# Pseudo-Mathematics in the Mental and Social Sciences^{[1]}

### H. M. Johnson

The admirers of a certain very famous psychologist often credit him with having said: "If anything exists, it exists in some degree, and therefore can be measured." "Moreover," say these disciples, "it should be measured. No fact of observation can become a scientific fact until it be measured, and its measure expressed by a number. Only quantitative facts are scientific facts, and we intend to be scientific."

So they say; and so they have set out to measure many kinds of *personal
traits, *which they call “intelligence,'' emotionality,' 'self-reliance,'
'dominance,' 'introversion,' etc., as well as persistent *social habits *of
the individual, which they call 'aptitudes,' 'opinions,' 'attitudes,'
'systematic prejudices,' 'fears,' 'superstitions,' etc. The list of these
so-called measurables is long, and it becomes longer whenever a psychological
journal issues a new number.

Unfortunately, many psychologists dislike to examine and analyze the fact-finding methods which they employ, fearing, perhaps, that if they should do so, then some one might suspect them of being logicians and philosophers. Hence, some of them have published many studies in a form which they believe to be mathematical, and therefore scientific. Many of their colleagues, being lazy, preoccupied in their own work, fearful of authority, or desirous of peace, have allowed this kind of activity to go unprotested until its results became a nuisance. It is now time to examine them.

Given a set of observational data, it is anti-scientific to subject them to
illicit mathematical treatment, just as it would be non-scientific not to apply
mathematical treatment if the latter is appropriate. But some kinds of
psychological, educational and social data lend themselves to mathematical
expression, while other kinds do not. Hence we should always begin by
determining what kind of data we have. Thus we must apply a set of universal
rules, which belong to elementary logic and elementary algebra, and make sure
whether the mathematical treatment that is proposed satisfies *all *these
rules or not. If it fails *any *of them it is invalid. Fortunately, these
rules are very few and also very simple.

If a collection of observational facts is to be treated mathematically, it is *
necessary, *though not *sufficient,
*that each fact can be perfectly denoted by a number. But three kinds of
numbers lend themselves to denoting, and only one kind to counting: hence,
before we perform any operations on the fact-denoting numbers we should
determine their kind.

*Nominal numbers. *We use these numbers each to denote a particular
member of the collection, and thus as substitutes for ordinary *names, *as
of football players, factory employees, prisoners, railroad cars. If two nominal
numbers stand in some particular relation to each other, the fact does not imply
that the objects which they denote stand in any corresponding relation to each
other. Thus, as Cohen and Nagel remark,**[2]**
from the fact that one prisoner is called "500" and another prisoner

( 343) "100" one could not infer that the first prisoner is "five times as dangerous or wicked" as the second prisoner. Neither could one infer that the first prisoner was sentenced to a longer term than the second prisoner, or even that "500" entered the prison later than "100," since if a convict leaves the prison his number can be reassigned to a newcomer.

Sometimes nominal numbers denote sentences. Thus, in the flag-code of a merchant marine the number 373 might denote, "Yellow fever on board; keep off." By convention, an instructor might write on a student's theme the number 95 to denote, "If you hand in many other papers like this one, I shall recommend you for honors;" or the number 55 to denote, "Unless you go to work effectively I shall exclude you from a continuation-course;" or the number 19 to denote, "You ought to be ashamed to hand in this paper for criticism." Certainly this code does not imply that the first paper exhibited 19/11 times as much information as the second and 5 times as much information as the third; any other set of three numbers could be made to denote the same three sentences as these three numbers respectively denote, if only the code were agreed to. Obviously, no meaning attends the result of any operation performed on nominal numbers, such, for example, as taking their mean.

*Ordinal numbers. *These numbers, such as "the first," "the second," ...
the nth," are assigned to individual objects, not to denote them as individuals,
but to denote the *places *which they severally occupy in some
*ordered series. *Note that if two objects in the ordered series should
exchange places, they would have to exchange ordinal numbers. To arrange the
objects in a collection into an ordered series, one first chooses some one
property which all the objects have in common. Thus, Mohs chose the property *
of hardness,
*which is shared by all the members of his collection of minerals. In
general, call the common property *x*. Next, one has to choose some *
operation *which defines the relation "x-er than." Thus, to define the
relation 'harder than,' Mohs chose the operation of *scratching. *If any
mineral *A *scratches another mineral *B, *and if also *B *does
not scratch *A, *then by definition *A *is 'harder than' *B. *
Thus by the defining operation the relation 'harder than' is *asymmetrical. *
Moreover, experiment is required to show that the relation "x-er than" is
*transitive: i.e. *if *A gt; B *
and *B > *C, then *A *> *C. *
(Here the symbol > means *"x-*er than.") Thus, Mohs showed that if *A *
scratches *B *and *B *scratches *C*, then *A *scratches *C*.

If any relation is asymmetrical and also transitive, it permits the objects
which it connects to be arranged in an ordered series, otherwise it does not.
Thus, Mohs arranged his minerals in the order of their hardnesses, placing
diamond, which scratches all the others, at one end, and talc, which is
scratched by all the others, at the other end. To diamond he assigned the number
10, to sapphire 9, to topaz 8, ... and to talc the number 1, assigning to each
mineral a number larger than the number of any mineral that it scratches, and
smaller than the number of any mineral that scratches it. Thus, carborundum,
which scratches sapphire, but which is scratched by diamond, would receive a
number greater than 9 but less than 10. These numbers are ordinal numbers: they
denote nothing except the *places *in which the several minerals stand in a
series that is ordered with respect to the relation *harder than.*

Note that Mohs chose these numbers *arbitrarily. *If he had assigned to
diamond

(
344) the number 3 and to talc the number 1; or if he had assigned to diamond
the number 10 and to talc the number 7, he could still have denoted perfectly
the place that each mineral occupies in the hardness-series, simply by following
the general rule that we mentioned above. Hence it is meaningless to say that
diamond is 10 times as hard as talc, or three times as hard as talc, or 10/7
times as hard as talc, or that its hardness bears any other ratio to the
hardness of talc. The relation *harder than, which is *identical with *
scratches, is
*perfectly denoted by *any *of an infinite number of series of ordinal
numbers; and operations on ordinal numbers give meaningless results.

Similarly, if we were to subject say 100,000 persons to the so-called
National Intelligence-Test, count the conventional answers that each individual
returns, and then ascertain that 70% of the population answered fewer of these
questions conventionally than John Walker answered conventionally, and that 35%
of the same population answered fewer of these questions conventionally than did
William Carpenter, then we might assign to Walker the number 70 and to Carpenter
the number 35 to denote their respective centile ratings in the population of
tested individuals. But then these numbers would be *ordinal *numbers, and
it would be absurd to say that because Walker's place in the series is denoted
by a number which is twice as great as the number which denotes Carpenter's
place, therefore Walker has twice as much National Test Intelligence as
Carpenter has, or that the relations between these two ordinal numbers denote
any relation between the 'amounts of' National Test Intelligence which these
individuals may be thought of as possessing.

In general, if a property can be *adequately *denoted by nominal numbers
or by ordinal numbers, then it is non-additive and non-measurable. Among such
proper-ties are hardness, shape, structure, generosity, dominance, radicalness,
intelligence and the like.

*Cardinal numbers. *These and only these numbers express the result of *
counting, *and they express nothing except that result. In counting the
objects in a collection, one treats each object as if it were interchangeable
with every other. Thus one disregards their individualities, which nominal
numbers denote, and also their several places in the collection, which ordinal
numbers denote. The cardinal number of any collection denotes *how many *
members it contains; if its members are *units *of some *distributable *
property, then its cardinal number denotes *how much *of the property is
distributed among the objects that are considered.

Consider the property called *weight *or *heaviness.***[3]**
Given a suitable scale-balance, we place in one of its pans a pile of sand and
in the other pan two bodies *B _{1} *and

*B*successively. If we should find that

_{2}*B*causes its pan to sink while

_{1}*B*does not, we assert that

_{2}*B*'heavier than'

_{1}is*B*If we should find that the same pile of sand exactly balances another body

_{2}.*B*and also still another body

_{3}*B*when

_{4},*B*replaces

_{4}*B*we assert that

_{3},*B*, and

_{4}*B*are 'equally heavy.' If we should put both

_{3}*B*

_{4}

*and*

*B*into one pan and add sand to the pile in the other pan until it counterbalances the first, we say that the second pile is 'twice as heavy as' the original pile. Thus, we can define a set of standard weights, and by means of this procedure we can say that

_{3}( 345) any member of the set bears some specified relation to any other member. Of course, we are limited by the capacity of the instrument and also by its construction. We feel sure that if these limitations could be removed, or if we could replace this instrument by other instruments in certain ranges of weight, or if we could correct for friction, distortion, etc., then these principles would hold perfectly both within and without our range of experience. But this is pure assumption.

These operations suggest that heaviness differs from hardness in being an *
additive *property. Nevertheless, we must not be too hasty in assigning
numbers to two or more bodies according to their operationally defined weights,
and in assuming that the result of adding these numbers expresses the result of
placing these bodies in the same pan of the balance. In general, if a property
is truly additive, and measurable, it must satisfy all the following criteria.

(1) The relation 'x-er than' must be unequivocally defined by the behavior of
the detector of the property x, and in such wise that this operational
definition shows the relation to be *asymmetrical. *Given a collection of
objects which have the property x, the detector must show that between any two
objects in the collection, such as *B _{i} *and

*B*one and only one of these relations holds:

_{j},*(a) B*(b)

_{i}> B_{j};*B*with respect to x-ness. Thus, if x is weight, we assert (a), (b), or (c) according as

_{i}= B_{j}; (c) B_{i}< B_{j};*B*counterbalances more sand than

_{i}*B*the same sand as

_{j},*B*or less sand than

_{j},*B*counterbalances, when

_{j}*B*and

_{i}*B*are interchanged in the detector. Outside a determinable range of uncertainty these judgments are not confused; if this range is greater than is permitted by the degree of precision

_{j}*which*we demand, then the operation of counterbalancing on this instrument does not define the relation 'heavier than.' But if the detector permits the unequivocal judgment

*B*then it always yields the judgment

_{i}> B_{j},*B*and not

_{i}≢ B_{j}*B*≯

_{j}*B*and thence the judgment

_{i},*B*Thus the relation 'heavier than' is

_{j}< B_{i}.*asymmetrical*if it is

*determinate.*

(2) The defining operations must show that the relation 'x-er than' is *
transitive: e.g. *if *B _{i} > B_{j} *and

*B*then

_{j}> B_{k},*B*

_{i}> B_{k}.(3) The collection of objects which have the property x must in its turn have
the so-called
*group-property.*'**[4]**
For example,
if two objects *Bi *and *B _{j} *severally excite the detector
of the property x, then the collection must contain another member

*B*such that

_{k},*B*and

_{k}> B_{i}*B*and

_{k}> B_{j}*B*in the sense that

_{k}= B_{i}+ B_{j}*Bk*excites the detector of the property x in exactly the same manner as

*B*and

_{i}*B*together excite it. In other words, if the effects of

_{j}*B*and

_{i}*B*on the detector are additive, then the detector must show an effect which is equivalent to their sum. This implies that the number of objects

_{j}*which*have the property is

*infinite.*

**[5]**

(4) The addition in the detector of bodies that have the property x must
satisfy the *commutative law *of the addition of numbers. Thus, if *B _{i}
+ B_{j} = B_{k}, *then

*B*=

_{j}+ B_{i}*B*.

_{k}( 346)

(5) The operation of empirical addition must satisfy the *associative law *
of addition of numbers. Thus, *(B _{i} + B_{j}) + B_{k}
= B_{i} + (B_{j} + B_{k}), *etc.

(6) The operation of empirical addition must satisfy the *axiom of equals *
which holds for the addition of numbers. Thus, if *B _{i} = B_{i}*'

*and*

*B*then

_{j}= B_{j}',*B*'

_{i}+ B_{j}= B_{i}*+B*and

_{j}'*Bi +B*'

_{j}' = B_{i}*+ B*

_{j}.In particular, if the property x has an antagonist which we agree to call the
negative *of x, *then the operation of empirical addition must satisfy two
additional criteria; namely:

(7) The collection of objects, ordered with respect to the relation 'x-er
than,' must contain a member *B _{0}
*which does not excite the detector of the property x, and which, if placed
in the detector along with another object

*B*will always yield the result

_{i}*B*Hence

_{0}+ B_{i}= Bi + B_{0}= B_{i}.*B*is called the

_{0}*neutral member*of the collection.

(8) Finally, if the collection contains an object *B _{i} *which
excites the detector of the property x in one manner, it must contain a
corresponding object

*B*which excites the detector in the opposite manner, and such that if

_{-i},*B*and

_{i}*B*are placed together in the detector, they do not excite it. Since the sum of their several effects on the detector is imperceptible, as is also the effect of

_{-i}*B*upon it, we say that

_{0}*B*+

_{i}*B*

_{-i}*= B*

_{0}.*If a property is measurable it satisfies all these criteria; if it fails
any of them it is non-additive and non-measurable. *In other words, the
addition of 'quantities' of the property x has to satisfy all the axioms of
addition of cardinal numbers: other-wise it cannot be expressed by the result of
numerical addition.

It is at least doubtful whether any observable property exists which *
satisfies *all these axioms. For example, the class of all weights that are
detectible by means of the scale-balance does not have the group-property (3);
and its members do not, in general, obey the commutative law of addition (4) ;
both criteria remain unsatisfied if the balance is overloaded. Nevertheless,
there is a range of weights, which we can determine empirically, and within
which we can *pretend *that scale-balance weight is additive, and prove
that we have not so introduced an uncertainty which exceeds some specified
standard of tolerance. Now some mental and social data are of this kind: they *
approximate *the criteria of measurability within a given standard, al-though
they do not *satisfy *the criteria. But other kinds of mental and social
data fail them utterly. We shall mention examples of both kinds.

*Are perceptible brightnesses measurable? *Perceptible brightnesses are
not identical with what the physicist calls the brightnesses or luminosities of
surfaces. To deter-mine the physical brightness of a surface, one first
ascertains the wavelengths of the radiation which the surface emits, transmits,
or reflects toward the detector. Next, with respect to each minute range of
wavelengths, one determines (a)its surface-rate of power, and (b) its so-called
'visibility factor,' taken from a standard wavelength-luminosity curve. Taking
the product of (a) and (b) for each minute range of wavelengths, one then
summates the products. These operations define the physical brightness of the
surface. It is imperfectly correlated with perceptible brightness within certain
limits, but it is not identical with the latter.

Consider a surface illuminated by two sources *S _{1}* and

*S*in succession. When

_{2}*S*is used alone, the observer perceives a brightness of the surface which we may call

_{1}*B*

_{1}

*;*when

*S*. is used alone, he perceives a brightness

_{2}*B*on the same surface. Using the flicker-method of photometry, or else the method of direct comparison,

_{2}(
347) let us balance *B _{1} *against a comparison-field

*B*and also balance

_{1}',*B*against another comparison-field

_{2}*B*Now, expose the surface to both sources

_{2}'.*S*and

_{1}*S*at once. If we agree that in so doing, we add

_{2}*B*

_{1}

*and*

*B*then by the axiom of equals (6),

_{2},*B*

_{1}

*+ B*

_{2}= B_{1}

*' + B*'

_{2}*,*and

*B*

_{1}

*- B*

_{2}'= B_{1}

*+ B*. But this is not generally true. Suppose, for example, that the sources which produced

_{2}*B*

_{1}

*and*

*B*respectively, emitted only lithium light (λ = 671 mµ), while the sources that produced

_{2}.*B*

_{1}

*'*and

*B*respectively, emitted only thallium light (λ = 535 mµ). Suppose moreover that

_{2}',*B*

_{1}

*= B*is high, while

_{1}'*B*'

_{2}= B_{2}*is low. Then*

*B*

_{1}

*+ B*is the sum of a bright red and a dim olive-green, while

_{2}'*B*is the sum of a bright olive green and a dim red. Although observation yields the separate equations

_{1}' + B_{2}.*B*

_{1}

*=B*it is very likely to yield

_{1}', B_{2}= B_{2}' ,*B*

_{1}

*+ B*≢

_{2}'*B*

_{1}'*+ B*It may also yield

_{2}.*B*

_{1}

*+ B*≢

_{2}*B*The operations may not satisfy or even approximate the axiom of equals. Some have suggested that heterochromatic brightnesses

_{1}' + B_{2}' .*would be*additive

*if*it were not for Purkinje's effect, and they may be right: anything probably would be different from what it is if only it did not remain the same. But since Purkinje's effect characterizes the brightnesses that we perceive, anything that lacked it would not be a perceptible brightness.

But are perceptible brightnesses measurable if they are *homochromatic? *
No, their addition is not
*commutative (4). *Suppose that direct comparison yields *B _{1} >
B_{0}, B_{2} > B*

_{1}

*.*Suppose also that

*B*is

_{2}*much*greater than

*B*Then, direct comparison may yield

_{1}.*(B*

_{1}+ B_{2}) > (B_{0}+ B_{2}) ;(B_{2}+ B_{1}

*)*=

*(B*whence, if

_{2}+ B_{o}) ;*(B*then

_{0}+ B_{2}) = (B_{2}+ B_{o}),*(B*≢

_{1}+ B_{2})*(B*

_{2}+ B_{1}

*).*Some have surmised that the axiom of equals would be satisfied, together with the commutative law, if it were not for the fact that brightness-perception goes by jumps; nevertheless, differential thresholds characterize brightness-perception, and anything that is not infected with them is not perceptible brightness.

Again: has the class of all perceptible brightnesses the so-called *
group-property? *If so, then the addition of any perceptible brightness *B _{i} *
to any other perceptible brightness

*B*must be equivalent to another perceptible brightness

_{j}*B*such that (3)

_{j},*B*and

_{k}> B_{i}*B*By definition of perceptibility, if

_{k}> B_{j}.*B*and

_{i}*B*are perceptible while

_{j}*B*is not, then

_{o}*(B*and

_{i}+ B_{j}) > (B_{i}+ B_{o}),*(B*But if either

_{i}+ B_{j}) > (B_{o}+ B_{j}).*B*or

_{i}*B*is near the ‘terminal threshold' of perceptible brightness, direct comparison is very likely to yield that

_{j}*(B*(

_{i}+ B_{j}) =*B*)

_{i}+ B_{o}*or that*

*(B*whence

_{j}+ B_{i}) = (B_{j}+ B_{o});*B*or

_{i}= B_{o}*B*which contradicts the hypothesis.

_{j}= B_{o},Hence, perceived brightnesses are not, in general, additive or measurable. Like scale-balance weights they fail some of the criteria of measurability, although, within certain limits, they approximate these criteria within some determinable standards. If we are content with these standards, then we can set limits to a range of brightnesses within which we can pretend that brightnesses are measurable and outside which we cannot pretend that they are. But we now have to consider some classes of mental and social data of which not even this is true.

*Are perceptible hues measurable? *First, can we arrange all perceptible
hues in the order of their resemblance to some standard hue, such as Helmholtz's
primary chlorine green? Yes; if we exclude neutral gray from the collection, we
can so arrange them, but not in a rectilinear series, as axiom (1)* *
requires. The arrangement must correspond to a closed curve, on which the purple
that complements the standard chlorine green will be at the pole opposite the
latter, since it resembles

(
348) the chlorine-green less than any other hue resembles it. But the
relation 'resembles' is non-transitive, and so violates axiom (2) : for example,
one may find a red and a blue green which resemble the chlorine green, but which
(being complementaries) do not resemble each other. We have noted that neutral
gray is not in this ordered series. But unless it belongs in the collection,
axiom (3) is not satisfied, for the rule that the addition of one hue to another
gives a third hue requires that the neutral hue be in the collection, otherwise
the rule does not provide for the addition of complementary hues in certain
proportions. Moreover, unlike the series of natural numbers, the number of
distinguishable hues is finite. For example, there is a chlorine green which is
"chlorine-greener" than any other hue, and there is a purple which is less
chlorine-green than any other hue. The same assertion holds in principle for any
primary hue that we may select. Axioms (1), (2) and (3) are violated: *
perceptible hues are non-additive and non-measurable.*

Consider next a class of *social facts, *which some authors falsely have
treated as measurables: *e.g. *attitudes, interests, intelligences,
aptitudes, skills, drives, and the like. Thurstone,**[6]**
for example, has asserted that attitudes can be measured. Let us look into this
question.

We say that a person holds one attitude or another toward some specified
change (now occurring or in prospect), according as he is predisposed to make
one kind of response or another kind of response to the change. Thus an attitude
is a preparation for action. We suppose that it is correlated with a
corresponding pattern of tensions and conductivities in the nervous system, and
thus determines the *equilibrium state *of the latter. Just as solid carbon
obeys one set of thermodynamic laws when it is in the form of soot, another set
when it is in the form of graphite, and still another set when it is in the form
of diamond: so a person, while he holds the 'attitude' of a pacificist toward a
war, will make certain responses to banners, posters, pageants, poems, and to
appeals to enlist, to buy bonds, to encourage the government, etc., which he
would not make if he were holding the attitude of a patriot toward the war. Of
course, we cannot describe his attitude physically, probably we would not so
describe it even if we could. Rather, we *name it after *the course of
action which we think it predisposes the person to execute, but the properties
of the attitude ebb not depend on our way of describing it.

These assertions together imply that there are as many attitudes as there are
distinguishable patterns of action, or distinguishable patterns of
brain-tension. Each attitude predisposes the subject to behave according to its
own corresponding set of laws. It is meaningless to ask *by how much *one
attitude differs from another as it is meaningless to ask *by how much *the
molecular arrangement of carbon in the form of graphite differs from its
arrangement in soot or in diamond.

Thurstone does not make the mistake of supposing that these questions are
meaningful, but he does assert that it is possible to measure some of the
properties of attitudes, such as their *predisposing tendency *toward a
given course of action, such as actively participating in a war. He assumes,
moreover, that this tendency is additive, in the sense that by taking the sum,
or else the mean, of the tendencies of two or more attitudes toward a given
course of action, he gets the equivalent of

( 349) the predisposing tendency of another attitude toward the same course of action. This assumption I invite you to examine.

Before we can proceed to *measure *any property of any attitude, we have
to *detect *the attitude itself. But why is it proposed to detect a
person's attitude toward a given change, in the first place? Most
attitude-testers are practical men: they wish to know, for example, what one
needs to do to a person to cause him to execute some course of action, such as
joining the church, voting against a bond-issue, buying accident-insurance,
enlisting in the army, or the like. Or, if they believe that some tendencies
toward action are not amenable to propaganda, then they wish to know what means
of persuasion they must avoid. Hence they wish to know what the person is
already set or tending to do about the issue; or, in other words, to detect his
attitudes. But, unless they intend to start working on him as soon as they have
finished the test, if they use the results of the test, they pre-suppose that he
will be holding on another occasion the attitudes which he held when they tested
him. Unless the second occasion is dated, they presuppose that the test
indicates what attitudes the person most probably will hold on *any *other
occasion within a reasonably short time. In other words, they presuppose that
the result of a single test will indicate what the person's attitudes are, *
habitually.*

This presupposition is not, in general, plausible. The sinner today may become a missionary before tomorrow night; a man who is ready to give all for the love of his lady today may be glad within a fortnight that she quit him. But if the practical man denies the presupposition, he thereby denies that the detection of attitudes is useful to the propagandist.

How do the attitude-testers try to detect what a person is tending to do
about a given change at the moment?
*They ask him questions!***[7]**
Does
what he *says *indicate what he is tending to *do? *Thurstone says
merely that
*if *it does, *then *the tester, by taking the average of the numbers
which many judges set down to denote 'how favorable' toward the question they
guess his statement of opinion to be, can deter-mine 'how strong' is his
predisposing tendency with respect to a corresponding *action. *The
'predisposing tendency' is not operationally defined; those who adopt this
hypothesis thereby assume that the average of the guesses of the individual
judges indicates the result of a set of operations that are either
inconceivable, infeasible, or else merely not yet performed. Thurstone
explicitly refuses to make this assumption, but his procedure is pointless
unless what a person *says, *at one instant, about an hypothetical
situation reliably indicates what he is most likely to *do *if the
situation should *ever *become actual; this he *implicitly *
presupposes.

Thurstone assumes that all possible opinions toward a given question (and
with them, the corresponding tendencies to action) can be arranged in a series
that is ordered according to the relation *more favorable than; *that
'favorableness' is additive according to the laws of algebraic addition; and
moreover, that if *B _{i} *and

*B*

_{k}(
350) are *any *two opinions in the ordered collection, the latter
contains another opinion *B _{j}, *such that

*B*is more favorable toward this course of action than is

_{i}*B*and less favorable than

_{i}*B*toward it. This assumption implies that between any two ordered opinions there are an infinite number of similarly ordered opinions, so that the numbers that express their 'favorableness' constitute a 'dense set.'

_{k}Consider these: If war should be declared, I would fight,

(A) under all conditions;

(E) under no conditions;

(I) under some conditions; namely, (a), (b), (c), :

(O) not under some conditions; namely, (a'), (b'), (c'),

Obviously the classes (A) and (E) of opinions contain only one member each;
classes (I)and (O) may each contain many opinions. Note especially that (A) and
(O) contradict each other, as do (E) and (I). The axiom of excluded middle
implies that between the sole opinion in class (A) and the least unfavorable of
the opinions in class (O), no opinions can be inserted in the ordered series;
and that between the sole opinion in class (E) and the least favorable opinion
in class (I) no opinions can be put into the ordered series. *Opinions do not
stand in a continuum. *If opinions are the transforms of attitudes, then *
attitudes do not stand in a continuum.*

We cannot now review the elaborate procedures which Thurstone employs**[8]**
for ordering statements of opinion respecting any given issue, and for assigning
scale-numbers to the statements. The careful reader will discover clearly that *
each procedure merely defines a rule for assigning *ordinal *numbers *to
the statements. Nothing in the procedure provides a means of determining 'how
much' favorableness any statement indicates, or 'how much more' favorable to a
given decision one statement of opinion is than another.

From another direction let us examine the assertion that a person who
endorses two or more statements expresses the sum of the 'favorablenesses' of
each. Is any statement more favorable to the prospective war than the sole
statement in (A) ? No, for it includes all the statements in (I). Hence the
property of 'favorableness to war' is non-additive, because the objects which
have it do not form a collection that has the
*group property. *If a person endorses (A)* *and also, for example,
(I. c), he has not indicated more 'favorableness' than if he endorsed only (A).
But if both are indicated by finite scale-numbers, their sum would be greater
than the scale-number of (A). If one averages the scale-numbers, as Thurstone
recommends because of the possibility of ambiguity in both question and answer,
then the individual who endorsed (A) and also every member of (I) separately
would earn a lower patriotic score than a person who endorsed only (A). Perhaps
the procedure should be worked over.

In brief, no property of any attitude can he measured unless the attitude can
be first detected. It is not evident that it can be detected from the person's
own assertions of 'opinion.' Nor does the procedure which we mentioned provide a
means of measuring the 'favorableness' of opinions themselves. It yields a
counterfeit measure of the strength of a person's tendency toward any course of
action. Similar defects inhere in the so-called measurement of interests,
drives, skills,**[9]**
X-O emotional
stability, intelligences, and the like.

( 351)

But why bother about principles? If these procedures are unsound, surely the 'Movements' will die of themselves! Why not let them alone?

The logician may answer, "Because if an invalid method leads one to factual truth, it is only by accident." The practical man may add, that the time and effort that are wasted in futile endeavor are subtracted from the amount that is available for effective work. There are questions concerning human traits which are interesting; they may be important; they may be answerable; but they probably will remain unanswered as long as they are attacked by attempts to measure what is intrinsically non-measurable.

For example, consider the attempts made during the past 30 years to determine
certain psychological effects of drugs, partial asphyxiation, fatigue, insomnia,
and sleep. The plan of nearly every experiment presupposed that the effect on
some arbitrarily selected function would be *graded *according to the
magnitude or the duration of the agent of impairment. In nearly every instance
no such gradation was found; whereupon the investigator drew the conclusion that
the agent was ineffective. The conclusion was false, because it rested on the
false presupposition, that if the agent had an effect, the effect would be of
the kind that he was seeking. But if the experimenter, like Dunlap,**[10]**
and Bagby**[11]**
and others in their study
of asphyxiation, had asked, not 'how much' the function was affected, but *in
what manner *it was performed under the compared conditions, he probably
would have had to draw a different and valid conclusion. Elsewhere,**[12]**
I shall present this field of investigation in detail. For the present, it is
enough to say that it illustrates, clearly and almost tragically, the practical
waste that results from reliance on false presuppositions and on
pseudo-mathematics. Those data should be measured which can be measured; those
which cannot be measured should be treated otherwise. Much remains to be
discovered in scientific methodology about valid treatment and adequate and
economic description of non-measurable facts. Their detection as such, however,
is logically simple.

H. M. JOHNSON

American University