Foundations of Measurement. Representation, Axiomatization, and Invariance

361 Pages·1990·18.481 MB·English

by  Patrick Suppes (Auth.)

Frontispiece by Ruth Weisberg
FOUNDATIONS OF
MEASUREMENT
R. DUNCAN LUCE DAVID H. KRANTZ
UNIVERSITY OF CALIFORNIA, COLUMBIA UNIVERSITY
IRVINE
PATRICK SUPPES AMOS TVERSKY
STANFORD UNIVERSITY STANFORD UNIVERSITY
VOLUME HI
Representation, Axiomatization, and Invariance
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
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This book is printed on acid-free paper. @
COPYRIGHT © 1990 BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24-28 Oval Road, London NW1 7DX
Library of Congress Cataloging-in-Publication Data
(Revised for vol. 3)
Foundations of measurement.
Bibliography: v. 1, p. 551-563.
Contents: v. 1. Additive and polynomial representa-
tions / David H. Krantz ... [et al.] — —v. 3.
Representation, axiomatization, and invariance /
R. Duncan Luce ... [et al.]
1. Mensuration. 2. Physical measurements.
I. Krantz, David H., Date.
QA465.F68 530.Γ6 72-154365
ISBN 0-12-425401-2 (v. l)(alk. paper)
ISBN 0-12-425402-0 (v. 2)(alk. paper)
ISBN 0-12-425403-9 (v. 3)(alk. paper)
PRINTED IN THE UNITED STATES OF AMERICA
90 91 92 9 8 7 6 5 4 3 2 1
... For whatever ratio is found to exist between intensity and intensity, in relating
intensities of the same kind, a similar ratio is found to exist between line and line,
and vice versa. For just as one line is commensurable to another line and
incommensurable to still another, so similarly in regard to intensities certain ones
are mutually commensurable and others incommensurable in any way because of
their continuity. Therefore, the measure of intensities can be fittingly imagined as
the measure of lines, since an intensity could be imagined as being infinitely
decreased or infinitely increased in the same way as a line.
Again, intensity is that according to which something is said to be "more such
and such, " as "more white** or "more swift. " Since intensity, or rather the intensity
of a point, is infinitely divisible in the manner of a continuum in only one way,
therefore there is no more fitting way for it to be imagined than by that species of a
continuum which is initially divisible and only in one way, namely by a line. And
since the quantity or ratio of lines is better known and is more readily conceived by
us—nay the line is in the first species of continua, therefore such intensity ought to
be imagined by lines and most fittingly by those lines which are erected
perpendicularly to the subject.
—Nicole Ores me, De configurationibus qualitatum et motuum
(14th century A.D.)
Preface
Sometime toward the end of 1969 or the beginning of 1970 we came to
realize that the material covered by this treatise was of such extent that the
originally projected single volume would require two volumes. Because
most of the work in Volume I was by then complete, we concentrated on
finishing it, and it was published in 1971. There (pages 34-35) we listed the
chapter titles of the projected Volume II, which we expected to complete by
1975. Fifteen years later, when we finally brought the work to a close, our
plan had changed in several respects. First, the total body of material far
exceeded a reasonably sized second volume so, at the urging of the
publisher, the manuscript expanded into two volumes, for a total of three.
Second, the chapter on statistical methods was not written, largely because
the development of statistical models for fundamental measurement turned
out to be very difficult. Third, we decided against a summary chapter and
scattered its contents throughout the two volumes. Fourth, Volume III
came to include two chapters (19 and 20) based on research that was
carried out in the late 1970s and throughout the 1980s. Volume II discusses
or references many results that have been obtained since 1971 by the large
number of persons working in the general area of these volumes.
As in Volume I, we attempt to address two audiences with different levels
of interest and mathematical facility. We hope that a reader interested in
the main ideas and results can understand them without having to read
xiii
XIV PREFACE
proofs, which are usually placed in separate sections. The proofs themselves
are given somewhat more fully than would be appropriate for a purely
mathematical audience because we are interested in reaching those in
scientific disciplines who have a desire to apply the mathematical results of
measurement theories. In fact, several chapters of Volume II contain
extensive analyses of relevant experimental data.
Volume II follows the same convention used in Volume I of numbering
definitions, theorems, lemmas, and examples consecutively within a chapter.
Volume III deviates from this with respect to lemmas, which are numbered
consecutively only within the theorem that they serve. The reason for this
departure is the large number of lemmas associated with some theorems.
Volumes II and III can be read in either order, although both
should probably be preceded by reading at least Chapters 3 (Extensive
Measurement), 4 (Difference Measurement), and 6 (Additive Conjoint
Measurement) of Volume I. Within Volume II, Chapter 12 (Geometrical
Representations) should probably precede Chapters 13 (Axiomatic Geome-
try), 14 (Proximity Measurement), and 15 (Color and Force Measurement).
Within Volume III, much of Chapter 21 (Axiomatization) can be read in
isolation of the other chapters. Chapter 22 (Invariance and Meaningfulness)
depends only on the definability section of Chapter 21 as well as on
Chapters 10 (Dimensional Analysis and Numerical Laws), 19 (Nonadditive
Representations), and 20 (Scale Types). Chapter 19 should precede Chap-
ter 20.
A cknowledgments
A number of people have read and commented critically on parts of Volume III. We are
deeply indebted to them for their comments and criticisms; however, they are in no way
responsible for any errors of fact, reasoning, or judgment found in this work. They are
Theodore M. Alper, Ching-Yuan Chiang, Rolando Chuagui, Newton da Costa, Zoltan
Domotor, Kenneth L. Manders, Margaret Cozzens, Jean-Claude Falmagne, Peter C. Fishburn,
Brent Mundy, Louis Narens, Reinhard Niederée, Raymond A. Ravaglia, Fred S. Roberts, and
Robert Titiev.
In addition, each of us has taught aspects of this material in classes and seminars over the
years, and students have provided useful feedback ranging from detecting minor and not-so-
minor errors to providing clear evidence of our expository failures. Although we do not name
them individually, their contribution has been substantial.
Others who contributed materially are the many secretaries who typed and retyped portions
of the manuscript. To them we are indebted for accuracy and patience. They and we found
that the process became a good deal less taxing in the past eight years with the advent of good
word processing.
With chapters primarily written by different people and over a long span, the entire issue of
assembling and culling a reference list is daunting. We were fortunate to have the able
assistance of Marguerite Shaw in this endeavor, and we thank her.
Financial support has come from many sources over the years, not the least being our home
universities. We list in general terms the support for each of the authors.
Luce: Numerous National Science Foundation grants to The University of California,
Irvine, and Harvard University (the most recent being IRI-8602765; The University of
California, Irvine, and Harvard University); The Institute for Advanced Study, Princeton,
New Jersey, 1969-1972; AT&T Bell Laboratories, Murray Hill, New Jersey, 1984-1985; and
The Center for Advanced Study in the Behavioral Sciences, Stanford, Cahforaia, 1987-1988
(funds from NSF grant BNS-8700864 and the Alfred P. Sloan Foundation).
Tversky: The Office of Naval Research under Contracts N00014-79-C-0077 and N00014-84-
K-0615 to Stanford University.
XV
Chapter 18 Overview
This chapter describes briefly and somewhat informally the several topics
covered in this volume, relates them to Volumes I and II, and discusses the
advantages of the axiomatic representation and uniqueness approach taken
in this work.
The theme of Volume I was measurement in one dimension. Each
measurement structure assumed both a qualitative ordering of some empiri-
cal objects and a method of combining those empirical objects, either by
concatenation, or by set-union, or by factorial combination. Measurement
involved the assignment of numerical representations to such empirical
structures. The ordering of the representing numbers always preserved the
qualitative ordering of the empirical objects. Moreover, there was always
some numerical combination rule, involving addition or multiplication, or
both at once, which could be used to calculate the number assigned to any
allowable combination of the empirical objects from the numbers assigned
to the various constituents of that combination. In Chapters 3, 5, and 6 the
principal numerical combination rule was addition; in Chapter 4, subtrac-
1
2 18. OVERVIEW
tion; in Chapter 8, weighted average, or expectation; and in Chapters 7 and
9, polynomial combination rules with certain restricted forms.
Volume I unified measurement theory by reducing the proofs of most
representation theorems to Holder's theorem. This reduction showed that a
common procedure underlies different methods for assigning numerical
values to empirical objects. One constructs sequences of equally spaced
objects, called standard sequences. The object to be assigned a number is
bounded between successive terms of a standard sequence. One counts the
number of terms needed to reach the target object, and the number needed
to reach an object that serves as a unit of measurement. The ratio of these
two counts approximates the desired numerical value.
Much of Volume II is devoted to geometrical representations that have
two or more dimensions, although Chapters 16 and 17 returned to one-
dimensional representations of fallible data. To some degree, additivity
continued to play a significant role in that volume; for example, consider-
able use was made of the additivity of segments along single dimensions.
Chapters 19 and 20 of this volume go substantially beyond the frame-
work of Volume I by considering one-dimensional representations that are
essentially nonadditive. Chapter 19 deals with two kinds of structure: an
ordering with a binary operation that is monotonie but can violate associa-
tivity, and a two-factor conjoint ordering that is monotonie but can violate
the cancellation (Thomsen) axiom. (The associativity and cancellation
axioms each force additivity.)
Chapter 20 goes further to consider in very general terms the possibilities
for measurement when the primitives of the structure are specified only to
the extent that one is a weak order and the others are all relations of finite
order. However, the structure is restricted by an assumption of homogene-
ity that implies a certain interchangeability of elements. In contrast to
axioms previously encountered, which are formulated in terms of the
primitives of the system, homogeneity is formulated somewhat indirectly.
For such homogeneous structures, our understanding is moderately com-
plete concerning the possible types of uniqueness that can arise, the
so-called scale types. This knowledge, coupled with fairly weak structural
constraints, is useful because it yields a detailed understanding of the types
of numerical representations that exist for homogeneous concatenation
operations and conjoint structures. Moreover, a good deal is also known
about numerical representations of structures that have neither a binary
operation nor a factorial structure.
Chapter 21 is devoted to issues (largely treated by methods of mathemati-
cal logic) concerning the formal axiomatization of measurement structures.
It also explores axiomatic issues that have arisen in all three volumes, such
as the significance of Archimedean axioms, and general logical topics,
18.1. NONADDITIVE REPRESENTATIONS (CHAPTER 19) 3
including the notion of definability, that are of particular interest to
measurement theorists.
The work on scale types in Chapter 20 and on definabihty in Chapter 21
leads naturally to the study in Chapter 22 of meaningful assertions within a
measurement context. It includes, among other things, a modification and
extension of parts of Chapter 10. Meaningfulness in the homogeneous case
is taken to mean at least invariance under the transformations that describe
the scale type. Various arguments are given for the plausibility of this
criterion, including the equivalence of several possible definitions of mean-
ingfulness. How the concept of invariance is best restricted to capture
meaningfulness in the more restricted sense of definabihty (which is criti-
cally important for nonhomogeneous structures) is an apparently difficult
and unresolved problem.
18.1 NONADDITIVE REPRESENTATIONS (CHAPTER 19)
A natural question raised by Chapters 3 and 6 is: What can be said about
concatenation and conjoint structures with nonadditive representations?
When we wrote Volume I, nearly 20 years ago, virtually nothing was known
about such structures aside from the polynomial models of Chapter 7 and
the weighted average models (which are closely related to additive conjoint
structures) arising both from bisymmetry (Section 6.9) and from the ex-
pected-utility axioms (Chapter 8). Since then, the nonadditive theory has
been developed rather fully for concatenation operations and conjoint
structures.
As in Volume I, if a structure has any numerical representation, then it
has one for each strictly increasing function on the range of the given
representation. That is, if Θ is a numerical operation that represents a
structure on a subset R of the positive real numbers, and A is a strictly
increasing function from R onto R, then the operation Θ defined by
xOy = h-l[h(x) <&h(y)]
is an equally good representation. Thus the mere fact that there is a
nonadditive representation does not imply that an additive one does not
exist. But if an additive one does exist, then the nonadditivity is said to be
inessential. In this case, where, for example θ = +, then any operation
involved in an alternative representation is necessarily associative, i.e.,
xG (yO z) = (xQy)Qz.
We are interested here in essential nonadditivity, i.e., cases for which no