Table Of ContentCAMBRIDGE STUDIES IN PHILOSOPHY
General editor Sydney shoemaker
Advisory editors j. e. j . altham, simon blackburn,
GILBERT HARMAN, MARTIN HOLLIS, FRANK JACKSON,
JONATHAN LEAR, JOHN PERRY, T. J. SMILEY, BARRY STROUD
JAMES CARGILE Paradoxes: a study in form and predication
PAUL M. CHURCHLAND Scientific realism and the plasticity of mind
N. M. L. N H A N Evidence and assurance
WILLIAM LYONS Emotions
PETER SMITH Realism and the progress of science
BRIAN LOAR Mind and meaning
DAVID HEYD Supererogation
JAMES F. ROSS Portraying analoypy
PAUL HORWiCH Probability and evidence
ELLERY EELLS Rational decision and causality
HOWARD ROBINSON Matter and sense
E. J. BOND Reason and value
D. M. ARMSTRONG What is a law of nature?
Theory
and measurement
Henry E. Kyhurg Jr
Professor of Philosophy, University of Rochester
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Cambridge University Press 1984
First published 1984
Printed in Great Britain at the University Press, Cambridge
Library of Congress catalogue card number: 82-17905
British Library cataloj^uin^ in publication data
Kyburg, Henry E.
Theory and measurement.—(Cambridge studies in
philosophy)
1. Mensuration
I. Title
530.8 T50
r ISBN 0 521 24878 7
'N
Contents
Preface vii
1 Introduction 1
2 Machinery 23
3 Relative length 36
4 Measurement of length 61
5 Direct measurement 90
6 Indirect measurement 113
7 Systematic measurement 143
8 Reduction of dimensions and fundamental units 159
9 Error: random and non-random 183
10 Interpersonal agreement 213
11 Theory and measurement 234
Bibliography 261
List of main notation 269
Index 271
Preface
Although there is a sizable literature on measurement theory, rel
atively little has been published in recent years on the epistemologic
al foundations of measurement, and on the complicated connection
between measurement and the testing of theories. The present work
is addressed to precisely these questions, though of necessity certain
more general considerations concerning theory change, linguistic
change, and epistemology are addressed along the way.
It is difficult to give a precise accounting of the time that went into
this book, but part or all of the time released by grants SOC 77-26021
and SES 8023005 from the National Science Foundation clearly went
into it, and the freedom provided by a fellowship from the Gug
genheim Foundation for the academic year 1980-1, though it was
mainly devoted to a more general topic, helped in the development
of the last two chapters. 1 am also grateful to the University of
Rochester for providing both leave and financial support during that
year.
Conversations and correspondence with a number of people, and
comments by a number of anonymous referees, have helped greatly
in getting my ideas clear. In this connection, I wish particularly to
thank Ernest Adams and Phillip Ehrlich, with whom I was lucky
enough to have long conversations during a trip to California, and
Zoltan Domotor who provided both unremitting criticism as a
referee and unfailing support as a fellow seeker after the truth in
measurement.
To my colleagues at Rochester I am grateful for moral support,
and to the department secretaries 1 am not only grateful for un
bounded patience and skill at typing and retyping sections of the
manuscript, but more importantly for constant understanding and
encouragement in a project that must not only have seemed myste
rious but unending.
Two notational ambiguities should be mentioned here. First, 1
have used the symbol in two senses, which arc distinct enough (I
Vll
hope) not to generate confusion. 1 have used it to modify predicates
of the object language to indicate that they have become somewhat
‘theoretical’; thus ~ represents a relation of observational indisting-
uishability, while represents the somewhat theoretical relation
in terms of which we generate equivalence classes of objects. I have
also used as an operator representing the multiplication of
(signed) quantities.
Second, I have followed two well-established conventions in my
use of square brackets. The equivalence class (under the equivalence
relation suggested by the context) determined by the object a is
denoted by [a]. But square brackets are also used to denote dimen
sions: thus [feet] is the dimension of length, as is [meters] and
[length] itself. Both of these conventions are commonly followed,
and it seemed better to admit the ambiguity than to generate still more
notation.
There are two approaches to reading this book. One, which led to
its present structure, is better for those who like to see concrete
instances before they ascend to more general abstractions. Chapters
2, 3, and 4, which provide a detailed development of the measure
ment of length on the basis of corri^^ihie comparative judgments,
provide the paradigm in terms of which the measurements of other
quantities can be construed. The other approach, which is also natu
ral, is better for those who want to see where they are going before
being faced with a lot of formal details. This approach would post
pone Chapters 2, 3, and 4 until after Chapter 8. This also has the
advantage that the general treatment of error in Chapter 9 then
follows conveniently on the specific treatment of errors of measure
ment of length found in Chapter 4. I have attempted to ensure that
the book may be read in either way.
1
Introduction
Measurement is so fundamental to the physical sciences and to en
gineering that it is difficult to know where we would be without it.
The psychological and social sciences, as currently practiced, involve
measurement in two quite distinct ways. In the first place, they
involve ordinary physical measurements: distances, reaction times,
voltages, and the like. But they also involve the development of
procedures and scales of measurement that are peculiar to their sub
ject matters: IQ as a measure of intelligence, indices of extroversion/
introversion, measures of manual dexterity or the ability to deal with
spatial relations. Given the prevalence of measurement in all the
branches of science, we would expect discussions of measurement to
play an important role in writings on the philosophy of science.
Our expectation is frustrated. This could mean either of two
things: that measurement is so well understood and so easy to under
stand that not much need be said about it; or that not even the
problems of measurement are sufficiently appreciated. Brian Ellis
inclines toward the latter view. He argues that while there is a ‘cli
mate of agreement’ concerning measurement, ‘One can only believe
that the agreement is superficial, resulting not from analysis but
from the lack of it.’ (Ellis, 1968, p. 2) Writing in 1968, he finds
Campbell (1920, 1928) and Bridgman (1922) to be the only ‘major
works of a primarily philosophical nature dealing with measure
ment’ (Ellis, 1968, p.l).
Arnold Koslow (1981), shares this point of view to some extent,
though he cites Mach (1960) and Helmholtz (1977) among the great
classical writers on measurement, and he refers to the ‘enormous’
contemporary literature on measurement, much of it written before
1968. It seems that even the question of whether or not there is a large
literature on measurement is controversial.
The opposite point of view — that measurement is really perfectly
well understood — is embodied in the monumental work of Krantz,
Luce, Suppes & Tversky (1971). There is no anomaly in the fact that
1
a well-understood subject should be the topic of a monumental
work. Although the subject matter is well understood, there are
many technical questions, both deep and difficult, that require to be
explored. As Krantz et al. write, ‘Scattered about the literatures of
economics, mathematics, philosophy, physics, psychology, and sta
tistics are axiom systems and theorems that are intended to explain
why some of the attributes of objects, substances, and events can
reasonably be represented numerically. These results constitute the
mathematical foundations of measurement.’ (1971, p. xvii) The goal
of the monumental work is to establish a general treatment in which
these scattered results will appear as special cases, to collect and unify
and render coherent a fragmentarily established body of doctrine.
We see here a clue as to the controversy concerning the magnitude
of the literature of measurement. Much of this litera
ture — especially that appearing in psychological journals — is con
cerned with a very specific sort of technical problem. We begin by
supposing that a certain attribute is measurable in roughly a certain
sort of way, and on a certain sort of scale. To ‘establish the mathema
tical foundations’ of the measurement of this attribute is to find
axioms concerning this attribute (and generally concerning a certain
way of combining objects with that attribute) such that we can prove
two important theorems: a representation theorem, which shows
that the attribute can be represented by a certain structure of real
numbers; and a uniqueness theorem, which shows that any two
functions from objects to real numbers that represent the attribute in
question are related in certain ways — for example by multiplication
by a constant, by a linear equation, by a monotonic transformation,
etc.
To illustrate this approach, consider the measurement of weight.
We begin by knowing that weight is represented by a ratio scale
(e.g., we know that we can convert a weight in grams to a weight in
pounds by multiplying by a constant), that weights combine addi-
tively (the weight of a pair of objects taken together is the sum of
their weights taken individually) and that of two objects, one is
heavier than the other, or they have the same weight. We compose
axioms (e.g. that ‘being heavier than’ is transitive) and attempt to
define a function (() from ponderable objects into the real numbers
such that for any x and y, ifx is heavier than y, then (J)(x) > (t)(y), if y is
heavier than x, then (J)(y) > <J)(x), and if x and y are equally heavy,
<()(x) = (|)(y), and furthermore such that if x ° y is the ‘combination’ of
X and y, then (j)(x ® y) = <|>(x) -I- cj)(y). To do this generally requires
existence axioms that we might not have discovered intuitively; for
example, we might need to assume that for any weights x and y,
where x is heavier than y, there is a natural number n such that n
replicas of y in combination are lighter than x, and « H- 1 replicas of y
in combination are heavier than x.
Having defined a function (|) that represents the attribute of
weight, we then go on to prove that if (|) and are any two such
functions, there is a real number a such that for all x, (j)(x) = ii\|)(x).
Now this represents an important, useful and enlightening piece
of analysis. But it leaves untouched many of the questions which
were of concern to writers such as Ellis, Campbell, and others. It is
all very well to say that if a certain set of the attributes of objects and a
certain operation on them obey certain axioms, then those attributes
can be represented by a function of a certain sort from those objects
to the real numbers. But we also want to know that the axioms are
satisfied; we must at least face the classical problem of inductive or
scientific inference. Furthermore, it is not clear that the range of the
function (j) is real numbers; engineers and scientists often take the
weight of an object to be so and so many grams: they write as if there
were abstract entities — magnitudes — that are approached through
measurement. Particularly troublesome in this regard is the nature of
the operation on objects which is required for the development of
useful and interesting scales. As Ellis notes, we might be tempted to
suppose that the operation by which we combine rigid rods were
rectilinear rather than collinear juxtaposition. Finally, we all know
that measurement is imperfect: no procedure of measurement is
perfectly accurate. But we are alleged to test scientific theories and
hypotheses by devising experiments which, if the theory or hypoth
esis is true, will lead to certain measurable results. We cannot expect
to get exactly the right result; but if the result is too far off — not
‘within experimental error’ — we have to reject the theory or
hypothesis. What does this phrase ‘within experimental error’ mean,
and how do we determine in a given case what counts as ‘ex
perimental error’? In general, how are our procedures of measure
ment coordinated with the development of a theory of errors of
measurement?
Norman Campbell, writing in 1920, laments the fact that the
theory of measurement and the study of the foundations of physics
have been taken over by mathematicians. ‘And as the physicist pos
