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Numbers as quantitative relations and the traditional theory of measurement.


Any account of measurement presupposes answers to other philosophical questions and a shift on these may affect the account of measurement given. For example, when empiricist conceptions of number seemed untenable, following Frege's [1884 critique of Mill [1843], philosophical opinion shifted from the traditional e.g (Clifford [1882, 1885]) to the representational (e.g. Campbell [1920]) theory o measurement. The representational and operational accounts of measurement have been the ones most commonly given this century. The representational account ha developed from Russell [1903] to the three volumes of the Foundations of Measurement, by Krantz, Luce, Suppes, and Tversky [1971, 1989, 1990]. The operational account (e.g. Dingle [1950]) has been widely espoused in some of th sciences, especially psychology. Less impressive has been the development of th traditional view. The presentation given by Whitehead and Russell [1913] appear to be the last systematic attempt. The view is still widely espoused by scientists (e.g. Beckwith and Buck [1961] and Massey [1986]) but not often defended philosophically. By the traditional account. I mean the view that measurement is the discovery or estimation of numerical relationships between quantities of some kind and a unit quantity of the same kind.

The differences between the various understandings of measurement are displayed most dramatically in the semantics of measurement statements. Taking a simple measurement statement, such as

(1) The length of object A is 10.25 metres,

these differences may be illustrated. According to the traditional view, (1) asserts a numerical relation between two properties: the length of object A and that length conventionally adopted as the metric unit, the metre. These two lengths are thought to be related numerically. The former is some number of times greater than the latter, specifically 10.25. Thus, (1) is a statement about how two properties relate. It is completely objective, in the sense that its subject and predicate terms both have objective reference.

According to the operational conception of measurement, statements such as (1) are about the outcomes of our measurement procedures. There is nothing in the world independent of us that is described by the phrase, 'the length of object A[prime]. Instead, this phrase denotes an operation or measurement procedure that we perform. The result of applying this procedure to object A is the reading '10.25 metres'. This, likewise, does not describe a property or relatio existing independently of us. It is simply what our procedure delivers when applied to A. Our operations (in particular, our measurement procedures) are grids or frameworks that we impose upon reality. We cannot penetrate the veil o method.

Between these extremes lies representationalism. The purpose of numbers in measurement, according to this view, is to represent empirical reality. Thus, the number, 10.25, in (1) represents the empirical entity, the length of object A. Since other numbers could do the job equally well, the particular function used in this instance, the metric scale, must be mentioned. Statement (1) does not describe A in the sense of reporting some natural fact about it. Rather, it informs us about a measurement convention. However, behind that convention stands a fact (a qualitative or non-numerical relationship between A and the standard metre) and (1) can be understood as representing that fact numerically So while measurement statements are not literally statements about empirical reality, numbers not being empirical entities, they will have empirical (qualitative) truth conditions. On this view, the numerical language of quantitative science is a framework we impose upon reality, but it is one that we can penetrate. Thus, the representational theory has the special problem of meaningfulness, which is that of specifying the qualitative content of the quantitative statements employed in science (Narens [1985]).

Metaphysical realism has recently been defended by a number of philosophers (e.g. Armstrong [1978]) and realist theories of number have been revived (Kitcher [1983]; Forrest and Armstrong [1987]; and Bigelow [1988]). No attempt will be made here to argue for realism in either sense. Instead, I am intereste in where this revival of interest in realism leaves the traditional theory of measurement. If it is granted that things have properties and stand in relations, and that some of these relations are genuinely numerical, can a coherent presentation of the traditional understanding of measurement be worked out? In rehabilitating the traditional account then, it must be explained, firstly, what quantities are and, secondly, how they sustain numerical relations. To a large degree these tasks have been completed. For example, Swoyer [1987] gives an excellent realist treatment of quantity, and Forrest and Armstrong [1987] sketch the numbers as quantitative relations. However, the tas of linking these two strands together into a realist theory of measurement and relating such a theory to recent developments within the representational framework has not been undertaken.


I begin with Forrest's and Armstrong's [1987] notion of a 'strongly particularizing property'. A property, X is strongly particularizing iff for an objects. A and B (A [is not equal to] B), if A is X and B is X then A and B are discrete (i.e. have no parts in common). For example, being an electron is such a property, while being a piece of chocolate is not. Henceforth, let X denote only strongly particularizing properties.

Let A and B be any two aggregates of Xs. A and B are equally numerous with respect to X iff there is a one to one correspondence between the Xs in A and the Xs in B. A is one more X than B iff A is entirely composed of discrete part C and D, where C is an X, and D and B are equally numerous with respect to X.

Any aggregate of Xs, A, is numerable iff either

(i) A is an X; or

(ii) A is one more X than some numerable aggregate of Xs.

Let [K.sup.X] be the class of all numerable aggregates of Xs. Let [[K.sup.X]] b the partition of this class into equivalence classes with respect to the relation being equally numerous with respect to X. Corresponding to each equivalence class in [[K.sup.X]] there is the property common to each aggregate within that equivalence class and belonging to none outside it, being an aggregate of Xs of just this size. Let [Q.sup.X] be the set of all such properties.

Let a, b, c, . . ., be any elements of [Q.sup.X]. a = b + c iff any aggregate which is a is entirely composed of discrete parts, one of which is b and the other of which is c. The set, [Q.sup.X], and this relation of additivity (or summation) upon it is a numerable quantity. Any such set and relation is a numerable quantity iff

(1) for all a and b in [Q.sup.X], a + b = b + a;

(2) for all a, b, and c in [Q.sup.X], a + (b+c) = (a+b) + c; and

(3) for all a and b in [Q.sup.X], one and only one of the following is true,

(3.1) a = b

(3.2) there exists c in [Q.sup.X] such that a = b + c;

(3.3) there exists c in [Q.sup.X] such that b = a + c.

Of course, [Q.sup.X] is ordered (for any a and b in [Q.sup.X], a [is greater than] b iff 3.2) and the order is transitive, asymmetric and connected (i.e. a strict simple order). While, for any X, there may be a greatest element in [Q.sup.X] (i.e. some properties are only present in a finite number of objects and, perhaps, all are), it is easy enough to consider the properties [Q.sup.X] would have if, for some X, there was no greatest element in [Q.sup.X].

A numerable quantity, [Q.sup.X], is unbounded above iff for any a in [Q.sup.X] there exists b and c in [Q.sup.X] such that b = a + c.

Of course, any numerable quantity, [Q.sup.X], is bounded below by the property, being an X. I denote this element of [Q.sup.X] by the symbol 'e'. Each property in [Q.sup.X] stands in a relation to this, the least element in [Q.sup.X]. The unbounded sequence of such relations is the sequence of natural numbers. These relations exist because any element, a, of [Q.sup.X] is a sum of es, that is either

(i) a = e; or

(ii) a = b + e (where b is a sum of es).

Hence, the unbounded sequence of relations may be specified as follows:

(i) a stands in the relation 1 to e (i.e. a is one e) iff a = e;

(ii) a stands in the relation (n + 1) to e (i.e. a is (n + 1)es) iff a = b + e and b stands in the relation n to e.

Why identify this sequence of relations with the natural numbers? After all, this sequence of relations is perfectly isomorphic to the sequence of propertie in [Q.sup.X]. Any formal properties the former possess. the latter have as well This is true, but the properties in [Q.sup.X] are indexed with respect to X and so are different to the natural numbers. Being nXs is not the same as being n.

If X and Y are two different strongly particularizing properties. and e and e[prime] the properties of being an X and being a Y respectively, then the sequence of relations in the situations, a is n es (for all a in [Q.sup.X]) and a[prime] is n e's (for all a[prime] in [Q.sup.y]) is the same sequence. Thus, these relations are general across properties, but so is the sequence of properties, being an n-membered aggregate. This sequence of properties has sometimes been thought to be the sequence of natural numbers (e.g. Mill [1843]) but it encounters the problem identified by Frege [1884], which is that the sam thing may be both an n-membered and an m-membered aggregate (e.g. the aggregate of volumes on my desk is both n books and m chapters of books), while being n and being m (n [is not equal to] m) are mutually exclusive. An aggregate may be both n and m but only in relation to different units. In the same way, a person may be both a sister and a mother, but only in relation to different individuals. If aggregate A is nXs and aggregate B is nYs, the feature common t both situations (the n-ness) is just how being n things of some kind is constituted of the unit property, being one thing of that kind, and this is a relation between these two properties. If by a ratio is meant the magnitude of one thing relative to another, then it is pretty clear that the natural numbers are ratios. They are the magnitude of an aggregate's size relative to some unit

However, there are thought to be problems with this identification of natural numbers with empirical relations. Some of these are considered by Forrest and Armstrong [1987]. Two are of particular interest here. Obviously, if the natura numbers are to be used in identifying the real numbers then they must be infinite, so the first problem is the possible boundedness of the size of empirical aggregates. This is not an insurmountable problem because the theory of the natural numbers (i.e. natural number arithmetic) is a system of hypothetical statements and, so, may describe more possibilities than are, or could be, realized. How unrealized possibilities are to be conceptualized by th realist is a philosophical problem in its own right (e,g. Armstrong [1989]).

The second problem is the fact that some empirical aggregates do not always behave in accordance with the laws of arithmetic. If the statements of arithmetic, like 1 + 1 = 2, are statements about aggregates, then they can only be true if they describe all aggregates. Some aggregates of things (e.g. raindrops, amoebas, rabbits) do not behave as pure arithmetic is thought to predict. Mackie had a sufficient reply to this problem. Such a proposition of arithmetic 'does not state that an occurrence of one sort is regularly followed by an occurrence of another sort, so it cannot be used on its own to license predictions. It states what Mill calls a uniformity of coexistence' (Mackie [1966], p. 27). This insight of Mackie's is extremely perspicacious and I shall resort to it again in relation to measurable quantities and real numbers. On this view of number, 1 + 1 = 2 does not mean that any aggregate of one X combined with any other aggregate of one X gives an aggregate of two Xs. That may or may not happen depending upon the other properties Xs have and the kind of combining operation employed. What 1 + 1 = 2 means is that any aggregate of two Xs is entirely composed of two discrete parts, each of one X. The former interpretation is about the behaviour of aggregates, the latter is about the numerical structure of quantities. Of course, we may only get to know about the latter via the former, so under certain physical conditions the behaviour of some aggregates of Xs may reflect the numerical structure of quantities. However. it need not always do so.


Having sketched an interpretation of the traditional view of numerable quantity and natural number, the way is opened for a traditional account of continuous quantity, real number, and measurement, for the real numbers presuppose the natural numbers. Amongst the difficulties for such an account are these. The connection between continuous quantity and the real numbers is not as simple as painted by Bostock [1979], Bigelow [1988], Forrest and Armstrong [1987], and Stein [1990]. While for any continuous quantity (such as length) there is an isomorphism between ratios of elements of that quantity and the positive real numbers, there is really no uniquely defined ratios for any two such elements. As it turns out, there are infinitely many such ratios. The implications of thi problem have never been faced by the traditional account. A second problem is that relations of additivity between quantities of the same kind are not always directly reflected in the behaviour of objects and, so, extensive measurement cannot be taken as the model by which all quantities are identified. When, in science, a variable is hypothesized to be quantitative, all that is being hypothesized is something about its structure. How that structure is manifest i the behaviour of objects is another matter.

The measurable quantities are those included within the study of physics. The paradigms are length and time, and the other measurable quantities are hypothesized to have a structure similar to these. Other sciences also postulat their own measurable quantities (e.g. the psychological quantities, such as the various mental abilities, etc.), but their quantitative structure is yet to be demonstrated. Length and time, like aggregate size, possess an additive structure. What distinguishes them from aggregate size is the fact that they ar taken to be continuous. That is, the order upon the elements (i.e. upon lengths or time intervals) from smaller to greater contains no gaps.

To be more precise, length and time, as quantities, are held to satisfy conditions (1), (2), and (3). As well, continuous quantities satisfy conditions (4) and (5). Let Q be the set of properties or relations constituting some quantity (such as length) and + a ternary relation upon Q. Then (Q, +) is a quantity if

(1) for any a and b in Q, a + b = b + a;

(2) for any a, b, and c in Q, a + (b + c) = (a + b) + c;

and (3) for any a and b in Q, one and only one of

(3.1) a = b,

(3.2) there exists c such that a = b + c, and

(3.3) there exists c such that b = a + c

are true.

Given (1), (2), and (3), Q is ordered by the following relation. For any a and in Q, a [is greater than] b iff (3.2).

In this context what is meant by '+' needs to be explained. For the moment, thinking of length, let + be defined as follows. For any a, b, and c in Q (the set of all possible lengths), a + b = c iff any distance of length c is entirel composed lengthwise of two discrete but undivided and adjoining parts, one of length a and the other of length b. Whether objects of length c can always be divided into discrete parts of lengths a and b, or objects of lengths a and b always concatenated into objects of length c are practical matters, logically distinct from this definition of additivity. As before, (1), (2) and (3) are like Mill's uniformities of coexistence (though this is an inapt term in the case of time). As will be made clear later, this definition of + gives but one of an infinite number of possible additive relations upon the class of all possible lengths.

A quantity, (Q, +), satisfying (1), (2) and (3) is order dense iff for all a an b in Q such that a [is greater than] b there exists c in Q such that

(4) a [is greater than] c [is greater than] b.

A quantity, (Q, +). satisfying (1), (2), (3) and (4) is continuous (or Dedekind complete) iff

(5) every non-empty subset of Q that has an upper bound has a least upper bound

(where, for any non-empty subset, Q[prime], of Q, any a in Q is an upper bound of Q[prime] iff for all b in Q[prime], a [is greater than or equal to] b; and any upper bound, a, of Q[prime] is a least upper bound iff a [is less than or equal to] b, where b is any upper bound of Q[prime]).

Ratios of pairs of Q may be defined as follows. For any a and b in Q, the ratio of a to b (a/b) [is less than] n/m iff ma [is less than] nb and a/b[is greater than or equal to] n/m iff ma [is greater than or equal to] nb (where m and n ar natural numbers and for any c in Q, 1c = c and (n + 1) c = nc + c).

Any two ratios, a/b and c/d, of elements of Q are identical iff for all natural numbers, n and m,

a/b [is less than] n/m iff c/d [is less than] n/m and a/b [is greater than or equal to] n/m iff c/d [is greater than or equal to] n/m.

A quantity, (Q, +), is unbounded iff for any a in Q and any natural number, n, na is also in Q.

Let R be the set of all ratios of elements of Q, a continuous quantity, then if Q is unbounded R is isomorphic to the positive real numbers (Holder [1901]).

In order for ratios of elements of (Q, +) to be isomorphic to the positive real numbers considerable structure has been added, viz., order density, continuity, and unboundedness. This may seem conceptually extravagant and one is tempted to seek some empirical warrant for these additional conditions. Whether or not suc warrant can be found is of no material interest here. We are concerned, as with the natural numbers, with the complete set of elements that Q could possibly have. Of course, that raises the prospects of non-standard quantities (i.e. quantities containing infinitesimal elements). They are not prospects to be dismissed, but considerations of simplicity lead first to the standard kinds of cases considered here. (For a consideration of non-standard quantities see Narens [1985].) Furthermore, the concept of quantity employed in the theories o physics is that of continuous quantity.

Also, the real numbers are not simply the positive real numbers, so Q must be augmented in some way to allow negative ratios. One way is to consider the set of differences between values of Q, [Q.sup.D], which allows both negative and zero elements to be identified and, then, ratios involving these new elements will constitute a complete ordered field (i.e. a system isomorphic to the real number system).

The philosophical thesis that these ratios are the real numbers is based on mor than this isomorphism. For if the system of ratios is isomorphic to the reals then so is [Q.sup.D], the set of all differences between elements of Q. Why identify ratios of lengths with real numbers but not identify lengths with them Clearly, to identify lengths themselves with real numbers would be unacceptable Not only would it do violence to our metaphysical preconceptions, but we recognize that the association of individual lengths with real numbers is arbitrary and varies with the choice of unit. That is, if lengths a and b are associated with real numbers r and s, this is only relative to a unit, c. Hence it is a relative to c, and b relative to c which are really r and s. And a relative to c is the ratio of a to c.

That is one argument but it does not clinch the case. As is well known, the reals can be mapped into themselves in ways that preserve order but transform additive relations into multiplicative relations. For example, for some real number r [is greater than] 1, and any real number x, let

f(x) = [r.sup.x].

Then, of course, for any reals x, y and z,

x+y=z iff f(x)xf(y)=f(z).

So any association between ratios and real numbers is also arbitrary and can be replaced by another in which the additivity between ratios is represented by multiplication between numbers instead of by numerical additivity. So, to push the same form of argument employed in the last paragraph, it is not a relative to c that is associated with a real number but a relative to c relative to something else. This turns out to be an alternative relation of additivity on Q In assigning real numbers to lengths relative to a unit in the normal way, one is in effect saying that, for any a, b, c, and d in Q

a + b = c iff a/d + b/d = c/d iff d(a) + d(b) = d(c)

(where d(a) is the real number assigned to a relative to d as unit). Now, if fo some real number, r [is greater than] 1,

d(a)+d(b)=d(c) iff [r.sup.d(a)] x [r.sup.d(b)] = [r.sup.d(c)]

then, relative to r, another relation of additivity, [symmetry] may be defined on Q as follows. For any a, b, and c in Q

a[symmetry]b=c iff [r.sup.d(a)] + [r.sup.d(b)] = [r.sup.d(c)].

Of course, + and [symmetry] are quite different. That is, it is not the case that for all a, b, and c in Q

a+b=c iff a[symmetry]b=c.

Yet [symmetry] is a relation of additivity (i.e. satisfies conditions of the same form as (1), (2), and (3)). Thus, if [less than] Q, + [greater than] is a continuous quantity, then so is [less than]Q, [symmetry] [greater than]. Since [symmetry] is relative to r it should be indexed, [[symmetry].sub.r], which emphasizes the point that for any quantity, Q, there are infinitely many structures of the form [less than] Q, [[symmetry].sub.r][greater than].

Now it is obvious why I equivocated earlier about defining + for length. Additivity for length is simply defined by form (conditions (1), (2), and (3)), as for any continuous quantity. The 'definition' given earlier is not a definition really. It is simply a hypothesis linking one of the infinitely many additive relations between lengths to an observable relation. This will not satisfy those for whom the additivity of lengths just is the conventionally understood composition of a distance out of discrete parts. However, the temptation to insist that additivity upon a continuous quantity is absolutely defined, as for numberable quantity, meets a counter-example in the case of relativistic velocity. In the case where the velocity of X relative to Z is b and Z relative to Y is d, it seems 'natural' to presume that the velocity of X relative to Y, a, equals b + d. There is nothing wrong with this presumption. Also, it seems 'natural' to define velocity as the ratio of distance to time. There is also nothing wrong with that definition. However, within relativistic physics, this definition is not consistent with that presumption. These two different views of velocity yield two different classes of velocity measurement scales: the 'non-additive', preferred scale whereby

v(a) = (v(b) + v(d))/(1 + (v(b)[center dot]v(d)))

(where v (a) is the measure of a with the velocity of light as the unit and velocity defined as the ratio of distance to time): and the 'additive', but not preferred, scale whereby

v[prime](a) = v[prime](b) + v[prime](d)

(where v[prime] (a) is the measure of a based upon the presumption that velocities add together under the kind of conditions described above). These tw scales are related by the non-linear but monotonic transformation

v[prime](a)=[tanh.sup.-1] (v(a)).

It should be noted that in the so-called 'non-additive' scale, additive relations still exist over the class of velocities. However, they are not generally recognized as such because they do not correspond to any easily observed operations of concatenation.

Thus, relative to any continuous quantity, there are infinitely many additive relations. The one used as a basis for measurement will be determined by some mixture of practical and theoretical considerations. The implication of importance here is that any association between the elements of Q and real numbers is not simply relative to a unit, it is also relative to a relation of addivity upon Q. A similar point is made by Ellis [1966] and Krantz et al. [1971].

Hence, for any pair, a and b, in Q, the ratio or b to a does not exist. What exists is the ratio of b to a relative to some additive relation on Q. This point may be made another way. Ratios may be thought of as based on pairs of infinite standard sequences. An infinite standard sequence for any a in Q is an unbounded ordered set, [S.sub.a]+, of elements of Q, such that

[S.sub.a]+ = [is less than][a.sub.1], [a.sub.2,] ... [is greater than]

(where for all natural numbers i, [a.sub.i]=ia). Then, for all i and j and any two standard sequences. [S.sub.b]+ and [S.sub.a]+, there is the set of relationships, {[b.sub.i] [is less than] [a.sub.j] and [b.sub.i][is greater tha or equal to] [a.sub.j]}, obtaining between [S.sub.b]+ and [S.sub.a]+ which determines the ratio of b to a. So ratios are relative to standard sequences, which in turn are determined by an additive relation upon Q. Thus, ratios in this context are really ternary and not binary relations, the third term being an additive relation. Equivalently, a ratio may be thought of as a binary relation between infinite standard sequences.

For any a and b in Q, the infinite standard sequences, [S.sub.a]+ and [S.sub.b][symmetry], are co-additive iff + and [symmetry] are identical additiv relations (i.e. for all a, b, and c in Q, a + b = c iff a[symmetry]b = c). The ratio between any two co-additive, infinite standard sequences falls between tw sets of rational numbers. Letting R ([S.sub.b]+,[S.sub.a]+) denote the ratio between [S.sub.b]+ and [S.sub.a]+, it is at least as great as any rational number in the set

[]={j/i[where][b.sub.i][is greater than or equal to][a.sub.j]}

(for all natural numbers i and j) but no greater than any rational number in th set

[]={j/i[where][b.sub.i][is less than or equal to][a.sub.j]}

(for all natural numbers i and j). Hence, it falls between these two sets. That is, if h/i is any element of B+ ba and j/k is any element of A+ ba then

h/i[is less than or equal to]R([S.sub.b]+,[S.sub.a]+) [is less than or equal to]j/k.

In this way, R([S.sub.b]+,[S.sub.a]+), is defined via a Dedekind cut. Thus, ove all [S.sub.b]+ and [S.sub.a]+ for any continuous, unbounded quantity Q, these ratios are the positive real numbers. (If Q is augmented by the complete set of differences between elements of Q, as suggested above, then the resulting ratio constitute a complete ordered field.)

In special cases, R ([S.sub.b]+,[S.sub.a]+) is a rational or natural number. If for some i and j, [b.sub.i]=[a.sub.j] then


and if for some i, b = [a.sub.j] then


So, the rational and natural numbers are subsets of the real numbers under this definition and all, alike, are ratios. It would be a trivial matter to redefine the naturals and rationals as ratios between infinite standard sequences based upon the numerable quantities. Such a definition would be equivalent to the one already given, for with numerable quantities only a single relation of additivity is possible.

The formal identity between such ratios and the positive real numbers, is, of course, necessary but not sufficient for the ontological identification suggested here. Arguments for making it have been advanced by Bostock [1979]. Such an identification makes it easy to explain why real numbers are applicable to ratios of continuous quantities. This is the primary application of real numbers in science. However, isomorphism is, again, sufficient to explain that fact. An argument must be advanced on ontological grounds (Bigelow [1988]). I presume that a theory according to which the real numbers are the kind of thing that exist in space and time is superior to its alternatives.

The advantage of defining the real numbers as ratios between infinite standard sequences of most relevance here is the extent to which it facilitates an understanding of measurement. On this view, measurement is the attempt to discover or estimate such ratios. Because human operations are finite, measurement deals only with finite standard sequences, rather than with the underlying, infinite standard sequences. A finite standard sequence for any a i Q, is a finite, ordered set

[S.sub.a,n]+[is less than][a.sub.1], [a.sub.2], ...., [a.sub.n][is greater than

(where n is a natural number). Relative to any two co-additive finite standard sequences. [S.sub.a,n]+ and [S.sub.b,m]+, R[[S.sub.b]+, [S.sub.a]+] is at least as great as any rational number in

[,mb] = {j/i[where][b.sub.i][is greater than or equal to] [a.sub.j]}

but no greater than any rational number in

[,mb] = {j/i[where][b.sub.i] [is less than or equal to] [a.sub.j]}

(for i=1, ..., m and j = 1, ..., n). Since both sets are finite they will contain maximum and minimum values and, so,

max ([,mb]) [is less than or equal to] R([S.sub.b]+,[S.sub.a]+) [is less than or equal to] min ([,mb]).

Therefore, the measure of b relative to a, +, and level of accuracy (n,m) is

[is less than] max ([,mb]), min ([,mb])[is greater than].

Some comments about this definition of measurement are in order, for it accommodates a number of well-known facts, neglected by most other definitions. The first is that measurement procedures generally give rise to interval rather than point estimates of ratios. This is a consequence of the logic of measurement itself and not just of observational error. Because n and m are finite, all irrational and some rational ratios will not be precisely specified by measurement procedures. The second is that measurement deals in the first instance with rational numbers rather than reals. While the ratios of infinite standard sequences are real numbers, only the rational approximation to these are attainable through measurement. Of course, calculation based upon measurements may yield real estimates of ratios.

Given that this is what measurement is, it becomes clear that there are two steps involved in devising procedures for the measurement of quantities. The first and logically prior step is establishing that the variable involved is really quantitative. The second is to develop procedures for discovering or estimating ratios between quantities relative to some additive relation. These two steps may be completed together. However, procedures which purport to measure a variable when the variable has not been shown to be quantitative may not measure anything. If such procedures are not sensitive to the presence or absence of additive relations then their use by itself does not amount to measurement. This fact was overlooked in many of the newer sciences (e.g. psychology) and the quantitative procedures there developed (e.g. for the measurement of intelligence) may not be measurement procedures at all, for the variables involved may not be quantitative.


Showing that a variable is quantitative is showing that the set of properties o relations involved has a special kind of structure. Now, as already mentioned, the laws specifying the quantitative structure of a variable (e.g. (1), (2) and (3), above) are like Mill's uniformities of coexistence in that they do not in isolation entail predictions about the behaviour of any objects. They simply show how some values or levels of the variable relate to other values or levels of the variable. What is required in order to test for quantity are specific hypotheses relating the allegedly quantitative properties or relations to the behaviour of objects possessing those properties or entering into those relations. Typically, these hypotheses will be specific to certain boundary conditions stating what other properties the objects must possess or relations into which they must enter in order for quantitative structure to be manifest i behaviour.

For example, we know that with a set of rods, and equipment for making new rods the conditions (1), (2), and (3) can be put to the test. Rods of different lengths can be put together in such a way that the length of the composite equals the sum of the lengths of the rods added. Letting 'XoY' denote the composite formed by concatenating X and Y end to end linearly and 'X[is similar to]Y' denote the relation between X and Y when each spans the other, additivity between lengths is then related to the behaviour of rods of manageable dimensions (under appropriate physical conditions) in the following way. If any rods X, Y, and Z are of lengths a, b, and c respectively (where a, b, and c are any humanly manageable lengths in Q, the set of all lengths) and if X and Y are concatenated and the span of XoY matched against Z, then

XoY[is similar to]Z iff a+b=c

(where a + b = c means that any rod whose length is c is entirely composed of discrete parts, whose lengths are a and b). The concatenation operation upon rods gives access to a particular additive relation between lengths.

Quantities like length, where an additive relation upon the quantity is more or less directly reflected in some operation of concatenation (e.g. time, electrical resistance, etc.), are known as extensive quantities. It was not clear how an additive relation upon a quantity could be tested for, in the absence of a suitable concatenation operation, until the theory of conjoint measurement was developed (Luce and Tukey [1964]). This theory shows how, in special circumstances involving one or more variables, the structure of an ordering enables a test for additivity. It shows how it is that non-extensive physical quantities (like density and velocity) are, none the less, additive an promises, in sciences where extensive quantities are non-existent, tests for quantity where hitherto such tests have not been made.

Conjoint measurement can be understood as a way of identifying standard sequences on one or more quantities, without prior identification of a relation of additivity. Consider the three variable case as the paradigm. If two quantities (call them A and X) are non-interactively related to a third (P) the differences between elements of A and between elements of X may be equated in terms of their effects upon P. Thus, given a fixed difference between values of X, say, a standard sequence on A may be identified via contiguous differences o A, all equal to the same difference on X. This can be done provided that elements of A, X, and P can be identified as equal or unequal, and elements of can be ordered. It requires no prior measurement upon any of the quantities involved.

The ideas can be explained more simply via an example. Suppose one could not measure mass, volume, or density, but only judge equality on each and, as well, order masses. Volumes, for example, could be judged equal or not by assessing the height of a column of liquid displaced in a standard beaker: density, by identifying the kind of substance involved; and mass could be ordered via a simple beam balance. Then, if the mass determined by density and volume, [d.sub.1] and [v.sub.2], equalled the mass determined by [d.sub.2] and [v.sub.1], and both were greater than that determined by [d.sub.1] and [v.sub.1], it could be concluded that the difference between [d.sub.2] and [d.sub.1] has the same effect upon mass as the difference between [v.sub.2] and [v.sub.1]. Similarly, if the mass determined by [d.sub.3] and [v.sub.1] equalle the mass determined by [d.sub.2] and [v.sub.2], then the difference between [d.sub.3] and [d.sub.2] has the same effect upon mass as that between [v.sub.2] and [v.sub.1], and hence, as that between [d.sub.2] and [d.sub.1]. What one is beginning to identify here is a standard sequence upon density differences, [is less than][d.sub.2]-[d.sub.1], [d.sub.3]-[d.sub.1], ... [is greater than]. Now, it would be interesting to know what conditions the ordering upon masses must satisfy for such standard sequences to exist and be consistent across changes i the unit volume difference ([v.sub.2] - [v.sub.1]) against which they are matched.

It would seem obvious that it must be the case that for any a in A and x, y in X, there exist b in A such that (a,x)=(b,y) (where (a,x) and (b,y) denote the elements of P determined by those conjoined elements of A and X). And, similarly, for any a and b in A and x in X there must exist B in X such that (a,x)=(b,y). If this were not true then there might be gaps in or upper bounds upon standard sequences. This condition is called solvability for obvious reasons.

Secondly, no differences on either A or X can be infinitesimally small or infinitely large relative to others, for then there would be no ratios between some pairs of infinite standard sequences. This is an Archimedean condition. Th statement of an Archimedean condition usually requires specifying an additive relation. This is here done for A and the specification for X is precisely analogous. For any [a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4], [a.sub.5], and [a.sub.6] in A

([a.sub.2]-[a.sub.1])+([a.sub.4]-[a.sub.3])=([a.sub.6]-[a.sub.5])iff for any [x.sub.1], [x.sub.2], and [x.sub.3] in X,

([a.sub.2],[x.sub.1]) = ([a.sub.1],[x.sub.2]), ([a.sub.4],[x.sub.2]) = ([a.sub.3],[x.sub.3]) and ([a.sub.6],[x.sub.1]) = ([a.sub.5],[x.sub.3]).

The Archimedean condition, then, is that if [a.sub.1], [a.sub.2], and [a.sub.3] are any three elements of A, there exists a natural number n such that

n ([a.sub.2]-[a.sub.1])[is greater than or equal to] ([a.sub.3] - [a.sub.1])

(where, of course, for any [a.sub.1] and [a.sub.2] in A, 1([a.sub.2]-[a.sub.1])=[a.sub.2]-[a.sub.1] and (n + 1) ([a.sub.2] - [a.sub.1]) n ([a.sub.2] - [a.sub.1]) + ([a.sub.2] - [a.sub.1])).

Thirdly, these sums of differences on one factor must retain whatever parity th part differences have with differences on the other factor. Given our definitio of additivity, this condition reduces to a condition now known as double cancellation. That is, for any [a.sub.1],[a.sub.2], and [a.sub.3] in A, [x.sub.1], [x.sub.2], and [x.sub.3] in X, if

([a.sub.2],[x.sub.1])[is greater than or equal to]([a.sub.1],[x.sub.2]) and ([a.sub.3],[x.sub.2]) [is greater than or equal to] ([a.sub.2],[x.sub.3]) then ([a.sub.3],[x.sub.1])[is greater than or equal to] ([a.sub.1],[x.sub.3]).

Krantz et al. [1971] show that these three conditions are sufficient for differences within the three variables A, X, and P to be quantitative.

The solvability and Archimedean conditions cannot be directly tested because they assert the existence of something which, even if not found by finitary procedures, may still exist. A theorem proved by Scott [1964] gives an equivalent set of conditions, each of which is directly testable, but because the set is infinite, it can never be tested in its entirety. These conditions are an infinite hierarchy of cancellation conditions, similar to double cancellation, but varying in the number of antecedent inequalities involved. So while the fact that a variable is quantitative cannot be proved from any finite set of data (hardly a surprising state of affairs), the hypothesis that a variable is quantitative can be put to the test.

The discovery of the theory of conjoint measurement is interesting for a number of reasons. In the first place, it means that quantitative science is not dependent upon extensive measurement, and that is important for sciences like psychology, which aspire to be quantitative but lack extensive concatenation operations. Secondly, it shows how the additive relation identified in quantification is, in part, a function of the capacities of the scientist. Imagine a creature with different sensory-motor capacities. An intelligent slug-like creature that extends its body to cover exactly the area of a surface could sense which of two areas is the larger via the tension involved in so stretching its body. This enables it to order the areas of rectangular surfaces In doing so, it notes that area seems to increase with two other dimensions, length and breadth: for rectangles of the same length, increasing area changes breadth and our creature infers that this must mean an increase in breadth. Similarly, it reasons that for constant breadth, length increases with area. Armed with that knowledge, the slug inquires into the contribution of length an breadth to area. It finds that increasing length from, say, [l.sub.1] to [l.sub.2], for fixed breadth, [b.sub.1], has the same effect upon area as increasing breadth from [b.sub.1] to [b.sub.2] with length, [l.sub.1]. Hence, i concludes [l.sub.2] - [l.sub.1] = [b.sub.2] - [b.sub.1], and has begun the journey toward the construction of standard sequences for differences in length breadth, and area and, hence, toward the measurement of such differences.

Interestingly, however, the additive relation for length which it would discove by this method is not the same as the one we have discovered via concatenating rods. For ours leads to the relations, area = length x breadth, while the slug would discover that area = length + breadth. Of course, as has been noted earlier, both relationships are correct relative to their respective additive relation and the slug-measures are simply logarithmic transformations of ours.

So the quantitative laws we discover are, in part, determined by our sensory-motor capacities, the features of the environment that we are sensitive to and the objects that we are able to manipulate. Practical considerations, however, are not the only ones. When, in conjoint measurement, an additive relation is defined upon differences as above, another element of arbitrariness is involved. That definition is quite proper, and the relation so defined not only exists but also satisfies conditions (1), (2), and (3) (given solvability, the Archimedean condition, and double cancellation). But a different definition could have been given. For any [a.sub.1], [a.sub.2], ..., [a.sub.6], in A,

([a.sub.2]/[a.sub.1]) x ([a.sub.4]/[a.sub.3])=[a.sub.6]/[a.sub.5] iff for any [x.sub.1], [x.sub.2], and [x.sub.3] in X

([a.sub.2],[x.sub.1])=([a.sub.1],[x.sub.2]), ([a.sub.4].[x.sub.2])=([a.sub.3],[x.sub.3]) and ([a.sub.6],[x.sub.1])=([a.sub.5],[x.sub.3]).

These ratios of elements of A presuppose different relations of addition to tha defined earlier and if any ratio [a.sub.2]/[a.sub.1] (where [a.sub.2] [is greater than] [a.sub.1]), is set equal to r (any real number [is greater than] 1) then a specific relation of addition is identified. Theory may guide one her as much as practice. A prior belief that area = length x breadth, no matter if it has a practical basis or not, would have led to the identification of a different additive relation to that found by our slug. It preferred a theory in which differences in area (for fixed breadth) directly reflected differences in length. Either theory does the same work as the other, but differently.

The practice of measurement, in which an instrument or procedure is used to estimate ratios (or intervals within which ratios fall), cannot then be devised independently of a commitment to a specific additive relation between elements of the quantity involved. Hence, it is unlikely that measurement of some quantity can sensibly proceed from the development of instruments to the identification of additive relations. Needless to say, certain sciences have proceeded in this manner. In attempting to become quantitative they emulated th quantitative sciences in the superficial respect of developing instruments that produced numerical outputs. However, they did not emulate them in identifying additive relations upon the variables to be measured. This was the basis of Campbell's [1940] criticism of psychological measurement. It still has some force (see Michell [1990]).

Department of Psychology University of Sydney


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Author:Michell, Joel
Publication:The British Journal for the Philosophy of Science
Date:Jun 1, 1994
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