Norman Campbell's Theory of Fundamental Measurement

as described in
"The application of the Theory of Physical Measurement to the Measurement of psychological Magnitudes, with Three Experimental Examples."
By Thomas Whelan Reese. Psychological Monographs. Vol. 55:3, 251, 1943. pp. 6-20.

SECTION A. INTRODUCTORY

T.W. Reese: RECENTLY a number of physicists (and logicians) together with a group of psychologists have leveled a particularly vigorous attack against the theoretical concepts upon which psychologists have based their practice of measurement. The criticism has come to a head with the recently published Final Report of the Committee appointed to consider and report upon the possibility of Quantitative Estimates of Sensory Events (14). The members of this committee were drawn from Sections A (Physics) and J (Psychology) of the British Association for the Advancement of Science.

In the following sections the criticisms of both the physicists and the psychologists will be examined in some detail. Suffice it to say here that the physicists have claimed that measurement in any true sense is impossible in psychology. They base this conclusion on what they consider to be the fact that none of the attempts at measurement in psychology meet the necessary logical requirements for fundamental measurement.

They argue that psychologists must then do one of two things. They must either say that the logical requirements for measurement in physics, as laid down by the logicians and other experts in the field of measurement, do not hold for psychology, and then develop other principles that are logically sound; or they must admit that their attempts at measurement do not meet the criteria and both cease calling these manipulations by the word "measurement" and stop treating the results obtained as if they were the products of true measurement

For example Guild, who seems to have taken the most extreme position against the possibility of measurement in psychology, says, "To insist on calling these other processes [i.e., the method of equal appearing intervals and the method of just noticeable differences] measurement adds nothing to their actual significance, but merely debases the coinage of verbal intercourse. Measurement is not a term with some mysterious inherent meaning, part of which may be overlooked by the physicists and may be in course of discovery by psychologists. It is merely a word conventionally employed to denote certain ideas. To use it to denote other ideas does not broaden its meaning but destroys it: we cease to know what is to be understood by the term when we encounter it; our pockets have been picked of a useful coin" (13).

The Final Report of the committee of the British Association for the Advancement of Science holds out hope for a third solution, when in paragraph to it states, "Some members, perhaps all, admit that their opinion might change if new facts were established; but the facts that would be necessary for this purpose are not of the kind that can be established by any experimental method at present in general use" (14). {This was before the Rasch model.}

For convenience the discussion will be divided into sections. The logical criteria which the physicists claim that all measurement must meet will be presented and discussed in Section B; the practical operations necessary for the fulfillment of these criteria will be discussed in Section C; the position of Stevens, who has given the most vigorous reply to the logicians' criticisms, will be discussed in Section D; the specific criticisms of the psychological operations for measurement which have been raised by the physicists will be presented and discussed in Section E; the criticisms of psychologists of their own methods will be presented and discussed in Section F; the special problem of zero subjective magnitudes will be discussed in Section G; it will be shown in Section H that measurement in psychology does not depend on the prior measurement of any other magnitude; Section J contains a brief summary of the discussion up to that point.

SECTION B. THE LOGICAL REQUIREMENTS OF MEASUREMENT

[In this section the author has borrowed liberally from Campbell (5, 6, 7), Guild (13) and Cohen and Nagel (9). No references are given except in those cases where an author is quoted directly or an illustrative example employed by the author is used in the text.]

A distinction must be made at the outset between "measurement defined as the construction of a scale" and "measurement defined as the use of the scale after it has been constructed." Utter confusion will result from the confounding of these two definitions. The use of a measuring scale after it has been constructed is a more or less simple matter involving the comparison of the object to be measured with the standard scale. The word measurement is never used in this sense in this paper. As here used it refers to the more fundamental problem of scale construction.

Measurement, according to Campbell, is the assignment of numerals to systems according to scientific laws. [Campbell apparently uses the term "system" to mean any objects with which the physicist deals. The term would seem to include anything from pieces of wood or rock to electric lamps and voltage dividers.] The scientific laws spring from the relations demonstrated between the systems with respect to a certain magnitude.

The first requirement for measurement is that it must be possible to arrange the systems to be measured in respect of a given magnitude, in an order, with respect to that magnitude. The result of this operation is known as an ordinal scale. To do this it must first be demonstrated by some operations that the relation between the systems is transitive and asymmetrical.

If the symbol -> means "bears a certain relation to" and # means "does not bear that relation to," it must then be shown experimentally that the relation in question is asymmetrical, that is, if

A -> B then B # A   I (I)
and transitive, that is, if
A ->B and B->C, then A->C   II (I)

If the above symbols are replaced by > (greater than) and <= (not greater than) or by the converse < (less than) and >= (not less than), it would be necessary to show that, if

A > B then B <= A   I (II)
and that if
A > B and B > C, then A > C.   II (II)

It will be seen that the relation = does not exist in such a series. An example of such a series without the relation =, given by Campbell, is the direct line of male descent. The generating relations are "ancestor of" and "descendant of."

The relation =, as between A and B, is associated with the following propositions:

A = B if, and only if,
1) A>=B and A<=B   III
2) if A > C then B > C   IV
3) if A < C then B < C   V

The relation = defined in this manner is always transitive, that is, if

A = B and B=C then A=C   VI

and is always symmetrical, that is, if

A = B then B=A.   VII

On casual inspection it might seem difficult for a given relation to satisfy III and yet fail to satisfy IV and V but, if the example of Campbell's, given above, is examined it will be seen that it would be possible for A to be neither the ancestor nor the descendant of B and thus satisfy III, and yet A cannot then satisfy IV or V. To quote Campbell "... no two males can have all the same ancestors and descendants" (6).

According to Campbell a magnitude must have the relation =, for, as will be seen later, this relation is necessary in the construction of an additive scale.

He sums up the first conditions for measurement as follows: "The first condition of measurement, namely that a magnitude must be capable of order, can now be stated formally as follows. The systems measured must, in virtue of the property concerned, be a field of a pair of converse T.A. [transitive asymmetrical] relations and the T.S. [transitive symmetrical] relation associated with them; every system must be > or < or = every other, and must be = at least one other. The first law of measurement is the statement that this condition is fulfilled" (6).

The rule for assigning numerals to represent a series in which the above relations have been established is: if A > B then the numeral assigned to A must be greater than the numeral assigned to B; conversely, if B < A, then the numeral assigned to B must be less than the numeral assigned to A. If A = B, then the numeral assigned to A must be the same as the numeral assigned to B. [The problem of how a numeral, which has been defined as a symbol, can be greater than another numeral is taken up later.]

According to Campbell, the existence of the relation = is one of the things that distinguishes the order characteristic of magnitudes from that which is characteristic of numerals. Numerals, by which is meant simply a group of conventional signs or marks on a piece of paper, obtain their order by convention. The order is not determined by facts such as the order existing in the family tree. If only one of the many numeral series is used, every member is either greater or less than every other member. There is no relation =. (Naturally if several series were combined, such as the decimal and fraction series, than it would be possible to find two that were equal to each other, as 1.5 = 1½.)

However the most important difference between numerals and magnitudes is that the order of the numerals is conventional while the order of the systems in respect of the magnitude is determined by experimental operations.

Numerals have by convention a transitive, asymmetrical relation, Now if they are going to be used to represent the order of the systems in respect of a certain magnitude, it must be shown experimentally that the relation between the systems which they represent is also transitive and asymmetrical. If it is impossible to show this then the numerals that have been assigned are meaningless in as much as the conventional relations between them do not express the relations between the systems.

There is nothing in the experimentally established relation A > B that tells what numeral is assigned to A. The rule simply states that it must be greater than that assigned to B. As yet there are no operations to determine by how much A > B, so the assigned numerals cannot reflect a relation that has not been established. In other words, if 2 is assigned to B and 4 to A, it is impossible to say that A is twice as great as B because it has not yet been shown experimentally that A is twice B. In other words, the numerals can express only those relations that have been shown experimentally to exist between the systems to which they are assigned.

An interesting example of an ordinal scale is the Mohs scale of hardness. Mohs, in developing his scale for the hardness of minerals, tried to ordinalize minerals according to the relation "scratches." The operation by which the relation of the minerals was to be determined was to attempt to scratch one mineral with another. He selected ten minerals to represent particular points on the scale and assigned numerals to them. The numerals ranged from 1 to 10, where the numeral 1 was assigned to that mineral which could be scratched by every other mineral and which could scratch no mineral; and 10 to was assigned to that mineral which could scratch every other one and be scratched by none. It should be noted that there is nothing in the operations adopted by Mohs that tells how many more times as hard one mineral is than another. It simply tells that one mineral is harder than another, as defined by the operation of scratching, and therefore should be assigned a higher numeral.

Later attempts at measuring hardness, defined by other operations, such as microscopic measurement of the depth of a scratch made by a diamond under constant pressure or the amount of work done in grinding away a certain weight or volume of material, have shown that the interval between Mohs' hardness of 10 and 9 was greater than the interval between 9 and 1 (37).

Mohs assumed that his relation of "scratches" was transitive and asymmetrical and that the = associated with it was transitive and symmetrical. This has been shown to be false. Some minerals have been found that satisfy III but not IV or V, that is they cannot scratch each other, and yet have different powers of scratching a third mineral. Because, of this, according to Campbell, hardness as defined by the operation of scratching is not a magnitude at all.

In order to determine what numeral should be assigned to A if A > B or, in other words, in order to be able to construct an additive or extensive scale, the property being scaled must be capable of being "added." The second requirement for measurement, then is that it must be possible to find some operation by which the magnitudes of two = systems may be combined to form systems that are <>. In order to be additive the proposed method of combination must meet the following conditions: if

A = A' and B > 0, then A + B > A'   VIII
A + B = X then B + A = X   IX
A = A' and B = B' then A + B = A' + B'   X

and

(A + B) + C = A' + (B' + C').   XI

To quote Campbell again, "The statement that these conditions are fulfilled by any proposed method of addition defined by + and ( ), applied to systems possessing any magnitude defined by >, <, and =, is the second law of measurement of that magnitude" (6).

It is now necessary to have a rule for the assignment of numerals to the systems to represent the new relations that have been obtained by the operation of addition.

In discussing the assignment of numerals it is well to stress again that it is numerals that are assigned and not numbers. As Campbell says, "... it would be difficult to avoid the impression that the conception of number and the rules of arithmetic were concerned in the matter, Actually they are not concerned. Of course, they are closely connected with measurement; but if we fail to recognize that they are not essential we shall not understand the connection" (6).

The first operation for the construction of an additive or extensive scale is to select some system that belongs to the series of the magnitude. The selection is entirely arbitrary. Then another system is found that is = to this first system. If the numeral A is assigned to the system first selected, then the numeral A' is assigned to that system that is = to A. Since the systems must be identifiable, that is since it is absolutely necessary that they can be told apart by some method, it will be convenient to call the second system A' to indicate that it is of the same magnitude as the system A but is a different system. The ' is not to be taken as an indication of a different numeral but of the same numeral assigned to a different system.

The next step is to "add" the systems A and A' and seek a system that is = to their combination. To this system the numeral B can be assigned. Another system that is = to B is then found and assigned the numeral B', B and B' are "added" and another system C is found such that B + B' = C, and so on.

Using the ordinary numeral series instead of the alphabet and arbitrarily assigning 1 to the systems previously assigned A, B would equal 1 + 1 or 2. Likewise C = 2 + 2 or 4. Naturally numerals may be assigned to intermediate systems. If a system A" is found that = A' = A, then it is theoretically possible to find a system = A + A' + A" to which the numeral 3 would be assigned.

It is easily seen that a great advance has been made when it is possible to construct an additive scale. The ordinal scale did not tell by how much A > B because there was nothing in the relations established by experiment that determined by how much A > B. But now, once having chosen a standard, all the other magnitudes are uniquely determined. It is known for example that C is not only > B but also that C = B + B', because the operations performed on the systems have determined this relation experimentally.

It is now time to examine more closely the rules for the assignment of numerals. Before this is done it will be necessary to clear up some questions of terminology. Much of the difficulty of the subject of measurement seems to stem not only from the confusion of the meanings of the words number and numeral, but also from the fact that the word number itself has several meanings. The following definitions have been adopted in this paper.

Numeral. A numeral is a sign or symbol that may be conventionally used to represent a number. In other words it is simply a black mark on a piece of paper. There are several conventional numeral series, 1, 2, 3, etc., or A, B, C, etc.

Number. {Bertrand} Russell's definition of number is not used in this paper. Number is here regarded as a discriminable characteristic of systems that may be measured as any other discriminable characteristic. The term is used in much the same way as Stevens (40) has used the term "numerosity." "Numerosity," he says, "is a property defined by certain operations performed upon groups of objects." In his discussion he begins by saying that it is possible to establish a rank order of groups of objects (beans for example) in respect of numerousness [subjective number] simply by looking at the piles of beans and judging which of the piles is largest, etc.

"We know from experience, however, that greater reliability can be had if we rank-order the groups by pairing successively one bean from each group until one group is exhausted. Then if any beans remain in the other group, that group is said to have the greater numerosity. ... If the pairing exhausts both groups simultaneously, their numerosity is equal ...," etc. It can be seen that number defined in this way can be measured fundamentally (see also Campbell, 6).

As this is not the usual definition of number it has been decided to call this kind of number "objective number" to distinguish it from number as a logical concept as used by Russell and from subjective number (numerousness).

It will be remembered that systems arranged in an experimentally established order had numerals assigned to them by the following rule: If A > B, then the numeral assigned to A must be greater than the numeral assigned to B and conversely if B < A then the numeral assigned to B must be less than the numeral assigned to A. Furthermore if A = B, the numeral assigned to A must be equal the numeral assigned to B.

It is now possible to ask the question how can a numeral, which has been defined as a mere sign or symbol, have a magnitude? In short how can one numeral be > or < or = any other numeral? One possible answer has already been implied, when it was assumed that the numerals used were those that are conventionally arranged in an order. In other words > with respect to numerals, means "following" in the numeral series. Thus 2 > 1 because it follows 1 and D > B because it follows B in the numeral series.

However it is extremely important to note that it is not necessary to use a conventional numeral series in the construction of the scale. Any other group of numerals would do as well. In the event that a group of numerals which did not have a conventional order were used in the construction of the scale, the numerals would be arbitrarily assigned to the various systems in the experimentally established ordinal series. But once having been assigned, their order, so far as measurement of the particular magnitude is concerned, would be uniquely determined by the order of the systems to which they were assigned. In other words, though the numerals did not originally possess an order, once they have been assigned to a group of systems, they represent the relations that have been determined experimentally between the systems in the ordered series. If the relation between the systems is transitive and asymmetrical, then the numerals express this transitive and asymmetrical relation; if the relation between the systems is intransitive and symmetrical, the numerals express an intransitive symmetrical relation. Whatever relations the numerals express, they express only by virtue of the fact that these relations were shown to exist between the systems to which they have been arbitrarily assigned.

It should be noted that this statement is also true if numerals with a conventional order are used. It so happens that by convention such a series as i, 2, 3, etc., is transitive and asymmetrical and it so happens that the relation that must first be established between the systems is also transitive and asymmetrical. In other words both are an ordered series, one is ordered by convention and the other by experimental operations. For convenience, then, we assign numerals to the systems so that if A > B, the numeral assigned to A is greater that the numeral assigned to B, always remembering that the relation "greater than" when applied to numerals is a matter of convention.

There is nothing in the size of the numeral assigned to A that makes A greater than B, For example if A > B and the numeral 2 is assigned to B, then any numeral that is conventionally greater than 2 may be assigned to A, say 4. But suppose that A > B and the numeral 2 is arbitrarily assigned to B and the numeral 1 is assigned to A, The fact that 1 is assigned to A does not now make A less than B, In fact it works the other way; the fact that A has been shown to be greater than B means that the numeral 1 assigned to A must be interpreted as greater than the numeral 2 assigned to B. A new series is brought into being by these operations in which the numeral 1 is "greater than" the numeral 2. This means that, that so far as this particular magnitude is concerned, the numeral series 1, 2, 3, etc,, cannot be used or interpreted in the usual way, i.e., 2 > 1, etc.

The seeming absurdity of 1 > 2 arises because one is accustomed to think of 1 and 2 as numbers, not numerals. The number conventionally represented by the numeral 1 is certainly not greater than the number conventionally represented by the numeral 2, but the numeral 1 may be regarded as either greater or less than the numeral 2 depending on the magnitudes of the systems to which these numerals are assigned. This would be highly inconvenient and it is much more reasonable to use the conventionally ordered series. But it is also important to see that it is not necessary to use the conventional series, in order to make clear the fact that the numerals add nothing to the experimentally determined relations.

The same reasoning that applies to the use of nonconventional numerals to represent systems in an ordered series applies to their use for representing the systems in an additive series. For example it was said. that if B = A + A, and the numeral 1 is assigned to A, then the numeral 1 + 1 or 2 would be assigned to B. But what meaning can be assigned to the statement, "the numeral 1 + the numeral 1 = the numeral 2"? There is nothing about the numerals 1 and 2 qua numerals [e.g., on car number plates] that would justify the conclusion that 1 + 1 = 2. However as convention numerals, 1 + 1 = 2 because the numeral 1 has been conventionally assigned to a certain objective number and the numeral 2 has been assigned to another objective number. Furthermore if we call the number to which the numeral 1 is assigned X and the number to which 2 has been assigned Y, and it is possible to show by a series of operations involving addition that Y = X + X', then it is possible to state that 1 + 1 = 2. But it must be carefully borne in mind that so far as measurement is concerned this numerical statement has no meaning apart from the experimentally established relations between the objective numbers X and Y. When the criteria for measurement have been met the relations between the systems are analogous to the relations between objective numbers which are represented by the ordinary numeral series 1, 2, 3, etc. It is then possible to use the ordinary numeral series in its conventional sense and apply the powerful tool of arithmetic to the symbols with the knowledge that these arithmetic manipulations represent, with only a small margin of error, the actual physical operations that might be performed on the systems themselves.

But it is not necessary to use the conventional numeral series, it is only convenient to do so. Any other numeral series, or any other group of numerals could be used, though it would be necessary to construct a new arithmetic, i.e., new laws for the manipulation of numerals, if it was necessary to manipulate these numerals rather than perform actual operations on the systems.

The procedure outlined above for the construction of an additive scale results in what Campbell names an A-magnitude, also sometimes called a fundamental magnitude. An A-magnitude (or fundamental magnitude) is one for which a practical operation of addition may be found. There is another larger and very important group of magnitudes which are called by Campbell, B-magnitudes, also sometimes called derived magnitudes. A B-magnitude (or derived magnitude) is one that is measured in terms of an A-magnitude. It cannot be measured directly because it is impossible to find a practical operation for addition that will meet VIII, IX, X, XI. B-magnitudes may be of two kinds. To quote Campbell (6), "The property measured in this manner may be nothing but that of being subject to the numerical law, and may be indefinable apart from that law. But it has often happened that the discovery of a numerical law, and of the constants associated with it, has enabled us to measure in this way a property that had previously been suspected of being a magnitude, but had not been actually measured."

The example usually given to illustrate B-magnitudes is density. Density = mass/volume. Both mass and volume are A-magnitudes. As Campbell shows, it is suspected that density is a magnitude because liquids might be arranged in an order of magnitude by defining "denser than" by floats on. It could be shown that this relation is transitive by showing that if A floats on B and B floats on C, then A will float on C. It could also be shown that the relation is asymmetrical. If A floats on B, then B does not float on A. Equality could be defined as that state in which neither liquid will float on the other permanently.

If liquids are then arranged in an order defined by the quotient mass/volume and this order is identical with the order obtained by defining density by flotation, it is possible to say that the property measured by the quotient mass/volume is the same as the property measured by flotation. To quote Campbell, "... the discovery of the law [i.e., the fact that mass/volume is a constant for given liquids under defined conditions] has enabled us to measure a property previously immeasurable"(6).

It may be asked why the quotient mass/volume is thought to measure the same magnitude that is measured by flotation. Campbell lays down the general principle that "the conception of a magnitude is inseparable from that of the order characteristic of it. It is natural, therefore, to regard as the same magnitudes, or as magnitudes of the same kind, properties that invariably have the same order"(6).

Temperature is often mentioned as an example of a magnitude which is measured without a practical operation for addition. Guild (13) defines temperature as "the condition of an object in virtue of which it may feel hot or cold to the touch ...." He states further that "Experiment has shown many observable relations of a general kind between the temperature of bodies and their measurable properties. The length and electrical resistance of a given rod, for example, are usually greater when the rod feels hot than when it feels cold."

In order to measure temperature as a B-magnitude it is only necessary to choose some measurable property which varies continuously with it. This having been done it is possible to postulate a law relating the property chosen to temperature. For example it is possible to postulate the law that equal increments in the chosen A-magnitude represent equal increments of temperature.

The mercury thermometer is an example of such a postulated law, Equal increments of the volume of mercury are deemed to represent equal increments of temperature. It is obvious that the relation of temperature to volume of mercury must be constant. If this were not true the scale would be useless. But how is it possible to tell whether the relation is constant? As Guild points out, it is impossible to determine the constancy of this relation by finding out whether various other phenomena bear a constant relation to temperature as defined by the mercury scale, That reasoning "is based on an a priori assumption of the constancy of natural laws" (13). Guild's answer to this question is, "The point is that the constancy of the law defining our scale does not require confirmation. It is not an assumption which may or may not be true, it is a postulate forming part of the conventional framework of physical measurement. The postulated law is necessarily always true for the simple reason that it serves the purpose of defining temperature as `the thing for which this law is true.' There is no criterion of the magnitude of a temperature (nor of any B-magnitude) other than the law by which we choose to define it. It would therefore be meaningless to ask whether the temperature to which our scale assigns the numeral n is in fact the same temperature at all times and places"(13).

The physicists seem to wish to restrict the term measurement to those magnitudes that may be measured fundamentally, i.e., A-magnitudes, For example Guild says (13), "The fact that there is no operation of addition applicable to temperature qua temperature, prevents it from being measurable in the true sense of the term," But it should be noted well that he also says (13), "When once we have defined some such scale of temperature, temperature becomes `measurable' in the broad sense in which this word is generally used; and the laws relating other physical variables with temperature as so defined become open to empirical investigation."

The author of this study thinks that the words "open to empirical investigation" might also have been put into italics.

Before closing this section it might be well to give Campbell's definition of zero magnitudes although any discussion will be postponed to Section C.

The system B has the magnitude 0, when A = A', if

A + B = A'   XII

SECTION C. THE OPERATIONS NECESSARY FOR FULFILLING THE PHYSICISTS' REQUIREMENTS FOR MEASUREMENT.

In the section above no stress was laid on the necessary operations for meeting the criteria for measurement that were discussed, The discussion of the criteria and the discussion of the operations have been separated only for convenience. Actually the criteria and the operations by which the criteria are satisfied are inseparable. It would be possible to describe the criteria that must be met, it would also be possible to describe the operations that ought to be performed on a group of systems in order to measure them, and still it might be impossible to measure the systems in respect of the given magnitude, because the operations could not be carried out in practice. The criteria are not theoretical, they are practical. The relations stated in them must be shown to hold empirically.

In glancing back at the criteria it will be noted that the following symbols were used: >, <, >=, <=, =,<>, () and +. The parentheses refer to a single system = to the sum of the systems included within them. Of these symbols, () and + may be described as operations and the rest as relations. That is >, <, <=, >=, =, and <> state that the relation "greater than," "less than" or "equals" has been found to exist or not to exist between any two systems. While it is true that these symbols do not represent operations, they imply that certain operations have been performed on the systems so that this relation could be determined.

For example, in order to determine whether the relation > existed between two systems with respect to any magnitude it would be necessary to:

1) State the operations by which > is to be defined.
2) Actually perform these operations.
3) Judge whether the operational criterion has been met.

If the magnitude were weight these three steps might be applied as follows:

1) Heavier than is defined by placing two objects on a balance, one on each pan. If one of the pans sinks, the weight in that pan will be deemed to be the greater; or, in other words, the system in that pan will be greater than the system in the other pan in respect of the property weight.

2) It is now necessary to find a balance, obtain a group of systems, place pairs of them on the balance; or, in other words, actually carry out the operations used to define the magnitude.

3) There must, of course, be some way of determining whether the pan sinks or does not sink. There may be several ways of determining this fact, but all of them will ultimately rest upon a judgment made by the experimenter. The usual judgments are those of "difference" (which can be either "greater than" or "less than," or "no difference.") It should be noted again that the judgment of "no difference" is not the same as the judgment of equality. The judgment of no difference is implied in III. But in, order to establish equality, it must be demonstrated that IV and V also hold.

The question may now be asked, why does one choose one operation rather than another? How, for example, does one know that the weights should be put in opposite pans of the balance? Why not place one weight in the pan of the balance and the other on the floor? Could not this operation define the relation "greater than" for the magnitude weight?

The answer is, simply, "Try it." Suppose "greater than" is defined in this fashion. It would soon be apparent that the relation established between the systems by these operations is not asymmetrical, though it is transitive. In other words, if the operations are actually tried, it will soon become evident that it is impossible to obtain both an asymmetrical and a transitive relation. In short, weight as defined by this operation is not a magnitude. The "correct" operations are those by means of which the necessary relations may be experimentally demonstrated. The "correct" operations may be found if the experimenter is ingenious and patient. Furthermore some magnitudes now thought to be fundamentally measurable may turn out not measurable; others now not measurable will turn out to be measurable when some ingenious experimenter discovers the "correct" operations.

>, <, etc., refer to relations that are based upon operations for their establishment but + refers directly to an operation. () also refers to an operation, or, perhaps better, to the result of several operations.

() is used in the sense that (A + B) = that single system that is equal to the combined systems A and B. It is clear that () refers to the result of several operations one of which is addition.

The operation for "addition" is the greatest single stumbling block to measurement in physics and psychology. [In the following discussion it must be remembered that addition may refer to a logical concept or to an actual set of physical operations. We are interested in addition in this latter sense. When the word is used in this sense it will be set in quotes.] In fact the physicists claim that measurement of sensation must almost always fail because the psychologist can hardly ever find a proper operation for +. Smith (36), Cohen and Nagel (9) and Johnson (28) have stressed the fact that this is not only true for sensation but also for mental testing, the measurement of attitudes, etc. {Ingenious inventors have now found additive operations for addition in mental testing, see "Rasch Model from Campbell Concatenation."}

It will be well worth while to examine the objections raised by the physicists, as it will shed a good deal of light on the criteria for additivity. Comment on the application of these criteria to the measurement of sensation and to the field of mental testing will be withheld until a later section.

The most important criterion that the psychologists fail to meet is that of "physical juxtaposition" {concatenation}. An example of physical juxtaposition would be placing the systems end to end in measuring length; placing the weights on the same pan of the balance in measuring weight; connecting resistances in series in measuring electrical resistance, etc. Campbell (14, 7) would admit that the simultaneous presentation of two auditory stimuli to different ears would satisfy this criterion of physical juxtaposition. So too brilliance might be "added" by allowing the light from two sources to fall on the same surface. Having met this first criterion of physical juxtaposition it would then be necessary to show that VIII, IX, X and XI held. As Johnson (28) points out, they certainly do not hold for brilliance in all cases.

It seems that it was this criterion of "physical juxtaposition" that was the stumbling block to any agreement between the physicists and psychologists on the Committee of the British Association for the Advancement of Science. The Committee reported that agreement seemed unattainable on the question of whether it was possible to make quantitative estimates of sensory events because, to quote Bartlett (8), "If all measurement must conform to the Laws of Measurement enunciated by Dr. Campbell, and, in particular, if the second law can only be satisfied by the physical juxtaposition of equal entities, then sensation-intensity cannot be measured."

Where has this new requirement for measurement come from? It will be remembered that it is not discussed with the logical requirements for measurement. Although this requirement might be called an "operational requirement," there certainly must be some logical basis for its inclusion as one of the criteria that the operation of addition must meet.

It is impossible to find a clear concise statement of what is meant by "addition" in any of the publications of the physicists that have been mentioned. It impossible to find a single italicized sentence beginning "Addition is ..." However it is possible to combine several statements made by Campbell and obtain a good idea of what he considers addition to be. But first it can be said at it is obvious that "addition" is no single operation that may be applied to any and all systems. The operation of adding lengths will not apply to weights.

Following is a selection of relevant phrases from Campbell:

1) ... the systems to be measured must be capable of a certain kind of combination, which will be termed addition ... (6).

2) ... A + B means the composite system formed by combining the systems A and B in a particular way; thus, if A and B are rigid bodies, A + B may mean the body obtained by connecting them rigidly (6).

3) The conditions that any proposed form of combination must satisfy in order that it shall be addition, and shall be suitable for the fundamental measurement of any magnitude, can then be expressed in a series of positions involving the symbols +, () and >,<,= characteristics of the magnitude (6).

4) The following are the chief of these conditions [mentioned above]. They are similar to the arithmetical "laws" of commutation and distribution in addition ... (Campbell here sets out the equivalent of VIII, IX, X, XI [6].)

5) So much for the properties of Numbers in virtue of which addition and subtraction are applicable to them. What is the similarity between these properties and the properties of bodies in respect of weight which enable us to apply to weight the process of addition? The similarity is between the relation denoted by the sign of addition and a relation which can be established experimentally between bodies in virtue of the fact that they have weight; the propositions which are true of one relation are true of the other.... Then corresponding to the arithmetical proposition that, if a = b and b = c, then a = c, we shall state that, if a certain body A balances another body B and if B balances another body C, then A must balance C; corresponding to the distributive law, a + (b + c) = (a + b) + c, we shall state that if P is a body which balances B and C on the same pan and Q a body which balances A and B on the same pan, then A and P on the same pan must balance C and Q on the same (plan, sic) pan; and so on for the other laws.

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 (5).

6) The only properties measurable directly by means of this rule are those (roughly termed quantities) which are additive - that is to say, which are such that, given two things A and B having the property, it is possible to produce by a precisely determined operation (combination) a thing C which is greater in respect of the property than either A or B ... (6).

It is possible to gather quite clearly from the above quotations that Campbell believes that no one operation is necessarily "addition" but only those operations that experimentally fulfill the criteria VIII, IX, X and XI. In other words, an operation is "additive" if it fulfills the criteria for additivity, just as a satisfactory operation for producing the relation "greater than" is one that satisfies I (II) and II (II).

But still there is no answer to the question "why the criterion of physical juxtaposition?" Since there is no mention of it in Campbell's treatment of the logical requirements, the suspicion arises that this criterion rose after the physicists had successfully measured a number of characteristics of their systems. If the physicists found that the operations that fulfilled the necessary requirements for additivity seemed to involve "physical juxtaposition," it is possible to imagine that they induced that "physical juxtaposition" was a necessary requirement. It should be pointed out however that it is possible for this to be true for the systems with which the physicists deal without being a general law of measurement.

Campbell says that "Addition is a process which is peculiarly characteristic of Numbers"(5). Objective number can be scaled fundamentally (6). The result is the numeral series 1, 2, 3, etc., associated with the objective numbers 1, 2, 3, etc. Numerals and numbers have become so inseparable in our thinking that statements like this are very confusing. Perhaps it might be better to say that numerals 1, 2, 3, have been assigned to the objective numbers in the following groups of objects ·, · ·, · · ·; thus, the number of this many objects, ·, has the numeral 1 assigned to it; this many objects, · ·, has the numeral 2 assigned to it; this many objects, · · ·, has the numeral 3 assigned to it, etc.

The defining operations for establishing the relation "greater than" for objective numbers are: i) select two groups; 2) pair off object for object; 3) if one group is exhausted before the other, the group that has not been exhausted is termed "greater" with respect to the magnitude "objective number." If both groups are exhausted simultaneously they are called "equal," etc. It can readily be seen that these operations establish an ordinal scale which fulfills all the necessary criteria.

But how are the groups added with respect to the magnitude "objective number" when an extensive scale is constructed? Suppose we had a group with this number of objects, ·, called A, and have found a group with an equal number of objects, ·, called A', and we wish to combine these two groups. How do we "add" them, · + ·? It is of course possible to place them in physical juxtaposition. So suppose that we place them on the table so that they are touching and they look like this, ··. This added group, A + A' is now called B. We can go ahead and find a group that is equal to B, which we call B', etc. Now it is absurdly obvious that by using this definition of "addition" and by using the operations outlined above for obtaining the relations of >, <, etc., an extensive scale can be constructed for the magnitude "objective number." All of the criteria of additivity can be met.

It is equally absurdly obvious that we do not have to place the objects in physical juxtaposition to reach the same result. We can show that · + · = ··, that · + · = · + ·, etc. In fact if some of the objects happen to be in New York and others in London the scale could still be constructed and the necessary criteria could be met. It is not physical juxtaposition that solves our problem, it is the simple fact that the operations that have been adopted allow its to meet the necessary criteria.

It may immediately be argued: "It is unfair to think of physical juxtaposition in such a literal fashion. Does not Campbell say that it is possible to place one weight in the pan of the balance and hang the other weight underneath the same pan? It is not literal physical juxtaposition that is demanded but it is an operation that allows the combined effect of the magnitude to be exerted in one direction. In the case of objective number it is true that the scale may be constructed if the objects or systems are not literally placed side by side but it is necessary for the two systems to be taken as a group, in other words, that the two groups B and B' are paired off against the group C as if they were a unitary or single group."

Suppose this is so, what are the criteria for combining the systems? The above argument has not answered the problem - but has restated it. In measuring length, why cannot one system be placed on two uprights and the other be hung beneath it? This would certainly be physical juxtaposition and would equally certainly meet none of the necessary criteria.

The answer can only be that the operation of placing of one weight in the pan and hanging the other underneath the pan meets the necessary logical criteria while the operation of placing one length between two uprights and hanging the other underneath it does not meet the necessary criteria. When > is defined by the usual set of operations for constructing an ordinal scale of length, this operation for + would not meet the criteria A + B > A'.

The suspicion seems to be partially confirmed that the physicists have induced this extra, operational, criterion of physical juxtaposition because it has commonly occurred in the operations they have found it necessary to use in order to meet the criteria for additivity.

It would indeed be curious if physical juxtaposition was found to be necessary for "addition" in the measurement of almost all magnitudes except objective number.

It has been found in discussing this point that it was extremely difficult to find examples that do not appear facetious or absurd. In the example quoted above for the measurement of length it was asked why one object could not he placed on two uprights and the other hung beneath it. This procedure seems absurd. It seems like a logical contradiction. It is obvious that if lengths are to be "added" they should be placed end to end. But is it so very obvious? The author feels that it is obvious only because it is such a well known, common, everyday experience. If one takes a less familiar example it does not appear so absurd. If the question were asked "How should I `add' electrical resistance? Should I connect the resistance in series or in parallel?" The incorrect answer does not appear quite so unreasonable.

If one then applied the same principle of physical juxtaposition to the measurement of inductance and juxtaposed the coils and connected them in series-aiding, or series-opposing, the result would not be so happy. The result of these operations for addition would yield La = L1 + L2 + 2M for the series aiding case and Lo = L1 + L2 - 2M for the series-opposing case. The strict interpretation of physical juxtaposition breaks down in this case. Yet we know that by other operations inductance may be measured fundamentally.

It has been said that A + B > A' (VIII) must be demonstrated. Again it might be argued that it is obvious that the magnitude length, defined by the appropriate operations for obtaining the relations >, < and =, cannot be "added" by hanging one object under the other. VIII can obviously not be demonstrated. But it is this very obviousness that confounds the thinking concerning the underlying logic of the problem. Really the above method of "addition" should not be obviously false until it has been experimentally demonstrated to be false. It is also obvious to the electrician that electrical resistance, to be "added," must be placed in series, though it may be doubted whether this insight is inherited!

This does not mean that the experimenter may not save himself time, energy and embarrassment if he uses his intelligence and his past experience in selecting an operation for "addition" that has some hope for success. But it does mean that the final, in fact the only test, is an empirical one. Nothing but experiment can determine what operation shall be "additive" for any given magnitude.

It is now possible to attempt a definition of "addition." Given a magnitude previously defined by appropriate operations for the establishment of the relations >, < and = between the systems with respect to this magnitude: "addition" is that operation, or series of operations, performed upon the systems in such a fashion that it is possible to meet the logical criteria for additivity (VIII, IX, X, XI).

It should be stressed that the same operations for demonstrating >, < and = in defining the magnitude (constructing the ordinal scale) must be used in demonstrating the criteria for additivity. For example, if = is defined in one way for the construction of the ordinal scale, the same definition must be used in criteria VIII, IX, X, XI, for additivity, etc. Furthermore, as Guild points out (13) the same definitions must be applicable for all parts of the same scale. There cannot be one definition for the smaller values of the magnitude and another for the larger.

{Sections omitted here consider Stevens theory of measurement.}

BIBLIOGRAPHY

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