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The Vectors of Mind
L. L. Thurstone (1934)
Classics in the History of Psychology
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The Vectors of Mind
L. L. Thurstone (1934)
Address of the president before the American Psychological Association, Chicago meeting,
September, 1933.
First published in Psychological Review, 41, 1-32.
Posted March 2000
Under the title of this address, "The Vectors of Mind," I shall discuss one of the oldest of
psychological problems with the aid of some new analytical methods. I am referring to the old
problem of classifying the temperaments and personality types and the more recent problem of
isolating the different mental abilities.
Until very recently the only attempt to solve this problem in a quantitative way seems to have
been the work of Professor Spearman and his students. Spearman has formulated methods for
dealing with the simplest case, in which all of the variables that enter into a particular study can
be regarded as having only one factor in common. The factor theory that I shall describe starts
without this limitation, in that I shall make no restriction as to the number of factors that are
involved in any particular problem. The resulting factor theorems are quite different in form and
in their underlying assumptions, but it is of interest to discover that they are consistent with
Spearman's factor theory, which turns out to be a special case of the present general factor
theory.
In this paper I shall first review the single-factor theory of Spearman. Then I shall describe a
general factor theory. Those who have only a casual interest in the theoretical aspects of this
problem will be more interested perhaps in the applications of the new factor theory to a
number of psychological problems. These psychological applications will constitute the major
part of this paper.
It is thirty years ago that Spearman introduced his single-factor method and the hypothesis that
intelligence is a central and general factor among the mental abilities. The literature on this
subject of factor analysis has tended temporarily to obscure his contribution, because the
controversies about it have frequently been staged about rather trivial or even irrelevant
matters. Professor Spearman deserves much credit for initiating the factor problem and for his
significant contribution toward its solution, even though his formulation is inadequate for the
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multidimensionality of the mental abilities.
Spearman's theory has been called a two-factor method or theory. The two factors involved in it
are, first, a general factor common to all of the tests or variables, and second, a factor that is
specific for each test or variable. It is less ambiguous to refer to this method as a single-factor
method, because it deals with only one common or general factor. If there are five tests with a
single common factor and a specific for each test, then the method involves the assumption of
one common and five specific factors, or six factors in all. We shall refer to his method less
ambiguously as a single-factor method.
We must distinguish between Spearman's method of analyzing the intercorrelations of a set of
variables for a single common factor and his theory that intelligence is such a common factor
which he calls "g". If we start with a given table of intercorrelations it is possible by Spearman's
method, and also by other methods, to investigate whether the given coefficients can be
described in terms of a single common factor plus specifics and chance errors. If the answer is
in the affirmative, then we can describe the correlations as the effect of (1) a common factor,
(2) a factor specific to each test, and (3) chance errors. In factor theory, the last two are
combined because they are both unique to each test. Hence the analysis yields a summation of
a common factor and a factor unique to each test. About this aspect of the single-factor method
there should be no debate, because it is straight and simple logic.
But there can be debate as to whether we should describe the tests by a single factor even
though one factor is sufficient. It is in a sense an epistemological issue. Even though a set of
intercorrelations can be described in terms of a single factor, it is possible, if you like, to
describe the same correlations in terms of two or three or ten or any number of factors.
The situation is analogous to a similar problem in physical science. If a particle moves, we
designate the movement by an arrow-head, a vector, in the direction of motion, but if it suits our
convenience we put two arrowheads or more so that the observed motion may be expressed in
terms that we have already been thinking about, such as the x, y, and z axes. Whether an
observed acceleration is to be described in terms of one force, or two forces, or three forces,
that are parallel to the x, y, and z axes, is entirely a matter of convenience for us. In exactly the
same manner we may postulate two or more factors in a correlation problem instead of one,
even when one factor would be sufficient. To ask whether there "really" are several factors
when one is sufficient, is as indeterminate as to ask how many accelerations there "really" are
that cause a particle to move. If the situation is such that one factor is not adequate while two
factors would be adequate, then we may think of two factors, but we may state the problem in
terms of more than two factors if our habits or the immediate context makes that more
convenient.
Spearman believes that intelligence can be thought of as a factor that is common to all the
activities that are usually called intelligent. The best evidence for a conspicuous and central
intellective factor is that if you make a list of stunts, as varied as you please, which all satisfy
the common sense criterion that the subjects must be smart, clever, intelligent, to do the stunts
well, and that good performance does not depend primarily upon muscular strength or skill or
upon other non-intellectual powers, then the inter-stunt correlations will all be positive. It is quite
difficult to find a pair of stunts, both of which call for what would be called intelligence, as
judged by common sense, which have a negative correlation. This is really all that is necessary
to prove that what is generally called intelligence can be regarded as a factor that is
conspicuously common to a very wide variety of activities. Spearman's hypothesis, that it is
some sort of energy, is not crucial to the hypothesis that it is a common factor in intellectual
activities.
There is a frequently discussed difficulty about which more has been written than necessary. It
has been customary to postulate a single common factor (Spearman's "g") and to make the
additional but unnecessary assumption that there must be nothing else that is common to any
pair of tests. Then the tetrad criterion is applied and it usually happens that a pair of tests in the
battery has something else in common besides the most conspicuous single common factor.
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For example, two of the tests may have in common the ability to write fast, facility with
geometrical figures, or a large vocabulary. Then the tetrad criterion is not satisfied and the
conclusion is usually one of two kinds, depending on which side of the fence the investigator is
on. If the investigator is out to prove "g," then he concludes that the tests are bad because it is
supposed to be bad to have tests that measure more than one factor! If the investigator is out
to disprove "g" then he shows that the tetrads do not vanish and that therefore there is no "g."
Neither conclusion is correct. The correct conclusion is that more than one general factor must
be postulated in order to account for the intercorrelations, and that one of these general factors
may still be what we should call intelligence. But a technique for multiple factor analysis has not
been available and consequently we have been stumbling around with "group factors" as the
trouble-making factors have been called. A group factor is one that is common to two or more
of the tests but not to all of them. I use the term common factor for all factors that extend to two
or more of the variables. We see therefore that Spearman's criterion, limited as it is to a single
common factor, is not adequate for proving or disproving his own hypothesis that there is a
conspicuous factor that is common to all intelligence tests. If his criterion gives a negative
answer it simply means that the correlations require more than one common factor. We do not
need any factor methods at all to prove that a common factor of intelligence, is a legitimate
postulate. It is proved by the fact that all intelligence tests are positively correlated.
There is only one limited problem for which Spearman's method is adequate, namely, the
question whether a single factor is sufficient to account for the intercorrelations of a set of tests.
The usual answer is negative. His criterion that the tetrads shall vanish is rarely satisfied in
practice. One might wonder then why it is that numerous examples have been compiled in
which the tetrad criterion is satisfied. The reason is simply this -- that in order to satisfy the
criterion, the tests must be carefully selected so as to have only one thing in common. Another
way by which the criterion is satisfied is to throw out of the battery those tests which do not
agree with the criterion. The remaining set will then satisfy it. The reason for these difficulties is
that Spearman's tetrad difference criterion demands more than his own hypothesis requires.
His hypothesis does not state that there shall be only one common factor or ability. He himself
deals with many factors. But his tetrad difference requires that there shall be only one common
factor.
Now it happens that one can readily put together several batteries of tests such that within
each battery the criterion is satisfied and therefore we have a common factor "g" in each
battery. But if we take a few tests from each of these batteries and put together a new
composite, then the criterion is not satisfied and we then require more than one factor. We are
then laced with the ambiguity that we have several batteries of tests each with its own single
common factor. Which of these common factors shall we call the general one? Which is it that
we should call "g"? Spearman's answer is that we should use a set of perceptual tests as a
reference and that if we are dealing with that particular common factor, then we should call it
general but that if we are dealing with one of the other common factors, then we should call it a
special ability, a group factor, a sub-factor of some sort. It would be more logical to assign a
letter or a name to each of these mental abilities and to treat them as related dependent
abilities. If we choose one of them, such as facility in dealing with perceptual relations, as an
axis of reference (Spearman's "g")· then it should be frankly acknowledged that the choice is
statistically arbitrary, for we could equally well start with verbal ability or with arithmetical ability
as an axis of reference. The choice of the perceptual axis of reference might be made from
purely psychological considerations, but it would not rest on statistical evidence.
The multi-dimensionality of mind must be recognized before we can make progress toward the
isolation and description of separate abilities. It remains a fact, however, that since all mental
tests are positively correlated, it is possible to describe the intercorrelations in terms of several
factors in such a manner that one of the factors will be conspicuous in comparison with the
others. But the exact definition of this factor varies from one set of tests to another. If it is this
factor that Spearman implies in his theory of intelligence, then his criterion is entirely
inadequate to define it, because the tetrad criterion merely tells us whether or not any given set
of intercorrelations can be described in terms of one and only one common factor.
Let us start with the assumption that there may be several independent or dependent mental
abilities, and let it be a question of fact for each study how many factors are needed to account
for the observed intercorrelations.[1] We also make the assumption that the contributions of
several independent factors are summative in the individual's performance on each one of the
psychological tests. If we do not make this assumption, the solution seems well-nigh hopeless,
and it is an assumption that is either explicit or implicit in all attempts to deal with this problem.
We may start our analysis of the generalized factor problem by considering the many hundreds
of adjectives that are in current use for describing personalities and temperaments. We have
made such a list. Even after removing the synonyms we still had several hundred adjectives. It
is obvious at the start that all these traits are not independent. For example, people who are
said to be congenial are also quite likely to be called friendly, or courteous, or generous, even
though we do not admit that these words are exactly synonymous. It looks as though we were
dealing with a large number of dependent traits.
The traditional methods of dealing with these psychological complexities have been
speculative, bibliographical, or merely literary in character. The problem has been to find a few
categories, called personality types or temperaments, in terms of which a longer list of traits
might be described. Psychological inquiry has not yet succeeded in arriving at a list of
fundamental categories for the description of personality. We are still arguing whether
extraversion and introversion are scientific entities or simply artifacts, and whether it is
legitimate even to look for any personality types at all.
In the generalized factor method we have one of the possible ways in which a set of categories
for the scientific study of personality and temperament may be established on experimental
grounds as distinguished from literary verbosity about this subject. It is our belief that the
problem can be approached in several rational and quantitative ways and that they must agree
eventually before we have a satisfactory foundation for the scientific description of personality.
The problem has geometrical analogies that we shall make use of. If we have a set of n points,
defined by r coordinates for each point, we may discover that they are dependent in that some
of these points can be described linearly in terms of the coordinates of the rest of the points. If
the adjectives were represented by these points, it is as though we were to describe most of
them in terms of a limited number of adjectives. But such a solution is not unique, because if
we have ten points in space of three dimensions, then it is possible, in general, to describe any
seven of the points in terms of the remaining three. It is just so with the personality traits, in that
a unique solution is not given without additional criteria.
But before demanding this sort of reduction it would be of great psychological interest to know
how many temperaments or personality types we must postulate in order to account for the
differentiable traits useful. If we have a table of three coordinates for each one of ten points and
if that matrix has the rank 2, then we know that all ten points lie in a plane and consequently
they can all be described by two coordinates in that plane instead of by three. The application
of this analogy would be the description of ten traits in terms of two independent traits which
would then become psychological categories or fundamental types. They would constitute the
frame of reference in terms of which the other traits would be described and in terms of which
interrelations could be stated.
But since we have no given frame of reference to begin with, we do not have the coordinates of
the points that might represent the adjectives. We therefore let each adjective represent its own
coordinate axis, so that with n traits we shall have as many axes. These coordinate axes will be
oblique, since the traits are known to be at least not all independent. The projection of any one
of these traits A on the oblique axis through another trait B is the cosine of their central angle,
but this is also the correlation between the two traits A and B. The correlations can be
ascertained by experiment, and then our problem becomes that of finding the smallest number
of orthogonal coordinate axes in terms of which we can describe all of the traits whose
intercorrelations are known.
The actual data that we must handle are subject to chance errors. It is therefore profitable to
see how the geometrical manner of thinking about this problem is affected by the chance
errors. It can be shown that each test may be thought of as a vector in space of as many
dimensions as there are independent mental factors. If the test is perfect, then it is represented
geometrically by a unit vector. If the test has a reliability less than unity, then the length of the
vector is reduced. In fact, the length of a test vector in the common factor space is the square
root of its reliability. If the test has zero reliability, then it determines nothing and this fact has its
geometrical correspondence in that the test vector is then a zero length so that it determines no
direction at all in the space of mental abilities.
The obtained correlation between two tests is the scalar product of the two test vectors. If the
two tests are perfect, then their scalars are both unity so that the true correlation between the
two tests is the cosine of the angular separation between the two vectors.
We shall consider next two of the fundamental theorems in a generalized theory of factors.[2]
Let us take a table of seven variables as an example. In Figure 1 we have shown such a table
toward the right side of the diagram. We may call it the correlational matrix. In a table of this
kind we show the correlation of each test with every other test. For example, the correlation
between the two tests A and B is indicated in the customary cell.
In the diagonal cells of a table of intercorrelations we are accustomed to record the reliabilities,
but that is incorrect in factor theory unless the tests have been so chosen that they contain no
specific factors. It is necessary for us to make a distinction between that part of the variance of
a test which is attributable to the common factors and that part of the variance which is unique
for each test. The part which is unique for each test may again be thought of as due to two
different sources, namely, the chance errors in the test and the ability which is specific for the
test. The reliability of a test is that part of the total variance which is due to the common factors
as well as to the specific factor. It differs from unity only by that part of the variance which is
due to chance errors. We need in factor theory another term to indicate that part of the total
variance which is attributable only to the common factors and which eliminates not only the
variance of chance errors but also the specific variance. We have used the term communality to
indicate that part of the total variance of each test which is attributable to the common factors. It
is always less than the reliability unless a specific factor is absent, in which case the
communality becomes identical with the reliability. It is these communalities that should be
recorded in the diagonal cells, but they are the unknowns to be discovered by the factorial
analysis.
At the extreme left of the diagram we have a table of factor loadings. Here the seven tests are
listed vertically and there are as many columns as there are factors. In the present example we
have assumed three factors, so that we have three columns and seven rows. Let us suppose
that the first factor represents verbal ability and that the second factor represents arithmetical
ability. These particular abilities are not independent, but we may ignore that for the moment.
Then the entry a
1
indicates the extent to which the test A calls for verbal ability and the entry a
2
indicates the extent to which it calls for arithmetical ability.
Since we have assumed three factors in this diagram we have three factor loadings for each
test. A table in which the factor loadings are shown for each test we have called a factorial
matrix, while the square table containing the intercorrelations we have called a correlational
matrix.
The experimental observations give us the correlational matrix, so that it may be regarded as
known. The object of a factorial analysis is to find a factorial matrix which corresponds to the
given intercorrelations. There is a rather simple theoretical relation between the given
correlational matrix and the factorial matrix. This relation constitutes the fundamental theorem
of the present factor theory. It is illustrated by an example under the correlation table and it can
also be stated in very condensed form by matrix notation in that the factorial matrix multiplied
by its transpose reproduces the correlational matrix within the observational errors of the given
correlation coefficients.
When we want to make an analysis of a table of intercorrelations, the first thing we want to
know is how many independent common factors we must postulate in order to account for the
given correlation coefficients. One of our fundamental theorems states that the smallest
number of independent common factors that will account exactly for the given correlation
coefficients is the rank of the correlational matrix. Although this theorem is of fundamental
significance it is not possible to apply it in its theoretical form, because of the fact that the given
coefficients are subject to experimental errors and the rank is therefore in general equal to the
number of tests. It is always possible to account for a table of intercorrelations by postulating as
many abilities as there are tests, but that is simply a matter of arithmetical drudgery and nothing
is thereby accomplished. The only situation which is of scientific and psychological interest is
that in which a table of intercorrelations can be accounted for by a relatively small number of
factors compared with the number of tests.
There is no conflict between the present multiple factor methods and the tetrad difference
method. When a single factor is sufficient to account for the given coefficients, then the rank of
the correlational matrix must be 1, but the necessary and sufficient condition for this is that all
of the second order minors shall vanish. Now if you expand the second order minors in a table
of correlation coefficients you find that you are in fact writing the tetrads. Hence the tetrad
difference method is a special case of the present multiple factor theorem.
It would be possible to extend the tetrad difference method by writing the expansions of the
minors of higher order and in that manner to write formulae for any number of factors which
correspond to the tetrads for the special case when one factor is sufficient. Such a procedure is
unnecessarily clumsy. In fact, there should be no excuse for ever computing any more tetrads.
Several better methods are available which give much more information with only a fraction of
the labor that is required in the computation of tetrads.
We return now to the multiple factor analysis of personality. In Table 1 we have a list of sixty
adjectives that are in common use for describing people. These adjectives together with their
synonyms were given to each of 1300 raters. Each rater was asked to think of a person whom
he knew well and to underline every adjective that he might use in a conversational description
of that person. Since it was not necessary for the rater to reveal the name of the person he was
rating it is our belief that the ratings were relatively free from the inhibitions that are usually
characteristic of such a task. With 1300 such schedules we determined the tetrachoric
correlation coefficient for every possible pair of traits. Since there were sixty adjectives in the
list we had to determine 1770 tetrachoric coefficients. For this purpose we developed a set of
computing diagrams which enable one to ascertain the tetrachoric coefficients with correct sign
by inspection.[3] Each coefficient can be ascertained in a couple of minutes by these computing
diagrams.
The table of coefficients for the sixty personality traits was then analyzed by means of multiple
factor methods[4] and we found that five factors are sufficient to account for the coefficients.
We reproduce in Figure 2 the distribution of discrepancies between the original tetrachoric
coefficients and the corresponding coefficients that were calculated by means of the five
factors. It has a standard deviation of .069. The average standard error of thirty tetrachoric
coefficients chosen at random in this table is .052.
It is of considerable psychological interest to know that the whole list of sixty adjectives can be
accounted for by postulating only five independent common factors. It was of course to be
expected that all of the sixty adjectives would not be independent, but we did not foresee that
the list could be accounted for by as few as five factors. This fact leads us to surmise that the
scientific description of personality may not be quite so hopelessly complex as it is sometimes
thought to be.
Next comes the natural question as to just what these five factors are, in terms of which the
intercorrelations of sixty personality traits may be described. Each of the adjectives can be
thought of as a point in space of five dimensions and- the five coordinates of each point
represent the five factor components of each adjective.
We shall consider a three-dimensional example in order to illustrate the nature of the
indeterminacy that is here involved (Figure 3). Let us suppose that three factors are sufficient to
account for a list of traits. Then each trait can be thought of as a point in space of three
dimensions. In fact, each trait can be represented as a point on the surface of a ball. If two
traits A and B tend to coexist, the two points will be close together on the surface of the ball. If
they are mutually exclusive, so that when one is present the other is always absent and vice
versa, then the two traits are represented by two points that are diametrically opposite on the
surface of the ball such as A and D. If the two traits are independent and uncorrelated, such as
A and C, then they will be displaced from each other in the same way as the north pole and a
point on the equator, namely by ninety degrees.
Now suppose that the traits have been allocated to points on the surface of the sphere in such
a manner that the correlation for each pair of traits agrees closely with the cosine of the central
angle between the corresponding points on the surface of the sphere. Then we want to
describe each of these traits in terms of its coordinates, but we should first have to decide
where to locate at least two of the three coordinate axes. This is an arbitrary matter, because
the internal relations between the points, that is, the intercorrelations of the traits, remain
exactly the same no matter where in the sphere we locate the coordinate axes. That is, when
the points have been assigned on the surface of the ball, we have not thereby located the
respective coordinate axes. It is possible to determine uniquely how many independent
common factors are required to account for the intercorrelations without thereby determining
just what the factors are. We may therefore use any arbitrarily chosen set of orthogonal axes.
It is psychologically more illuminating to investigate constellations of the traits. In the factor
analysis of the adjectives, constellations of traits reveal themselves in that the points which
represent some of the traits lie close together in a cluster on the surface of a five-dimensional
sphere. One or two examples will illustrate this type of analysis with respect to the personality
traits. We find, for example, that the following traits lie close together in a duster, namely,
friendly, congenial, broad-minded, generous, and cheerful. It would seem, therefore, that as far
as the five basic factors are concerned, whatever be their nature, these several traits are very
much alike as far as can be determined by the way in which we actually use these adjectives in
describing people.
Another such cluster of adjectives which are used as though they signified the same
fundamental trait are the adjectives patient, calm, faithful, and earnest. They cling close
together in the factorial analysis. Another small group is found in the three traits persevering,
hard-working, and systematic, which lie close together. Still another one is the cluster of traits
capable, frank, self-reliant, and courageous.
It is of psychological interest to note that the largest constellation of traits consists of a list of
derogatory adjectives. Such a cluster is the following: self-important, sarcastic, haughty,
grasping, cynical, quick-tempered, and several other derogatory traits that lie close by. It clearly
indicates that if you describe a man by some derogatory adjective you are quite likely to call
him by many other bad names as well. This lack of objectivity in the description of the people
we dislike is not an altogether unknown characteristic of human nature.
The schedules used in this study contained 120 adjectives, since every one of the 60 adjectives
in the principal list was represented also by a synonym. This enabled us to ascertain the
correlation between each pair of synonyms and we used it as an index of the consistency with
which the trait was judged. Let us consider this correlation to be an estimate of the reliability of
the judgments of the trait. By factor analysis we know the communality of each trait. It is that
part of its total variance which it has in common with the other adjectives, namely, the five
common factors. The difference between these two variances is the specificity. It shows the
magnitude of the specific factor in each adjective.
We have listed these differences for each adjective in Table 2 and they yield some
psychologically interesting facts. We find, for example, one adjective which has a surprisingly
high specificity of nearly .60, and we want to know, of course, what kind of trait it represents.
We find that it is the adjective talented. It seems reasonable to guess that this adjective refers
largely to the intellectual abilities which are not represented by this list. When this study is
repeated, we shall include several adjectives of this type, so that intelligence may be
investigated as a vector in relation to the personality traits.
Another item with a high specificity is the adjective awkward. This means that the trait awkward
has something about it which is unique in our list of sixty and which is not represented by the
five common factors. This trait is probably ease and facility in body movement, which is
certainly not represented in the rest of the sixty traits. When this study is repeated we may or
we may not include several adjectives of this type, depending on whether we want to include
this additional factor in a study of personality. Another adjective with high specificity is religious
and the explanation is undoubtedly along the same lines because this is the only adjective in
the list that refers to any kind of religiousness.
Studies of this sort should be repeated until every important trait is represented by several
adjectives. The analysis should yield as many independent factors as may be required. When
the factorial analysis is complete, the specifics should all vanish or they should be relatively
small. Then the communalities and the reliabilities will have nearly the same value. The
constellations to be found in such an analysis will constitute the fundamental categories in
terms of which a scientific description of personality may be attained.
There is no necessary relation between the number of factors and the number of constellations.
A system of tests might be found which can be accounted for by several factors even though it
contains no constellations. Fortunately the constellations can be isolated in a very simple
manner when the coefficients have been corrected for attenuation.
I turn next to a factor study of the insanities. I have used a very elaborate set of data which Dr.
Thomas Verner Moore of Washington, D.C., collected and investigated by other factor
methods. Dr. Moore worked with a list of forty-eight symptoms, thirty-seven of which are listed
in Table 3. He recorded the presence-absence, or a rating or test measure of each symptom for
each of several hundred patients. With these records it was possible to ascertain to what extent
any two symptoms tend to coexist in the same patient. For example, the extent to which the two
symptoms excited and destructive tend to coexist in the same patient is indicated by the
tetrachoric correlation of +.71. The records were sufficiently complete so that we could prepare
a table of intercorrelations of the tetrachoric form for thirty-seven symptoms or 666 coefficients.
These computations were also made by the computing diagrams.
The multiple factor method was then applied to the table of 666 coefficients and we found that
five factors are sufficient to account for the correlations, with residuals small enough so that
they can be ignored. The communalities were then computed for each one of the symptoms
and we found that about ten of the symptoms do not contain enough in common with the other
symptoms to warrant their retention in a factor study. In other words, about ten of the
symptoms are either so specific in character or so unreliable as to estimates that they do not
yield significant correlations with the several other symptoms. This left twenty-six symptoms
which are more or less related and for which the factorial clusters of symptoms could be
profitably investigated.
In Table 4 we have listed the psychotic symptoms which lie in each of several constellations.
We find, for example, that the following symptoms are functionally closely related, namely,
mutism, negativism, being shut-in, stereotypism of action, stereotypism of attitudes.
Stereotypism of words, and giggling. These seven traits are evidently related in that they tend
to be found in the same patients and we recognize the list as descriptive of the catatonic group.
Another constellation consists in the presence of logical fallacies, defect in memory, defect in
perception, and defect in reasoning. This is a constellation of symptoms that indicates a
derangement of the cognitive functions of the patient as contrasted with derangement