problems that beset the older psychophysics. For one thing, "measurement of perceptual responses" is a confused notion. It is not clear what such measurement consists in, or whether it is possible. At any rate, it is unclear that such measurement will satisfy any of the legitimate needs of a psychological science of perception. What is needed is a complete break with the past. The measurement of perception does not consist in "measuring responses." It consists, rather, in measuring stimuli, obtaining numerical and nonnumerical responses to them, and, by statistical manipulation of the results, expressing the perceptual abilities of perceiving organisms. Perceptual abilities are, of course, defined by perceptual responses to perceived stimuli, and for that reason the distinction between stimulus and response is crucial. But it is perceptual abilities that psychophysicists measure. This way of solving the problem of psychophysical measurement is quite different from that suggested by Stevens, and quite different from his way of treating the larger problem of the relation between mind and body. Additionally, as chapter 9 shows, Stevens and other contemporary psychophysicists, in spite of the behaviorist flavor of some of 18 S. S. Stevens, "The Operational Definition of Psychological Concepts," Psychological Review, 42 (1935), 523 i f .

500

PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

their suggestions, have not really abandoned the distinction between psychological and physical dimensions. They have not, in an unequivocal way, substituted for it the distinction between response and stimulus. The old distinction dies hard.

11 Perceptual Psychophysics Perceptual psychophysics is, in the traditional conception, the science of relations between perceptual psychological dimensions and physical stimulus dimensions. The psychophysical laws of perception are conceived as laws relating dimensions of these two kinds. More often than not, psychological dimensions of perception are construed as dimensions of the sensations supposedly involved in all perception: visual sensations, auditory sensations, tactual sensations, and so on. The traditional conception of perceptual psychophysics generates what may be called the problem of psychophysical measurement. If the psychophysical laws of perception are to be numerical, which is clearly desirable, then we must be able to measure not only the physical stimulus dimensions involved, but also the perceptual dimensions (dimensions of sensations, on one interpretation). But it is by no means certain that acceptable measurement of this sort is possible. Traditional psychophysics has produced more than a hundred years of frustration and controversy: the problem of psychophysical measurement has been endlessly discussed, but never really solved. The aim of this book is to destroy the traditional conception of perceptual psychophysics, and to offer an alternative way of construing the nature of perceptual measurement. Drawing upon previous chapters, this chapter provides a critical analysis of the traditional conception, traces its past and recent history, and briefly indicates what should replace it. THE MEASUREMENT OF MIND The question running through the pages of this book is whether sensation and perception are capable of measurement. It is an instance of the much larger question: Is mind capable of measurement? Can mental phenomena of the various kinds be measured? Is it possible to

502

PERCEPTUAL PSYCHOPHYSICS

measure feeling, emotion, attitude, imagination, memory, thought, belief, expectation, intelligence, aptitude, habit, tendency, personality? PREJUDICES AGAINST MENTAL MEASUREMENT

Nowadays it is a mark of an enlightened psychologist—perhaps even of an enlightened scientist—to answer this large question in the affirmative. And for fairly compelling reasons. Some application of measurement has been attempted for at least some aspect of virtually every mental phenomenon. It seems only a matter of time (and sufficient research grants) until every aspect of every phenomenon has been quantitatively treated. Some of the attempts made thus far have not proved useful, but there seems no reason to doubt that the future will bring forth better ones. In 1936, while it was in the process of formation, Guilford described what is now the attitude of virtually all experimental psychologists on the question of mental measurement: "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." 1 This attitude is, within the context of the history of science, a novel one; and it was formed in the face of a powerful countertradition. Many philosophers in the seventeenth and eighteenth centuries observed that mental phenomena are qualitative, not quantitative. Physicists have usually supposed that only the phenomena of their own science are capable of measurement in the strict sense. Even psychologists in the early decades of the twentieth century were dubious about making psychology a quantitative science. Undoubtedly one reason for this tradition is the relatively late occurrence of experimental psychology in the history of science. Measurement entered psychology, according to Boring, only after the middle of the nineteenth century. He lists four dates in as many areas to mark its emergence. Fechner measured sensitivity and, possibly, sensation in i860. Donders measured reaction time in 1862. Ebbinghaus measured remembering in 1.885. And, in 1883, Francis Galton devised the first mental tests.2 In contrast, mea1

J. P. Guilford, Psychometric Methods (New York, 1936), p. 3. Edwin G. Boring, "The Beginning and Growth of Measurement in Psychology," in Quantification, ed. Harry Woolf (New York, 1961), pp. 108-109. 2

PERCEPTUAL PSYCHOPHYSICS

surement in experimental physics has a history of at least five hundred years, and in chemistry longer than that. But the fact that psychological measurement is only a hundred years old is surely not the only reason for belief in its impossibility. Without doubt a major part of the explanation for the stubborn belief that mental phenomena cannot be measured is the strength of the dualist tradition in philosophical, religious, and scientific thought. Within this tradition emotion, belief, thought, and the like, are held to be attributes of a special substance called "mind" or "soul." Now the soul is thought to be immaterial, which seems to entail that neither it nor any of its attributes can be measured. We measure the weight of an object by placing it in one pan of a balance and comparing it with unit weights added to or subtracted from the other pan. But belief cannot be added or subtracted. Length can be measured by laying a unit rod in successive, adjacent positions along the rod to be measured. Such a procedure is possible only because length is one of the properties of extended objects, objects that have spatial dimensions. But emotion is not extended, not spread out of space. Because of considerations like these it has seemed, not merely philosophically mistaken, but impious to suggest that the mind and its attributes can be measured. If only material substance and its attributes can be measured, then to suggest that the soul is measurable is to say that the soul is material. And, to many theologians, this notion is heresy, for if the soul is material, there is no very deep distinction between soul and body, and, consequently, no important distinction between men and animals. If the soul is material, perhaps it is not immortal, but is destroyed when the body is destroyed. Perhaps we can overlook the fulminations of tender-minded theologians. But it is no longer a laughing matter when we find physical scientists, who in popular mythology are hardheaded and always right, adopting the same view for no better reason. In the view of some physical scientists it is virtual heresy to suggest that mental attributes can be measured. Their reasons are often as dogmatic as those of the theologian. Space and time are held to be the fundamentally measurable attributes, and all other measurable attributes are said to be defined in terms of these. Since mental attributes are nonspatial (if not nontemporal), they are incapable of fundamental measurement. Nor does it seem possible to define nonspatial attributes, like sensation and at-

PERCEPTUAL PSYCHOPHYSICS

titude, in terms of spatial ones, which prevents us from measuring mental attributes by means of some derived procedure. Furthermore, mental attributes are private, but procedures of measurement are public. One experimenter can confirm or disconiirm the results of another's measurement of volume. How can these public, confirmable procedures be applied to private attributes like emotion and thought? It is fair to say that a belief in the immeasurability of mind, when supported by arguments like those described above, is little better than superstition, whether it be held by theologian, scientist, or philosopher. Defended by such arguments, the belief rests, first, on a simpleminded and quite indefensible view of the nature of measurement, and second, on a crude and entirely problematic view of the nature of mind. There is a tendency, when reflecting on measurement, to think of what is done by grocers, pharmacists, seamstresses, carpenters, and the early Babylonians. Such persons measure what J. L. Austin called, in «mother connection, "moderate-sized specimens of dry goods." One's view of the matter will not be much improved by including in the list of examples the early Egyptians, who were surveyors and astronomers. Surveying and elementary astronomy deal principally with length and distance, the clearest and simplest examples of measurable dimensions, but by no means the only ones. If we are to understand the nature of measurement—or most concepts for that matter—we must avoid what Austin calls the scholastic habit of mind, the tendency to concentrate on only one or two clear instances of the concept. 8 We must think, not merely of length and weight, but also of density, friction, hardness, temperature, frequency, electric current, angle, duration, viscosity, acidity, and so on. All these dimensions are measured by chemists and physicists, and yet their measurement is in important respects unlike the measurement of length or weight. It is true that some of these dimensions—density, for example—can be measured only by defining them in terms of length or weight. But many besides length and weight—electric current, angle, for example—are fundamentally measurable. To get a clear view of our subject, we must also consider the mea3

J. L. Austin, Sense and Sensibilia, reconstructed by G. J. Warnock (Oxford,

1962), p. 3.

PERCEPTUAL PSYCHOPHYSICS

505

surements made in the economic and social sciences—of life expectancy, demand for goods, buying power, voter trends, preference for television shows, and so on. The measurement of these statistical dimensions consists in counting entities (deaths, purchases, dollars, votes, times watched, etc.) and obtaining from the count the value of some coefficient or index. These coefficients or indexes are normally applied to groups of subjects. But the same kinds of techniques are used to measure habit strength, intelligence, and other qualities and attributes of single subjects, which brings us pretty clearly into the sphere of mental phenomena, of mind. The dogmatic belief that mind is immeasurable rests also on a certain view of the nature of mind, usually called dualism. The dualist view is that there are two different sorts of world: the mental world and the physical world, the first containing mental phenomena, and the second, physical phenomena. There are as many mental worlds as there are sentient beings, each world private to the being whose world it is. There is only one physical world, which is public and open to everyone's inspection. In this view, the measurement of mental phenomena may indeed seem impossible. For measurement, like all other scientific processes, seems to be a public process, one that can be duplicated and confirmed by any number of observers. But if the mental world is private, then "measurement" of its phenomena can be performed only by the person whose world it is. Furthermore, mental phenomena, in the dualist view, are usually regarded as evanescent and fleeting, which seems to imply their "measurement" cannot be duplicated even by the person whose world they inhabit. More important, for our purposes, the dual world view tempts us to compare the contents of the one world with those of the other, and thus to apply the same categories to both. We speak, for example, of the substance of physical things. We ask what substance a physical thing is composed of, and we distinguish various kinds of substance: lead, bromine, mercury, sulfur. Suppose we ask, in a similar way, what substance mental things like perceptions, emotions, and beliefs are composed of? Mental substance, it seems clear, is quite unlike material substance: it is evanescent, ethereal, "insubstantial." And how can we possibly measure things whose substance is of this sort? Consider also that material substance is complex and divisible, whereas mental sub-

506

PERCEPTUAL PSYCHOPHYSICS

stance is simple and indivisible. Simple substance cannot be broken up into parts, cannot be added, subtracted, or similarly manipulated. How then can it be measured? The error in these arguments lies in the assumption that all physical concepts and categories can be applied to mental entities. Although it is proper to ask what substance a rock is composed of, it is improper to ask this question about an emotion or belief. Emotions are not composed of a substance, even mental substance. The same point seems to apply to the categories of "thing," "event," "state," and "process." There is something peculiar in calling an emotion a thing, or an event, or a state, or a process. There is less peculiarity in calling perception a process. But even here something seems wrong. (Gilbert Ryle has suggested that the error lies in failing to realize that perception verbs are "success verbs." 4 In Ryle's view, to say that a person sees a chair is like saying that he has won a race; it implies that an activity has been successful. But winning a race, or, by analogy, perceiving, is not properly called a process, or an event.) The dual world view presents us with a problem. We seem to have a clear understanding of the physical world. We conceptualize and categorize it with ease. But we have difficulty doing the same thing for the mental world, which presents itself to us in a kind of haze. So we are tempted to compare the two worlds, to draw an analogy between them, to understand the hazy one in terms of the clear one. But when we apply the categories of the physical world to the mental world, we find that they do not quite fit. Instead of drawing the correct conclusion—that the attempted application is improper—we suppose that "mental substances," "mental things," "mental events," "mental states," and "mental processes" are unlike physical substances, things, events, states, and processes. And then we come to feel that familiar physical procedures, such as measurement and experimentation, cannot be performed on mental phenomena. The adequacy of the dualist view of mind is one of the central, unsolved questions of philosophy. We will enter it here only a little way, just enough to suggest an alternative view and its bearing on the possibility of mental measurement. It is possible to regard attitude, emotion, belief, perception, intelligence, and other mental phenomena * Gilbert Ryle, The Concept of Mind (London, 1949), pp. 222-225.

PERCEPTUAL PSYCHOPHYSICS

as dispositions to behave in certain ways, or as abilities to perform certain actions. Thus we may treat fear of a lion as a disposition to run from the lion, to scream, to attack the lion, and so on. And we may regard mathematical intelligence as an ability to solve mathematical problems, construct formulas, find proofs, and so on. These are, in a perfectly good sense, mental dispositions and abilities, and may be contrasted with the disposition of a plate to break when dropped or the ability of a rock to break the plate. But the use of the terms "mental" and "physical" to make this contrast implies no distinction between separate worlds containing different kinds of phenomena. Physical and mental dispositions and abilities are both located in the same world, the only world there is, which is neither mental nor physical.5 What is given above is a sketch, not a defense, of a view of the nature of mind. It has not been adequately explained, little argument has been provided for it, and its difficulties have not been explored. I present it merely to suggest that one's view about the possibility of mental measurement may hinge on his view of the nature of mind. Viewed as dispositions or abilities, mental phenomena no longer seem incapable of being measured. The disposition of a plate to break is, obviously, much less complex than the disposition that we call fear. But the latter is no less measurable than the former. The ability of a rock to break a plate is much simpler than the ability we call mathematical intelligence. Nevertheless, there is no reason to think the one is measurable and the other immeasurable. CONCEPTUAL DIFFICULTIES IN MEASURING PERCEPTION

To become more specific, let us see how one's view of the nature of measurement and of mental phenomena can affect his appraisal of the possibility and the character of the measurement of perception. In order to measure any phenomenon we must identify it and isolate it from other phenomena, and decide what aspect of the phenomenon we wish to measure. We must conceptually prepare the specimen for examination and dissection, lay it out in thought on the operating table. For example, to measure lightning flashes, we must distinguish them from flashes of other kinds, and then must decide whether we wish to measure their length, intensity, frequency, location, or some 5

A view very like this can b e found in R y l e , The Concept

of

Mind.

508

PERCEPTUAL PSYCHOPHYSICS

other aspect. When we try to do the same thing for perception (and psychological phenomena in general), the object always seems to elude our conceptual grasp. It is not that we have any difficulty using the language of perception. We all know perfectly well how to employ the words "see," "hear," and the like, to describe the life of humans and animals. But this ability is not sufficient for the task at hand. Consider a few simple instances of perception, (i) Jones sees a large, black bird circling in the sky. (2) Smith hears a loud, high-pitched whine in the basement. (3) Brown feels a cold, heavy stone underneath his pillow. (4) Green tastes a dry, astringent wine. (5) Black smells thick, acrid smoke in his room. What, in these examples, would we measure in order to measure perception? It seems hardly necessary to point out that in measuring the size or color of the bird we do not thereby measure Jones's seeing of the bird, and that in measuring the loudness or pitch of the sound we do not measure Smith's hearing of a sound. Jones sees the bird, Smith hears the sound; but the seeing is not identical with the bird, nor the hearing with the sound. Perhaps the correct view is that, although perceiving and the object perceived are not identical, nevertheless the measurement of perceiving requires, among other things, measurement of the object perceived. This suggestion tells us very little. What else must we measure in addition to the object perceived? And how are measurements of the object to be used in measuring the perceiving? Furthermore, this suggestion seems inapplicable to instances of hallucinatory perception. When Jones sees a hallucinatory bird, there is no bird shape or bird color to be measured. And yet the seeing exists. How then is it to be measured? Let us try another suggestion. We can concede that the actual, objective size and color of the bird need not be measured in order to measure Jones's perception. What we do need to measure is the perceived, or subjective, size, and the perceived, subjective color. This suggestion is quite unclear. What is subjective size the size of? What is subjective color the color of? Not of the bird. And it makes no sense to speak of the size or the color of Jones's perception, just as it makes no sense to say that his emotions have size or color. Perhaps Jones has "sensations" when he sees birds, and subjective size and color are the size and the color of these sensations. But, in the first place, it would appear that size and color are properties of physical

PERCEPTUAL PSYCHOPHYSICS

things, like birds, and not of sensations. Second, it is not dear what is meant by "sensations," or whether Jones has any of these when he sees birds. The above difficulties arise because we are trying to find some dimension involved in perception, analogous to length or intensity of lightning flashes, which we can measure in ways analogous to those in which we measure length and intensity. We are trying to categorize perception as we categorize certain physical phenomena so that we can measure it in the way we measure physical phenomena. But of course we fail, since the categories we employ do not apply. Seeing should not be regarded as analogous to a lightning flash (event), or an electric charge (state), or a heat transfer (process). Perhaps we can regard it as an ability (or a cluster or abilities), but even here a strong antilogy with physical abilities, like the ability of a rock to break things, may be misleading. The difficulty is that perception is being improperly conceived, conceptually prepared in the wrong way by applying the wrong categories to it. Traditional perceptual psychophysics is the result of this kind of improper conceptual preparation of the object of measurement. It holds that to measure perception is to measure psychological dimensions of perception. In the area of hearing, loudness and pitch are available as candidates for the psychological dimensions to be measured; in the area of color perception, brightness and hue. Since no such ready-made dimensions are available for length and weight perception, traditional psychophysics creates some, calling them"psychological length" and "psychological weight." All this, however, is a mistake: loudness, pitch, brightness, and hue are not psychological, but physical, dimensions, and the nature of "psychological length" and "psychological weight" is quite obscure. They seem to be fictions, and useless ones at that. Instead of looking for psychological dimensions of perception to measure, psychophysics should be isolating perceptual abilities, and carrying out the quite straightforward task of measuring them. The traditional conception of psychophysics is dualistic, for it presupposes a distinction between psychological and physical dimensions. At the same time, it is firmly committed to the belief that perception and other mental phenomena are measurable. Such a view is bound to generate enormous tensions, since, as we have seen, dualism

5io

PERCEPTUAL PSYCHOPHYSICS

tends to support the view that mental phenomena are immeasurable. The tension could be relieved by adopting a nondualist view, and by regarding mental phenomena as dispositions or abilities of a certain kind. But the psychological science of perception has not adopted that solution, at least not in a clear and definite manner. It is still dominated by the traditional, dualist conception of perceptual psychophysics. TRADITIONAL PSYCHOPHYSICS Psychophysics, as defined by Fechner who founded it, is the science of the relations between mind and body, or, more generally, between mental or psychological phenomena and physical phenomena. A psychophysical relation is one between psychological and physical phenomena. A psychophysical law governs psychological and physical phenomena in their relations to each other. A psychophysical experiment is one that attempts to discover a psychophysical law or relation. Psychophysical laws or relations may be stated in numerical or nonnumerical terms. Psychophysics, as a full-fledged science, is the attempt to discover and explain numerical psychophysical laws and relations. When these laws relate psychological dimensions of perception (or, dimensions of sensations) to their physical correlates (or, stimuli), they are psychophysical laws of perception. Perceptual psychophysics is the enterprise of discovering psychophysical laws of perception. Psychophysical relations were widely believed, before Fechner, to be unfathomable and incapable of scientific treatment. There was a science—physics—which studied relations among physical phenomena. But a science of the relations between physical and mental phenomena was either thought to be impossible or not entertained as a possibility. Fechner, on the other hand, believed it was possible to measure mental phenomena and to discover laws relating them to physical phenomena. He proposed and developed several methods of psychological measurement which are extensively used today by experimental psychologists, and he formulated a general psychophysical law that bears his name and is, with modifications, still widely accepted. Most of Fechner's research dealt with perceptual psychophysics, with the relations between the "mental" phenomena of seeing, hearing, feeling, and so on, and their physical stimuli.

PERCEPTUAL PSYCHOPHYSICS

511

As it is traditionally understood, psychophysics presupposes a distinction between psychological dimensions and physical dimensions. Just as physical laws govern physical dimensions, psychophysical laws govern psychological and physical dimensions. A numerical psychophysical law is one relating some psychological dimension(s) and some physical dimension(s). Length, weight, hardness, acidity, frequency of sound waves, and energy of light waves are, obviously, physical dimensions. Standard examples of psychological dimensions are brightness of color, pitch of tones, sourness, size of afterimages, and intensity of pain. Psychophysics is, then, the attempt to discover numerical laws relating dimensions of the first type to dimensions of the second type. THE ANALOGY BETWEEN PHYSICAL AND PSYCHOPHYSICAL SCIENCE

The chief article of faith in traditional psychophysics is that mental or psychological phenomena are, like physical phenomena, capable of experimental, quantitative treatment. The central presuppositions of physical science are (1) that knowledge about physical phenomena is obtainable through observation, either perceptual or instrumental, and (2) that at least some of this knowledge can be expressed in numerical terms through the use of physical measurement. Analogously, the central presuppositions of traditional psychophysics are (1) that knowledge about psychological phenomena is obtainable through observation, if not by instruments then at least by some type of (perhaps inner) perception, and (2) that at least some of this knowledge can be expressed in numerical terms through the use of psychological measurement. Psychological science is thus held to be as much a possibility as physical science, even if it has been slower in developing and its gains have been less dramatic. Underlying the traditional conception of psychophysics is an analogy between psychological and physical science, part of which is between the object of measurement in the one and the object of measurement in the other. Just as there are measurable physical phenomena, so there are measurable psychological phenomena. In measuring physical phenomena we measure physical dimensions. Analogously, in measuring psychological phenomena we measure psychological dimensions. If we wish to measure emotion, for example, we must identify its psychological dimensions and try to devise procedures for their measurement. If we wish to measure perception, we must identify its psychological

5"

PERCEPTUAL PSYCHOPHYSICS

dimensions and devise procedures for measuring them. Commonly offered examples of the psychological dimensions of perception are perceived length, perceived weight, brightness of color, pitch of tone, and intensity of pain. It is for such dimensions as these that perceptual psychophysics must devise procedures of measurement. The other part of the analogy between physical and psychophysical science is between the laws that each attempts to discover. A physical law is one relating some physical dimension(s) to some other physical dimension(s), a law of the form

[1]

R $2,

where and $2 are physical dimensions, and R is some relation, numerical or nonnumerical, between them. A psychophysical law is, as the term suggests, one relating some psychological dimension(s) to some physical dimension (s); a law of the form

[2] Yi R $1, where is a psychological dimension, $>i a physical dimension, and R some relation, numerical or nonnumerical, between them. It is obvious, if we follow out this conception, that a third type of law is possible. A psychological law is one relating some psychological dimension^) to some other psychological dimension(s), a law of the form

[3] Y, R where Y i and T2 are psychological dimensions, and R is some relation, numerical or nonnumerical, between them. A proponent of the traditional conception will usually admit that there are differences between psychological and physical dimensions, and, consequently, between the various types of laws. Psychological dimensions are, he will say, "subjective": that is, they are dependent on a subject, dependent on his mind (if we admit that he has one) or on his nervous system or on both. And our knowledge about psychological dimensions is obtained in a different way from our knowledge about physical dimensions. Knowledge of physical dimensions is obtained through "outward perception" and the use of instruments; knowledge of psychological dimensions, through "inner perception." Or, if this introspectionist view of the matter seems unacceptable, at least the contrast can be made by pointing out that knowledge of psychological dimensions is obtained by examining the responses of subjects, whereas knowledge of physical dimensions is not. Whatever the differences be-

PERCEPTUAL PSYCHOPHYSICS

513

tween psychological and physical dimensions, they do not, in the traditional conception, mitigate against the analogy between physical and psychophysical laws. Both types relate two or more dimensions. Both may be numerical or nonnumerical. If they are numerical, a system of measurement is presupposed for the dimensions thus related. Let us examine the analogy between physical and psychophysical laws more closely. The temperature of a gas is directly proportional to its pressure: if volume is held constant, temperature increases as pressure increases. This law may be expressed as

[4] T dp P, where T is temperature, P is pressure, and dp designates the relation of direct proportionality. Now [4] is not a numerical, or quantitative, law. It tells us that gas temperature increases as the pressure increases, but it does not tell us by how much the temperature is increased when the pressure is increased by a given amount. Perhaps when the pressure is doubled the temperature doubles, or perhaps doubling the pressure triples the temperature; [4] does not tell us which, if either, of these is true. To discover a numerical law relating temperature and pressure, we must be able to measure both temperature and pressure, to assign to these two dimensions numerals that represent the quantity of temperature and the quantity of pressure. With these systems of measurement, we are able to formulate and verify laws of the form [5] T = MP), where T is temperature expressed in terms of the unit in some system of temperature measurement (degrees Kelvin, for instance), P is pressure expressed in terms of the unit in some system of pressure measurement (pounds per square inch, for instance), and fi is a numerical function determined by experiment (by varying pressure, measuring it, and then measuring the resulting temperature). Laws like [4] are useful, but not so useful as those like [5]. One reason is that laws of the latter type have more content, and thus enable us to express a larger number of facts about the world. The relation between the temperature and the pressure of helium gas is, like that between the temperature and the pressure of hydrogen gas, one of direct proportion. But this statement has relatively low factual content. Doubling the pressure of helium may double its temperature, and doubling the pressure of hydrogen may triple its temperature. Or

5^4

PERCEPTUAL PSYCHOPHYSICS

doubling the pressure may double the temperature for both gases. And so on. We will not be able to say which of these is true if we limit ourselves to statements like [4]. A closely related reason for preferring laws of type [5] is the possibility of placing them in more sophisticated deductive relations to one another. This gives them greater theoretical value, makes it possible to imbed them in finer theoretical nets. The relation between the pressure and the volume of a fixed amount of gas is one of inverse proportion. That is to say, [6] P ip V, where P is pressure, V is volume, and ip designates the relation of inverse proportionality. From [4] and [6] we can deduce that the temperature and the volume of a fixed amount of gas are in inverse proportion to each other. We can deduce, that is to say,

[7] T ipV. But suppose the numerical relation between the temperature and the pressure of helium is given in [5], and that the numerical relation between the pressure and the volume of helium is [8] P = f 2 (V). From [5] and [8] we can deduce [9] T = fi(MV)). From nonnumerical laws only nonnumerical laws can be deduced. From numerical laws we can deduce both numerical laws and nonnumerical laws, which gives the advantage to numerical laws. In the traditional conception of psychophysics, psychophysical laws are seen as analogous to physical laws like those discussed above. The psychological dimension allegedly involved in weight perception is sometimes called "psychological weight" ("subjective weight," "perceived weight"). It is obvious that psychological weight is directly proportional to physical weight ("objective weight"): as the physical weight of an object increases, so to the normal observer does its psychological weight. We can express this relation as [10] T . dp i>„, where is the psychological weight and the physical weight. Law [10] is a nonnumerical law. It does not tell us by how much the psychological weight is increased for a given increase in physical weight.

515

PERCEPTUAL PSYCHOPHYSICS

Its factual content is, therefore, less than that of its numerical counterpart, [11] = f,(*w), where ^w is the psychological weight expressed in psychological units, is physical weight expressed in physical units, and fi is a numerical function determined by psychophysical experiment. In addition to the familiar procedure for measuring physical weight, [ 1 1 ] presupposes some special procedure for measuring psychological weight. (Such procedures are discussed in chapters 8 and 9.) Numerical psychophysical laws are preferable to nonnumerical laws because they have more factual content and because they enter into more sophisticated deductive relations with other psychophysical and psychological laws. Suppose "psychological weight" is directly proportional in the normal observer to "psychological volume" ("perceived volume," "subjective volume"). Suppose, that is to say, that

[12]

dp

where is psychological weight and is psychological volume. (This relation in fact obtains, if it obtains at all, only within classes of objects of the same type.) From [10] and [12] we can deduce [13]

dp

a nonnumerical law relating psychological volume and physical weight. But suppose numerical laws relating psychological and physical dimensions are possible. Then, in addition to [1], we may discover that [14] Yw = M ^ v ) , where f2 is some numerical function determined by experiment. From [ 1 1 ] and [14] we can deduce [ l 5 ] MY*) = M*w), a numerical law relating psychological volume to physical weight. The psychological and psychophysical laws [10] through [15] are analogues of the physical laws [4] through [9], respectively. The belief that such psychological and psychophysical laws are possible is the belief that the experimental and theoretical procedures of the physical sciences can be extended across and into the domain of the psychological, into the realm of the mental. It is one form of the belief that psychology can be a science. The traditional conception of physics has tended to be that science stops at the edge of the physical realm,

5

i6

PERCEPTUAL PSYCHOPHYSICS

whereas the traditional conception of psychophysics holds that its own subject matter is also included in science. Science, according to this view, includes not only the usual physical laws, but psychophysical and psychological laws as well. All these are to be comprehended in one complete scientific theory that includes the entire empirical universe, not just the physical part of it. Universal science is, then, the dream of the traditional psychophysicists. THE PROBLEM OF PSYCHOPHYSICAL MEASUREMENT

If the aim of psychophysics, in the traditional conception, is to discover numerical laws relating psychological and physical dimensions, then the dimensions related by such laws must be measurable. Measurement is a certain kind of procedure for assigning numerals to the objects within a dimension. The simplest method of length measurement consists in applying a yardstick or a meterstick to the object being measured. After the object is aligned with the zero end of the yardstick, the numeral on the yardstick opposite the other end of the object is assigned to it. Only if we possess some such method for assigning numerals to the objects within a dimension can we express the magnitude of those objects in numerical terms. And only if we can express their magnitude in numerical terms can we discover numerical laws relating the magnitude to other magnitudes. Can the psychological dimension that is the correlate of physical length—namely, "psychological length"—be measured? How might we go about measuring it? Obviously we cannot do so merely by measuring physical length. It appears that some special, psychological procedure must be employed. But what is that procedure? Nothing comes immediately to mind in answer to this question. But if we cannot measure psychological length, the numerical laws relating it to physical length (or other dimensions) are impossible. This problem of devising procedures for measuring the so-called psychological dimensions may be called the problem of psychophysical measurement. Perhaps the reason for the difficulty in specifying methods for measuring psychological length and weight is that the nature of such dimensions is not fully understood. Let us turn, then, to some simpler and more familiar examples and raise the question again. Consider the hue and brightness of colors, or the loudness and pitch of sounds, which are commonly cited as examples of psychological dimensions. Laws

PERCEPTUAL PSYCHOPHYSICS

517

relating these dimensions to their physical correlates are commonly offered as examples of psychophysical laws of perception. Laws relating brightness to the intensity (energy) of light waves, and those relating hue to the frequency of light waves, are laws of visual perception. Laws relating loudness to the intensity of sound waves, and those relating pitch to the frequency of sound waves, are laws of auditory perception. The measurement of intensity and frequency, either of light waves or of sound waves, is normally accomplished by using electronic instruments that serve as intensity and frequency meters. But how are the brightness and hue of colors and the loudness and pitch of sounds to be measured? Sounds are objects of a different kind from sticks and stones. The length of a stick can be measured with a yardstick; the weight of a stone, by placing it on a balance. But yardsticks cannot be laid against sounds and sounds cannot be placed on a balance. Measurement by yardsticks and balances is of course measurement of the simplest, clearest sort. Since there is no reason to suppose that every procedure of measurement must be exactly like these, it is fallacious to suggest that sounds are incapable of measurement simply on the ground that they cannot be measured in exactly the same way as length or weight. But consider. Is it not true that every procedure of measurement requires the specification of units of measurement, and the use of an operation of physical addition to combine these units, so that combinations may be compared with the object being measured? If so, sounds seem incapable of measurement, for they cannot be added in such a way that the pitch of two of them is greater than the pitch of either alone. And it would appear that the same is true for loudness, that sounds cannot be added in such a way that the loudness of the combination sound is greater than the loudness of either sound alone. And even if there were operations of addition for loudness and pitch, how could we specify units for these dimensions? (Chapters 6 and 7 have shown that these arguments are mostly fallacious.) Considerations like these have led to contortions of the most violent kind on the part of theorists who wish to establish that loudness, pitch, and other so-called psychological dimensions are capable of measurement, who wish to solve the problem of psychophysical measurement. In the main, three "solutions" have been proposed. (1) Psychological dimensions have been identified with their measurable,

5i8

PERCEPTUAL PSYCHOPHYSICS

physical correlates. (2) It has been claimed that psychological dimensions are measurable indirectly, by means of their measurable, physical correlates. (3) The definition of measurement has been stretched so as to include procedures available for the psychological dimensions. Each of these proposals is discussed in turn, with loudness and pitch as the principal examples. (All points in this discussion have received extensive treatment in earlier chapters.) 1 ) The identification of psychological and physical dimensions.— Some theorists believe that sounds and sound waves are one and the same set of entities under different names, and that every dimension of sounds is identical with some dimension (or dimensions) of sound waves. Loudness of sounds is often identified with intensity of sound waves, and pitch of sounds with frequency of sound waves. If these identifications were correct, it would clearly be possible to measure loudness and pitch, for it is clearly possible to measure intensity and frequency by means of electronic instruments that serve as intensity and frequency meters. If loudness were identical with intensity, measuring intensity with an intensity meter would be measuring loudness. And if pitch were identical with frequency, measuring frequency with a frequency meter would be measuring pitch. There are several strong objections to the identification of dimensions of sounds with dimensions of sound waves (see chap. 3). At the same time, powerful arguments favor the identification. Without reviewing these arguments here, we note merely that the question of identifying sounds and sound waves is so difficult, and so controversial, that it is dangerous to rely on its outcome in deciding whether loudness and pitch are capable of measurement. A related difficulty is that of deciding which question has the higher logical priority: the measurability of psychological dimensions, or the identification of psychological dimensions with their physical correlates. If loudness is identical with intensity, and pitch with frequency, then, since intensity and frequency are measurable, loudness and pitch are measurable. On the other hand, if loudness and pitch are incapable of being measured, then, since intensity and frequency are capable of being measured, loudness is not identical with intensity nor is pitch identical with frequency. But which question is to be attacked first: the measurability of loudness and pitch, or their identity with intensity and frequency? A final objection to identifying psychological dimensions with their

PERCEPTUAL PSYCHOPHYSICS

519

physical correlates is that such identification destroys their "psychological" status. If loudness is identical with intensity, it is, like intensity, a physical dimension. And if pitch is identical with frequency, it is, like frequency, a physical dimension. 2) The indirect measurement of psychological dimensions.—Most writers on the subject distinguish two types of measurement: direct and indirect, under these or some other names. The measurement of length by means of a meterstick is an example of direct measurement. The measurement of temperature by means of a thermometer is an example of indirect measurement, accomplished by defining units of temperature in terms of units of mercury-column length. It is sometimes supposed that in much the same way the loudness of a sound can be measured by measuring the intensity of the correlated sound wave. Units of sound-wave intensity are taken as, or by some mathematical transformation made into, units of loudness. A parallel indirect method for pitch would measure this dimension by measuring the frequency of the correlated sound wave. The distinction between direct and indirect measurement is much more troublesome than it may seem (as shown in chap. 3). One question of special importance is whether indirect measurement is inconclusive, and to that extent unsatisfactory. It would seem that we can measure one dimension by means of another only if equal increments in the one correspond to equal increments in the other. But how can we know whether this condition is fulfilled without a method for directly measuring both dimensions? If we cannot, then the indirect measurement of one dimension by means of another, directly measurable dimension can be shown to be satisfactory only by devising a method for directly measuring the first. Thus, if loudness is incapable of direct measurement, we can never determine whether measuring it by means of intensity is satisfactory. Another point against the indirect measurement of a psychological dimension by means of its physical correlate is that this method precludes the possibility of discovering empirical laws relating the two. We should like to know the numerical, empirical relation between loudness of sounds and intensity of sound waves. If we indirectly measure loudness by stipulating that units of intensity are also units of loudness, the relation between loudness and intensity is established in the stipulation. Loudness then equals intensity; that is, L = I. But this law has no empirical content: it is true merely by stipulation.

520

PERCEPTUAL PSYCHOPHYSICS

Although such stipulations may (or may not) be useful, the formulas that describe them do not represent discoveries about the empirical world, and do not deserve the lofty designation of "laws." 3) Stretching the definition of measurement.—Under the classical definition, measurement is the assignment of numerals to objects within a dimension by means of an operation of physical addition of units. If this definition is correct, it seems clear that pitch and hue cannot be measured. N o operation of addition is available for these dimensions. It is not possible to add two colors so as to produce a combination color whose hue is greater than that of either of the two components. Nor is it possible to add two tones so as to produce a combination tone whose pitch is greater than that of either component. Many theorists have supposed that operations of addition for loudness and brightness are similarly impossible. (Chapter 6 has shown that this is incorrect.) Rather than conclude that psychological dimensions are incapable of measurement, modern psychophysicists have abandoned the classical definition of measurement and have proposed an alternative definition so wide that it includes the procedures of numeral assignment they employ. Thus, Stevens has proposed that measurement be regarded as the assignment of numerals to things according to some rule. Under this definition it can be argued that pitch can be measured by assigning numerals to just noticeable differences in pitch, and that loudness can be measured by assigning numerals to loudness so as to represent the consistent ratio or interval estimates of observers. These methods, although they do not conform to the classical definition of measurement, do consist in assigning numerals to things (psychological entities) according to some rule. The chief difficulty is that this solution adopts an unacceptably broad definition of measurement. Under the definition proposed b y Stevens, assigning numerals to houses—a different numeral to each house—counts as measurement. Indeed, if we were to assign numerals to length by spinning a roulette wheel, and assigning to different rods the numerals indicated at the end of different spins, we would be measuring length in accordance with the broad definition! And there are other difficulties. One of the two methods of psychological measurement mentioned above employs the just noticeable difference (jnd) as a unit of psychological magnitude. For example, the numeral obtained by counting off the number of loudness jnd's a given tone lies above

PERCEPTUAL PSYCHOPHYSICS

521

the loudness threshold is assigned to the loudness of the tone. It is surely improper to regard this procedure as a method for measuring loudness. We obtain just noticeable differences in order to measure the observer's sensitivity to loudness, not to measure loudness. The other method pretends to measure loudness by having the observer numerically estimate the loudness of the tone, using a numeral to indicate the magnitude he hears, and assigning the numeral supplied by the observer to the loudness of the tone heard. Surely this method does not measure loudness, but merely records observer estimates of loudness. Not all the alleged psychological dimensions are like loudness and pitch, or brightness and hue. They may be divided roughly into three classes: (a) loudness, pitch, brightness, hue, sourness, bitterness, and so on; (b) psychological length, psychological weight, psychological roughness, and so on; (c) intensity of pain, extent of tickling sensation, size of afterimage, and so on. Whatever the differences among these classes, the problem of how to measure psychological dimensions arises as quickly for (b) and (c) as for (a). Again the major solutions that may be attempted are the three discussed above. Physical weight is measured with a balance, but how is psychological weight measured? An attempt to solve the problem could take any of the following forms. (1) Psychological weight might be identified with some measurable dimension. Clearly it is not identical with physical weight, but it may be possible to identify it with some physiological process in the observer: some dimension of kinesthetic nerve impulses, for example. (2) Psychological weight might be measured indirectly, by means of some physiological dimension, or perhaps by defining it in terms of the observer's responses to the measurable dimension of weight. (3) Psychological weight might be interpreted as the intensity of weight sensations, and the definition of measurement broadened so as to make the observer's estimates of his own sensations qualify as their measurement. These solutions are subject to the objections previously described. Intensity of pain, a paradigm example of dimensions of sensations, is clearly psychological. The difficulty in its measurement seems especially acute. Pains, it would seem, cannot be added. Nor is it clear how to specify a unit of pain. How then can pain intensity be measured? Again, any one of the three solutions might be attempted. (1) Intensity

522

PERCEPTUAL PSYCHOPHYSICS

of pain might be identified with some measurable, physiological dimension, say, the rate of tissue destruction. (2) Pains might be measured indirectly, by allowing the numerals assigned to rate of tissue destruction to represent also the intensity of pain. (3) The definition of measurement might be broadened to include the method of just noticeable differences, and pains measured by counting off the number of pain jnd's they lie above the pain threshold. But these solutions are objectionable for the reasons previously given. The problem of psychophysical measurement is imposed on us by the traditional conception of perceptual psychophysics, which holds that the measurement of perception consists in measuring the psychological dimensions of perception. Such measurement is thought necessary to permit the discovery of psychophysical laws, laws relating psychological and physical dimensions. But it appears that psychological dimensions are incapable of measurement, and that the discovery of numerical laws is impossible. This problem has never really been solved. The reason is not lack of ingenuity or effort, but confusion in the conception of psychophysics within which the problem arises. THE DISTINCTION BETWEEN PSYCHOLOGICAL AND PHYSICAL DIMENSIONS

Chapters 2 through 7 of this book present, among other things, an extended argument to the effect that loudness and pitch of sounds can be measured, as length and weight can be measured, directly, and without identifying them with their physical correlates. The same kind of argument can easily be applied to at least some of the other so-called psychological dimensions—brightness and hue of colors, for instance— to show that they are measurable. It may be supposed that this result is a solution to the problem of psychophysical measurement, and a victory for traditional psychophysics. But it is not, for the same argument shows that loudness and pitch of sounds are physical dimensions, even if they are distinguished from their physical correlates of intensity and frequency of sound waves. This contention by itself deals a virtual deathblow to the traditional conception of psychophysics. Loudness and pitch are usually cited as paradigm examples of psychological dimensions, and laws relating loudness and pitch of sounds to intensity and frequency of sound waves are taken to be paradigm examples of psychophysical laws of perception. With these examples in mind, the traditional conception of psychophysics seems

PERCEPTUAL PSYCHOPHYSICS

523

simple, unobjectionable, and quite compelling. But, as chapter 2 has shown, loudness and pitch are physical dimensions; consequently, laws relating loudness and pitch to intensity and frequency are physical laws. What then are the psychophysical laws of auditory perception? What dimensions are related by these laws? With these questions, the traditional conception of psychophysics loses its simple, attractive character and begins to shatter. One proposal for repairing it takes the psychological dimensions of auditory perception to be dimensions of auditory sensations. But serious difficulties are raised by this proposal. Are auditory sensations always or even usually involved in auditory perception? What are auditory sensations? How do they differ from sounds? And how do sounds differ from sound waves? Do the psychophysical laws of auditory perception relate sensations and sounds, or sensations and sound waves? If these difficulties do not destroy the traditional conception of auditory psychophysics, they do at least make it seem much less tenable than before. The discussion of loudness and pitch is important because it exposes grave difficulties in the traditional conception of psychophysics. There are also other ways of revealing deficiencies in that conception. The dimensions listed as psychological may be roughly divided into the three classes described earlier: (a) loudness, pitch, brightness, hue, sourness, bitterness, and so on; (b) psychological length, psychological weight, psychological roughness, psychological hardness, and so on; (c) intensity of pain, size of afterimage, and so on. These classes do not seem to be on a par. The dimensions in (a) and (c) are observable, either by external perception or internal perception (introspection), but those in (a) are public and those in (c) are private. The dimensions in (b) are public but unobservable. Experimenters observe the observations of observers. Observers observe length, weight, roughness, and hardness, and by means of their observations the corresponding psychological dimensions are defined. The psychological dimensions are not observed. There are several ways of attempting to remove these disparities. Each of the three most important ways produces a different interpretation of the distinction between psychological and physical dimensions, and thus a different version of the traditional conception of psychophysics. i)The objectivist interpretation.—The items in class (a) are the models for psychological dimensions. Loudness and pitch, and intensity

524

PERCEPTUAL PSYCHOPHYSICS

and frequency, are dimensions of one and the same set of entities (sounds and sound waves are identified). All are public dimensions, and all are observable. Loudness and pitch are as "real" and as "objective" as intensity and frequency. Loudness and pitch, however, are observed by means of hearing, and intensity and frequency by means of instruments or other operational procedures. In general, psychological dimensions and their correlated physical dimensions are dimensions of the same entities and to this extent are on a par. The difference between them lies in the methods for observing them. Psychological dimensions are, one might say, perceived dimensions. This interpretation does not work well for dimensions in classes (b) and (c). Rods surely do not possess two separate dimensions, length and perceived length. Surely there is only one dimension here, which is sometimes merely seen and at other times is measured with a yardstick. As for the items in class (c), can we really suppose that the size of a visual afterimage and the area of retinal stimulation accompanying it are dimensions of the same thing? Can the intensity of pain and the rate of tissue destruction which is its physiological correlate be considered dimensions of the same thing? It does not seem so. To these difficulties we may add that the objectivist interpretation makes psychological dimensions a type of public, physical dimension. z)The constructionist (behaviorist) interpretation.—The items in class (b) are the models for psychological dimensions. Psychological length and psychological weight are not empirically real, nor are they the objects of observation. Length and weight are the observable dimensions here; it is to them that observers perceptually respond. Out of their perceptual responses psychological length and psychological weight are constructed by means of definitions like the following. An observer, O, sees one rod to be longer than another if and only if it is psychologically longer than the other. O sees one rod to be twice as long as another if and only if it is psychologically twice as long as the other. Thus defined, psychological dimensions are theoretical constructs, entities brought into existence by definition to serve some scientific purpose, and are not empirically real. It is difficult to apply this interpretation to the dimensions in classes (a) and (c), for they are observable dimensions. They are empirically real; they have not been brought into existence by definition.

PERCEPTUAL PSYCHOPHYSICS

525

If we say that loudness of sounds is a construct from observer responses to intensity of sound waves, the implication is that observers perceive sound waves and that they do not perceive sounds, both of which seem false. Perhaps we should distinguish among sound waves, sound, and psychological sound. Psychological loudness will then be a construct from observer responses to loudness. But it is difficult to see a distinction between loudness (as distinct from intensity of sound waves) and psychological loudness. Similar difficulties arise for intensity of pain. If we say that this dimension is a construct from observer responses to some physiological dimension, such as rate of tissue destruction, we imply that observers perceive the physiological dimension and not the pain. And the attempt to distinguish psychological pain from pain, making the former a construct from responses to the latter, seems to produce plain nonsense. 3)The sensationist (introspectionist) interpretation.—The items in class (c) are the best models for psychological dimensions. Psychological dimensions are dimensions of sensations, and are private and observable only by introspection. There are as many examples of a given psychological dimension as there are observers having the sensations that possess the dimension. Psychological dimensions are empirically real, but are not objective. The psychophysical laws of perception relate sensations to their physical stimuli; they are, in a perfectly straightforward sense, laws of mind and body. This interpretation does not seem to apply to the items in classes (a) and (b). There is an inclination to think it applies nicely to sounds, which legions of thinkers have supposed to be sensations. But sounds are not sensations; they are public, perception-independent, and locatable in space. And the same is true of colors. We are forced, therefore, to posit auditory sensations, color sensations, and so on, to give the sensationist interpretation an application in these areas of perception. It is doubtful, however, that such sensations are always or even usually involved in the perception of sound and color. As for the items in class (b), there is so much obscurity in their nature that it seems easier to adopt a sensationist interpretation for them. Psychological length becomes some dimension of length sensations. But what dimension? Intensity? Length? Not the latter, since rods and not sensations possess length. Psychological weight becomes some dimension of

526

PERCEPTUAL PSYCHOPHYSICS

weight sensations. Gut what are weight sensations? Kinesthetic sensations? Pressure sensations? And what is their psychological dimension? Not weight, which is a dimension of physical objects. This brief discussion is enough to show that no single interpretation of the nature of psychological dimensions can be applied to every area of perception. Not only does it indicate that something is wrong in the distinction between psychological and physical dimensions, but it also helps to explain why no definite interpretation of this distinction has ever been adopted. Psychophysicists generally shift uneasily and unclearly back and forth among the three interpretations. The shifts are unclear to them because they have not explicitly separated the various interpretations. Had they done so, they might have become disenchanted with the distinction between psychological and physical dimensions and the traditional conception of psychophysics which presupposes it. Of the three interpretations—objectivist, constructionist, and sensationist—the latter receives most attention in this book. The reason is as follows. Psychophysicists tend to gravitate toward the sensationist interpretation (as chap. 9 shows) even while disavowing it. This preference probably stems from a dualist metaphysics lying deep in the consciousness of even those psychologists who wish to abandon it. The picture of two worlds, one mental and one physical, is best served by regarding psychological dimensions as dimensions of sensations. It is badly served by regarding them as public, objective dimensions, and ambiguously served by regarding them as constructions from observer responses to physical dimensions. The sensationist interpretation promotes the feeling that psychophysics is a science designed to explore the mysterious relation between mind and body. This feeling attended the birth of the science, and for a hundred years has made it an object of compelling interest to both psychologists and philosophers. The sensationist interpretation is, therefore, the kingpin of traditional psychophysics. If it can be defeated, the other interpretations will probably collapse of their own weight. THE SENSATIONIST CONCEPTION OF PSYCHOPHYSICS

Perception, according to one familiar theory, consists in having sensations. To see is to have visual sensations, to hear is to have auditory sensations, to feel weight is to have weight sensations, and

PERCEPTUAL PSYCHOPHYSICS

527

so on. If the sensation resembles the stimulus that produces it, the perception is correct or veridical; otherwise it is incorrect or nonveridical. The oldest version of this theory regards the sensations involved in perception as empirically real, private entities of which the owner can be aware through introspection. The perceiver infers the character of the stimuli producing his sensations from the character of the sensations. In this way he maintains contact with the external world. Although highly suspect, this theory of perception is not directly attacked in this book; my main concern is with the conception of perceptual psychophysics which normally accompanies the theory. The theory is designated the sensationist theory of perception; the conception that accompanies it, the sensationist conception of perceptual psychophysics. In the sensationist conception, the aim of perceptual psychophysics is to measure the sensations involved in perception and thus to discover numerical laws relating these sensations to their stimuli. In the area of vision, psychophysics aims to measure the visual sensations observers have when they see things, and to discover the numerical laws relating these sensations to the things seen. In the area of hearing, psychophysics aims to measure the auditory sensations observers have when they hear things, and to discover numerical laws relating these sensations to the things heard. And so on for every other department of perception. These numerical laws provide explanations of the various types of perception. They enable us to predict and control what observers see, hear, and feel. They constitute, in short, a scientific theory of perception. And perhaps they even provide an answer to the ancient philosophical question of whether and how observers know the external world. This conception of perceptual psychophysics produces one form of the problem of psychophysical measurement. How is it possible to measure the sensations involved in perception? We know how to measure length, weight, and other dimensions of physical entities. But sensations are mental. They are private, which makes them inaccessible to everyone but their owner, and makes it impossible for other persons to duplicate measurements of them. They are transitory, which makes it difficult if not impossible to establish units of measurement for them. They are insubstantial, and thus incapable of being manipulated and added like rods and weights. And if sensations are incapable of being

528

PERCEPTUAL PSYCHOPHYSICS

measured, perceptual psychophysics is impossible. Instead of drawing this bleak conclusion, however, psychophysicists assume that sensations are measurable, but measurable by procedures quite unlike those used for length and weight. Different types of dimensions require different types of measurement. Physical dimensions require physical methods of measurement, and psychological dimensions—dimensions of sensations—require psychological procedures of measurement. Various procedures of psychological measurement have been devised; but when they are examined, none seems acceptable if regarded as a procedure for measuring sensations. Thus the problem of psychophysical measurement remains. When we reflect on auditory perception, the sensationist conception virtually forces itself on us, as follows. When an observer hears a bell, sound waves generated by the vibrations of the bell produce sound sensations in him. These sensations possess loudness, pitch, and possibly other dimensions. To measure the observer's hearing we must measure the loudness and pitch of his sound sensations. Then perhaps we will be able to discover numerical laws relating dimensions of sounds to the intensity and the frequency of the sound waves that are their stimuli. These laws are the psychophysical laws of auditory perception. The sensationist conception is equally compelling when we try to understand the perception of color. Here, too, we are inclined to say that brightness and hue are dimensions of color sensations, and that the psychophysical laws of color perception are those relating brightness and hue to the intensity and frequency of light waves. The inclination to adopt a sensationist conception for the perception of sound and color makes it highly desirable to pay a great deal of attention to one or the other of these two varieties of perception. In this book I have seized on sound. I argue that sounds—those entities that possess loudness and pitch—are not sensations, but physical entities, and that this is true whether or not we distinguish sounds from sound waves. Sounds, as I have shown earlier, can be measured. But to measure them is not to measure the auditory sensations many theorists believe to be involved in hearing. Consequently, the psychophysical laws of auditory perception—the numerical laws relating auditory sensations to their stimuli—are not to be confused with laws relating sounds, which are physical entities, to sound waves. Once this point is made clear, it becomes extremely difficult to adopt a sensation-

PERCEPTUAL PSYCHOPHYSICS

5*9

ist conception of auditory perception. If sounds are not sensations, what are the auditory sensations allegedly involved in auditory perception? Sounds, and their dimensions of loudness and pitch, are familiar to all of us. If sounds could be viewed as sensations, we would all have a clear notion of the auditory sensations allegedly involved in hearing. But if sounds are not sensations, the notion of an auditory sensation becomes vague and problematic. Perhaps there are no such sensations. Or perhaps auditory sensations are not always involved in auditory perception. In either event, it is of course a mistake to regard the psychophysical laws of auditory perception as laws relating auditory sensations to their stimuli. This book does not settle the question of whether auditory sensations are necessarily or always involved in the perception of sounds. Nor does it decide whether the sensationist theory of perception is generally correct or incorrect. But by distinguishing sounds from sound sensations, we at least break the spell that theory exercises on us both in the department of hearing and generally. More to the point, by distinguishing the measurement of sounds, which are physical entities, from the measurement of sensations, we break the spell the sensationist conception of psychophysics exercises on us in this and other departments of perception. It is no longer possible to offer a law relating sounds and sound waves as a paradigm example of a psychophysical law of perception, as a law relating sensations to their stimuli. The ensuing difficulty in finding a clear example of such a law helps to clear the way for an entirely different conception of perceptual psychophysics, and an entirely different conception of psychophysical measurement. My central aim is to destroy the sensationist conception of psychophysics outlined above, and thus to dissolve the problem of psychophysical measurement generated by it. The measurement of perception does not consist in measuring sensations, but in measuring abilities of a special kind, perceptual abilities. It is highly doubtful that sensations are necessarily or always involved in perception. But if they are, the methods of measurement which psychophysidsts employ are not methods for measuring them. They are methods for measuring the abilities of observers to detect, to discriminate, to respond to the stimuli that impinge upon their sense organs. None of the psychophysical methods requires any procedure of measurement other than

53°

PERCEPTUAL PSYCHOPHYSICS

those available for measuring physical stimuli. Every procedure for measuring perception consists in (a) measuring a group of physical stimuli, (b) eliciting perceptual responses to these stimuli from observers, and (c) processing the results by statistical methods. Procedures of the type (a)-(fc)-(c) may be called "psychophysical" or "psychological" procedures. But they involve no procedure of measurement over and above the measurement of physical entities. In particular, they do not involve the measurement of sensations. An analogy may help to clarify the conception of psychophysics endorsed here. Human beings have the ability to lift objects within a certain limited weight range. To discover how far this ability extends is a simple matter. We weigh a large number of objects and then require subjects to try to lift them. But someone may contend that when human subjects lift objects they have volitions—private, internal events that precede the lifting process and make it possible—and that the measurement of lifting ability consists in measuring these volitions and discovering the numerical laws relating them to the objects lifted. Obviously, this view is absurd. In the first place, it is doubtful that volitions necessarily or always accompany lifting. Second, the relation between volitions (if there be any) and lifting reveals nothing about lifting ability. Consequently, the measurement of lifting ability does not consist in measuring any events called volitions. Similarly, the measurement of perceptual ability does not consist in measuring private, internal events called sensations. This analogy is far from exact, since perceptual abilities are importantly different from motor abilities such as lifting. But it does serve to indicate, in bold outline, the rationale behind the conception of perceptual psychophysics recommended in this book. First, it is highly doubtful that sensations are necessarily or always involved in perception. But second, if they are, the relation between sensations and stimuli has no bearing on perceptual abilities. Consequently, the measurement of perceptual abilities does not consist in measuring any sensations that allegedly accompany the exercise of these abilities. And the measurement of perception should be viewed as the measurement of perceptual abilities. PSYCHOPHYSICS: OLD, NEW, AND RADICAL Reference to a "traditional" conception of psychophycics is likely to be misleading, for it may be taken to imply that the conception

PERCEPTUAL PSYCHOPHYSICS

531

has been abandoned by contemporary psychophysicists in favor of a "modern" one. This implication is not intended, and it is not true. Virtually all contemporary psychophysicists still hold the traditional conception, in one or another of its versions. This is not to say that the conception is generally shared by contemporary psychologists of perception, most of whom do not regard themselves as psychophysicists. Many psychologists regard psychophysics with puzzlement and suspicion, probably because the science retains to this day the essential feature of the traditional conception, and thus has an aura of obsolescence. The essential feature of traditional psychophysics is its dualism between mind and body, between sensations and stimuli, between the mental and the physical. Psychophysics is traditionally defined as the science of the relations between psychological dimensions (magnitudes, phenomena) and physical dimensions. Now it is initially quite difficult to think of another conception to replace this one. What other definition would warrant the use of the term "psycho-physics"? Under what other definition would a science obtain and attempt to explain the facts dealt with by traditional psychophysics? A major purpose in this book is to suggest an alternative conception, one that defines perceptual psychophysics as the science of perceptual abilities. As of today, almost no psychophysicist subscribes to this characterization of his science, not even those practioners of the "new" psychophysics who follow Stevens. Although virtually all psychophysicists conceive their science as that of the relations between psychological and physical dimensions, there are important differences among them. They differ, first of all, in their preferences among the three versions of the traditional conception distinguished earlier: the objectivist, the constructionist, and the sensationist. These differences are, however, not clear or consistent enough to create "schools" of psychophysics. Few psychophysicists have explicitly distinguished the three versions above, and nearly every psychophysicist seems to have embraced each of the three at some point in his work. Psychophysicists differ, in the second place, in their choice of methods for measuring psychological dimensions, and in the psychophysical laws and theories of perception they endorse. Here the differences are consistent enough to create schools. Fechner and his followers employed just noticeable differences to measure psychological

532

PERCEPTUAL PSYCHOPHYSICS

dimensions, and they argued for a logaritmic law relating psychological and physical dimensions. Stevens and his followers have employed units defined by ratio and interval scales to measure psychological dimensions, and have contended for a psychophysical power law. Thus we have the "old" psychophysics of Fechner, and the "new" psychophysics of Stevens. Both are attacked in this book as schools within traditional psychophysics. Thurstone and his followers employed discriminal dispersions as psychological units, and advanced a law called the law of comparative judgment.6 Thurstone qualifies as a traditional psychophysicist, since he assumes a distinction between psychological dimensions (dimensions of "discriminal processes," as he calls them) and physical dimensions. Although I do not deal with Thurstone's theory, my attack can be applied to it as well.7 FECHNER's PSYCHOPHYSICS

G. T. Fechner founded psychophysics with the publication in i860 of his Elemente der Psychophysik. In this work he proposed a law to describe the general relation between sensations and their stimuli. His derivation of this law was based chiefly on two premises. The first was Weber's experimental law, which states that the difference between one stimulus and another that is just noticeably different is a constant fraction of the first; that is, [16]

= c

where d $ is the just noticeable difference between $ and some other stimulus. The second premise was an assumption, the assumption that sensation differences corresponding to just noticeable differences among stimuli (sensation jnd's, for short) are equal. From these premises (and certain additional assumptions) Fechner deduced his famous logarithmic law: [ i 7 ] T = fclog In words, this formula says that a sensation equals the logarithm of its stimulus multiplied by a constant. This is a psychophysical law, relating psychological dimensions to physical dimensions. And it is, 8

L. L. Thurstone, The Measurement of Values (Chicago, 1959), chaps. 2-4. S. S. Stevens, ' T o w a r d a Resolution of the Fechner-Thurstone Legacy," Psychometrika, 26 (1961), 35-47, argues that Thurstone is a latter-day Fechnerian. 7

PERCEPTUAL PSYCHOPHYSICS

535

like the best scientific laws, numerical. It was the realization of Fechner's dream of a true science of mind and body. He predicted that it would "take on for the field of mind-body relations just as general and fundamental a meaning as the law of gravitation in the field of celestial movement." 8 Since [17] is a numerical law, it presupposes a method for measuring not only stimuli, but also sensations. For most stimuli, such methods are available. But how are we to measure sensations? It seems obvious that we cannot, in the fullest sense, measure a dimension without selecting and employing dimension units, for the measurement of a dimension consists in expressing the magnitude of entities within the dimension in terms of dimension units. It seems equally obvious that we cannot, in the fullest sense, measure a dimension without specifying a dimension zero. Without a zero, we can employ units to determine whether a difference between entities is greater than, less than, or equal to another difference, and also how much greater or less it is. But we cannot, without a zero, say that one entity is twice, thrice, one-half, or one-third as large as another. Units make it possible to express equality and inequality of differences; a zero is required in addition to express equality and inequality of ratios. Fechner is generally credited with having provided, or at least suggested, a method of sensation measurement which satisfies the above requirements. The units he suggested are sensation differences corresponding to just noticeable differences among stimuli (sensation jnd's), and the zero sensation he stipulated is the sensation produced by the just noticeable stimulus (or, perhaps we should say, the just not noticeable stimulus). Fechner's proposals were immediately subjected to a barrage of objections. Critics objected to the premises that formed the basis of his derivation. Weber's law, they complained, did not rest on adequate experimental evidence; Weber had performed few experiments and most of those he did perform were haphazard. And later experimentation did not verify the law. Furthermore, the assumption that sensation jnd's are equal was, they said, just that. There is no apriori reason to suppose that sensation jnd's are equal, and Fechner himself provides no empirical evidence that they are. If they are unequal, then 8

G. T. Fechner, Elements of Psychophysics, trans. Helmut E. Adler (New York, 1966), I, 57.

534

PERCEPTUAL PSYCHOPHYSICS

not only does one of Fechner's premises collapse, but the units in his method of measurement, being unequal, are unacceptable. There were further attacks. The method of measurement suggested by Fechner is apparently indirect, since it attempts to measure sensations by measuring their stimuli. Indirect procedures like this one are inferior to direct ones. What is needed is a method for directly measuring sensations, one that does not require the prior measurement of stimuli, and does not define sensation units in terms of stimuli. Whether such a procedure would provide evidence for Fechner's law is uncertain. More important, Fechner seems to beg the principal question at issue: Can sensations be measured? He assumes that they can be measured, and proceeds, by using Weber's law and certain other doubtful premises, to hypothesize a law relating sensations and stimuli. But there is excellent reason to suppose that sensations cannot be measured, and, consequently, that a numerical law governing them is meaningless. Sensations (the critics said) are not additive. A larger sensation is not a sum of smaller ones, and sensations cannot be added to produce larger ones. Entities that are not additive surely cannot be measured. Surely it makes no sense to say that one of them is twice, thrice, one-half, or one-third as large as another. Many of the critics who insisted that sensations are not measurable, and that Fechner had not measured them, offered an explanation of why he thought he had. Fechner had committed the "stimulus error": he had confused the sensation with the stimulus, and had ascribed one of the properties of the latter—its measurability—to the former. We can say that a rod, Ri, is ten times as long as another, Rs. But we cannot meaningfully say that the sensation of Ri is ten times as long (or, as great) as the sensation of R2. Unfortunately, our perceptual estimate of the rod is difficult to distinguish from our introspective estimate of the sensation of the rod. We tend, therefore, to confuse the two, which produces the illusion that the sensation is, like the rod, measurable. That, allegedly, was Fechner's error. There are two ways of assessing what he did. On the one hand, we may say that he provided a special, "psychological" method for measuring stimuli, a method for "rescaling stimuli" in accordance with observer responses. On the other hand, we may say that he hypothesized a law relating physiological dimensions to physical, stimulus dimensions, a

PERCEPTUAL PSYCHOPHYSICS

535

law relating excitation to stimulation. If we let © represent excitation, then the proper way to write Fechner's law is [18] 0 = k log Thus interpreted, the law may or may not be true. To verify it we must measure excitation, which is of course possible. But his law cannot be construed as one relating sensations to stimuli, since the former are incapable of measurement. Controversy of this kind swirled about Fechner's law for several decades after the publication of his Elemente. (It is examined in greater detail in chapter 8. The interested reader can survey its history in Titchener's Introduction to Experimental Psychology.9) The controversy was inconclusive and entirely frustrating. Fechner's critics were unable to destroy his proposals; his defenders were unable to vindicate them. As a result, Fechnerian psychophysics went into decline. William James in 1890 wrote what he intended to be its epitaph. The Fechnerian Massformel [measurement formula] and the conception of it as an ultimate "psychophysic law" will remain an "idol of the den," if ever there was one. . . . it would be terrible if even such a dear old man as this could saddle our Science forever with his patient whimsies, and, in a world so full of more nutritious objects of attention, compel all future students to plough through the difficulties, not only of his own works, but of the still drier ones written in his refutation. Those who desire this dreadful literature can find it; it has a "disciplinary value"; but I will not even enumerate it in a footnote. The only amusing part of it is that Fechner's critics should always feel bound, after smiting his theories hip and thigh and leaving not a stick of them standing, to wind up by saying that nevertheless to him belongs the imperishable glory of first formulating them and thereby turning psychology into an exact science (!).10

This judgment, even if true, is a harsh one, because it fails to give due credit to part of Fechner's work. Fechner suggested a method for measuring sensations, and a law relating these to their stimuli. But he also devised and used several precise methods for measuring the just noticeable difference (now called the differential limen): the method of limits, the method of constant stimuli, and the method of average error (all discussed in chapter 8). These methods have been 9 E . B. Titchener, Experimental Psychology (New York, 1905), II, pt. 2: Instructor's Manual, pp. xiii-clxxi. 10 William James, The Principles of Psychology (New York, 1950), I, 549.

536

PERCEPTUAL PSYCHOPHYSICS

modified and refined, and are presently used to determine the differential limen, the absolute limen, sense distances, and other quantities of interest to contemporary psychologists. They can be regarded as methods for measuring, not sensations, but sensitivity. So regarded, they are not employed to discover psychophysical laws of the form Y = /(i>). So regarded, they have been accepted by and absorbed into experimental psychology, and constitute Fechner's enduring legacy. What was rejected in Fechner, or held in suspicion, was his method for measuring sensations and his law governing them. 11 The decline of Fechner's psychophysics was due in large part to the rise of behaviorism in modern psychology. Fechner assumed a dualism between mental and physical phenomena. Early behaviorism rejected all such dualistic psychologies on the ground that only physical phenomena are capable of scientific treatment. Fechner was generally regarded as having proposed a method for measuring and a law governing "introspectional" sensations—mind-dependent entities that can be perceived (introspected) only by their possessor. Behaviorism rejected, or at least excluded from consideration, all such entities on the ground that they are subject neither to measurement nor to experimentation of any kind. From the behaviorist viewpoint, the only legitimate "psychophysical" laws are those relating stimulus and response. If a so-called private, mental entity cannot be defined m terms of publicly observable responses, it is not capable of scientific treatment. And the sensations in Fechner's psychophysics did not seem to be thus definable. Behaviorist criticism was sufficient to elicit the following comment by Boring in 1 9 2 1 : There is no doubt . . . that Fechnerian psychophysics stands or falls according to its success in distinguishing between measurements of mind and measurements of body, or between sensation and the object of sensation, the stimulus. . . . Now that behaviorism has come into vogue, it is not apparent that we do not have two kinds of psychophysics—a psychophysics of process that gives, as Fechner wanted, the correlation between mental and physical data, and a psychophysics of behavior that seeks to identify response with its stimulus.12 11

Edwin G. Boring, A History of Experimental Psychology (New York, 1942),

p. 293.

12 Excerpted and rearranged from Edwin G. Boring, "The Stimulus Error," American Journal of Psychology, 32 (1921), 452.

PERCEPTUAL PSYCHOPHYSICS

537

It is possible to interpret Fechner as having measured, not introspectional sensations, but rather sensations defined in terms of observer responses to stimuli.13 But this possibility was not taken very seriously either by Fechner's critics or by his sympathizers, and has played almost no role in the development of his psychophysics. STEVENS' PSYCHOPHYSICS

Psychophysics is currently enjoying a renaissance, the credit for which belongs principally to S. S. Stevens and his co-workers in the Harvard Psychological Laboratory. One's initial impression of his work is that it has given psychophysics an entirely new direction, both philosophically and scientifically. On the philosophical side, Stevens seems to have met the criticism of the behaviorists by defining psychological dimensions in terms of observer responses to physical dimensions. In addition, he has proposed a new and liberal definition of measurement, a definition on which measurable dimensions need not be additive. On the scientific side, he has rejected the just noticeable difference as the unit of psychophysical measurement. And he has proposed, in place of Fechner's logarithmic law, a power law relating psychological and physical dimensions. One important thesis in this book is that these appearances are deceptive, and that Stevens is a psychophysicist in the traditional, Fechnerian sense. Let us begin with Stevens' attempt to liberalize the concept of measurement. The classical definition insists that measurement is a process of assigning numerals to objects by means of a physical operation of adding units (see chap. 4). Stevens finds this definition too narrow, and proposes that we define measurement as a process of assigning numerals to objects according to rule (see chap. 5).14 The important question then becomes one of distinguishing different types of rule for assigning numerals. Instead of asking whether a particular process of numeral assignment is measurement, we should ask what kind of measurement it is, that is, what kind of rule for assigning numerals it employs. If we assign numerals so as to represent merely 13 H. M. Johnson, "Did Fechner Measure Introspectional Sensation'? " Psychological Review, 36 (1929), 257-284, offers such an interpretation. 14 S. S. Stevens, "Mathematics, Measurement, and Psychophysics," in Handbook of Experimental Psychology, ed S. S. Stevens (New York, 1951), pp. 21 ff., esp. p. 29.

PERCEPTUAL PSYCHOPHYSICS

53»

the order of objects, we create an ordinal scale. Mohs's scale of hardness for minerals is an example. If we assign numerals so as to represent the intervals between objects, we create an interval scale. The Fahrenheit scale of temperature is an example. If we assign numerals so as to represent ratios between objects, we create a ratio scale. The centimeter scale of length is an example. The examples above are all scales for physical dimensions: hardness, temperature, and length. But, according to Stevens, scales of the various kinds are also possible for psychological dimensions. Let us consider an example of an interval scale. Suppose we present an observer, O, with a number of weights and ask him to assign numerals from x to 1 0 to the weights in such a way that the numerals represent felt equality of intervals. For example, O is to assign 3 to $ 1 , 4 to $2, and 5 to $3 only if $3 feels as much heavier than as $2 feels heavier than The numerals thus assigned represent subjective (psychological) weight, weight as felt by O, and not the objective, physical weight of the objects. Suppose O's assignments are those in table 1 7 . TABLE 17 PSYCHOLOGICAL WEIGHT

10 9 8 7 6

5 4 3 2 1

PHYSICAL WEIGHT

100 88 76 62 50 41 31 22 15 7

This table contains an interval scale for psychological weight in relation to physical weight. We might, alternatively, have asked O to assign numerals to weight so as to represent their felt ratios. For example, he is to assign 2 to and 4 to $2 only if $2 feels twice as heavy to him as In that event, the table of results would contain a ratio scale for psychological weight in its relation to physical weight.

PERCEPTUAL PSYCHOPHYSICS

539

It is possible to stipulate psychological units for these scales so that any scale for a given psychological dimension can be compared with any other. We might, for example, stipulate that the psychological magnitude associated with a weight of 50 grams is 20 psychological units. By selecting units in this manner, and by constructing ratio scales (which provide the best sort of measurement), Stevens and others have obtained data for a general law relating psychological and physical magnitudes, namely, [19] Y = i»\ where $ is the stimulus in physical units, Y is psychological magnitude in psychological units, and n is an exponent that varies with the dimension being scaled. Exponents have been experimentally determined for more than a dozen psychological dimensions. Table 1 8 contains a partial list of these exponents.15 TABLE 18 PSYCHOLOGICAL

PHYSICAL

DIMENSION

DIMENSION

Loudness Brightness Taste Seen length Seen area Felt weight Subjective duration

Intensity of sound Intensity of light Solution concentration Measured length Measured area Measured weight Clock time

EXPONENT

UNIT

03

Sone Bril Gust Mak Var Veg Chron

0.3-0.5 1.0 1.1 0.9-1.15 1-45

1.05-1.2

Stevens claims that his scaling methods of measurement are superior to Fechner's jnd method in many respects. And yet, if the matter is examined closely, every objection brought against Fechner can, with some modification, be directed against Stevens. Some of these objections are merely sketched here (they are presented fully in chapter 9). To begin with, Stevens assumes that psychological dimensions can be measured, an assumption that is entirely questionable. Stevens 15

Taken, with modifications, from S. S. Stevens, "On the Psychophysical Law,"

Psychological

Review, 64 (1957), 166.

54°

PERCEPTUAL PSYCHOPHYSICS

tries to persuade us of their measurability by liberalizing the definition of measurement, but his definition is unacceptably broad. He says, for example, that any assignment of numerals to objects which represents their ratios is measurement. This definition neglects the manner in which numerals are assigned, and focuses only on the result of the assignment. If a person assigns numerals to rods at random, we do not say that the rods have been measured, even if by happy accident the numerals correspond to length ratios. The manner in which the numerals are assigned is critical. Perhaps, as Stevens argues, an operation of physical addition is not necessary; but some duplicatable, physical operation is. This neglect of the manner of numeral assignment causes Stevens to overlook the fact that the observers in his scaling experiments assign numerals to psychological length or weight by estimating ratios or intervals. And estimation is not measurement. Stevens insists that his methods for measuring psychological dimensions are, in contrast with Fechner's, direct. In one perfectly good sense, this is obviously false. Stevens' methods depend on the prior measurement of stimulus magnitudes. His scales can be constructed only for those psychological magnitudes that correspond to measurable physical magnitudes. He denies this objection by offering an experiment dealing with a stimulus dimension that, he claims, has no underlying metric. But the dimension is roughness of sandpaper, and the grit numbers of the paper are used as measures of physical roughness.18 Furthermore, Stevens' methods rest on almost as many assumptions as Fechner's, and are in this sense indirect. For example, in the construction of ratio scales the following assumption is made: If O estimates that the ratio between ratio between Y i and ¥ 2 is n:m.

and $2 is n.m, then the

This assumption is just as problematic as Fechner's assumption that differences between sensations corresponding to just noticeable differences between stimuli are equal. This point is intimately related to the question of unit equality. The most popular objection against Fechner was that his psychological units—sensation jnd's—are unequal, or, at least, that he had no reason to assume their equality. Stevens confirms this objection in the 16

82.

S. S. Stevens, ' T o Honor Fechner and Repeal His Law," Science, 133 (1961),

PERCEPTUAL PSYCHOPHYSICS

541

discovery of his power law. For if [19] is true, then Fechner's law, [17], is false, and sensation jnd's (Stevens calls them "subjective jnd's") are unequal. But the objection of unit inequality can also be brought against Stevens. He defines his units for psychological ratio scales as follows: If O estimates that is twice the magnitude of $2, and if it is stipulated that i>i is n psychological units, then $2 is n/2 psychological units. This definition entails the assumption formulated in the preceding paragraph, and is equally problematic. It is even more tempting to charge that Stevens has committed the stimulus error than that Fechner did so. Stevens asks his observers to estimate the intervals and the ratios between physical entities—stimuli —and he then takes these estimates as indicative of the intervals and ratios between psychological entities. But is he not confusing a physical estimate with a psychological estimate, and confusing physical dimensions with psychological dimensions? Physical, stimulus dimensions are clearly measurable, and quantitative stimulus estimates are meaningful. But we cannot thereby infer that psychological entities are measurable, or that quantitative estimates of psychological entities are meaningful. Stevens seems to have drawn this invalid inference. There are two ways of reconstructing his procedures so as to avoid the above objection. On the one hand, we might say that he has provided a special, "subjective" method for measuring physical dimensions, that he has "rescaled stimulus dimensions" in a way that conforms to observer estimates. On the other hand, we might say that he has hypothesized a law relating physical, stimulus dimensions to physiological dimensions, dimensions of excitation. Letting 8 represent excitation, Stevens' law then becomes [20] e = c Unlike Stevens' original law, [20] is verifiable because it is possible to measure excitation. However we reconstrue Stevens' law, it cannot be construed as one relating sensations to stimuli, since the former are incapable of measurement. The above discussion shows that Stevens' psychophysics is vulnerable in much the same way as Fechner's. The explanation is that for both scientists it is psychophysics in the traditional sense. Psycho-

542

PERCEPTUAL PSYCHOPHYSICS

physics is conceived by both as the science of the relation between psychological and physical dimensions, between mental and physical phenomena, between mind and body. Stevens would undoubtedly balk at the terms "mental" and "mind," since he pretends to be free of the introspectionist philosophy ascribed to Fechner. He claims to have defined psychological dimensions in terms of observer responses to physical dimensions, and thus to have avoided the suggestion that psychological dimensions are private and introspective, and therefore incapable of scientific treatment. As we found in chapter 9, however, Stevens' claim here is questionable. Even when psychological magnitudes are behavioristically defined, they become substitutes for introspective magnitudes, for magnitudes of sensations, and take on many of the objectionable features of the latter. Stevens, like Fechner, still believes that psychophysics is the science of mind and body, though mind has become in his theory an uncertain and pallid replica of its old original self. RADICAL PSYCHOPHYSICS

It is only by replacing the traditional conception of perceptual psychophysics with a radically different one that we can eliminate, once and for all, the difficulties that beset Fechner and Stevens. This requires abandoning the distinction between psychological dimensions and physical dimensions, and replacing the concept of a psychological dimension with the concept of a perceptual ability. Instead of viewing psychophysics as the attempt to measure psychological dimensions, we must regard it as the attempt to measure perceptual abilities. Thus conceived, psychophysics does not require "psychological units," or any special methods of "psychological measurement." Every procedure for measuring perception is then seen to consist in (a) measuring a group of physical stimuli, (b) eliciting perceptual responses to these stimuli from observers, and (c) processing the results by statistical methods. We may call such a procedure "psychological" if we choose, although it does not, strictly speaking, involve any procedure of measurement over and above the measurement of physical dimensions. The measurement of "psychological dimensions" is not involved. In this radical conception, perceptual psychophysics does not seek laws of the form, Y = /($). Weber's law, [16] d$ = c has

PERCEPTUAL PSYCHOPHYSICS

543

the form, $1 = f($2), and contains only physical variables. Its verification requires no psychological unit, no special psychological method of measurement. And yet it is a perfectly acceptable psychophysical law. A s chapter 8 shows, Weber's law contains all the empirical information of Fechner's law, [17] V = k log Every legitimate scientific use that can be served by Fechner's law can also be served by Weber's. In chapter 9 it is suggested, similarly, that every legitimate scientific use served by Stevens' law, [19] Y = can also be served by a law of the form, = and that Stevens' law is therefore dispensable. These arguments are meant to illustrate the point that perceptual psychophysics can proceed by discovering laws that do not have the traditional form. Every legitimate discovery in perceptual psychophysics, every psychophysical law, can be expressed in some nontraditional form which requires no concept of a psychological dimension, and no special psychological method of measurement. Perceptual psychophysics thus conceived generates no philosophical problems and no insuperable methodological problems. The conception of psychophysics recommended here is not totally new. It is vaguely suggested in some of the objections to Fechner, particularly in the objection that he has committed the stimulus error. It was almost adopted in 1892 by Fullerton and Cattell, who concluded that sensation cannot be measured, at least not directly, and suggested a different, attainable task for the psychophysicist. If an observer can, in fact, estimate quantitative amounts of difference in sensation, apart f r o m association with k n o w n quantitative differences in the stimuli, a relation between mental and physical intensity can be determined. T h e writers, h o w e v e r , agree in finding that they cannot estimate such quantitative differences in sensation in a satisfactory manner. 1 7

They go on to say that all acceptable psychophysical methods "determine the error of observation under varying circumstances, and [do] not . . . measure at all the quantity of sensation." 18 These methods, they claim, "determine the error of observation. This is a physical quantity. Its correlation with other physical quantities (for example, 1 7 G. S. Fullerton and J. M. Cattell, On the Perception of Small (Philadelphia, 1892), p. 20. 18 Ibid., p. 24.

Differences

544

PERCEPTUAL PSYCHOPHYSICS

the magnitude of the stimulus) depends on psychophysical conditions, and offers an important subject for psychological research. A mental quantity is not, however, directly measured."19 Describing the work of Fullerton and Cattell in 1 9 2 1 , Boring said that "a quantitative psychology of error is of necessity a psychology of capacity—of the capacity of the organism to respond correctly to stimuli." A psychology of capacity "give[s] up the measurement of mind, substituting the measurement of sensitivity or of capacity-fordiscrimination."20 The quantitative psychology of capacity admits the quantity objection [the objection that sensations cannot be measured] and denies—or at least ignores—mental quanta. This psychology sees no distinctively mental measurement, but undertakes the physical measurement of bodily response as a function of the physical quantities of the stimulus. There is no sharp epistemological line discernible between this sort of measurement and other physical measurement, and it thus meets the requirement of modern behaviorism that psychology interpenetrate physical science without sensible demarcation. This psychology of capacity is also the psychology of mental tests and of Urban's psychophysical experiments.21

Boring's description of capacity psychology could serve as a description of the radical conception of psychophysics recommended in this book. But since he applies it to Fullerton and Cattell and certain other psychophysicists, it is important to understand the subtle way in which their conception differs from my own. The "capacity psychologists" mentioned by Boring imply that it would be desirable to measure the sensations involved in perception, if it were possible to do so. They conclude, resignedly, that it is not, and turn instead to the measurement of perceptual capacities, such as errors of stimulus observation. They tend to regard this method as an indirect, second-best way of getting at sensations. To propose such a solution to the problem of psychophysical measurement is to suggest that sensations may, after all, be measurable, if only someone could discover an acceptable method for measuring them. It is to issue a challenge to later psychophysicists to provide the method, to succeed where others have failed. Stevens accepts the challenge and, after numerous experiments, says: 19 20 21

Fullerton and Cattell, On the Perception of Small Differences, p. 1 5 3 . Boring, "The Stimulus Error," pp. 459-460. Ibid., p. 460.

PERCEPTUAL PSYCHOPHYSICS

545

"When I see with what assurance and consistency some Os [observers] (I might even say most Os) assess subjective magnitudes, I am puzzled that so much bitter ink should have been spilled over the so-called 'quantity objection' to psychophysics."22 Stevens claims to have done what Fullerton and Cattell would have liked to do, to have measured sensations, or, as he more often says, subjective magnitudes. My view, unlike that of Fullerton and Cattell, unlike that of Stevens, unlike that of virtually every psychophysicist, is that the measurement of sensations, of psychological magnitudes, is not a desirable or proper goal of perceptual psychophysics. Sensations may or may not be produced whenever stimuli are perceived. But if they are, perception does not depend on them. Observers do not perceive stimuli by introspecting sensations produced by these stimuli. They perceive the stimuli directly, without intermediaries. Sensations, when involved in perception, are not an operative part of the perceptual mechanism: they are epiphenomenal. Consequently, the task of perceptual psychophysics is not to measure sensations, or to discover laws relating sensations to their stimuli. Rather, it is to measure perceptual abilities, and to discover the laws governing these. Stevens, like every psychophysicist who preceded him, yearns to measure sensation magnitudes; it is just that he calls them "psychological magnitudes." And he yearns to discover laws relating psychological magnitudes to physical, stimulus magnitudes. His aim is not to disavow Fechner's conception of psychophysics. It is, in the words of the title of one of his articles, "To Honor Fechner and Repeal His Law." It is to honor Fechner's conception of psychophysics, but to repeal the particular psychophysical law he proposed. In contrast, I regard the repeal of Fechner's law as a question of secondary interest. The primary question is whether his conception of psychophysics is acceptable. I suggest that it is not, that the traditional conception of perceptual psychophysics, which Fechner, Stevens, and virtually every other psychophysicist share, must be abandoned. Another difference between my radical conception and existing ones emerges when we ask whether mine is a "behaviorist" conception. In one sense it is. Perceptual abilities, which are the proper objects of psychophysical measurement, are defined in terms of behavior, in terms of perceptual responses to stimuli. But in another sense such a 23 S. S. Stevens, "The Direct Estimation of Sensory Magnitudes—Loudness," American Journal of Psychology, 69 (1956), 13; see also pp. 2,18.

546

PERCEPTUAL PSYCHOPHYSICS

characterization is misleading. Behaviorist psychologists are often frustrated introspectionists, who fundamentally believe that private, introspective entities are the proper subject matter of psychology, and are trying to devise respectable scientific ways of dealing with such entities. As chapter 9 shows, this is especially true of behaviorist psychophysicists, like Stevens. Magnitudes of sensations, when taken to be the private entities of older psychology, seem not to be measurable. So Stevens suggests that they can be defined in terms of public responses to physical stimuli, and labels them "psychological magnitudes." In this way he tries to get at the apparently inaccessible subject matter of psychophysics. In contrast, I hold that the measurement of sensations is not the proper task of perceptual psychophysics, whether the sensations are defined behavioristically or introspectionistically. The concept of a sensation, however defined, must be replaced by the concept of a perceptual ability. When this is done, psychophysics can no longer be viewed as the measurement of psychological dimensions in order to discover their numerical relations to physical dimensions. In this book I have discussed only the psychophysics of perception, the psychophysics of seeing, feeling, hearing, and so on. Even if my radical conception can be employed in these areas, there is a serious question as to whether it will serve in other areas of interest to psychophysicists, for example, in the study of hallucinations, and the study of sensations such as pains and aftersensations. When O sees or feels the weight or the length of objects, his perceptual responses are responses to public, physical entities—to stimuli. These responses define his perceptual abilities, and the abilities can be measured by measuring stimuli to which he responds. But when O hallucinates a rod or a weight, no public, physical entity—no rod or weight— exists to which he perceptually responds. Physical or physiological entities of some kind may be the cause of his hallucination, but they are not stimuli to which he perceptually responds. In a hallucination, O perceptually responds—if it is correct to speak this way—to a private, psychological entity. He exhibits no abilities to see and feel rods and weights, but only disabilities to do these things. His perceptual abilities—if it is correct to speak this way—are defined by responses to private, psychological entities, and can be measured only by measuring these entities. The psychophysics of hallucination thus

PERCEPTUAL PSYCHOPHYSICS

547

seems to require the measurement of psychological dimensions. The concept does not here seem dispensable. Or, again, consider pains, which are paradigm examples of sensations. There is the strongest kind of inclination to say that psychophysical laws of pain are laws relating sensations to their physical (or physiological) causes, laws of the form, Y = /($). It does not seem that the concept of a perceptual ability can be substituted here for the concept of a psychological dimension. What perceptual ability would be measured? It cannot be the ability to perceive some public, physical entity, since pain is not public and not physical. To feel pain is to feel a private, psychological entity. Public entities—a pin, for example —may cause O to feel pain. But to feel the pin is not to feel the pain. It is not the ability to feel pins we wish to measure here, but the ability to feel pains. And this ability cannot be measured without a method for measuring pains, for measuring sensations. The question of whether my radical conception of psychophysics can be made to apply in these areas is not examined in this book. There is, however, one point that can be made independently of such an examination. Even if the traditional conception must be employed for the psychophysics of hallucinations and the psychophysics of sensations like pains, it does not follow that it must or should be employed for the psychophysics of perception, for the seeing of rods, the feeling of weights, and the hearing of sounds.

Bibliography Adrian, Edgar Douglas. The Basis of Sensations. New York: Norton, 1929. . The Physical Background of Perception. Oxford: Clarendon Press, 1957Andrews, T. G., ed. Methods of Psychology. New York: John Wiley and Sons, 1948. Angell, J. T., and Warner Fite. "The Monaural Localization of Sound," Psychological Review, 8 (1961), 225-246. Armstrong, D. M. Bodily Sensations. New York: Routledge and Kegan Paul, 1962. . "Is Introspective Knowledge Incorrigible?" Philosophical Review, 72 (1963), 417-418. Perception and the Physical World. New York: Humanities Press, 1961. Austin, J. L. Sense and Sensibilia. Reconstructed by G. J. Warnock. Oxford: Garendon Press, 1962. Baker, K. E., and F. J. Dudek. "Weight Scales from Ratio Judgements and Comparisons of Existent Weight Scales," Journal of Experimental Psychology, 50 (1955)/ 293-308. Bartlett, R. J. "Measurement in Psychology," The Advancement of Science: The Report of the British Association for the Advancement of Science, 1 (1939-40), 422-441. Beck, Jacob, and William A. Shaw. "The Scaling of Pitch by the Method of Magnitude Estimation," American Journal of Psychology, 74 (1961), 242-251. Beebe-Center, J. G., and D. Waddell. "A General Psychological Scale of Taste," Journal of Psychology, 26 (1948), 517-524. Beecher, Henry K. Measurement of Subjective Responses. New York: Oxford University Press, 1959. . "A Method for Quantifying the Intensity of Pain," Science, 1 1 8 (i953)/ 322-324. Beranek, Leo L. Acoustic Measurement. New York: John Wiley and Sons, 1949. Bergmann, Gustav. "The Logic of Measurement," State University of Iowa Studies in Engineering (1956). Bergmann, Gustav, and K. W. Spence. "The Logic of Psychophysical Mea-

55°

BIBLIOGRAPHY

surement," Psychological Review, 51 (1944), 1-24- Repr. in Readings in the Philosophy of Science, ed. H. Feigl and May Brodbeck (New York: Appleton-Century-Crofts, 1953), pp. 1 0 3 - 1 1 9 . Bergson, Henri. Time and Free Will. Trans. F. L. Pogson. London: Allen and Unwin, 1910. Berkeley, George. The Works of George Berkeley. Vol. I. Ed. A. A. Luce. London: Thomas Nelson and Sons, 1948. Vol. II. Ed. T. E. Jessop. London: Thomas Nelson and Sons, 1949. Berry, William. "The Flight of Colors in the After-images of a White Light," Psychological Bulletin, 19 (1922), 307-337. Blanshard, Brand. "Behaviorism and the Theory of Knowledge," Philosophical Review, 37 (1928), 328-352. Boring, Edwin G. "The Beginning and Growth of Measurement in Psychology." In Quantification, ed. Harry Woolf (New York: Bobbs-Merrill, 1961), pp. 108-127. . "Did Fechner Measure Sensation?" Psychological Review, 35 (1928), 443-445. "Fechner: Inadvertent Founder of Psychophysics," Psychometrika, 26 (1961), 3-8. . A History of Experimental Psychology. New York: AppletonCentury-Crofts, 1942. . " A History of Introspection," Psychological Bulletin, 50 (1953), 169189. . "Is There a Generalized Psychophysical Function?" American Journal of Psychology, 35 (1924), 75-78. . "The Psychologist's Circle," Psychological Review, 38 (1931), 1 7 7 182. . Sensation and Perception in the History of Experimental Psychology. New York: Appleton-Century-Crofts, 1942. . "Size Consistency and Emmert's Law," American Journal of Psychology, 53 (1940), 293-295. . "The Size of the Differential Limen for Pitch," American Journal of Psychology, 53 (1940), 450-455. . "The Stimulus Error," American Journal of Psychology, 32 (1921), 449-471. Boring, E. G., et al. Symposium on Operationism, Psychological Review, 52 (1945), 241-294. Boring, E. G., and A. H. Holway. "The Apparent Size of the Moon on a Function of the Angle of Regard: Further Experiments," American Journal of Psychology, 53 (1940), 537 ff. . "The Dependence of Apparent Visual Size upon Illumination," American Journal of Psychology, 53 (1940), 587 ff. . "The Moon Illusion and the Angle of Regard," American Journal of Psychology, 53 (1940)/109 ff-

BIBLIOGRAPHY

551

Brain, W. R. Mind, Perception and Science. Oxford: Basil Blackwell, 1951. Braithwaite, R. B. Scientific Explanation. New York: Harper, 1953. Bridgeman, P. W. The Logic of Modern Physics. New York: Macmillan, 1927. . The Nature of Physical Theory. Princeton: Princeton University Press, 1936. . "Operational Analysis," Philosophy of Science, 5 (1938), 1 1 4 - 1 3 1 . Broad, C. D. Perception, Physics, and Reality. Cambridge: Cambridge University Press, 1914. Brown, Warner. "The Judgement of Difference," University of California Publications in Psychology, I (1910), 1 - 7 1 . Campbell, Norman Robert. An Account of the Principles of Measurement and Calculation. London: Longmans, Green, 1928. . Physics: The Elements. Cambridge: Cambridge University Press, 1920. Reprinted as Foundations of Science. New York: Dover Publications, 1957-

. "Symposium: Measurement and Its Importance for Philosophy," Proceedings of the Aristotelian Society, Supplement, 17 (1938), 121-142. . What Is Science? London: Methuen, 1921. Campbell, Norman Robert, and J. O. Irwin. "On the Sone Scale," App. I, pp. 337-338, in "Quantitative Estimates of Sensory Events," Advancement of Science, no. 2 (1940), 331-349. Canter, R. R., and J. Hirsch. "An Experimental Comparison of Several Psychological Scales of Weight," American Journal of Psychology, 68 (1955), 645-649. Carnap, Rudolf. Philosophical Foundations of Physics. Ed. Martin Gardner. Esp. Pt. II. New York: Basic Books, 1966. Churchman, C. West, and Philburn Ratoosh. Measurement: Definitions and Theories. New York: John Wiley and Sons, 1959. Clausen, John, and H. E. King. "Determination of the Pain Threshold on Untrained Subjects," Journal of Psychology, 30 (1950), 299-306. Clemens, Samuel L. The Writings of Mark Twain. Vol. XIII. New York: Harper, 1918. Cobb, Percy W. "Weber's Law and the Fechnerian Muddle," Psychological Review, 39 (1932), 533-551. Cohen, Morris R., and Ernest Nagel. An Introduction to Logic and Scientific Method. New York: Harcourt, Brace, 1934. Comrey, A. L. "Mental Testing and the Logic of Measurement," Educational and Psychological Measurement, 1 1 (1951), 323-334. . "A Proposed Method for Absolute Ratio Scaling," Psychometrika, 1 5 (1950), 317-325Coombs, Clyde H. "A Theory of Psychological Scaling," Engineering Research Bulletin, no. 34 (1952), 1-94. . A Theory of Scaling. Ann Arbor: University of Michigan Press, 1952.

55*

BIBLIOGRAPHY

Cowdrick, M. "The Weber-Fechner Law and Sanford's Weight Experiment," American Journal of Psychology, 38 (1917), 585-588. Crombie, A. C. "Quantification in Medieval Physics." In Quantification, ed. Harry Woolf (New York: Bobbs-Merrill, 1961), pp. 13-30. Crookes, T. G. "The Apparent Size of After-images," American Journal of Psychology, 72 (1959), 547~553Dallenbach, Karl M. "Pain: History and Present Status," American Journal of Psychology, 52 (1939), 33*"347Delboeuf, J. R. L. Éléments de Psychophysique générale et spéciale. Paris: Librairie Germer G. Baillière, 1883. Denes, P., and F. A. Carter. "Difference Limina for Quality of Pure Tones: An Exploratory Study," Quarterly Journal of Experimental Psychology, 3 (1951)/ 89-91. Dennis, Wayne. Readings in the History of Psychology. New York: Appleton-Century-Crofts, 1948. Eccles, J. C. The Neurophysiological Basis of Mind. Oxford: Clarendon Press, 1953. Eccles, John C., ed. Brain and Conscious Experience. New York: SpringerVerlag, 1966. Edwards, Ward. "Apparent Size of After-images under Conditions of Reduction," American Journal of Psychology, 66 (1953), 449-455. . "Emmert's Law and Euclid's Optics," American Journal of Psychology, 63 (1950)/ 607-612. Edwards, Ward, and E. G. Boring. "What Is Emmert's Law?" American Journal of Psychology, 64 (1951), 416-422. Eisler, H. "Empirical Tests of a Model Relating Magnitude and Category Scales," Scandinavian Journal of Psychology, 3 (1962), 88-96. Ekman, Gösta. "Is the Power Law a Special Case of Fechner's Law?" Perceptual and Motor Skills, 19 (1964), 730 ff. . "Some Aspects of Psychophysical Research." In Sensory Communication, ed. W. A. Rosenblith (New York: John Wiley and Sons, 1961), pp. 35-47. . 'Two Generalized Ratio Scaling Methods," Journal of Psychology, 45 (1958)/ 287-295. Ekman, Gösta, and Lennart Sjöberg. "Scaling," Annual Review of Psychology, 16 (1965), 451-474Ellis, Brian D. Basic Concepts of Measurement. Cambridge: Cambridge University Press, 1966. . "On the Nature of Dimensions," Philosophy of Science, 31 (1964), 357-380. . "Some Fundamental Problems of Direct Measurement," Australasian Journal of Philosophy, 38 (i960), 37-47. . "Some Fundamental Problems of Indirect Measurement," Australasian Journal of Philosophy, 39 (1961), 13-29.

BIBLIOGRAPHY

553

Evans, R. M. An Introduction to Color. New York: John Wiley and Sons, 1948. Fechner, G. T. Elemente der Psychophysik. 3d ed. Vols. I, II. Leipzig: Breitkopf und Härtel, 1907. i s t ed., i860. . Elements of Psychophysics. Vol. I. Trans. Helmut E. Adler. New York: Holt, Rinehart and Winston, 1966. Ferguson, A., et al. "Quantitative Estimates of Sensory Events," Report of the 108th Annual Meeting, British Association for the Advancement of Science (1938), pp. 277-335. . "Quantitative Estimates of Sensory Events," The Advancement of Science: The Report of the British Association for the Advancement of Science, 1 (1939-40), 331-349. Femberger, S. W. " O n Absolute and Relative Judgements in Lifted Weight Experiments," American Journal of Psychology, 43 (1931), 560-578. Fletcher, Harvey. "Physical Measurements of Audition," Journal of the Franklin Institute, 196 (1923), 289-326. Fletcher, Harvey, and W . A. Munson. "Loudness: Its Definition, Measurement, and Calculation," Journal of the Acoustical Society of America, 5 (1933), 82-108. Fullerton, G. S., and J. M. Cattell. On the Perception of Small Differences. Philadelphia: University of Pennsylvania Press, 1892. Fulton, J. F. Textbook of Physiology. 16th ed. Philadelphia: W. B. Saunders, 1950. Gage, F. H. " A n Experimental Investigation of the Measurability of Visual Sensation," Proceedings of the Royal Society, 116 (1934-35), 123-138. . "The Measurability of Auditory Sensations," Proceedings of the Royal Society, 116 (1934-35), 103-119. Galanter, E. H. "Contemporary Psychophysics." In New Directions in Psychology, ed. Roger Brown (New York: Holt, Rinehart and Winston, 1962), pp. 87-156. Galifret, Yves, and Henri Pieron. "The Increase of the Differential Fraction with Increased Light," Année Psychologique, 52 (1952), 1-37. Garner, W. R. "Some Statistical Aspects of Half-Loudness Judgements," Journal of the Acoustical Society of America, 24 (1952), 153-157. . " A Technique and a Scale for Loudness Measurement," Journal of the Acoustical Society of America, 26 (1954), 73-88. Gasking, D. A. T. "Clusters," Journal of Philosophy, 38 (i960), 1-36. Gibson, Eleanor J., Richard Bergman, and Jean Purdy. "The Effect of Prior Training with a Scale of Distance on Absolute and Relative Judgements of Distance over Ground," Journal of Experimental Psychology, 50 (1955), 97-105. Gibson, James Jerome. The Perception of the Visual World. Boston : Houghton Mifflin, 1950.

554

BIBLIOGRAPHY

. The Senses Considered as Perceptual Systems. Boston: Houghton Mifflin, 1966. Goude, Gunnar. On Fundamental Measurement in Psychology. Stockholm: Almquist and Wiksell, 1962. Graham, C. H. "Sensation and Perception in an Objective Psychology," Psychological Review, 65 (1958), 65-76. Guilford, J. P. "Fluctuations of Attention with Weak Visual Stimuli," American Journal of Psychology, 38 (1927), 534-583. . " A Generalized Psychophysical Law," Psychological Review, 39 (1932)/ 73-85. "Psychological Measurement a Hundred and Twenty Years Later," Psychometrika, 26 (1961), 109-127. . Psychometric Methods. New York: McGraw-Hill, 1936. Rev. ed., I 954Guilford, J. P., and Milton Cotzin. "Judgement of Difficulty of Simple Tasks," American Journal of Psychology, 54 (1941), 38-52. Guilford, J. P., and H. F. Dingman. " A Validation Study of Ratio-Judgement Methods," American Journal of Psychology, 67 (1954), 395-410. Gulliksen, Harold. "Linear and Multidimensional Scaling," Psychometrika, 26 (1961), 9-26. Gulliksen, Harold, and Samuel Messick, eds. Psychological Scaling: Theory and Applications. New York: John Wiley and Sons, i960. Hamlyn, D. W. Sensation and Perception. New York: Humanities Press, 1961. Hanes, R. M. "The Construction of Brightness Scales from Fractionation Data: A Validation," Journal of Experimental Psychology, 39 (1949), 719-728. . " A Scale of Subjective Brightness," Journal of Experimental Psychology, 39 (1949)/ 438-452Hardy, James D., Harold G. Wolff, and Helen Goodell. Pain Sensation and Reactions. Baltimore: Williams and Wilkins, 1952. Harper, R. S., and S. S. Stevens. " A Psychological Scale of Weight and a Formula for Its Derivation," American Journal of Psychology, 61 (1948), 343-351Harris, William P. " A Revised Law of Comparative Judgement," Psychometrika, 22 (1957), 189-198. Hartnack, J. "Remarks on the Concept of Sensation," Journal of Philosophy, 56 (1959)/ 111-117Hastorf, A. H., and J. L. Kennedy. "Emmert's Law and Size Consistency," American Journal of Psychology, 70 (1957), 114-116. Helson, Harry. "Adaptation-Level as Frame of Reference for Prediction of Psychophysical Data," American Journal of Psychology, 60 (1947), 1-29. Helson, Harry, Walter C. Michels, and Artie Sturgeon. "The Use of Comparative Rating Scales for the Evaluation of Psychophysical Data," American Journal of Psychology, 67 (1954), 321-326.

BIBLIOGRAPHY

555

Hempel, Carl G. Fundamentals of Concept Formation in Empirical Science. International Encyclopedia of Unified Science. Vol. II, no. 7. Chicago: University of Chicago Press, 1952. . "Geometry and Empirical Science," American Mathematical Monthly, 52 (1945), 7-17. Repr. in Readings in Philosophical Analysis, ed. H. Feigl and W. Sellers (New York: Appleton-Century-Crofts, 1949), pp. 238-249. . "On the Nature of Mathematical Truth," American Mathematical Monthly, 52 (1945), 543-556. Repr. in Philosophy of Mathematics, ed. P. Benacerraf and H. Putnam (Englewood Cliffs, N. J.: Prentice-Hall, 1964), and in Readings in Philosophical Analysis, ed. H. Feigl and W. Sellers (New York: Appleton-Century-Crofts, 1949)Henmon, Vivian Allen Charles. "The Time of Perception as a Measure of Differences in Sensations." Archives of Philosophy, Psychology and Scientific Method. Vol. 8. New York: Science Press, 1906. Henmuller, F., and K. Menger. "What Is Length?" Philosophy of Science, 28 (1961), 172-177. Hirsh, Ira J. "Binaural Summation—A Century of Investigation," Psychological Bulletin, 45 (1948), 193-206. . The Measurement of Hearing. New York: McGraw-Hill, 1952. Holway, A. H., and C. C. Pratt. "The Weber-Ratio for Intensitive Discrimination," Psychological Review, 43 (1936), 322-340. Householder, A. S. "A Neural Mechanism for Discrimination," Psychometrika, 4 (1939), 45-58. Householder, A. S., and Gale Young. "Weber Laws, the Weber Law, and Psychophysical Analysis," Psychometrika, 5 (1940), 183-193. James, William. The Principles of Psychology. Vols. I, II. New York: Dover Publications, 1950. First published, 1890. Jastrow, Joseph. "A Critique of Psycho-Physic Methods," American Journal of Psychology, 1 (1887), 271-309. Johnson, H. M. "Did Fechner Measure 'Introspectional Sensation'?" Psychological Review, 36 (1929), 257-284. . "Pseudo-Mathematics in the Mental and Social Sciences," American Journal of Psychology, 48 (1936), 342-351. "Some Fallacies Underlying the Use of Psychological Tests," Psychological Review, 35 (1928), 328-337. . "Some Properties of Fechner's 'Intensity of Sensation/ " Psychological Review, 37 (1930), 1 1 3 - 1 2 3 . Jones, F. N. "Scales of Subjective Intensity for Odors of Diverse Chemical Nature," American Journal of Psychology, 71 (1958), 305-310. Kellogg, W. N. "An Experimental Comparison of Psychophysical Methods," Archives de Psychologie, 106 (1929), 1-86. Kenneth, J. H., and R. H. Thouless. "Relationship between the Absolute and Differential Thresholds for an Auditory Stimulus," American Journal of Psychology, 42 (1930)/ 389-398-

556

BIBLIOGRAPHY

King, William A., and Howard E. Gruber. "Moon Illusion and Emmert's Law," Science, 135 (1962), 1 1 2 5 - 1 1 2 6 . Kuhn, T. S. "The Function of Measurement in Modern Physical Science." In Quantification, ed. Harry Woolf (New York: Bobbs-Merrill, 1961), pp. 31-63. Kiilpe, O. Outlines of Psychology. New York: Macmillan, 1908. Künnapas, Theodor M. "Scales for Subjective Distance," Acta Psychologica, 19 (1961), 214-216. Kuntze, J. E. Gustav Theodor Fechner. Leipzig: Breitkopf und Härtel, 1892. Landahl, H. D. " A Contribution to the Mathematical Biophysics of Psychophysical Discrimination," Psychometrika, 3 (1938), 107-125. Lemer, D., ed. Quantity and Quality. Glencoe: Free Press, 1961. Lewis, D. R. "Psychological Scales of Taste," Journal of Psychology, 26 (1948), 437-446. Locke, John. An Essay Concerning Human Understanding. Vol. I. Ed. A. C. Fraser. New York: Dover Publications, 1959. Luce, R. D. "On the Possible Psychophysical Laws," Psychological Review, 66 (1959), 81-95. Luce, R. D., R. R. Bush, and E. H. Galanter, eds. Handbook of Mathematical Psychology. Vol. I. New York: John Wiley and Sons, 1963. Luce, R. D., and W. Edwards. "The Derivation of Subjective Scales from Just Noticeable Differences," Psychological Review, 65 (1958), 222-237. McGill, William J. "Loudness and Reaction Time: A Guided Tour of the Listener's Private World," Acta Psychologica, 19 (1961), 193-199. Mackay, D. M. "Mentality in Machines," Proceedings of the Aristotelian Society, Supplementary Volume, 26 (1952), 61-86. . "Psychophysics of Perceived Intensity: A Theoretical Basis for Fechner's and Stevens' Laws," Science, 139 (1963), 1213-1216. Malcolm, Norman. "Direct Perception," Philosophical Quarterly, 3 (1953), 301-316. Repr. in Norman Malcolm, Knowledge and Certainty (Englewood Cliffs, N.J.: Prentice-Hall, 1963), pp. 73-95. Margolis, Joseph. "After-Images and Pains," Philosophy, 41 (1966), 333340. Marx, Melvin H., ed. Psychological Theory. New York: Macmillan, 1951. Meissner, W. W. "Intervening Constructs—Dimensions of Controversy," Psychological Review, 67 (i960), 51-72. Metfessel, M. F. "A Proposal for Quantitative Reporting of Comparative Judgements," Journal of Psychology, 24 (1947), 229-235. Miller, Dayton C. The Science of Musical Sounds. New York: Macmillan, 1934. Miller, G. A. "Sensitivity to Changes in the Intensity of White Noise and Its Relation to Masking and Loudness," Journal of the Acoustical Society of America, 19 (1947), 609-619. Mitra, S. "Some Experiments on Fechner's Paradox," Indian Journal of Psychology, 1 (1926), 153-164.

BIBLIOGRAPHY

557

Moran, H., and C. C. Pratt. "Variability of Judgements on Musical Intervals," Journal of Experimental Psychology, 9 (1926), 492-500. Murphy, Gardner. Historical Introduction to Modern Psychology. 2d ed. New York: Harcourt, Brace, 1949Nagel, Ernest. "Measurement," Erkenntnis, 2 (1931), 315-333. . " O n the Logic of Measurement." Unpublished Ph.D. dissertation, Columbia University, 1930. Nelson, T., and H. Bartley. "Numerosity, Number, Arithmetization, Measurement, and Psychology," Philosophy of Science, 28 (1961), 178-203. Newman, E. B. "The Validity of the Just Noticeable Difference as a Unit of Psychological Magnitude," Transactions of the Kansas Academy of Science, 36 (1933), 172-175. Nutting, P. G. "The Complete Form of Fechner's Law," Bulletin, Bureau of Standards, 3 (1907), 59-64. Oberlin, K. W. "Variations in Intensitive Sensitivity to Lifted Weights," Journal of Experimental Psychology, 19 (1936), 438-455. Oldfield, R. C. "Apparent Fluctuations of a Sensory Threshold," Quarterly Journal of Experimental Psychology, 7 (1955), 1 0 1 - 1 1 5 . Pap, Arthur. An Introduction to the Philosophy of Science. New York: Free Press of Glencoe, 1962. Peirce, C. S., and J. Jastrow. " O n Small Differences of Sensation," Memoirs, National Academy of Science, 3 (1884), 75-83. Peters, R. S., ed. Brett's History of Psychology. London: Allen and Unwin, 1962. Place, U. T. "Is Consciousness a Brain Process?" British Journal of Psychology, 47 (1956), 44-50. Price, H. Review of J. J. Gibson, The Perception of the Physical World, in Mind, 62 (1953), 406-410. Prichard, H. A. Knowledge and Perception. Oxford: Clarendon Press, 1950. Purdy, Jean, and Eleanor J. Gibson. "Distance Judgment by the Method of Fractionation," Journal of Experimental Psychology, 50 (1955), 374380. Ramul, Konstantin. "The Problem of Measurement in the Psychology of the Eighteenth Century," American Psychology, 15 (i960), 256-264. Reese, E. P., T. W. Reese, J. Volkmann, and H. H. Corbin, eds. Psychophysical Research Summary Report. N A V E X O S P-1104, SDC 1 3 1 - 1 - 5 . 1953. (Under contract with Psychophysical Research Unit, Mt. Holyoke College, South Hadley, Mass.) Reese, T. W . "The Application of the Theory of Physical Measurement to the Measurement of Psychological Magnitudes," Psychological Monographs, 55 (1943)Richardson, M. W. "Multidimensional Psychophysics," Psychological Bulletin, 35 (1938), 659-660. Riesz, R. R. "The Differential Intensity Sensitivity of the Ear for Pure Tones," Physical Review, 31 (1928), 867-875.

558

BIBLIOGRAPHY

. "The Relationship between Loudness and the Minimum Perceptible Increment of Intensity," Journal of the Acoustical Society of America, 4 (1933), 2 1 1 - 2 1 6 . Rosenblith, Walter A., ed. Sensory Communication. New York: John Wiley and Sons, 1961. Rosenblith, Walter A., and K. N. Stevens. "On the DL for Frequency," Journal of the Acoustical Society of America, 25 (1953), 980-985. Ruch, T. C. "The Nervous System: Sensory Functions." In Textbook of Physiology, ed. J. F. Fulton (16th ed.; Philadelphia: W. B. Saunders, 1949), p. 311. Ruckmick, C. A. "The History and Status of Psychology in the U.S.," American Journal of Psychology, 23 (1912), 517-531. Ryle, Gilbert. The Concept of Mind. London: Hutchinson, 1949. Savage, C. Wade. "Introspectionist and Behaviorist Interpretations of Ratio Scales of Perceptual Magnitudes," Psychological Monographs, 80 (1966), 1-32. . "The Measurement of Loudness and Pitch." Unpublished Ph.D. dissertation, Cornell University, 1963. Schneider, Bruce, and Lane Harlan. "Ratio Scales, Category Scales, and Variability in the Production of Loudness and Softness," Journal of the Acoustical Society of America, 35 (1963), 1953-1961. Scott, Dana, and Patrick Suppes. "Foundational Aspects of Physics," Journal of Symbolic Logic, 23 (1958), 1 1 3 - 1 2 8 . . Foundational Aspects of Theories of Measurement. Technical Report no. 6, 1957, NR 171-034. Office of Naval Research, Behavioral Sciences Division, Applied Mathematics and Statistical Laboratory, Stanford University, Stanford, California. Skinner, B. F. Verbal Behavior. New York: Appleton-Century-Crofts, 1957. Smart, J. J. C. "Measurement," Australasian Journal of Philosophy, 37 (1959)/ 1 - 1 3 . "Sensations and Brain Processes," Philosophical Review, 68 (1959), 141-156. Smith, B. B. " A Contribution to the Problems of Apparent Size and Shape," Bulletin, British Psychological Society, 26 (1955), 14-15. Smith, J. E. K. "Stimulus Programming in Psychophysics," Psychometrika, 26 (1961), 27-34. Smythies, J. R. Analysis of Perception. New York: Humanities Press, 1956. Stevens, J. C., and B. Scharf. "The Form of the Loudness Function Near Threshold," Proceedings, Third International Congress of Acoustics. Pp. 80-82. Amsterdam: Elsevier, 1961. Stevens, J. C., and S. S. Stevens. "Physiological Zero and the Psychophysical Law," Acta Psychologica, 19 (1961), 192-193. Stevens, S. S. "The Attributes of Tone," Proceedings, National Academy of Science, 20 (1934)/ 457~459. "Biological Transducers," Convention Record of the Institute of

BIBLIOGRAPHY

559

Radio Engineers (1954), pp. 27-33. . "The Calculation of the Loudness of Complex Noise," Journal of the Acoustical Society of America, 28 (1956), 807-832. . "Concerning the Psychophysical Power Law," Quarterly Journal of Experimental Psychology, 16 (1964), 383-385. . "Cross Modality Validation of Subjective Scales for Loudness, Vibration and Electric Shock," Journal of Experimental Psychology, 57 (1959), 201-209. . "The Direct Estimation of Sensory Magnitudes—Loudness," American Journal of Psychology, 69 (1956), 1-25. . "Matching Functions between Loudness and Ten Other Continua," Perception and Psychophysics, 1 (1966), 5-8. . "Mathematics, Measurement, and Psychophysics." In Handbook of Experimental Psychology, ed. S. S. Stevens (New York: John Wiley and Sons, 1951)/ PP- i-49. "Measurement, Psychophysics, and Utility." In Measurement: Definitions and Theories, ed. C. West Churchman and Philburn Ratoosh (New York: John Wiley and Sons, 1959), pp. 18-63. . "Measurement and Man," Science, 127 (1958), 383-389. . "A Metric for the Social Consensus," Science, 1 5 1 (1966), 530-541. . "On the New Psychophysics," Scandinavian Journal of Psychology, 1 (i960), 27-35. . "On the Operation Known as Judgement," American Scientist, 54 (1966), 385-401. . "On the Psychophysical Law," Psychological Review, 64 (1957), (1957), 1 5 3 - 1 8 1 . . "On the Theory of Scales of Measurement," Science, 103 (1946), 677-680. . "On the Validity of the Loudness Scale," Journal of the Acoustical Society of America, 31 (1959), 995-1003. . "The Operational Definition of Psychological Concepts," Psychological Review, 42 (1935), 517-527. . "Operations or Words?" Psychological Monographs, 80 (1966), 3 3 38. (A reply to Savage, 1966.) . "Pitch Discrimination, Mels and Kock's Contention," Journal of the Acoustical Society of America, 26 (1954), 1075-1077. . "Problems and Methods of Psychophysics," Psychological Bulletin, 55 (1958)/ 177-196. "Psychology and the Science of Science," Psychological Bulletin, 36 (1939), 221-263. Repr- in Psychological Theory, ed. M. Marx (New York: Macmillan, 1951), pp. 2 1 - 5 1 . . "The Psychophysics of Sensory Function," American Scientist, 48 (1960), 226—253. Repr. in Sensory Communication, ed. W. A. Rosenblith (New York: M.I.T. Press and John Wiley and Sons, 1960), pp. 1 - 3 3 . . "The Quantification of Sensation," Daedalus, 88 (1959), 606-621.

560

BIBLIOGRAPHY

. "Ratio Scales, Partition Scales, and Confusion Scales." In Psychological Scaling: Theory and Applications, ed. Harold Gulliksen and Samuel Messick (New York: John Wiley and Sons, i960), pp. 49-66. . "The Relation of Pitch to Intensity," Journal of the Acoustical Society of America, 6 (1935), 150-154. . " A Scale for the Measurement of a Psychological Magnitude: Loudness," Psychological Review, 43 (1936), 405-416. . "Sensation and Psychological Measurement." In Edwin G. Boring, Foundations of Psychology, ed. H. S. Langfeld and H. P. Weld (New York: John Wiley and Sons, 1948), chap. 2. . "The Surprising Simplicity of Sensory Metrics," American Psychologist, 1 7 (1962), 29-39. . 'Tactile Vibration: Dynamics of Sensory Intensity," Journal of Experimental Psychology, 57 (1959), 210-218. . ' T o Honor Fechner and Repeal His Law," Science, 133 (1961), 8086. . 'Toward a Resolution of the Fechner-Thurstone Legacy," Psychometrika, 26 (1961), 35-48. Stevens, S. S., A. S. Carton, and G. M. Shickman. " A Scale of Apparent Intensity of Electric Shock," Journal of Experimental Psychology, 56 (1958), 32fr-334. Stevens, S. S., and Hallowell Davis. Hearing: Its Psychology and Physiology. New York: John Wiley and Sons, 1938. . "Psychophysiological Acoustics: Pitch and Loudness," Journal of the Acoustical Society of America, 8 (1936), 1 - 1 3 . Stevens, S. S., and E. H. Galanter. "Ratio Scales and Category Scales for a Dozen Perceptual Continua," Journal of Experimental Psychology, 54 (1957)/ 377-4«Stevens, S. S., and M. Guirao. "Loudness, Reciprocality, and Partition Scales," Journal of the Acoustical Society of America, 24 (1962), 14661471. . "Subjective Scaling of Length and Area and the Matching of Length to Loudness and Brightness," Journal of Experimental Psychology, 66 (1963), 177-186. Stevens, S. S., and R. S. Harper. "A Psychological Scale of Weight and a Formula for Its Derivation," American Journal of Psychology, 61 (1948), 343-351Stevens, S. S., and Judith Harris. "The Scaling of Subjective Roughness and Smoothness," Journal of Experimental Psychology, 64 (1962), 489-494. Stevens, S. S., R. C. Jones, and M. H. Lurie. "Three Mechanisms of Hearing by Electrical Stimulations," Journal of the Acoustical Society of America, 1 2 (1940), 281-290. Stevens, S. S., and J. D. Mack. "Scales of Apparent Force," Journal of Experimental Psychology, 58 (1959), 405-413. Stevens, S. S., C. T. Morgan, and J. Volkmann. "Theory of Neural Quantum

BIBLIOGRAPHY

561

in the Discrimination of Loudness and Pitch," American Journal of Psychology, 54 (1941)/ 315-355Stevens, S. S., E. B. Newman, and J. Volkmann. "On the Method of Bisection and Its Relation to a Loudness Scale," American Journal of Psychology, 49 (1937)/ 134-137Stevens, S. S., and E. C. Poulton. "The Estimation of Loudness by Unpracticed Observers," Journal of Experimental Psychology, 51 (1956), 71-78. Stevens, S. S., and G. S. Reynolds. "The Binaural Summation of Loudness," Journal of the Acoustical Society of America, 32 (i960), 1337-1344. Stevens, S. S., M. S. Rogers, and R. J. Hernstein. "The Apparent Reduction of Loudness: A Repeat Experiment," Journal of the Acoustical Society of America, 27 (1955), 326-328. Stevens, S. S., and J. C. Stevens. "Warmth and Cold: Dynamics of Sensory Intensity," Journal of Experimental Psychology, 60 (i960), 183-192. Stevens, S. S., J. C. Stevens, and J. D. Mack. "Growth of Sensation on Seven Continua as Measured by Force of Handgrip," Journal of Experimental Psychology, 59 (i960), 60-67. Stevens, S. S., and G. Stone. "Finger Span: Ratio Scale, Category Scale, and JND Scale," Journal of Experimental Psychology, 57 (1959), 91-95. Stevens, S. S., and J. Volkmann. "The Quantum of Sensory Discrimination," Science, 92 (1940), 583-585. . "The Relation of Pitch to Frequency: A Revised Scale," American Journal of Psychology, 53 (1940), 329-353. Stevens, S. S., J. Volkmann, and E. B. Newman. "A Scale for the Measurement of the Psychological Magnitude Pitch," Psychological Review, 43 (1936), 405-416. Suppes, Patrick. "Measurement, Empirical Meaningfulness, and ThreeValued Logic." In Measurement: Definitions and Theories, ed. C. West Churchman and Philburn Ratoosh (New York: John Wiley and Sons, 1959). "A Set of Independent Axioms for Extensive Quantities," Portugalia Mathematica, 10, fasc. 4 (1951), 163-172. Suppes, Patrick, and Joseph L. Zinnes. "Basic Measurement Theory." In Handbook of Mathematical Psychology, ed. R. D. Luce, R. R. Bush, and E. H. Galanter (New York: John Wiley and Sons, 1963), I, 1-76. Swartz, P. "A New Method for Scaling Pain," Journal of Experimental Psychology, 45 (1953), 288-293. Swartz, R. J., ed. Perceiving, Sensing, and Knowing. New York: Doubleday, 1965. Swets, J. A. "Detection Theory and Psychophysics: A Review," Psychometrika, 26 (1961), 49-64. . "Is There a Sensory Threshold?" Science, 134 (1961), 168-177. Szasz, Thomas S. Pain and Pleasure. New York: Basic Books, 1957. Taylor, D. M. "The Location of Pain," Philosophical Quarterly, 15 (1965), 53-62.

562

BIBLIOGRAPHY

Thomson, G. H. " A Comparison of Psychophysical Methods," British Journal of Psychology, 5 (1912), 203-241. Thurstone, L. L. The Measurement of Values. Chicago: University of Chicago Press, 1959. . Multiple Factor Analysis. Chicago: University of Chicago Press, 1947-

. "Psychophysical Methods." In Methods of Psychology, ed. T. G. Andrews (New York: John Wiley and Sons, 1948). Titchener, E. B. Experimental Psychology. Vol. II. Pt. 1 : Students' Manual. Pt. 2: Instructors Manual. New York: Macmillan, 1905. Torgenson, W. S. "Distance and Ratios in Psychophysical Scaling," Acta Psychologica, 19 (1961), 201-205. . "Multidimensional Scaling: I. Theory and Method," Psychometrika, 17 (1952)/ 401-419. Theory and Methods of Scaling. New York: John Wiley and Sons, 1958. Treisman, Michael. "Laws of Sensory Magnitude," Nature, 198 (1963), 914915. "The Nature of the Psychophysical Law and Its Significance for the Scaling of Sensory Magnitudes," Acta Psychologica, 19 (1961), 213-214. . "Neural Effects and the Psychophysical Law," Nature, zoo (1963), 1241-1242. . "Noise and Weber's Law: The Discrimination of Brightness and Other Dimensions," Psychological Review, 71 (1964), 314-330. . "Psychological Explanation: The Private Data Hypothesis," British Journal of Philosophy and Science, 13 (1962), 130-144. . "Sensory Scaling and the Psychophysical Law," Quarterly Journal of Experimental Psychology, 16 (1964), 1 1 - 2 1 . . "What Do Sensory Scales Measure?" Quarterly Journal of Experimental Psychology, 16 (1964), 387-391. Vernon, M. D. The Psychology of Perception. Baltimore: Penguin Books, 1962. Vesey, G. A. "The Location of Bodily Sensations," Mind, 70 (1961), 25-35. Von Fieandt, Kai. 'Toward a Unitary Theory of Perception," Psychological Review, 65 (1958), 315-320. Von Helmholtz, H. Physiological Optics. Ed. J. C. P. Southall. New York: Dover Publications, 1962. Warren, R. M. "A Basis for Judgements of Sensory Intensity," American Journal of Psychology, 71 (1958), 675-687. Warren, R. M., E. A. Sersen, and E. B. Pores. "A Basis for Loudness Judgements," American Journal of Psychology, 71 (1958), 700-709. Warren, R. M., and R. P. Warren. "A Critique of Stevens' 'New Psychophysics/ " Perceptual and Motor Skills, 16 (1963), 797-810. . "Effect of the Relative Volume of Standard and Comparison Object

BIBLIOGRAPHY

563

on Half-Heaviness Judgements," American Journal of Psychology, 69 (1956), 640-643. Weber, Ernst Heinrich. De Pulsa, Resorptione, Auditu & Tactu: Annotationes Anatomicae et Physiologicae. 1834. Wilks, S. S. "Some Aspects of Quantification in Science." In Quantification, ed. Harry Woolf (New York: Bobbs-Merrill, 1961), pp. 5-12. Wittgenstein, Ludwig. The Blue and the Brown Books. Oxford: Basil Blackwell, 1958. Wood, A. B. A Textbook of Sound. London: G. Bell and Sons, 1957. First published, 1930. Woodger, J. H. The Technique of Theory Construction. International Encyclopedia of Unified Science. Vol. II, no. 5. Chicago: University of Chicago Press, 1939. Woodworth, Robert S. Experimental Psychology. New York: Henry Holt, 1938. Woodworth, Robert S., and Harold Schlosberg. Experimental Psychology. New York: Holt, Rinehart and Winston, 1954. Woolf, Harry, ed. Quantification. New York: Bobbs-Merrill, 1961. Wyburn, G. M., R. M. Pickford, and R. J. Hirst. Human Senses and Perception. Toronto: University of Toronto Press, 1964. Young, F. A. "Boring's Interpretation of Emmert's Law," American Journal of Psychology, 63 (1950), 277-280. . "Concerning Emmert's Law," American Journal of Psychology, 64 (1951), 124-128. . "The Projection of After-images and Emmert's Law," Journal of General Psychology, 39, 2d half (1948), 161-166. Zajac, J. L. "Spatial Localization of After-images," American Journal of Psychology, 73 (i960), 505-522. Zinnes, Joseph L. "Scaling," Annual Review of Psychology, 20 (1969), 447478.

Index (In name entries no distinction has been made between text and notes.) Abnormal perception, 32 Absolute: threshold, 1 1 - 1 2 , 250-251, 332, 338; scale, 64, 253-254, 265, 271; measurement, 75, 87; zero, 115-122, 165, 225-227, 248, 266-267 Accuracy: of perception, 1 1 ; in numerical laws, 269; of observer estimates, 378, 390, 410-414; circularity in tests of, 379; of sensation reports, 382 Acidity: and sourness, 60, 87; measurement of, 87; tests for, 460 Addition: of weights, 7; of tones, 7,214, 257,520; method of, 70,139,181, 2 1 1 ; physical operation of, 100-101, 125; arithmetical, 105; vertical and horizontal, 109; of unit objects, 1 1 3 ; operations of, 133, 136, 214; of brightness, 134, 244; of temperature, 135; extraneous factors in, 135; of sweetness, 135,245; and measurement, 137, 197, 297, 520; of spaces, 139; objectivity of, 139; of sensations, 163, 281, 296, 329, 446, 554; and interval equality, 257; of intervals, 273; in ratio scales, 377; of loudness, 517. See also Axioms of addition Additivity axiom: and measurement, 1 1 1 ; statement of, 1 1 1 , 210, 212; and loudness, 236; for sensations, 327; and jnd measurement, 329, 361 «-determinations: and ^determinations, 142 ff.; objectivity of, 143; and estimates, 145; observer agreement in, 147; sensitivity of, 1 5 1 ; and operations, 155 Afterimages: measurement of, 18, 56, 279, 376; perception-dependence of, 19; perception of, 21, 24, 29; privacy of, 25,144; misperception of, 29; location of, 38; and brain processes, 61; as sensations, 374

Aftersensations: visual, 38; location of, 38; auditory, 38, 53; as examples of sensations, 469. See also Afterimages Alignment, 140, 445 Alphabet used in scaling, 1 0 0 , 1 1 2 , 1 6 4 Amplification, 12, 72, 80 ff., 449 Amplitude: of vibrations, 77-78; particle, 79-81 Anesthesia and perception, 28 Apparent magnitudes, 370, 409, 458, 484 ff. See also Psychological magnitudes Appearance: as test for equality, 143 ff.; reports of, 143-147; and reality, 440, 442. See also a-determinations Arbitrary zero. See Conventional zero Arithmetical: equality, 105; addition, 105; laws, 1 1 3 ; operations, 1 1 3 ; statements, 124-127, 188, 238; concepts, 155 Armstrong, D. M., 28 Associativity axiom, 102,108-110 Asymmetrical distinction, 445 ff. Asymmetry axiom, 102-103 Auditory perception: psychophysical laws of, 8, 278, 283, 475-477» 523< 528; sensatlonist theory of, 47-54; object and cause of, 5 1 ; depth in, 53 Auditory sensations : and sounds, 7 , 1 8 59, 90, 278; in different perceivers, 26; and perception, 52, 523; aftersensations as, 53; and sound waves, 90; psychophysical laws of, 280; description of, 469 Austin, J. L., 504 Average error, method of, 340, 535 Awareness: of pain, 21; unconscious, 23 Axioms of addition, 108-123, 239, 274 Axioms of measurement: list of, 102103; contingency of, 105-108, 274; particular and abstract, 107,183; and

566 experimental laws, 125; in loudness measurement, 239 Balances: in measurement, 79,108; and gravity, 134; in tests for equality, 140-141, 178; certainty of, 150, 353; sensitivity of, 1 5 1 , 358; types of, 153, 460; conditions affecting, 193; threshold for, 352 Balancing mechanism, neurological, 353 Basilar membrane, 90,94-9;, 479 b-determinations. See a-determinations Beats, 33, 449 Beaufort scale, 199 Behavioral dimensions, 465 Behaviorism: in modern psychophysics, 365 ff., 536, 545-546; laws of, 396; exclusion of sensations in, 536; psychological dimensions defined in terms of, 542 Bel, 240 Beranek, L., 74, 76, 78 Bergmann, G., 64, 6 5 , 1 0 2 , 1 0 5 , 1 1 1 , 1 3 5 , 136,166,329,384,423,443 Berkeley, G., 20, 47,147, 436, 437, 460 Bias in perceptual estimates, 388 Bodily sensations, 38 Body. See Mind and body Boring, E. G., 83, 296, 384, 391,493, 502, 536/ 544 Brain processes: and pains, 61, 91; and afterimages, 61, 91; and sensations, 88, 477; logic of, 89; and sounds, 94, 482; and psychological dimensions, 480 Braithwaite, R. B., 132 Brentano, F., 294 Brightness: measurability of, 7, 173, 244, 520; and sensations, 57-58; and intensity, 87; addition of, 134, 244; psychophysical laws of, 517 Bril, 539 Buoyancy and negative weight, 120 Bweight: and weight, 120; zero for, 122 Calculation and measurement, 1 1 2 114 Calibration: of instruments, 73, 81; of intensity meters, 80; of frequency meters, 80; of thermometers, 179 Campbell, N. R., 63, 64, 70, 71, 99,101, 102,104,105,116,118,124,125,137, 1 3 8 , 1 6 1 , 1 9 9 , 388, 432, 443

INDEX

Candlelight, 285 Cattell, J. M., 289, 336, 543, 544, 545 Centigrade scale: use of, 64, 1 6 ; , 226; zero in, 116, 122, 165; and ordinal scale, 166; numeral assignment for, 168, 179; as ratio scale, 1 7 1 ; transformation of, 255; and dependent measurement, 272 Centimeter scale, 168 Chain balance, 9 3 , 1 5 3 , 1 9 5 , 206 Charles's law, 255, 265-269 Chron, 539 Circularity: in measurement, 66, 69; in Stevens' definition of measurement, 173 ff., 182; in using thresholds as units, 345; in accuracy tests, 379 Classical definition of measurement, 1 0 0 , 1 0 7 - 1 0 8 , 1 2 3 , 1 3 4 , 1 5 7 , 273, 433434/ 520 Cochlear potential, 400, 401 Cohen, M. R., 64, 102 Color: physical correlates of, 87; measurability of, 99; quantitative comparisons for, 431; subjectivity of, 441 Colorimeter, 449 Combination operation, 100-101, 123, 138 Commutativity: definition of, 102-103; axiom of, 108-110; of horizontal addition, 1 1 0 Comparative judgment, law of, 332 Comparative measurement: and absolute measurement, 75-76; of intensity, 80; of frequency, 80; of acidity, 87 Comparison: method of, 70; concept of, 196; and addition, 197; of jnd's, 293; stimulus, 334 f¥.; sensations, 341; in length measurement, 376 Compounds: tones as, 243; sensations as, 243, 296, 328 Confirmation of observations, 33-34, 450-451, 494 Congruency, 1 4 1 Conservation of energy, law of, 169 Constants in psychophysical laws, 286288, 293 Constant stimuli, method of, 340,535 Constructs. See Theoretical constructs Continuous function: sensations as, 347 ff. Contours: equal-frequency, 83; equalloudness, 85

INDEX

Conventional zero: and absolute zero, 116; definition of, 121; in sensation measurement, 301 ff. See also Zero Conventions of measurement, 67, 161 Corbin, H. H., 368, 370 Correct perception, 32, 307, 472, 527. See also Errors in perception Correlation: of loudness and intensity, 60; of physical stimuli and psychological dimensions, 60; of hue and frequency, 60, 87; of sourness and acidity, 60, 87; of pain intensity and tissue destruction, 60, 94; of brightness and intensity, 87; of heat and molecular motion, 87; of sounds and sound waves, 93-94 Correspondence principle, 370-373, 378-382, 385-390, 402-404 Corrigibility: of sound reports, 31-36, 48, 94; of weight reports, 33; of adeterminations, 143; of appearance reports, 144; of determinations of relative loudness, 220 Counting as measurement, 71 Cross-modality comparisons: psychological magnitude in, 404; and Stevens' law, 408; and psychophysical laws, 425-426 Davis, H., 74, 80, 83, 84, 85, 86, 96, 228, 229, 233, 238, 250, 400, 491 Decibel, 240 Degree, 272 Density: measurement of, 62, 68-69; definition of, 67 Dependent measurement: and independent measurement, 63, 65, 79; definition of, 65, 68, 200, 270; inconclusiveness of, 68; of weight, 73; of loudness, 73; of pitch, 73, 246; of frequency, 76, 81; of particle amplitude, 78; extraneous factors in, 136; Stevens' definition of measurement as, 176; of temperature, 176, 272; Kelvin scale as, 272; of sensations, 311-312. See also Indirect measurement Depth: perception of, 53 Derived magnitudes, 159, 432. See also Fundamental magnitudes Derived measurement: and fundamental measurement, 63, 443, 504. See also Indirect measurement

567 Differential limen. See Relative threshold Differential sensitivity: jnd as index of, 355; and sensations, 355, 413 Dimensions: of perception, 1 ; of sensations, 2-3, 6, 235, 244; concomitance of, 65; extensive and intensive, 102, 37°- 375; intervallic, 275, 434; fundamental, 277; empirically real, 378, 383, 416; psychological and physical, 428 ff.; sensory, 445; perceived, 445, 461; theoretical, 462; macroscopic and microscopic, 462; molecular and molar, 462, 466, 468. See also Subjective magnitudes Dingman, H. F., 379 Direct measurement: and indirect measurement, 60,62-74, 519; of intensity, 61; of frequency, 61; of pain, 61; jnd method as, 309; ratio scaling as, 364; of sensations, 474; Stevens' method as, 540; Fechner's method as, 540. See also Independent measurement Direct perception: of physical objects, 62; as test for equality, 143 ff.; of equality of intervals, 175; of stimuli, 319; ideas as objects of, 437 Discontinuous function, 349 Discriminal dispersions, 552 Distance: measurement of, 243 Donders, F. C., 502 Dualism, 90, 91, 365, 373, 392, 480, 503, 505, 531 Dynamometer, 405 Ebbinghaus, H., 296, 502 Ekman, G., 365, 397, 402, 403, 404 Electrical currents: intensity of, 80; estimation of, 425 Electrons, 486 Empirical law: Fechner's law as, 305; of perception, 474; possibility of, 519 Epiphenomena: sensations as, 319, 545; sounds as, 477 Epistemology: and psychophysics, 1617; and ratio scaling, 383 Equality: of jnd's, 4, 166, 287, 294296, 316-329, 364; empirical tests for, 106,140; direct perception as test for, 143 ff.; observer agreement in, 148150; certainty in tests for, 150 ff.; sensitivity of tests for, 151; numeral assignment for, 167, 175; estimation of, 175; of units, 211-213, 329»' test

568 for loudness, 219; physiological conditions o f , 228; of pitch intervals, 262-263; senses of, 323-327; psychological, 324; of sensation intervals, 325; of vegs, 326; transitivity o f , 350 Equality of intervals: in interval scale, 1 6 5 ; numeral assignment for, 167, 1 7 5 ; direct perception o f , 1 7 5 ; tests for, 176, 177, 1 7 8 ; and addition, 257; and subjectivity, 258; ability to distinguish, 260 Equality of ratios: in ratio scale, 1 6 5 ; numeral assignment for, 167, 1 7 5 , 1 7 6 ; tests for, 176, 1 7 7 , 178 Errors in perception: of sensations, 2830, 325; of sounds, 31-32, 34; correction o f , 32,144. See also Perceptual error Errors of use, 2 9 , 1 4 4 Estimates: verification o f , 1 1 5 ; and appearance reports, 144; definition o f , 1 4 5 ; accuracy o f , 232,410-414; of relative loudness, 234; of pitch ratios, 256; objective and subjective, 357; psychological curves for, 369; of psychological ratios, 378; nerveimpulse, 380; external and internal, 380, 386; of psychological magnitudes, 381; explanation o f , 382 ff.; inconsistency between, 387-388; of stimuli, 392; of value, 393-394; and psychological magnitudes, 399; of apparent magnitudes, 409; of sensations, 409, 414, 543; theoretical constructs defined b y , 413 Estimation: and measurement, 5, 1 1 4 1 1 5 , 1 9 6 , 235, 377, 380, 423, 540; and verification, 6 6 , 1 7 5 Evans, R. M., 450 Evidence, prima facie, 27-28 Excitation: measurement o f , 535. See also Physiological dimensions Experience: privacy o f , 413 Experimental laws, 125 Experimenter and observer, 15, 494-497 Fahrenheit scale, 1 1 6 , 1 2 1 , 1 6 5 , 226, 538 Fatigue, 463 Fechner, G . T., viii, 1 7 , 137, 161, 243, 372, 502, 510, 531-537. 545; psychophysics o f , 1 , 10, 278-279, 366, 419. See also Traditional psychophysics Fechner's l a w : derivation of, 286 ff., 302-304, 532; as rule of numeral assignment, 290 ff.; as hypothesis, 292,

INDEX 296; and Stevens' law, 295-296, 424; meaningfulness o f , 297-298; verification o f , 310-313; and W e b e r ' s law, 5 1 2 - 3 1 4 ; empirical content o f , 313; as law of sensations, 5 1 5 , 362; and equality of units, 329-330; and axiom of additivity, 330; alternatives to, 361-363; and ratio scaling, 403; assumptions underlying, 404; and cross-modality comparisons, 407; " r e p e a l " o f , 525 Fechner's method of measurement: as direct or indirect, 298, 403, 534, 540; stimulus error in, 534, 543; units in, 540 Fechner's paradox, 350 Ferguson, A . , 63, 64, 1 1 2 , 163, 257, 388 Fictions: psychological entities as, 594, 509; grammatical, 485-487. See also Theoretical constructs Finger span, 390 Fletcher, H., 85 Flicker photometer, 449 Focal length, 79 Fractionation method, 5, 367 ff., 404 Frequency: of light waves, 60-61, 8 7 88; of sound waves, 74 ff., 267-268; meters, 80, 9 1 , 9 3 , 2 6 3 Fullerton, G . S., 289, 336, 543, 544, 545 Fundamental magnitudes, 7 1 , 277. See also Derived magnitudes Fundamental measurement: and derived measurement, 63, 443, 504; Hempel's definition o f , 195; of space and time, 53 Galanter, E. H., 200, 235, 265, 370, 382, 410, 458 Galton, F., 502 Garner, W . R., 380, 393 G a s laws, 267, 289, 513 Genuine location, 37-39 Goude, G., 386 Gravitation: and balances, 1 3 4 ; and psychophysical laws, 553 Guild, G., 1 0 2 , 1 1 2 Guilford, J. P., 103, 166, 336, 340, 370, 379/ 502 Gust, 539 Halftones: as pitch units, 247, 257; in measurement of pitch, 247 ff.; equality of, 262; and octaves, 268 Hallucinations: auditory, 1 2 , 94; and

569

INDEX

perceptual abilities, 546; psychophysics of, 546-547 Handgrip, 425 Hardness: measurement of, 100, 175177, 185; tests for, 140 ff., 175, 175, 177,189-190. See also Mohs's scale of hardness Harper, R. S., 371 Hearing: abnormal, 261; physiological processes in, 90, 94-95, 401, 479-480 Heat: physical correlates of, 87 Hempel, C. G., 135,137, 141, 157, 188, 189-196 Henning's odor prism, 431 Hirsh, I. J., 364, 365, 366, 415 Horizontal addition, 1 0 8 - 1 1 3 , 1 2 6 Hue: as sensation dimension, 57-58; and frequency of light waves, 60, 87; as intervallic, 275; psychophysical laws of, 517 Ideas: perception-dependence of, 20; and qualities, 435-437. See also Locke, J. Identity: of loudness and intensity, 4748, 61, 82, 232, 276; of pitch and frequency, 47-48, 276; of hue and frequency, 61; of sourness and acidity, 61; of pain intensity and tissue destruction, 61; of loudness and frequency, 87; of sounds and sound waves, 88; of sensations and sounds, 90-94; of brain processes and sensations, 91 Illusory perception, 28, 546-547. See also Errors in perception Immediate perception. See Direct perception Impression reports, 454 ff. Incorrigibility: of sensation reports, 2731; Armstrong on, 28; concept of, 2831; Malcolm on, 29; of pain reports, 94, 144; of a-determi nations, 143; of afterimage reports, 144 Independent measurement: and dependent measurement, 63, 65, 79; defined, 69, 270; rareness of, 73; of loudness, 82, 270, 277; of pitch, 82, 277; and absolute zero, 117, 122; of psychological dimensions, 277; of sensations, 311. See also Direct measurement Indirect measurement: of psychological dimensions, 60; and direct measure-

ment, 60, 62-74, 519; of density, 62; Fechner's method as, 298, 403, 534; objections to, 299; of weight sensations, 302-304; of length, 306; fractionation methods as, 404; of sensations, 473. See also Dependent measurement Inferred length, 484 Infinite regress. See Regress Instruments: for sounds, 13, Bo; for sound waves, 76, 91; specification of, 78; in tests for equality, 141; certainty of, 150; advantages of, 151; use of, 155; piano, 247; psychophysical laws of, 423; observation without, 492 Intelligence: measurement of, 502; as an ability, 507 Intensity: of sound waves, 47-48, 60-61, 74 ff., 232 ff., 276, 288, 481, 490; of light waves, 87-88; meter, 80, 91, 93 Intensive dimensions, 102, 370,385 Interpretation: of axioms of measurement, 107,183; of arithmetical statements, 124, 126, 127; perception as, 478; of psychological dimensions, 442 ff., 523-525 Intervallic dimensions, 274-275, 281,

434

Interval of uncertainty, 338 Intervals: of pitch, 247, 257-260, 274; as units, 257-260, 273; jnd's as, 320; between stimuli and sensations, 325; size and number of, 399. See also Equality of intervals Interval scale: as scale of equal intervals, 165-166; assignment of numerals in, 167; invariance transformation for, 169; creation of, 174; and ratio scale, 388; Fahrenheit scale as, 538; for psychological dimensions,

53»

Intervening variables in psychophysical laws, 403 Introspection: of sensations, 24, 525, 472, 542; of intermediate sensations, 507; of compound sensations, 528; in verification of sensation statements, 467. See also Perception, of sensations Introspectlonism in psychophysics, 9, 371, 383, 417, 542, 546 Invariance of scales, 169-172 Isomorphism between mathematical and physical relations, 104

57° James, W., 243, 244, 296, 299, 300, 304, 328, 535 JND. See Just noticeable differences Johnson, H. M., 537 Just noticeability, 333 ff. Just noticeable differences: equality of, 166, 287, 294 ff., 316 ff., 533; as sensation units, 282, 291, 299, 315, 359, 533' 537/ between sensations and stimuli, 294, 308, 360; as units of sensitivity, 300, 355-356, 361; in sensation measurement, 300 ff., 398; between sensations, 306; as perceptual intervals, 320, 324; vagueness in, 322; methods for determining, 334 ff.; numerical expression of, 335; variability in, 336; unclarity in concept of, 359; in scales, 398 Just noticeable stimulus, 333-334 Juxtaposition, 101, 106, 1 1 5 , 134, 158, 180, 230 Kant, I., 17 Kelvin scale, 64,117, 254-255, 265, 271272 Kries, J. von, 296 Kulpe, O., 296 Laws: true by definition, 299; descriptive versus explanatory, 395; of stimulus and response, 536; of physiological dimensions, 541. See also Numerical laws; Psychophysical laws Leibniz, G. W., 441 Length: perception of, 14; zero for, 120, 305; tests for equality of, 140 ff.; ratio scale for, 179; and pitch discrimination, 261; estimation curves for, 369; estimated and actual, 394; visual and tactual, 460; perceptual and physical, 483-484; as measured dimension, 484, 490 Length measurement: independent, 72; addition in, 1 0 1 , 138, 180-187, 198; use of zero in, 179; and Stevens' definition, 179-184; methods of, 197198; unit intervals in, 259-260; and sensation measurement, 305; indirect, 306; comparison in, 376 Levers, theory of, 109 Lifting test, 150-151 Lightning: as electrical discharge, 462; as psychological phenomenon, 464; measurement of, 507

INDEX

Limen. See Absolute threshold; Relative threshold Limits: of perception, 1 1 , 1 5 - 1 6 , 3 3 1 ff.; method of, 336 ff., 535 Location: of sensations, 7, 36-39; of pains, 36-38; of afterimages, 38-39; of sounds, 39-46 Locke, J., 10, 435-439, 442, 462 Logarithmic: interval scale, 167; law in psychophysics, 532, 537. See also Fechner's law Loudness : and sensation dimensions, 67, 235, 244; corrigibility of statements about, 31-36, 55; meaning of, 47-48, 61, 92; measurement of, 54-57,73,82, 159/173-174/ 214-245/ 277/ 520; and intensity, 60-61, 82 ff., 232 ff., 276, 481; and frequency, 85, 87; zero for, 123, 225, 227; addition of, 215, 517; memory of, 216-219; t e s t s f ° r relative, 219-220; threshold for, 227, 240; sensation of, 229-231, 236, 238; estimates of, 232, 234; fractionations, 233; objective and subjective, 235236; as heard dimension, 276; physical laws of, 279; and cochlear potential, 400; as irreducible dimension, 477; psychophysical laws of, 517; as physical dimension, 522 Mack, J. D., 405, 406 Magnification, 12, 72 Magnitudes: estimation of, 367. See also Apparent magnitudes; Dimensions; Intensive dimensions; Physiological dimensions; Psychological magnitudes; Subjective magnitudes Mak, 200, 326, 368, 539; experiment, 367 ff. Malcolm, N., 29-30 Materialism, 61, 90-91. See also Dualism Mathematical: relations, 104; addition, 105; equality, 105 Measurability: of brightness, 7, 1 7 3 174, 244; of loudness, 54-57/ 244; determination of, 107-108, 160, 172, 181 ; of sensations, 162-163, 299, 376, 473; of psychological dimensions, 244, 276, 518; of pitch, 246; of quantitative dimensions, 432 Measurement: of sensations, 2-3, 18, 54-57/162-163, 280, 283 ff., 291, 293, 296, 298, 301 ff., 305, 376, 530, 533, 544; of pains, 2-3, 55-56, 279, 521;

INDEX

and estimation, 5, 114-115,196, 235, 377/ 380« 4^3/ 54»; nature of, 7, 99101; Stevens' definition of, 7, 160, 164, 168, 172-173, 175-176, 178-189, 377/ 520, 537, 540; of perceptual abilities, 10-11, 421, 529, 542; of afterimages, 18, 56-57, 279, 376; of psychological dimensions, 54-60, 73,159, 516-517; duplicability of, 55; direct and indirect, 60, 62-74; of temperature, 64, 66, 79, 175-176, 178-179; dependent and independent, 65, 79; conventions of, 67, 161; of volume, 68; of density, 68-69; of length, 70, 72, l o i , 177-184, 305; of frequency, 74-83; of intensity, 74-85; of sound waves, 74-88; absolute and comparative, 75; simple and complex, 77, 81; of amplitude, 77-79; definitions of, loo, 158, 161, 196 if., 277, 516, 520; axioms of, 102-105; a s operational, 106, 133-156; purpose of, 112, 123133, 187-188, 210; classical view of, 125, 157, 273; arbitrariness in, 129 ff., 165, 167; and addition, 131, 137, 297, 520; of extraneous dimensions, 136; in physics, 137, 503; technical and ordinary uses of, 157-138, 158159, 162, 187-188, 191, 503-505; ordinary meaning of, 158, 198; of time, 159; of derivative dimensions, 159; of loudness, 159, 173-174, 239, 520; of pitch, 159, 246-282; as philosophical question, 161, 172-173; of brightness, 173-174,520; of hardness, 175/ 177; as test, 178, 207; method versus result of, 184; scaling as, 189, 311; as univocal assignment of numerals, 190; Hempel's definition of, 190, 225; comparison in, 196; duplication of, 203, 207, 209, 212; precision in, 205, 222, 225, 265; logical and empirical requirements of, 210; of weight, 222; of distances, 245, 244; immediate perception in, 259; training for, 261; of jnd's, 300; of thresholds, 331 ff.; of sensitivity, 357, 502, 521, 536; of physiological processes, 576,481; of shadows, 377; in accuracy tests, 379; of perceptual magnitudes, 489; of responses, 499, 544; of mind, 501 ff.; history of, 502; of mental phenomena, 502; of reaction time, 502; and dualism, 505; of space, 503; of time, 503; of lightning, 508; of

571 perceived object, 508; in traditional psychophysics, 509; of lifting ability, 530; of perception, 530; of volitions, 530; of excitation, 535; of observation errors, 543; mental, 544. See also Psychological measurement; Scales; Units Memory: and loudness equality, 216219; and pitch equality, 252; measurement of, 502 Mental: world, 10; phenomena, 286, 502, 507, 542; tests, 502, 544; substance, 505; measurement, 544 Micron, 78, 82, 224 Microphone, 75, 81, 92 Middle C: as zero pitch, 247, 251; scale, 254-255, 266 Mill, J. S., 429 Mind: measurement of, 501 ff. See also Mind and body Mind and body: laws of, 280; relations between, 280, 284, 362-363, 373, 499, 526, 542; psychophysics as science of, 362, 372, 498; philosophical issue of, 365 Minerals: scaling of, 190, 193 ff. See also Hardness Minimal differences, 520 ff. Mohs's scale of hardness, 164,170-172, 187-190, 199, 432-433» 538 Molar: dimensions, 10, 263, 462, 466; descriptions, 439 Molecular: motion, 87-88; dimensions, 468 Moment, 109 Monads, 441 Munson, W. A., 85 Muscular ability, 422 Nagel, E., 64, 102, 103, 134 Naming and measurement, 186 Natural zero, 301 Negative weight, 119-121 Nerve impulses: and sounds, 94; estimation of, 580; psychological dimensions as, 480 New psychophysics, 1, 364-366, 409, 411, 419, 531 Newton, I., 169 Nominal scale: nature of, 164; rules for, 167; invariance transformation for, 169; for football players, 186-187; and measurement, 186-187 Nothing: concept of, 117 Noticeability: of magnitude increase,

572

INDEX

285; of stimulus differences, 3 1 7 318; and sensation differences, 318 Numbers: measurability of, 71-72; as properties of classes, 72; and numerals, 99,104,124; and phenomenal structure, 112 Numeral assignment: and measurement, 99, 126, 132, 160; to football players, 164, 172, 185, 189; arbitrariness of, 165; rules for, 166-182, 291; and scales, 167; empirical test for, 175; to temperature, 178 ff.; manner and result of, 184; to hardness, 185; univocality of, 190-194; to sensations, 291, 303, 314, 361; in thought, 361; to perceptual magnitudes, 488 Numerals: and numbers, 99, 104, 124 Numerical identity of sounds, 35, 45-46 Numerical laws: and measurement, 124-133, 159; as fortuitous result, 1 3 1 ; and Mohs's scale, 188; of loudness, 241; of pitch, 252, 264-268; accuracy in, 269; and nonnumerical laws, 279, 513; in psychophysics, 279, 514; theoretical value of, 514 Numerosity, 71

Odor, 431 Old psychophysics, 1,366, 419, 532, 542 Operational: definitions, 141, 154-155; verifiability, 445 ff.; observations, 492-493 Operationism in modern psychophysics, 366 Operations: for loudness, 13; for length, 14; and measurement, 1 3 3 156; of addition, 133, 258; for equality, 141; and perceptual determinations, 142,156; instrumental, 154155/ 445-447; senses of, 154-156; verification by, 155, 447; of thought in sensation measurement, 314-315; of scratching, 447 Ordinal scale: nature of, 164; rules for, 167; invariance transformation for, 169; Mohs's scale as, 538 Ordinary use: of "loudness," 47-48; of "pitch," 47-48; of "brightness," 5758; of "hue," 57-58; of "indirect," 62; of "measurement," 138, 158, 187, 188, 191; vagueness of, 162; Mohs's scale as measurement in, 188 Oscilloscope, i, 80, 380

Oberlin, K. W., 355, 356 Objectivity: meaning of, 139; in a- and ^-determinations, 143; of loudness, 237; of estimates, 357; of confirmation, 451 Observations: optimal point of, 42; types of, 91; continuity between, 92; instrumental, 92; argument on, 258; psychological and physical, 491; noninstrumental, 492; of star transits, 493; of experimenter and observer, 494-497; as defining stimulus and response, 497; measurement of errors in, 543 Observation statements: incorrigibility of, 27, 94; of pain, 94; of pitch, 258; verification of, 44;, 450, 456; about psychological dimensions, 452, 454; language difficulties in, 453; as impression reports, 454; agreement in, 455-457 Observer: and experimenter, 15, 496; agreement, 147-15°/ 381, 455~457; competency of, 263; normal, 467. See also Estimates; Observation statements Occam's maxim, 418 Octaves, 266-269, 399

Pains: measurement of, 2-3, 18, 55-56, 61, 279, 521; zero for, 18; failure to notice, 21-22; perceivability of, 2425; privacy of, 25, 55-56; mistakes about, 28; location of, 36-38, 94; and tissue destruction, 60, 94; and brain processes, 61, 91; incorrigibility of reports of, 94, 144; psychophysical laws of, 547; and perceptual abilities, 547 Pap, A., 67, 68,102,106,111, 432 Particle amplitude, 79-81 Partition scales, 389, 398 Perceived dimensions. See Perceptual dimensions Perceived length, 484 Perception: causes of, 20, 51-52; of sensations, 24-25, 27, 31, 314-319, 325, 347-353, 545; normal and abnormal, 32; veridical, 32, 527; object of, 51-52, 437-439, 508; sensations in, 52, 468-478, 486, 523, 526-529, 549; and operations, 142; of stimuli, 319, 351, 471; limitations of, 331332; of psychological entities, 372; inner, 412; causal theory of, 439; verification of, 447-448; aids to, 450;

INDEX

psychophysical laws of, 474-479; as interpretation, 478; and perceptual dimensions, 483 ; measurement of, 507 fif., 530; psychophysics of, 546547. See also Auditory perception; Direct perception; Sensationist theory of perception Percepts, 486-487 Perceptual: reports, 143-147; determinations, 156; error, 307, 472; intervals, 320, 324; mechanisms, 378, 383, 386, 545; dimensions, 482-491; observations, 492-494; phenomena, 509 Perceptual abilities: perception as, 10; measurement of, 10-11, 421, 499, 529, 542; jnd as unit of, 355; psychophysics as science of, 419, 542; physiological mechanisms underlying, 420; comparison of, 420; defined by perceptual responses, 499, 545; and motor abilities, 530; and sensations, 546-547 Phantom limb, 44 Phonautograph, 76 Physical : phenomena, 10, 441, 491, 502, 505 £f.; objects, 19-21, 36, 41-42, 5152, 62, 281; correlates, 87-88, 392; addition, 100, 123-132, 134-135, 139, 275; relations, 104; laws, 169, 279, 512; ratios, 378,381; dimensions, 423, 458, 522; science, 440, 511 fif.; measurements, 443; observations, 492493 Physiological: theory of lifting, 149; mechanisms of perception, 352, 400, 420; processes, 375-37*/ 379» 479/ 481; dimensions, 400, 465, 479-482, 541 ; laws, 481 Piano: in pitch measurement, 247; tuners, 252, 449 Pitch: as physical dimension, 6, 522; corrigibility of statements about, 3136, 55; and sound-wave dimensions, 33/ 83, 87-88, 276; meaning of, 4748, 92; zero for, 123, 247-249, 273; perfect, 247, 252; lowest audible, 248251; laws of, 252, 255, 264, 267-268, 279, 517; equality of intervals of, 252, 258, 260, 262-263; ratio estimates of, 256; addition of, 257-258; observations of, 258-259, 261; as intervallic, 274, 434; as psychological dimension, 276, 445; as quantitative, 434; subjectivity of, 441 Pitch measurement: dependent methods

573 of, 73, 246; independent methods of, 82, 270, 277; possibility of, 100,159, 246; zero in, 123, 247-249, 273; halftone system of, 246 ff.; scales of, 253255, 271; units of, 257, 262, 266, 277; and numerical laws, 264 ff. Place, U. T., 88, 92 Pores, E. B., 391 Power law, 5, 326, 532, 537. See also Stevens' law Precision in measurement, 202-203, 222-225, 265 Pressure: as empirically real magnitude, 383; gas laws of, 513-514 Primary qualities, 435-438 Principle of correspondence. See Correspondence principle Privacy: of pains, 3, 25; of sensations, 7/ 25, 315, 342, 366, 415, 453, 546; contingent versus necessary, 26; concept of, 26 ff.; defined by incorrigibility, 31; of psychological entities, 371, 375; of physiological processes, 375/ 379; psychological magnitudes, 381, 416; of immediate experience, 415; of mental world, 505 Private dimensions, 55 Projection of sensations, 42-44 Proportionality, 66 Prothetic continua, 319, 364 Psychological: measurement, 138, 388, 403, 443, 510; equality, 319, 324; similarity, 320; scales, 364 ff., 402; entities, 371-376/ 380, 390-394, 486; ratios, 378, 381; explanation, 396; laws, 512, 516 Psychological dimensions: measurement of, 1 ff., 6, 57-61, 73,179, 244, 276-277, 516-519; and physical dimensions, 6,10, 58-59, 428, 518, 521 ff.; examples of, 6, 10, 60, 459, 512, 521, 523; dimensions of sounds as, 18-19, 445; physical correlates of, 6o, 88; ratio scales of, 171-172; sameness of, 369; empirical reality of, 383; as quantitative, 429, 434; dependent on organism, 444; as sensation dimensions, 446, 461, 467; statements about, 452-457; as molar dimensions, 462; as behavioral dimensions, 465; as physiological dimensions, 465, 479-482, 521; as brain processes, 480; as perceptual dimensions, 482-491; as fictions, 509; subjectivity of, 512; interpretations of, 523-525; as de-

574 fined by observer responses, 542, 557. See also Physical dimensions Psychological magnitudes: and physiological magnitudes, 368, 374 ff., 400; scaling of, 371; empirical reality of, 372,416; as perceived magnitudes, 374; estimation of, 381, 387; privacy of, 381, 416; definition of, 385-386; as theoretical constructs, 394, 416, 426; arguments for retaining, 397408; and psychophysical laws, 398; in Stevens' law, 399, 404, 422 Psychologist's circle, 384, 498 Psychology: subject matter of, 439441; as quantitative science, 502; of capacities, 544 Psychophysical experiments, 15, 24; standard stimulus in, 334 ff.; introspectionist interpretation of, 371; behaviorist interpretation of, 373; sensations in, 375; instructions in, 375, 382; intervening assumptions in, 404; objective of, 4 1 1 ; of private sensations, 412; types of, 495; requirements of, 496-497 Psychophysical laws: traditional conception of, 1 ff.; nature of, 8,283,286, 314, 363, 510, 512; of auditory perception, 278, 283, 475-477/ 5^5/ 527; possibility of, 279; of sensations, 280, 474; as explanatory, 396; and psychological magnitudes, 398; and physiological mechanisms, 400; intervening variables in, 403; governing instruments, 423; and cross-modality functions, 426; of perception, 477479; as physiological laws, 481; of psychological dimensions, 517; and gravitational laws, 533 ; of pain, 547. See also Fechner's law; Stevens' law; Weber's law Psychophysical measurement: traditional conception of, 1 , 6, 9 - 1 1 , 509; problem of, 1 ff., 501, 516-522, 527; as rescaling of stimuli, 423, 534, 541. See also Measurement; Psychophysical laws Psychophysical methods: for determining thresholds, 533-341 Psychophysics: possibility of, 1 1 ff.; Fechner's conception of, 278-279; as science of mind and body, 362, 372, 498; behaviorism in, 365-366, 546; opera tionism in, 366; of perceptual abilities, 419, 542; of stimulus and

INDEX

response, 499; as science, 510, 555; sensationist conception of, 526; schools of, 531; of sensations, 546547. See also New psychophysics; Old psychophysics; Psychophysical experiments; Psychophysical laws; Psychophysical methods; Radical psychophysics; Traditional psychophysics Qualitative dimensions: and psychological dimensions, 429, 434; and quantitative dimensions, 450 ff.; derived magnitudes as, 432 Qualitative identity of sounds, 34-36, 45-46 Quality: ordinary meaning of, 429-430; and quantity in measurement, 435; and idea, 435, 437; sensible, 438; perceived, 508. See also Secondary qualities Quantitative: relations, 106; concepts, 155; dimensions, 430, 432; comparisons, 431 Quantity: technical use of, 429-430; and quality in measurement, 433 Quantity objection, 296-297, 544-545 Radical psychophysics, 6, 542 ff. Ranking: of shape and color, 99-100; measurement in, 123; and sensitivity, 152 Ratio: comparisons, 100; estimates, 378, 388. See also Psychological ratios Ratio scale: absolute zero in, 1 1 6 , 1 6 5 ; nature of, 165; numeral assignment in, 167; invariance transformation for, 169; and centigrade scale, 1 7 1 ; of psychological dimensions, 1 7 1 - 1 7 2 ; and loudness, 173; creation of, 1 7 3 174; of length, 179; of pitch, 255 ff.; methods of producing, 326, 367; advantages of, 364; as direct measurement, 364; sample of, 367-373; principles for constructing, 371; addition and, 377; purpose of, 382 ff., 395 ff.; epistemological justification of, 383; and interval scale, 388, 398; and jnd scale, 398 Reaction time, 493, 502 Reality: and appearance, 440, 442; monad as basic constituent of, 441 Reese, T. W., 1 4 1 , 368, 370, 371 Regress: in perception of sensations, 318; in sensationist explanation of thresholds, 347, 549

INDEX

Relative limen. See Thresholds Relative magnitude: numerical expression of, 100; methods for determining, 140; memory of, 2 1 7 - 2 1 8 ; sensitivity of determinations of, 222 Relative threshold: definition of, 333, 338; as perceptual difference, 339; experiments to determine, 355 Relativity: of perception, 20; of precision in measurement, 265 Reliability of perception, 1 1 , 1 5 - 1 6 Rescaling of stimulus, 334, 541 Responses: and stimuli, 366, 498-499, 536; scales of observer, 385; psychological length as magnitude of, 385; sensations in terms of, 4 1 ; ; observations defining, 497; measurement of, 499, 544 Retinal image, 38 Roughness: observations of, 92; measurement of, 540 Ruch, T. C., 42 Ryle, G., 24, 506, 507 Scales: Stevens' classification of, 1 1 6 , 164; addition in, 1 3 7 ; types of, 1 6 3 164, 168, 1 8 5 , 1 8 7 - 1 8 9 , 1 9 2 - 1 9 6 , 364; for hardness, 1 6 4 , 1 7 0 - 1 7 2 , 1 9 9 , 432; invariance in, 169-170; defined, 1 7 0 1 7 1 ; nominal, 186-187; and measurement, 189, 3 1 1 ; quantitative and qualitative, 199; jnd, 364, 3 7 1 ; validity of, 389; numerical, 399; of muscular ability, 422. See also Interval scale; Nominal scale; Ordinal scale; Pitch measurement, scales of; Ratio scale Scaling: Stevens' definition of, 188; and measuring, 189, 3 1 1 ; of minerals, 190; method of ratio, 326; methods of, 367; of psychological magnitudes, 371 Schlosberg, H., 288, 289, 336, 431 Scientific: theories, 1 3 1 ; laws, 329; descriptions, 440 Scratch test for hardness, 1 4 1 , 1 7 3 , 1 7 5 , 189-190, 447 Secondary qualities, 10, 435-438, 442 Sensation dimensions: loudness and pitch as, 6, 235, 244; psychological dimensions as, 446, 461, 467; list of, 468 Sensationist : theory of perception, 3-4, 47-54, 283, 471, 527; explanation of thresholds, 346 ff.; conception of psychophysics, 526

575 Sensations: measurement of, 2 - 3 , 18, 54-57, 162-163, 280, 283 ff., 291 ff., 301 ff., 311-312, 357/ 37«/ 473-474/ 527 ff., 544; in perception, 3-4, 5 2 53, 468 ff., 470-475, 478, 486, 526, 529, 545; misperception of, 9, 28, 307, 325/ 347; and sounds, 18-59,474-475, 525, 529; perception-dependence of, 19 ff., 301, 307, 322, 342, 348; perception of, 21-25, 27» 31/ 3°7/ 314-315» 318-319, 325, 351, 375, 534; privacy of, 25 ff., 315, 342, 366, 412, 415, 453, 527, 546; reports of, 25 ff., 382; numerical identity of, 26-27, 453; nonIocatability of, 36-46; and brain processes, 88, 477; logic of, 89; of sound, 90; addition of, 163, 281, 296, 446, 534; as compound, 245, 296, 328; as intervallic, 281; units of, 281-282, 291, 294, 299-300, 309-310, 3 1 5 - 3 1 8 , 326, 331, 359; zero for, 281-282, 301 ff., 310 ff., 330-331, 333, 354; absolute threshold of, 282, 308, 341, 344; jnd's between, 282, 316-329, 294 ff., 306-308, 344, 360; and stimuli, 287, 331/ 345, 347, 355, 360; Fechner's law of, 291, 315, 362; Weber's law of, 294, 318; psychophysical laws of, 300, 326, 474, 546-547; and sensitivity, 300, 357-358; intermediate, 307; as theoretical constructs, 315; and stimulus judgments, 3 1 7 - 3 1 9 , 358; as epiphenomena, 319, 545; additivity of, 327-329; detection of, 342; continuum of, 343; concept of threshold for, 345, 352-354; as continuous function, 347 ff.; equality of, 350; of muscular strain, 358, 405; psychological entities as, 374; aftersensations as, 374, 469; estimates of, 409, 414, 543; substitutes for, 413, 416, 546; defined by responses, 4 1 3 - 4 1 6 ; kinesthetic, 455, 467; descriptions of, 469-474; as grammatical entities, 486; and external world, 527; and perceptual abilities, 546 Sense perception. See Perception Sensitivity: jnd as index of, 9, 356, 361; of tests for relative magnitude, 1 5 1 , 222; and ranking, 152; and precision, 222; and sensation, 500; measurement of, 357, 502, 521, 536; of balances, 358 Sersen, E. A., 391

576 Shape: measurability of, 99; visual and tactual determinations of, 147; quantitative comparisons of, 431 Shape solid, 432 Sight and touch, 147 Similarity: argument from, 471. See also Psychological similarity SjSberg, L., 365, 402 Smart, J. J. C., 88, 91, 92 Smell, 87 S-method of numeral assignment, 192 ff. Solipsism, 384 Sone, 539 Sound pressure, 81 ff. Sounds: and sound waves, 7 , 1 3 , 50-51, 83-90, 93, 475-476; measurement of, 7, 18, 54-57, 474, 517; and auditory sensations, 7, 18-59, 9°» 278»' causes of, 21, 40, 51, 477; publicity of, 26 ff.; numerical identity of, 26-27, 45-46; as processes, 36, 281; travel of, 3941, 44-46, 89, 94; direction of, 39-46; source of, 40-44, 94; projection of, 42-44; analogy of, to winds, 44-45; logic of, 89; velocity of, 89; ordinary meaning of, 89-90; as surd, 9 1 ; ideal laws of, 267; as heard, 443; least audible, 446; and sensations, 474, 475, 525, 529; as eplphenomena, 477 Sound waves: and sounds, 7 , 1 3 , 50-51, 83-9°/ 93. 475-476; theory of, 40, 75 ff.; velocity of, 41, 89; measurement of, 74-88; and vibrations, 75, 477; logic of, 89; ordinary meaning of, 89; concept of, 89; and auditory sensations, 90; instruments for detecting, 91; travel of, 94 Sourness and acidity, 60-61, 87-88 Space: addition of, 139; measurement of, 503 Spence, K. W., 64, 6 5 , 1 0 2 , 1 1 1 , 1 6 6 , 329, 384, 423» 443 Spring balance, 80 Square root law, 289 Standard meter, 168, 201 Stevens, J. C., 405, 406, 409 Stevens, S. S., 5, 71, 74, 80, 83, 84, 85, 86, 96, 1 1 2 , 1 1 3 , 1 1 6 , 137, 138, 157, 199, 200, 228, 229, 233, 235, 237, 248, 250, 294, 295, 300, 320, 326, 445, 458, 489, 491, 499, 545; psychophysics of, 1 , 10, 364-427, 537 ff.; defines mea-

INDEX

surement, 7, 160, 163-189, 377, 520, 537/ 540 Stevens' law: and Fechner's law, 295296, 424; truth of, 363; evidence for, 365, 539; in cross-modality experiments, 408; psychological magnitudes in, 422; theoretical significance of, 424; experiments in, 539 Stimuli: psychological dimensions and dimensions of, 10, 60; laws of sensations and, 287, 300, 315; and sensations, 294, 308, 325, 331, 343, 345, 355/ 36«; perception of, 317-319, 348, 351, 358; perceivable and unperceivable, 352; standard, 334 ff.; continuum of, 343; and responses, 366, 498-499, 536; observations defining, 497 Stimulus error, 391, 534, 541, 543 Stimulus thresholds: and sensation thresholds, 287, 294, 308, 310, 3 1 7 320, 331, 345, 351, 355, 360; equality of, 317-319 Stone, G., 370 Stumpf, C., 328 Subjective jnd's, 360 Subjective magnitudes: scales for, 199; examples of, 200; loudness as one of, 235; and objective magnitudes, 236, 370; as subject matter of psychology, 441; and physical magnitudes, 458. See also Psychological magnitudes Subjectivity: unclarity in meaning of, 139; and addition methods, 139 ff.; of a- and b-determinations, 143; and operations, 154; of loudness determination, 220; of weight measurement, 221; of equality, 258; of color and pitch, 441; in verification, 448 Subliminal perception, 22 Summation: of loudness, 229, 256-258; of units, 231; of intensity, 232, 238 Summativity: definition of, 102, 103; axiom of, 1 1 5 , 1 2 1 , 134 Suppes, P., 103, 1 1 1 , 167 Sweetness: measurability of, 100; addition of, 135-136; additivity of, 245 Swets, J. A., 333 Symmetry, 102-103 Tactual observation, 54 Taste: physical correlates of, 87; quantitative comparisons for, 431

INDEX Technical use: of "indirect," 62; of "measurement," 1 3 7 , 1 5 8 - 1 5 9 , 1 6 2 , 1 8 7 - 1 8 8 , 1 9 1 ; of "operation," 1 5 5 Temperature: measurement o f , 63-66, 79, 1 7 5 - 1 7 9 ; absolute scale of, 64, 265 ; centigrade and Fahrenheit scales of, 1 1 6 ; absolute zero for, 1 1 7 , 266; zero for, 1 2 2 , 1 7 9 ; tests for equality of, 1 4 0 £f.; thermometer test for, 1 7 1 , 1 7 5 ; unit of, 272; sensations of, 469; and pressure of gases, 5 1 3 Theoretical: use of measurement, 1 3 1 1 3 2 ; terms, 1 5 5 , 439; constructs, 4 1 3 , 426, 462, 486; descriptions, 439-440. See also Ordinary use Thermometer: in temperature tests, 1 4 1 , 1 7 0 , 1 7 5 - 1 7 6 ; description of, 1 7 8 ; calibration of, 1 7 9 ; construction o f , 271 Thorndike, E. L., 502 Thresholds: absolute and relative, 3 3 1 ; measurement of, 3 3 1 ff.; upper and lower, 332; methods for determining, 3 3 3 - 3 4 1 ; numerical expression o f , 3 3 9 - 3 4 1 ; concept of, and sensations, 3 4 1 , 344-345, 352-354; of stimuli and of sensations, 345, 355; sensationist explanation of, 546 ff.; vagueness in, 350; for balances, 3 5 2 - 3 5 3 ; physiological explanation of, 352-354. See also Absolute threshold; Just noticeable differences Thurstone, L. L., 333, 532 Time: measurement of, 159, 503 Tissue destruction and pain, 60-61, 94 Titchener, E. B., 284, 286, 290, 294, 296, 391, 447, 535 Tone deafness, 261 Tones: addition of, 2 1 4 - 2 1 5 , 229-230, 520; units for, 2 1 5 , 237; identified by causes, 2 1 6 ; equality of, 228; summation of, 237; as compound, 243; axioms of addition for, 274. See also Loudness; Pitch Touch and sight, 147 Traditional psychophysics : unacceptability of, 1 , 6; measurement in, 9 - 1 1 , 509, 5 1 6 - 5 2 2 ; sensationist conception in, 24, 283, 526-530; psychological and physical dimensions in, 428 ff., 522-526; scientific analogy in, 5 1 1 5 1 6 ; and new psychophysics, 530— 542; dualism in, 5 3 1 ; decline of, 535.

577 See also Fechner, psychophysics of Transformation of scales, 169 ff., 2 5 4 256 Transitivity, 1 0 2 , 1 0 3 ; of sensation equality, 350 Treisman, M . , 403 Tuning fork, 5 1 , 74, 92, 94, 252 Uncertainty: in measurement, 1 5 0 , 3 5 3 ; in thresholds, 3 3 5 - 3 3 6 , 3 5 0 - 3 5 1 ; in observations, 454 Unconscious perception, 22-23 Units: of length, 7 0 - 7 1 , 168, 259-260; f o r sound waves, 75 £f.; simple and complex, 77, 8 1 ; jnd's as, 1 6 1 , 282, 284, 291, 299-300, 309-310, 345, 359, 537; rules for, 1 6 8 , 1 7 5 , 1 7 9 ; examples o f , 1 9 8 - 2 0 0 ; intervals as, 2 0 1 202, 2 5 7 - 2 6 0 , 2 7 3 ; nature of, 2 0 1 - 2 1 3 ; as conventional, 202-205, 3 68 < 402; precision as function of, 202-204, 220; changes in, 204 ff., 230; fractional, 2 1 1 ; equality o f , 2 1 1 - 2 1 3 , 329 ff.; of loudness, 2 1 5 ff., 539; summation o f , 2 3 1 , 237; of pitch, 257, 262,266, 277; of temperature, 272; for sensations, 281-282, 291, 299-300, 309-310, 3 1 5 - 3 1 6 , 326, 359; of sensitivity, 300; of perceptual ability, 3 5 5 ; psychological, 402, 532, 540-541; of perceptual magnitudes, 488 Unperceived sensations, 3 0 1 , 307, 342, 348-349 Urban, F. M., 544 Vacuum: weighing in, 1 1 8 Validity of scales, 389, 402, 421 Var, 539 Veg, 326, 539 Velocity: definition o f , 68; of sound waves, 75, 89; of particles, 8 1 ; of sounds, 89 Verification: of observation statements, 445; by sense perception, 445, 447; by operations, 446-447; subjectivity in, 448; methods o f , 460; of sensation statements, 467 Vertical addition, 1 0 8 - 1 1 0 , 1 1 2 - 1 1 4 , 1 2 6 Vibrations : amplitude of, 77-78 Visual: estimation, 1 4 0 ; field, 374-375 Volkmann, J., 166, 368, 370, 445 Volume: measurement o f , 68; and weight, 292; and pressure, 5 1 4

578 Warmth, 436 Warren, R. M., 391, 411 Warren, R. P., 411 Weber, E. H., 137, 284-289, 294, 298, »99/ 302, 304, 311-314, 318, 330, 532, 533/ 543; fraction, 284, 288 Weber's law: statement of, 284 ff.; as psychophysical law, 286, 314; criticism of, 288-289; applicability of, to sensations, 294, 302, 318; verification of, 299, 311-312, 533; and Fechner's law, 312-314; empirical content of, 314, 543; and equality of sensation jnd's, 330 Weight: measurement of, 68, 72-73, 79, 108-111, 138, 221, 293; addition of, 108-110; absolute zero for, 118 ff., 227; tests for equality of, 140 ff., 152;

INDEX

jnd's in, 285 ff.; sensations, 301-304, 34«/ 35«; psychological, 515, 521 Weight/volume coefficient, 64, 67, 69 Wood, A. B., 76 Woodger, J. H., 132 Woodworth, R. S., 288, 289, 336, 431, 444/ 446 Zero: for loudness, 4; for temperature, 64,116,121-122,165,179; for length, 103, 305; in centigrade scale, 116, 121-122,165; mathematical laws for, 117-119; for weight, 118; rules for, 168,175; for pitch, 247, 251, 273; for sensations, 282, 301-304, 310-314, 33°-33*/ 343/ 354/ 533; stimulus, 310, 343. See also Absolute zero Zinnes, J. L., 365