The Growth
of Logical Thinking
from Childhood to Adolescence
BOOKS BY JEAN PIAGET
AVAILABLE IN ENGLISH TRANSLATION
The Language and Thought of the Child (1926)
Judgment and Reasoning in the Child (1928)
The Child's Conception of the World (1929)
The Child's Conception of Physical Causality (1930)
The Moral Judgment of the Child (1932)
The Psychology of Intelligence (1950)
Play, Dreams, and Imitation in Childhood (1951)
The Child's Conception of Number (1952)
The Origins of Intelligence in Children (1952)
Logic and Psychology (1953)
The Construction of Reality in the Child (1954)
[WITH BARBEL INHELDER]
The Child's Conception of Space (1956)
The Growth of Logical Thinking from Childhood to
Adplescence (1958)
DATES GIVEN IN PARENTHESES ARE
THE FIRST PUBLICATION DATES OF THE ENGLISH TRANSLATIONS
The Growth of
Logical Thinking
FROM CHILDHOOD TO ADOLESCENCE
__
B
COPYRIGHT 1958 BY BASIC BOOKS, INC.
PRINTED IN THE UNITED STATES OF AMERICA
LIBRARY OF CONGRESS CATALOG CARD NUMBER: 58-6439
DESIGNED BY SIDNEY SOLOMON
FIRST PRINTING MARCH 1958
SECOND PRINTING FEBRUARY 1959
Published in France
under the title De La Logique de T enfant a la logique de T adolescent
by Presses Universitaires de France
Translators' Introduction:
A Guide For Psychologists
SOME of those who are drawn by the psychological title to choose
this book from the display shelf may be tempted to put it down
when they discover how many pages are filled with v's and D 's or
p's and q's. But their interest is not misplaced, for this is not a work
on logic. It is a book which should be relevant both to the experi-
mental psychologist interested in cognition and to the clinical
psychologist or psychiatrist who deals with children or adoles-
cents and who would like to know more about ego development.
Logic does appear, both in that it is the more strictly logical
aspects of the child's and the adolescent's thinking which make
up the subject matter of the book and in that logical notation is
used to provide a structural model of their thought processes. But
this, we think, is not sufficient reason for putting it down, even
for the person whose traumatic experiences with high-school
mathematics have erected barriers around that part of the cogni-
tive field labeled "abstract symbolism."
Nevertheless, the book poses a number of problems for the
reader who is not familiar with the authors* methods and basic
assumptions and who has no formal training in logic. On the one
hand, it is a new installment in a long series of empirical works on
the child's mental processes: it goes one step more up the genetic
scale and covers the transition to adolescence. But in addition
viii TRANSLATORS' INTRODUCTION
and in this respect it goes far beyond most of the earlier works
it is an attempt to isolate and describe the mental structures on
which these reasoning processes are based. It is here that logic
comes in. The empirical material is complemented by a structural
analysis which uses symbolic logic as a tool. The set of mental
structures which characterize the reasoning of the /-ii-year-old
child is isolated and differentiated from the structures which char-
acterize the reasoning of the 12-is-year-old adolescent.
As the authors state in the preface, this is a collaborative work
based on an after-the-fact convergence. Professor Inhelder is pri-
marily an experimental child psychologist and, at the time the
work was conceived, was engaged in the study of adolescent
reasoning. Professor Piaget, on the other hand, is an interdisci-
plinary thinker, and, besides the better known work on the
thought of the child, he has also done independent work in logic.
In comparing results of their respective recent work, it was dis-
covered that Prof. Piaget's logical analysis provided the appro-
priate structural model for the data on adolescent reasoning col-
lected by Prof. Inhelder.
Their collaboration rests on a view of the relationship between
logic and psychology which Piaget has exposed elsewhere. 1 To
sum it up briefly, logic and psychology are two independent dis-
ciplines: the first is concerned with the formalization and refine-
ment of internally consistent systems by means of technically
purified symbolism; the second deals with the mental structures
that are actually found in all human beings, independent of
formal training or the use of particular notational symbols and
regardless of consistency or inconsistency, truth or falsehood.
But, although the formalization of systems as an activity in its
own right belongs to logic alone, logic may be applied as a the-
oretical tool in the description of the mental structures that gov-
ern ordinary reasoning, as is done here. As an attempt to describe
such structures, the present work is obviously of interest to the
psychologist. But, since Piaget's work presupposes some under-
standing of the methods and concepts of two fields (and he does
not attempt to translate concepts across academic boundaries for
the uninitiate) his innovations are not easily assimilated. For this
iSee Piaget, Logic and Psychology (New York: Basic Books, 1957).
TRANSLATORS INTRODUCTION IX
reason we will try, in this "Guide/* to furnish a few landmarks for
the psychologist, experimental or clinical, by defining in terms
more familiar to American psychology a few of the basic concepts
used.
Although, over the last few decades, child psychology has on
the whole been a more prominent focus of attention in the United
States than in Europe, the work of the Piaget school has had little
significant influence on this side of the Atlantic. The failure of
the concepts to spread can be explained partially from the fact
that, both theoretically and methodologically, Piaget occupies a
sort of midway point between the main currents in American
psychology. His direct plunge into complex human functioning
and his neglect of tables of statistical significance or systematic
response variation in favor of running commentary on selected
protocols in the presentation of data have separated him from
those groups which most emphasize methodological rigor. But his
work is equidistant from that of the clinically-oriented psycholo-
gists and those currents which touch on sociology, social psy-
chology, or anthropology, since he has no grounding in motivation
theory and for the most part has chosen problems relative to
cognitive functioning taken in isolation from any motivational
variables. Moreover, since both sides of the American psycho-
logical world tend to divorce themselves from any philosophical
tradition, his rationalist framework and ventures into philosophy
have not been easily assimilated by either. Perhaps one could say
that Piaget uses logic in a way analogous to the American use of
theories of motivation (either reinforcement theory or psycho-
dynamics) as an external frame of reference for study of the learn-
ing process.
But within its own framework, the Piaget method is both flexible
and coherent and in a sense reconciles clinical and experimental
approaches. Its basis is genetic i.e., intelligent behavior is ana-
lyzed with respect to the growth continuum. It is experimental in
that constant problems or questions are presented to children
of varying ages in samples large enough for general significance
to be attributed to the differences found between age levels e.g.,
over 1500 subjects were tested for this work. Sometimes tests have
been used; in these cases, success at a particular age is judged
X TRANSLATORS INTRODUCTION
on the basis of a statistical norm. 2 Other quantitative devices, such
as counting the occurrence of certain types of logical connectives
relative to age in the spontaneous speech of children, have been
used. For further problems, where response types may only par-
tially correlate with age, medians have demonstrated that the age
variable accounts for at least some of the systematic variation. 3
Thus some measurable growth in intelligence has been isolated.
But Piaget has not restricted his work to the gathering of meas-
urable data. Like most clinicians, he has been very much con-
cerned with some of the limitations and systematic biases inherent
to quantifiable tests. Given the goal of describing the spontaneous
intelligence of the child rather than his intelligence as seen through
adult eyes, he usually chooses questions and evaluates answers
in a way which the psychiatrist will find more sympathetic than
will the experimental psychologist. Some works in particular
The Child's Conception of the World 4 are based almost entirely
on an intuitive attempt to explore an inner world and use no
methodological paraphernalia beyond the skillful choosing of
questions and evaluation of answers.
But, actually, the bulk of the research is neither experimental
nor clinical in these two polar senses. It depends little on quan-
tification of specific responses since intelligence is considered as
a whole and often, as in the present study of prepositional logic,
in its most complex and highly integrated forms. Nevertheless, it
is systematic and empirical in that various aspects of the child's
intelligence have been taken up in turn and examined through
the presentation of the same well-defined questions to large sam-
ples of subjects. And since a continual effort has been made to
test hypotheses and reformulate the theoretical whole during
more than twenty-five years of work, a solid body of knowledge
2 A test is not considered passed until 75 per cent of the children tested suc-
ceed. See Piaget, Judgment and Reasoning in the Child (Humanities Press,
1952)-
& See Piaget, The Moral Judgment of the Child (Free Press, 1948). It was
discovered that children's expectations in regard to punishment differ sig-
nificantly with age and at least to some extent independently of family social-
ization methods,
* Piaget, The Child's Conception of the World (Humanities Press, 1951).
See the introduction to this work for a view of the relationship between testing
and clinical method and a conception of the latter as used for the understand-
ing of the child's mentality.
TEANSLATORS' INTRODUCTION 3Ci
has been obtained. The criticism has been made that neglect of
motivational factors detracts from the significance of the results.
But, as Piaget states in the Genetic Epistemology, 5 the difference
between science and philosophy is that the former tries to relate
everything to everything else whereas the latter tries to delimit
problems and find specific methods for dealing with them. The
problems concerning validity of this schema across cultural lines,
variation in function of motivational factors, etc., are still open. 6
In the last chapter of this work, the authors go beyond the purely
cognitive and attempt to draw out the consequences of intellec-
tual development for the affective and social psychology of the
adolescent.
Over the series of works which attack intelligence at different
points on the growth continuum and focus on different functions,
the over-all aim has been to trace the development of intelligence
as it comes to deal with increasingly complex problems or as it
deals with simple problems in increasingly more efficient ways.
The following are the four major stages of growth which have
been delineated; the present work deals primarily with the transi-
tion from the third to the fourth. 7
The first, covering the period from birth to about two years, is
the sensori-motor stage. This is when the child learns to coordinate
perceptual and motor functions and to utilize certain elementary
schemata (in this context, a type of generalized behavior pattern
or disposition) for dealing with external objects. He comes to know
that objects exist even when outside his perceptual field and
coordinates their parts into a whole recognizable from different
perspectives. Elementary forms of symbolic behavior appear, as
5 Piaget, Introduction & r&pistemologie gntique (Presses Universitaires
de France, 1950), Vol. I, p. 7.
6 Some attempts have already been made to combine the findings of Piaget
with those of psychoanalysis. Piaget himself takes up problems relative to
affectivity and discusses the relationship between his own theory and that of
Freud in Play, Dreams, and Imitation in Childhood (Norton, 1952), and
Charles Odier, in Anxiety and Magic Thinking (International Universities
Press, 1956), attempts to relate the psychoanalytic theory of regression to
Piaget's model of the early stages in ego formation. See also Rapaport, David,
ed., Organization and Pathology of Thought (Austen Riggs Foundation, 195 1 ).
pp. 154-92.
7 Concise definitions of the stages can also be found in Tanner and Inhelder,
eds., Discussions on Child Development (Tavistock Publications, 1956), and
Piaget, The Psychology of Intelligence (Routiedge & Kegan Paul, 1956).
Otii TRANSLATOKS INTRODUCTION
for example in the child who opens and shuts his own mouth while
"thinking" about how he might extract a watch chain from a half-
open matchbox. Expressive symbolism is also seen, as when
Piaget's daughter at one year and three months lies down and
pretends to go to sleep, laughing as she takes a corner of the
tablecloth as a symbolic representation of a pillow. 8 From the
behavioral standpoint, this period is covered in The Origins of
Intelligence in Children 9 and from that of the organization of
the perceptual field and the construction of the permanent object
in The Construction of Reality in the Child. 10
The preoperational or representational stage extends from the
beginnings of organized symbolic behavior language in particu-
laruntil about six years. The child comes to represent the external
world through the medium of symbols, but he does so primarily
by generalization from a motivational model e.g., he believes
that the sun moves because "God pushes it" and that the stars,
like himself, have to go to bed. He is much less able to separate
his own goals from the means for achieving them than the opera-
tional level child, and when he has to make corrections after his
attempts to manipulate reality are met with frustration he does so
by intuitive regulations rather than operations roughly, regula-
tions are after-the-fact corrections analogous to feedback mecha-
nisms (cf. note p. 246). In the balance scale problem (Chap. 11),
for example, we see that the preoperational subjects sometimes
expect the scale to stay in position when they correct a disequilib-
rium by hand. They may, from an intuitive feeling for symmetry,
add weight on the side where it lacks but may equally well add
more on the overloaded side from a belief that more action leads
automatically to success.
Protocols on this stage are found throughout the works, includ-
ing this one. The Child's Conception of the World gives it the most
attention from the standpoint of thought content. The Child's
Conception of Space n takes up where The Construction of Reality
leaves off in dealing with the perceptual aspect and the structuring
of the spatial field.
8 Play, Dreams, and Imitation in Childhood, p. 96.
$ International Universities Press, 1952.
10 Basic Books, Inc,, 1954.
n Piaget and Inlielder, The Child's Conception of Space (Routledge &
Kegan Paul, 1956).
TRANSLATORS' INTRODUCTION xiii
Between seven and eleven years, the child acquires the ability to
carry out concrete operations. These greatly enlarge his ability to
organize means independently of the direct impetus toward goal
achievement; they are instruments for dealing with the properties
of the immediately present object world.
The stage of concrete operations has probably been more exten-
sively studied than any other, but it is also that at which the
greatest gaps are found in the list of English translations. The
Child's Conception of Physical Causality 12 is especially interest-
ing in that it covers many of the experiments used here, but it was
written before the major phase of theoretical development. The
Child's Conception of Space is theoretically closer to the present
work and is in a sense its complement in covering the transition
from the preoperational stage to the stage of concrete operations;
it presents more fully some of the logical formulations used as the
base line for discussing the transition to adolescence.
The fourth and final phase, preparatory to adult thinkuag, takes
place between twelve and fifteen years and involves the appear-
ance of formal as opposed to concrete operations. It is covered for
the first time in detail in this work. Its most important features
are the development of the ability to use hypothetical reasoning
based on a logic of all possible combinations and to perform con-
trolled experimentation.
Both the third and the fourth stages are operational as distin-
guished from the first two. The concept of operation has been
elaborated gradually since Piaget's early work, partly in response
to criticisms from Anglo-Saxon psychology 13 that the verbal
aspects of intelligence had been overemphasized at the cost of
actions. An operation is a type of action: it can be carried out
either directly, in the manipulation of objects, or internally, when
it is categories or (in the case of formal logic) propositions which
are manipulated. Roughly, an operation is a means for mentally
transforming data about the real world so that they can be organ-
ized and used selectively in the solution of problems. An opera-
tion differs from simple action or goal-directed behavior in that
it is internalized and reversible. According to the authors: 14
12 Harcourt, Brace and Co., 1930.
18 Logic and Psychology, p. ist.
14 Definition given by the authors.
xiv TRANSLATORS' INTRODUCTION
An operation is a reversible, internalizable action which is bound up
with others in an integrated structure.
From the equilibrium standpoint [see Chap. 16], a transformation is
reversible when it gives rise to complete compensation. From the struc-
tural standpoint [see Chap. 17], it is reversible when it can be canceled
by an inverse transformation (as for example the direct and inverse
operations comprising the "group" of transformations found in formal
thinking) .
The simplest definition of a reversible operation as it can be
observed in concrete stage behavior is an action already performed
which is symmetrically undone: e.g., the child who puts a weight
on the balance scale and realizes that it tips too far can take it
off and search systematically for a lighter one, rather than add
more weight simply for the sake of corrective action. With the
advent of operations, the margin of trial-and-error is greatly de-
creased because the child selects means on the basis of an internal
structure (in this example the structure is a serial order of weights).
But even the most complex operations of prepositional logic are
seen as having their beginnings in actions which when internalized
develop into highly differentiated mental structures.
From the theoretical standpoint, it is the structural integration
of concrete and formal operations which is the principal concern
of the present work. Although the number of intelligent acts of
which a child is capable at any given age obviously depends on
learning in a quantitative sense and on the situations which he
happens to confront, the range of available operations can be
described in terms of a limited number of interdependent struc-
tures. The structures found and the way in which they are in-
tegrated depends on the stage of development considered; each
set of structures can be related to a particular group of logical
forms. Thus the concrete structures depend on the logic of classes
(for class-inclusion operations) and the logic of relations (for
serial-ordering), whereas the adolescent's propositional logic de-
pends on the integration of "lattice" and "group" structures in
the structured whole. This obviously does not mean that the child
or the adolescent acquires class logic or propositional logic in a
formal sense. Rather, it means that in dealing with concrete prob-
lems he arrives at answers which imply the presence of certain
logical forms although he does not isolate them from content as
TRANSLATORS INTRODUCTION XV
does the logician. The logical forms both are present in the child's
reasoning and serve as a structural model for analysis of it
The structures are integrated at each stage in the sense that
each partial operation is used in relation to the totality of those
which are available: 15
Structural integration occurs when elements are brought together in a
whole which has certain properties as a whole and when the properties
of the elements depend partially or entirely on the characteristics of
the whole. Some examples are: classifications, serial orders, correspond-
ences, matrices, "groups," lattices, etc.
Each set of structures has its limitations in terms of the field which
can be covered. The concrete stage protocols show the point at
which the limits of the "grouping" structures are reached, neces-
sitating the development of a new form of integration.
Below are some definitions of the operations based on the "group-
ing" structure which develop during the concrete stage (see Mays'
introduction to Logic and Psychology for a more formal set of
definitions):
i. Class inclusion operations. These relate to the child's abil-
ity to manipulate part-whole relationships within a set of cate-
gories. In order to define the operations for class inclusion,
logicians use the terms addition, subtraction, and multiplication
in a qualitative sense which is analogous to their use in defining
arithmetical operations. Two classes can be added up so that they
are included in a larger one: boys + girls = children; children +
adults = people i.e., A -f- A' = B. By the same token, a part can
be subtracted from the whole: people adults = children. When
the child can do this systematically, reversibility is present in that
when the child needs to generalize or discriminate he can pass
from the part to the whole and back again.
Likewise, classes can be multiplied. The child obtains four sub-
classes by discriminating between objects or properties accord-
ing to two intersecting criteria: A (geometric figures) divided into
B (squares) and B' (circles) and multiplied by (X) AI (their colors)
B (red) and B\ (green) gives BBi + BB'i + B'Bi + B'B'i (red
squares, red circles, green squares, and green circles). These are
the double-entry tables referred to frequently in the text. The
i 5 Definition given by the authors.
XOi TRANSLATORS INTRODUCTION
class multiplication operation is the means the 7-n-year-old
child uses for discriminating between two (or more) independent
variables.
Such relationships seem obvious, but experiments have shown
that they are not made systematically when the preoperational
child uses categories. Before the age of about seven, for example,
children, given a box containing about eighteen brown and two
white beads, all wooden, and asked whether there are more brown
or more wooden beads, reply that there are more brown ones
because only two are white, 16 That the categories are available
and observations correct is shown by the fact that the younger
children, when asked the questions separately, give correct an-
swers as to the relative proportions of brown, white, and wooden
beads. However, without class inclusion operations they cannot
deal with the parts and the whole at the same time, and thus they
make a false generalization.
2, Serial ordering operations. These operations relate to abil-
ity to generalize along a linear dimension or to arrange objects
(or their properties) in series. They are based on the logic of rela-
tions rather than class logic: the signs are > and < (greater than
or less than). At about seven years, when the child is given a set
of sticks to arrange in order of size, he proceeds by taking the
smallest first (or the largest), then the smallest of those which are
left, and so on, rather than beginning at random and rearranging
when discrepancies are noticed. Mentally he is able to conclude,
from A > B and B > C, that A > C. Other empirical factors are
ordered in the same way at different points during the concrete
stage e.g., weights are ordered later than lengths, at about nine
to ten years. When he has acquired serial ordering operations, the
child is able to register in detail the changes in magnitude of a
given variable e.g., in the angle problem (Chap, i), he sees that
the ball's course changes in direct relation to his angle of firing.
Actually, in the mental operation he puts the angles into a series
of increasing magnitude of which each member corresponds to
a trial.
Secondly, given two independent series, the child learns to find
correspondences between them (sign <-). In other words, he be-
gins to relate two variables accurately by observing concurrent
16 The Psychology of Intelligence, p. 133.
TRANSLATORS' INTRODUCTION xvii
changes. In the angle problem he sees that the series of angles of
firing corresponds to the series of angles in the ball's course to-
ward the goal, thus he is able to adjust his firing correctly. As
opposed to the preoperational child, he comes to know which
means goes with which end.
In sum, the concrete operations are based on the logic of classes
and the logic of relations; they are means for structuring immedi-
ately present reality. During the formal stage, on the other hand,
the adolescent comes to control formal logic. Rather than reason-
ing with directly given data alone, he begins to reason with propo-
sitions and with hypotheses. The concept of the "concrete opera-
tion" was developed as a means of applying logical analysis to the
child's operations when he is dealing directly with objects and
thus of avoiding the fallacy of judging the child's intelligence in
terms of that of the logician. But in the study of adolescent rea-
soning, which is much closer to that of the logician than that of
the seven-year-old child, the concept of operation has by no means
been abandoned. Rather, formal logic is also conceived of as a
set of mental operations, although they are based on a different
structure. From one standpoint the formal operations differ from
the concrete in that they are performed on propositions rather
than directly on reality: 17 they are a set of transformations which
can be made as a way of generalizing from the data at hand. As
opposed to concrete operations which are limited to the empiri-
cally given, they make it possible for the subject to isolate vari-
ables and to deduce potential relationships which can later be
verified by experiment.
The propositions on which formal operations are performed
refer both to variables hypothesized as causal and to the effects
they produce in the experimental situation. In the flexibility prob-
lem, for example (Chap. 3), where the adolescent is able to isolate
and combine variations in flexibility (which depends on a number
of factors), what the subject does from the behavioral standpoint
is to ascertain a number of facts and formulate them as proposi-
tions e.g., "This rod is steel; it is also long," "That rod is steel,
but it is shorter," etc. These refer to the potentially causal prop-
erties of the rods he has chosen to test. He also ascertains the
17 This does not mean that f onnal logic is verbal and concrete logic is non-
verbal. (See pp. 252-254.)
xviii TRANSLATORS' INTRODUCTION
results of his experiments e.g., a long steel rod bends, a short
brass one does not, a short steel one does, etc. The formal opera-
tions enable him to combine these propositions mentally and to
isolate those which confirm his hypotheses on the determinants
of flexibility. The combinatorial system is the structural mechanism
which enables him to make these combinations of facts.
In other words, formal operations are ways of transforming
propositions about reality so that the relevant variables can be
isolated and relations between them deduced. The operations de-
scribed (see p. 103, the sixteen binary operations) are different
kinds of combination, any one of which may be appropriate de-
pending on the particular situation observed. The frequently
recurring term "association" refers to an observed conjunction of
facts e.g., "this rod is steel" and "this rod (the same) bends enough
to touch the water." The kind of relationship formulated depends
on the particular association of facts observed: e.g., implication
means that every time one variable appears in the experimental
situation, a particular result is present every time the string of a
pendulum is lengthened, the amplitude of the swing increases
and if this variable is not present, that result is never present the
amplitude cannot be increased without lengthening the string.
The formula for implication is p D q. Another type of combina-
tion is disjunction e.g., the situation in the flexibility problem
when the subject sees that sometimes short rods bend but at other
times they do not (p.q v p.q v p.q v p.q) you can have short rods
that bend, or short rods that do not bend, or long rods that bend,
or long rods that do not, with v symbolizing "or/* The p's and q's
with their negations p and q stand for the observation that a given
variable or its result is or is not found in the experimental situa-
tion. When the disjunctive relationship is observed, the subject
concludes that length alone does not determine flexibility. Since,
unlike the concrete level child, he does not have to limit his con-
sideration to a single relationship at a time, he can then proceed
to the consideration of other variables which might determine it.
He "feeds" his information into a general mechanism the com-
binatorial system or structured whole which assimilates the facts
in the form of propositions and arranges them according to all
possible combinations (the logical term is composition). He can
move around among these possibilities (there is reversibility and
TBANSLATORS' INTRODUCTION XIX
complete compensation) so as to select a situation that would tell
him which other variables are involved and which of a number of
potential explanations in fact explain what he saw: for example,
that length alone does not determine flexibility so that another
factor was involved for the short rod that bent; length does par-
tially determine flexibility so that, in the situation in which the
long rod did not bend, there was a counteracting factor, etc. Thus
the prepositional operations always operate as a whole and as a
whole which is structured internally. The adolescent both dis-
criminates between parts (variables or specific events which occur,
such as the rod's bending beyond the degree required for flexibil-
ity, etc.) and generalizes to an over-all explanation of the results
and to other potential situations. As in Gestalt psychology, the
development of thought is seen as moving toward the construc-
tion of wholes, but, as is emphasized to a greater extent, it also
moves toward a finer discrimination of elements within the whole.
The structured whole is structured precisely in the sense that the re-
lationships between its parts are separable as well as integrated. 18
In reading the book, various aspects of the structural model are
best seen in individual experimental problems. Thus, the com-
binatorial system is presented in its purest form in the coloring
liquids problem (Chap. 7) where the experiment itself calls for
the systematic combining of a number of variables given as dis-
crete; the differences between the adolescent method, which goes
around the full circle of possibilities each time, and the child's
method of one-by-one combination, which always leaves some
18 In the translation "structured whole" we have had to sacrifice some of
the connotations of the original in the interest of securing a meaningful and
communicable equivalent. The French term is ensemble des parties, where
ensemble means both "whole" (with the implication of integration as used by
Gestalt psychology) and "set" as used by mathematical set theory. For logicians
the term should be translated "the set of all sub-sets." Readers of Mays' trans-
lation of Logic and Psychology will notice a terminological difference in that
he has used "structured whole" for the more general structures ffensemble.
There are structures d'ensemble, which we have translated as "structural inte-
gration" or "integrated structures," at both concrete and formal stages, but
the type of structure found differs between the two. The combinatorial system
or "structured whole," on the other hand, is the particular type of structural
integration which characterizes formal thinking. Moreover, the term structure
d'ensemble is employed by a school of mathematicians, in particular the
Bourbaki, whose structural research lies at the foundation of this work. Here
also, ensemble means "set" as used in set theory as well as "whole" or "inte-
XX TBANSLATORS INTRODUCTION
steps untouched, become obvious in the protocols. The contrast-
ing process in which discrete variables have to be extricated from
a situation where they appear in combination is described in
Chap. 3, the multivariate flexibility problem. There one sees that
at the concrete stage the variables somehow tend to stick together
once given combinations have been observed in sharp contrast to
the easy detachability from the given which they acquire at the
formal level. In sum, the structured whole, by virtue of which the
subject is able both to combine parts into a whole and to separate
them from it, might be impressionistically characterized as a sort
of mental scaffolding held up by a number of girders joined to
each other in such a way that an agile subject can always get from
any point vertically or horizontally to any other without trap-
ping himself in a dead end. The other structural forms the 'lat-
tice*' and the "group** we had better leave to the logicians.
ANNE PABSONS
Preface
THE DOUBLE TITLE of this work [on the original French edition, it
was De La Logique de I'enfant & la logique de Tadolescent: Essai
sur la construction des structures operatoires formelles] does not
indicate simply that the authors collaborated in a new way, or
simply a desire to distinguish between their respective contribu-
tions to a common task. Actually it reflects the twofold nature of
the questions which they have asked, and it in no way impairs the
ultimate unity of the conclusions. With reference to the choice of
this title, there is an anecdote worth citing, particularly because
it is a good example of how experimental and deductive methods
can at the present time converge in the area of the operational
analysis of intelligence, given a deductive analysis based on pre-
cise logical techniques. It happened that the second author left
his experimental work for a while in order to complete his Genetic
Epistemology* Treatise on Logic, and an Essay on the Transfor-
mations of Logical Operations. The aim of the two latter works
is to furnish a possible symbolic model of the actual processes of
thinking. During this period the first author and her assistants
undertook a systematic empirical study of the induction of physi-
cal laws in children and adolescents. But this genetic study of
experimental induction led to two unexpected results.
ipiaget, Introduction d rpi$tmologie gntique (Presses Universitaires
de France). Volume I will appear shortly in English translation.
XXti PREFACE
First, in our earlier works we had repeatedly stressed the impor-
tance of the stage of development beginning at 11-12 years. In
our new study it became increasingly clear that this stage was not
simply the culmination point of the /-n-year stage (when con-
crete operations are worked out by the child) but also involved a
period of new structuring leading to another level of equilibrium
at about 14-15 years. So it seemed possible to describe the adoles-
cent's thought in terms of the structuring of certain methods of
experimental induction, and above all in terms of those methods
of systematic verification not found in the child.
As to the second result, the methods of discovery and experi-
mental proof found in the adolescent but not in the child were
shown to be bound up with an entirely new set of operational
structures. These are based on prepositional logic and a "formal"
mode of thought as distinguished from the "concrete" operational
thought found between 7 and 11 years. (The latter requires only
a limited number of operations taken from the logic of classes
and relations.)
The second author intervened at this point, and for the follow-
ing reasons, which well illustrate the convergence referred to
above between the results of experiments and those of deduction.
It became clear that the well-known techniques of prepositional
logic were inadequate to analyze the integrated structures of
operations found in the adolescent's formal thinking. For we also
had to make use of the group of four transformations (inversions
and reciprocities) which one of us has described as necessary to
the functioning of the mechanisms of formal thought. 2
Now, the outstanding feature of the data of the empirical in-
vestigation was that they showed that formal thought is more
than verbal reasoning (prepositional logic). It also entails a series
of operational schemata which appear along with it; these include
combinatorial operations, propositions, double systems of refer-
ence, a schema of mechanical equilibrium (equality between
action and reaction), multiplicative probabilities, correlations, etc.
But in trying to explain how these operational schemata and prep-
ositional logic develop together, we found it was not enough to
2 See Piaget, Trait6 de Logique (Colin, 1949), pp. 264-^286, and especially
Essai sur les transformations des operations logiques (Presses Universitaires
de France, 1952), Chap. II. Neither of these works is available in English.
PREFACE
refer only to the specific operations of propositional logic. For in
addition and this is most important we have to refer to the
"integrated structures" on which they are based-f.e., to the dual
structure of the lattice and the group of four transformations
(Klein group or Vierergruppe) analyzed by the second author in
his work on the transformation of propositional operations.
In other words, while one of us was engaged in an empirical
study of the transition in thinking from childhood to adolescence,
the other worked out the analytic tools needed to interpret the
results. It was only after we had compared notes and were making
final interpretations that we saw the striking convergence between
the empirical and the analytic results. This prompted us to col-
laborate again, but on a new basis. The result is the present work.
But this is not all. The operational structures of adolescent logic
are not only interesting in themselves; they also cast a backward
light on an earlier set of structures, those of the child's concrete
logic. Actually, the only logical operations the child can handle at
the concrete level are the "elementary groupings" of classes and
relations; the class groupings are based on a form of reversibility
which can be called inversion (negation), and the groupings of
relations on another such form, called reciprocity. At this stage
there is no general structure to integrate transformations by inver-
sion and transformations by reciprocity into a single system. But
analysis of the set of four transformations found in the proposi-
tional logic of the adolescent shows how the two forms of con-
crete operational reversibility finally do come to be coordinated
into a single system. 3 Meanwhile, the combinatorial system of
propositional lattices develops as a result of a generalization of
classification. In other words, it seems clear that the twofold struc-
ture found in formal thought is the end product of a series of co-
ordinations as they attain a final level of equilibrium. (This is no
bar to new integrations and continual growth in adult thinking.)
Therefore, an analysis of the mechanisms of formal thought is
indispensable if we are to draw up an operational theory of intel-
ligence which aims at a step-by-step explanation of the successive
and hierarchical organization of thinking as it develops.
This book has two aims: to set forth a description of changes
3 The four transformations: inversion; reciprocity; inversion of the re-
ciprocal, or reciprocation of the inverse; and identical transformation.
XXfa PREFACE
in logical operations between childhood and adolescence and to
describe the formal structures that mark the completion of the
operational development of intelligence. To tie these together the
authors have tried to present the material in a way that would
stress the close relationship between the two. Each of the first
fifteen chapters (Parts I and II) includes an experimental part by
the first author and a brief final analysis by the second author.
This analysis aims to isolate the formal or prepositional structures
found in each case. 4 Chapters 16 and 17 (beginning of Part III)
are the work of the second author, whereas Chapter 18 is a joint
production. In addition, the specific problems of experimental
induction analyzed from a functional standpoint (as distinguished
from the present structural analysis) will be the subject of a
special work by the first author. 5
BAKBEL INHELDER
JEAN PIAGET
* For a more detailed presentation of the symbolism of prepositional opera-
tions employed in the chapter conclusions and in Chap. 17, see Trait de
Logique, Chap. V.
translators' note: The experimental diagrams from this work have been
included in this translation.
Contents
Translators' Introduction: A Guide for Psychologists vii
Preface xxi
PART I
The Development of Propositioned Logic
1. The Equality of Angles of Incidence and Reflec-
tion and the Operations of Reciprocal Implication 3
2. The Law of Floating Bodies and the Elimination
of Contradictions 20
3. Flexibility and the Operations Mediating the
Separation of Variables 46
4. The Oscillation of a Pendulum and the Operations
of Exclusion 67
5. Falling Bodies on an Inclined Plane and Opera-
tions of Disjunction So
6. The Role of Invisible Magnetization and the Six-
teen Binary Prepositional Operations 93
PART II
The Operational Schemata of Formal Logic
7. Combinations of Colored and Colorless Chemical
Bodies 107
8. The Conservation of Motion in a Horizontal Plane 123
CONTENTS
g. Communicating Vessels 133
10. Equilibrium in the Hydraulic Press 148
11. Equilibrium in the Balance 164
12. Hauling Weight on an Inclined Plane 182
13. The Projection of Shadows 199
14. Centrifugal Force and Compensations 210
15. Probable Dispersions and Correlations 224
PART III
The Structural Integration of Formal Thought
16. Formal Thought from the Equilibrium Standpoint 245
17. Concrete and Formal Structures 272
18. Adolescent Thinking 334
Index 351
Part I
THE DEVELOPMENT
OF PROPOSITIONAL LOGIC
IF WE are to explain the transition from the concrete thought of
the child to the formal thought of the adolescent, we must first
describe the development of propositional logic, which the child
at the concrete level (stage II: from 7-8 to 11-12 years) cannot
yet handle. Experimentation shows that after a long period dur-
ing which only operations appropriate to class and relational
groupings and to the numerical and spatiotemporal structures
which resulted from them are used, the beginnings of stage III
(substage III-A, from 11-12 years to 14-15 years; and substage
III-B, from 14-15 years onward) are distinguished by the organ-
ization of new operations performed on the propositions them-
selves and no longer only on the classes and relations that make
up their content.
To study the questions raised by this development, we must
analyze how children or adolescents at stage III go about solving
problems -which appear purely concrete but which experiments
indicate can be resolved only at stage III and which actually pre-
suppose the use of interpropositional operations. Part I of the
present work will be devoted to this analysis.
The Equality of Angles
of Incidence and Reflection
and the Operations
of Reciprocal Implication 1
OUR AIM in this chapter, and in the remainder of Part I, is not
a systematic study of the concept of the equality of two angles.
Actually, we already know how the concept is constructed: that
it is first acquired at the level of concrete operations. 2 But it is
precisely the fact that the concept is akeady so well known by
the time the formal level (stage III) is reached that makes the
reasoning process involved in the discovery of the equality be-
tween the angles of incidence and reflection so instructive. One of
the aims of this study, then, is to isolate the operational mecha-
nisms involved in the formal reasoning process itself, when this
reasoning rests on notions already constructed at the concrete
level.
The experimental apparatus consists of a kind of billiard game.
Balls are launched with a tubular spring device that can be piv-
oted and aimed in various directions around a fixed point. The
1 With the collaboration of H. Aebli, former research assistant, Laboratory
of Psychology, Science Faculty, University of Geneva, professor, ficole
normale sup&ieure, Zurich; L. Muller, former research assistant, Institut des
Sciences de 1'fiducation, University of Geneva; and M. Golay-Barraud, stu-
dent, Institut des Sciences de 1'fiducation.
2 See Piaget and Inhelder, The Child's Conception of Space (Routledge &
Kegan Paul, 1956), Chap. XII, and Piaget, Inhelder, and Szeminska, La
Geometrie spontange de ? enfant, Chap. VTH. (Not transl.)
3
4 THE DEVELOPMENT OF PROPOSITTONAL LOGIC
ball is shot against a projection wall s and rebounds to the interior
of the apparatus. A target is placed successively at different
points, and subjects are simply asked to aim at it. Afterwards, they
report what they observed.
But the equality between the angles of incidence and reflection
is discovered only at stage III-A (11-12 to 14 years) and is often
not formulated until stage III-B (14-15 years). Our problem is to
understand why a concept as familiar after 7-9 years as that of
the equality of two angles is utilized in the induction of an ele-
mentary law only at this late date and, especially, why formal
operations are necessary for its use. We shall try to answer this
question by retracing briefly the ground covered by the child be-
fore his arrival at the formal level, then by examining the latter
more closely.
Stage I
In the course of stage I (up to about 7-8 years) subjects are most
concerned with their practical success or failure, without consid-
eration of means; often even the role of rebounds is overlooked.
The result is that, except toward the end of the stage, the trajec-
tories are not generally conceived of as formed of rectilinear seg-
ments but rather as describing a sort of curve:
DAN (552)* succeeds at first: "I think it works because it's in the same
direction' 9 He adjusts the plunger by himself, but proceeds by em-
pirical trial-and-error. Then he asks spontaneously: "Why do you have
to turn the plunger sometimes? . . . No, you have to put it there
[he fails]. If it could be pushed a little further" [he does this and suc-
ceeds]. But, although he knows how to control the rebounds success-
fully, DAN has no idea that they are made up of angles: the curve he
describes with his finger is not tangent to the wall; he takes into ac-
count the starting point and the goal but not the rebound points.
WIRT (5 ; 5): "It came out here and it went over there. . . . I'm sure to
make it" etc. He succeeds occasionally but describes the trajectories
3 With a rubber buffer.
* Figures within parentheses indicate age in years and months i.e., five
years; two months.
FIG. 1 . The principle of the billiard game is used to demonstrate the
angles of incidence and reflection. The tubular spring plunger can be
pivoted and aimed. Balls are launched from this plunger against the
projection wall and rebound to the interior of the apparatus. The
circled drawings represent targets which are placed successively at
different points.
6 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
with his finger only in the form of curves not touching the walls of the
apparatus; he considers only the goal as if there were no rebounds.
NAN (5 ; 5), on the other hand, is astonished by the detour made by
the ball which first touches the walls. "It always goes over there." But
he does not succeed in adjusting his aim: "Oft, it always goes there.
, . . it witt work later."
PIT (5 ; 5) notes about one of his tries [a failure] : "It was straight [as
if this were an exception]. Why didn't it hit it? I thought I hit it"
[no comprehension].
ANT (6 ; 6) becomes aware of the existence of rebounds at the same
time that he notices the rectilinear character of the trajectory seg-
ments: "It [the ball] hits there, then goes over there" [his gesture in-
dicates straight lines],
PER (6 ; 6), in contrast, in spite of his age, resorts to the curvilinear
model: "It goes there and it turns the other way" [gesture indicating
a curve].
The reactions of this stage are extremely interesting, for al-
though the children demonstrate by their behavior that they know
how to act in the experimental situation, sometimes success-
fully, they never internalize their actions as operations, even as
concrete operations, In a general sense, by concrete operations we
mean actions which are not only internalized but are also inte-
grated with other actions to form general reversible systems.
Secondly, as a result of their internalized and integrated nature,
concrete operations are actions accompanied by an awareness on
the part of the subject of the techniques and coordinations of his
own behavior. These characteristics distinguish operations from
simple goal-directed behavior, and they are precisely those char-
acteristics not found at this first stage: the subject acts only with
a view toward achieving the goal; he does not ask himself why
he succeeds. In the experiment under consideration he is not
aware of either the rectilinear nature of the trajectory segments
or the existence of rebounds except toward the end of the stage
(toward 6 or 6-7 years); consequently he cannot take note of the
presence of angles at the rebound point.
Stage II (Substages II- A and II-B)
Substage II-A is distinguished by the appearance of concrete
operations in the sense just defined:
VIR (7 ; 7) succeeds after several attempts. He points out and then
draws trajectories with two distinct rectilinear segments, saying: "To
aim more to the left, you have to turn [the plunger] to the left!'
TRUF (7 ; 10): *7 know about where it will go"; in fact, he shows by
his gestures that he realizes that the angle of rebound is extremely
acute when the plunger is raised and extremely obtuse when it is low-
ered. Thus, he shows us that he has a vague global intuition of the
equality between the angles of incidence and reflection. But he does
not make it explicit, since he fails to divide the total angle indicated by
his gesture into two equal angles,
BEND (8 ; o): "It's the corner [the angle of rebound] that makes it turn;
you change the contour [the size of the angle] when you change the
plunger" [inclination of the plunger]. He demonstrates as did the pre-
ceding subject that the angle is extremely acute when the plunger is
slightly inclined and extremely obtuse when it is sharply inclined. We
ask him what he means by the contour, and he points to the opening
of the angle with a gesture indicating that he is thinking of the very
generation of the angle by the progressive rotation of the plunger and
of the rebound of increasing amplitude which results.
DESI (8 ; 2): "The ball always goes higher when the plunger & higher"
Then: "The ball will go there [further] because the plunger is tilted
more; I put my eyes high up [ = I pinpoint the rebound point] and
from the rubber [ = the rubber band attached to the wall on which
the ball rebounds] I look at the round pieces" [ = the disks serving
as goals].
At substage II-B the preceding operations, which give rise to a
model that includes straight lines and angles, are complemented
by an increasingly more accurate formulation of the relations be-
tween the inclination of the plunger and that of the line of re-
flection:
MC (9 ; 4): Tow have to move the plunger according to the location
of the target; the ball has to make a slanting line with the target."
8 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
(9 ; 6): "The more I move the plunger this way [to the left i.e.,
oriented upwards], the more the ball will go like that [extremely acute
angle], and the more I put it like this [inclined to the right], the more
the ball will go like that" [increasingly obtuse angle]. KAR reaches the
point of discovering that the ball returns to the starting point when
the plunger is "straight" i.e., perpendicular to the rebound wall.
BAER (9 ; 6): "How do you explain it?" "I* has to be at the same distance
as the target" [he points out the angle increasing with the withdrawal
of the target and not the length of the line between plunger and re-
bound point or between the latter and the target].
ULM (9 ; 8) : "As you push the plunger up, the ball goes more and more
like that [acute angle], and the more I put it like that [inclined to the
right], the more the ball will go like that" [obtuse angle] ."But, tell
us more about what you are looking at." "I am still looking at that
[the goal], and that's all, because it turns with the plunger" [because
the direction of the path between the rebound and the goal changes
with the inclination of the plunger],
DOM (9 ; 9): "It hits there > then it goes there" [he points out the equal
angles, repeating his phrase for different inclinations of the plunger].
Thus we see that the subjects succeed in isolating all of the
elements needed to discover the law of the equality of the angles
of incidence and reflection, yet they can neither construct the law
a fortiori nor formulate it verbally. They proceed with simple
concrete operations of serial ordering and correspondences be-
tween the inclinations of two trajectory segments (before and
after the rebound), but they do not look for the reasons for the
relationships they have discovered. And they do not consider the
segments except from the standpoint of the directions taken; thus
the idea o dividing the total angle made up of the two segments
into two equal angles (incidence and reflection) fails to occur to
them.
However, in contrast to the stage I subjects, substage II-A and
II-B subjects no longer limit themselves to overt performance but
internalize their actions in the form of operations of placing or
displacing: thus they have become aware of the facts that the
plunger can be adjusted to specific slopes, that the trajectory of
the ball is composed of two rectilinear segments, and, above all,
THE OPERATIONS OF RECIPROCAL IMPLICATION 9
of the fact that these two segments form an angle (whose peak
coincides with the rebound point) whose size varies according to
these slopes. They manage to order serially these latter inclina-
tions, distinguishing between "sharper" or "more to the left?
"higher" etc., and "less sharp" etc., which amounts simply to a
translation of the more or less well-ordered operations that they
know how to execute beginning with stage I into coordinated
operations of serial ordering. Similarly, they succeed in ranking
the degrees of incline or the directions of the trajectory segments
included between the rebound point and the goal ("the ball keeps
going higher" or lower, it "will go here" or there, etc.). Finally,
and particularly important, they establish a correspondence be-
tween the slope or direction of the plunger (and consequently of
the first segment of the ball's trajectory) and the inclinations or
directions of the second segment: "The more the plunger is (in-
clined, etc.), the more the ball will go (downwards, etc.)."
If the increasing inclinations of the plunger (and of the first seg-
ment of the trajectory for the ball leaving the plunger) are symbol-
ized by the letters a, /?, y, etc. and the inclinations of the second
segment (between the rebound point and the goal) by the signs
a', /?', y', etc., the serial ordering and correspondence operations
which subjects of this second stage can perform are as follows:
a < )8 < . . . or of < p* < y' . . . and (i)
a - a', ft <H> /3', y <H> y', etc. (2)
(where the sign <- stands for the correspondence).
Why, then, does the correspondence between the two rank or-
derings fail to lead to the discovery of the law of the equality of
the angles of incidence and reflection? It is because the subjects
stick to the concrete rank ordering and correspondences without
looking for the reasons behind this correspondence, just as the
subjects at stage I knew what to do to attain their goal but did
not look for the reasons behind their reactions (displacement of
the plunger, etc.). The stage II subjects stick to dealing with facts
whose accuracy is due to serial ordering and correspondence
operations, but they do not seek to explain these facts further in
terms of the formal operations of implication, etc., which are the
conditions of hypothetico-deductive thought.
Since they do not seek an explanation of the observed facts,
10 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
they must remain at the level of rough, global observation, cer-
tainly a great advance over that of stage I but still too global to
lead to an analytic breakdown of the observed angles. Thus,
because they are content to point out slopes of directions and to
deal with the total angle composed of the two segments of the
trajectory (BEND: the "contour"; ULMS "It turns"; etc.), they do not
divide this total angle into the two equal parts that would give
us the angle of incidence and the angle of reflection. That is why,
although the subjects are very close to the discovery of the law
and already possess all of its elements, it is not yet discovered;
the formal operations needed for the quest for an explanatory
hypothesis are lacking.
The Formal Stage (Substages III- A and III-B)
At this last stage the subjects finally discover the law of the equal-
ity of angles. At first the discovery is slow and partial, including
verification or rejection of several specific hypotheses, then com-
plete and rapid because subjects are oriented by the hypothesis
that there is a necessary equivalence between two successive seg-
ments of the trajectory.
First, let us look at a case typical of substage III-A:
BON (14 ; 8) first invokes the launching force, then realizes that the
trajectories are the same whether the balls are shot hard or soft. Next
he invokes the role of the "distances, how you have to place the rod."
Then he establishes concrete correspondences in the same way as the
stage II subjects: "Ifs the position of the lever [of the plunger]: the
more you raise the target, the more you raise it here" [the lever]. He
uses a ruler to mark the trajectory of the ball between the rebound
point and the target in such a way as to verify its correspondence with
the orientation of the plunger. Then he hypothesizes that the angle is
always a right angle: "It has to make a right angle with the lever"
But after several trials he concludes: "No, above [ = when the plunger
is straightened] it won't work" "It isn't ever a right angle?'* "Yes,
that's correct for one position" "And without that?" <f When you turn,
one should be smaller, the other larger. Ah! They are equaF [he points
out the angles of incidence and reflection].
THE OPERATIONS OF RECIPROCAL IMPLICATION II
Thus, in the later stages there is a search for a general hy-
pothesis which can account for the concrete correspondences be-
tween the inclinations as soon as they are found. Subject BON
thinks first of the right angle, then ascertains that the total angle
is sometimes acute, sometimes obtuse; then he breaks it down to
form two equal angles (incidence and reflection).
But the hypotheses found at substage III-A are still very close
to concrete correspondences in that they attempt only to express
the general factor which the correspondences contain. Substage
III-B, on the other hand, is distinguished by a new exigency
which is absent at substage II-B and still implicit at III-A: the
need to find a factor which is not only general but also necessary
i.e., which will serve to express beyond the constant relations
the very reason for these relations.
In other words, the subjects at substage III-B are not com-
pletely satisfied with the establishment of a correspondence be-
tween the inclinations of the plunger and the line included be-
tween the buffer and the target, as are those of II-B. Nor are they
satisfied with the search for a single constant factor which trans-
lates these correspondences, as are those of III-A. Initially they
ask themselves why a certain difference in inclination Xi of the
plunger necessarily corresponds to a difference x 2 in the buffer-
target line. This pursuit of a necessary reason, in certain cases
going as far as an immediate appeal to the concept of necessity,
is what distinguishes formal thinking, with its operations of impli-
cation or equivalence ( = reciprocal implication), from concrete
thinking, with its simple statements of constancy. This is demon-
strated by the following subject, who begins, like BON, with the
hypothesis of the right angle but soon afterwards turns to a search
for necessity:
DEF (14 ; 8) imagines at first that the two trajectory segments always
form a right angle. But after three trials he says: "The more the target
approaches the plunger, the more the plunger must [necessity] also
approach the target" [which signifies evidently that the two inclina-
tions of the plunger and of the line between the target and the buffer
imply each other reciprocally]. "What do you mean by *must also
approach the target?* ""For example, if there were a line here [he
indicates a line perpendicular to the buffer], the ball would come back
12 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
exactly the same way" Then he puts the plunger at 45: "That makes
a right angle here and you have about the same distance as there"
[ = the two angle openings]. Then he continues with several angles
chosen at random and again verifies his law of equality. We object that
the law may not be a very general one: "It depends on the buffer too;
it has to be good and straightand also on the plane it has to be com-
pletely horizontal. But if the buffer were oblique, you would have to
trace a perpendicular to the buffer and you would still have to take
the same distance from the plunger [to the line and from it] up to the
target: the law would be the same" The buffer is turned around, re-
placing the rubber by wood: "Perhaps the wood is less elastic: the
ball would be sent back with less force'' "Then what about your
law?" "The law doesn't vary."
Even in this first case we see several new factors appear which
psychologically distinguish formal from concrete thinking: the
requirement of necessity ("the plunger must also, . . ." "it would
have to be the same width here and there/' "you still would have
to take the same distance again," etc.); the ability to formulate
hypotheses or hypothetical constructions not given by direct ob-
servation (trace an ideal perpendicular from the buffer, etc.); con-
fidence in the generality of the law because it is conceived of as
necessary, thus as holding true even if conditions are modified
("the law doesn't vary," etc.).
It is clear that new operations appear at this level (after a pre-
paratory period beginning with substage III-A) which are super-
imposed on the concrete operations. Specifically, of what do these
new operations consist? This should be made clear in the course
of the examination of the following protocols, to be considered
jointly with the preceding one.
GUG (14 ; 4), after several trials, says: "The more you go toward the
right angle [i.e., the more the plunger approaches a position perpen-
dicular to the buffer] the closer to the starting point the ball comes
back"- "Is that always true?" "Yes, or at least I think so: ijou'd have
to check" He continues his trials and, when there is chance dispersion
due to deficiencies of the apparatus, he concludes: "There must be
something wrong." After several new trials he concludes: "You have
to find the angle" and he looks first for equality in the complementary
angles included between the walls of the apparatus and the plunger or
the line buffer-goal. At last he discovers that: "You have to trace the
THE OPERATIONS OF RECIPROCAL IMPLICATION 13
perpendicular" [in relation to the buffer]. He then realizes the constant
equality between the angles of incidence and reflection.
MUL (14 ; 3) begins with a series of correspondences: "I was here and
it went in this direction," etc.; "You change the angle to see how it
goes." By systematically diminishing the total angle, he discovers the
fundamental proposition: T/ I shoot it straight, at a right angle [i.e.,
when the plunger is perpendicular to the buffer], it will come right
back." Then he inclines the plunger progressively, according to the
angles ai, /3 9 yi, etc., and ascertains that, as these angles increase,
their complementaries ai' fti' 9 yi' decrease [of, ft standing for the
angles included between the plunger and the buffer]: "The smaller
you make the angle here [ai', /V, etc.], the larger the angle there"
[ai, $L, yi]. Then he perceives the equality which he had been seek-
ing from the time he understood that, in the case where the plunger
is perpendicular, the ball returns to its starting point. "This angle [ai']
is the same as that one fag']; you have to make it parallel to that one
[a2 / ] I o/m going to see [he checks for several different angles]. Yes,
I think that's it. You have to carry over exactly that angle" [the com-
plementaries ai', and c^'j etc.].
POM (15 ; 5) also begins by noting the correspondences between the
angles: "I look a bit at an angle. . . . The higher up you want to aim,
the under the angle has to be" [he calculates on the complementary
as did MUL]. In order to verify this hypothesis, he spontaneously places
the plunger perpendicular to the buffer: "If the lever is straight, the
ball returns exactly." Afterwards he adjusts the plunger in three differ-
ent positions, but without moving the target and without firing, and
concludes immediately: "You have to have two angles: the inclination
of the lever equals the angle that the trajectory of the ball makes"
[from the buffer to target],
LAM (15 ; 2): "The rebound depends on the inclination [of the
plunger], . . . Yes, it depends on the angle. I traced an imaginary
line perpendicular [to the buffer] : the angle formed by the target and
the angle formed by the plunger with the imaginary line will be the
same."
(15 ; 4): "tfs a right angle [several trials]. No, this slant has to be
the same as that one." When there are chance misses due to the
apparatus, he says, *7 didn't move; the gadget isn't fair"
GOD (15 ; 9), after several fruitless trials: "You would have to find the
rebound angle." First he indicates ai' = az as did MUL. Then he traces
14 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
the perpendicular and points out the angles of incidence and reflec-
tion: "The two must be equal."
FORT (16 ; o) begins with several trials: "You have to move the lever
according to the target position and vice versa [reciprocity]. "You must
have an angle there, but it isn't always the same [he continues his
trials]. It's obvious that everything changes." Then: "You have to think
in straight lines. To the extent that the lever is displaced, you find the
same distance in the other direction: you have to displace it according
to the mean [ = the perpendicular from the rebound point to the buff-
er which he has spontaneously designed]. The two distances [the
angle openings], the two sides, always indicate the angles" [of inci-
dence and reflection].
JAN (16 ; 4): "You have to find the corresponding angle: the more
acute the target angle, the more the lever goes towards the middle and
vice versa." "Can you measure it?" "It's more or less a right angle.
No, it varies here the same way as there" [same design as FORT].
BERG (16 ; 6), after analogous explanations, is shown a wooden buffer
rather than the rubber one: "I think that it's the same law. Yes, Fm
sure of it. I take the perpendicular and I focus on the distances. Yes,
now the angles have to be equal."
Although they differ from each other in a number of respects,
these examples of reasoning have in common several essential ele-
ments which must be differentiated before we can describe the
differences between formal and concrete thinking as they relate
to the experimental problem under consideration.
In the discovery of the law, the general starting point for these
subjects seems to be the fact that the establishment of the con-
crete correspondences between the inclines of the plunger and
the path of the ball after it strikes the buffer seems to lead auto-
matically to the idea of a necessary reciprocity i.e., each incline
implies the other and vice versa. For example, this is expressed
by FORT: "You have to move the lever according to the target and
vice versa." But this reciprocity, which adds the idea of mutual
implication to that of one-to-one reciprocal correspondence, does
not in itself entail the realization that the two angles are equal
(as FORT, who is at first struck by the variation of the angles,
demonstrates).
THE OPERATIONS OF RECIPROCAL IMPLICATION 15
The bridge from the idea of reciprocity to that of equality
and this is the second point common to all of the answers is
actually furnished by the assertion (explicit, or in certain cases
purely mental) that the ball returns to the starting point when the
plunger is perpendicular to the buffer. Then it follows that if the
null incline of the plunger implies the null incline of the ball's
return course, any inclination of either implies an equal inclina-
tion of the other.
Once in possession of this double assertion (mutual implication
of inclines and return of the ball to its starting point in the case of
null incline), the subject will either imagine a perpendicular to
the buffer from the rebound point, which leads him to discover
the equality of the angles of incidence and reflection; or he will
look for the complementary angles (located between the plunger
and the buffer or between the former and the trajectory of the
ball after the rebound), which step also leads him to the idea of
equality.
In either case, the construction of the law is due to the quest
for a necessary explanation of the observed inclinations; the serial
orders and correspondences established prior to this point are
not in themselves sufficient for the subject to discover the rela-
tionship between the angles, or even for him to break up the total
angle included between the two successive segments of the trajec-
tory into two partial angles.
Conclusion: The Transition from (Concrete) Cor-
respondence to (Formal) Reciprocal Implication
In spite of what we have just said about the discoveries of our
stage II subjects, we have yet to understand just what formal
thought adds to concrete operations in the specific case, since sub-
jects at stage II seem a posteriori so close to the formulation of
the law. What is the contribution of formal operations to the solu-
tion of a problem that at first glance seems to require nothing
more than correspondences and equalization? Actually, the con-
text of stage III reactions is quite different from that of preceding
stages: reasoning by hypothesis and a need for demonstration
have replaced the simple stating of relations. In other words,
16 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
henceforth thought proceeds from a combination of possibility,
hypothesis., and deductive reasoning, instead of being limited to
deductions from the actual immediate situation.
The distinction between the one-to-one correspondence of the
angles of incline (at stage II) and the reciprocity leading to the
idea of the equality of angles (discovered at stage III) is ex-
tremely fine as long as we are not in a position to state exactly
what the differences are between the operations used at these two
stages. Nevertheless, there is a difference. And though it is slight
in this first case, it does give us an example of the general opposi-
tion of concrete and formal operations that we shall encounter
again in increasingly clearer form in the following chapters.
The difference can be stated as follows: Although concrete
operations consist of organized systems (classifications, serial or-
dering, correspondences, etc.), they proceed from one partial link
to the next in step-by-step fashion, without relating each partial
link to all the others. Formal operations differ in that all of the
possible combinations are considered in each case. Consequently,
each partial link is grouped in relation to the whole; in other
words, reasoning moves continually as a function of a "structured
whole."
Stated in symbolic terms, when two classes, AI and A 2 , with
their complementaries, A\ and A' 2 , are taken, concrete class
logic furnishes only four elementary products (AiA 2 + AiA' 2 +
A'iA 2 + A'iA'2). On the other hand, formal logic, taking the two
propositions p and q with their negations p and q, furnishes six-
teen possible combinations derived from the four elementary
prepositional conjunctions (p.q) v (p.q) v (p.q) v (p.q), which de-
fine respectively relations of implication, disjunction, etc., de-
pending on whether the conjunctions are taken one-by-one,
two-by-two, three-by-three, the four together, or none at all. The
implication of q by p, for example, corresponds to the sum of
three conjunctions, (p-q) v (p*q) v (p.q)', the implication of p by q
corresponds to the sum of (p.q) v (p.q) v (p.q); and the equiva-
lence of p and q (or reciprocal implication) corresponds to the sum
of the two conjunctions (p.q) v (p.q). But, in order to affirm the
truth of one of these three links, p D q or q D p or p = q, one also
has to establish the respective falsehood of (p.q) for p^q, of
(p.q) for qOp, and of (p.q) as well as of (p.q) for p = q.
THE OPERATIONS OF RECIPROCAL IMPLICATION 17
In other words, the difference between the concrete level sub-
jects (who do not go beyond the formulation of term-by-term cor-
respondences between the inclinations of the plunger and the
course of the ball from the buffer to the target) and the formal
level subjects (who look for necessary reciprocity immediately)
can be wholly accounted for by distinguishing the step-by-step
operations based on simple correlations found in class and rela-
tional groupings from the combinatorial operations based on the
"structured whole" which constitute prepositional logic. Thus,
subjects at stage II are limited to stating successively the corre-
spondences in question and to constructing from the resulting
table that the more the plunger is inclined, the more the course
of the ball between buffer and target is inclined. Certainly this
could be called a law, but it is a law which is a simple summary
of formulations made one by one.
In contrast, stage III subjects view the experiment from the
start both in terms of the total number of possibilities and in terms
of necessary relations, since they possess operations which both
are combinatorial and contain the potential assurance of deductive
necessity. In their first correspondence operations they do not
merely take note of the empirical relationships but immediately
proceed to search for an explanation i.e., they consider the cor-
respondences as implications. Of course, in a sense the implica-
tion p D q is still a statement of fact, equivalent to establishing
that the case (p.q) never occurs. Still, in order to establish this it
is necessary to consider the four possibilities (p.q) v (p>q) v (p*q) v
(p.q); in any case, the implication is nothing more than the addi-
tion of three possibilities (the first, the third, and the fourth) com-
bined by the operation (v) which signifies "or" i.e., it is an addi-
tion of what is possible and not of "realities."
Actually, when faced with a correspondence p.q (let p be the
term for a certain angle of incline of the plunger and q the term
for the corresponding angle of incline of the course of the ball
between the buffer and the target), the stage III subjects are not
restricted to pointing out the existence of the conjunction, as are
those of stage II, who are satisfied at this point. They exclude the
possibility (p.q) i.e., they introduce by hypothesis an implied link
between p and q; but they also exclude (p.q) i.e., they also in-
troduce by hypothesis an implied link between q and p. Thus
IS THE DEVELOPMENT OF PROPOSITIONAL LOGIC
they proceed immediately from stating the conjunction p.q. to
stating the hypothesis of a reciprocal implication p i q, with the
assumption that this reciprocity p q or p = q (which is not in
itself an equality of content but a simple equivalence from the
point of view of the truth of the propositions) covers the equality
of some real factor.
At this point, the reasoning process of the i4-i6-year-old sub-
jects, based from the start on the twofold consideration of pos-
sible combinations and necessary links, is elaborated into a true
hypothetico-deductive construction. Unlike stage II subjects, who
are limited to noting the occurrence of various correspondences, 4
the adolescents at stage III sooner or later (and often very early)
try to uncover the general principle underlying the special case of
null inclination. Having established that the ball returns to its
starting point, they immediately draw the conclusion that the
corresponding inclinations must be equal and consequently the
angles which determine them must also be equal; after verifica-
tion with one or two they generalize the conclusion to all cases. 5
In symbolic terms, the subject's reasoning at substage III-B is
approximately the following (see as an example the extremely
clear case of DEF):
p q, because (p.q) v (p.q) are true and (i)
(p.q) v (p.q) false where p and q state corresponding inclinations
having the respective values x and y. But
(* = o) i (y = o), and (2)
(* = a )i(0 = ) (3)
where a is a determinate inclination > o. Therefore,
* c V, and (4)
AagAy (5)
where A x and A y are the angles of incidence and reflection (or
their complementaries).
4 Which may include the case in which the plunger is not inclined and the
ball returns to the starting point, but from which they do not abstract the gen-
eral principle.
5 Note that the elementary reasoning by recurrence is itself accessible at
the concrete level (see La G$omtrie spontanSe de I* enfant, Chap. IX, no. 4).
It appears so late in this case because all of the subject's deduction is directed
by preliminary reciprocal implications.
THE OPERATIONS OF RECIPROCAL IMPLICATION 19
In sum, the discovery of the equality of the angles is the result
of the reciprocal implication between the corresponding inclina-
tions postulated from the start and not the inverse; this reciprocal
implication differs from simple concrete correspondence by the
fact that it results from a calculation of possibilities and not
merely from an account of the empirical situation.
The Law of Floating Bodies
and the Elimination
of Contradictions 1
A GIVEN NUMBER of disparate objects are presented to the subject,
who is asked to classify them according to whether or not they
float on water. Then (the classification completed) he is asked to
explain the basis of his classification in each case. Next, the sub-
ject himself experiments, having been given one or several buck-
ets of water; finally, he is asked to summarize his observations,
this latter request suggesting that he is to look for a law, if this
has not already spontaneously occurred to him. 2
Unlike the law considered in the problem presented in the first
chapter, the law of floating bodies cannot be derived from con-
cepts which are entirely accessible at the level of concrete opera-
tions. Neither the conservation of volume nor, consequently, of
density, is worked out in systematic fashion before substage III-A
( 11-12 years); however, the conservation of weight and certain
schemata preparatory to the concept of density are acquired at
substage II-B.
1 With the collaboration of J. Nicolas, former research assistant, Laboratory
of Psychology, and M. Meyer-Gantenbein, former research assistant, Labora-
tory of Psychology.
2 With the older subjects, in addition to the objects to be classified we
present three cubes of equal volume having different densities and an empty
cube with "plexiglass" or plastic walls (with a density of about one) to facili-
tate accurate comparisons with the density of water.
20
THE ELIMINATION OF CONTRADICTIONS 21
But given that the law to be found is that objects float i their
density or specific gravity is less than that of water, two rela-
tionships are essential to the solution of the problem: density
i.e., the relation of weight to volumeand specific gravity i.e.,
the relation between the weight of the object (its density if it is
solid, or the weight of its matter plus that of the air which it
contains) and an equivalent volume of water.
In addition, the problem requires the construction of a classifi-
cation including both the class of bodies which float on water and
the class of bodies which do not float plus two other eventual
classes that of bodies which may float in certain situations and
not in others (such as empty bodies which can be either full of
air or full of water) and that of bodies which remain suspended.
The law ultimately to be discovered states a relation between only
two large classes; that of bodies whose density is less than the
density of water and that of bodies whose density is greater.
Thus the law states a single and noncontradictory relationship.
But in order to construct it empirically the subject first has to
eliminate a series of contradictions that frequently characterize
the early stages. For example, at first the explanation may be
formulated in terms of weight alone, although in fact it is some-
times the heavier, sometimes the lighter bodies which will float.
Secondly, the element common to several different explanations
(weight, volume, air, etc.) must be isolated. Although the simplest
contradictions can be overcome by means of concrete operations
alone, the elimination of the more subtle ones, and particularly
the formulation of a unified explanation, requires the use of im-
plications i.e., the intervention of formal prepositional operations.
In the light of these considerations we feel that the problem of
floating bodies, like that of the equality of angles, is an appropri-
ate choice for a preliminary analysis of the transition from con-
crete to formal thinking.
Stage I (Substages I- A and I-B)
The stage I reactions (until about 7-8 years) are very interesting,
for they are far from demonstrating that a search for a single
and noncontradictory explanation is primitive. We find instead
that the youngest children are satisfied with multiple and often
22 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
contradictory formulations. Furthermore, although the problem
calls for classification of objects into two groups (floating and non-
floating bodies), a substage I-A must be distinguished during
which the subject does not even formulate this elementary di-
chotomy because successive judgments in time relative to a single
object are still contradictory.
Although the children of this first substage, once they have de-
termined whether or not a particular object floats, may come to
predict that it conserves its properties, because they lack a general
frame of reference they do not extend the same properties to
other analogous objects. Furthermore, they do not always conceive
of these properties as constant even for an identical object:
IEA (4 years) says, for example, of a piece of wood that "it stays on top.
The other dag I threw one in the water and it stayed on top." But a
moment later: <e Wood? It will swim anywhere.' 9 "And this one?" [a
smaller piece] .-"The little wood will sinfc."-"But you told me that
the wood would swim." "No, I didn't say so" On the first presenta-
tion of a wire, he says, "The wire goes to the bottom" [he has not done
the experiment] .-"And this weight?" [metal] .-'It will swim."- 'The
wood?" "It will swim anywhere" 'The wire?" [third presentation].
"It will swim" Finally, for two metal needles of identical appearance
he says the opposite: "This one?"-*!* wtU float, "-" And that one?"-
"lt will sink." We must add that although IEA generalizes little, his
explanations can be reduced to the format: "The pebble?" "It will
$ink"-'"Why?"-"Because it stays on the bottom"
MIC (5 years) predicts that a plank will sink. The experiment which
follows does not induce him to change his mind: [He leans on the
plank with all his strength to keep it under the water.] "You want to
stay down, silly!" 'Will it always stay on the water?" "Don't know."
-"Can it stay at the bottom another time?"-"Y0s."
Classification cannot follow from such responses. First, there is
no basis for sorting the objects into floating and nonfloating
classes. One way to construct these two classes would be to in-
voke a constant quality which would in itself furnish the explana-
tion of the fact that a given object floats or fails to float However,
at this stage the subjects do not yet use such explanations and are
restricted to looking for the cause in the description of a particular
case.
THE ELIMINATION OF CONTRADICTIONS 23
One could maintain from another standpoint that the subjects
could classify the objects a posteriori after having observed their
properties in the experiment. However, in this case as well the
properties would have to be perceived as constant. Thus classifi-
cation is no less impossible, since for these subjects: (i) The same
object does not necessarily conserve its properties over time (cf.
the wire for IEA and the plank for MIC); (2) Different properties
may be attributed to two identical objects (cf. the needles for
EGA); (3) Analogous objects may also be given different properties
(cf. the small piece of wood which sinks and the large one which
floats IEA).
The reader could object that the child's reasoning is not actually
contradictory in that it is analogous to the reasoning of the mete-
orologist who knows that the same cloud may send forth rain at
a given moment and not at another, or that of two similar clouds
one might produce rain and the other not. Here the fluctuations
or lack of constancy appear in reality itself and not in the think-
ing of the observer. Nevertheless, the meteorologist seeks to gen-
eralize. In spite of the risks inherent in his profession, he goes on
to assume that one can fit deductions to the empirical world; he
ascribes discrepancies to the operation of chance factors and con-
cludes that under constant conditions his predictions would be
accurate. In contrast, the substage I-A child does not try to fit de-
ductions to the situation and does not yet know how to distin-
guish the deductible from the random and he does not assume
that results will be similar under equivalent conditions. Rather, he
assumes invariance and deductibility only in certain cases (cf. the
wood which floats because it floated "the other day" IEA). But
that is exactly the point; since his assumptions vary from case to
case he is not able to discern either the reasons for invariance or
the reasons for variation. 8
Thus, to attribute the probabilistic reasoning of the meteorolo-
gist to the subjects at substage I-A would be to commit the "fal-
lacy of the implicit." As additional evidence we can say that, since
from substage II-A on the subjects do seek invariances and do
s Translators' note: For further experimental evidence concerning the fail-
ure of the substage I-A child to understand probabilistic reasoning, see Piaget
and Inhelder, La Gdnese de ride de hasard chez I'enfant (Presses Universi-
taires de France, 1951).
24 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
construct classes having general qualities, there is no reason why
we should not interpret the successive reactions as they appear
genetically. Thus, beginning with substage II-A an attempt to
eliminate contradictions can be assumed, whereas for the substage
I-A subjects, to assume either an attempt to discover contradic-
tions or a radical impermeability to them would be equally mis-
leading; rather, we must speak of indifference before contradic-
tions when the causal problem cannot be resolved. (The problem
is not resolved at substage II-A either, but the child does assume
the possibility of a coherent solution and this expectation is suf-
ficient to urge him to try to resolve contradictions.)
At substage I-B the child tries to classify the objects in a stable
way into floating and nonfloating, but he does not achieve a
coherent classification for the following three reasons (the first of
these is logically legitimate, but the other two relate to preopera-
tional thinking): (i) Since the law is not discovered (although he
begins to look for it), the subject is satisfied with multiple explana-
tions or a series of subclasses difficult to arrange hierarchically;
(2) In the experimental situation, he finds new explanations and
thus adds new divisions to his classification but does not reformu-
late the whole; (3) There are contradictions between some of
these classes.
TOSC (5 ; 6) divides the objects presented into two classes prior
to the experiment: class B (objects remaining above water) and
class B' (objects which sink). Class B includes seven subclasses:
(Ai) Objects which "swim" or float because it is their nature:
boats and ducks ("My little duck that swims like the real ones").
(A 2 ) Small objects ("little tiny pebbles" tokens, needles). (A 3 )
Light objects (small pebbles float "because they arent heavy"
and thus belong simultaneously to A 2 and A 3 , but an aluminum
plate floats because it is light although it is not small). (A 4 ) Flat
objects (example: "This pebble, because it's so flat 99 ). (A 5 ) Thin
objects (a wooden blade). (A 6 ) Objects which are the same color
as the receptacle ("Why will this plank stay on top?" "Because
they are both the same color" [the plank and the bucket]).
(A 7 )~ Objects which have already floated (example: a piece of
wood '"because it stayed up before").
Class B' includes the following subclasses: (Ai) Objects "that
don't belong on the water* by nature (for example, a piece of a
THE ELIMINATION OF CONTRADICTIONS 25
candle: "Where will it go?"-"To the bottom."- t Whyr'--"Because
the candle doesn't belong on the water." We put it in the water:
"It floats. Why?" "Because it swims on the water!' Thus the
candle is classified parallel to subclass A! of class B.). (A 2 ) Large
objects. (A 3 ) Heavy ones (with the same difficulty in identifica-
tion and the same interference as for A 2 and A 3 of class B).
(A 7 ) Those that "went to the bottom before." (A 8 ) Long objects
(a copper wire sinks "because it's long"). (A 9 ) Those which have
been shoved (example: a metal cover).
We notice in this classification some effort at assimilation
(small = light, and sometimes thin = flat and flat = small), but
it fails because the criteria adopted are inadequate. Initially the
child assumes that class B is composed of only two subclasses
which, moreover, are heterogenous; there are the objects which
float by function or nature (Ai) and the "small ones" (A 2 ). The
subject does not seek the common quality which defines the first
category. For the second, he thinks that the quality "small" in-
volves other properties, such as light, etc. However, before the
experiment, when he enumerated the objects thus collected, the
subject felt he had to specify (without either order or hierarchy)
the connotations of the concept "small": light, flat, and thin. In
addition, new criteria unrelated to the preceding were brought in
at particular points i.e., the color. Finally, a global category
analogous to that of AI was constructed (but after the fact): that
of objects which have already stayed above water.
As for the objects in B', those which sink, we find three sub-
classes which correspond in the negative to AI, A 2 , and A 3 . But
the categories derived from flat, thin, or color have no negatives;
reciprocally, two new subclasses (A 8 and A 9 ) have no correspond-
ing class in B.
Summing up, this type of classification (of which there are other
analogous examples) can be defined by the following character-
istics: (i) The subclasses are not all disjunctive; (2) They do not
all have negatives; (3) They do not allow for grouping either by
simple hierarchy (inclusions and complementarities) or by mul-
tiple hierarchy (double- and triple-entry tables). Thus, for the
child, the experiment complicates rather than simplifies matters.
For TOSC, for example, it is responsible for the post facto introduc-
tion of subclasses A 8 and A 9 in class B' without correspondence
26 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
in B. It is true that 'long" could be the opposite of "small," but
that is the point; the subject predicted that the wire would float
because it was "thin" (A 5 ) and afterwards that it sinks because it
was 'long" (A 8 ), although long is not conceived as contrary to
"thin" for "thin" derives from "small" by specification.
Thus the diversification of subclasses without hierarchy must
sooner or later introduce a contradiction. Actually, in TOSC'S
reasoning the contradiction was present from the start, for two
interfering classes, "small" and "light," were implicitly regarded
as identical or included within each other; classification by simple
class inclusion was not differentiated from the double-entry table
system. This type of confusion of two potential sorts of classifica-
tion is accentuated as each new addition is made until the contra-
diction becomes explicit e.g., 'length" makes an object sink and
"thinness" makes it float, but the copper wire is both long
and thin; by the same token, "smallness" makes an object float and
"heaviness" makes it sink, but several objects are both small and
heavy.
Several typical examples of the types of contradiction found
at substage I-B are presented below:
TOSC (5 ; 6, same subject), after having said in reference to the plank:
"It goes to the bottom." "Why?" "Because it is heavy" adds a little
while later "because it is big." Then he sees that the plank floats and
explains the fact as follows: "It's too big and then there's too much
water" [to touch the bottom]. A moment later he tries to hold it at the
bottom with another plank and a wooden ball; the two come back up
"because this plank is bigger and it came back up." "And why does
the ball come up?" "Because it's smaller" "And this cover?""!* will
come back up" "Why?" "Because it is smaller than this piece of
woody than the plank" "Try" "It stayed down because I pushed too
high up"
BEZ (5 ; 9) explains the floating by the weight [inversely to TOSC] :
"Why do these things [previously classified] go to the bottom?" "They
are little things"- *Why do the little ones go to the bottom?"-"B0-
cause they aren't heavy, they don't swim on top because it's too light."
"And these?" [class of floating objects]. "Because they are heavy, they
swim on the water." We go on to the experiment: the key sinks "because
it's too heavy to stay on top" whereas the cover sinks "because it f $
light." Comparing two keys: the larger does not stay above water
THE ELIMINATION OF CONTRADICTIONS 27
"because it's lighf-'And the little one?"-"!* will go to the bottom
too."-"W}iy?"-"Because it's too light."
GIO (6 ; o) "These things [previously classified] go to the bottom?"
"Yes, that one" [the wooden ball]. "Why?" "Because it's heavy."
"And these?" [the class of floating objects]. "That one swims because
it's light." We do the experiment with the cover. It floats "because it's
light." 'And if you push it?" [It sinks.] "It's because it's light, and
light things never stay on fop." "And that plank?" "It will stay on
top"-"Why?"-"Because it's heavy. "-"WhyF-'Because it's big"-
"And if you lean on it?" [He does.] "It comes back up because it's
light." "And this?" [large needle]. "It goes to the bottom because it's
big."-" And that [metal plate] if you push?"-"!* will stay at the bot-
tom"-"Why?"-"Because it's light"
ELI (6 ; 10): "That?" [candle].-*!* goes to the bottom"-"Wl)y?"-
"Because it's round."-" And that?" [ball] .-"It stays on top /-"Why?"
"Because it's round too." Thus the contradiction does not relate only
to the weight. "And that needle?" [placed on the water].-"!* floats
because it's light."-" And if you push?"-"!* will go under/ -"Why ?"-
"Because it will be heavy." Here contradiction goes with nonconserva-
tion.
In reference to analogous observations, a logician once main-
tained that such assertions are not contradictory, just because the
same result can be due to either of two opposite causes; for
example, persons who pay taxes that are low proportionate to
their wealth may be either those who are very rich or those who
have hardly anything. But children at the present stage are far
from such a subtle schema, and it seems to us that three lands of
considerations demonstrate that they remain indifferent to con-
tradictions or, more accurately, that they do not perceive their
continual contradictions:
i. From the beginning, the subjects predict a simple distribu-
tion but according to two contrary explanations (which already
reveal the ambiguity of the concepts used). The bodies that float
are either those that are light because they are small, or those
that are heavy because they are large. However, each of these
two explanations already includes a number of implicit contradic-
tions, for light and heavy do not coincide with small and large
and the floating is due to relative and not to absolute weight.
28 THE DEVELOPMENT OF PROPOSIHONAL LOGIC
2. The experiment does not set the child right, but he tries to
reconcile the whole by adopting either explanation alternately
without perceiving the incompatibility; bodies float or sink
equally well if they are large, small, heavy, or light (or even, by
association, because they are round, long, etc.). Thus the con-
tradiction is made explicit, but it is not any closer to being
noticed, doubtless because of the initial ambiguity of the pairs
small X light and heavy X large.
3. The same assertions judged mutually compatible by the
subjects at substage I-B would appear irreconcilable beginning
with substage II-A. Here we find the best proof that they do not
constitute the reflection of an implicit coherence but rather of a
thinking process in a state of disequilibrium for lack of instru-
ments of coordination (operational classification, etc.) which will
attain equilibrium only at the point when concrete operations are
structured.
Stage II (Substages II-A and II-B)
The behavior of the 7-g-year-old subjects is marked by an effort
to remove the main contradiction to which they have submitted
previously without reaction: that certain large objects can float
and certain small ones sink without, however, barring the pos-
sibility that in general the light ones float and not the heavy. The
contradiction tends at this point to be surmounted by a revision
of the concept of weight, now seen in relation to that of volume
i.e., the child begins to renounce the notion of absolute weight in
order to look toward density and, above all, toward specific
gravity.
Specific gravity refers to the relationship between the weight of
a given volume of a body and that of an equal volume of water,
and density refers to the weight of a cubic centimeter of the body
considered. But we will take these two concepts in a more ele-
mentary sense. We will speak of density when the subject ex-
plicitly relates the weight and the volume i.e., when the concept
is understood as a relationship and of specific gravity when the
subject understands that for the same volume each substance has
a characteristic weight. (In the latter case the subject does not
THE ELIMINATION OF CONTRADICTIONS 29
refer explicitly to the volume.) Thus, substage II-A children
acquire the beginnings of the conception of specific gravity and
try to resolve the earlier contradictions by invoking it.
The two problems that arise at this stage in the development
of logical operations are: (i) Do the stage I contradictions tend to
disappear of themselves because the subject grasps the notion of
specific gravity, or is it in trying to surmount these contradictions
that he constructs the concept in question? (2) If the latter is true,
how does the child come to perceive the contradictions with the
aid of concrete operations alone?
KER (7 ; 6) classifies as floating objects wood, matches, corks, a cover,
metal clamps, an eraser, small nails, and a small cylinder of hollow
metal; as nonfloating a key, some stones, a metal disk, a needle, and a
heavy wooden ball. After the experiment he constructs a third class,
that of objects which float or sink depending on whether they are
empty or filled with water the cover and the needle [whose eye may
permit the water to pass]. The first two classes are defined by 'Tight"
and "heavy," but notice that KER wavers between two possible mean-
ings of these notions; the earlier or absolute sense [the small nails for
the 'Tight" and the large ball of wood for the "heavy"] and the new or
relative sense-i.e., the specific gravity. "The little pebble goes to the
bottom?"-"yes/-"But isn't it light?"-"No, ifs stone *-*And the
nail?" [He does the experiment] ''Ifs because it's iron." 9
BAR (/ ; 11) first classifies the bodies into tibree categories: those which
float because they are light [wood, matches, paper, and the aluminum
cover]; those which sink because they are heavy [large and small keys,
pebbles of all sizes, ring clamps, needles and nails, metal cylinder,
eraser]; and those which remain suspended at a midway point [fish].
"The needle?"-"!* goes down because ifs iron."-"And the key?"-
"It sinks too ""And the small things?" [nails, ring clamps]. "They
are iron too" "And this little pebbleP" Tfo heavy because ifs stone. 9 '
-"And the little nauT-"/^ just a little heavy ."-'And the cover, why
does it stay up?" "It has edges and sinks if ifs filled with water"
"Why?"~-"Because ifs iron."
DOT (7 ; 6): "That baDP"-*7* stays on top. Ifs wood; ifs light "-'
this key?"-"G0$ down. Ifs iron; ifs heavy "-"Which is heavier, the
key or the ballP '"The ball."-"Wby does the key sink? 9 '- <e Because
it is heavy."- 'And then the nail?"-*7ft light but it sinks anyway. Ifs
iron, and iron always goes under?
30 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
The principal difference between these reactions and those of
substage I-B lies in the real effort made to resolve the contradic-
tions. This is done by improving the classification system by the
utilization of class inclusion operations. These permit the subject
to distinguish systematically between "all" and "some" by means
of the reversible addition A + A' = B and B - A' = A, from
which it follows that A < B. As soon as this operation permits
him to determine the accurate inclusion of the part in the whole,
the subject is led to the most significant discovery of the first
operational level; that small objects do not always weigh less than
large ones or, in other words, that it is false to consider all small
objects as light and all large ones as heavy. In the case of floating,
in particular, all the objects which float are not small and all those
which sink are not large. Thus the child succeeds in making a
double-entry classification with reference to weight and volume
which gives four possibilities: the small light objects, the small
nonlight objects, the large light, and the large heavy. As a result
of the operation of class inclusion, the subject becomes sensitive
to contradiction and, by coordinating two classes now perceived
as distinct from each other, can separately formulate a double-
entry table. There would be contradiction if weight and volume
were identified in the presence of these four subclasses.
Thus the child is led to revise his notion of weight and to place
the concept of absolute weight .#., of weight equal to the volume
or to the quantity of matter in opposition to a new concept of
weight perceived as relative to the matter under consideration
i.e., of weight as a quality of distinct types of matter which is a
rough approximation of specific gravity. But we must insist that
the way they achieve this rough restructuring of the concept of
weight one which avoids the inconsistencies of earlier formula-
tionscannot be understood without considering the new logical
apparatus composed of concrete class and relational operations.
Actually, even reporting on the experimental data relative to
weight and to the quantity of matter presupposes that the parts
are distinguished from the whole for a given class ("all" and
"some") i.e., the presence of a coherent structure is indispensable
in order to avoid contradictions.
However, the notion of weight approximated in this way re-
mains insufficient. As yet it is no more than a quality inherent in
THE ELIMINATION OF CONTRADICTIONS 31
various types of matter, not a relation between the weight and
the volume. The reason for this is simple. As we have seen else-
where, at this level the child cannot yet conceptualize either the
conservation of weight or the conservation of volume, and the
only invariant available to him is the quantity of matter; thus he
is not able to make any accurate composition of the relationship
between weight and volume from the standpoint of the relations
between bodies or their internal configurations, It would serve no
purpose to refer back to earlier experiments, which are com-
pletely confirmed by the present results. 4 It is sufficient to note
that, without conservation or composition of the relationships
referring either to weight or to volume, specific gravity could not
be conceived as other than a simple quantity inhering in each
respective substance.
Moreover, given the incompleteness of the concept of specific
gravity, the failure to distinguish between the concepts of abso-
lute weight and of specific gravity naturally persists at this level.
This is a residual source of contradictions in spite of the visible
effort of the child to overcome them. (See, for example, in KER'S
report, the large wooden ball which he sometimes conceives of
as heavy, sometimes as light; likewise the nail, etc.) Furthermore,
the subject vacillates between the two concepts he applies to
weight because he is not entirely aware of the fact that he is
dealing with two concepts, though he can distinguish them to
some extent. Actually, in order to distinguish the two explicitly,
he would have to possess the operational means for such a distinc-
tion. But we have just seen that he does not possess them. Even
the serial ordering of weights between objects of the same volume
is not acquired until substage II-B, 5 Thus, the nascent notion of
specific gravity marks only the beginning and not the completion
of the separation of the variables of weight and volume. It is the
expression of the discovery that not all the small objects are light
nor all the large ones heavy; but the concept remains at this stage
of preliminary classification and does not yet reach a higher level
of organization.
4 Piaget and Inhelder, Le Dfoeloppement des quantitfe chez I'enfant (De-
ladhaux and Niestle, 1940). See Chaps. I-III and especially VIII-IX on the
composition of relationships between weight and volume.
5 DSveloppement des quantits, p. 233.
32 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
At about 9 years or 9 ; 6, weight begins to be conserved by the
child. 6 Thus, from then on he knows how to apply to weight the
concrete operations of serial ordering and equalization and even,
up to a certain point, of measuring. As for specific gravity and
density, he is no longer limited to qualifying the various materials
in terms of simple weight: iron is heavy, wood light, etc. Instead,
he introduces a new general explanatory scheme: the objects with
high specific gravity are more "full" than the others. 7 But, since
volume is not automatically conserved at this stage, we do not
yet see the formulation of an operational relationship between
the two.
For this reason, in comparing the weights of specific bodies to
the weight of water (which begins at this stage), the child does
not relate the object's weight to that of an equal volume of water
but rather to the water contained in the entire receptacle. 8
It is on this point that a new residual core of contradiction can
be observed during this stage. 'Although several of the contradic-
tions of the preceding level are eliminated as a result of the
progress which we have just described, in contrast the subjects
still assimilate the weight of the body, compared with that of the
total volume of water, to a substantial force or to a motor activity,
giving rise to a new group of dynamic explanations which are
mutually contradictory. In addition, the notion of "filled," in spite
of the fact that it permits the unification of the explanations relat-
ing to solid homogenous matter, gives rise, in the case of hollow
objects (boats, covers, etc.), to the hypothesis that the latter float
because they are filled with air. However, without being wrong,
this explanation provides another source of possible difficulties.
The following examples illustrate these various types of reac-
tion:
6 The term "conservation" is used in a sense specific to the authors* mean-
inga particular empirical factor (weight, volume, etc.) remains an invari-
ant in the child's mind throughout observed changes of state. The timing
of the appearance of conservation for various factors differs, but those dis-
cussed here all appear during the concrete stage.
7 Dveloppement de$ quantites, p. 173.
8 Cf. Piaget, The Child's Conception of Physical Causality (Harcourt Brace
and Co., 1930), Chap. VI: at this stage a boat which can float on a lake would
be too heavy for the Rhone, etc.
THE ELIMINATION OF CONTRADICTIONS 33
BAR (9 years): [class i] Floating objects: ball, pieces of wood, corks,
and an aluminum plate, [class 2] Sinking objects: keys, metal weights,
needles, stones, large block of wood, and a piece of wax. [class 3]
Objects which may either float or sink: covers. Later BAR sees a needle
at the bottom of the water and says: "Ah! They are too heavy for the
water, so the water cant carry them." "And. the tokens?" "I don't
know; they are more likely to go under." "Why do these things float?"
[class i]. "Because they are quite light." "And the covers?" "They
can go to the bottom because the water can come up over the top"
"And why do these things sink?" [class 2]. "Because they are heavy"
-"The big block of wood?"-"!* will go under"-"Whyr~"There is
too much water for it to stay up" "And the needles?" "They are
lighter" "So?" "If the wood were the same size as the needle, it
would be lighter" "Put the candle in the water. Why does it stay up?"
"I don't know" "And the cover?" "It's iron, that's not too heavy
and there is enough water to carry it" 'And now?" [it sinks]. "That's
because the water got inside" "And put the wooden block in."
"Ah! Because it's wood that is wide enough not to sink" "If it were a
cube?" "I think that it would go under." 'And if you push it under?"
"I think it would come back up" 'And if you push this plate?"
[aluminum]."/* would stay at the bottom." "Why?" "Because the
water weighs on the plate." "Which is heavier, the plate or the
wood?" "The piece of wood" "Then why does the plate stay
at the bottom?" "Because it's a little lighter than the wood, when
there is water on top there is less resistance and it can stay down. The
wood has resistance and it comes back up." "And this little piece of
wood?" "No, it will come back up because it is even lighter than the
plate" "And if we begin again with this large piece of wood in the
smallest bucket, will the same thing happen?" "No, it will come back
up because the water isn't strong enough: there is not enough weight
from the water."
BRU (9 years): "The water can't carry the pebbles. The wood can be
carried" "And if it is pushed under?" "I* will come back up because
the water isn't strong enough: it doesn't have enough weight" [ = this
time the weight operates to maintain it at the bottom and no longer
to carry it!]. And a moment later, "The wood comes up when you let
go because it springs up"
The case of BAR clearly illustrates most of the characteristics of
this stage. In the first place, he classifies the objects according to
34 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
specific gravity and not absolute weight; there were three excep-
tions, two of which are due to ignorance (wax and aluminum),
and one (the large piece of wood) to the fact that it is related to
the total volume of water in the bucket. At one point the subject
even gets at an explicit relation between the weight and the vol-
ume of the body; the needle is heavier than the piece of wood
because "if the wood were the same size as the needle, it would
be lighter." Why, in this case, after such a favorable beginning, is
the subject unable to find the law, at the end losing himself in an
increasing number of contradictions? His failure results from the
fact that in relating weight and volume, he has not yet found a
general operational form (logical multiplication for equal volumes
and different weights or for equal weights and different volumes)
and has stayed within the limits of the particular case of the com-
parison of iron and wood. In addition, whenever the principal
relationship relevant to the formulation of the law appeared
i.e., that between the weight of the body under consideration and
that of the water he did not compare weights with equal volumes
(body and water) but the weight of the object and that of the total
quantity of water; "heavier than water" signified "too heavy for
the water to be able to carry them." But once he began to con-
ceptualize the relationship between weights in terms of active
forces, all explanations became possible as his observations pro-
ceeded and sooner or later he was bound to entangle himself in a
contradiction. This is brought out at the end of our questioning
of BAR (as well as the text cited from BRU) up to the point where
BAR returns to the explanation in absolute weight, which is easier
to reconcile with his dynamic imagination.
These initial efforts at unification and internal consistency
which, for lack of adequate operational instruments, are not
crowned with final success, reappear in the cases which are most
difBcult from the standpoint of an integrated explanation; the
case of hollow objects, where the air plays a part, and that of the
needle, which floats in certain cases because of the surface ten-
sion. Thus, certain subjects who explain the specific gravity by the
notion of more or less filled generalize the case of covers or boats,
which float when they are empty (but supported by the air) and
sink when they are filled with water, up to the point of using it
as a prototype for the specific gravity of all sorts of objects.
THE ELIMINATION OF CONTRADICTIONS 35
RAY (9 years): "The wood isn't the same as iron. It's lighter: there are
holes in between." "And steel?" "It stays under because there aren't
any holes in between' 9
DUM (9 ; 6): The wood floats "because there is air inside"; the key does
not "because there isn't any air inside''
But the analogy cannot be considered valid except on condition
that the "holes" in the wood stay closed. This leads to the follow-
ing type of explanation given in the case of needles poised deli-
cately on the surface of the water:
RAY (9 years, same subject): "The needle pricks and goes in the water
because it is thin and heavy." "Look" [it floats]. "AW It's because
there was a hole in the other needle that went under." "But this one
has a hole too."- . . . -"And that ring clamp?' '-"It will go to the
bottom because there are holes; the water comes in."
AND (10 ; o): "The needle floats because there is a little hole." "And
if it were big? 9 '-"It falls "-"How can you tell beforehand?"-"/* de-
pends on whether it is big or small. If the water doesn't come over the
top of it, it stays up"
On the whole, substage II-B shows significant progress in the
direction of internal consistency and in the search for a single
explanation based on the preliminary relating of the weight to the
volume presupposed by the schema of more or less "filled." How-
ever, since the volume of water envisaged is not that of the dis-
placed water but rather of the total quantity of water contained
in the receptacle, the relationship between the weight of the body
and that of the water remains one between active forces, thus
reintroducing a complexity rich in contradictions. The probability
that they will appear is greater when the air is seen as intervening
and holes, open or closed, are assigned a role. In short, for lack
of operational relations sufficiently worked out to dominate the
sum of the relationships between weight and volume, the explana-
tion, although vaguely intuited, is not clearly discovered, and a
coherent system is not as yet formulated.
Stage III
We have put a great deal of emphasis on the preoperational levels
and the concrete operational stages with two purposes in mind;
36 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
in order to point out, first, what a long road thought processes
must tread before even the attempt to find a single noncontradic-
tory explanation appears, and second, why the completion of a
model for such explanations cannot be achieved without the aid
of formal operations even in the present case, in which the law to
be found can be stated by using purely concrete operations. We
must now try to analyze the role of the formal operations needed
to discover the law. But the problem is somewhat more complex
than that in the case of the law of equality of angles studied in the
preceding chapter. Actually, in the latter case, in themselves the
correspondences between the inclinations found at substages II-A
and II-B gave a first approximation of the law; only the reason for
these correspondences remained beyond the subject, and formal
thought introduced nothing more than an element of necessary
implication to a set of relations which were already exactly formu-
lated. In the present example, on the other hand, the law is not
completely discovered at substage II-B, and formal thought is
indispensable to its formulation in a complete form. This differ-
ence can be given two explanations, which are as follows:
In the first place, even if the relation between densities, once
it is found, can be expressed in a purely concrete form, formal
schematization is still needed to work out the relevant concepts.
The concept of density in fact presupposes that of volume. How-
ever, we have stated before that the conservation of volume is
not worked out conceptually before the beginnings of the formal
level i.e., toward 11-12 years. 9 Without a doubt the reason for
this is that, in contrast to simple forms of conservation, which the
subject masters by simple additive compensations, the conserva-
tion of volume throughout changes of form presupposes the ability
to handle proportions. 10 However, we shall see in the course of
chapters 11-14 of the present work why the concept of proportion
does not itself appear before the formal level, when it arises in
connection with certain general properties of the group structures
characteristic of prepositional operations.
In the second place, formal operations are particularly impor-
9 Le DSveloppement des quantits chez I* enfant, Chap. Ill and La G6-
om&rie spontanSe de I'enfant, Chap. XIV.
10 If the three dimensions of a volume, x, y, and z, are transformed into
xf, y' 9 and d, there is conservation when xyftftf = tf/z, from the formula
xyz = xV^. These are multiplicative compensations, thus proportions.
THE ELIMINATION OF CONTRADICTIONS 37
tant in the case of the law of floating bodies in order to make
possible both the exclusion of the too-simple interpretations of
stage II and the purely imaginative construction of a hypothesis
which does not correspond to any of the directly observable con-
crete data. The stage II explanations are not actually absurd and
do not directly contradict the facts, and, if they are to be excluded,
the fact that they are not coherent enough must be felt. But this
can be done only by a thought process able to deduce the conse-
quences of simple hypotheses with necessity. On the other hand,
to relate the weight of the body under consideration to the weight
of an equal volume of water is to invent a situation which has no
empirical correlate, because only the total volume of the water in
the receptacle is actually observed, whereas the conceptualization
of a volume of water equal to that of the object to be compared is
the product of a subtler separation of variables which once more
requires hypothetico-deductive thought
Doubtlessly the subjects we are going to examine now are much
more likely to use acquired knowledge, for they are approaching
the academic level where they deal with such questions. But when
this acquired knowledge does not correspond to the mental struc-
tures indispensable to their assimilation this is immediately rec-
ognized in the questioning, and we have not used the cases
prematurely influenced in this way. In addition, we have seen
how, as early as 9 years of age, subject BAB compares wood and
iron at equal volumes. The generalization of the same mental
operation to the water itself is made so naturally in the course of
stage III that it is hard not to allow for the role of spontaneity
in the progressive structuring of the data, even if it is hastened by
the surrounding social environment. The following examples, be-
ginning with two intermediate cases, illustrate this stage:
FRAN (12- ; i) does not manage to discover the law, but neither does he
accept any of the earlier hypotheses. He classifies correctly the objects
presented but hesitates before the aluminum wire. "Why are you hesi-
tating?" "Because of the lightness, but no, that has no effect." "Why?"
"The lightness has no effect. It depends on the sort of matter: for ex-
ample, the wood can be heavy and it -floats. 99 And for the cover: *I
thought of the surface" "The surface plays a role?" "Mat/be, the
surface that touches the water, but that doesn't mean anything 9 ' Thus
he discards all of his hypotheses without finding a solution.
38 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
FIS (12 ; 6) also, in the transition phase between stages II and III,
comes close to solution, saying in reference to a penny that it sinks
"because it is small, it isn't stretched enough. . . . You would have
to have something larger to stay at the surface, something of the same
weight and which would have a greater extension."
ALA (11 ; 9): 'Why do you say that this key will sink?" "Because it is
heavier than the water." "This little key is heavier than that water?"
[the bucket is pointed out]. "7 mean the same capacity of water would
be less heavy than the key" "What do you mean?"-"Y0w would put
them [metal or water] in containers which contain the same amount
and weigh them."
JIM (12 ; 8) classifies floating or sinking objects according to whether
they are "lighter or heavier than water ." "What do you mean?" "Yow
would have to have much more water than metal to make up the same
weight" "And this cover?" "When you put up the edges, there is
air inside; when you put them down, it goes down because the water
comes inside and that makes more weight" "Why does the wood
ftoati^-TJecawe it is light" "And that little key?"-"No, this piece of
wood is heavier ""So?"- "If you measure with a key [ = with the
weight of a key], you need more wood than lead for the weight of the
key" "What do you mean?" "If you take metal, you need much more
wood to make the same weight than metal"
MAL (12 ; 2): "The silver is heavy, that's why it sinks" "And if you
take a \xee?""The tree is much heavier, but it is made of wood"
"The silver is heavier than that water?" [bucket]. "No, you take the
quantity of water for the size of the object; you take the same amount
of water" "Can you prove that?" "Yes, with that bottle of water. If
it were the same quantity of cork, it would float because the cork is less
heavy than the same quantity of water. 9 ' And again: "A bottle full of
water goes to the bottom if it is full because it's completely filled with-
out air, and that bottle stays at the surface if you only fill it halfway."
We see how, rejecting any suggestion that they relate the
weight of the objects in question to the weight of all the water in
the receptacle, these subjects reach the point of comparing the
first weight to that of an equivalent volume of water. FRAN begins
by assigning a possible role to the contact surface; FIS believes
that a piece of metal would float if, without adding to its weight,
its "extension" could increase; then ALA, JIM, and MAL are able
THE, ELIMINATION OF CONTRADICTIONS 39
to reason about the amount of water equal to the volume of the
object. "You take the quantity of water [equal to] the size of the
object/' says MAL, for example. Thus the "more or less full" schema
used at substage II-B is transformed into a relationship between
the weight and the correct volume (for FIS, the "stretch" becomes
the relationship of weight to "extension") and finally between the
weight and volume of the object in question and the correspond-
ing weight and volume of water displaced by that object.
These facts raise three related problems: (A) How does the
subject start to discard the hypotheses he has had up to this
point? (B) How does he go about constructing the new hypothesis?
(C) How does he go about verifying it?
A. On the first question, it is worth noting that from this point
on the subject discards only crude hypotheses without verifica-
tion, whereas he is more and more likely to verify the superior
hypotheses. He even discards the first almost without explicit
reasoning, as when FRAN says, "That has no effect," or "That
doesn't mean anything." In other words, he finds that in order to
refute an explanation it is sufficient to invoke verbally or mentally
a case where the purported factor is associated with the opposite
effect, Thus FRAN eliminates absolute weight in saying, "For ex-
ample, wood can be heavy and it floats." Likewise, ALA and MAL
discard all comparison with the total volume of water in the
receptacle, knowing well that the variations of this volume leave
the floating or nonfloating of the bodies in question unchanged. In
comparison with stage II, the innovation is the same in the case
of floating bodies as in the problem of the equality of angles
studied in Chap, i; the subject views the problem in terms of all
possible combinations in such a way as to draw out their implica-
tions or nonimplications instead of noting the empirical links
simply in order to draw tables of correspondences or classifica-
tions from them. We have seen in the first chapter how implica-
tion is substituted for simple correspondence, and in a moment
we will return to the subject of implication in reference to the con-
struction of the new hypothesis characteristic of this stage. But
the elimination of the hypotheses deemed inadequate is accom-
plished by the following three procedures, each of which supposes
comprehension of nonimplication.
In the first place, if we call p the assertion that the bodies will
40 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
float and let q be any factor associated with p for example,
lightness (absolute) the subject will find that to state the occur-
rence of the association p.q is all that is needed to discard the
factor q; for example, the large block of wood is heavy, neverthe-
less it floats. It is known, in effect, that p.q is the negation of the
implication p D q:
(i)
In the second place, the subject may note the two possibilities
combined (p.q) v (p.q) i.e., of p. (q v q) which constitutes the
operation we may speak of as the affirmation of p independently
of the truth or falsity of q. But this operation contains p.q and
amounts also to discarding p D q. This is what FRAN, for example,
says when he declares that "the wood can be heavy (or light) and
it floats":
p.(qvq) = (p.q)v(p.q). (2)
Finally, the subject may not retain a factor because he knows
that all the possible combinations are true. For example, if p states
that the bodies float and q that there is a large quantity of water
in the receptacle at the present level, the subjects do not attribute
any more importance to statement q because they know well
that one can observe the occurrence of all four combinations,
(p.q) v (p.q) v (p.q) v (p.q) i.e., one object may float equally well
on much or little water, another may sink in the same two situa-
tions. But this operation with four conjunctions, which is called
"tautology" or "complete affirmation," again contains the non-
implication (p.q):
(p*q) = (p.q) v (p.q) v (p.q) v (p.q) . (3)
B. As for the explanation of how the child gets to the hypothe-
sis found at stage III according to which 'lighter (or heavier)
than the water" signifies also "at equal volumes," it can be ex-
plained in terms of what we have said for both the operations and
the concepts themselves.
As for the concepts, the child at stage II has already learned
that one body may be heavier than another with equal volume
(see BAR'S comments on the wood and on the metal of which the
needle is made), but he believes that the weight of the body in
THE ELIMINATION OF CONTRADICTIONS 41
question is to be compared to the total volume of the water in
the receptacle. The stage III child, on the other hand, rejects the
latter hypothesis. All that remains for him to do is to relate the
weight of the body to that of a quantity of water no longer any
quantity whatsoever but a quantity equal to the volume of the
body itself. In other words, the discovery distinctive of stage III
is nothing more than the generalization of the mode of compari-
son roughly formulated at stage II for two solid bodies, but
henceforth it is applied to the water itself as well as to the object
judged heavier or lighter than it. That this comparison should be
more difficult when it involves a solid body and water than when
it involves two solid bodies should be obvious, since the volume
of water equal to the volume of the immersed solid has no visible
contours and can be conceptualized only after a preliminary
abstraction. But this discovery is nothing less than the resultant
of all the previous conceptualization of relative weight or specific
gravity.
Thus, from the standpoint of the relevant operations this com-
parison with a hypothetical equal volume touches on a reasoning
process of which we will find numerous examples later and which
consists of considering the variation of a single factor "all other
things being equal." If we let p be the assertion that a given
object floats and p the assertion that it does not, q the assertion
that its volume is equal to that of a certain quantity of water,
r the assertion that it is lighter than that quantity of water, and
r the assertion that it is heavier, the relationship which the subject
establishes is the following:
p.q.r. v p.q.r , (43)
which is in fact the schema of proof based on the assumption "all
other things being equal." But this expression is itself equivalent
to the product of two formal operations, (p g r) and (q.r) v (q.r)
i.e., the reciprocal implication (or equivalence) between the
floating of the body and its weight (relative to the same volume of
water) and the assertion that weight and volume vary independ-
ently. The operation (p.q) v (p.q) i.e., proposition (2) by means
of which the subject shows that a given factor does not play a
causal role, can also be recognized in the operation (q.r) v (q.r ).
Thus the explanation discovered at stage III covers all possible
42 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
cases including hollow objects without requiring the assignment
of a causal role to the opposing forces of water and air or to any
sort of hole. In the case in which the weight of a body equals that
of the water (at the same volume) we have, of course (if p =
neither floats nor sinks and r = neither heavier nor lighter than
water):
p .q.r vp .q.r . (4b)
For example RAY (12 ; 7) says, in reference to a very thin cube
of plastic (whose density is approximately equal to that of water),
that if it were filled with water "It would stay in the middle, in the
liquid, because the weight is the same."
After the multiple attempts at unification seen throughout stage
II, the subjects finally attain a unified noncontradictory explana-
tion; the two principal previous sources of contradiction (absolute
weight and active forces) are eliminated by the single hypothesis
of density or of the relation between weight and volume.
C. If we study the verification processes used by the subjects,
we find they confirm completely what has been said earlier and,
in particular, allow us to verify the fact that the subjects' reason-
ing no longer operates in a simple formulation of relationships or
concrete correspondences but requires a formal combinatorial sys-
tem. Whereas at the first preoperational level the subject is not
capable of any proof, at the level of concrete operations (sub-
stages II-A and especially II-B) he does not feel spontaneously
the need for it, but he can furnish it if asked. However, in keeping
with the entire logic of concrete operations, which is simply a
matter of organizing the reading of the raw experimental results
(by classification, setting up of relationships, etc.), at this point
the only method of verification of which he conceives is to accu-
mulate facts until more or less complete certainty is reached but
without going beyond the general i.e., without introducing neces-
sary links by isolating these facts from their contextual interde-
pendence and deducing the relations thus isolated.
BON (11 ; o) wants to prove that "all wooden objects float" Therefore
he puts two in the water [wooden cube and the ball] : "1 only have to
put the two things in the jar. They both float. All wooden objects
float."
THE ELIMINATION OF CONTRADICTIONS 43
But how valid is this jump from "some'' particular cases to "all/*
which reminds one of the amplifying induction that classical logic
wanted to regard as a fundamental reasoning process? In the
absence of probabilistic reasoning (excluded at substage II-B),
it is only a worthless extrapolation, for p.qO(pDq) gives
(pDq)v (p.q) 9 therefore p.q D (p * q)-i.e., "some wooden objects
float" could imply any one of several results in a particular case.
At the level of formal thought (from substage III-A on), on the
other hand, proof consists in demonstrating the truth or f alsehood
of a particular or general assertion which takes into account (or
tries to take into account) the total number of possible combina-
tions, thus permitting the subject to group combinations in a
demonstrative fashion. However, grouping these combinations is
exactly the same as selecting the cases where a single factor
varies (the others being held constant) so as to isolate universal
relationships from simple contingent conjunctions and above all
so as to be able to discover necessary relationships between vari-
ables. Such a composition of relations requires that we must
resort to what we have called in Chap, i the "structured whole"
(the combinations having o, i, 2, 3, and 4 conjunctions); thus it
contrasts to the simple additive and multiplicative class inclu-
sion i.e., "some" and "all" characteristic of concrete operations.
In other words, the verification process found at stage III makes
explicit use of the schema "all other things being equal" to which
we have just compared the explicative hypothesis of which the
subject conceives [4a, 4b], Two further remarks should be made
with regard to both the difference between substages III-A and
III-B and the relationship between the present problem and prob-
lems which will be taken up in the following chapters.
As a general rule and in its authentic form, the schema "all
other things being equal" appears only at substage III-B, as we
shall see later in reference to flexibility: but then the problem is,
given n factors A, B, C, D, . . . independent of each other, to
vary A leaving B, C, D, etc., unchanged. But in the present case
the two factors, weight and volume, are not independent in this
sense, since the subject is trying to determine the relation between
them and to link them in a new concept i.e., density. Thus, in
comparing an object to the water it floats on, it is easier to vary
weight and leave volume constant than to hold several independ-
44 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
ent factors constant (such as temperature, pressure, etc.), as one
would in studying the role of weight. This is why the schema "all
other things being equal/' in the elementary form it takes for the
present problem, appears as early as substage III-A:
GER (12 ; 7) to prove that the coin has a higher density than water
says: "If there were a jar filled with water and another just like it filled
with pennies . . . [the latter would be] heavier and would go to the
bottom of the lake' 9
AL (12 ; 8): "With the same volume, the water is lighter than that key."
To prove it: T would take some modeling clay, then I would make an
exact pattern of the key and I would put water inside: it would have
the same volume of water as the key . . . and it would be lighter."
But it is still true that it is only at substage III-B that this
schema acquires its general value. Actually, for the present ques-
tion, it is only toward 13-14 years of age that it gives rise to the
search for a common metric unit. Some of the objects we used in
our experiments were (among others) a cube of wood, a cube of
iron, and an empty plastic cube (density about i), all three of the
same volume. But it is striking to see the subjects at substage
III-B turn sooner or later to these units; they are the only subjects
who do so spontaneously:
LAMB (13 ; 3) correctly classifies the objects that sink: *7 sort of felt
that they are all heavier than the water. I compared for the same
weight, not for the same volume of water" "Can you give a proof?"
"Yes, I take these two bottles, I weigh them. . . . Oh! [he notices
the cubes] I weigh this plastic cube with water inside and I compare
this volume of water to the wooden cube. You always have to compare
a volume to the same volume of water. ""And with this wooden ball?"
"By calculation." "But otherwise?" "Oh, yes, you set the water level
[in tie bucket]; you put the ball in and let out enough water to main-
tain the original level.' 9 'Then what do you compare?" "The weight
of the water let out and the weight of the ball."
WUR (14 ; 4): "I take a wooden cube and a plastic cube that I fill with
water. I weigh them, and the difference can be seen on the scale
according to whether an object is heavier or lighter than water."
Here we notice that the factor left invariant, as well as the
common unit sought, is always seen in terms of volume, although
THE ELIMINATION OF CONTBADICTIONS 45
theoretically it would be equally possible to say that for equal
weights of water and of the object in question, the latter would
float if it had a greater volume (cf . the case of FIS and his piece
of money which would float if it had greater extension). But
experimental verification would be more difficult in this case.
Thus, generally, verification at stage III consists of two pro-
cedures: (i) separating out variables according to combinations
not given by direct observation, and (2) the composition of these
relationships according to operations of conjunction and implica-
tion such as those of proposition (4). It is in this respect that, in
the present problem as in the case of the equality of the angles of
incidence and reflection, in the end the required law must be
worked out formally even though the discovery of this law has
been prepared by a long process of concrete structuring. But,
without a doubt, in neither case are the possible combinations
numerous enough for the role of formal operations to be clearly
distinguished from that of concrete operations and particularly
for the schema "all other things being equal" to acquire all its
general significance. For this reason we need to pass on to the
analysis of more complex problems.
Flexibility and the Operations
Mediating the Separation
of Variables 1
THE FLEXEBUJTY of a rod depends on the material it is made of, its
length, its thickness, and the form of its cross-sections. All other
things being equal, the degree to which it bends varies as a func-
tion of the weight that is placed at its tip. To study the reasoning
processes mediating the separation of variables and the verifica-
tion of their respective roles, it seemed worth while to give our
subjects a problem involving much greater empirical difficulty
than the earlier ones, though not requiring for its solution con-
cepts essentially more complex. In the case of floating bodies, we
have just had a glimpse of the importance which the schema "all
other things being equal" plays in hypothetico-deductive think-
ing. But the interference of five distinct variables, as in the flexi-
bility problem, furnishes a situation particularly favorable for the
study of the formation of this experimental schema and of the
formal operations which it presupposes, for if a complete solution
is to be attained each factor must be varied independently and
the others held constant. 2
1 With the collaboration of A. M. Weil, former research assistant, Institut
des Sciences de 1'fiducation, and J. Rutschmann, research assistant, Labora-
tory of Psychology.
2 The experimental technique is as follows: The experimenter presents the
subject with a large basin of water and a set of rods differing in composition
(steel, brass, etc.), length, thickness, and cross-section form (round, square,
46
THE OPERATIONS OF THE SEPARATION OF VARIABLES
47
Stage I
If we are to understand in what way formal operations comple-
ment concrete operations at stage III, we must first find out what
the latter contribute to the separation of variables; but in order
n
O
FIG. 2. Diagram A illustrates the variables used in the flexibility ex-
periment. The rods can be shortened or lengthened by varying the
point at which they are clamped (see B for apparatus used). Cross-
section forms are shown at the left of each rod; shaded forms represent
brass rods, unshaded forms represent non-brass rods. Dolls are used
for the weight variable (see B). These are placed at the end of the rod.
Maximum flexibility is indicated when the end of the rod touches the
water.
rectangular). Three different weights can be screwed to the ends of the rods.
In addition, the rods can be attached to the edge of the basin in a horizontal
position, in which case the weights exert a force perpendicular to the surface
of the water. The subject is asked to determine whether or not the rod is
flexible enough to reach the water level. His methods are observed and his
comments on the variables he believes influence flexibility are noted; and
finally, proof is demanded for the assertions he makes.
48 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
to do that we must start by describing responses at the preopera-
tional level (until about 7 years). The reactions of this stage are
simple; in all his explanations the child is limited to describing
what he sees. As neither classifications nor organized operations
of serial ordering are yet available, he fills in his observations
with precausal linking (finalism, animism, moral causality, etc.):
BIG (5 ; o) puts 200 grams on the 40 cm. square steel bar: "It doesn't
touch the water" [the rods to be compared represent mobile "bridges"
attached to a plank which in turn represents a "road"]. At the end of
these bridges are found small dolls, or "fishermen," which reach the
edge of the water if the "bridges" bend enough. Next he takes up the
round brass rod 7 mm. 2 in diameter which, unlike the steel bars above,
touches the water. "Why?" "Because it is lower down." For the round
steel rod, 22, cm. long, diameter 16 mm. 2 : "Why doesn't it touch the
water?"-"Bec<mse the bar is too high." "Why does it stay too high?"
"Because it's on a plank [ = attached to a plank, but so are all of the
rods!]. "But why with that one [brass] and not with this one?"
"Because it's too smalF [ = too short]. "And why didn't it work with
the first one?" [40 cm.]. "ft didn't work because there is wood [the
attachment plank] . . . [attached] to the second there is wood too.
I am going to try again [he begins again]. No, that doesn't work."
"Why?' 9 'Because it's heavier and it goes down in the water." 'And
this one?" [new rod]. "It doesn't work because ifs too high . . . /*
etc.
(5 ; 5) after a number of trials puts 100 grams on a rod and waits
as if it were going to descend in a moment. "Why don't all the sticks
go down the same way?" "Because the weight has to go in the water"
Then he places 200 grams on a thick rod and 100 grams on a fine
one: 'Which one bends the mostP'-'Tto one" [the fine rod].-"Why?"
'The weight is bigger here [he points out 200 grams on the other
one]; it ought to go into the water" [We put 200 grams on the thin
one, which then touches the water. He laughs.] "Why does it touch
now?" "Because it has to"
We see that these subjects are generally limited to a simple
report of what they perceive; the rod does not touch the water
because it remains too high or it touches the water because it
descends too low, etc. Finalism and moral causality ("It has to"),
etc., are added. They also start to formulate relations, which
process has a certain logical interest for us in that the child is
THE OPERATIONS OF THE SEPARATION OF VARIABLES 49
satisfied with undifferentiated, overly-general classes. Just as,
when making the transition from definitions which correspond in
form to the finalism and moral causality of primitive explanations,
he defines by generic classes which lack internal differentiation
(as when he states that "a mama" is "a lady" without referring
to her children), so at the present level (which immediately fol-
lows the level of precausal explanations) objective relational
processes do appear but in terms of generic rather than specific
inclusions. Thus RIG declares that a certain bar does not touch the
water because it is attached to the plank, although those that do
touch the water are similarly attached. 3 An instant later he takes
up the same explanation again; "Because there is woocF ( = the
plank), but adds spontaneously "There is wood (attached) to the
second too." In explaining why a thin rod bends more than a
heavy one, HUG limits himself to noting that the heaviest weight
is on the rod that bends the least, as if to imply that both
"should" (in the moral rather than the logical sense) touch the
water. There is still a great gap between this kind of inclusion,
exclusively generic because its form is even more primitive than
that of concrete operations, and the formal type of implication
that will eventually succeed the latter.
Stage II (Substages II- A and II-B)
With the appearance of concrete class and relational operations,
it becomes possible to report on raw empirical data through the
use of classificationscoherent and differentiated serial ordering
and correspondences but this is not in itself sufficient to assure
the separation of variables i.e., to assure the organization of a
valid experiment.
MOR (7; 10), after having put the weight on a narrow rod which
reaches the water, says: "It won't fall the same way with this one
[thick] "because the other one is thinner.* 9 Then he changes the weight:
"This one isn't so heavy as the other one 9 '; he places the heavy weight
on a short rod and the light one on a long rod, predicting that the
3 Cf. Piaget, The Moral Judgment of the Child (Free Press, 1948): The sub-
ject SCHMA (6 ; 6) thinks that a little liar fell in the water because he lied,
but that if he had not lied he would nevertheless have fallen in '^because the
bridge was old."
50 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
curve will be sharper "because the other weight is lighter than this
one." The experiment does not confirm his prediction, and then he
lengthens the short rod: "Oh! with the [thick] one you have to do
that . . . " etc. The subject is asked to summarize what he has dis-
covered up to that point by ordering the rods serially according to
flexibility: "Which one bends most?" "This one because it is the thin-
nest:'-"Next?"-"That one" [long and thin, metal]. -"Next?' '-'That
one" [short, wood].-"Next?"-"T7iw one [thicker]; it goes with the
weight" [heavy]. -"Next?' '-"That one [heavy, metal]; it didnt go in
the water because 1 had to do that* [lengthen it],
BAU (952): "Some of them bend more than others because they are
lighter [he points out the thinnest] and the others are heavier."
"Show me that a light one can bend more than a heavy one [he is
given a short thick rod, a long thin one, and a short thin one]. [He
places 200 grams on the long thin rod and 200 grams on the short thick
one without noticing the fact that the thin rod that he has chosen is
also the longest.] "You see." "Show me that the long one bends more
than the short.'* Again he puts 200 grams on the same two rods and
this time the result is supposed to demonstrate the role of length. "If
I take away the long one, can you compare again to find out whether
it's the lightest rod that bends more?" "Yes, this one and that one"
[the two short rods, one thick, one thin]. "Which is better, to compare
these two or to compare the way you did before?" "These two" [long
and thin, short and thick]. "Why?" "They are more different"
These two cases are sufficient to show us both the progress
made over stage I and the inability of the subjects at substage
II-A to separate out the experimentally relevant variables.
As before, the advance over stage I lies in the fact that the sub-
ject becomes capable of systematically registering the raw data
i.e., the facts as directly observed though not as they might be
selected with the question in mind of the verification of a hypothe-
sis or the separation of variables. The registration of data is sys-
tematic in that, instead of depending on the formulation of a
simple global relationship (such as the unspecified generic class
inclusion found at stage I), the subject is capable of differentiated
classification, serial ordering or equalizations, correspondences,
etc., which are all accurate when considered independently. For
example, MOR manages to compare lengths, thicknesses, weights,
etc., by serial ordering and even to set up a series of five rods
THE OPERATIONS OF THE SEPARATION OF VARIABLES 51
arranged in order of observed inclinations. Furthermore, each one
of these operations is correct including the last one, which is the
most complicated. But taken together they prove nothing when
the subject is left to his own initiative. When the experimenter
chooses two terms of comparison with respect to a certain factor,
all other things being equal, the operation of comparison that the
subject accomplishes seems meaningful. But when the subject is
left to himself, everything is mixed up. Thus, the series of five
inclinations that MOR arranges to summarize what he has observed
by himself is a multifactor confusion from which nothing can be
deduced. Likewise, in order to demonstrate the role of the width
(which moreover he confounds with the weight), BAU compares
two rods, one of which is the narrowest but also the longest. After-
wards he chooses the same elements to demonstrate the role of
the length, and when we try to encourage him to separate the two
factors he answers that it is best to compare the terms which differ
most widely.
In such cases the difference is clear between (i) the formal
operations that would enable the subject to separate out the
variables by use of the indispensable combinatorial system and
(2) the concrete operations needed to report the facts but insuf-
ficient to structure an experiment which could utilize this separa-
tion. Less clear are the reasons why concrete operations are
insufficient. Before analyzing this problem further, let us reexam-
ine the reactions of substage II-B; these add explicit multiplica-
tive schemas to the operations used at substage II-A, which appeal
only to implicit logical multiplication (BAU knows that his rod
is "more different" because "at the same time" thinner and longer,
but he does not say so and proceeds by simple addition of
relations).
The only change found at substage II-B is the successful use
of multiplications between asymmetrical relations. While the sub-
jects at substage II-A do not use logical multiplications except
under the elementary form of one-by-one correspondences, at
substage II-B subjects use double-entry tables with orders ori-
ented serially in different directions 4 as well as multifactor group-
ings (several links for the same result):
* Cf. the coordinate axes for space which also appear towards 9-10 years
(The Child's Conception of Space, Chaps. XIII-XW).
52 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
OT (9 ; 3) begins by referring to length: "You see that because the bar
is longer it can go down better" "And if you take two bars of the
same length?" [he is given a thin and a thick one]. "There it goes
down further, because it is thinner than the other which is fat, and
that one isn't." Next he determines the influence of weight and pre-
dicts for a short rod: "That wont work: the rod is too short and the
weight is too light for the rod. 9 '
HAE (10 ; 9) discovers the roles of the material the rods are made of,
thickness, and length. "Can you tell me without trying whether that
[weight] will reach the water with this rod?"'7 could 9 but by pulling
it in [only] a little; it's made of the same metal as the other one but
it is thicker, so you wouldn't have to pull it in as much as the other."
Thus [A same metal as B] X [thicker] X [longer] = [same inclina-
tion]. In addition there is understanding of the compensation between
two relations oriented in opposite directions: [less thin] X [longer] =
[thinner] X [shorter].
This last example indicates the appearance of both double or
triple-entry tables (condition of the multiplication of transitive
asymmetrical relations, with or without compensations) and mul-
tifactor multiplication (several causes are possible for the same
effect).
Still, subjects at this level are unable to verify the action of one
factor by leaving all of the other known factors constant. Likewise,
although they understand the compensation of length and thick-
ness for identical matter, they do not know how to generalize the
concept of compensation to the mutual compensation of all known
factors. Why this should be so raises a problem. The subjects'
failure to generalize is even more difficult to explain when we
consider the fact that they seem to be in possession of all the
requisite operational instruments. But although the subject ox,
for example, when given two bars of the same length, well under-
stands that the thinner one will bend more, when asked to demon-
strate the role of thickness he compares bars of unequal width
without assuring equivalence among the other factors and does
not realize that his verification is worthless. Likewise HAE
does not proceed any more skillfully, in spite of his discovery of a
potential compensation between two specific factors. Everything
seems to indicate that we have come across two different systems
of thought: one, the concrete, permitting simply the composition
THE OPERATIONS OF THE SEPARATION OF VARIABLES 53
of relations and of classes which depend on the immediate data,
and the other, the formal, permitting a restructuring of necessary
links.
The concrete system consists of tables of associations or corre-
spondences either of classes or of relations. For purposes of simpli-
fication we can express this in the language of classes. Let us call
Ai the class of rods which are 50 mm. or more long and A'i that
of rods < 50 mm.; A 2 the class of weights of 300 grams or more
and A' 2 that of weights < 300 grams; A 3 will be the class of brass
rods and A' 3 that of non-brass rods; etc. Finally, X will be the
class of rods touching the water and X' that of rods which do not.
Remember that when joined together the two classes A and A'
give the total class B.
In this case we have, for the two couples of classes (a single
factor and its result X or X'), the double-entry table:
(BO X (X + XO = AiX' + AiX + A'i + A'iX'. (i)
For two factors and their result there are eight combinations
(triple-entry table):
(Bx) X (B a ) X (X + X') = AiA 2 X + AaAaX' + A X A' 2 X + , ,
AnA'sX' + A'iA 2 X + A'iA 2 X' + A'iA' 2 X + A'^X'. w
Similarly, from three factors and their results, sixteen combina-
tions can be derived, from four factors, thirty-two combinations,
and finally from five factors and their results X or X' there are sixty-
four combinations. In the course of the experiments, in a more or
less empirical way (depending on his level) the child executes
these sixty-four combinations fully or partially; they allow him to
correlate the variation of factors with the result X or X' (the com-
binations are in fact even more numerous, since the factors of
length, thickness, weight, and inclination themselves give rise to
more variations than A or A' and since the class of A' 3 of non-brass
rods is in fact subdivided).
First, we must find out whether or not the subject at substage
II-B can construct such tables. It is likely that he can when one
or two factors are involved, because he uses reasoning based on
congruent structures and we know how easy it is for subjects of
7-10 years to structure serial correspondences. For three to five
factors they can proceed by addition of new elements in succes-
54 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
sion (doubling the preceding table each time). But it is obvious
that a complete calculation would not be possible, and, further-
more, it is not needed as long as the subject works with immedi-
ate correspondences from one element to the next.
Furthermore, in the presence of a single factor and its result,
X or X', it is generally sufficient for the child to find an immediate
correspondence in order to establish the relation between AI and
X: if AiX + A'iX' occurs and the combinations A'iX and AiX' are
null ( = not given in observation because nonoccurring), it is clear
the subject will conclude that AI influences X. But if all four com-
binations or the three combinations AiX -f A'iX + A'iX' occur
empirically, the method of formulating simple correspondences
between AI and X (or between their negations) will no longer
suffice; the subject must hypothesize a second factor which oper-
ates in the case A'iX (i.e., a factor which produces X for causes
other than AI). However, although he knows how to do this for
the simple cases (when the width compensates the length, etc.),
his efforts are less and less systematic when the number of factors
increases and the experimenter does not simplify the experiment
in his successive presentations of the factors to the child. In other
words, at the stage of concrete operations (II-A and II-B) the
subject knows how to observe the experiment in terms of the vari-
ous correspondences which actually occur, which means that he
can construct increasingly more complex tables from the empirical
associations (positive and negative). But he does not know how
to interpret his tables except when immediate correspondences
are sufficient. And he does not know how to separate variables
when they are too thoroughly mixed.
The reason for this failure is that in order to separate variables
one needs to vary each in turn while holding the others constant
("all other things being equal"). To do so, it no longer is enough
to consider the table as a whole in which all the correspondences
are simultaneously given; the associations AiX, etc., must be
analyzed situation by situation so that one may see which are
linked and which are mutually exclusive. But to arrive at this
analysis the subject would have to use a complete combinatorial
system, one which is no longer the mere construction of a table of
associations such as tables (i) with its 4 associations or (2) with
its 8 associations. This complete combinatorial system involves
THE OPERATIONS OF THE SEPARATION OF VARIABLES 55
considering the associations one-by-one, two-by-two, etc., so that
16 combinations can be derived from table (i) or 256 combina-
tions from table (2) instead of the 4 or 8 derived from the double-
or triple-entry tables. In other words, while tables (i) and (2)
constitute simple wholes composed of 4 or 8 parts (or associations)
brought together, the combinatorial system necessary for the for-
mal analysis of the associations is based on what can be called a
"structured whole," which here is composed of 16 combinations
in case (i) and of 256 in case (2).
Moreover, we have already seen in chapters i and 2 that this
complete combinatorial system is precisely the mark of formal
thought, for its structure goes beyond additive or multiplicative
groupings of classes and relations (with their simple concrete
inferences founded on the transitivity of class inclusions (or of
relational linkings) and engenders the structuring of prepositional
logic. Actually, for two propositions p and q the 16 possible opera-
tions (conjunction, disjunction, implication, incompatibility, etc.)
correspond exactly to these 16 combinations which can be derived
from table (i), and for three propositions p, q, and r the 256 pos-
sible operations (which, moreover, are all reducible to composi-
tions of binary operations) correspond to the 256 combinations
that can be derived from table (2).
In other words, if the substage II-B subjects do not yet isolate
the variables but simply establish the empirically given corre-
spondences, it is because they have not acquired the combina-
torial system which constitutes prepositional logic. The result is,
on the one hand, that they do not know how to combine empirical
results in such a way as to demonstrate which among the possible
associations of variables actually occurs and, on the other, that
they do not know how to reason by implication, etc., in such a
way as to combine the various factual data that they observe in a
form that is both necessary and conclusive. However, these two
failures can actually be reduced to a single one, since the same
combinatorial system will permit the stage III subjects to devise
experiments for separating variables and to deduce the results of
these experiments by the means of interpropositional operations.
Moreover, we must emphasize that it is because this same prop-
ositional logic is not available that the reasoning process which
would be used to prove the empirical hypotheses remains inacces-
56 THE DEVELOPMENT OF PROPOSITTONAL LOGIC
sible at substages II- A and B. Assuming A -> B and B - C (where
- can be an inclusion, an equality, or a transitive asymmetrical
relation), the concrete reasoning consists in concluding A -> C.
This sort of inference is found in the reactions of HAL and OT.
But this utilization of transitive relations does not give rise to an
interpropositional operation such as an implication. It amounts
only to combining classes or relations among each other on the
basis of a certain order of class inclusion.
In contrast, propositional operations consist in combining vari-
ous empirical associations on which multiplicative classes are
based in all possible ways: implication, for example, would be
defined as deriving from the combination AiX + A'iX + A'iX' the
affirmation that p D q (if p = the affirmation of AI and q = that of
X), for if only (p.q) v (p.q) v (p-q) occur and never p.q (corre-
sponding to AiX'), then q is always true when p is true. These are
the new combinations which, as we shall see later, distinguish the
thinking typical of the stage III subjects and which at the same
time give rise to both deductive capacity for demonstrative rea-
soning and experimental capacity for the isolation of the relevant
variables as each one may influence the end result.
Stage III (Substages III- A and III-B)
This level is characterized both by the incipient formal thinking
revealed in the appearance of hypothetico-deductive reasoning
and by an active attempt at verification. However, at first the sub-
ject is not able to handle the complete range of interpropositional
operations; as a result, even though we may observe the genesis
of implication, exclusion, etc., we do not yet find him able to
organize a systematic proof conforming to the schema "all other
things being equal" except in certain cases and even then not for
all of the relevant factors.
PEY (12 ; 9) speculates that if the rod is to touch the water it must be
'long and thin." After several trials, he concludes: "The larger and
thicker it is, the more it resists" "What did you observe?" *TAfe one
[brass, square, 50 cm. long, 16 mm. 2 cross-section with 300 gram
weight] bends more than that one [steel; otherwise the same condi-
tions which he has selected to be equal]: it's another metal. And this
THE OPERATIONS OF THE SEPARATION OF VARIABLES 57
one [brass, round] more than that one" [brass, square; same condi-
tions for weight and length, but 10 and 16 mm. 2 cross-section]. "If
you wanted to buy a rod which bends the most possible?""! would
choose it round, thin, long, and made of a soft metal."
AULE (12 ; 10) wants to prove that a long rod bends more than a short
one. He takes the two steel bars, one round and 22 cm. long, the other
square and 50 cm. but, not noticing that they do not have the same
cross-section form, he adjusts both of them to 22 cm. for length; "This
one [round] bends more because it is thin" [they have the same
width, but one is round, the other square]. "What have you proved?"
"I don't think I've proved anything. Oh! Yes, that the round ones
bend more than the square!'
DUR (11 ; 10): "There are flat ones, wider ones, and thinner ones and
longer ones. If they are both long and thin, they bend still more."
"Could you show me that a thin rod bends more than a wide one?"
[He puts 100 grams on the round steel rod 50 cm. long and 16 mm. 2
cross-section and 200 grams on the round steel rod 50 cm. long and
10 mm. 2 cross-section.] "That one bends more" [10 mm. 2 cross-sec-
tion]. "I would like you to show me only that the thin one bends
more than the wide. Is that way right?" [He takes off the loo-gram
weight and puts 200 grams on the 16 mm. 2 rod.] "You see, this is the
right way."
KRA (14 ; i): "Can you show me that a wide one bends less than the
narrow?" [He puts 200 grams on the round steel bar 50 cm. long
and 10 mm. 2 cross-section and 200 grams on the square brass rod
50 cm. long and 16 mm. 2 cross-section.] "This one [thin steel] goes
down more" "Why?" "It is round, more flexible, the steel is less
heavy, it is round and narrower." "Fine, but I would like a rigorous
proof that it's because it is narrower/' [He places 200 grams on tihe
round steel rod 50 cm. and 16 mm. 2 and 200 grams on the round
steel rod 50 cm. and 10 mm. 2 .] "You see, this one bends more because
it is less wide""Bi8LVO. Can you demonstrate the same thing with
others?" "Yes. [Steel, square 50 cm. and 16 mm. 2 instead of round
steel of 16 mm. 2 ; thus the comparison is no longer exact.] This one
[narrow and round] bends more, it is less heavy" 'And can you dem-
onstrate the role of the form?" He puts 200 grams on the rectangular
brass rod, 50 cm. long and 16 mm. 2 cross-section. "Why does this
one [round, steel] bend more?" "Because it is round." "Is that the
only reason?" "The brass is also heavier" [he then spontaneously dis-
cards the steel rod and takes a square brass rod 50 cm. and 16 mm. 2 ].
58 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
The subjects' set at stage III is essentially new in comparison
to the set that characterizes concrete operations; it consists of not
being satisfied with empirical events as directly given but in re-
garding them from the start as one aspect of a larger domain, the
domain of the possible. In effect, stage II subjects are limited to
recording the successive data in terms of all the relations and
classes required by their diversity, but they neither separate out
variables nor elaborate hypotheses or proofs. On the other hand,
substage III-A subjects from the start conceive of reality as a
product of various factors arranged in a set of possible combina-
tions. This results in the appearance of two formerly insignificant
behavior patterns: the formulation of hypotheses, which consists
of the restructuring of these possible combinations as they might
occur empirically, and attempts at proof, which consist of deter-
mining which of the possibilities in fact do occur.
No doubt, the initial reactions of each of the preceding subjects
do not seem to differ from those observed at stage II. They con-
sist of describing empirical givens by means of relations and clas-
sifications. But, whereas the stage II subject accepts everything
pell mell, believing that in this way he has gotten to reality itself,
stage III subjects use preliminary concrete descriptions only as
material for setting up hypotheses and proofs; the result is a
more active set.
This new behavior can be observed in the choice of rods to be
compared i.e., in the tendency to compare them only from a
standpoint bearing on a delimited question. Whereas the stage II
subject compares any rod whatever to any other, limiting himself
to a statement of the most obvious relations, the stage III subject
understands that if he is to establish a given relationship, it is
important to select certain pairs of rods rather than others. It is
this choice, which is the most easily observed new reaction at
stage III, that allows us to demonstrate the nature of the logical
operations utilized.
The most important of these operations, or at least the one
which nearly always orients the substage III-A subjects' analysis
at the beginning of the experiment, is the formal operation of
implication by which the subject assumes that a determinate fac-
tor produces the observed consequences in all cases. At stage II,
a comparable causal relationship was established by simple corre-
THE OPERATIONS OF THE SEPARATION OF VARIABLES 59
spondence-for example, the longer the rod, the more flexible
but this type of reasoning cannot be legitimately generalized to all
cases. The operation of implication takes a similar statement of
correspondence as its starting point (the conjunction p.q translat-
ing AX). However, at substage III-A two new forms of behavior
appear, resulting in three types of statement that distinguish the
formal operation of implication from concrete correspondence.
First, a more or less systematic effort is made to determine the
consequences of eliminating or diminishing factor A, as compared
to the simple search for association between factor A and its result
X, which we found at the concrete level (although this effort is not
completely systematic before substage III-B); thus subjects at the
formal level can determine that in certain cases effect X is itself
eliminated or diminished (the association A'X' is found), whereas
in certain others it is conserved because it can be produced by
factors other than A (association A'X). The unified relation
AX + A'X + A'X' (or in propositions: p.q v p.q v p.q) thus consti-
tutes a system of interpretation which is broader than simple
correspondence because it integrates three possibilities simultane-
ously (either AX or A'X or A'X') and because in this way it can
bring into a single whole the results of several different groupings
of classes and relations.
In this case we have an elementary example of the combina-
torial system discussed above that handles "structured wholes":
in the case of implication the three parts (or associations) AX,
A'X, and A'X' are integrated in the manner presented above,
whereas in the case of disjunction or incompatibility they are
linked in other ways. Moreover, the ability to handle the combina-
torial system which appears at substage III-A is manifested not
merely in the appearance of this or that operation but in their
system as such-z.e., by all of the sixteen binary operations and
by the possibility of linking a determinate number of them in such
a way as to give rise to operations of an advanced sort.
Moreover, and this is the second behavior pattern new to sub-
stages III-A and III-B, the formation and the utilization of this
total system are manifested in the development of proof and nota-
bly in the schema "all other things being equal." The latter as-
sumes the utilization of a set comprising several distinct types of
implications integrated with other operations. From substage
60 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
III-A on, we observe a search for demonstration which is oriented
toward proof and the control of experimental conditions, but the
difficulties which prevent its realization are equally evident. In
fact, the subjects are not capable of more than partial proof. For
example, in order to give proof of the influence of the type of
metal, PEY compares two bars, one copper, the other steel, while
holding all the other factors constant; but in order to prove the
influence of width he compares rods of 10 and 16 mm. 2 cross-
section with unequal section forms (round and square) without
realizing that he is then faced with two independent factors.
Likewise DUR varies width and the weight simultaneously before
correcting for the lack of equivalence between the two. We are
certainly dealing here with a search for equivalence in the condi-
tions of comparison, but we still find difficulties in achieving it.
In order to have a better grasp of the nature of the operations
required by the schema "all other things being equal" and of the
dependence of these operations on the total combinatorial system
referred to above, let us begin by comparing the reactions found
at substage III-A with those of substage III-B; during this latter
stage proof becomes rigorous for the experiment under consid-
eration.
One good illustration will suffice:
DEI (16 ; 10): "Tell me first [after experimental trials] what factors
are at work here." "Weight, material, the length of the rod, perhaps
the form" "Can you prove your hypotheses?" [She compares the
200 gram and 300 gram weights on the same steel rod.] "You see, the
role of weight is demonstrated. For the material, 1 don't know"
"Take these steel ones and these copper ones." "I think 1 have to take
two rods with the same form. Then to demonstrate the role of the
metal I compare these two [steel and brass, square, 50 cm. long and
16 mm* 2 cross-section with 300 grains on each] or these two here
[steel and brass, round, 50 and 22 cm. by 16 mm. 2 ]: for length I
shorten that one [50 cm. brought down to 22]. To demonstrate the
role of the form, I can compare these two" [round brass and square
brass, 50 cm. and 16 mm. 2 for each.] "Can the same thing be proved
with these two?" [brass, round and square, 50 cm. long and 16 and
7 mm. 2 cross-section] .-"No, because that one [7 mm. 2 ] is much nar-
rower." "And. the width?" "1 can compare these two" [round, brass,
50 cm. long with 16 and 7 mm. 2 cross-section].
THE OPERATIONS OF THE SEPARATION OF VARIABLES 61
Our problem is to understand how the subject acquires such a
systematic method one whose apparent simplicity should not mis-
lead us, for only at 14-15 years can subjects spontaneously organ-
ize and utilize it without error.
If we refer back to proposition (2) (on page 53), which gives
the eight basic associations possible for two factors and their
results X or X', we must first assume that the subject begins by
establishing the facts, as at stage II, by means of concrete classi-
ficatory and correspondence operations. For example, for the
factor B 2 (weight) and the factor B 3 (metal), he may obtain the
following table of observations:
A 2 A 3 X = 300 gr., brass, inclination X (maximum);
A 2 A 3 X' = 300 gr., brass, inclination X' (because it is too short,
etc.);
A 2 A' 3 X rz 300 gr., steel, inclination X (sufficiently thin, etc.);
A 2 A' 3 X' = 300 gr., steel, inclination X 7 ;
A' 2 A 3 X rz 200 gr., brass, inclination X (sufficiently long or thin,
etc.);
A'2A' 3 X' = 200 gr., brass, inclination X 7 ;
A' 2 A' 3 X = 200 gr., steel, inclination X (sufficiently long, etc.);
A'2A' 3 X' nr 200 gr., steel, inclination X 7 .
Such observations show the subject from the start that the fac-
tors A 2 and A 3 are not the only relevant ones, since the same com-
bination A 2 A 3 may give either X or X'. This table is actually
extracted from the table of sixty-four associations corresponding
to the subject's potential observations. But the innovation found
at stage III is that, having organized a complex situation by means
of concrete operations, the subject does not consider his sets of
facts as a final ordering from which it would be sufficient to ex-
tract such and such relations and correspondences. Instead he
views them as a starting point for new combinations such that,
in associating each one of these eight base associations one-by-one,
two-by-two, three-by-three, etc., he can extract a new set of oper-
ations corresponding to the "structured whole" of the initial tables.
These are the new operations that make possible the separation of
variables, owing to the utilization of a set of implications in com-
bination with the simple conjunctions.
To state the new reasoning process in prepositional terms, let
62 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
us call p, q, r, s, and t the propositions which affirm the presence
of factors AI, A 2 , A 3 , A 4 , and A 5 respectively and p, q, r, s, and f
the propositions which deny their presence, and let us designate
by x and by x the propositions which affirm the results X and X'
respectively. The verification schema "all other things being
equal" thus amounts to nothing more than varying one of the
factors corresponding to p or q, etc., and leaving the others un-
changed. For example, for AI corresponding to p, we have
(p.q.r.$.t.x) v (p.q.r.s.t.x), (3)
which amounts to saying that for two rods supporting 300 grams
( =1 q\ of brass ( = r), thin ( = s), and with round section forms
( = *), it is sufficient to shorten the initial length of 50 mm. suffi-
ciently (p transformed to give p) if we are to modify the result
(x transformed to give x).
Thus we see that PEY compares two rods such that (p.q.r.s.t.x) v
(p.q.r.s.t.x) ( = r): in order to demonstrate the role played by the
factor metal in this case the equivalence (r.x) v (r.x) acquires a
demonstrative value because the other propositions (p.q.s.t) re-
main unchanged i.e., that (p.q.r.$.t.x) v (p.q.r.s.t.x) are chosen
among the totality of possible propositions. Furthermore, this
choice presupposes an understanding of the fact that, if proposi-
tions (p.q.s.t) are not kept unchanged (thus if the facts that they
express are not held constant), the effect x could result from a
cause other than r: the equivalence (reciprocal implication)
(r.x) v (r.x) is thus actually derived from an implication (r.x) v
(r.x) v (r.x) and consequently is integrated with all the implica-
tions conceived of as possible between p, q, r, s, t, and x. In other
words, to hold four out of five factors constant is equivalent to
granting that each one could in turn give rise to the same com-
binations. That is why the process of verification based on the
schema "all other things being equal'* is so complex and actually
involves the whole interpropositional combinatorial system.
The proof of this is that at substage III-A this type of demon-
stration is still only partially understood. The subject we have just
cited, PEY, later reasons unsystematically when he tries to verify
the role of the section form (round = i or square = t). He varies
t and t and the width ( = s) simultaneously i.e. he sets up the
proposition (p.q.r.s.t.x) v (p.q.r.s.t.x). In this case it is clear that
THE OPERATIONS OF THE SEPARATION OF VARIABLES 63
nothing concerning the role of t alone can he deduced from
(s.t.x) v (s.t.x). This sort of error is found again and again through-
out substage III-A. On the other hand, at substage III-B the proof
is rigorous; for example, subject DEI applies the same schema (3)
separately to all the factors which she distinguishes and does it
without any error. In other words, the formal combinatorial sys-
tem based on the "structured whole" (in contrast to the one-by-
one multiplications which furnish it with its base associations) is
under construction during substage III-A but is completed only
at III-B.
Another acquisition that these same studies show to be specific
to the formal level (substages III-A and B) is the capacity to deter-
mine qualitatively certain compensations between heterogeneous
relations. We have already seen the operation of certain logical
calculations for compensation which are based on the multiplica-
tion of concrete relations. Thus HAL (10 years) discovers that a
rod made of the same metal as another but thicker rod may bend
an equal amount providing it is lengthened. In this case, the
compensation is explained by the following operations: if F
designates the transition from thicker to thinner and L the transi-
tion from shorter to longer and F and L the inverse transitions,
we have:
This compensation is easy to understand in terms of simple
multiplication of inverse relations because these relations are
homogeneous. Both thickness and length are spatial dimensions,
and, since they work in opposite directions, it is easy to multiply
them by each other in a compensatory manner to obtain the same
product. We have already observed the same phenomenon in
reference to the conservation of quantities: 5 a tall and narrow
beaker may contain the same quantity of water as a low wide
beaker because the increase in width may compensate for the loss
of height. Although in both cases three dimensions (including two
which are distinct) and an operation of logical multiplication are
involved, in the case of quantities the relevant operation is more
5 See Piaget and Szeminska, The CMcFs Conception of Number (Routiedge
& Kegan Paul, 195*)* Chap. I.
64 THE DEVELOPMENT OF PROPOSITTONAL LOGIC
additive than multiplicative. This is because of the subject's im-
pression that it is possible to displace certain parts of the object
taken from the width in the direction of the length and vice versa,
thus leading him to an additive equalization of products.
Let us now examine the following cases of compensation:
1. For equal lengths, a round thin steel bar has the same
flexibility as a round thicker brass bar;
2. For equal lengths, a round thin steel bar has the same flexi-
bility as a flat brass bar with a larger cross-section surface;
3. For equal lengths, a round thick steel bar has the same
flexibility as a square narrower steel bar.
These pairs of bars are shown to the child, who is asked to
explain only why the rods bend equally for the same weights.
Nevertheless, these three problems are correctly explained only
at the formal level; problem i in substage III-A and problems
2 and 3 in III-B, with maximum difficulty for 3. Why this disparity
among the three problems?
In problem i the fact that the first rod is thin compensates for
the lesser flexibility of its steel composition. But since these two
factors are dissimilar, the subject must first separate out the
relevant variables. At the same time he must perceive them as
acting concurrently if he is to multiply the concrete relationships
between them. Here we see an analogy between this double re-
quirement and the verification schema "all other things being
equal/' Actually, in both cases the subject must cancel the effect
of one of the factors in order to determine the effect of the other.
Since the two factors are always present simultaneously, in both
cases he must limit himself to holding constant only the factor to
be canceled out (mentally or experimentally). Thus, he actually
cancels not the effect itself but rather possible variations in the
effect.
However, at the concrete level relationships between the
metallic composition of a rod and its flexibility or between thick-
ness and flexibility are formulated in rough form simply by noting
the data in varied situations without equalizing other factors. The
result is that in situations where compensations are exact, such as
in problem i, stage II subjects cannot be certain that the differ-
ence in flexibility due to metallic composition is being compen-
sated by thickness alone. On the contrary, ascertaining that the
THE OPERATIONS OF THE SEPARATION OF VARIABLES 65
lengths and degree of curvature are the same, they are led to
believe that the metallic factor (or eventually thickness) is less
important than they had formerly believed. So it is only when the
factors are both separated and integrated at the same time i.e.,
at the level where implication replaces simple concrete corre-
spondencethat the subject is able to conceive of two factors as
compensating each other exactly, even though he does not know
how to determine the quantitative influence of each factor. When
this equivalence is achieved and the subject has worked out the
separation of variables, his thinking turns to the variations that
are possible under pure, unmixed conditions, and it is not limited
to actual and mixed variations. It is from this that formally de-
duced compensations derive ("if it were . . . that should be this
case ..."). They arise in cases where compensation by corre-
spondences or concrete multiplications is inadequate.
The same holds for problems 2 and 3, but since in these cases
form and thickness compensate each other while the forms them-
selves also differ, the thickness (section surface) is not given per-
ceptually but must be formulated as a hypothetical possibility.
Thus the greater difficulty of these latter problems, problem 3 in
particular, is accounted for. Nearly everything must be deduced
by the subject. In problem 3 the section surface is hard to discern.
As for the intellectual operations, there is (aside from implica-
tions) a sort of proportion mediating the subject's understanding
of these compensations which is interesting because, since we
have not given our subjects any metrical or numerical data, it is
a pure qualitative or logical schema. The starting point is a double
implication (which we write for statements p and q, which desig-
nate any two factors, it being understood that in the case of metal
and thickness, r and s are used, or that in the case of thickness
and section surface, s and t, etc.):
p.q D x and p.qDx or x D (p.q) v (p.q) . (5)
This double implication signifies that the presence of factor p 9
in combination with the absence (or diminution) of factor q gives
the same result (designated by x) as the absence (or the diminu-
tion) of factor p and the presence of that designated by q.
In this case, tihe formulation is as follows: (i) conjunctions
(p.q) and (p.q\ which individually express a relationship of re-
66 THE DEVELOPMENT OF PROPOSITTONAi, LOGIC
ciprocal exclusion between p and q (let p wq = p.q v p.q\ lead
to the same consequence, in the present case x: (z) thus, not only
are they reciprocal but factors p and q can be substituted for
each other without influencing the result. The notion of a certain
logical proportion by reciprocity (R) follows. It is general for
(p.q) v (p.g). But here it serves as a schema for compensation
itself. For in this case reciprocity signifies an operation whose
value is equal but which is oriented in the opposite direction
(diminution or reinforcement):
This expression signifies (depending on whether it is read
diagonally, vertically, or horizontally): (a) that p.q = R(p.qr);
(b) that p v q = E.(p v q); (c) that p.q = R(p.?); and (d) that
p.p = R(q.q) since o = Ro.
We will come across many similar examples of logical propor-
tionality either independent of all metrical data or prior to
numerical determination. For the moment it is enough to note
that the problem involves not only prepositional reasoning but,
in addition, a formal structuring of the elements themselves. This
formal structuring is the subject matter of the second part of
this work.
4
The Oscillation of a Pendulum
and the Operations
of Exclusion 1
WE HAVE JUST SEEN how the subject goes about separating out
factors in order to determine their respective effects in a multi-
factor experimental setup. The present chapter takes up the
reactions of the child and adolescent in an analogous situation 2
with the difference that only one of the possible factors actually
plays a causal role; since the others have no effect they must be
excluded after they have been isolated. Such is the case for the
pendulum. The variables which, on seeing the apparatus, one
might think to be relevant are: the length of the string, the weight
of the object fastened to the string, the height of the dropping
point ( = amplitude of the oscillation), and the force of the push
given by the subject. Since only the first of these factors is actually
relevant, the problem is to isolate it from the other three and to
1 With the collaboration of A. Morf, research assistant, Laboratory of Psy-
chology and Institut des Sciences de Ifiducation; F. Maire, former research
assistant, Laboratory of Psychology; and C. Levy, former student, Institut des
Sciences de Ffiducation.
2 The technique consists simply in presenting a pendulum in the form of
an object suspended from a string; the subject is given the means to vary the
length of the string, the weight of the suspended objects, the amplitude, etc.
The problem is to find the factor that determines the frequency of the oscilla-
tions.
67
FIG. 3. The pendulum problem utilizes a simple apparatus consisting
of a string, which can be shortened or lengthened, and a set of varying
weights. The other variables which at first might be considered relevant
are the height of the release point and the force of the push given by
the subject.
THE OPERATIONS OF EXCLUSION 69
exclude them. Only in this way can the subject explain and vary
the frequency of oscillations and solve the problem.
Stage I. Indifferentiation Between the Subject s Own
Actions and the Motion of the Pendulum
The preoperational stage I is interesting because the subjects'
physical actions still entirely dominate their mental operations
and because the subjects more or less fail to distinguish between
these actions and the motion observed in the apparatus itself. In
fact, nearly all of the explanations in one way or another imply
that the impetus imported by the subject is the real cause of the
variations in the frequency of the oscillations:
HEN (6 ; o) gives "some pushes" of varying force: "This time it goes
fast . . . this time it's going to go faster"- er fhsit's true?" "OW Yes"
[no objective account of the experiment]. Next he tries a large weight
with a short string: Ifs going faster [he pushes it]. It's going even
/aster." "And to make it go very f ast?" "Yow have to take off all the
weights and let the string go all by itself [he makes it work but by
pushing], I'm putting them all back, it goes fast this time" [new
pushes]. As for the elevation: "If you put it very high, it goes fast"
[he gives a strong push]. Then he returns to the weight explanation:
"If you put on a little weight, it might go /aster/* Finally we ask him
if he really thinks that he has changed the rate. "No, you can't; yes 9
you can change the speed"
DUG (7 ; 3) is a little more advanced in that he finds several
(nonsystematic) correspondences between the lengthening of the
string and the increase in frequency. But he cannot prevent him-
self from pushing constantly and he counts the oscillations badly,
always influenced by his expectations.
One can see, then, that because of the lack of serial ordering
and exact correspondences the subject cannot either give an
objective account of the experiment or even give consistent ex-
planations which are not mutually contradictory. It is especially
obvious that the child constantly interferes with the pendulum's
motion without being able to dissociate the impetus which he
gives it from the motion which is independent of his action.
70 THE DEVELOPMENT OF PROPOSITTONAL LOGIC
Stage II. The Appearance of Serial Ordering and
Correspondences but Without Separation of Vari-
ables
Stage II subjects are able to order the lengths, elevations, etc.,
serially and to judge the differences between observed frequencies
objectively. Thus they achieve an exact formulation of empirical
correspondences but do not manage to separate the variables,
except insofar as the role of the impetus is concerned.
At substage II-A serial ordering of the weights is not yet
accurate:
JAC (8 ; o) after several trials in which he has varied the Length of the
string: "The less high it is [ = the shorter the string], the faster it
goes." The suspended weight, on the other hand, gives rise to inco-
herent relationships: <t With the big ones [ = the heavy ones] it falls
better, it goes faster, for example, It's not that one [500 grams], it's
this one [100 grams] that goes slower' 9 But after a new trial, he says
in reference to the xoo-gram weight: "It goes faster, 3 ' <4 What do you
have to do for it to go faster?" "Pwfr on two weights .""Or else?"
"Don't put on any: it goes faster when it's lighter." As for the dropping
point: "If you let go very low down, it goes very fast'' and "It goes
faster if you let go high up" but in the second case JAC has also short-
ened the string.
Since the ordering serially of the other factors is accurate, the
subject discovers the inverse relationship between the length of
the string and the frequency of the oscillations at this and suc-
ceeding levels, However, since he does not know how to isolate
variables, he concludes that the first variable is not the only
relevant one in the problem. Moreover, if he attributes causal
roles to the weight and the dropping point as well, it is because
he varies several conditions simultaneously.
In spite of the marked progress seen at substage II -B, which is
due to an accurate ordering of the effects of weight (in the raw
data), the factors cannot always be separated:
BEA (10 ; 2) varies the length of the string [according to the units two,
four, three, etc., taken in random order] but reaches the correct con-
clusion that there is an inverse correspondence: *7f goes slower when
THE OPERATIONS OF EXCLUSION 71
it's longer." For the weight, he compares 100 grams with a length of
two or five with 50 grams with a length of one and again concludes
that there is an inverse correspondence between weight and frequency.
Then he varies the height of the drop without changing the weight or
the length [without intending to hold them constant, but by simplifi-
cation of his own movements] and he concludes: "The two heights go
at the same speed." Finally he varies the force of his push without
modifying any other factor and again concludes: "It's exactly the
same."
CRO (10 ; 2), Likewise, cannot separate weight and length. However, in
contrast to BEA, he does vary the dropping point. He begins with a
long string and 100 grams, then shortens the string and takes 200
grams that he drops from a higher point: "Did you find out anything?"
"The little one [100 grams] goes more slowly and the higher it is
[200 grams with a short string] the faster it goes" But afterwards he
puts 50 grams on the same short string: "The little weight goes even
faster" However, the subject neglects this last case: "To go faster,
you have to pull up the string [diminish the length] and the little one
goes less fast because it is less heavy" Then: "Do you still wonder
what you have to do to make it go faster?" "The little weight goes
faster" 'How can you prove it?" "You have to pull up the string"
[diminish the length].
PER (10 ; 7) is a remarkable case of a failure to separate variables: he
varies simultaneously the weight and the impetus; then the weight, the
impetus, and the length; then the impetus, the weight, and the eleva-
tion, etc., and first concludes: "It's by changing the weight and the
push, certainly not the string" "How do you know that the string has
nothing to do with it?" "Because it's the same string" He has not
varied its length in the last several trials; previously he had varied it si-
multaneously with the impetus, thus complicating the account of the ex-
periment. "But does the rate of speed change?" "That depends, some-
times it's the same. . . . Yes, not much. . . . It also depends on the
height that you put it at [the string]. When you let go low down, there
isn't much speed." He then draws the conclusion that all four factors
operate: "It's in changing the weight, the push, etc. With the short
string, it goes faster" but also "by changing the weight, by giving a
stronger push" and "for height, you can put it higher or lower"
"How can you prove that?" "Yow have to try to give it a push, to
lower or raise the string, to change the height and the weight" [He
wants to vary all factors simultaneously].
J2 THE DEVELOPMENT OF PKOPOSITIONAL LOGIC
MAT (10 ; 6) goes so far as to set up the simultaneous variation of fac-
tors as a principle. "How do you know that it goes faster when there
is more weight?" <f When you put on a big weight, it goes faster.''
"Did you find that out?" "Y es, by raising the string [ = by diminish-
ing its length], then you put on the big weight at the same time."
These cases are extremely instructive because they show the
difference between concrete and formal operations. From the first
standpoint, the subjects can handle all the forms of serial ordering
and correspondence which make the variation of the four factors
possible and assure the reporting of the result of these variations,
but they know how to draw from these operations nothing more
than inferences based on their transitiveness (from the model
A < C if A < B and B < C). They remain inept at all formal rea-
soning. From the second standpoint, they commit the following
two errors: (i) In varying several factors simultaneously so that
AI A2 A 3 A 4 are transformed to A'i A's A'a A' 4 and in ascertain-
ing the change from the result X to X', they think they have
shown that each one of the factors in turn implies X'. Put into
prepositional language, the error amounts to concluding from
(p.q.r.s)Dx that (pDac).(g Dx).(rDx).(sDx), without suspecting
the existence of other possible combinations (see MAT for two
factors: p.qDx therefore qO#); (2) Reciprocally, subject PER,
having varied all of the factors except one (the length of the
string) and not being very sure whether or not the result has been
modified, concludes that the single constant factor must be in-
effective ("Anyway it's not the string . . . because it's the same
string!"). In other words, from p.q.r.s (x) v p.q.r.s (x v x) he con-
cludes pDx.
Thus it is evident that the subjects still lack some logical instru-
ment for interpreting the experimental data and that their failure
to separate out the factors is not simply the result of mental lazi-
ness. Just as BAU (Chap. 3) varied two factors simultaneously in
the comparison of the flexibility of rods so that the results would
be "more different," so do MAT and the preceding subjects ex-
plicitly propose to modify all factors simultaneously so as to
accomplish more impressive transformations. At this stage the
subjects lack a formal combinatorial system. Since they are accus-
tomed to operations of classification, serial ordering, and corre-
THE OPERATIONS OF EXCLUSION 73
spondences, they are limited to simple tables of variation and do
not conceive of the multiplicity of combinations which can be
drawn from them. Since they have no combinatorial system based
on the "structured whole/' they do not even begin to isolate the
relevant variables.
Substage III-A. Possible but not Spontaneous Sepa-
ration of Variables
At the lower formal level, substage III-A, the child is able to
separate out the factors when he is given combinations in which
one of the factors varies while the others remain constant. In this
case he reasons correctly and no longer according to the kinds of
inference of which we have just seen several examples. But he
himself does not yet know how to produce such combinations in
any systematic way i.e., formal operations are already present
in a crude form, making certain inferences possible, but they are
not yet sufficiently organized to function as an anticipatory
schema.
JOT (12; 7) believes that "you have to putt down [lengthen] the
string'' He suspends 20 grams and varies the length: "It goes more
slowly when you lower [lengthen] the string and faster when it's high
up." "That's all?" "Maybe the weight does something' 9 But to verify
this, he takes 100 grams and lengthens and shortens the string, then
50 grams, lengthening and shortening the string again: "Yes, it goes
faster high up [ = when the string is short]; it's the string." In other
words, he varies the string instead of the weight. Then he changes the
weight while again varying the string in the same way. This process
makes it possible to draw a conclusion, providing that the respective
frequencies are remembered from one situation to another, but it com-
plicates the matter uselessly. When the subject is asked to give proof
of the influence of length, he is satisfied with a pure deduction: "When
the string is long, it takes more time to go from one end to the other.
When it is short, it takes less time*'
ROS (12 ; 8) immediately discovers the role of length by lengthening
and shortening the string with the same weight. Then he reduces the
weight, but at the same time shortens the string [200 grams with the
long string and 20 grams with the short string]. His conclusion is that
74 THE DEVELOPMENT OF PHOPOSITIONAL LOGIC
"the weight has an effect too." He proceeds in the same manner to
control for the role of impetus and concludes that the "impetus plays
a part too! 9 But he is doubtful about the weight and half sees the need
to leave the other factor, that of length, invariant; he shortens the
string, attaching 50 grams and 100 grams successively. The result does
not change and his doubts grow: "I have to do it over again to be sure
it's right." Then he begins again, but once more he varies both weight
and length. This time he doubts the role of the length and takes 20
grams while lengthening and shortening the string. "When it's smaller
[ = shorter], the weight goes faster. It's because I didn't put on the
same weight; that's why [why nothing is proved]. Now I'll put on the
same weight." Nevertheless, he still believes that the weight has an
influence. Then we change the weights and lengths in front of him
simultaneously: "Does that prove anything?" "No, because you have
to put on the same weight." 'Why?" "Because the weight makes it
go faster" [!].
LOU (13 ; 4) also compares 20 grams on a short string to 50 grams on
a long string and concludes that "it goes faster with the little weight."
Next, rather curiously, he performs the same experiment but reverses
the weights [50 grams with a long string and 100 grams with a short
one]. However, this time he concludes that "when it's short it goes
faster" and "I found out that the big weight goes faster 9 '; however, he
does not conclude that the weight plays no role. "Does the weight
have something to do with it?" "Yes [he takes a long string with 100
grams and a short one with 20 grams]. Oh, I forgot to change the
string [he shortens it, but without holding the weight constant]. Aft,
no, it shouldn't be changed." "Why?" "Because I was looking at [the
effect of] the string."-'But what did you sQe?"-"When the string is
long, it goes more slowly." LOU has thus verified the role of the length
in spite of himself but has understood neither the need for holding
the nonanalyzed factors constant nor the necessity for varying those
which are analyzed.
These transitional cases are of an obvious interest. They demon-
strate, even better than the examples from substage II-B, the
difficulty which arises in distinguishing factors and in the method
"all other things being equal." In the first place, as among the
substage II-B subjects, we find the tendency deliberately to vary
two factors simultaneously, and even (as for LOU) the tendency
not to vary the particular factor under consideration. But almost
in spite of themselves and under the influence of nascent formal
THE OPERATIONS OF EXCLUSION 75
operations, these same subjects feel that in proceeding as they
do they are not proving anything, so they manage either actually
to transform the factor which they want to leave unchanged (as
LOU) or to vary all factors by turns without knowing how to focus
their analysis on the point being analyzed (as JOT). In such cases,
the conclusion is accurate insofar as it relates to the factor of
length, the only effective factor; but because the subjects lack
combinations which would make exclusion possible, it is not
accurate for weight or impetus, etc. In other words, the formal
logic in the process of formation is for these subjects superior to
their experimental capacity and has not yet adequately structured
their method of proof; consequently, they are able to manipulate
the easiest operations, those which state that which is and estab-
lish the true implications. But they fail in the case of the more
difficult ones, those which exclude that which is not and deny the
false implications.
Substage III-B. The Separation of Variables and
the Exclusion of Inoperant Links
For the pendulum problem, as for flexibility (Chap. 3), substage
III-B subjects are able to isolate all of the variables present by
the method of varying a single factor while holding "all other
things equal." But, since only one of the four factors actually plays
a causal role in this particular problem, the other three must be
excluded. This exclusion is a new phenomenon that contrasts
sharply with substage III-A, where such an operation was- still
impossible, and with the flexibility experiment where it was
unnecessary.
EME (15 ; i), after having selected 100 grams with a long string and a
medium length string, then 20 grams with a long and a short string,
and finally 200 grains with a long and a short, concludes: "It's the
length of the string that makes it go faster or slower; the weight doesn't
play any role" She discounts likewise the height of the drop and the
force of her push.
EGG (15 ; 9) at first believes that each of the four factors is influential.
She studies different weights with the same string length [medium]
and does not notice any appreciable change: "That doesn't change the
76 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
rhythm" Then she varies the length of the string with the same 200-
gram weight and finds that "when the string is small, the swing is
faster! 9 Finally, she varies the dropping point and the impetus [suc-
cessively] with the same medium length string and the same 200 gram
weight, concluding for each one of these two factors: "Nothing has
changed"
The simplicity of these answers is in contrast to the hesitation
found at substage III-A, but this must not mislead us. The answers
are the result of a complex elaboration whose operational mecha-
nism must now be isolated.
Let p be the statement that there is a modification in the length
of the string and p the absence of any such modification; q will be
the statement of a modification of weight and q the absence of
any such modification; likewise r and s state modifications in both
the height of the drop and the impetus and r and s the invariance
of these factors. Finally, x will be the proposition stating a modi-
fication of the result i.e., of the frequency of the oscillationsand
x will state the absence of any change in frequency.
When EME varies the length of the string with equal weights
(and successively for three different weights), she states the truth
of the following combinations:
(p.q.x) v (p.q.x) v (p.q.x) v (p.q.x) . (i)
This is to say that the modification of the length corresponds,
with or without modification of weight, to a modification of the
frequency and that the absence of the first transformation corre-
sponds, with or without modification of weight, to the absence of
the result x.
On the other hand, none of the four combinations (p.q.x) v
(p.q.x) v (p.q.x) v (p-q.x) is verified because when p is present
x is never present and reciprocally when x is present p is never
present.
But expression (i) can be broken down into two operations.
First, when the subject says: "It's the length of the string which
makes it go faster or slower," he expresses the reciprocal implica-
tion between p and x i.e., p x. Secondly, between q and x there
is no single linkage, since the four possible combinations
(q.x) v (q.x) v (q.x) v (q.x) all occur, (This can be written in the
form (q * x), in which case we say there is a tautology or "com-
THE OPEBATIONS OF EXCLUSION 77
plete affirmation.") This is what the subject expresses when he
says: "The weight has no effect." As for the relationship between
p and 9, it can be written p.(q v q) or, abbreviated, p [g]-t.0.>
there is affirmation of p with or without q\ likewise, we have
p.(q v q)i.e., negation of p with or without q. (The affirmation
and negation brought together are the same as p * q).
Thus expression (i) can be written:
c
* *) = P-(q v q) g x , or, abbreviated,
p[q] g*.
We see in these formulae that the exclusion of weight as a cause
of variation in the frequency of oscillations results simply from
the subject's realization of (p * ac) i.e., from the fact that all of the
combinations possible between q and x occur: to exclude weight
means to exclude the choice of any particular linkage between
q and x .
The reasoning process is the same for the exclusion of height of
the drop and impetus. However, since the subject takes both the
length and the weight into account when he analyzes the role of
the height of the drop (r and f), there are eight true combinations:
(p.q.r.x) v (p.q.f.x) v (p.q.r.x) v (p.q.r.x) v (p.q.r.x) v
(p.q.f.x) v (p.q.r.x) v (p.q.f.x) = (p c x).(q * x). (r * *) (3)
= p[qvr] g x,
where the expression p [q v r] stands for p.(q v r) v p.(q.r).
Furthermore, when he studies the role of the impetus ($ or s)
the subject also takes into account the length, the weight, and the
height of the drop. In this case, he finds sixteen true combinations:
(p.q.r.s.x) v (p.q.r.$.x) v (p.q.r.c.x) v (p.q.r.$.x)
v (p.q.r.s.x) v (p.q.r.s.x) v (p.q.r.s.x) v (p.q.r.s.x)
v (p.q.r.s.x) v (p.q.r.s.x) v (p.q.f.s.x) v (p.q.r.s.x) (4)
v (p.q.r.s.x) v (p.q.r.s.x) v (p.q.r.$.x) v (p.q.r.s.x)
(P* x).(q*x).(r*x).(s*x)=p[qvrvs] x.
Thus we see that the exclusion of the three inoperant factors
(which at first seemed so simple) as well as the reciprocal impli-
cations of the length and the result x actually presuppose a com-
plicated combinatorial operation which the subject cannot master
except by ordering seriately the factors which are to be varied
78 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
one-by-one, each time holding the others constant. For example,
in expression (4), the first two combinations (p.q.r.s.x) v (p.q.r.s.x)
are sufficient for the subject to deduce that frequency does not
imply the operation of impetus (s.x) and it is sufficient that he add
the last two combinations to conclude (s * x)i.e., to exclude
completely the role of this factor. But it goes without saying that,
in order to choose the conclusive combinations in this way, he
must have at least an approximate idea of all of the rest. This fact
explains why the isolation of variables by the method "all other
things being equal" and the exclusion of inoperant factors appear
at such a late date, being reserved for substage III-B.
The best proof that such a combinatorial system is needed is
that the substage III-B subject is not satisfied with drawing exact
conclusions from the demonstrative combinations that he con-
ceives of in the course of the experiment with such apparent sim-
plicity. He avoids as well all of the paralogisms that we have
noted at substages II-B and III-A. But, in comparing the correct
inferences found at substage III-B with the earlier false ones, we
see that the choice is again dictated by the presence of one or two
conclusive combinations. Once more they presuppose a degree of
mastery of the system of all possible combinations.
For example, in the case of the hypothesized influence of weight
(q), the subject may hesitate between operation (3), p [q] x and
the operations (p v q) or (p.q) x . . . assumed at substage
III-A and signifying that the change of frequency is due either to
the length or the weight or to both at once (p v q) or else that it is
always due to both at once (p.q)- In such cases, we would have:
[(p v qr) g *] = (p-q.x) v (p.q.x) v (p.q.x) v (p.q.x), and (5)
[(p.q) g x] = (p.q.x) v (p.q.x) v (p.q.x) v (p.q.x). (6)
Here we see that expression (5) does not differ from expressions
(i) and (2), themselves mutually equivalent, except for the pres-
ence of (p.q.x) and the absence of (p.q.x). And expression (6) does
not differ except for the presence of (p.q.x) and the absence of
(p.q.x). But the adolescent at substage III-B certainly knows how
to exclude (p.q.x) and (p.q.x), since he verifies accurately the false-
hood of p.x and p.x ( = changes of frequency without modification
of length or the reciprocal) even while admitting the truth of q.x
and of q.x ( = simultaneous variation of frequency and weight or
THE OPERATIONS OF EXCLUSION 79
invariance of both) when the length factor operates at the same
time.
It should be clear that the fact that a mode of reasoning which
was freely accepted up to substage III-B is then rejected again
presupposes a certain choice among the possible combinations
i.e., among those which are to be excluded as well as the true ones.
To refer to a concrete case, the reader will recall that ROS (in
III-A) varies weight and length simultaneously and concludes that
the first is operant: from the combinations (p.q.x) v (p.q.x) he
extracts q D* or x Oq. But the distinctive feature of EME'S experi-
ment (in III-B) is that she is not satisfied with these two combina-
tions and thus retains the truth of the four combinations contained
in expression (i), notably, (p.q.x), which excludes x D q (for q.x =
variation of the frequency without modification of weight) and
(p.q.x), which excludes q D x (for q.x = variation of weight with-
out result for the frequency). Of course a similar selection is
found in connection with the height of the drops and impetus.
Analyzing all the inferences accepted by a substage III-B subject
and all those which he rejects, one must assume that he has knowl-
edge of the combinations of expression (4), This knowledge itself
presupposes a knowledge of the sixteen other rejected combina-
tions i.e., a choice among thirty-two basic combinations. 8 Such
choices imply, after all, a selection among a set of basic combina-
tions. Once more we see that this selection implies the operation
of the formal combinatorial system based on the "structured
whole/* "whereas concrete operations amount simply to construct-
ing correspondences from which these basic combinations are
composed.
8 In the case of flexibility (five factors and the result) tibere are even more
i.e., sixty-four basic combinations. But to give proof of the influence of each
factor it is sufficient to retain them separately by couples of combinations,
whose model is furnished by operation (3) presented in Chap. 3, which can
be taken in turn.
Falling Bodies
on an Inclined Plane
and the Disjunction Operations 1
THE EXPERIMENTAL APPARATUS consists of a plane adjustable to
various angles of incline. A ball can be rolled down the plane; it
bounds when it hits a springboard at the base. The problem is to
find the relationship between the height of the point from which
the ball is released and the length of its bound. Naturally the
subject will not be able to calculate the parabolic form of the
curve the ball describes, but he will be able to discover that its
length varies only as a function of the height of the release-point
(learning to exclude the effects of the mass or weight of the ball).
In part, the solution of the problem depends on the way in which
the factors are presented.
Stage I. Global Intuition Without Operational
Registering of the Experimental Data
Even before 7 years, the correspondence between the angle of
incline and the length of the bound is perceived intuitively, but
the height at which the ball is released is not separated from the
angle of incline and weight is constantly assigned a role. However,
this latter role is not always consistently formulated,
i With the collaboration of H. Aebli and L. Miiller.
80
THE OPERATIONS OF DISJUNCTION
81
VER (5 years): "That one goes to 2 [the second compartment from the
lower extremity of the plane] because it is too small. If it were big like
that [gesture], it would go here" [8],
STU (6 ; 5) discovers that a marble reaches the fourth compartment
for a given slope, then the second "because the gadget was lowered."
'What are we going to do to make it go there?" [6]. "Lower it more
[failure], -No, you have to put it higher up. I want it to go here [8]:
I have to put it way up [approximate success]. Yes; to go near you
have to put it way down and to go far you have to put it higher up"
As for the mass, he believes that a small ball will not go as far.
PIT (6 ; 6): "Where will this ball go?"-"Way down to the bottom: if$
T^flttfer/''Watch [we take a small ball which goes to the same place],
"It's because it's high up"
MIC (6 ; 10): to make it go far, you have to "raise up the trough."
"And if you can't?" "Yow have to throw it hard [it reaches the third
compartment]. Ifs because it isn't high up, it doesn't go fast."
FIG. 4. The inclined plane can be raised or lowered by moving the
peg on which it rests to different holes in the board. These also serve
as an index for measuring height. Marbles of varying sizes are re-
leased at different heights on this plane, hit a springboard at the
bottom, bound in parabolic curves, and come to rest in one of the com-
partments (numbered 1 to 8). These are the subject's index to the length
of the bound.
82 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
VAL (7 ; i): "Because it rolls fast and it still has force." "Tlnis one?"
"It will have enough force to get there [4: failure]. But it had force
anyway; you have to go up a little more"
WAG (6 ; 7): "I'm going to put on that big one; I'm going to put it
further down, otherwise it will go too far because it has more weight;
when it is heavy, it goes too fast and it goes too far; it is heavy: that
makes force." 'And. that one?" "The very tiny little one won't go so
far because it won't have any force, it isn't heavy." Experiment: it
does go far. "Because it was far! It should have fallen faster than the
others because it's small. I am going to try a big one: maybe it will go
all the way to the bottom [experiment]. Yes, because it is big it went
far. I have to watch a middle-sized one [it falls at the same point as
the last two]. Yes, it's because it's heavy: it falls faster [new experi-
ment: idem]. It's because it's small, it's not heavy, so that's why! It
didn't go very far" [now he denies the fact]. Another ball: "Because
it is heavy, it falls faster because it has a lot of force. I am going to
put on the big one: it wants to go far because it [the slope] is very
steep."
In each one of these cases we find some intuitive understanding
obviously drawn from the child's experience (slides, sleds, small
vehicles, etc.): the steeper the slope, the further and more quickly
an object falls. But the height at which the ball is released is not
separated from the angle of incline, and the weight (judged pro-
portional to the size) is attributed a systematic role. But the spe-
cific role assigned to weight changes; in general a heavier marble
is thought to roll further but if necessary this can also be the case
for the smaller ones. In this respect inconsistent observations do
not yet correct the subject, and when he is in a difficult spot he
either contradicts himself or denies the facts (WAG uses the two
processes alternately). This is the case because neither serial
ordering nor correspondence operations, which can integrate
separate statements coherently, are as yet organized. Restricting
oneself to an untalkative subject like STU, one could gain the
impression that an exact correspondence is formed between the
angle of incline and the length of the bound. But when we con-
sider a subject who says all that he thinks, or even a little more,
we can see that this intuition does not go beyond the global level
because it appears in a general form without differentiating
operations.
THE OPERATIONS OF DISJUNCTION 83
Stage II. Attempts at Operational Correspondences
and Usual Exclusion of Weight
Beginning at substage II-A, correct formulations of correspond-
ences can be observed, but they are not yet systematic and of
course they lack the formal procedures essential to the separation
of variables. However, even at this point, depending on the way
in which the balls are presented, the subject often manages to
exclude the factor of weight insofar as it is incompatible with any
serial correspondence:
GUI (752). To make the ball roll further "You have to put it higher
up." "And to get [down] here?'' [extremity]. "Way up [experiment].
Ah! Yes. It's the last" [compartment] ."And for this one?" [first com-
partment]. "Yo have to lower it [notch] because it slides less."
"And here?" [toward the middle]. "Higher up, because it slides
faster," etc.
LAU (8 ; 2) indicates same correspondences for the angles of incline.
As for the sizes, LAU declares spontaneously: "The balls will go in the
holes [at greater and greater distances] in order of size" [expected
serial ordering]. "What do you mean by *in order of size'?" "The
smallest goes to the nearest and the biggest goes to the furthest hole;
those in the middle go to the middle" [he does the experiment] ."So?"
"They go all over the place. Size has nothing to do with it; they were
all about the samef 9 At the end: "According to where you put the
slide [inclined plane] they go in the holes. You put it way up to make
the marble go further: it depends on the height of the slide" "And
the size of the marbles?" "The size doesn't do anything"
sera (8 ; 8), likewise, <e You have to lower it, raise it" etc. At the end:
"It depends on the size?" "Oh! No. They go in any old box, and then
you raise it to make them go further" etc.
Here there is exact serial ordering of the slopes and lengths of
the bounds, with approximate correspondence between the two
("the more . . . the more") approximate because the subject
does not think of the elevation and does not even consider the
possibility of separating the distance covered on the downward
path from the slope of the plane. But, since the deviations are not
large, the correspondence works in a rough way.
84 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
But what is remarkable is the exclusion of weight, an exclusion
which, though not commonplace, is easy to obtain, as is shown by
the very clear cases of LAU and SCHI. But you will recall that in the
pendulum problem weight was excluded only at substage III-B;
the 12-14-year-old subjects (III-A) were not able to separate the
relevant variables. On the other hand, the hypothesis that weight
plays a role in the fall is very natural and is common even at the
adult level among those who have forgotten their physics courses.
Thus, the exclusion of this factor at substage II-A, when subjects
are unable to make use of any formal prepositional operation,
poses a problem for us.
It seems to us that the explanation lies in the fact that in this
particular case the factors of weight and slope dissociate them-
selves from each other without the subject's having to supply any
operational activity. Actually, when LAU wants to verify his expec-
tation that there is a correspondence between the size of the
marbles and the length of their bounds, the idea does not occur
to him to vary the slope at the same time because the slide is
immobile unless it is intentionally moved. But in the case of the
pendulum, where the problem is to estimate the frequency of
oscillations and where the subject must adjust the weights to the
strings, he will always be tempted to change the weight and
the string at the same time as a way of obtaining clearer results
("more different," as LAU said). In addition, he has to use a system-
atic method to separate out the variables. The factors of slope
and weight, however, are automatically dissociated. Conse-
quently, in this problem it is easy for the child to see that balls
of varied sizes may reach nearly the same place, in direct con-
tradiction to his expectations.
The second reason for the ease with which weight is excluded
has to do with the obvious lack of correspondence between weight
and the length of the bound. In the pendulum problem, on the
other hand, even after negative observations the subject could
still ask himself whether or not the weight plays some role. The
systematic experiments which result in a selection of crucial com-
binations among the total number of possibilities and which do
not appear before substage III-B are needed for the exclusion
of this factor.
In addition to serial ordering and more systematic correspond-
TOE OPERATIONS OF DISJUNCTION 85
ences, substage II-B is distinguished by the beginning of disso-
ciation of the height of the release-point from the slope. It is
interesting to note that it is at this same level in other experiments
as well (cf. the dumping apparatus in Chap. 13) that the height
factor is first differentiated and formulated in terms comparable
with the others. But this nascent differentiation does not go far
enough to allow the subject to exclude slope in favor of the height
alone; to do so would presuppose a systematic active verification
procedure designed to determine whether the two factors are
actually independent or not
JEA (8 ; 10) orders the angles of incline systematically: "Now 3 be-
cause I just tried 2.? etc., then says, "The more it goes down, the faster
it goes." Afterwards, he ascertains that with a gentler slope [4 instead
of 7]: "If you put it further [ = higher], it's as if you moved it a
notch" Thus, the attempt at systematic serial ordering forces him to
discover that the factor of height is distinct from the factor of angle of
incline.
(9 ; 9) : "It' 8 combined; if you raise it [he has successively raised
the slide to 3, 4, and 5], it makes a bigger jump here. Tm going to
watch the bound. [He takes a smaller ball and begins again: 3, 4, 5,
6, 7, and 8.] Ifs the same for the big one and the little ones; it's the
height that does it [determines the length of the bound]. The lightness
has nothing to do with it" But he does not dissociate height and slope
further.
BLI (10 ; 2) varies the slope: "If the slope is steeper, the batt goes jur-
ther"-"AiLd the sizes?"-"AZZ the balls will go in the same hole; that
can't change all of a sudden" He checks on a little one, then returns to
the slope, and, after an error in prediction, he says: *7 put it too far
backwards [ = too high], so I have to put it further down [experi-
ment]. Ifs too low down [new trial, still without varying the slope
again]; you have to put it higher up because it has less force when it
slopes less" [ = he compensates for the small angle of incline by releas-
ing the ball at a greater distance, thus at a greater height]. After sev-
eral new trials: *7 know now. It always goes behind the same door
[ = in the same compartment] for the same height" He tries to for-
mulate a correspondence between the slope and the length of the
downward path so that he can reach the same hole each time [3]:
25 cm. for the incline 10, 30 cm. for 8, 35 cm. for 6, and 40 cm. for 4.
86 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
The experiment confirms his expectations and he concludes: "The
more you raise it, the more holes there are" [ = the longer the
bound].
We see above that general correspondences of the type "the
more . . . the more" no longer suffice at this stage; rather, the
subjects become interested in organizing systematic correspond-
ences, for example in following the ascending order i, a, 3, ...
for the angle of incline so that they are able to note the corre-
sponding order of lengths for the bounds. (Moreover, neither of
these series is numbered on the apparatus; the angles of incline
are determined by a succession of holes in which the peg which
fixes the slide is inserted and the compartments are distinguished
by means of varied designs house, pine tree, etc.)
Although the correspondence is accurately formulated at this
stage, there are three reasons why it cannot be verified completely.
In the first place, as was our intention, the holes determining the
slope do not correspond exactly to the compartments. In the sec-
ond place, there are possible chance fluctuations (due to jigglings,
etc.). Thirdly, if he is not careful, the subject may vary the dis-
tances involuntarily; at the interior of the slide is a centimeter
scale of such a sort that for a given slope one can still put the ball
at either 25, 30, 35 cm., etc., thus varying the height of the release-
point independently of raising or lowering the slide. Hence, an-
other source of possible deviation from the initial correspondences.
Faced with these variations in the correspondence between the
slope and the length of the bounds, the subject tries to determine
which factors have influenced the result and in which ways. First,
weight occurs to nearly all the subjects with very few exceptions
(such as BLI). But this factor is discarded in the course of the ex-
periment for the same reason as at substage II-A: absence of any
observed correspondence (see MID).
The factor of height remains; at this substage the subjects gen-
erally discover its role as a result of the greater precision of their
attempts at correspondence. For example, when JEA encounters
irregularities in his correspondence, he notices the fact that 'If you
put it further away (thus higher), it's as if you moved it up a
notch" i.e., that for a slope of 4 you can give the ball a higher
starting point and obtain the same result as for a slope of 7 with
a lower starting point. As for BLI, he goes so far as to determine a
THE OPERATIONS OF DISJUNCTION $7
series of metrical equivalents according to the logical formula
higher X less slope = lower X greater slope, thus reaching the
same compartment every time.
However, these subjects are far from the discovery that height,
not slope and distance, is the only relevant factor, although height
can be calculated from slope and distance combined (according
to BLI'S formula). The problem of the exclusion of slope in favor
of height is quite different at substages II-B, III-A, or III-B
from that of the exclusion of weight or amplitude in favor of
length in the pendulum problem. For it is a question not of ex-
cluding one independent factor in favor of another, but rather of
excluding a particular relationship in favor of another of which
it is a part. Actually, at equal heights neither slope nor distance
plays a role if it is varied; there is not, on the one hand, a factor
slope and, on the other, a factor distance, or height; there is a
logical multiplication, "slope X distance height," in which only
the product (height) counts. The two multiplicands, in fact, never
operate as separate factors. But this fact does not yet occur to the
subject and it is understood only with difficulty at stage III. In
other words, the subject at substage II-B thinks of slope and dis-
tance as if two independent factors were involved, one of which
has a role that seemed obvious from the beginning, the other a
role which he has just discovered. Moreover, he conceives of them
as two factors that can compensate each other (of. BIJ). He has yet
to see that height alone counts, and that in order to find a corre-
spondence between the length of the bounds of the ball and the
determinant causal factor, the height, it is sufficient to consider
the latter without regard to slope or distance. It is true that the
child sometimes seems to have understood ('It's the height that
does it,'* MID), but we have here no more than inadequately dif-
ferentiated statements.
Stage III. Necessary Compensations Between Angle
of Incline and Distance (III-A) Followed by Discov-
ery of Height as the Sole Determining Factor (III-B)
Substage III-A (12-14 years) hardly differs from II-B for this
problem except in the method used. Subjects at substage II-B
begin by finding systematic correspondences between slope and
88 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
length of the bounds and discover the role of the distance only
secondarily. But the preadolescents of substage III-A produce
hypotheses more easily and from the start try to catalogue the
factors. They do this in such a way that they are able to separate
slope and distance as coexistent factors more quicldy. But they do
not discover (any more than the preceding subjects) the role of
the height as the single sufficient factor because they fail to pro-
ceed according to the habitual method used at substage III-B
isolation of factors by one-at-a-time variation, "all other things
being equal." The result is that, in reading the responses of sub-
stage III-A, one is especially struck by the widespread appearance
of the idea of compensation between slope and distance, an idea
which has already been found at substage II-B:
ROU (12 ; i): "The highest possible and it will get here" [the furthest
compartment]. But the slope continues to play a separate role: *7
thought that it would go with less force because it fell off steeply. 9 ' He
then discovers the compensation: "When it was higher [angle of in-
cline], you had to put it one lower down [distance], and when it's
lower [angle of incline], you have to put it one higher up" [distance],
and "if you go up [distance], you have to take down the slide 5 or 10
degrees, and when you raise it you have to start lower down 9 [dis-
tance]. He then takes a slope of 4 and [lowers] his starting point 5 cm.
at a time to aim for progressively nearer compartments: "When it
stays fixed [slope], you have to lower [the starting point] in steps of
5 cm." Conclusion: "Each time the angle gets smaller by 5 degrees
you have to go down 5 cm."
STRO (12 ; 6): "The more the slide is horizontal [ = less inclined], the
more you have to put the ball aside" [ = increase the distance]. Next
he makes some complicated calculations: "You can base it on the
points [slope] and the intervals [distance]; you multiply each hole."
-"How?" "A little more, a little less" [actually he does not get beyond
the qualitative concept of compensation].
HER (13 ; 6) first tests the role of weight and concludes: "That doesn't
have too much to do with it; it's as if they were the same" "Sure?"
-"Quite sure" Then, like the preceding subjects, he realizes that in-
creasing the distance is equivalent to raising the slide.
As ROU, in particular, shows us when he analyzes the role of
distance at equal angles of incline, as soon as these subjects pro-
THE OPERATIONS OF DISJUNCTION 89
ceed to a systematic study of the variables combined with the
concept of compensation which is general at this level, they are
led to the hypothesis that height is the single relevant factor. The
hypothesis is actually proposed and verified at substage III-B:
SAL (13 ; 3) begins with the hypothesis that mass is the determining
factor: "The little one will certainly go faster" But the facts do not
confirm his expectations. "Does size have an effect?" "No, I don't
think so. The large one would naturally go further, but since the little
one goes faster on the downgrade, they compensate each other.' 9 He
goes on to the variations in slope, then proposes "to take the same
slope with a higher starting point.' 9 Next he varies the two simul-
taneously and discovers the compensation: "Now I am going to vary
the height [ = slope] and the distance; they compensate each other!"
"And with extreme variations, would you get something?" "Yes
[trials]. That makes me think that it always has to take off from the
same heightfrom the same horizontal point" [ = thus height inde-
pendently of slope and distance!]. "Are you sure or is it a hypothesis?*'
"Whatever slope you take, a large or a small ball gets there [ = to
the same compartment] if it takes off from the same height" The ex-
periment that he devises as a control consists of taking the same height
for slopes of 3 and 9: "There you really have extremes!"
HOW (16 ; 4) begins by discarding the weight hypothesis: "I would
have expected the difference in weight to have changed the distance"
[ = the length of the bound]. Then he studies the role of slope, then
distance: **Yo have to make the ball start less high up," etc. Next he
ascertains the possible compensation: "If you raise [the slide] you
have to start from lower down." Finally lie is asked to formulate the
law: "It depends on where you start the ball. The line is constant, but
the angle moves" -"What line?" [He points out the guiding points
which make it possible to determine common heights for different
slopes.] "The balfs starting point is constant." *What do you mean?"
-Tfc* height."
As usual, substage III-B subjects differ from intermediate sub-
stage III-A subjects in that they try to separate out the variables.
In this task III-A subjects fail to dissociate them for two reasons
aside from the usual ones. First, at equal slopes distance and
height vary concurrently; thus they do not distinguish the two
factors from each other and generally call "higher* or 'lower"
what they actually measure in distance covered on the inclined
90 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
plane (cf. ROU and STRO). Thus, they believe that they have ac-
counted for height when in fact they have not formulated a clear
relationship. In the second place, in asserting that distance and
slope compensate for each other, they actually limit themselves to
a statement of covariance without looking for the invariant that
results from it, since they partly confuse this invariant, height,
with the distance itself. In contrast, substage III-B subjects try to
separate out the variables by the usual method of varying each
factor in turn while holding all of the other factors constant. In
this case, where there is mutual compensation of slope and dis-
tance, they vary each relationship separately before varying them
together (cf. SAL: "Take the same slope and start from higher up";
then, "Now I am going to vary height and distance"). Finally, this
allows them to distinguish clearly among all three factors slope,
distance, and height and not just between two of them as they
have done up to this point. In addition, as the first two factors
compensate each other, the subjects immediately look for the in-
variant that the compensatory mechanism presupposes; they are
no longer satisfied with simple covariance.
But how do they come to decide that the constant is height and
not either of the other two factors? Of course the discrimination
is a result of their experimentation, but, as SAL shows, a prelimi-
nary deduction is involved. The starting point of this deduction is
the compensation itself. The subject sees that if a given slope is
conserved, distance and height increase or decrease simultane-
ously; if, on the other hand, the height is conserved, the slope
increases while the height decreases or vice versa in such a way
that the height, product of the compensation, is at the same time
the invariant postulated to account for the occurrence of identical
results even when the other two factors are modified, "Whatever
the slope is," says SAL, you have to look for "the same height."
Even more forcefully, HOW states, "The line (height) is constant"
even though "the angle moves." These seem to be the reasons for
the discovery of the height factor; they also explain its late
appearance.
In analyzing the reasoning of these adolescents, as usual we
come first across a selection of the true combinations among the
possible ones. Furthermore, since the subject does not make a
trigonometric calculation but is restricted to observing the empiri-
THE OPERATIONS OF DISJUNCTION 91
cal covariations of the factors (both among themselves and with
the experimental result), the combinations found will bear on the
covariations as much as on the effect produced. (In fact, one may
consider this as the innovation of the present experiment com-
pared to those found in Chaps. 3 and 4.)
Let us call p the statement of the conservation of slope and
p the statement of a variation in this factor; call q and q the
same statements made in reference to distance; r and f the same
in reference to height; finally, we can designate by X Q and the
statements affirming or denying invariance in the result obtained
(length of the ball's bound).
In this case, the true combinations that the subject states (from
the standpoint of invariance or variation of each factor in relation
to the others) are the following:
(p .9 .r ) v (p .q f ) v (p Q .q Q .r Q } v (po.q Q .r Q } v (PO-^O^O). (1)
Thus the excluded combinations are: p q r (when slope but not
distance varies, the height must vary as well), p q r (reciprocally,
if distance varies without slope, height also varies), and p q Q r (for
if slope and distance do not change, height must also remain
constant).
But from combination (i) a twofold consequence results which
is correctly drawn by the subject when he is able to utilize the
disjunction operation:
ft D (p v q )] v [r D (p .g ) v (po.go)] ; (a)
i.e. ? a modification of height (of the release-point) presupposes a
modification of either slope or distance or both, whereas mainte-
nance of the same height presupposes either variation of both
slope and distance at the same time or conservation of both.
But it is clear that we also have:
[p D (q v F )] v [p D (qfo-fo) v (ft-fo)], and (3)
[q, D (po v Fo)] v [q D (p .r ) v (p .q<>)]> fea)
But the subject assumes that height alone (r .F ), not either of the
two other possible factors, actually plays the causal role. The rea-
son for this is that the three implications r D (po-qO v (p -9o);
p D (q.r Q ) v (^ .F ) and q D (p .r ) v (p .f ) contained in expres-
sions (2) and (3a) are no longer isomorphic if the direction of the
92 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
sign of the relevant changes is taken into account. Thus proposi-
tions p , 9< and f can be broken down into two pairs of propo-
sitions fiat we shall call p, q, and r when they state respectively
an increase in slope, in distance, and in height; p, q, and f when
they state respectively decreases in the same factors. In this case
we have:
p D[(q.r)v(q.f)v(q .r )]
9 D[(p.r)v(p.f)v(p r )] (4)
r<>3[(p.q)v(p.q)v(p Q .q )]',
i.e., the conservation of height r can be assured by the compensa-
tions (p.q) or (p.q) as well as by the absence of change (p .g ), as
is not true for either p or g -
At this point, the subject hypothesizes (see SAL):
r i x, or (f g So), (5)
which amounts to saying that either f Q .x or r Q .x must be true
(height and result either vary together or are both conserved).
The experiment then gives the true combinations:
(Po><7o**o) V (po>q<>.X ) V(p ,<7 .*o) V (po-9o.*o) V (po^o^o). (6)
The following combinations are excluded: p .q .x (since change
in slope without modification of distance transforms height and
does not lead to the same result ff ); p^q^x* (for reciprocally, p .q Q
implies a change in height) and p .</o.*o (for conservation of slope
and distance could not produce the change aJ ).
We see above how the true combinations (6) coincide with com-
binations (i); therefore the role of the height as the single neces-
sary and sufficient factor is verified. It is worth noting that the
above subjects are not satisfied with controlling the result of the
variations in height (p .q Q .x ) or (p ^ ^o) or (p .<7o.*o)> but also
demonstrate the validity of (p<>.q .x Q ) as counterproof , Subject SAL
even varies slopes from 3 to 9, concluding: "Here you really have
extremes!"
The Role
of Invisible Magnetization
and the Sixteen Binary
Propositional Operations 1
THE EXPERIMENTAL PROBLEMS set for the subjects in Chaps, i to
5 were designed to show a gradation in the sorts of difficulties
overcome by the combinatorial method inherent in formal think-
ing and adolescent prepositional logic. To conclude the first sec-
tion, we should like to examine briefly another rather simple
problem, one which has already been used in one of our previous
studies; 2 it will serve to show how the stage III subjects utilize
disjunctions and exclusions in integration with the entire set of
binary operations. The problem is to determine why a metal bar
attached to a nonmetallic rotating disk stops with the metal bar
pointing to one pair of boxes instead of any other boxes placed
around the disk; actually, the crucial pair contain several magnets
concealed in wax. (Everything is placed on a board which is
divided into sectors of different colors and equal surfaces.)
1 With the collaboration of M. Denis-Prinzhorn, former research assistant,
Laboratory of Psychology.
2 J. Piaget and B. Inhelder, La Gendse de ?id4e de hasard chez Tenfant,
Chap. m. (Not transl.)
93
THE DEVELOPMENT OF PROPOSITIONAL LOGIC
Stage I. Preoperational Disjunctions and Exclusions
We need not refer to the responses of the youngest subjects (sub-
stage I-A), for they have been described in our previous study.
FIG. 5. One pair of boxes (the starred ones) contains concealed mag-
nets, whereas the other pairs contain only wax. The large board (A)
Is divided into sectors of different colors and equal surfaces/ with
opposite sectors matching in color. A metal bar is attached to a non-
metallic rotating disk (8); the disk always stops with the bar pointing
to one pair of boxes. The boxes (which are matched pairs as to color
and design) can be moved to different sectors, but they are always
placed with one of a pair opposite the other. The boxes are unequal
in weight, providing another variable.
THE SIXTEEN BINARY PROPOSITIONAL OPERATIONS 95
But at substage I-B, a rough sketch of what at stage II we will
call concrete disjunctions and exclusions (for example, the disk
stops "here or there," "it's not that one/' etc.) already appears in
the form of intuitive representations:
voi (6 ; 5) thinks that the disk will stop on the blue "because the
blades [on the disk, which act as brakes] are blue," and, since it stops
on the green [the magnet boxes have for the moment been placed on
the green sectors of the board], he explains that "green goes well with
yellow" [the boxes are yellow]. Next he predicts "on the green, because
it always stops on the green" He opens the magnetized boxes: "There
is wax" but he does not find this fact helpful in explaining the phe-
nomenon and adds, "It doesn't come from that either" [the designs
decorating the boxes]. The magnets are put on the red sector. "Where
will it stop?" "I don't know; here or there" [red or green]. In the end
he is limited to the explanation "There is something in the boxes" but
without saying why some of them stop the disk while others fail to.
WEB (6 ; 9): "Maybe there, because there is a star" [decorating the
magnet box]. Then: "It will always stop on the red."-"Why?"-'I don't
know. It's too heavy here" [the blades serving as brakes]. "What
could be done to see if it's really that?"-"Tafc0 them off 9 [this is done].
"Where is it going to stop?' -"On the &Ze."-(Experiment: red.) 'It
was really that?" [the blades]. "No." 'Then how does it happen that
it stops here?" 'You push too hard [force of the disk]. I'm going to try
gently [it again stops on the red]. Ah! I see. I'd say it's too heavy [he
opens the boxes and compares them]. That one is the heaviest" [the
box with stars containing the magnets; it is not actually the heaviest].
"And like that?" [the experimenter puts the magnets on the blue and
the disk stops there]. "Ah! I know, I'm happy I found out. Maybe the
star is more useful, so it's heavier [ = strong], because at night it lights
the streets. Even between two houses you always see the star" "But is
this one a real star?" "No, it's made of paper, but maybe it sue-
The elementary form of the operation which later becomes
interpropositional disjunction is based on the observation that two
classes are partially or entirely disjunctive. The subject need not
possess concrete class operations before he realizes intuitively that
the needle can stop on any color: "here or there," as voi says. His
phrase expresses the beginnings of a development which leads
both to the inclusion of partial classes in a total class and to the
96 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
notion of the possible. The needle, he means to say, will stop on a
color (B) which will be green (A) or red (A')-i.e., an intuitive
addition A + A' = B with disjunction (either A or A'), although
systematic operations are not yet present.
As for exclusion, it appears in its most elementary form as lack
of correspondence. In this case as well, intuitive correspondences
and noncorrespondences (based on perceptual configurations) may
appear before operational correspondences (in which equivalences
are conserved when the configuration changes). 3 But when voi
rejects the explanation based on the content of the boxes (because
not only the boxes in front of which the disk stops but all of them
are filled with wax) or on the designs which distinguish the boxes
("it doesn't come from that either") he can do no more than note
the lack of correspondence; he cannot organize his observations
in a detailed way.
WEB'S behavior is more advanced when he proposes to take off
the disk brakes (blades) to see whether or not they have anything
to do with the disk's stopping; this proposal is a preliminary type
of verification. He discards other causes (force, etc.) in the same
way because their removal does not eliminate the effect. We see
here the beginnings of correspondence or the perception of non-
correspondence with consequent reversals of behavior which
forecast the transformation of this behavior into reversible opera-
tions. But the end of the interrogation shows that this nascent
structuring is not carried very far yet. In the first place, when
WEB tries to explain the stops as a result of weight, he does not
compare all of the weights and is satisfied with two or three com-
parisons whose results are erroneous. Later, and more important,
he goes so far as to attribute the stopping to the star design which
he sees as "useful*' ( = efficacious) and "heavy" ( = strong); he
does this because the star has "succeeded," although it appears
only symbolically as a paper representation.
Substage II-A. The Beginnings of Concrete Disjunc-
tions and Exclusions
When concrete operations are organized by reversible coordina-
tion of behavior, the rough forms of disjunction and exclusion
3 See The Chil&s Conception of Number, pp. yoff.
THE SIXTEEN BINARY PROPOSITTONAL OPERATIONS 97
which we have just noted begin to be systematically structured as
a function of the nascent groupings of classes and relations:
KEL (7 ; 3) first says that the needle may stop "here or there or there;
you cant tell in advance" Then he rapidly discovers that the needle
always stops on the same color [violet]. Next the experimenter puts
the box with the star design on the red sector; the needle stops there.
"It's because you changed that" [blades serving as brakes for the disk].
The experimenter repeats the trial. "No, it's the boxes! The stars were
on the violet before, now on the red," etc. But he does not allow him-
self to say any more.
MAMB (8 ; 3). First: "It depends on whether you turn it -faster or
slower." He holds to this idea for a long time: "Maybe you turned it
too hard" etc. Finally, since the needle always stops on the star: "It's
because the [starred] boxes are heavier" "And these?" [the heavier
boxes]. "Maybe they are too heavy, so it doesn't work'"
BER (959): "One of them is light, the other heavy, and one a little less
heavy" He realizes that the starred boxes are the same weight as those
marked with a circle, heavier than the squares, and less heavy than the
diamonds. **Yes, maybe ifs the weight" 4
(9 ; 10) has weighed all of the boxes: "The square is lighter, then
come the star and the circle" "So why does it stop in front of the
star?" "Because the square is next to the star and it is lighter."
KER (10 ; o) also hypothesizes that it "has to do with the weight in the
boxes and in the disk," attempting to reconcile the ambiguities by
using a notion of mean weight, defined in terms of the over-all dis-
tribution of the individual boxes. "Do you want to see if you are
right?" He compares equal boxes: stars and circles. "This one is pretty
heavy; it's the same weight as the star. 9 ' "But where does the needle
stop?' '-"On the star "-"Then it's the weight anyway?"-"!* must be
the weight because they are in order [ = the weight is distributed in
a certain manner which he describes in pointing out the boxes]. That
one [diamond] is heavier. The two round ones are the same weight;
the two square ones too [but lighter]. So Tm sure it must be the
weight." "Which are the heaviest?" He indicates the diamonds.
"Then it stops there?"-"No/-"And the weight still has an effect?"-
4 See La Gendse de ?id6e de hasard chez T enfant, p, 96 (Dan, Desp, and
Tos), p. 97 (Ful, intermediate), and pp. 100-101 for similar cases.
98 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
"Yes, of course' [he is thinking of an equilibrium between all of the
weights which results in the needle's stopping at the mean weights'].
In our work on the child's notion of chance we have discovered
a number of analogous cases in which subjects hypothesized that
only the mean weights stop the disk e.g., "It's only the middle
ones which stop it" (FUL 8 ; 4); 'It's the weight of these boxes,
because they are neither very heavy nor very light" (DUF 9 ; 4).
In these cases we see a curious mixture of concrete disjunctions
and exclusions (or nonexclusions); it serves to show both the
progress made over the preoperational level and the deficiencies
of nonformal operations in comparison with the true exclusions
of prepositional logic.
We have already observed the child's behavior for disjunction
in studying chance and lotteries. 6 At this level, when he draws
elements from a collection B, he discovers that he may sometimes
come across representatives of subclass A and at other times rep-
resentatives of subclass A', although at the preceding level he
generally believed that he would come across one rather than the
other. In the present problem we can see this elementary form of
operational disjunction, based on the structuring of classes and
relations, in part revealed by the way in which the notion of
weight is employed. The subject believes that the choice of stop-
ping point can be explained conclusively only in terms of weight;
on the other hand, he discovers the diversity of weights in the
experimental situation. Consequently, he assumes that the weight
may act in one of three ways; the effect results from the heaviest
or the lightest or from an intermediate value (not "too heavy," as
MAMB says or "neither too heavy nor too light'* as DUF says). We
see that this type of disjunction is based on a simple approxi-
mated serial ordering. As KER expresses it: 'It's in order" of three
categories, ranging from the heaviest to the lightest.
This solution of the problem, based on the disjunction of rela-
tions, is a subtle one; still, it raises a delicate point with regard to
exclusion. The fact that there is only correspondence between
some of the weights and the disk's stopping points rather than
between the weights as ordered serially and the degree of fre-
quency or exactness of its stopping does not induce the subject
*lbid.
THE SIXTEEN BINARY PROPOSITIONAL OPERATIONS 99
to exclude the weight factor at this stage. However, at substage
II-B and subsequently, weight is excluded for this reason. At sub-
stage II-A, on the other hand, absence of complete correspond-
ence simply limits the effect of weight in the subjects' eyes to the
effect of a hypothetical optimum weight which is assumed to be
in the middle range. But why is this weak hypothesis maintained
at substage II-A when it is rejected at II-B?
One very simple explanation can be offered. It is of particular
interest from the standpoint of the psychology of exclusion; more-
over, it relates to facts which we have studied extensively in an-
other work. 6 At substage II-A conservation of weight is not yet
organized and none of the concrete operations of serial ordering,
equalization with transitiveness, correspondence, etc., which have
already been acquired in a number of other areas, are as yet ap-
plied to it. 7 In contrast, at about 9-10 years (the beginning of
substage II-B), all of the relevant operations are organized simul-
taneously and conservation is assured. Since at substage II-A
weight is not always structured from the standpoint of concrete
operations, it may still be conceived of as an active force giving
rise to multiple and inconsistent effects. (We have observed ex-
actly the same phenomena and the same inconsistencies in Chap. 2
for the floating bodies problem.) In other words, the child cannot
make systematic exclusions because weight has not yet been given
a place in the system of operations essential to the formulation of
accurate correspondences.
In reading the responses of BER and especially of KER, we dis-
cover the surprising fact that they have recognized that the
weights of the starred boxes ( = those where the needle stops)
and the circles ( = where the needle does not stop) are equal,
even though this discovery does not shake their belief in the effect
of weight. However, if one remembers that they do not order the
weights exactly and that the equalities are not transitive, the fact
is less astonishing. A particular weight can be seen as possessing
a potentiality for attracting the needle which cannot be replaced
by any other equal weight.
* J. Piaget and B. Inhelder, Le D^veloppement des quaniiUs chez I'enfant,
Chaps. II, VI, X and XI.
i We know, for example, that serial ordering of five distinct weights with
equal volumes is accomplished only at the mental age of 10 as defined by the
Binet-Simon tests.
THE DEVELOPMENT OF PROPOSITIONAL LOGIC
VOG'S arguments are also interesting; after he has weighed all
of the boxes, he maintains that the starred one stops the disk
because of its medium weight but adds the provision that it is
placed beside the square which is lighter. His combination of two
weights has none of the distinctive characteristics of an opera-
tional composition; rather he describes an interplay of forces
which cannot be translated into terms of conservation. The same
holds for the notion of total action of all of the weights "in order"
to which KER refers; moreover, HER is unable to make his concep-
tion more explicit.
But even if the weight explanation does not result in an accu-
rate exclusion, the substage II-A subjects can still utilize concrete
exclusions for various other factors which they first suppose to be
causal the disk brakes, force of pushes, etc. In such cases the
incorrect hypotheses are more or less rapidly abandoned as soon
as the lack of correspondence is seen.
Substage II-B. The Concrete Exclusion of Weight
At substage II-B concrete operations for handling weight have
been structured (a delay of two to three years beyond the develop-
ment of such operations for lengths and simple quantities); the
result is that the subjects have quite a different attitude toward
the present problem:
DUP (10 ; 9) begins with the hypothesis that weight is a causal factor:
"It depends on how the weight is placed. This box [magnets] is the
heaviest [he weighs all of them]. Oh! It's the middle one! The heaviest
is the diamond; that one [square] is empty and these two here [the
starred box containing the magnets and the box marked with a circle]
are the same [he weighs them again]. Yes, about the same." Then he
spontaneously puts the circled boxes in the position of the starred
boxes in order to verify his hypothesis, but again the disk stops in front
of the star: "You can't do anything about it, it's always the same thing!
It's complicated." He finds no better explanation but does abandon
weight.
SAN (10 ; 8) "Does the weight do anything?" "Oh, no. That one is
heavier than the star [than the box containing the magnets], and it
goes on the star?'
THE SIXTEEN BINARY PROPOSITIONAL OPERATIONS 101
PAU (11 ; 11): "The round one and the star are the same weight, so it
may fall on either one of the two [he performs the experiment a sec-
ond time]. Yes, but there is something that does it because it always
falls here" [stars] . He weighs all of the other boxes and concludes by
reasoning [in a form already hypothetico-deductive]: "If it were the
weight, it would fall on the heaviest and not the medium ones."
The effects on these subjects of operations of serial ordering or
serial correspondence as well as those of equalization with transi-
tiveness, newly acquired for weight, are evident. From the fact
that there is no term-by-term correspondence between the weight
and the disk's stopping points, SAN and PAU conclude that the
weight plays no role. Similarly, from the fact that two equal
weights do not produce the same effect, DOT and PAU also con-
clude that this factor is ineffective. They both try one of two
counterproofs. They either replace the magnet box with one of
equal weight (though, of course, not containing a magnet), or they
repeat the experiment to be certain that the disk stops only in
front of one of the two. In sum, once operations are applied sys-
tematically to structuring a particular dimension such as that of
weight, concrete operations are adequate to assure the possibility
of excluding factors when there is neither correspondence be-
tween classes and relations nor transitiveness.
Stage III. Prepositional Disjunctions and Exclusions
Although the exclusion of weight is already possible at substage
II-B by the utilization of concrete operations, the formal opera-
tions of disjunction (pvq~p.qv p.q v p.q) and simple (p.q v
p.q) or reciprocal (p.q v p.q) exclusion present additional advan-
tages. First, they allow some variety in the selection of disjunc-
tions or exclusions; but more important, they locate these various
possibilities in the total set of combinations. The combinatorial
power of the structured whole then in itself determines implica-
tions and nonimplications or incompatibilities. To observe these
advantages, we can turn to a case which will be referred to again
in a later work by one of the authors from the standpoint of
inductive strategy:
THE DEVELOPMENT OF PROPOSITIONAL LOGIC
GOU (14 ; 11): "Maybe it goes down and here it's heavier [the weight
might lower the plane, thus resulting in the needle's coming to rest at
the lowest point] or maybe there's a magnet" [he puts a notebook
under the board to level it and sees that the result is the same].
"What have you proved?"-"There is a magnet [he weighs the boxes].
There are some that are heavier than others [more or less heavy], 1
think it's more likely to be the content" [in substance]. 'What do you
have to do to prove that it isn't the weight?" He removes the diamond
boxes which are the heaviest. "Then 1 changed positions. If it stops at
the same place again, the weight doesnt play any role. But I would
rather remove the star boxes. We'tt see whether it stops at the others
which are heavier [experiment]. It's not the weight. It's not a rigorous
proof, because it does not come to rest at the perpendicular [to the dia-
mond boxes]. The weight could only have an effect if it made [the
plane] tip. So I'll put two boxes, one on top of the other, and if it
doesnt stop that means that the weight doesn't matter: [negative ex-
periment]. You see ""And the color?" "No, you saw when the posi-
tions of the boxes were changed. The contents of the boxes have an
effect, but it's especially when the boxes are close together; the boxes
are only important when they are close [he puts half of the boxes at
a greater distance]. Ifs either the distance or the content. To see
whether it's the content Tm going to do this [he moves the starred
boxes away and brings the others closer]. It falls exactly between the
round ones which are near and the stars which are far off. Both things
have an effect and it's the result of two forces [experiment in which
the star is moved away by successive steps]. It's more likely to be dis-
tance [new trial]. It seems to be confirmed, but I'm not quite sure.
Unless it's the cardinal points [he takes off the stars]. No, it's not that.
The stars do have an effect. It must be the content. If it isn't a mag-
net, I don't see what it could be. You have to put iron on the other
boxes. If the magnet is there [disk], it will come [to] these boxes. If it
is in the boxes [stars] there is iron under the disk [he removes the
starred boxes]. I'm sure that it's the boxes."
We see here the great difference between substage II-B sub-
jects, who are limited to serial correspondences or transitive
equalities, and the stage III subject, who utilizes the formal
combinatorial system and as a result does not experiment until
he has made deductions from his preliminary hypotheses. Like
the II-B subjects, GOU hypothesizes the relevance of weight, but
he reasons from a set of possibilities as to ways in which it would
THE SIXTEEN BINARY PROPOSITIONAL OPERATIONS 103
be manifested if in fact weight had an effect (tilting the appara-
tus). This hypothetical reasoning not only gives him the idea of
verifying whether the plane is horizontal but even the idea of
placing two boxes together in order to increase the weight.
Moreover, GOU uses prepositional rather than concrete opera-
tions. Most important, they are based on the set of sixteen binary
combinations in continuous transition from one to the next; their
consistent integration is demonstrated with particular clarity.
The following operations can be distinguished in his protocol:
(1) Disjunction (p vq) = (p.q) v(p.q) v(p.q): 'It's either the
distance or the content (or both)";
(2) Its inverse, conjunctive negation (p.q): changing the posi-
tion of the boxes verifies the hypothesis that neither weight
nor color is the determining factor;
(3) Conjunction (p.q): both content and distance are effective;
(4) Its inverse, incompatibility (p.q) ~ (p.q) v (p.q) v (p.q): the
effect of the magnet is incompatible with moving the boxes
from the center for the needle may stop without the boxes
being moved and vice versa, or neither occurs.
(5) Implication (p^q) = (p.q) v(p.q) v(p.q): if a magnet is
attached to the disk, it will stop in front of the boxes contain-
ing iron;
(6) Its inverse (p.q): when it does not stop, nonimplication is
shown;
(7) Converse implication (qDp) = (p.q) v (p.q) v (p.q): if there
is a magnet in the box, it will stop the disk;
(8) Its inverse (p.q) operates in (i), (4), (10), etc.;
(9) Equivalence (p q) = (p.q] \v(p.q): to assert that weight
has an effect is equivalent to asserting that the needle stops
because of inclination of the plane;
(10) Its inverse, reciprocal exclusion (pvyq) = (p-q)v(p.q): the
fact that the plane is horizontal excludes the weight factor,
for either the plane is horizontal and weight has no effect or
weight has an effect and the plane is not horizontal;
(11) Independence of p in relation to q-i.e., p [q] = (p.q) v (p.q):
the stopping point may coincide either with a color or with
its absence; thus color is excluded as a variable;
(12) Its inverse (which is also its reciprocal) p [q] = (p.q) v (p.q):
failure to stop may also coincide with the color or its absence:
104 THE DEVELOPMENT OF PROPOSITIONAL LOGIC
( 1 3H 1 4) Independence of q and q in relation to pi.e., q [p]
and q [p]: tbese operations are found in (15);
(15) Complete affirmation or tautology (p * q ) = (p.q) v (p.q) v
(p.q) v (p.q): all possible combinations, thus absence of par-
ticular links, for example between the box which contains
the magnet and the colored sector on which it has been
placed;
(16) Its inverse, complete negation or contradiction ( ): to deny
that weight has an effect and to reassert it would be a con-
tradiction.
The above examples all come from the protocol of a single sub-
stage III-B subject; thus we are not exaggerating when we claim
that it is possible for subjects at this level to work in turn with
each of the sixteen binary combinations of prepositional logic. Of
course, at substage III-A the keyboard is not yet complete (for
examples of this intermediate substage, see our previous study on
the problem of magnets). 8 But when formal equilibrium has been
attained the combinatorial system which characterizes the "struc-
tured whole" pays off in full, and the subject is no longer satisfied
with reasoning based on simple concrete correspondences. For
example, when GOU has observed the noncorrespondence of the
stopping points with weights, he does not feel that his proof is
adequate ("rigorous") because he realizes that if weight acted to
produce an inclination of the plane, it could be combined with
other factors.
In sum, even in a problem as simple as the present one (chosen
to conclude the first section because of its very simplicity), the
transition from concrete to formal operations is distinguished by
the appearance of a complete combinatorial system whose vari-
ous types of disjunction and exclusion are continuously linked to
implications. Lacking in even the most advanced children at
substages II-A and II-B, this is what gives the hypothetico-
deductive new look to the responses of stage III subjects; it
manifests itself even in the small details of experimentation.
8 La Gen&se de Fidde de hasard, pp. 101-106.
Part H
THE OPERATIONAL SCHEMATA
OF FORMAL LOGIC
THE TRANSFORMATIONS of thought that characterize the first stages
of adolescence during stage III (notably at substage III-B) can
in no way be reduced to the formation of prepositional operations
of the sort we found in connection with the sixteen binary opera-
tions (after having analyzed more complicated examples involv-
ing ternary operations, etc.) On the contrary, the analyzable facts
of the growth of experimental reasoning are interesting because
they show us that a number of new operations and concepts
emerge in close linkage with the establishment of prepositional
logic; they require intellectual capacities greater than those of
the concrete level and derive from the operational transformations
entailed by the total structures ("groups" and lattices") inherent
in prepositional logic rather than the prepositional operations
themselves.
Thus far we have seen that prepositional logic is always bound
up with a combinatorial system based on the "structured whole"
as opposed to the simple class inclusions that make up the "group-
ings" of classes and relations of concrete logic. But this "struc-
tured whole" and the combinatorial system it presupposes form
more complex structures which, in contrast with these elementary
groupings, fuse the two great modes of reversibility into a single
whole i.e., inversion (or negation) characteristic of "groups'* and
reciprocity (or symmetry) characteristic of lattices." Thus the
operations or new notions -which we have just mentioned and
which we are going to study in this second section have this com-
mon characteristic of deriving from specific properties of these to-
tal structures as such i.e., their general transformations and no
105
J06 THE FORMAL OPERATIONAL SCHEMATA
longer only from the particular operations to which they give rise.
Thus we will designate by the term "operational schemata" those
operations and new notions which are relative to total transforma-
tions of a system as opposed to the particular operations analyzed
in the first section. In the first instance they will be the combina-
torial operations themselves, no longer conceived of in their purely
prepositional aspect but in their general form. Next, they will in-
clude the notions relative to inversion and reciprocity which
appear in all the problems that relate to the physical notion of
equilibrium or of action and reaction. In addition, they will con-
sist of certain notions of conservation, whose discovery requires
the use of formal thought. They also include the notions of pro-
portions, whose mathematical form derives from a more general
qualitative logical form. And last we have the notions of correla-
tion, in certain respects close to the notions of proportion.
In sum, we are dealing with a set of schemata whose dual
nature stems from the fact that, whereas their structuring pre-
supposes formal reasoning, they also derive from the most general
characteristics of the structures from which this same formal
thought arises.
7
Combinations
of Colored and Colorless
Chemical Bodies
WE HAVE constantly seen that the formation of prepositional
logic, which itself marks the appearance of formal thought, de-
pends on the establishment of a combinatorial system. The struc-
tured whole depends on this combinatorial system which is
manifested in the subjects' potential ability to link a set of base
associations or correspondences with each other in all possible
ways so as to draw from them the relationships of implication,
disjunction, exclusion, etc. Faced with an induction problem in
which subjects at concrete stage II would be limited to classifica-
tions, serial ordering, equalizations, and correspondences, the
sub stage III-B adolescents combine all of the known factors
among themselves in terms of all of the possible links. But the
problems given the subjects up to this point have involved factors
which can be disassociated and combined at will or simply made
to correspond without going beyond the level of observation or
of "raw" experiment. One may wonder what would happen if we
posed a problem that involved combinations directly i.e., one
that involved elements or factors whose combination is indis-
pensable if variable results are to be obtained. Will subjects at
1 With the collaboration of M. Noelting, research assistant, Laboratory of
Psychology, and doctor in chemistry.
107
108
THE FORMAL OPERATIONAL SCHEMATA
substage II-B or even II-A discover a combinatorial system to
meet the requirements of the experiment, one which would
demonstrate the independence of this combinatorial system in
relation to propositional logic? Must one await the formal stage
for the establishment of this experimental combinatorial system,
and will the stage II children accomplish nothing more than scat-
tered empirical combinations without a total system such as we
have seen elsewhere (in studying the formation of the mathe-
matical operations of combinations, permutations, and arrange-
ments)? 2
The best technique with regard to this matter is to ask subjects
to combine chemical substances among themselves. In experiment
1 + 3 2 1+3+g
FIG. 6. This diagram illustrates Experiment I in the problem of
colored and colorless chemicals. Four similar flasks contain colorless,
odorless liquids: (1) diluted sulphuric acid; (2) water; (3) oxygenated
water; (4) thiosulphate. The smaller flask, labeled g, contains potassium
iodide. Two glasses are presented to the subject; one contains 1+3,
the other contains 2. While the subject watches, the experimenter adds
several drops of g to each of these glasses. The liquid in the glass con-
taining 1+3 turns yellow. The subject is then asked to reproduce the
color, using all or any of the five flasks as he wishes.
2 La Gen&se de ?id4e de hasard chez Venfant, Chaps. VII to IX.
COLORED AND COLORLESS CHEMICAL BODIES 109
I, the child is given four similar flasks containing colorless, odor-
less liquids which are perceptually identical. We number them:
(i) diluted sulphuric acid; (2) water; (3) oxygenated water;
(4) thiosulphate; we add a bottle (with a dropper) which we will
call g; it contains potassium iodide. It is known that oxygenated
water oxidizes potassium iodide in an acid medium. Thus mixture
(i + 3 + g) will yield a yellow color. The water (2) is neutral, so
that adding it will not change the color, whereas the thiosulphate
(4) will bleach the mixture (i + 3 + g)* The experimenter pre-
sents to the subject two glasses, one containing i -f- 3, the other
containing 2. In front of the subject, he pours several drops of g
in each of the two glasses and notes the different reactions. Then
the subject is asked simply to reproduce the color yellow, using
flasks i, 2, 3, 4, and g as he wishes.
A second experiment (II) made use of combinations which were
not between substances alone but between some substances and
an indicator. Take Ac = a burette containing sulphuric acid N/4;
B = a burette containing caustic soda N/4; E = three glasses of
pure water, and Ind = a little phenolphthalein in three other
glasses of water. The combinations in this case are:
(Ind X B)
(Ind X Ac) = colorless
(E X B) = colorless
(E X Ac) = colorless
(Ind X B X Ac) = colorless
(Ind X B X E) = pink
(E X Ind) = colorless
(B X Ac) = colorless
In practice the ternary combinations are rare and serve only as
a counterproof for the older children; they are not needed to
produce color. As for B X Ac, this combination is excluded in
practice because two burettes are involved.
The result obtained by means of these two experiments demon-
strates that a systematic combinatorial system appears only at
substage III-A. At substage II-A the subject is limited to multi-
plying all of the factors i to 4 by g. At substage II-B a preliminary
attempt at combination by trial-and-error is observed, but it is
unsystematic.
120 THE FOKMAL OPERATIONAL SCHEMATA
Stage I. Empirical Associations and
Precausal Explanations
At the preoperational level subjects are limited to randomly asso-
ciating two elements at a time and noting the result in explaining
it by simple phenomenalism or by other forms of prelogical
causality:
NOD (5 ; 5): [Ind X B] e< SyrupI"-[Ind X Ac] "Wofer."--"Can you make
some more syrup?" "Yes. "You have to do this [he shakes the water,
then reproduces Ind X B]. It's syrup again."-"Caa you [change it
back] to water?"- Tea [Ind X Ac], Its water again:'^ t Vfhy?"- e The
syrup has gone away' [he points to a bottle of methylorange one meter
away].
MAM (5 ; 9): [Ind X B] "Ifs like wine. [Ind x Ac] It's like water."-
"Is there any color?" *7tf went down to the bottom, it went away like
that [gesture, then Ind X B]. Some red [Ind X Ac]. The red runs
away in the glass. The color disappeared at the bottom. 'You don't see
it any more. It melted."
EG (6 ; 6): [Ind X B] "It turned pink. Maybe there is paint in the glass.
[Ind X Ac] Maybe the piece melted. Maybe the paint flattened out
completely in the glass. [Ind X B] Rose. Maybe it's because the water
changes. Maybe it changes at the surface of the water [Ac X Ind X E].
Maybe it's because when you have taken some white water in the tube,
there is a bar that stops it and ifs the pink water that runs out."
AR (6 ; 9): [Ind X B] "This time you put some red water inside. [Ac X
Ind] It cant get red because the red has gone away in the water over
there [the first]. [B X Ind] It's formed, it's getting colored. It can come
back better in that water over there than in this. [E] It can't ever come
back there [E] because it's the same color but ifs not the same water."
Preoperational thinking of this type contains neither proof nor
even hypothesis. The apparent hypotheses of EG "maybe it's
because . . ."are nothing more than fictions, for they make no
reference to verification and she simply replaces or fills in the real
world with imagery. Since they are not placed in a precise con-
text of actions, these representations remain precausal; the color
is a sort of active element that emanates from the water (it's the
COLORED AND COLORLESS CHEMICAL BODIES Hi
water that "changes") but may "go away," "go down to the bot-
tom," "flatten out" to the point where it becomes invisible, or fly
away to a beaker more than one meter away. The color can also
"come back" but only to certain beakers of "water" and not to
others. Appropriately, the subject may even shake the uncoopera-
tive beakers (subject NOD, for example).
Substage II-A. Multiplication of Factors by "g
At the time concrete operations appear it is interesting to note the
extent to which subjects spontaneously and systematically asso-
ciate the element g with all of the others (in the case of experi-
ment I) but without any other combination. If the subject is
directly encouraged to combine several factors simultaneously, a
few tentative empirical procedures are elicited but they are not
followed up:
REN (7 ; i) tries 4 X g, then 2, X g, i X g, and 3 X g: "I think I did
everything. ... 7 tried them all! 9 "What else could you have done?"
*7 don't know." We give him the glasses again: he repeats i X g,
etc. **You took each bottle separately. What else could you have
done?" "Take two bottles at the same time" [he tries i X 4 X g, then
2 X 3 X g, thus failing to cross over between the two sets (of bottles),
for example i X 2, i X 3, a X 4, and 3 X 4]. When we suggest that
he add others, he puts i X g in the glass akeady containing 2 X 3
which results in the appearance of the color: "Try to make the color
again." "Do I put in two or three? [he tries with 2 X 4 X g, then
adds 3, then tries it with i X 4 X a X g]. No, I don't remember any
more/' etc.
GAY (7 ; 6) also limits himself to 4 X g, i X g, 3 X g, and z X g, and
discovers nothing else. "Could you try with two bottles together?"
[Silence. ]-"Try."-[4 X i X g] "It doesn't work."-"Try something
else."-[3 X i X g] "There it &r-"And that one [a], do you think
that it will be as yellow?"-[No trial ]-"What do you think makes the
color, the three together or only two?" "Here" [3]. "And that one?"
[i]. "There isn't any color. "-"And that one?" [g].-"Y0s, it's there
inside." "Then what good are i and 3?" "There isn't any color"
IM (7 ; 6) also begins with 4 X g, 2 X g, 3 X g, and i X g, but since
nothing happens he adds more drops to 3 X g, then to the entire series.
THE FORMAL OPEBATIONAL SCHEMATA
After this [which is new] he mixes the four together, but in the se-
quence 3X2X1X4, adding the drops each time: "It didn't come.
It's gone away again" [the color appeared after 3 X 2 X i X g, but
he did not stop at this point and the color disappeared with 4].
"What made it go away?" "Because I put in too much water"
[ = liquid from the four bottles. ) Til take away that bottle [4]. Begin
again/* Again he makes the whole mixture, without understanding the
suggestion about exclusion. "It didn't come back because I put in too
much
CUR (8 ; 11) also proceeds one by one with g: "Nothing happens. You
cant do it unless you put everything in the same glass." He mixes the
four without success, then hypothesizes not that he has put in too
much but that he should have chosen another order: "Nothing hap-
pens. I should have started with that one" [a]. He does this, but since
he does not control the permutation operations any better than the
combination operations, he follows the sequence 2, 3, 4, i, g, then he
adopts any sequence whatsoever: "The color doesn't come because I
did it in reverse." Finally [always with the intention of blending the
four] he follows the sequence i X 2 X 3 X g: "It's becoming yellow!"
But immediately he adds 4 and has to begin all over again. "Put in as
few as possible." "The fewest possible, that's two."
The reactions at this stage are of interest because, although
these subjects are in possession of logical multiplication opera-
tions of one-by-one correspondence, the idea of constructing
combinations two-by-two or three-by-three, etc., does not occur
to them.
From the standpoint of combination operations, the only spon-
taneous reactions of the subject are either to associate each one
of the bottles i to 4 in turn to the dropper g or to take all four at
the same time. In both cases combinations are involved, but only
the elementary and limited combinations that operate in multipli-
cative "groupings" of classes and relations (i.e., associations or
correspondences between one term and all the others).
Even when he sees he has failed, the subject does not use two-
by-two combinations without prompting by the experimenter. On
the other hand, his two hypotheses are either purely quantitative
("too much water" or not enough, the result being a new distribu-
tion of drops) or have to do with serial ordering (CUE). But this
appeal to order is also prompted by a grouping structure, since
COLORED AND COLORLESS CHEMICAL BODIES
the serial ordering which is acquired from 7 years on rests on
sequences. But here again the subject's reaction is to introduce a
single change in order or to invert this sequence; he fails to try
all of the possible sequences that combinatorial permutation
operations allow. In sum, no true combinatorial operation has
appeared as yet, but only correspondences and serial ordering
i.e., first-degree combinations based on fixed class inclusions.
Another interesting point is that, with the color already formed,
the subject is well aware that liquid 4 is "a kind of water which
takes away the color.* 9 But when he is in the process of bringing
together the four elements iX^X3X4Xg and the color,
which has appeared after he has combined the first three ele-
ments with g, disappears under the influence of 4, he no longer
has the idea of a possible exclusion between 4 and the color; he
simply declares that the color has disappeared for various reasons.
Because it helps to point up the opposition between noncom-
binatorial and combinatorial structures, we should note that at
this level the child does not think of attributing color to the com-
bination of several elements as such. Rather, he thinks in terms
of such and such an element taken by itself, whether or not it
combines with others. For example, GAY thinks that the color is
in 3; then he withdraws it from 3 in order to assign it to g, as if it
could be linked to only one liquid at a time.
The experiments in which indicators are used differ from those
involving solutions in that in the former the two-by-two combina-
tions are sufficient for a complete classification of the color-pro-
ducing cases (base B X Ind) and of the three nonproductive
cases. Can we then say that the concrete operation of logical
multiplication can alone furnish the solution of the problem?
Interestingly enough, it is adequate for the extraction of the law
(favorable and unfavorable cases) but not sufficient for its expla-
nation. This is so for exactly what we have just seen in reference
to the color attributed to a single liquid.
At substage II-A the elements of this second apparatus give rise
to the following associations. In certain cases, the glasses (E and
Ind conceived of as identical) are associated successively with the
base B and the acid Ac. But from 7 years on complete tables of
four cells, (B X Ind) + (B X JE) + (Ac X Ind} + (Ac X E), are
obtained just as often, thus making possible discovery of the law:
H4 THE FORMAL OPERATIONAL SCHEMATA
SEHNE (7 ; 5) associates [Ind X B] pink, [Ind X Ac] colorless, [E X B]
colorless. "There isn't any more pink"; again [E X B]: "The pink
doesn't come any more'' [Ind X B] "Pink again" "Can you take
away the red?" "You put in a little white" [he associates Ind x Ac and
E X Ac]. "They're all the same, these four glasses?" [E and Znd]-
"No. . . . ?es [he tries Ind X B and E X B]. N0/-"Why?"-<7*
comes only in two glasses. These two are pink and these two white"
[correct]. "And the burettes?" "No, here pink [B] and here white"
[Ac]. "Can you take the pink away from this glass?" [Ind X B].
"Yes" [he pours in some acid].
However, these multiplicative operations differ from complete
combinatorial reactions in two ways. First, it does not occur to
the subject to combine the two glasses between themselves
(E X Ind) even after having established their differences, nor to
combine the two burettes: thus he is restricted to the four base
combinations (glass E X burette B) -+- (glass Ind X burette B) +
(glass E X burette Ac) + (glass Ind X burette Ac). Secondly,
even in discovering the law (Ind X B = pink), the subject does
not conclude that the color is due to the combination; rather, he
thinks that the base and the indicator contain it, thus making use
of the notion of the potential in the sense of a "disposition to pro-
duce pink" and not yet in the sense of a possible combination, with
a resultant which is distinct from the effects linked to each of the
elements of the combination.
Substage II-B. Multiplicative Operations with the
Empirical Introduction of n-fct/-n Combinations
The substage II-B reactions are analogous to the preceding ones
but with a visible progress, namely, the appearance of n-by-n
combinations. However, the subject does not as yet discover any
system; only tentative empirical efforts are involved:
KIS (9 ; 6) begins with [3 X g] + [i X g] + [2 X g] + [4 X gj, after
which he spontaneously mixes the contents of the four glasses in an-
other glass; but there are no further results. "O.K., we start over
again." This time he mixes 4 X g first, then i X g: "No result." Then
he adds 2 X g, looks and finally puts in 3 X g. "Another try [i X g,
then 2 X g, then 3 X g]. Ah! [yellow appeared, but he adds 4 X g].
COLORED AND COLORLESS CHEMICAL BODIES 115
Oh! So that! So that's [4] what takes away the color. 3 gives the best
color ."-"Can you make the color with fewer bottles?"-"No." "Try"
[he undertakes several 2, by 2 combinations, but at random].
ALB (10 ; 4) begins with iX2X3X4Xg, then changes the order:
3XiX4X2Xg. "It's different, because the first time I went in
order and this time I didn't. [He puts 2X4XiX3Xg.] Gives
nothing' 9 [he tries several more permutations at random, then aban-
dons the effort]. "Do you have to take all of them?" "No, you can
take 2 or 3 if you want [he tries unsystematicaliy and succeeds by
chance]. It changes!"
TUR (11 ; 6) begins with i X g, etc. "That doesn't work. You have to
mix all four [he does this]. That doesn't work either [he changes the
order several times without success, then tries two-by-two combina-
tions: iX4Xg, 2X3Xg, 3X4Xg, then 2 X i X g]. I wonder
if there isn't water in all of them!" Then he spontaneously moves on to
three-by-three combinations [ X g],but without order: 3 X 4 X i X g,
then 2 X 3 X 4 X g, then i X 4 X 2 X g, then 3 X 1 X 2 X g.
"That's #."-"What do you have to do for the color?"-"Pw* in 2."-
"All three are necessary?" On# at a time [always with g] it doesn't
work. It seems to me that with two it doesn't work; a liquid is missing."
"Are you sure that you have tried everything with two?" "Not sure
[he tries in addition 2 X i X g, already attempted, then 3 X i X g].
It works! It's i and 3!" "Tell me what effect the bottles have."
"i is a colorant, 2 prevents the color; no it doesn't prevent it because
it worked. 3 takes away the effect of 2, and 4 doesn't do anything."
We see that, as at substage II-A, these subjects begin by mul-
tiplying each element by g or by taking them all at once, but
finally they spontaneously use two-by-two or three-by-three com-
binations (each tune with g). This is the true innovation of this
substage, since at substage II-A this type of combination had to
be called forth by the experimenter. But the fact that these com-
binations are not systematic defines the upper limit of this sub-
stage: TUR, who is the most advanced of the cases cited, does not
even attain the six possible two-by-two (X g) combinations.
As for the cause of the color, it is still sought in particular
elements rather than in their combination; TUR locates the color
in i only and misinterprets the roles of 2, 3, and 4. Others discover
the negative effect of 4 but by direct (and fortuitous) formulation
and without having a specific method of proof.
126 THE FORMAL OPERATIONAL SCHEMATA
In connection with the indicators (experiment II), the only
notable innovation is the appearance of combinations between
two glasses (E X Ind); this shows that there is no longer only a
double-entry multiplicative table but a search for all of the pos-
sible combinations. But the explanation remains the same:
M:ER (9 ; 3) tries E X B; Ac X Ind; Ind X B; E X Ac: "'There [B],
it makes it get red and in that glass [Ac] it stays white." "And in the
glasses, is it the same?" "The water isn't the same" "Which one
gives the color?"-'7n both of them, in the glass [Ind] and in there"
[B].-"Are they the same?" [B and Ind].-'Yes."-"Can you show me?"
[He combines Ind X B (red), then Ind X E (colorless)] "Oh! No,
it's not the same and that [B] isn't the same as that" [Ac]. "In the
four, is it different?"-"Y0s.--"Tell me what there is in that one" [B].
"It makes it get red."- 'And there?" [Ac]. "That bleaches the
water. "-"And that?" [E]. -"There isn't any pellet" [with red dye].-
"And there?" [Ind]. -"There is a pellet."
Notice here the new combination X Ind, devised to see
whether B and Ind are similar (a proof which, moreover, is not
complete). But the explanation remains the same as at substage
II-A: the color is thought to be virtually contained in B and in
Ind (potential) or that there is a "color pellet" hidden (invisible
content).
Substage III-A. Formation of Systematic
n-by-n Combinations
The two innovations which appear at the formal level are the
systematic method in the use of n-by-n combinations, and an
understanding of the fact that the color is due to the combination
as such:
SAR (12 ; 3): "Make me some more yellow." "Do you take the liquid
-from the yellow glass with all four?* "I won't tell you/' [He tries
first with 4 X % X g, then 2 X g X 4 X g] "Not yet. [He tries to smell
the odor of the liquids, then tries 4 X i X g] No yellow yet. Quite a
big mystery! [He tries the four, then each one independently with g;
then he spontaneously proceeds to various two-by-two combinations
but has the feeling that he forgot some of them.] Td better write it
down to remind myself: i X 4 is done; 4 X 3 is done; and 2X3.
COLORED AND COLOBLESS CHEMICAL BODIES 117
Several more that I haven't done [he finds all six, then adds the drops
and finds the yellow for i X 3 X g]. Ah! it's turning yellow. You need
i, 3, and the drops ."-'Where is the yellow?"- . . . -"In there?"
[g]-"No, they go together "-'And 2?"-' 7 don't think it has any effect,
ifs water" "And 4?" "It doesn't do anything either, it's water too.
But I want to try again; you can't ever be too sure [he tries 2 X 4 X g].
Give me a glass of water [he takes it from the faucet and mixes
3 X i X water X gi.e. 9 the combination which gave him the color,
plus water from the faucet, knowing that iX2X3X4Xg pro-
duce nothing]. No, it isn't water. Maybe it's a substance that keeps it
from coloring [he puts together i X 3 X a X g, then i X 3 X 4 X
g] Ahl There it is! That one [4] keeps it from coloring. 99 "And
that?" [2] .-'It's water."
CHA (13 ; o): "You have to try with all the bottles. I'll begin with the
one at the end [from i to 4 with g]. It doesn't work any more. Maybe
you have to mix them [he tries i X 2 X g, then i X 3 X g]. 1* turned
yellow. But are there other solutions? Til try [i X 4 X g; 2 X 3 X g;
sX4Xg; 3X4Xg; with the two preceding combinations this
gives the six two-by-two combinations systematically]. It doesn't work.
It only works with" [i X 3 X g]. "Yes, and what about 2, and 4?"
"2 and 4 don't make any color together. They are negative. Perhaps
you could add 4 in i x 3 X g t o see if it would cancel out the color
[he does this]. Liquid 4 cancels it all. You'd have to see if 2 has the
same influence [he tries it]. No, so 2, and 4 are not alike, for 4 acts on
1X3 and 2 does not." "What is there in 2 and 4?" *7n 4 certainly
water. No, the opposite, in 2, certainly water since it doesn't act on
the liquids; that makes things clearer." 'And if I were to tell you
that 4 is water?" "I/ this liquid 4 is water, when you put it with
i X 3 it wouldn't completely prevent the yellow from forming. It isn't
water; it's something harmful."
We see the complete difference in attitude between these sub-
jects and those at substage II-B, in spite of the fact that the latter
attempt some n-by-n combinations. The new attitude found at
substage III-A can be noticed both in the combinatorial methods
adopted and in the reasoning itself.
From the point of view of method, two achievements are worthy
of note. The first is the establishment of a systematic n-by-n com-
binatorial system complete for the numbers involved in this ex-
periment. For example SAR, who is afraid of forgetting certain
associations, makes out a written list, and CHA works out the six
118 THE FORMAL OPERATIONAL SCHEMATA
two-by-two combinations without hesitation. We again encounter
(though in a form which is all the more significant since it is
more spontaneous) what we have seen in another work in study-
ing the operations of combination with instructions which them-
selves suggest the operation. 3 The second achievement is just as
important from the point of view of the utilization of these com-
binations (for it is obviously the needs linked to this use or, in
other words, functional considerations which determine the com-
pletion of the corresponding structure): once the combination
i X 3 X g which brings about the color is found, the subject, not
satisfied with a single solution to the problem, does not stop there
but looks for others. Thus his main interest is not success by the
intermediary of a particular combination but an understanding of
the role which this combination plays among the total number of
possible combinations.
This leads us to the advances made in reasoning. The way sub-
jects use combinatorial operations demonstrates that they are not
concerned with particular mathematical operations at this point
(moreover, the required operations have not yet been taken up
in class by these subjects); but certainly we are dealing with a
general logical structure, analogous to that of the multiplicative
groupings utilized at substage II-A and tending to round out the
structure after substage II-B.
At the same time as they combine the factors involved in the
experiment among themselves (the liquids presented in the four
flasks), stage III subjects form their judgments according to a
combinatorial system having the same form, that of the sixteen
binary propositions (combinations one-by-one, two-by-two, three-
by-three, four, or zero of the four base possibilities p.qvp.qv
p.q v p.q). In other words, when these subjects combine factors
in the experiment, by the same token they generate a combina-
torial system which corresponds to the observed facts. This is
how they determine the links of conjunction, implication, exclu-
sion, etc., by means of which they interpret the experimentally
established combinations. Moreover, this fact explains the prog-
resscorrelated with that of the combinatorial operations them-
selveswhich is noted in their deductive reasoning and in the
formulation of verbal statements.
*Ibid., Chap.VH.
COLORED AND COLORLESS CHEMICAL BODIES 119
This reasoning bears on elements 2 and 4 in particular. Element
2 is judged neutral because it is sometimes present, sometimes
absent, in a colored combination as well as in others. If p desig-
nates the presence of color and q the presence of element 2, then
the subject sees that one can have:
(p.q) v (p.q) v (p.q) v (p.q) = (p * q) , (i)
thus excluding the possibility of any positive or negative effect
for 2: "It's water/' conclude SAR and CHA. On the other hand, be-
tween liquid 4 and the color there is reciprocal exclusion or
incompatibility, as CHA says clearly:
(p.q) v (p.q) = (p w q\ or (2)
(p.q)v(p.q)v(p.q) = p/q (3)
(where q now designates liquid 4).
But, from the fact that he has formulated the association p.q
(in combinations i X 4; 3 X 4; etc.), at first SAR believes that 4 is
neutral, just as is 2, so he replaces 2 with 4 in a combination
(iXsX^Xg) and perceives that i X 3 X 4 X g fades, whence
the associations (p.q) v (p.q) which characterize reciprocal exclu-
sion.
Secondly, this formal mode of reasoning i.e., founded on the
combinations of factors and consequently on combinations of the
statements themselves naturally leads the subject to a new con-
ception of the cause of the color. This cause is no longer sought in
one or another of the elements but in their being brought together
or, more precisely, in the very fact of their combination. For
example, SAR refuses to locate the color in g because "they go
together* ( = it's the whole [mixture] i X 3 X g as such which
is the cause); CHA refers to elements which make "or don't make
any color together"; and another subject, SEE (12 ; 6), declares:
"This one (3), joined to i and to g, gives the color: 3 all alone does
nothing and x alone does nothing either." From this, if p, q and
r = the statements concerning the effects of i, 3, and g and if
x = the statement that the color appears:
acD(p.qr.r) and no longer xDr. (4)
As for the reactions of the subjects at this level to experiment II
(in which indicators are used), they add nothing new to the pre-
120 THE FORMAL OPERATIONAL SCHEMATA
ceding. Nevertheless, it is interesting to note that even after hav-
ing carried out no more than the four base combinations cor-
responding to a double-entry multiplicative table, the substage
III-B subject already concludes that the color is a result of the
combination as such, according to the schema which we have just
described:
VIR (13 ; 4) associates Ind X B, E X Ac, Ind X Ac, and E X B:
"What do you think about it?" "Simply that there is chemical water
in two glasses and ordinary water in the other two . . . with one col-
umn [burette] it turned red and with the other nothing happened."
~"So where does the color come from?" "It's only the contact of the
two waters . . . when they touch each other the color appears." Then
he passes on to combinations Ind X E and even Ac X B and to three-
fold combinations to study successive reactions.
It is evident that even before passing on to the combinations
beyond his initial double-entry table schema, vm already had a
combinatorial interpretation of the color.
Substage III-B. Equilibration of the System
In experiment I the difference between substages III-A and III-B
is only one of degree, actually it is not at all necessary in this case
to apply the method "all other things being equal/' since the fac-
tors are already presented in a dissociated state. Thus, the only
innovations of substage III-B are that the combinations, and more
particularly the proofs, appear in a more systematic fashion i.e.,
this level appears as a point of equilibrium in relation to the pre-
ceding level which is a phase of organization:
ENG (14 ; 6) begins with 2 X g; l X g; 3 X g; and 4 X g: "No, it
doesn't turn yellow. So you have to mix them." He goes on to the six
two-by-two combinations and at last hits i X 3 X g: "This time I
think it works. 99 "Why?" "It's 3. and 3 and some water" *You think
it's water?" 'Yes, no difference in odor. I think that it's water."
"Can you show me?" He replaces g with some water: i X 3 X water.
"No, it's not water. It's a chemical product: it combines with i and 3
and then it turns into a yellow liquid [he goes on to three-by-three
combinations beginning with the replacement of g by 2 and by 4 i.e.,
1X3X2 and 1X3X4]. No, these two products arent the same
COLORED AND COLORLESS CHEMICAL BODIES 121
as the drops: they cant produce color with i and 3 [then he tries
i X 3 X g X 2]. It stays the same with 2. I can try right away with 4
[i X 3 X g X 4]. It turns white again: 4 is the opposite of g because
4 makes the color go away while g makes it appear" "Do you think
that there is water in [any of the] bottles?" '7'ZZ try [he systematically
replaces i and 3 by water, trying i X g X water and 3 X g X water,
having already tried i X 3 X water]. No, that means 3 isn't water
and i isn't water." He notices that the glass i X 3 X g X 2 has stayed
clearer than i X 3 X g. *7 think 2, must be water. Perhaps 4 also? [He
tries i X 3 X g X 4 again] So it's not water: 1 had forgotten that it
turned white; 4 is a product that makes the white return"
Thus the results are the same as in III-A (save that the neutral
character of 2, had not been established systematically at the
earlier level). But they are discovered by a more direct method
because, from the start, the experiment is organized with an eye to
proof. This method may be described as a generalization of sub-
stitution and addition. For example, having established the fact
that the color is due to i X 3 X g> the subject replaces g by 2
then by 4 to see if they play equivalent roles; then he immediately
goes back to i X 3 X g and adds 2 and 4 alternately to the mix-
ture in order to determine the effects of these additions. But it
should be understood clearly that substitution as well as addition
is already operating in the stage III-A combinatorial system.
When the subject constructs the combinations i X 2, i X 3, and
iX4 ? the very construction of these associations implies the sub-
stitution of 3 and then of 4 for 2; and when he makes the transi-
tion from two-by-two to three-by-three combinations, he adds the
alternative elements 3 and 4 to a given couple (for example i X 2)
i.e., 1X2X3 and 1X2X4 Moreover, as we have seen, sub-
stage III-A subjects already use these substitutions and additions
to prove certain effects. Thus, the only innovation appearing at
substage III-B is the greater speed with which the subject under-
stands the use he may make of these substitutions and additions
in the determination of the respective effects of the elements dur-
ing the actual construction of these combinations. Thus, first and
foremost, progress is to be sought in the organization of the proof
and in the integration of methods of discovery and methods of
proof. From the start, the combinatorial system becomes an instru-
ment of conclusive deduction.
122 THE FORMAL OPERATIONAL SCHEMATA
On a more general level, the lesson to be drawn from this ex-
periment is that it points up the close correlation that exists be-
tween the mode of organization or the over-all structure of the
combinatorial operations on the one hand and those of the formal
or interpropositional operations on the other. At the same time that
the subject combines the elements or factors given in the experi-
mental context, he also combines the prepositional statements
which express the results of these combinations of facts and in this
way mentally organizes the system of binary operations consisting
hi conjunctions, disjunctions, exclusions, etc. But this coincidence
is not so surprising when we realize that the two phenomena are
essentially identical. In other words, the system of prepositional
operations is in fact a combinatorial system, just as from the sub-
ject's point of view the only purpose of the combinatorial opera-
tions applied to the experimental data is to make it possible for
him to establish such logical connections. Nevertheless, we had
to show empirically that such an intimate relationship between
the combinatorial operations and the prepositional operations
does exist, and in order to do this we have had to examine the
reactions of the child and the adolescent to an experimental situa-
tion that did not impose either kind of operation by any sort of
instructions but in which they would have to be discovered and
organized in a completely natural and spontaneous way.
8
The Conservation of Motion
in a Horizontal Plane 1
THE FIRST formal operational schema we described had to be the
schema of combination operations, since the lattice structure
which characterizes the system of prepositional operations implies
a combinatorial system. On the other hand the second opera-
tional schema, which we are now going to study, derives from the
group structure and the reversibility by inversion which is its
distinctive feature. As we will elaborate at greater length in the
following discussion, the system of formal operations constitutes
both a lattice and a group and thus unites transformations by
reciprocity and transformations by inversion into a single cluster.
The experimental problem involves a ball 2 launched by a
spring device and rolling on a horizontal plane. If no external
obstacle interferes, it will maintain a uniform rectilinear motion
(principle of inertia). Actually, a number of factors prevent the
free operation of inertia friction, which slows the ball down as
a function of weight, air resistance, which slows it down as a func-
tion of volume, the irregularities of the plane, etc. As a result, two
interesting problems arise which must be resolved by formal
thought: (i) the problem of what is ideally or theoretically pos-
i' With the collaboration of A. M. Weil and J. Bal, fonner student, Institut
des Sciences de Education.
2 The material consists of a set of balls of various weights and volumes.
123
124
THE FORMAL OPERATIONAL SCHEMATA
sible i.e., not realizable in fact. In other words, how does the
subject come to understand the conservation of motion by inertia
given that it is never observable? From the physical and mathe-
matical viewpoint, conservation of motion is a group invariant;
but we would like to know whether an understanding of conserva-
tion also presupposes mediation of the reversibility by inversion
that characterizes the groups of transformations from the purely
logical and qualitative viewpoints of our subjects. We will try to
show that this is the case. (2) The problem of the relative pos-
sible^., of the possibilities which are realizable in fact modi-
fication of the movement by retarding factors and interferences
among these factors explaining the irregularities and fluctuations
of the course of a particular ball.
The subject's task is to predict the stopping points while vary-
ing the size and weight of the balls and to explain the observed
movement. Our interest in the problem lies in the fact that, if
concrete operations of serial ordering and correspondence forma-
tion allow the establishment of some relationships between the
FIG. 7. Conservation of motion in a horizontal plane is demonstrated
with a spring device which launches balls of varying sizes. These roll
on a horizontal plane/ and the subjects are asked to predict their
stopping points.
CONSEKVATION OF MOTION IN A HORIZONTAL PLANE 125
properties of the balls and the stopping points, the idea o con-
servation o movement by inertia escapes the realm of the "con-
crete/' for such conservation cannot actually be achieved under
ordinary experimental conditions.
Stage I. Absence of the Operations "Necessary -for
an Objective Account of the Experiment and the
Use of Contradictory Explanations
The very young subjects react to this experiment as they react to
the problem of floating bodies (Chap. 2) i.e., with a group of
precausal predictions and explanations possessing certain regu-
larities but mutually contradictory: the light balls will go further
because they are easier to set in motion and the large ones because
they are stronger; or there is no motion without force (the force
residing in the moving body or the force of the mover) and the
motion stops of itself by extinction of the force imparted by the
initial push, by fatigue, or by a tendency to rest.
RA (5 ; 4) tries to prolong or to stop the motion of the ball by framing
it with his hands, which are placed parallel to it without touching.
Sometimes the small and sometimes the large balls are supposed to go
the furthest, the first because they are light and the second because
they are heavy, but when a heavy one does not go far, it is "because
it's too heavy."
BREI (6; 4): "Will they all go the same distance?" "No, there are
some that will go further." "Which ones?" 'That one" [small wooden
baXL].~"Why?"-'Because tfs smaller "-"Axe there others which will go
further?" "That one [also a small wooden ball], because its smaller 9
and that one" [large, copper]. "Why that one?' "Because it's bigger 9
and that one [large, aluminum] because it's big" We ask the child
to show where these four balls will stop and he answers: "There [7-8
units for the small wooden ball], because ifs smaller. That one [large,
aluminum] there" [13-14]. The small aluminum ball is also placed at
13-14 as is the small copper one; the large wooden ball at 5-6 "because
it's bigger, and that one there [large, aluminum, at 19-20] because it's
big. This one here [small wooden, at 2,4] because it's smaller" It is
evident that the small ones are expected to go near or far [from 7-8
to 24] because they are small and the large ones near or far [from 5-6
126 THE FOBMAL OPERATIONAL SCHEMATA
to 19-20] because they are large. Next we ask for explanations, which
we find similar but with a certain note of finality about them: "It didn't
get very far because it didn't have a flag."
MEY (6 ; 8). The little wooden ball "won't go very far because ifs
smatt^-^And that one?" [large wooden ball]-"!* can't go very far
because it's big." Then: "The two big ones will go less far because
they're big. . . . The three little ones won't go as far as the big ones."
The contradictions among the predictions bear witness to the
absence of any law in the child's mind. His explanations do not
achieve a greater coherence but relate all types of motion to a sort
of animated force.
Substage II-A. Attempts to Eliminate Contradictions
and Corrections after the Experiment
Although the conservation of motion may not always be seen (the
motion is regarded as being due to a force in the Aristotelian
sense, and the cessation of movement is spontaneous) and al-
though the predictions are based on variable factors (false or
correct), henceforth there is a certain internal consistency in the
assertions as well as in the utilization of experimental results:
pm (7 ; 6): "Some of them will go further than others"-'Why? > '- e This
one will go -further because it is big and that one less far because it is
small [the first one is put in motion]. Ifs less far than I thought ."
"Why?" "Because ifs heavy."
NIC (8 ; o): "The big one will go further because the little ones have
more weight." 9 And "that one won't go as far because ifs big, heavy,
and made of iron."
HAL (8 ; 3): "The big ones won't go as far because the little ones are
lighter" When a ball comes to rest close to the starting point: t lfs
because it is heavier than I thought" and, comparing a small copper
ball to a large aluminum one, "They go to the same place because they
have the same weight"
But the difficulty with an explanation in terms of force, such as
used at this level, is still that of reconciling the force with which
CONSERVATION OF MOTION IN A HORIZONTAL PLANE 127
the object is launched with the force of the moving body and
under conditions when the latter is heavy and when it is light.
HOR (8 ; 6): "This one [large, aluminum] will go further because it is
heavy' [force itself is tied to the weight]. She rolls the copper ball. "It
doesn't go as far because it is sm0ZZ/ > -"And the other?" "I didn't push
it hard enough" Next the large wooden ball: "It wiU go all the way to
the bottom because ifs light."
In spite of the effort to eliminate them, a residue of contradic-
tions is left from the fact that the heavy balls have a greater force
when in motion but are less easily set in motion, whereas the light
ones have less force but are more easily launched.
Substage II-JB. The Beginning of the Reversal of
the Problem in the Direction of the Causes of Slow-
ing Down
The explanations used at this level are not different from the pre-
ceding ones, in spite of the increasing but fruitless effort to unify
the factors. However, since the child is increasingly sensitive to
chance variation in the results, he exhibits a tendency to reverse
the problem and to explain the causes of the slowing down rather
than the cause of motion. He is not aware of this tendency. More
particularly, little by little weight ceases to be perceived as a
cause of motion and comes to be thought of as the (indirect) cause
of the balls' coming to rest. Moreover, to the extent that subjects
understand that the variability of the stopping points is due to
the factors of volume, weight, and force of launching, they are
more likely to think that weight and volume have a braking effect
and even less likely to maintain that light weight and small size
are causes of the prolongation of motion. These two kinds of
assertions seem to be equivalent; the following will show that this
is not true in the least:
JAD (10 years), referring to a zone of dispersion of about 20 cm., says
of one ball, "It is too heavy to go any further" [than the extreme point]
but at the same time "it is too light* to come to rest before the zone.
This kind of assertion shows clearly that the subject tends to
invert the problem of motion. But he is that much less likely to
128 THE FORMAL OPERATIONAL SCHEMATA
suspect that his explanations remain the same as at substage II-A.
In particular, he thinks of the air as promoting the motion by
current backlash (dvTwrepi<n-ao-i) rather than as an obstacle.
Substage III-A. Explicit Reversal of the Problem
of Motion During the Experiment
The great difference between this level and the preceding ones
is that from this point on the objective of the explanation is re-
versed; the problem is no longer to understand why the ball
advances but what blocks its movement at a given moment,
As we have just seen, this reversal begins in substage II-B, but
unconsciously. In contrast, although at first the III-B subjects are
preoccupied in their predictions with motion, the experiment
immediately leads them to focus their attention on the causes of
the balls* slowing down or stopping. Thus, for these subjects the
cessation of motion is no longer a positive state, the repose or
aim of movement; instead, it becomes a negative state which
must be explained by the intervention of new factors working
in opposition to the positive state of motion.
MAL (12 ; 3); "For a ball to go far?" "You have to pull the trigger
[spring] hard 9 [experiment], "So, why didn't it go further?" 'Yes,
but it's a bad stretch [plane insufficiently smooth]; it won't go so far"
CHAP (13 ; 3) predicts that the large ones will go further because they
are heavier. After the experiment, he reverses his explanation. "Why
do the light ones go further?"--"!* depends on whether there is wind."
"What?" "If s the wind [ = air] that stops them from going on.
When there isn't any wind, the light ones go far because nothing stops
them."-' And the heavy ones?"- 7 don't know."
(13 ; 3): "The air keeps it back and it doesn't go as far."
Thus, starting with substage III-A, subjects touch on two
causes of the cessation of motion: friction (terrain) and air resist-
ance.
Without doubt the progress involved in reversing the explana-
tion is due to the need to unify nascent formal thought Since
neither weight nor volume are causes of motion and (in contrast
CONSERVATION OF MOTION IN A HORIZONTAL PLANE 129
to explanations based on this conception) the ball goes further in
proportion as it is both small and light, it follows that there is
no simple cause for the continuation of motion. But it is more
difficult to acknowledge multiple causes for motion itself (which
may be considered the prototype of any simple phenomenon)
than for the factors relating to the cessation of motion. However,
even here the subject begins by looking for a unified explanation.
He does so in spite of having seen the spread in results and chance
fluctuations, which themselves were one of the reasons for his
reversal of the question. That is why he does not succeed at first
in this new line of attack Time after time he fails to determine
all the relevant variables simultaneously. Thus, CHAP discovers
the factor of air resistance but fails to think of the friction for the
heavy balls. MAL does the opposite, etc.
Substage III-B. Conservation of Motion
Finally, substage III-B leads to the fundamental explanation
which results from the reversal of the positions taken at substage
III-A: the conservation of motion by inertia. It should be said
that all of the subjects do not solve the problem. Naturally, cul-
ture plays its diffuse role here. ( Society had to wait for Galileo
and Descartes with the "intellectual mutation," as A. Koyr called
it, which resulted from their discovery.) But for certain subjects
the rediscovery of the principle of inertia seems quite spontane-
ous, whereas for others there is, at least, a personal reconstruction
of what they had learned:
DEV (14 ; 6) from the first experiment [large wooden ball]: "It stopped
because the air resists" "And this one?" [a small wooden ball, pre-
diction]."!^ about the same, but the ball Is smaller: there is less re-
sistance from the air and it will go further." "Is it the same for all of
the balls?" "No, the bigger they are, the stronger the air resistance."
"And for the small, heavy one?" "A heavier baft takes of less easily,
but goes further because it has force in itself 9 [weight = force!].
[Experiment] "So?" "That comes from the surface and the friction.
The resistance varies with the substance the balls are made of: the
wood is rougher, it scrapes more; the metal balls are smooth and witt
scrape less." [Experiment: small aluminum and large wood.] "Air
TPT5 FORMAL OPERATIONAL SCHEMATA
resistance is proportional to size and weight [l]."- tt And if you com-
pare this large aluminum ball with the small brass one?" "Oh! No,
they take off with the same force. Only air resistance and friction come
into play. . . . This ball [brass] is heavier and there will be more
friction." Conclusion: "And if there were no air resistance, the ball
would continue to roll.' 9
RAS (14 ; 4) "Theoretically it should go to the end, but it's completely
illogical* [he means by illogical that which is contrary to the facts of
direct experience]. Comparing a small and a large ball, he says again:
"The friction is less for the little one. Air resistance also plays a role.
Theoretically, you would have to move it in a vacuum."
DESB (14 ; 9): "If you send them off with a push of the same strength,
it [resting point] depends on weight, friction, and volume" Next, he
doubts that volume plays any role, but in comparing a small and a
large ball, he says: "The small one will go better because it has less
friction, less air resistance^-'Th&t's all?"- 7/ ifs truly horizontal" *
The protocols show that the reasoning which leads to the con-
servation of motion is extremely simple and is furnished in the
most explicit form by DEV. The first stage consists of establishing
the causes of the balls' slowing down or stopping. If we let p be
the statement concerning slowing down or stopping, and let
q, r, s, t, etc., be statements of friction or air resistance, irregu-
larities of the track, of an eventual lack of (perfect) horizontality,
etc., then:
. . .). (i)
Inversely, at the second stage the subject asks himself what
should be the result of the negation of all of these factors, this
negation implying a corresponding negation of statement p, that
of slowing down. This is equivalent to the assertion of the con-
tinuation of motion:
q.r..t ... Dp. (2)
It is interesting to compare this form of conservation, which is
specific to formal thinking, with numerous concrete forms of con-
servation (wholes, lengths, weights, etc.; conservation of volume
3 See other cases of this stage or other protocols from the same subjects in
the third section of Chap. 15.
CONSERVATION OF MOTION IN A HORIZONTAL PLANE 131
and surface area imply formal thought only because of the pro-
portions). In both cases, conservation is achieved because of the
role played by reversible operations (reversible by inversion or
negation). When a modification arises as the result of the experi-
mental actions, they allow a correction to be made for it by a
transformation in the opposite direction (and thus a return to the
null transformation). But in the case of concrete thinking this
inverse transformation, even if it occurs only mentally, is of the
same order as experimental modifications which alter the system
and could in fact be carried out by the subject. For example,
the transformation of a stretched-out section of modeling clay can
be annulled by pushing it into a more compact mass, for what the
object has gained in length it has lost in thickness. Thus it is pos-
sible to restore it by actions involving inverse modifications.
In contrast, in the case of the conservation of motion, opera-
tional reversibility occurs at the mental level only and does not
correspond to any transformation which can be realized in full
by the subject even in a laboratory situation. Even if one could
eliminate all the causes of slowing down (though it is in fact
impossible), one would still have to make use of an infinite amount
of space and time to verify the principle of inertia completely.
Nevertheless, the substage III-B subject manages to discard
mentally the causes of stopping by thinking in terms of what is
theoretically possible (but which cannot occur in fact) or, in
other words, in terms of purely hypothetico-deductive implica-
tions.
Having done this, once more a reversible operation (i) and
(2) suffices; here it is the counterposition (equivalence of p D q
and q D p\ but in this case it rests on the double negation of
(pvqvrv . . .) resulting in (p.q.r. . . .) (thus of p or q or
r . . . resulting in neither p nor q nor r . . .) and of p result-
ing in p.
One may, if one wishes, say that this reversibility comes back
to the famous principle toUitur causa, tollit effectus, but on the
one hand in order to eliminate the causes in the particular case
the subject must think in terms of what is theoretically possible;
on the other hand, since these causes cannot be eliminated in fact,
the operation amounts to inverting an implication to give its con-
verse by changing signs. Thus, the subject is proceeding on the
132 THE FORMAL OPERATIONAL. SCHEMATA
basis of pure implications and no longer on the basis of trans-
formations which can actually be effected.
We now see both the similarity and the differences between
the several forms of conservation: all are based on a group prin-
ciple ('which is qualitative or logical before becoming quantita-
tive or metrical), but conservation may be achieved either by
concrete operations of classes and relations 4 (or at an even earlier
stage by the integration of parts into a whole object) or, as at the
formal stage, by the use of interpropositional operations alone.
*In this case the group aspect corresponds to the reversibility of the
"grouping*" i.e., to the nontautological transformations (identical with those
of Boolean algebra).
Communicating Vessels 1
IN THE PROBLEM of the conservation of motion, we encountered
the simplest form of the operational schemata relating to group
structure, for the construction of this notion by the adolescent
rests directly on formal reversibility by inversion. In the equi-
librium problems, of which the problem of communicating ves-
sels gives us a first example, we come to a more complex variety
of schema resting on group structure. In every equilibrium the
two possible forms of reversibility operate simultaneously: inver-
sion., which corresponds to the additions or eliminations effected
in the parts of the system which come into equilibrium, and reci-
procity, which corresponds to the symmetries or compensations
between these parts (thus to actions which are both equivalent as
regards their respective products and oriented in opposite direc-
tions). But, inversions and reciprocities also form a group between
themselves.
In order to illustrate our point and, more particularly, in order
to understand more clearly in what way the operational schema
corresponding to the notion of equilibrium is at the center of the
mechanisms of formal thought, we have to remind ourselves that
beyond the operations themselves in the strict sense of the term
iWith the collaboration of F. Pitsou, former research assistant, Institut
des Sciences de Education, and A. M. Weil.
133
134 THE FORMAL OPERATIONAL SCHEMATA
(or "operators")-^., the operations of prepositional logic, such
as disjunction (pvq), implication (pDq), etc. there are more
general transformations which transform particular operators into
others. Thus an operator such as pvq can be transformed by
inversion or negation into p.q, a transformation that we may desig-
nate by N, so that N(p v q] = p.q. But (p v q) can also be trans-
formed by reciprocity R, so that R(p v q) = p v q = p/q. Again
(p v q) can be transformed, by correlativity C (i.e., by permuting
the v and the .), so that C(p v q) = p.q. Finally, the operator
(p v q) may be transformed into itself by identical transformation
I, so that I(p v q) = (p v q). Thus, one can see that I, N, R, and G
form a commutative group of four transformations among them-
selves, for the correlative C is the inverse N of the reciprocal R,
so that C = NR (and C = RN as well). Likewise, we have
R = CN (or NC) and N = CR (or RC). Finally, we have I = RCN
(or CRN, etc.).
This group is of psychological importance because it actually
corresponds to certain fundamental structures of thought at the
formal level, for inversion N expresses negation, reciprocity R
expresses symmetry (equivalent transformations oriented in oppo-
site directions), and correlativity is symmetric with negation. This
explains why the notion of equilibrium, which at a very early age
gives rise to certain rough intuitions (balance, etc.), is not really
understood before the formal level, when the subject can both
distinguish and coordinate inversions, reciprocities, and correla-
tivities (inversions: for example, increase or diminish a force in
one of the parts of the system; reciprocities: compensate for a
force by an equivalent force, thus assuring symmetry between the
parts; correlativities: reciprocity in negation).
Although they may be relatively simple in certain concrete
cases, these transformations actually require dunking and state-
ments of a very abstract sort in most problems involving action
and reaction, for here the difficulty is to grasp that X is at the
same time equal to Y and acting in the opposite direction from it.
In such cases, the instruments necessary for thinking go beyond
prepositional logic to include its fundamental group I N R C.
This is what we shall see in the following chapter in reference to
the problem of the equilibrium between the pressure of a piston
and the resistance of liquids; but at this time take note of the
COMMUNICATING VESSELS 135
same question as it relates to the preliminary problem of the
equilibrium of communicating vessels.
In the case of communicating vessels, reciprocity serves to
express the compensatory actions between separate vessels; trans-
formations by inversion express the rise and fall of the water level
(Changes in water level are brought about not by adding or taking
away water but by raising and lowering the receptacles.) In appa-
ratus A, the subject raises or lowers the vessels by hand by adding
or taking away the stands on which they rest. In apparatus B, he
raises or lowers the two vessels with levers, and in apparatus C,
he can only move one of the vessels, the other being stationary,
Since the receptacles have neither the same shape nor the same
volume, in some cases one has to exclude these two factors to find
the law. But air pressure can be disregarded, for it is equivalent
for the two columns of liquid. 2 On this last point acquired knowl-
edge may intervene, but we still want to know how well the
adolescent can understand and make use of this knowledge, so
the problem of formal operations remains decisive here and the
influence of school is no bar to our analysis.
Stage I. Lack of Differentiation Between the Actions
of the Subject and the Transformations of the Object
and Absence of Reciprocity
The stage from 4-5 to 7-8 years is highly interesting from the
point of view of the development of operations. No operation is
yet possible at this stage, for the child fails to dissociate his sub-
jective action from objective transformations and there is no
reversibility between successive actions. The result is that the
subject succeeds neither in predicting nor in understanding the
2 The liquid used was only water and there was no difference in density
between the contents of the two vessels. In systems of communicating vessels,
the pressures are proportional to the weights of the liquids (pressure is the
quotient of force divided by surface area). The fundamental principle in-
volved is the following: the difference between two pressures pi pa exerted
at two points by liquid of density d in equilibrium is equal to the weight zd
of a cylinder of liquid having as a base a unit of surface area and for height
the vertical distance between the two points: pi p 3 = zd (where zd repre-
sents the pressure force measured in grams).
THE FORMAL OPERATIONAL SCHEMATA
symmetry of the objective effects in the relations between the two
containers:
GUY (5 ; 6). Apparatus B: He pulls the lever at the two sides alter-
nately and concludes: "If 1 puU there [I] the water goes away, and
then it comes up there [II]. If you pull there [II] the water goes
here" [I]. Apparatus A: He lowers the vessel to the left and holds the
one on the right in an inclined position: "There is more water here
[right]. Look, there is more water there [left]: I lowered it here and
the water went there! 9 "Why does it go down?" "I don't know; it's
because it doesnt want to go back up" Apparatus C: "I get it: the
water has to go there" [when he pulls at the other side].
We see that the child has a perfect understanding of the fact
that, when he pulls on one side, thus raising the receptacle, the
water passes to the other side; but he does not understand that a
difference in height is involved. For apparatus A, designed so as
to make the differences in water level more visible, he thinks that
it is enough to tilt a beaker to make the water flow into the other.
And when we insist that the water level goes down, he is limited
to saying that it does not want to go back up. Furthermore, it
should be noted that this reaction is not specific to the problem
of communicating vessels. For example, during this same stage the
child does not know that the water in rivers always flows down-
ward. 3 But in this particular case this lack of understanding is
reinforced by the subject's failure to dissociate his own actions
(pulling, tilting, etc.) from the objective process. The effect is that
reciprocity is understood only when it occurs between the act of
pulling and its results and not when it occurs between the rise and
fall of the liquid which tends towards an equalization of levels.
Substage II-A. The Translation of Actions into Objec-
tive Operations and the Discovery of the Elevation
Relationships
The role of elevation is discovered. The higher one beaker is in
relation to the other, the more the water level rises in the latter as
it falls in the former. This observation is certainly based on the
3 See Piaget, The ChMs Conception of Physical Causality, pp. 104-114.
COMMUNICATING VESSELS 137
subject's actions as he raises or lowers the beakers with levers or
directly, but these actions are translated into operations which
bear on the results obtained and describe those results in terms of
objective relations dissociated from the subject's own activity:
NEL (7 ; 11): "It always goes up more in that one when I raise this one
and it goes down when I go down."
TAG (8 ; 10). Apparatus B: We hide the right beaker and we ask the
subject to bring the water level up to the third marker. The subject
succeeds but only in a rough way and says, "I looked about here' 9 [she
looks at the height of the other beaker]. Apparatus A: "Before you had
to pull; now you have to put on the stands and that raises the beakers."
Then she draws the same conclusions as NEL.
But as far as the explanation itself goes, it does not at all deal
with the equilibrium between weight and pressure; instead, sub-
jects assume that water merely descends in the higher beaker to
enter the lower one simply because that one is lower. As for
explaining how the water level rises in this latter beaker (since it
enters at the lower end of the receptacle), at this stage subjects
refer to the impetus, rate of speed, air, etc.:
MIC (7 ; 10): "Did you understand how the water moves?"-'T0u have
to lower [the beaker in which the water is to rise], then the water
comes . . . it flowed."
TEA (8 ; i): "The water went through the pipe and into the other one!"
"What did you see?" To see the speed. The water doesn't diminish
as fast [in the small one] as in the big tube, because the tube is thinner
and has less air than the big one. Tm surprised that the big tube y which
has more air, works, and that one which has less works too: maybe it's
because it's longer. If there is a big block of air, the water would move
less quickly"
Thus the air aids or blocks depending on whether it pushes or
already occupies the place.
The progress made at this level over stage I is quite clear. The
subject now describes the rise and fall of the water level and no
longer only his own actions of pulling the lever or moving the
beaker. On the other hand, it is hard to see the difference between
these reactions and those of substage II-B unless we refer to the
J3S THE FORMAL OPERATIONAL SCHEMATA
spatial structures which are available to the subjects. As we have
demonstrated elsewhere, 4 at substage II-A the child is not yet able
to represent the horizontality of the water level in a tilted recep-
tacle because he does not try to base his observations on reference
points outside this receptacle and limits himself to the interior
relations. It is only toward 9-10 years, at the stage when co-
ordinate systems are structured, that the horizontal and the verti-
cal acquire a precise representative meaning. But it so happens
that in the present experiments the subjects of substage II-A do
not yet perceive the equality of level attained by the water in the
two receptacles; they discover simply that the water goes down
in one as it rises in the other until it stops moving in both. Further-
more, they know that the water level drops in the beaker in the
higher position and rises in the lower one, but these relations of
elevation are applied only to the receptacles themselves and do
not always imply that the two water levels will finally be equal
i.e., that the line that unites the two levels will be horizontal. That
is why, when one of the beakers is hidden and the child is asked
to attain a certain elevation (see TAG: the third stage for appara-
tus B), he can only succeed in a rough way and focus on the
height of the other beaker and not its water level.
All in all, then, subjects have just about begun to get a glimpse
of the notion of system equilibrium. And what notion they have
boils down in essence to raising and lowering the beakers with a
view toward raising or lowering the water level. Doubtless, a
preliminary inversion (raising and lowering the beakers) and reci-
procity (the water goes down in one vessel as it rises in the other)
are present. But lacking is the condition of equivalence which
alone would allow the child to coordinate these transformations
the final equality of the two water levels.
Substage II-B. Final Equality of Water Levels
but Without an Explanation
As we know, subjects become able to handle concrete opera-
tions at substage II-B. This substage also marks a kind of upper
limit in the structuring of the equiKbrium schema, that is, insofar
* Piaget and Inhelder, The ChilcFs Conception of Space, Chap. XIII.
COMMUNICATING VESSELS 139
as the subject does not bring in any formal operations. Owing to
the construction of systems of spatial reference (natural coordinate
axes) the child discovers the law of the equality of water levels
in the two receptacles for conditions of system equilibrium (the
water line in both beakers falls on a single horizontal line). But in
this way subjects can only enunciate the law without discovering
its causes, for a statement of the law depends on class and rela-
tional operations alone, and these are sufficient to determine the
relevant correspondences, but an explanation of the law requires
the intervention of the four groups of interpropositional transfor-
mations cited at the beginning of this chapter.
xi (8 ; 9): e< When I pull here, the other [beaker] fills up; when I pull
there, the other one fills up too/' "The water is at No. 2; put it at
No. 3." He succeeds. "How did you do it?" "I saw that when the
water goes up here it goes down over there; so I did the opposite . . .
[etc.]/' At one point, he makes a mistake [the point to be reached is
hidden each time]. He then takes the ruler and measures [the dis-
tance] from the table to the number indicated. Then he refers to the
same elevation on the visible beaker at the other side in order to deter-
mine the water level. Another time, he places the ruler horizontally
to assure the equality of the levels.
MIC (9 ; 11) succeeds in determining the water level correctly when
one beaker is hidden: "How did you know?" "Because 1 calculated
the height here and I looked there for the same thing"
soc (10 ; 9): "The water is at the same level When I raise it here, the
water goes up there, but there is always the same capacity, even if it
goes up"-~ "What do you mean by capacity?" "Tftere is always the
same amount of water [he knows well that the volumes differ]: the
water stays at the same height on both sides."
DOM (11 ; 4): "The level is exactly the same. The water rises quickly
in this tube and -falls less quickly in the bottle. That comes -from the
volume of the beakers, but in contrast the water will always stay at
the same level." Apparatus A: "And the level?*' "I* wiH change. No,
rather it will always be the same [on both sides], but the [absolute]
height will change' 9
GAS (10 ; 6) measures the elevations and verifies the horizontal level.
He is presented with a long tube communicating with a very large
crystallizer: he predicts that the levels will still be identical.
140 THE FORMAL OPERATIONAL SCHEMATA
Thus, one can see that these subjects discover both the equality
of water levels and the means of verifying this equality once a
reference system based on the coordinates of immediate physi-
cal space (vertical and horizontal) is established. Verification is
effected either by checking whether the line uniting the two sur-
faces is horizontal (xi and GAS) or by measuring their respective
heights (xi, MIC, and GAS). This transition from the qualitative to
the metrical shows us well enough how preoccupied the subject is
with coordinate axes and with substituting the concept of equi-
librium based on the equality of water levels for the concept of
equilibrium found at substage II-A i.e., based on rising and fall-
ing. In this case transformations by inversion amount to the raising
or lowering of the level in one of the beakers, whereas, henceforth,
reciprocity includes the whole set of transformations which relate
the level of liquid in one receptacle to the level of the liquid
in the other.
What is the nature of the mechanism of these transformations?
The concrete operations available to the subject at this stage do
not allow him to answer this question; by their temporal and spa-
tial serial ordering and correspondences, they allow him to deter-
mine the conditions of equilibrium, but by no means do they allow
him to grasp the play of forces involved. In the course of proceed-
ing from a statement of the law to its explanation, some subjects,
like soc, invoke an equality of "capacities" or "amounts," but since
it is evident that the volumes differ, in the final analysis this quan-
tity amounts to nothing more than the equality of the elevations
themselves i.e., the equality of water levels. A ten-year-old sub-
ject specified that the water always goes "as low as possible." This
demonstrated that henceforth the tendency of the water to fall is
accounted for in terms of its weight (as we had known as the result
of other experiments); but weight itself is no longer often called
upon as an explanation of the equilibrium, since the volumes
involved are clearly different. In sum, from this point on the equi-
librium is well described, thanks to the concrete operations which
make it possible for spatial and temporal inversions and reciproci-
ties to be established. But by no means is it explained, for the
child fails to make use of inversions and reciprocities bearing on
the actions and reactions themselves.
COMMUNICATING VESSELS 141
Substage III-A. Preliminary Explanation and
Formal Structuring
At this first formal level we can observe, in contrast, an important
reworking of the operations and the explanation. The conservation
of volume is finally acquired and the volume is finally distin-
guished from the quantity of matter and the weight. 5 This leads
to the paradoxical fact that substage III-A subjects seem to find
it impossible to accept a situation that did not bother substage
II-B subjects at all i.e., the equality of levels when the volumes
(as well as the shapes of the receptacles) differ. Far from gen-
eralizing the law to all cases, as at substage II-B (where, by the
way, the generalization is limited, since it bears on the levels
alone), substage III-A subjects start by restricting the scope of the
law to those cases in which shapes and volumes are equal. They
expect that the equality of levels will no longer hold for unequal
forms and volumes. When the experiment contradicts their expec-
tations, they limit their conclusions to the cases actually observed
and refuse to make any generalization that would admit of what
seemed to them to be an exception. We have here a neat example
of mutual interference between the operations constitutive of the
law and explanatory or causal operations. More precisely, in be-
coming explanatory the stage III operations lead the subject to
limit the generalization based on concrete or legitimate operations
(legitimate because here they bear on the levels alone and not yet
on the equilibrium of actions and reactions).
What goes into these new operations? At this stage equilibrium
in communicating vessels is no longer conceived of as the simple
flow of water from a higher level to a lower one until equality of
levels is achieved, but as a system of actions and reactions whose
inversions and reciprocities are stated in mechanical and not
merely in spatiotemporal terms. That is why the subjects require
equality of weight based on equal volumes before they are willing
to talk about equality of levels, and that is why they deny that two
vessels of unequal capacity can verify the law. They have failed
5 See Piaget and Inhelder, Le Dfoeloppement des qwntitis chez Venfant,
Chaps. Ill and VIII-IX.
142 THE FOKMAL, OPERATIONAL SCHEMATA
to understand the compensation resulting from the relationship
between the weight of the vertical column of water and the
surface area of the base of this column.
AND (12 ; 9) establishes the equality of levels in all positions. "And if
instead of this one we used a bottle with a conic shape?" "It wouldn't
work because the shape is conic." "But you told me that the levels
were always the same?" "They're the same providing the diameter is
the same at all heights of the beaker. Here [with conic and cylindrical
formsl the elevations . . . you couldn't manage to compensate [one
with the other]. It's the shape of the beaker that plays a role." He
does the experiment and is astonished to discover the same level,
"How do you explain it?' "Probably that the shape of the vessel
doesn't matter"-"Why 'probably ?"-"Because the facts are therer
[cf. the opposition between facts and theory so characteristic of formal
thinking]. "Why is it the same level?" "Because it [the conic beaker]
widens toward the top" [the opening is at the bottom]. "But if it
were?" [turned upside down]. "I* wouldn't be the same level [!]."
"What happens when the water is in the pipe?" "It isn't the same level
because the tube is thinner than the beaker"
BON (12 ; 8) affirms that the levels are the same. With apparatus b,
he takes exact vertical measurements when one of the beakers is hid-
den: "Whenever you lift one container, the water rises in the other, so
the water should rise or fall in both" "Does it always happen like
that?" "Yes, always. . . . No, not in all cases > not when the beakers
are not of the same width." "But in this apparatus [C] are the beakers
of the same width?" "No, but the length of the pipe and the width
of the beaker can contain the same quantity of water" "And here
[long tubular beaker to the left and large crystallizer on the right (cf.
the case of GAS at substage II-A)]?" "The water here (crystallizer)
will only go up to here" [much lower level than the other]. "Why?"
"Because the beaker is larger"
BAN (13 years). Apparatus A: He raises both beakers "to see if by rais-
ing the two together I get the same level in both as when they were
down below" Next: "To add some water in one, you have to take out
the same amount in the other. When I put the beakers in different
positions [in relation to each other], I can always see [that the level
is] the same for both." "And if you put a narrow bottle in that one's
place?" "No, if I have a large one and a small one because the vol-
ume is larger . . . the level is always higher"
COMMUNICATING VESSELS 143
PIE (14 ; 3), identical beakers: "If one bottle is [placed] higher than
the other, the water goes into the other 9 because "for the quantity, it's
the same thing." But with unequal beakers, "When there is the same
quantity in the two glasses, the level in the tube will be higher, because
it is thinner" When one of the beakers is hidden, he answers: "I can't
figure it out, because the diameter of the two glasses isn't the same"
We see the difference between these subjects and those of sub-
stage II-B even though they often use the same words. Henceforth
"compensation" (AND) is a matter of "quantities" understood in
the sense of sources of equal forces (because of the equality of
weight and volumes) as if it were a balance scale. Thus unequal
levels should correspond to unequal quantities. When they per-
ceive facts to the contrary, the subjects resign themselves to them,
as for example AND ("because the facts are there"), but refuse to
generalize to other cases. In sum, they do not know the details of
the explanation. But if we consider only what they do know, we
find that they reason in a coherent manner and furnish a very
revealing example of the logical subordination of a general law to
the concrete case and of the assimilation of that case to the formal
transformations of inversion and reciprocity projected into the
real world. In fact, one might say that these subjects are of inter-
est to us precisely on account of their ignorance of the exact expla-
nation: though they have received no academic instruction about
communicating vessels, they still sketch out an interpretation
based on compensation (as AND says) i.e., on the fact that each of
the two quantities of liquid exerts a pressure on the other, the two
pressures being, by this very fact, oriented in opposite directions.
Certainly we have here a differentiation and a coordination of
the transformations of inversion (raising or lowering the levels)
and reciprocity (actions and reactions of one of the quantities of
water on the other). The only limitation of this explanation is that
the subject does not yet know how to generalize it to the case of
unequal quantities; still, the principle is accurate. Before trying
to give a precise statement of this reasoning, let us examine the
reactions of substage III-B, which we have not yet considered.
Unlike the earlier reactions, these are influenced by academic
knowledge (which, moreover, has been assimilated only to the
extent that it fits into the schema whose development we have
just noted).
THE FORMAL OPERATIONAL, SCHEMATA
Substage III-B. Formal Generalization of
Acquired Knowledge
Finally, at substage III-B, the spontaneous schema of explanation
outlined during substage III-A is filled in with information gained
through education; thus the contradiction between the equality
of water levels and the eventual inequalities of the amounts of
liquid is eliminated. But one can easily see that this contribution
from without does not modify the structure of the reasoning:
PIC (13 ; 6): "These levels are always equal because the forces com-
pensate each other"; according to PIC these forces are air pressure and
the weight of the water.
MIN (14 years): "I/ you have two beakers of the same sizes or different
sizes, the water will come up to the same level in both, because the
larger the beaker, the more air presses on a large surface; and vice
versa, the smaller the beaker, the more the water will act on a small
surface, so an equilibrium is reached" "Always?" "No. When you
have two chambers, if there were more air pressure in the chamber
where you put the bottle on the left and less pressure in the chamber
where you put the bottle on the right, the level at the left would be
lower."
In other words, having a more or less clear understanding of
the fact that the pressure of the liquid is relative to the surface
area of the vertical column at its base, the subject explains the
phenomenon of communicating vessels in a fashion analogous to
that used at substage III-A, but generalizes to the case of unequal
quantities. Thus the essential point in the explanation is that even
in the case of unequal volumes the pressures compensate each
other in function of the height of the columns "so equilibrium is
reached," as MIN says, this time referring to beakers having differ-
ent capacities.
The Notion of Equilibrium and the Group of Four
Interpropositional Transformations I N R C
In order better to understand the nature of the formal structuring
which culminates in the operational schema of equilibrium, it
COMMUNICATING VESSELS 145
seems worth while to compare what can be called the concrete
reciprocities of stage II with the formal reciprocities of stage III.
The preliminary form of reciprocity appears for the first time at
substage II-A with the discovery (inaccessible to stage I subjects)
that the higher one of the beakers (I) is in relation to the other
(II), the more the water rises in the second beaker. The discovery
of this relationship entails the following operations:
Serial ordering of heights:
A! < B! < Ci < . . . (i)
taking the other beaker as a point of reference;
Serial ordering of the levels of elevation as they increase in the
lower beaker (with interior references to this beaker):
A 2 < B 2 < C 2 < . . . ; (2)
An (ordered) correspondence between the two sets of serial
orderings:
Ai < Bi < Ci < . . . ,
$ $ $ (3)
A 2 < B 2 < C 2 < . . . ;
In the case of reversal of the situation, the elevations occupied
by beaker I may be ordered serially in descending order:
. . Ci > Bi > Ai > , (4)
as may be done for the levels in II:
. . . C 2 > B 2 > A 2 . (43.)
The correspondence is in this case established in reverse order.
Thus concrete reciprocity consists of a symmetry between the
two correspondences:
(A! -> A 2 ) <F (A 2 <- Ai). (5)
At substage II-B, a system of external reference is added to
these relationships, allowing the introduction of the notions of the
horizontal and of the equality of levels in terms of rate of flow.
We have treated the operations needed to construct this spatial
system elsewhere. 6 Here we may limit ourselves to noting that the
6 Piaget and Inhelder, The Child's Conception of Space, Chap. XIII.
THE FORMAL OPERATIONAL SCHEMATA
serial orderings and correspondences (i)-(s) are replaced by a
correspondence between the actual water levels in the two beak-
ers. If we call + A, + #, + C, etc., the increasing elevations in-
cluded between the horizontal (the line of final equality) and the
levels in the higher beaker, and A, - B, - C, the correspond-
ing increasing heights included between the line of equality and
the levels in the lower beaker, the substage II-B child establishes
the correspondence:
Thus the reciprocity depends on the equality of the differences
+ X and X and their continuous compensation, which occurs
until the difference is zero (when the line which eventually unites
the two levels is horizontal). So reciprocity boils down to a spatial
symmetry (but without an adequate causal explanation). As for
the inversion operations, they consist of increasing or diminishing
the differences A, B, zt C, etc. This operation may be
effected by addition or by elimination of quantities of liquid in
one of the two vessels; this is accomplished easily by raising or
lowering this beaker. Hence (if A' is the difference between the
increasing heights A and B):
A + A' = B, etc., and B - A' = A . (7)
But no total operational system as yet allows the subject to fuse
reciprocities by correspondence and increases or decreases of dif-
ferences into a single whole. That is why the subject is limited
to describing the equilibrium and cannot manage to understand
it as a single causal system. When the notion of compensatory
actions and reactions appears at stage III, two innovations come
into play: the spatial reciprocity of levels becomes a reciprocity
of pressures; this constitutes a single operational system with
inversion operations.
This coordination of inversions and reciprocities can be formu-
lated in the following manner: Let us call p and q the statements
concerning the effects of any two pressures exerted at separate
points on the liquid contained in beaker B. Let us call p and q
the statements that these effects are canceled out, either by inver-
sion of their causes (thus of diminutions in beaker A until the
COMMUNICATING VESSELS 147
initial elevations are eliminated) or by compensation under the
influence of pressures operating in opposite directions.
Thus there are four possibilities:
I(pvq), (8)
direct transformation or effects of the pressures exerted by liquid
A on liquid B;
N(p v q) = p v q = p.q , (8a)
inverse transformation or elimination of the effects p and q;
v q) = p v q , (8b)
reciprocal transformation or effects of the pressures exerted by
liquid B on liquid A;
C(p vq)=pvq = p.q, (Be)
correlative transformation i.e., inversion ( = negation) of the re-
ciprocal, thus canceling out the opposite (negative) effects p v q,
which is equivalent to the simultaneous assertion of p and q
i.e., p.q.
Such is the reasoning schema which the stage III subject uses.
He understands that the point of equilibrium is reached when
values x and t/, corresponding respectively to the pressures repre-
sented by p v q and by p v q, are equal. As long as one has x > y
or x < y, the liquids are actually still in motion in tubes A and B.
On the other hand, any movement ceases when the liquids reach
the same level (represented by r) because:
r=)[x(pvq) = y(pvq)]. (9)
Although substage II-B subjects perceive the horizontal level
common to tubes A and B, they are unable to explain it. Stage III
subjects interpret it as due to an equality between pressures,
stated by the double reversibility of transformations I N R C.
10
Equilibrium
in the Hydraulic Press 1
THE INCREASINGLY more advanced explanation which the subjects
give of the phenomenon of communicating vessels has just shown
us the importance of the formal transformations of inversion and
reciprocity and of the I N R C group that they form among
themselves, according to the possible combinations, for the estab-
lishment of the operational equilibrium schema. But the drawback
of the experiment with communicating vessels is that the pressure
intrinsic to the liquid is completely overlooked by our subjects.
A detailed account of the explanation is not found until it is given
in terms of acquired knowledge. In the apparatus dealt with in
this chapter, two communicating vessels again appear, but one of
them is provided with a piston which may be loaded with varying
weights; thus, the pressure exerted on the liquid is directly pro-
portional to the weights. (It is to be noted that the piston is
propelled not by an external force but by its own -weight.)
Now, there is a reaction of the liquid corresponding to the
action of this weight (the displacement of the liquid under pres-
sure is inversely proportional to its resistance), but here, too, the
resistance reaction can be made tangible by varying the density
1 With the collaboration of A. M. Weil; A. Tissot, former research assistant,
Institut des Sciences de I'fiducation; and M. Wikstrom, &l&oe dipldmee,
Institut des Sciences de Tfiducation.
148
FIG. 8. The equipment used for this problem
in equilibrium involves two communicating
"vessels' 7 of different sizes and shape. Vessel A
is provided with a piston that can be loaded
with varying weights. The amount of pressure
exerted by the piston (which is dropped into
the vessel by the subject) is varied by adding
weights.
150 THE FORMAL OPERATIONAL SCHEMATA
of the liquid i.e., by using water, alcohol, or glycerine in turn,
To the question o equilibrium is added that of the actual trans-
mission of forces; the problem is to understand that the force
exerted by the piston is transmitted in a uniform manner through
the entire liquid and that the equilibrium between action and
reaction relates not only to the surface of the liquid (or the lowest
point in the apparatus) but to the entire system.
Therefore, in this particular case the problem of the relation-
ship between concrete and formal thought is to understand how
the subject makes the transition from simple observed correspond-
ences between the weights and the displacements of the liquid to
an explanation expressing the complete transmission of force as a
function of weight and inversely as a function of density. Thus
it is not only the general operational schema for equilibrium which
reappears here but, more particularly, the equality of action and
reaction. The subject matter is especially promising for the study
of the role played by the I N R C group in thinking.
Stage I. Lack of Understanding of the Role of Weight
(Substage I-A) Followed by Global Understanding
Without Either Serial Ordering or Operational Cor-
respondences
At substage I-A, the subject does not even make an unequivocal
prediction that the water will rise in the thin tube (B) as a result
of the weight of the piston because, if the "heavy one" has force
^eaned on 7 * or "pushed" by assimilation to his own action), the
Tight one" is likely to "rise" (by assimilation to raising itself).
Moreover, no conception of conservation of quantities is present
at this level. Thus the water in the tubes will not necessarily be
conserved and the tubes may be filled or emptied without ade-
quate reason.
KOT (5 ; 6): "The water will go up [he points out tube B]; it has to fitt
tip to go up "-"If I use this piston?"-"!*'* going to fitt up ."-"And if I
put on this box?" [500 grams] "It could fill up more [this is done].
If 3 because it [the box] presses "-"If I take off tie box?* '"ft witt go
down." ee WheTQ? >9 t< Like before." He puts on the s-kflogram box.
"A little higher." He replaces the box by the i,5oo-gram box. "A little
EQUILIBRIUM IN THE HYDRAULIC PRESS 151
lower. It's the heavy ones that go the lowest [in B!]. The little ones
cant lift anything high. 2 The big one and the fat ones can bring it up
close [small displacement]. The heavy ones cant, but the little ones go
high up" [contradiction]. 'Take the little ones."[He tests their
weight.] "This one must be the smallest [1,500 grams; he puts it on],
It went up a little anyway! I think the lightest is this one [500 grams]
and I think that this one is the heaviest [2 kilograms; the two extremes
are evaluated accurately, but the rest of the serial ordering is wrong:
1,000 > 1,500 grams, etc.]. Oh! This one [2 kilograms] is the smallest
[he puts them all on in turn]. This one goes the lowest" [500 grams],
-"Why?' '-"It's the biggest [he puts on 2 kilograms]. Oh! But that one
that goes up highest: you see that it's the smallest "'Why the small-
est?' '-"It's the lightest, and also the smalkst."-'It's the lightest?"-"!
feel it when I carry it [he lifts them in turn without comparing them
two by two], Yes" "Why does the water go up most with that one?"
"Because it presses, it makes the water go up" "It you put on a
small one?** "That makes the water go up very high." 'And a light
one?" "It's like the fish in the water: they can raise themselves [he
puts 1,500 and 1,500 grams together]. It goes up high" 'Why?'*
"Because that [he points out one of the two i,5oo-gram boxes] is the
heaviest" [!].
JOG (6 years): "It's the water that went up again!" "Why?" 'Because
the tube [piston] was put on top; that made the water go up." ""And
if you put this box on top, will it change?" "It will go up more" [he
points out a level that is too low]. We do the experiment and point
out the predicted level. "Why not there?" "It was too heavy" [thus
the weight is seen as the cause of the rise]. "Next?" He uses 1,500
grams and 1,000 grams and sees that the first results in a greater rise
in the water level. 'Why?*'-- "Because it's less heavy than the yellow"
[this is not true; moreover he now sees the lack of, rather than presence
of, weight as causing the rise!].
These cases were worth citing because they show the initial
incoherence of the reasoning which later reaches the level of for-
mal logic. In the first place, we see that these subjects can neither
serially order the weights correctly nor allow for the equality of
size of the boxes which contain them (although they are all exactly
alike). Secondly and in connection with this preoperational per-
spectivethe heavy weight may cause the water to rise more or
2 The boxes (from 500 to 2,000 grains) all have the same volume.
252 THE FOBMAL OPERATIONAL SCHEMATA
less and the light one may in turn fail to press on the water or,
on the other hand, cause it to rise. It may even happen that two
superposed weights cause the water to rise quite high because
one of the two is erroneously judged lighter than the other; thus,
nonadditivity is an additional source of contradictions. In sum,
the child cannot possibly formulate any law yet, for he lacks
operational coherence.
At substage I-B, the child understands in general terms that the
heavier the weight, the more the water rises in tube B, but he is
not yet using serial ordering, correspondence, or exact predictions,
for completed operations are still lacking:
JAC (6 years), after several trials, predicts that the 1,500-gram box will
produce the same result as the 2-kilogram box "because it's just as
heavy." The water does not rise as high. Next the experiment is done
over again with the 2-kilogram box. He predicts once again that the
level will be the same "because they are the same." The water rises
higher. "Why?"~-' u Because it's heavier." "And that one [500 grams]
will go where?" "I don't know." 'To the same place?" "No, because
it is less heavy." "And that one [1,500 grams]; can you guess where it
will make the water go?" "No" [however he weighs it]. "Why does
the water come up here?" [B]. "Because there is something that
touches the water, the pipe [piston] presses on it. It presses more or
less and it always makes it go up."
MON (6 years): "Because there is more weight and that makes it go
higher up. The piston sinks in more [in A] and there will be more
water there" [inB],
DEL (7 years) gives the same explanations but also tries to recognize
the weights by the sizes. But the goo-gram box seems to him larger
than the i,5oo-gram box, whence spring several contradictions:
"heavy" goes with "big"; the "heavy one" makes the water rise more,
but the 'less big/' too.
Thus there is progress in comparison to substage I-A in the
sense that as a general rule the weight becomes the cause of the
rise of the water. However, since the weights are badly ordered
and the equalizations badly established, a certain amount of in-
determinacy often renders the predictions inaccurate.
EQUILIBRIUM IN THE HYDRAULIC PRESS 153
Substage II-A. Exact Serial Ordering of the Weights
and Correct Correspondence with the Water Levels
With the appearance of concrete operations, the weights and the
water levels are made to correspond; the weights being found to
be either equal to or different from 500 grams, their serial ordering
does not cause any more difficulty (which is not the case before
substage II-B when the differences are less perceptible). But
progress stops at this point. For, as regards the density of the
liquid, up until 9 years (on the average) the child thinks that the
heavier the liquid, the higher it will rise, because its weight is
added to that of the piston which makes the water rise. Thus, at
this first operational level there is no notion of liquid resistance
i.e., of a reaction oriented in the opposite direction from the pres-
sure action thus no notion of reciprocity in the realm of forces:
SOL (7 ; 11): "If the piston sinks in further, the water goes up more. 9 *
The 2-kilogram box will cause the water to rise more than the one of
1,500 grams "because it's loaded.' 9 "What does that depend on?"
"On the weight that's inside "-"If it is heavier?"-"!* goes up higher."
"And lighter?" "Lower. If you had a very big weight, it would go
[gesture], it would gush out from up above"
COR (7 ; n) same reactions with water, after which we substitute alco-
hol: ''Why doesn't this box go up as high now?"-'The box got lighter. 9 '
-"Why?"-"It'$ not the same liquid"-*WhB.t difference does that
make?" "Maybe the liquid goes up higher. Ifs because the liquid is
heavier, it has more weight" "This [alcohol] is heavier?'* "Yes, it's
heavier [since] it goes up higher; that makes the weight and then it
makes it go up."
PAL (8 years). Experiment with glycerine: "The water is heavier, so
everything will go a little higher because it's heavy." 'Why doesn't
it go as high with the glycerine?" "Because the glycerine is lighter."
"But weigh it yourself* [two equal volumes]. "I was wrong, it's the
opposite' 9 "So why does it rise when it's heavier?" "Because it doesn't
have enough force to rise higher." Thus there is a contradiction; weight
involves force, but the glycerine does not come up to the level.
154 THE FORMAL OPERATIONAL SCHEMATA
These facts are interesting from two points of view: the child's
physics and his logical operations. From the first point of view, he
does not perceive action and reaction, but thinks that the force of
the pressure exerted by the piston and the boxes releases in the
liquid a force due to their own weight and oriented in the same
direction. We have here a new example of the influence, which is
general at this age, of the two motors schema. 3 The external motor
sets off the action of the internal motor, and the two work together
for the execution of the same movement. (This Aristotelian schema
is frequently used in the child's explanations, even for the move-
ment of projectiles.) These same subjects do have a notion of
action and reaction in the case of balance, but it is intuitive (for
it is simply linked with visible displacements) and lacks opera-
tional generality. In the present situation, they have none, al-
though they may compare the increasing weights of alcohol,
water, and glycerine at equal volumes and see the inverse corre-
spondence between the weights and the rise of the liquid in
tube B (with the same boxes placed on the piston of tube A).
The reason for this is that the principle of action and reaction
cannot be understood in terms of concrete operations alone. Class
groupings presuppose inversion and relational groupings presup-
pose reciprocity, but neither taken by itself provides any mecha-
nism for the integration of the two in a single operational system
such as the I N R C group. But the principle of action and reac-
tion is based on this group: it entails the intervention in the sub-
jects* reasoning of logical transformations which include both
reciprocal and inverse operations and coordinations between
them. Inverse operations alone may assure the coordination of
operations in the same direction (e.g., understanding that taking
off weight diminishes the pressure), but reciprocal transforma-
tions are required for equating operations which are oriented in
opposite directions (e.g., for understanding that a greater liquid
density compensates rather than adds -force to the weight on the
piston so that the more dense the liquid, the less the rise of the
water in tube B). The integrated group by definition implies
formal operations or a "structured whole" as opposed to elemen-
tary groupings of classes and relations.
Thus, because he lacks formal operations, the subject at this
8 See J. Piaget, The Chtttfs Conception of Physical Causality, Chaps. I-V.
EQUILIBRIUM IN THE HYDRAULIC PRESS 155
level does not come to understand the relations between action
and reaction, just as in the case of communicating vessels he gets
as far as a spatial reciprocity of rises and falls without discovering
the reciprocal relationship of operative forces themselves. It is
clear that, at this substage (II-A), equilibrium between liquid and
a piston of variable weight pressing on it is not conceived of as an
equilibrium process but as a one-way process; the piston exerts its
force on the liquid in the departure tube (A), and the liquid acts
not in return but (even in tube B) in the direction of the piston
itself and adds its own force (because of its weight) to that of the
piston. But the transmission of force does not raise any problem
because, strictly speaking, it is not a transmission. It still consists
of releasing or activating the force of the liquid with the force of
piston and weight placed on it. Thus the serial ordering of weights
and the correspondence between them and the attained levels are
still a long way from being an expression of the law or even
a fortiori an understanding of it.
Substage II-B. Intuitive Formulation of the
Notion of Resistance
The stage from 9 to 11 years is a transition stage, during which
the subject begins to get a glimpse of the fact that the liquid
resists as a function of its density. Some children say even at this
point that in tube B the water rises less than the alcohol because
the alcohol is lighter. But that does not yet mean that they have
mastered the problem from the point of view of action and re-
action. In particular, they wait for the piston to fall to the bottom
of large tube A 5 as if it fell to the bottom of any receptacle what-
ever and as if counter pressures were irrelevancies. Thus, the
column of water in narrow tube B is always conceived of not as
exerting a reaction in a direction opposite to that of the piston
pressure and coming into equilibrium with it but only as resisting
the rise somewhat as a function of the liquid's weight.
HED (10 ; 3) predicts that "it's going to rise here [B], and here [A] it
witt go doww.~--Experiment.-~ "Oh! I thought it would go up higher."
-"Why?" "Because the piston didnt go aU the way [as he expected]
and the water didn't go all the way up. 9 ' "And with that box?" [i kilo-
THE FORMAL, OPERATIONAL, SCHEMATA
gram] .-"Way up there" [higher]. -"And if we take it off ?"-"! will go
down again."-' And with that box?" [2 kilograms ].-"!* will go higher
because it's heavier." The water is replaced by alcohol; he expects the
same levels: "It teas higher "-'Why? 9 -' 'Because it [the alcohol] is
lighter"->"And with that box?"~'There" [a little lower than with
water],
FRA (10 ; 10): "The tube [piston] is going to fall and the water will
overflow because when you put something heavy in a container full of
water, then there is more volume and that makes it overflow."
HAF (11 years), contradicting HID, attributes to the lightness of the
water the fact that it comes to rest lower than the alcohol: "It's not
the same liquid now"- <e Why?"- <e Maybe it weighs less" [the water;
we began with alcohol].- c Why?"-"But it doesn't go as high. . . . It's
the liquid that weighs less. It's surely the liquid since the boxes are
the same. The first time [alcohol] the liquid was heavier, since it went
higher. When the liquid is heavy, it has more weight, more pressure: it
goes down faster here" [in A]. "And here [in B] the liquid doesn't
press?" "No, since it's this one [in A] that goes there" [in B].
This reasoning is extremely clear and less contradictory than
it seems, for sometimes the subject thinks of the liquid which
drops in tube A, sometimes of the liquid that rises in tube B.
When considering tube A, he expects the heaviest liquid to fall
the most easily, with the aid of the piston, and consequently the
level to be highest in B (HAF). On the other hand, in considering
tube B, he thinks that the heaviest liquid rises with the most diffi-
culty and consequently that the level will be lower (HID). In both
cases, it is less a question of resistance offered by the liquid in the
sense of an equal and opposite reaction in relation to the piston
than a variable resistance to the rise with a variable facility for
the descent; whence the idea that the piston "sinks in'* to the
bottom of tube A without any resistance, as HID and FRA expect.
Thus, at the last of the stages prior to formal operations, the
subject can make accurate predictions of the effect of the weights
on the piston and sometimes even of that of the densities of the
liquids used, but he does not as yet formulate (from his predic-
tions) any total explanation in terms of an equilibrium principle.
The reason is that he lacks the operational instruments which
would permit coordination of inversions and reciprocities (the
EQUILIBRIUM IN THE HYDRAULIC PRESS 157
I N R C group). But there is no doubt of the fact that weight is
still conceived of in absolute terms (as a "pressure" says HAF) and
is not adequately related to volume although its conservation and
rudimentary quantification are present. Even at this point, density
gives rise to the intuition of "filled" or full but not as yet to the
operational relations which will constitute the notion of compres-
sion-density found at stage III.
Stage III. Reciprocity of Action and Reaction
The best index of the appearance of the notion of reaction, or of
resistance oriented in the opposite direction from that of the pres-
sure of the weight, is the subject's attitude toward the drop of the
piston in tube A. Whereas up to this point the piston was seen as
sinking into a liquid without resistance and even as tending to
descend with it as a function of its own weight, from this point
on the descending piston is seen as meeting resistance propor-
tional to the density of the liquid. Density, in turn, comes to be
conceived of as a relationship between weight and volume i.e.,
as the result of a more or less great compression of matter into
an equal volume. But to conceive of a resisting force, capable of
equilibrating the force of the pressure according to a set of varied
compensations until it stops the piston in its descent, the subject
must introduce a reciprocity between the density (conceived of
as a compression capable of resistance) and the pressure of the
weights. In other words, the subject joins a reciprocal transforma-
tion in a single I N R C group with the inverse transformations
(of adding or subtracting weights). This integration is made
initially at substage III-A:
TRI (11 ; 2): "It's the weight of the boxes. It's that [piston] that pushes
the water .** "And if you change the liquid, will that have an effect?"
"Yes. Some liquids are heavier than others'' "If you use alcohol?"
"I think that alcohol is heavier [simple factual error]. So it wiU rise less
because it's harder to move" [resistance!]. He does the experiment
without comment on our part. "No, the alcohol is lighter' 9 "Why do
you think that now?" "Because the weight of the box makes it rise
more." "What does that?" "Because the weight of the box makes it
go up higher. The weight [of the box] can push better"
THE FORMAL OPERATIONAL SCHEMATA
DUM (11 ; 2). After the water, we perform the experiment with alco-
hol: "It will go tip less because the weight of the liquid is heavier."
"The alcohol is lighter than the water. So?"-"ft will go up higher."
"Why?"-"17i0 lighter the liquid, the more the piston will be
less . . ." [less resisted]. -"Less what?"- 'Since the alcohol is lighter
than the water, the piston will descend more" "You said less* ""No,
the piston descends more. The liquid is lighter, so the piston sinks in
more" "Why?" "Because the liquid is lighter than the water." 'And
it follows that the piston sinks in more?" 'Because the piston isn't
held back as much by the weight of the alcohol"
YA (11 ; 6). Same reactions. "Why doesn't the piston go all the way
to the bottom?" "Because the piston no longer has enough force to
bear down. It is held back because the liquid is heavier than the
piston"
RTV (13 ; o): The water goes up to there "because it has to come back
to the same weight in both tubes" *But why doesn't the water rise
any higher?' "Because the piston can't come down any more."
"Why?"-- <e Because the water holds it back." Having pressed down on
the piston with his hand, he sees that it returns to the initial position
and says, "If you press on the piston, the water has more force"
These responses show clearly that from this point on the subject
is aware of the existence of action and reaction. The weight of the
liquid or, more exactly, its density (for henceforth the subjects
speak only of weight relative to volume) is no longer a factor pro-
moting the pressure of the piston but, to the contrary, is an ob-
stacle to this pressure and thus a factor whose action is oriented
in the opposite direction; thus it is a reaction. The liquid, when
it is lighter than the water, actually rises more in B, because the
piston, reinforced by the weights, "can push better ' (TRI) or is
less held back" (DUM). With respect to tube A, on the other
hand, the piston "can't go down" below a certain point "because
the water holds it back" (RIV and YA).
Thus, the discovery characteristic of substage III-A is that the
system involved is an equilibrium of opposed forces and no longer
a one-way process. But before trying to formulate the reasoning
involved, let us examine the further problems which the subjects
have yet to solvethe way in which the force of the piston is
transmitted and the place in which the action and reaction attain
equality and come into equilibrium.
EQUILIBRIUM IN THE HYDKAULIC PRESS 259
The first question is solved from substage III-A on, since the
pressure of the piston is no longer considered a triggering device
or an excitation of the force intrinsic to the liquid (due to its
weight). Rather, it is viewed as an action exerted through the
entire liquid from the beginning (descent of the liquid in tube A)
up to the arrival (rise and coining to rest in tube B) and bringing
on a reaction in the opposite direction resulting in a final state of
equilibrium.
But at what point does equilibrium come about? Can it be
localized at a particular point, or is the total quantity of the liquid
involved by degrees? This particular question cannot be clarified
prior to substage III-B. The reactions of this latter substage are
roughly analogous to those of the preceding. As regards this ques-
tion, several substage III-B subjects still imagine that there is a
particular place where the opposing forces meet each other. This
would be at the bottom of the rubber tube connecting the two
glass tubes i.e. 9 the lowest point of curvature of the system. In
contrast, other subjects come to understand that, from the point
of contact between the piston and the liquid up to the level
reached by the liquid in tube B, there is action and reaction. On
the one hand, the pressure of the piston makes itself felt through-
out the liquid. On the other hand, since the reaction is a function
of the density (which is conceived of as a compression), it surges
at every point of the volume occupied by the liquid in such a way
that throughout there are both action and resistance, the latter
tending to repulse the action exerted on the liquid; the action and
reaction are thus equal at every point.
The following are examples of each of these response types:
BOI (14 ; 6): "If you put on that box?" "It wiU rise higher because of
the pressure" [experiment] . "Why not higher?*" "The pressures at the
bottom of the tube [of the rubber tube] are equal" "Row do you
know?" "Because the apparatus [piston] doesn't fall and doesn't rise,
and, reciprocally, because the water neither rises nor faZfc/* "And
when I put on a box?" *7t*$ heavier. The result is a higher column of
water [in B] and also a larger weight [in A], and it comes into equi-
librium at the bottom of the tube"*Why?'''-"Because the pressure is
the same at the bottom of the tube* "But that doesn't explain the com-
bination of the weights?" "Yes, it does, because the water is dislodged
by the weight [in A]: It comes into equilibrium at a certain moment be-
THE FOBMAL OPERATIONAL, SCHEMATA
cause the weight of the water [of the column of water in B] increases
when it rises. 9 ' "What makes the water rise?*' "The weight of the
boxes; it makes a greater pressure at the bottom of the tube and that
dislodges the water."
IAC (14 ; 6): "The pressure is the same [from both sides] at the bottom
of the tube [cf. BOX'S conception]. No, the water comes into equilib-
rium if it communicates by a tube and the pressure is transmitted in
full." "Does the elevation play a role?" "No, the water will transmit
the pressure the same way if both columns are high or low' 9 The re-
sistance is also conceived as the same throughout.
Thus we see that the substage III-B responses (in which the
influence of acquired knowledge may occasionally be perceived)
add little to those of substage III-A, which are more spontaneous.
Conclusion. Stage III Reasoning and the Formal
Operational Schema for Equilibrium
In order to analyze the formal operations needed to understand
the notions of equilibrium or equality between action and re-
action, one must first remember that causality is a system of opera-
tions applied to transformations in the real world in such a way
that each one of these transformations can be assimilated to an
operation of the subject while at the same time is conceived as
accomplished by the objects themselves. Thus, we must first estab-
lish the transformations which our subjects attribute to the system
under consideration and then look for the operations or logical
transformations to which these real modifications are assimilated.
But the principal transformations involved in the system are the
four following:
I. The action exerted by the pressure of the weight of the piston
and by the weights added to its own weight;
II. The suppression or diminution of this action by eliminating
the weights added or the weight of the piston itself;
III. The reaction due to the resistance of the liquid, which it-
self constitutes a pressure, one which, however, is oriented in the
opposite direction and which is dependent on the height and
density of the liquid;
EQUILIBRIUM IN THE HYDRAULIC PRESS 161
IV. The elimination or diminution of this resistance by elimi-
nating part of the liquid or by substituting a liquid of less density.
But we notice that although transformations (I) and (II) can be
assimilated to direct and inverse operations, transformations (III)
and (IV), in contrast, are symmetrical with the first two; they too
consist of operations which are direct and inverse in relation to
each other, but which act in the opposite direction from the first
two. Transformation (III) thus constitutes a reciprocal of transfor-
mation (I) and transformation (IV) a reciprocal of transformation
(II). Whereas (IV) is the inverse of (III), in contrast (III) is not
the inverse of (I), since it does not cancel it but simply neutralizes
the effect by compensation. In other words, the composition of
(I) and of (III) results not in the cancellation of the pressures but
in their equivalence, and thus in an equilibrium.
Consequently, here again we find a mechanism isomorphic to
the group of four transformations I N R C. Therefore we should
not be astonished if we find this same I N R C structure in the
very reasoning of the child in a form analogous to that which we
have already seen at work in the experiment of the communi-
cating vessels, although here we find it in slightly different form
since in this case the reasoning bears directly on pressures and
resistances:
I. The first operation consists of stating the intervention of a
pressure in tube A under the influence of one weight or the other:
let this be (p v q);
II. The inverse operation consists of stating the cancellation
of this action: let this be (p.q);
III. According to the stage III subjects, each pressure p or q
has a corresponding resistance which we may designate by
p' or q'i.e., (p'vg') and which is expressed in the column of
liquid B by the weight of the portion going beyond the level of
the liquid in column A;
IV. The inverse of III will consist in stating the cancellation
of p' and q'i.e, 3 (p'.q').
The discovery unique to stage III is that transformations (II)
and (III) are reciprocal i.e., that there is compensation between
them. But to hold that compensation occurs is tantamount to
regarding the intervention of a resistance in B (expressed by p'}
as equivalent to the elimination of a pressure in A. The adolescent
162 THE FORMAL OPERATIONAL SCHEMATA
realizes that, without the resistances in B, the pressures exerted in
A would cause the liquid to rise much higher and that to each
pressure p corresponds an equal resistance p'. Thus (p' v q') can
be written in the form (p v q). Whence:
l(pvq)
Thus one can see that transformation (III) is clearly the recipro-
cal R of transformation (I) and that (IV) is the correlative C
i.e. 9 the NR or the RN of transformation (I). Thus, the equivalence
between pressures and resistances is expressed in two ways (posi-
tive R and negative C). Since the reciprocal (p v q) of the opera-
tion (p v q) is the symmetrical operation in which the same com-
binations (p.q v p.q v p.q) and (p.q v p.q v p.q) are found, though
with a change of signs, it expresses the fact that equivalent forces
oriented in opposite directions are involved, and thus the opera-
tion is distinguished from the inverse N. In other words, the in-
verse cancels the direct operation while the reciprocal does not
cancel it; instead, it compensates it by a symmetrical operation of
the same value but with a change of signs.
This distinction between the reciprocal and the inverse causes
all the difficulty in the problem of action and reaction, and the
subject cannot make the distinction before the formal stage. Until
then, although the child clearly understands the inverse operation
(i.e., can cause the water in B to drop again by cutting down the
weight in A) he does not understand the reciprocal operation. He
conceives of it as a simple prolongation of the direct operation and
not as a symmetrical operation oriented in the opposite direction
and compensating for the direct operation ( = the liquid does not
resist the pressure but acts in the same direction). This is so be-
cause, if the subject is to understand the four transformations
(I-IV) when he has not intuitively distinguished between their
respective actions (as in the case of the balance), he must possess
an operational mechanism made up only of formal operations.
Thus, the late appearance of this discovery.
As for fixing the equilibrium at a particular point (stated in the
proposition r, subject RIV describes its conditions when he says
EQUILJEBTCTM: IN THE HYDRAULIC PRESS 163
that the water in B stops at a fixed level "because it has to come
back to the same weight in both tubes/* In other words:
ND[x(pv 9 )=y(pv9)] (a)
where x and y are the values assigned to (p v q) and to (p v q}.
Unlike those in communicating vessels, the equilibrium points
do not form a horizontal line from one tube to the other but vary
according to weights p v q and to the resistances, which them-
selves depend on the density of the liquids. With that exception,
the explanation of the phenomenon presupposes the same formal
schema. Thus, it is not by chance that the mechanisms of action
and reaction are discovered at the same substages, III-A and III-B.
We have seen this to be true with respect both to a piston acting
on a liquid and to communicating vessels. And we shall soon see
that it holds for the case of the balance scale. In these three cases,
an understanding of the physical processes presupposes an opera-
tional schema putting into effect simultaneously the inverse and
reciprocal transformations which remain separated at the level
of concrete groupings and which are linked into a whole only
through the I N R C group. But in this, as in all the other cases
examined up to this point, the difference between concrete and
formal thinking relates to the construction of the "structured
whole/' It is the double reversibility characteristic of this whole
(which at the same time allows for the lattice and group struc-
tures) -which constitutes the I N R C group, whereas the two forms
of reversibility remain separated in the elementary concrete
groupings inversion is found only in class groupings and reci-
procity in relational groupings.
11
Equilibrium in the Balance 1
IN A PROBLEM using a simple balance-type weighing instrument, a
seesaw balance, we again find the operational schema of equilib-
rium between action and reaction. But the experiment was set up
in a way that would force the question of proportionality. When
two unequal weights W and W are balanced at unequal distances
from an axis L and I/, the amounts of work WH and WH' needed
to move them to heights H and H' corresponding to these dis-
tances are equal. Thus, we have the double (inverse) proportion:
W/ W = U/L = H'/H
The result is that finding the law presupposes the construction
of the proportion W/W' = U/L and spelling out its explanation
implies an understanding of the proportion W/W = H'/H. It
seemed to us that it would be interesting to study how this pro-
portionality schema develops as it is linked with the equilibrium
schema. As a result of previous research we know that in all realms
(space, speed, chance, etc.) the notion of proportions does not
appear until formal substage III-A. Now we are going to find out
why this is so.
1 With the collaboration of F. Matthieu, former research assistant, Institut
des Sciences de Tfiducation, and J. Nicolas. In reference to the same subject,
see the previous study of Mme. Refia Ugurel-Semin, Istanbul University
Yayinlari (1940), no, pp. 77211.
164
s\
10
III
20|
o o o o " 3
,r. , .-*.* C a,,)
A
B
FIG. 9. The balance scale is here shown in two forms: (A) a conven-
tional balance with varying weights which can be hung at different
points along the crossbar; (B) a balance equipped with baskets which
can be moved along the crossbar to different points and in which dolls
are used as weights.
166 THE FORMAL OPERATIONAL SCHEMATA
Stage L Failure to Distinguish Between the Subject's
Action and the External Process (I-A) Followed by
Integration of Intuitions in the Direction of the Com-
pensation of Weights (I-B)
From about 3 to 5 years, subjects give responses which, given our
interests, are instructive. As we have said before, in general
causality is an assimilation of external processes either to the sub-
ject's own actions or to his operations, but with the delegation of
one or the other to reality itself. In the case of an apparatus such
as a balance scale, the notion of an equilibrium between one's
body weight and other weights is constructed very early, but the
notion is undifferentiated and extends beyond weight itself to
include the muscular force of an upward or even a downward
push. (Moreover, the weight is thought to be linked with the
actions of pushing up or pressing down.) The balance is first
assimilated to this sort of undifferentiated action and not to a sys-
tem of compensation operations between weights nor a fortiori to
weight X length. In fact, no form of concrete operations exists at
this level for there are only representational regulations i.e., in-
struments of global compensation without systematic reversibility.
The result of this situation is that the substage I-A subjects can-
not guarantee equilibrium simply by distributing weights but
intrude in the working of the apparatus with their own actions,
which they fail to distinguish from the actions of the objects that
they are trying to control.
MIC (4 ; 6), presented with two equal weights at distances of 14 and 9:
**Why is one way down and the other up high?" He continually raises
and lowers the arms of the apparatus, believing that they will maintain
the forces and positions he delegates to them. ''Can you make it
straight [horizontal gesture] so it will stay there all by itself ?" Neither
"yes" or "no."-"How was it before?"-"Ltfe? thaf [horizontal]-"You
can't do it with the weights?" He shakes his head and tries to main-
tain the horizontal position with two unequal weights, raising and
lowering the arms several times. "Can you do it without your hand?"
[We have him weigh the weights with his hands, then he works at a
new set of trials. We suggest that he add weight to one side or the
other, etc,] Conclusion: "You caritr [attain the horizontal position].
EQOTLJBRIUM IN THE BALANCE 167
MAR (4 ; 8) suspends two weights on one side without putting any-
thing on the other, with the aim of reaching horizontal equilibrium!
One can see that, in constantly interfering with the apparatus
in order to correct the position of the balance arm, the subject
expects the apparatus to conserve the results of his manipula-
tions. Thus, the instrument and his own actions are not distin-
guished. But, although barring the notion that the balance con-
stitutes an independent equilibrium, the lack of differentiation
does not preclude his making predictions about some more or less
constant effects. It is true that for our purposes the most striking
aspect of these predictions is their negative aspect. For example,
at this level the child does not yet think that equilibrium implies
the equality of weights (even at equal distances); thus MAE puts
two weights on one side and none on the other in order to attain
the horizontal. The heavy side moves upward and the light one
downward, as well as the reverse. The relationship between
weights is not formulated; the epithet "too heavy" may be applied
to a single weight suspended to one arm without its counterpart,
"too light," being used. There is no conservation of weight. The
subject tries constantly to repeat with new weights what he has
just accomplished by chance with others, without paying any
attention to differences in weight. However, through improved
regulation these subjects come to see that weight has a relative
influence. Generally they suspend at least one weight at each side
for purposes of symmetry. Often they add new weights to the
others to improve the equilibrium, but they add them not to
the side where weight is lacking (which would tend to equalize the
weights) but to the side where the weight already is largest with
the idea that several weights will improve the situation.
But the adding characteristic of this level is not yet operational.
Although it does constitute the beginning of the additive opera-
tion, the operation is not achieved because of lack of equalization
between parts (A + A') and the whole B (compensating A + A'
on the other arm). Most important, it is not an operation, because
reversibility is lacking; at this point elements are not removed
with the deliberate aim of equalizing the weights. When the sub-
ject removes a weight, it is only in order to try a new and different
course of action after earlier attempts have failed.
IQS THE FORMAL OPERATIONAL SCHEMATA
Generally, the subject is not concerned with the question of
the distances from the axis and does not look for any equality or
coordination between the distances and the weight. Nevertheless,
operations may start to take form here in that the subject may
establish a preliminary form of symmetry. However, once again
this is not an operation in the strict sense. First, coordination with
the weight is lacking; secondly, this symmetry is related generally
to the two extremities of the arms and does not include equaliza-
tions for the intermediate distances.
In contrast, from about 5 to 7-8 years (substage I-B) one can
see increasing integration of these intuitive representations mov-
ing toward reversible operations.
MAL (5 ; 8) notes that the arms are not horizontal: "You have to put
another [weight] on the other side. I know what has to be done; put
still another one there because there isn't any weight here [she adds
it]. These here must be lighter than those over there. You have to take
two that have the same weight." Next: "You could take one off' [be-
cause it is too heavy on one side]. MAL does not spontaneously dis-
cover the influence of distance, but when a weight is moved in front
of her she says: "You brought that one up closer 9 that makes more
weight. If it were at the end 9 it wouldn't work and there it makes
more weight''
GAS (5 ; 9): "You could put one at the other side: the same [he takes
a weight of the same color but having a very different weight]. That
doesn't work: maybe there is a little too much weight there."
Thus, from this point on the child understands that weight is
needed on both sides to achieve a balance and even that the
weights should be approximately equal. But he does not yet know
how to proceed toward this equalization in a systematic way.
Similarly, henceforth he succeeds in adding and subtracting, but
without accurate equalizations. His actions are successive correc-
tions, (thus regulations) and are not yet strictly reversible.
We see how these two sorts of regulations by equalizations
and by addition or subtraction furnish the starting point for
future transformations by reciprocity (symmetries) and by inver-
sion, relative to the weight. As for the distances, there is progress
in the tendency toward symmetry (the weights are no longer put
at equal distances only at the extremities but also in the region
EQUILIBRIUM IN THE BALANCE 169
close to the axis). Sometimes the subject discovers the role played
by changes in distance (cf. MAL). But there are as yet no sys-
tematic correspondences of the type further = heavier.
Substage II-A. Concrete Operations Performed on
Weight and Distance but Without Systematic
Coordination Between Them
From this point on, weights are equalized and added exactly,
while distances are added and made symmetrical. But coordina-
tion between weights and distances as yet goes no further than
intuitive regulations. The subject discovers by trial-and-error that
equilibrium between a smaller weight at a greater distance and
a greater weight at a smaller distance is possible, but he does not
yet draw out general correspondences: 2
(7 ; 7) begins with E 3 and D 3, then replaces them with G 3 and
F 3 (thus equal distance and an attempt to find equal weights), adds
two other weights, takes off some, then all, and finally weighs two
equal weights [E] in his hands, counts equal numbers of holes [14]
and places E 14 at each side. Afterwards he looks for other forms of
equilibrium; he adds the weights, moves them, takes off some, and
finally has GED on one side and P 3 on the other: "That's it [em-
pirical compensation of weight and distances]. It's just like when
there weren't any [when the arms were horizontal without weight];
it's the same weight on each side." He begins again with large weights
[for which there are no matched pairs], "I should have put one on
each side. Since there aren't any, I had to put three on one side and
two on the other. It stays straight because it's the same weight on each
side." He predicts that equal distances are necessary for two unequal
weights, but he does not find the law: Heavier ? nearer. "If you put
on C and E, where would you have to place them?""! would say one
hole and another hole [ = two different distances], but they shouldn't
go the same way [at equal distances] or it wouldn't make the same
weight."
2 From now on we will indicate objects of increasing weight by the letters
A, B, C, etc. Increasing distances (which are measured for the child at three
equidistant points where the hooks for the weights are attached) are indicated
by the numbers i, 2, 3, etc.
170 THE FORMAL OPERATIONAL SCHEMATA
NEM (7 ; 4) discovers empirically that C on the left at a distance of 10
balances E on the right at a distance of 5. We ask him to place C on
the right and E on the left, but he does not succeed in inverting the
distance relationship. After the experiment, he exclaims, "AW Yow
have to do the same thing as before but in the opposite way!"
Thus, from this point on the subject can order serially the
weights he comes across as well as determine whether they are
equal. He can add them in a reversible manner and correctly
compare one pair of weights with another pair. What is more, he
knows how to make use of the transitiveness of the relations of
the equality or inequality of the weights. Moreover, all these
operations reappear when he compares distances, but with the
additional correspondence between distances oriented in opposite
directions (symmetries relative to the axis).
Applied to the problems of the balance, these operations allow
subjects to obtain the following results (by logical multiplication
of relations):
Two equal weights BI and B$ situated at equal distances L a
come into equilibrium by symmetry:
(B! X L.) = (B, X L,). (i)
Thus, one of the weights is conceived of as compensating the
other by reciprocity. Two equal weights BI and Ba at unequal dis-
tances Ly and L v do not balance each other:
(Bi X L.) (B 2 X Ly) if x 3* y . (a)
Two equal weights BI and B 2 at unequal distances L x and L y
do not come into equilibrium either:
(Ax X L.) < (B 2 X L a ). (3)
Moreover, in each one of these relations the subject can sub-
stitute for one object an equivalent set of others through additive
operations:
Ci^Aa + A'a + B'a) (4)
and the same holds for distances.
On the other hand, in the case of unequal weights AI and Ba
and of unequal distances L^ and L y , coordination is not yet pos-
sible at substage II-A. Even when the subject discovers by expert-
EQUILIBRIUM IN THE BALANCE 171
mentation that a large weight at a small distance to the right of
the axis balances a small weight at a large distance to the left, he
does not know how to invert these relations from one side to the
other and discovers too late that he should have "done the same
thing but in the opposite way" (NEM).
Substage II-B. Inverse Correspondence of
Weights and Distances
The example just described (unequal weights and distances) is
resolved at substage II-B, not yet by metric proportions (with
the occasional exception of the relationship between i and 2) but
by qualitative correspondences bordering on the equilibrium
law: "The heavier it is, the closer to the middle."
FIS (10 ; 7) sees that P does not balance F "because it's heavy: that
one [F] is too Kg/rt."-"What should be doneP"-"Mooe it forward
[he moves P toward the axis and attains equilibrium], I had to pull it
back from 16 holes [arbitrary] to see if it would lower twice [arbitrary]
the weight' 9 "What do you mean by that?" "It raises the weight"
"And if you put it back over there?*' [moves P away]. "It would make
the other one go t/p." "And if you put it at the end?" [F]. "It would
go up still more 9 ' [F], etc. Conclusion: When you have two unequal
weights "you move up the heaviest" [toward the median axis]. But ns
does not measure the lengths even for the relations of i to 2,.
ROL (10 ; 10): "You have to change the position of the sack because
at the end it makes more weight.' 9 He moves the lightest away from
the axis: "No, it's heavier" He is presented with G at 2 and A at 14:
they balance "because that one is there [A at 14] and it is less heavy
than the other one."
The difference between these reactions and those of substage
II-A is clear. At the earlier stage, when the subject comes across
two weights which do not come into equilibrium, he works mostly
with substitutions additions or subtractions. In this way he
achieves certain equalizations by displacement, but only excep-
tionally and by groping about (regulations). On the other hand,
at the present stage the subject who comes to two unequal weights
tries to balance them by means of an oriented displacement on the
172 THE FORMAL OPERATIONAL SCHEMATA
hypothesis that the same object "will weigh more" at a greater
distance from the axis and less when brought closer to it. He is
working toward the law, but without metrical proportions and by
simple qualitative correspondences.
Thus, the new operation mediating the determination of the
conditions of equilibrium is a double serial ordering of weights
A < B < C < . . . and distances LI > L 2 > L 3 > . . . but with
bi-univocal inverse correspondences:
A < B < C < . . .
$ t $ (5)
Li > Lo > L 3 > . . .
which can be translated into reciprocities (expressed in the lan-
guage of relational multiplication):
(A X Li) = (B X L a ) = (C X Ls) = . . , etc. (6)
But it is clear that such qualitative operations are inadequate
to establish the law. The logical multiplications of type (6) allow
some inferences but leave certain cases indeterminate:
heavier X same distance = greater force,
less heavy X same distance = less great force,
same weight X further (from the axis) = greater force,
same weight X less far = less great force; but (7)
heavier X further = indeterminate,
less heavy X less far = indeterminate; and
heavier X less far = less heavy X further
(but only under certain metrical conditions).
However, at this level the subject can quantify the weights (he
knows that B = 2.A; etc.) as well as the distances (measurable by
the number of holes). Given these facts, why must we await
formal stage III before the schema of proportions is organized?
We might say (hat it is a matter of book-learning, but, in con-
tradiction, we are able to present some examples (analogous to
those which we have already published elsewhere 3 ) in which the
3 See Piaget and Inhelder, The Child's Conception of Space, Chap. XII;
No. 9; Piaget, Les Notions de mouvement et de vitesse chez Fenfant, Chap.
IX, nos. 2 and 3; Piaget and Inhelder, La Genese de Tid6e de hasard chez
Tenfant, Chap. VI, nos, 5 and 6.
EQUILIBRIUM IN THE BALANCE 173
proportionality schema is organized before any academic knowl-
edge enters. Thus, it is probable that this schema requires, as a
necessary and sufficient condition, a qualitative operational sys-
tem that is both differentiated and unified, analogous to the
I N R C group. This hypothesis is even more plausible when we
consider that in this particular case a set of balanced actions and
reactions is involved similar to that whose understanding we have
analyzed in Chaps. 9 and 10.
Stage III. Discovery and Explanation of the Law
When the experimenter restricts himself to a procedure such as
the foregoing, where the subject is allowed to hang the weights
simultaneously on the two arms of the balance, subjects start to
discover the law at substage III-A. It takes the form of the
proposition W/W L'/L (where W and W are two unequal
weights and L and U the distances at which they are placed);
this law is so immediately obvious that it does not give rise to a
particular causal explanation even during substage III-B. ('It's a
system of compensations," as CHAL will tell us.) But, when the
experiment proceeds by successive and alternate suspensions of
the weights, the subject's attention turns to the inclinations and
the distances in height to be covered; this may lead him to an
explanation in terms of equal amounts of work (displacement of
forces). It is true that, although this explanation is already pos-
sible at substage III-A, it only rarely appears before substage
III-B. Nevertheless we have observed it in several cases and
think it worth analyzing.
First, we will present a case of the discovery of the law at sub-
stage III-A:
HOG (12 ; 11): for a weight P placed at the very tip of one arm [28
holes], he puts C + E in the middle of the other arm, measures the
distances, and says: "That makes 14 holes. It's half the length. If the
weight [C + E] is halved, that duplicates" [P]. "How do you know
that you have to bring the weight toward the center?" [to increase the
weight]. "The idea just came to me, I wanted to try. If I bring it in
half way, the value of the weight is cut in half. I know, but I can't ex-
plain it. I haven't learned." "Do you know other similar situations?**
J74 THE FORMAL OPERATIONAL SCHEMATA
"In the game of marbles, if five play against four, the last one of the
four has the right to an extra marble" He also discovers that for two
distances of i and 1/4 you have to use weights i and 4; that for two
distances of i and 1/3 you need weights i and 3, etc.: "You put the
heaviest weight on the portion that stands for the lightest weight
[which corresponds to the lightest weight], going from the center."
The rapidity with which the subject makes the transition from
the qualitative correspondence to the metrical proportion seems
at first to indicate the presence of an anticipatory schema. How-
ever, the analogy that the subject established between the bal-
ance and the game of marbles shows that this schema is taken
from notions of reciprocity or of compensation. So we have to
examine how, starting with substage III-B, the subjects proceed
from the same conception to a search for an explanation in the
strict sense of the term (with the apparatus using alternate sus-
pensions):
CHAL (13 ; 6) quickly discovers that "the greater the distance, the
smaller the weight should be. It's staying up." e Why? > ' e It is com-
pensated there and there' 9 'What is compensated?" "The distances
and the weights; it's a system of compensations. Each one rises in turn.
For equal distances you need equal weights, and if it's inclined it
rights itself and goes down on the other side" [We propose a test
with two weight units at a distance of 5 and one at a distance of 10.]
"What will the angles be?" "Larger on one side [he points out the
two-unit side] and smaller on the other [experiment]. Oh! No, the
same angles!" He outlines them: "The distance compensates for the
ttf<3zg7it"-"What distances do they cover?" [heights H and H' are
pointed out]. "The smallest weight covers a greater distance and the
large weight a shorter distance" "And what forces are required?"
[strings which can be used to raise and lower the weights are pointed
out]. "For the smallest, there is more distance to pull, for the large
one, less distance" "So where is more force required?" '"Here [two
units]. Oh! No, its the same: the distance [he is speaking of height]
is compensated by the weight."
SAM (13 ; 8) discovers immediately that the horizontal distance is in-
versely rekted to weight "How do you explain that?" "You need
more force to raise weights placed at the extremes than when it's closer
to the center . . . because it has to cover a greater distance." "How
do you know?" "If one weight on the balance is three times the other,
EQUILIBRIUM IN THE BALANCE 175
you put it a third of the way out because the distance [upward] it
goes is three times less" "But once you referred to the distance [hori-
zontal gesture] and once to the path covered?" "Oh, that depends on
whether you have to calculate it or whether you really understand it.
If you want to calculate, it's best to consider it horizontally; if you
want to understand it, vertically is better. For the light one [at the ex-
tremity] it changes more quickly, for the heavy one less quickly"
TIS (13 ; 8) discovers the proportion i to 2 and shows the heights: "If
I replaced this weight [one unit] with that one [two units], it would
only go halfway up . . . [the distance in height] is much longer when
it is at the end of the arm than when it is in the middle" "Does com-
pensation take place?" "Yes, between the force and the height."
"How can you measure it?" "It* s easier to measure the height, but it's
really the same" [as the horizontal distance].
These reactions, found at both substages of stage III, bring us
back to the now familiar schemata of the I N R C group and in
the same form that we found in Chaps. 9 and 10. But, above all,
they show us how the general equilibrium schema is differentiated
in the present case by constructing the proportions W/W r = L'/L
and W/W' = H'/H. Thus we have two questions to discuss-
first, how is the proportional schema organized; second, how does
it relate to the I N R C group?
In these responses the I N R C group first appears in a form
which we could have described earlier when dealing with the
problem of the oscillations of a liquid in communicating vessels
(Chap. 9). One of the arms of a balance will lower when a weight
is hung on it at a given distance from the axis; when an equal
weight is placed on the other arm in a symmetrical position
( = at the same distance from the axis as the first weight), this
second arm will lower. "One goes up and then the other," says
CHAL, "and if it's inclined [below the horizontal plane] it comes
back to the middle and goes down on the other side."
In other words, a reciprocal relation operates in this case
(p D q) = R(qr Dp) in which p and q stand for the upward motion
of the arms. But there is something new in the case of the bal-
ance: two factors are operative and they compensate each other;
operating alone, a weight W at a distance L produces the same
inclination as a weight W = riW at a distance U == L/n.
J76 TEDS FORMAL OPERATIONAL SCHEMATA
is astonished by this fact ("the same angles"), then finds it quite
natural because "the distance is compensated by the weight"
The I X R C group reappears in the same form as in the prob-
lem of pressure and resistance in the equilibrium of liquids
(prop. [8], Chap. 9 and prop, [i], Chap. 10). Two kinds of opera-
tions for reestablishing equilibrium can correspond to the opera-
tion in which a weight is placed on one of the arms at a given
distance the inverse N which consists of taking off this weight
or the reciprocal R which consists of putting on equal weight at
an equal distance on the other arm of the balance. Moreover,
whereas the inverse N cancels the original operation, the recip-
rocal R compensates it without canceling it; still, N and R have
the same final result i.e., they bring the arms back into the
horizontal plane. It is not at all surprising that the transformations
described in this connection (prop. [8], Chap. 9, and prop, [i],
Chap. 10) reappear in the present context, for they are based on
an extremely simple intuition already acquired at stage II through
qualitative correspondences. But once again there is the addi-
tional fact specific to the balancedistances compensate weights.
Thus in both forms i.e., as it relates to pressures and resist-
ances, or to oscillations and inclinations the I N R C group
doubles as a proportional schema: an inverse proportion of hori-
zontal distances and weights W/W' = Z//L in the case of pres-
sures and resistances and an inverse proportion of heights and
weights W/W' = H'/H in the case of inclinations. There is a
third proportion it is direct rather than inverse (L/U = H/H'),
of a purely geometric character and obvious to our subjects
(cf. us: "It's really the same*' whether you measure horizontal
distances or height). So our problem is to establish how our sub-
jects construct the first two proportions. Is the construction done
independently and by a direct structuring of the empirical data,
or is it linked to the operational schema of equilibrium based on
the I N R C group?
The Proportional Schema and thelNRC Group
First, we should remember that an understanding of proportions
does not appear until substage III-A; this is true in all spheres
and not only in the balance scale experiments. During substage
EQUILIBRIUM IN THE BALANCE
177
II-B it has often been noted that subjects search for a common
denominator of the two relations that they compare, but this
common relation is thought to be additive. Thus, instead of the
proportion W/W = U/L one would have an equality of dif-
ferences W W = U L. Clearly, the formation of the notion
of proportions presupposes that simple relations of difference be
substituted for the notion of the equality of products WL - WU.
But we must also note that the transition from the difference to
the product rarely takes place from the start in a form that is
metrical. The numerical quantification of the proportion is usually
preceded by a qualitative schema based on a conception of logical
product i.e., by the idea that two factors acting together are
equivalent to the action of two other factors added together. "The
larger the distance, the smaller the weight," says CHAL, using
simple qualitative correspondence (cf. prop. [5]). But he adds,
"They go together."
In other words a small weight combined with a great distance
is equivalent to a large weight with a small distance. These logical
multiplications are outlined at substage II-B (cf. props. [6] and
[7]), but the subjects fail to generalize to all possible cases. Where
does the generalization found at substages III-A and III-B come
from? Without doubt, this is where the notions of compensation
and reciprocity connected with the INRC group come in.
It is clear that when the subject at stage III becomes able to
understand transformations by inversion (N) and reciprocity (R)
and to group them into a single system (I, N, R, and N R = C),
by the same token he becomes able to make use of the equality of
products in a more general form than in the multiplication of rela-
tions (6) and (7). Moreover, this form already implies the notions
of compensation and cancellation. The possibility of reasoning in
terms of a group structure I N R C indicates an understanding
of the equalities NR = 1C, RC = I N, NC I R, etc., the equal-
ities between the products of two transformations. The result is
that the I N R C group is itself equivalent to a system of logical
proportions: ^^ ^^
C* ~ N* r I* ~~ N*
since I N = R C (where x = the operation transformed hy
I, N, R, or C).
178 THE FORMAL OPERATIONAL, SCHEMATA
For example, let us examine the subjects' reasoning on the
changes of weight and horizontal distance (to simplify notation
we shall disregard the constant weights and distances). Let p be
the statement of a fixed increase of weight and q of a fixed increase
of distance; let us call p and q the propositions stating a corre-
sponding diminution of weight and distance on the same arm of
the balance. Propositions p' and q f correspond to p and q, and
p' and q' correspond to p' and q / on the other arm. By a process
isomorphic to prop, (i) of Chap. 10, the subjects understand the
following relations of inversion and reciprocity (the I N R C group
but with p.q chosen as the identical operation I):
I (p.q) = to increase simultaneously the weight and the
distance on one of the arms;
N (p v q) = (p.q) v (p.q) v (p.q) = to reduce the distance
while increasing the weight or diminish the weight while in-
creasing the distance or diminish both; (8)
R (p'-q') compensates I by increasing both weight and dis-
tance on tie other arm of the balance;
G(p'vq'} = (p'.q'}v(p'.q'}v (p'.q'} = cancels R in the
same way that N cancels I.
But, since R (p'.q f ) is equivalent to compensating action I (p.q)
with a reaction (symmetry) on the other arm of the balance, we
find that it can be written p.q; and since (p' v q') is also equiva-
lent to compensating the action N by symmetry, we can write it
(pvq). Therefore proposition (8) can be formulated as follows:
- .
( '
C(pvq).
The system of these transformations, which states only the
equilibrium of weights and distances, is in itself equivalent to the
proportionality: 4
4 This logical proportion signifies the following:
(p.q).(pvq) = (p.q).(pvq) =o forl.N = R.C (a)
(p.q) vjp v q) = (p.q) v (p v q) = (p * q) for I v N = R v C (b)
(p.q).(p^^(pvq).(pvq) =(p.q) for L(NR) = C.(NN) (c)
(p.qUp vq)= p.q.(p v q) = o for L(NC) = R.(NN) (d)
EQUILIBRIUM IN THE BALANCE 179
p.q pvq . Ix Cx
=
In other words, an understanding of the system of inversions and
reciprocities (8) and (8a) follows directly from an understanding
of this proportional relation; an increase of weight and distance
on one arm of the balance is to the symmetrical increase on the
other arm as an increase of weight or distance on one arm is to a
reciprocal operation on the other.
Undoubtedly, this qualitative schema of logical proportions cor-
responds to the global intuition of proportionality with which the
subject begins. And it is easy to pass on from this qualitative
schema to more detailed logical proportions (involving a single
proposition) and from there to numerical proportions.
In this respect, remember that, for a single proposition p, the
correlative C is identical with I and the reciprocal R identical
with N. From proportion (9) one can construct:
t) Q
*- = -, whence p v p = q v q . (10)
In other words, the increase of weight is to the increase of dis-
tance as the decrease of distance is to the decrease of weight.
Secondly, beyond the direct proportions of types (9) and (10)
the I N R C group includes what can be called reciprocal pro-
portions, where one of the cross-products is the reciprocal R of
the other:
R[(p.q).(pvq)=p.q].
Hence, by virtue of (10) and (11), the reciprocal proportion:
The formulae demonstrate that the two logical proportions
(11) and (12) are isomorphic to the numerical propositions which
can be obtained by giving the same coefficient n either to an
2SO THE FORMAL OPERATIONAL SCHEMATA
increase in weight (p) or to an increase in the distance (q). In
other words, if p = nW and q = nL, then:
'O Q , riW n:L f ,
~ ~ corresponds to =r = , for example
q p * 2 . 8 (13)
2X8 = 2^4 ;
and
p _ p _ nW W:n - .
= R corresponds to =r- = -= , for example
q q * nL L:n'
2 X 4 4:3
2X8~~8:2*
Formulae (9) to (14) may seem much too abstract to account for
the actual reasoning of our subjects. Actually, this is in part an
independent result of the symbolism which we have introduced;
nevertheless this is how proportions are discovered. Before intro-
ducing numbers as measurements for weight and distance, the
subject usually begins by assuming:
p.q = K (p.q) (15)
(increasing the weight and reducing the distance on one of the
arms is the same as reducing the weight and increasing the dis-
tance on the other arm).
However, proposition (15) is none other than proportion (12),
which then implies (10) and (9) and leads to metrical proportion
(14). Thus we are justified in considering the preceding formulae
symbolic expressions of the actual reasoning of our subjects.
As for the proportion between weight and height, as soon as
they encounter alternating suspensions in the apparatus all the
subjects understand that an increase in distance (q) implies a
determinate increase in height (r), thus:
9 gr. (16)
Consequently proportions (10) and (12) imply:
and
EQUILIBRIUM IN THE BALANCE 181
Finally, the transfer of a weight to a higher point constitutes
work. This is expressed by our subjects in their own words, as
they do not have the technical vocabulary of physics at their dis-
posal: "There is more distance to pull" (CHAL) or "More force to
raise the weight'* (SAM). Actually, if a heavy weight hung at a
small distance from the axis balances a weight n times smaller at
a distance n times larger, it is because the same amount of work
is needed to raise the first to a given height and to raise the second
to a level n times higher than that height. As FIS says, there is
compensation "between the force and the height." This idea of an
equivalent amount of work, half-understood during stage III,
provides the explanation of the phenomenon of equilibrium.
However, since the reaction of these subjects is not completely
spontaneous on this point, we must turn to the next experiment.
There we replaced the overly- simple apparatus of the balance
scale with one for hauling a weight on an inclined plane; we can
see from this experiment how the concept of work is elaborated
beginning with the concrete substage II -B; and we can see how
it is used in the explanations of the formal stage III.
12
Hauling Weight
on an Inclined Plane 1
OXCE AGAIN our subjects are given an equilibrium problem; one
not too different from the balance problem but especially de-
signed to bring out work relationships. A toy dumping wagon is
drawn along a rail whose inclination can be varied. The task is to
predict the movements or equilibrium position of the wagon as a
function of three variables the weight it carries, the counter-
weight suspended by a cable fastened to the wagon, and the incli-
nation of the track. This last variable is calculated not in terms
of its angle measured in degrees, but in terms of its sinei.e., of
the (variable) height ft. Thus, the law of equilibrium to be found
is W/M = h/H, where W is the (variable) counterweight, M the
weight of the toy wagon (which itself weighs 4 units, but -which
can be loaded with varying amounts of weight), and H the total
height (the unvarying length of the track assuming it is held
vertically).
Stage I. Failure to Distinguish Between Ones Own
Actions and Objective Processes
At stage I the subject is most likely to explain the situation in
terms of the totality of actions which he can perform on the
apparatus:
1 With the collaboration of A. Morf, H. Olivieri, former research assistant,
Institut des Sciences de Tfiducation, G. Mercier, and D. Royo.
182
HAULING WEIGHT ON AN INCLINED PLANE
183
BAG (6 years) pushes the wagon to make it descend: "It goes down?"
"No, it goes up." "Can you do anything else?"- "Push it by hand"
"And to make it go up?" "You drive in the train." "And with the
weights?" He loads the counterweight "I put something on." "Why
does it go up?""! don't know. Because it's heavy." "And to make it
go down?" "I don't know. You could push it.'"
HER (6 years): "What can you faff 9 "Make the wagon go/* "How?"
"With the chain" [he pulls] .-"And to make it go all hy itself?" "Take
off the weights [at W]; put the weights on the wagon' [he takes off
two units at W, replacing them at M]. "What else could you do?"
"Put the track up higher" [he puts it at 45].-"Will it go
FIG. 10. A toy dumping wagon, suspended by a cable, is hauled up
the inclined plane by the counterweights at the other end of the cable*
The counterweights can be varied and the angle of the plane is adjust-
able; weights placed in the wagon provide the third variable.
FORMAL OPERATIONAL SCHEMATA
it will go down [it goes up]. You have to push it.' 9 "If you put it
higher, will it go up?" "No, it will go down because it is slanted"
"What do you have to do to make it go up?" "Pull [the rail is low-
ered]. You have to push it" [he pushes it by hand].
BEL (6 ; 6): "That's [W] to putt with." "How can you make it go?"
"You have to lower that" [he reduces the inclination]. "And what will
the wagon do?" "It will go down [experiment]. It goes up!" "Why?"
"Because it's up high" "And to make it go down?" "You have to raise
the chain [it is raised, and one unit of weight is put on the wagon and
two units at W]. It goes up even morel It can't go down. . . "
For these subjects, the apparatus is not yet seen as an inde-
pendent set of causes and effects, but is still assimilated to the
actions which they perform. There are two complementary senses
in which this is true. First, the subject does not try to isolate rela-
tionships external to his actions, but locates his own roles in the
same dimension as objective causes; second, the causes them-
selves are still conceptualized by assimilation to a motivational
model. Thus, weight is conceived of as a force which can push or
pull, etc. But it is also true that, at all levels of development,
causality is an assimilation of transformations of reality to the
subject's actions or operations with delegation of their power to
the real world. In other words, when the subject has reached a
certain operational stage of development, modifications of reality
are conceived of as isomorphic to the operational transformations
effected. But when the subject's activity consists of irreversible
actions which are not as yet coordinated into systems of opera-
tions, then reality is represented as a set of equally uncoordinated
forces which cannot possibly be differentiated from one's own
actions.
Substage II-A. Determination of the Role of the
Weights Without Operational Coordination with the
Inclinations
The subject begins to relate the weights in the toy wagon to the
counterweight because the two are homogeneous factors. He is
also aware of the fact that the inclination of the track plays a role,
but he cannot as yet coordinate it with that of the weight:
HAULING WEIGHT ON AN INCLINED PLANE 185
GOD (7 years): "What do you have to do to make the wagon go?"
"Pull "-"And if you don't pulF'-'WeB, you can push it."-" And if you
don't push." "Take off the load? [he removes two units of weight from
the wagon which immediately goes up]. "Is there anything else?"
"Yes, you can put on a weight" [he puts the two units at W]. "What
else?" [the subject is shown that the rail can be moved]. "You have
to put it a little lower down. 9 ' "O.K. Now, what will happen if I add
here?" [two units at W]. "The wagon will go up/* "And if I put the
weights in the wagon?" "It won't move." [Experiment] "So what do
you do to make it go up?" "You have to add another weight" [he adds
one at W]. "And if I can't do that?" "Oh, you have to take some off
here" [M]. The equilibrium is achieved for a given inclination: "And
if the track is lowered?" "1 dont think it will move" [the weights
are in equilibrium independently of inclination]. Experiment: "It goes
up" "Why?" "Because there were several weights here" [W].
FER (7 ; 10): Same beginning; then, in order to make the wagon go up
one must "take off some weights [M] and put on some weights" [W].
"And if I take off only one weight without adding any?" "It won't
move [experiment]. It moves!" "And to make it go down, what can
you do?" "Raise the chain: lift up the chain and take off a weight"
[W]. But in making predictions FER takes only the weight relationships
into account, as if the equilibrium remained the same for given weights
independently of the inclination.
There is marked progress over stage I. A reversible composition
of weights appears; to add one unit to the counterweight W is
equivalent to removing a weight of the same value from the
wagon M, etc. But these are only simple compositions based on
the assumption that equilibrium between M and W is assured by
a simple weight equivalence, as in the case of a balance (when the
distances from the axis are equal). However, inclination is seen
as playing a partial role; m the subject predicts tihat steepening the
slope works in favor of descent. However, he does not understand
that steepening the slope automatically reduces the effect of the
counterweight and that more weight is required at W in order to
raise the wagon on a steeper slope than on a gentler one, Thus
inclination is a secondary factor which operates in certain special
cases, but it is not yet taken as a general factor which can be
combined with the others.
186 THE FORMAL OPERATIONAL SCHEMATA
Substage II-B. Discovery of the General Role of
Inclination and Beginnings of the Concept of Work
From the start the subjects of this substage see that three factors
are involved the weight of the wagon, the counterweight, and
inclination. But only gradually do they discover that the equi-
librium between the weights involves more than simple equalities
and varies according to inclination. As they discover this fact, they
try to find the relationship. This leads them to the conclusion that
more work is needed to pull a given weight along a steeper than
along a more gently sloping track:
sou (9 ; 6) notices the weights W and M from the start, then:
do you have this curve? [inclination]. Can you change it?*' "Would
you like to change the slope?" "Yes, to see how it works. Could I put
on another weight?" "Where? 99 'Here [W]: perhaps the wagon wiU
go faster. If the weight is heavy enough, the wagon will go up.
If it isn't heavy enough, it won't move." He has several trials. *7/
you put on more weight [at W] it goes up even faster [he adds on
up to 7 units]. It can't go down because there is too much weight
[he takes off some at W; the wagon descends]. It goes down if I put
on less." Next: *7 don't want to put any weight at all [at W] because
I want to see if it stays down below or if it goes up more slowly." The
wagon goes down; then the subject varies the weights at M to see
whether "it makes a difference if you put several weights here [he
loads M]. It still goes too fast [he takes off weights at W and adds
weights at M until they are equal]. That makes equal weight because
I put 4 weights here [W] and there are 4 weights on the wagon
[(which continues to move); lie adds 2, and 2, 0.5 and 0.5, then 3 and 3].
It doesn't move? No, it moves anyway!" He has now discovered that
equality of weight does not guarantee equilibrium. "At the beginning
you asked me what the curve was for; do you remember?"-"O/i/ Yes,
you can lower the track, then it will go up" "Are you sure?"-"ft wiU
go up [he lowers the rail to i and the wagon goes up], because now it
doesnt slant as much so it's easier for it to go up." Next, sou varies the
inclinations and realizes that when the track is vertical "the weight [W]
win putt the wagon" because the counterweight is sufficient, but he
does not vary it. At 45 and i unit at W he ascertains that the
wagon descends and laughs: "I thought it was going to go up because
it goes up even when it stands up straight" Then, realizing that it still
HAULING WEIGHT ON AN INCLINED PLANE 187
goes down at 30: "You have to add or take off weight. You have to
experiment when you see that it goes up or down too much! 9 But he
does not do so systematically.
JAN (10 ; 8): "To make it go up, you have to put a heavier weight
here [W] ."-"What else could you do?"-"(7nZoad the wagon ."-"And
for the wagon to stay at the same point?" He puts 4 units on the
wagon and 4 at W. "The weights are equal. No, it doesn't move."
"Can you do something with the rail?" "Maybe you could lower it;
it's easier for the wagon to go forward because the track isn't as high."
"If you lift the rail and add weight?" "It will stay poised because
it's harder for it to go up." Then he weighs the wagon and declares
that it is equal to 4 weight units. "So would it remain in equilibrium
if you leave the wagon empty and put 4 weights here?" [W]. "No, it
would go up" [thus he understands that the equilibrium depends on
inclination]. "And if you raise the rail?" "It's harder for it to go up."
"Why?" "Because the wagon gets heavier"
Two main advances in thinking about the problem occur at this
stage: (i) an understanding of the fact that the equilibrium is not
due to a simple equality between weights, and (2) an understand-
ing of the role of inclination Le., more work is needed to pull a
wagon up a steeper incline.
First the subject discovers that placing n units at counterweight
W and n in wagon M does not guarantee an equilibrium; then he
realizes that the wagon itself weighs 4 units, but that p = 4 + M
does not achieve equilibrium either (or does so only if the rail is
vertical). This discovery leads him to focus on the problem of
inclination.
Thus, the child discovers the role of the slope at this level either
because of the preceding reason or because he varies the slope
directly. "Now there is less slant/' says BOU, "so it's easier for it to
go up." "The wagon will go forward more easily," says JAN, "be-
cause the track isn't as high." Furthermore, he understands that
equilibrium will be conserved if inclination and counterweight
are increased simultaneously "because it's harder for it to go up."
Finally, he predicts that at equal weights on W and M the wagon
will go up and that if the slope is increased still more the wagon
"gets heavier."
These latter protocols are instructive in that they show us how
inclination gets to be thought of in the same terms as weight and
THE FORMAL OPERATIONAL SCHEMATA
is combined with weight in the form of work. Raising a large
weight a little is equivalent to raising a lighter weight to a higher
point; in other words, the amount of work is the same. JAN ex-
presses this fact directly when he declares that "the weight gets
heavier" if the slope is increased. Although he is not familiar with
the parallelogram of forces, 2 the subject arrives at a pretty accu-
rate intuitive understanding of the relationship between weight
and inclination. Thus, even at the concrete level the concept of
work is accessible in a qualitative form based on the multiplica-
tion of the relations between height and weight.
Now, it is remarkable that both the inverse relationship be-
tween weight and height found in the equilibrium of the wagon
and the notion of work as the upward displacement of weight are
structured at the same substage (II-B) as the discovery of the
inverse correspondence between weights and distances in the bal-
ance. Both deal with the same physical law, but the child does
not know this, since he thinks of neither work nor height in the
balance problem unless the problem is presented in the form of
an alternating suspension apparatus (see Chap. 11). Nor does he
think of the balance in connection with the relations between
weight and inclination dealt with in this chapter. Everything pro-
ceeds as if, at a certain level of development, the entire set of
concrete operations applicable to a given subject arises simultane-
ously in the structuring of that delimited area (an example of such
a delimited area would be the equilibrium of weight as a function
of height and distance).
But there is a gap; the subject does not come to state the law
in its entirety. He clearly takes the three factors into consideration
(W, M , and inclination) and successively compares them two-by-
two without changing the third. But it is not that he intends to
hold one factor constant each time; he is not trying to apply the
"all other things being equal" proof. Rather, in comparing any two
factors he simply forgets the third, thus leaving it invariant with-
out being aware of the fact. He does not get to formulate the law,
2 The parallelogram of forces states that the portion of weight supported
by the track (and making the wagon lighter by the same amount) increases
as the inclination decreases, whereas the portion not supported by the track
increases in direct proportion to the inclination.
HAULING WEIGHT ON AN INCLINED PLANE 189
for he lacks the means to coordinate the entire set of factors simul-
taneously.
It is easy to see why he has failed at total coordination there
are two main reasons and both are intrinsic to the nature of con-
crete operations. First, the relevant correspondences are too com-
plex to be handled by proceeding in successive pairs or trios. It is
true that the subject, since he possesses the required operations,
could make a sufficiently complete inventory of factors. There is
no need to describe the operations of serial ordering, equalization,
and addition of weights at this time, for they have already been
described in connection with the balance (Chap. 11). Obviously,
the subject can also order the inclinations serially. Thus, if he
wanted to, the subject could determine the correspondence be-
tween the weight of the wagon (M) and the counterweight (W)
for each inclination so that he would know when the wagon would
be in equilibrium and when it would go up or down. But one can
see how complex such an empirically constructed triple-entry
table would be. Besides, the idea does not occur to the subject,
and he is content to deal with a few individual cases; hence, the
first reason for his failure at total coordination of the relevant
variables.
The second reason is that if the subject, instead of proceeding
by successive correspondences, tries to utilize the form of logical
calculus available to him i.e., multiplication of relationships (and
he does in fact proceed in this way in determining the relation-
ships of work) then a certain number of products remain indeter-
minate. In other words, we find a parallel to the indeterminacy
already encountered in the case of the logical multiplication of
weights and distances in the balance scale (Chap. 11, prop. [7]).
Starting from an equilibrium point:
Inclination x <-> [Wy o Mz],
the subject can certainly conclude that, if slope x is increased and
y and z are left invariant, the wagon descends, whereas if x is
diminished, it mounts, etc. But if slope x is increased at the same
time as counterweight W ( > y} is increased, or if slope a: is in-
creased while the weight on the wagon M ( < z) is diminished,
the product is indeterminate: the double relation y
> ^ > or ^ /\ ^
190
THE FORMAL OPERATIONAL SCHEMATA
can give rise to a product > , <, or = as long as the given factors
are not extensively quantified.
If an increase, lack of change and a decrease in the weight to
be displaced upward are designated by + m = m ? an d m
respectively, and an increase, lack of change, and decrease in the
height itself by + h, = h, and h, we see that the double-entry
table which characterizes the concept of work, worked out by
means of logical multiplication at substage II-B, involves two
indeterminate products out of nine (these products expressed in
_[- 5, . 5, z= , or rh signifying "more work," "less work,* "same
work,** or "indeterminacy"):
m
= m
m
-h
= h
+ h
!_ _
=
+
dt =
+
+
(l)
In summary, as long as the subject is limited to using concrete
operations of classes and relations, he cannot determine the law
(even in the form of implicit qualitative proportions found at
substage III-A). The explanation is twofold; first, the correspond-
ences which must be empirically established are too complex, and
second, the products of the multiplications of relations are in part
indeterminate.
Substage III-A. Qualitative Coordination of the
Three Factors, but Without Proportion as a Function
of Height
It is the nature of formal thought to consider an entire set of pos-
sibilities and to deduce from them what is real. With its appear-
ance, subjects use a remarkably different approach to the problem.
Instead of getting lost in the inventory of actual cases an inven-
tory which is in fact inexhaustible because the correspondences to
be determined by successive experiments are much too complex
HAULING WEIGHT ON AN INCLINED PLANE 191
in trying to cover all possible cases, the subject very quickly
turns to a selection of crucial cases i.e., the extremes and the
middle. It occurs to him to place the rail horizontally and verti-
cally (an idea which is unexplainable without a preliminary expli-
cation of the possible transformations). Then he determines the
demonstrative intermediate positions.
Thus, from the start the subject seeks to coordinate the three
factors into a single law, itself a qualitative proportional schema.
But since the subjects think in terms of the angle expressed in
degrees and not in terms of its sine i.e>, height they still fall a
little short of discovering the law they are groping for.
LAV (10 ; 6 advanced): "To make it go up you have to put on more
weight [at W], to make it go down, less." "What else can you do to
make it go up?" "Lower the rail. 9 " "And to make it go down?'* "Take
off all the weights [W]. 'You can also put some weights in the wagon.
You can put the track higher up, too, because the wagon comes dowr?
[with more force]. "What do you have to do for the wagon to stay in
place?" "That depends on how you place the track and whether you
put more or less weight on the wagon' 9 [he puts the track at about 45
and finds that 3 units in W balance the wagon]. "Are there other
places where the wagon rests without moving?" "Put the track hori-
zontal and take off the weights here" [W]. Then he measures the
weights needed when the track is vertical after having announced
"4 or 5 [units] because the wagon has a weight of 4? And he discovers
the point of equilibrium for W = 4. *Tell me what you have to do so
the wagon won't move?" "You have to put the track way down with-
out any weight or way up with 4 weights and also put the rail halfway
with 2 weights" [he has not tried it at the midheight; ie., 33]. "Are
there other points?" "Yes, everywhere" [he finds one unit for 15].
"Did you understand everything?" "Yes, the higher up you go [in-
clination of the rail], the more weight you have to put on for the
wagon to stay where it is; the more you go down, the less you need"
END (11 ; 6): "You can take off a weight to make it go down, put on
one to make it go up or raise the track more." Then he experiments by
himself; he starts by lowering the track to the horizontal point, then lifts
it to the vertical and says, *7f you want to put the track straight up
[vertical], you have to put on more weight; you need 4 weights"
"Why 4?" "Because with 4 it doesn't go up. You can compare it with
a bdance-scale: on one side 400 grams and on the other 400 too."
j[92 THE FORMAL OPERATIONAL SCHEMATA
Next: "I want to see how many weights it takes at 45" [he finds
2.5]. Then he concludes: "The more you lower it 9 the more weights
you have to take off. The more you go up, the more you have to add''
scu (11 ; 12): "To make it go up you can lower the track or take off
weight in the wagon or put on some at" [W]. He seeks the equilibrium
positions by varying the slope: "When it is low, there isn't enough
weight in the wagon and the wagon goes up; the counterweight pulls
harder." He weighs the wagon [M = 4], then tries at 33 and estab-
lishes that W = 2 at the equilibrium point. "That's funny: a minute
ago I saw that the wagon weight was 4 and now it's 2 [W 2] and
it pulls a wagon that weighs 4. You have to raise it more to make it
equal [4 = 4] and calculate its relationship to the inclination" He
raises the track higher and higher: "Still not enough [he has reached
80]. That's almost it. It has to be straight [90]. When there is an
inclination the equilibrium changes 9 and when it is straight the rela-
tionship is one to one'' But for half of the inclination he tries at 45,
although he had already established 2 units for 33 and does not un-
derstand that height alone plays a role.
CLA (11 ; 6): "To make it go down, you can either pull up the line or
take off some weight from the counterweight [W] or add some in the
wagon" [M].
Unlike the subjects at the advanced concrete substage (II-B),
subjects of the first formal substage (III-A) immediately or very
rapidly coordinate the three factors into a single relationship. At
first this integration is the simple statement of factors in the form
of a ternary disjunction. If we call p the increase in weight at W
(and p its decrease), q the increase of weight at M (and q its
decrease); r the increase (or r the decrease) of the inclination, and
t the rise of the wagon (or t its descent), we see that scu and CLA
start out with reciprocities:
tD(pvqvr) and tD(pvqvr). (2)
In practice these ternary disjunctions can be distinguished by
the fact that the subject no longer modifies two of the factors
without thinking of the third but looks for covariations. From
then on he is quickly convinced that the equilibrium of the wagon
and the counterweight varies according to the inclination, and the
subject makes several tries. Most often the subject tests the ex-
HAULING WEIGHT ON AN INCLINED PLANE 193
treme positions almost at once the horizontal position where the
wagon is at equilibrium without any counterweight (W = o) and
the vertical, where it is in equilibrium as if on a balance when the
counterweight is equal to its own weight (W = M = 4). Hence,
the qualitative law: the more the inclination is increased, the
greater the counterweight required to bring the wagon into equi-
libriumuntil the upper limit (vertical inclination) where the
counterweight is equal to the weight of the wagon.
Since he possesses disjunctive operations (2) from the formal
standpoint, the subject also possesses the I N R C group in the
form:
I(p v q v f)
K(pvqv r)
and the utilization of the structure naturally furnishes the schema
of proportionality evident in several cases (in particular LAV):
4
p.q.r p.q.r '
In other words, all the subjects at this level understand the
possible compensations between p and f and between q and r.
If this group (for this proportionality) and these equivalences
are to result in the formulation of the law h/H = W/M, a fourth
variable corresponding to absolute elevation H (the length of the
rail measured when it is held in a vertical position) must come
into play. Only then would the prepositional reciprocities and
inversions express the reciprocities and inversions operant in the
equilibrated system being analyzed. But the subject calculates the
inclinations in degrees, which results in one constant for H (90)
whatever the apparatus chosen and which leads him to look for
half of the inclination at 45, where the counterweight does not
have the value of 2 but an intermediate value between 2, and 3.
Either the subject generalizes falsely or he is prevented from
discovering a simple law.
In spite of the appearance of formal implications and disjunc-
tions with the consequences that they imply (3) and (4) at this
stage the subject does not manage to exclude the angle (in de-
194 THE FORMAL OPERATIONAL SCHEMATA
grees) in favor of height. This fact may seem curious, since even
at substage II-B the child formulates the concept of work as a
function of the lifting of a weight But this is because at this stage
the subject is limited to qualitative reasoning and is not yet able
to separate the concepts of angle and height. On the other hand,
when the subject at substage III-A wants to go beyond this quali-
tative relation of inclination to find a metrical expression, he
thinks of the angle rather than the height, doubtlessly because the
apparatus governing the inclination of the track describes a
rotating movement.
Substage III-B. Discovery of the Law
As soon as the angle measured in degrees has been excluded in
favor of the height (sine), the subject discovers the proportionality
of heights and weights. But, curiously enough, this exclusion is
not easy (we suggested it to the second of the three following
subjects):
GIL (12 ; 7) is asked to find the equilibrium points and to extract the
law. He finds W = 4 at the vertical, then looks for the midpoint
[W = 2] at 45 and then at 60: "Why?"-*7 count half the dis-
tance [horizontal]. No, that doesnt work. You have to find the" [half-
way point]. He does the experiment for W = 2. "Ifs about 30. But
there is also that [height]. Here, with 2 weights, ifs 32. For 3
weights, you have to put it at 3 [in height]. Anyway, I think so [ex-
periment]. Yes, for i weight [W = i] you have to put the rail at i,
for 2 weights at 2, for 3 weights at 3 and for 4 way up at the top"
"Can you give a single rule?" "Yes, for 2 weights you put it halfway
up. For the halfway height it's half of the weight of the wagon, for
one-fourth ifs a quarter of the total weight," etc.
DEZ (14 ; 3): *7/ the rail is vertical, you have to put enough units here
[W] to make them equal [the weights and M: he finds 4]. The 2
weights are the same, one on each side. For the half, you have to put
on half the weight." "For half of what?"-"O/ the inclination: 45
[experiment: 2.5]. No! Maybe there is friction" 'It's possible, but
it doesn't play an important role." He finds that W = 2 corresponds
to 33. "33, that makes about two-thirds. Look here [height]. Ohl
The height! It isn't the angle that does to, but the height [he tries ele-
vations 1/2, 3/4, and 1/4]. The weight pulling the wagon [W] has to
HAULING WEIGHT ON AN INCLINED PLANE 195
be equal to the height; for example, if you have a height of 2 you need
2, over 4, if ifs i, then i over 4" [i and 2 in elevation are the fourth
and the half],
VUL (15 ; 6) determines 4W for M when the rail is vertical; then: "At
33 I find 2; at 15 Z find i; at 60 it should be 4 but it isn't. If it
isn't proportional to the angle, then. . . .""Is there something else
you might consider?" "The height corresponds to the angle. If I take
twice the height: height 2 corresponds to 2 weights. Let's see: eleva-
tion 3 gives 3 in weight. Good, ifs in proportion to the height. Each
time you increase the height by a certain amount, you have to add a
proportionate amount of weight" Summary: "The height is propor-
tional to the weight."
There are two complementary types of response at this stage.
For the first type (DIZ), the discovery of height is a result of sug-
gestion, but the subject immediately formulates the law of pro-
portionality once the suggestion is made. But, in the second type,
height is discovered as a factor because of the search for propor-
tionality i.e., the simple proportionality which is still beyond the
III-A subjects.
In both cases the law is discovered: h/H = W/M. If we let s
stand for an increase in the total height (vertical length of the
rail, which in fact, does not vary in our experiment), the logical
proportion is as follows:
where q.r stands for an increase in the work to be accomplished
and p.s for an increase in the work furnished by the counter-
weight as a function of the total height.
Given the already established equivalences between p and f or
between q and r and given the equivalence between q and s (the
result is the same when the weight of the wagon is increased or
when the value of H is decreased), proportion (5) can be deduced
from:
Thus, in the present case, the forms of the equilibrium schema
(I N R C group) and the proportionality schema are the same,
196 THE FORMAL OPERATIONAL SCHEMATA
mutatis mutandis, as in the balance-scale problem (Chap. 11). This
isomorphism raises an interesting problem for the psychology of
formal thinking. It is true that the structure of the two laws is the
same. If we call B the heavier of the two weights in equilibrium
on the balance and A the lighter, B corresponds here to the weight
of the wagon (M) and A to that of the counterweight (W). If we
call L the horizontal distance (from the axis) corresponding to
weight A (thus the greater of the two distances for the smaller
of the two weights) and I the distance that corresponds to weight
B (the smallest distance for the greatest weight), then in this case
L corresponds to H and I to h. The formula is:
Z/L = A/B, as h/H = W/M .
However, the intuitive content of the two laws is quite different
(so different that many psychology students take a great deal of
time trying to understand their identity). In the balance problem,
the relations between weight and lengths (I and L measured hori-
zontally on the two arms) are crucial, whereas the heights are
potentially effective factors only if the system is in equilibrium,
unless an alternate suspension apparatus is used (as in Chap. 11);
thus, the concept of work is not immediately elicited. But, in the
case of the wagon, the horizontal distances do not influence the
system; the inclination is intuitively given and the concept of
work is elicited as soon as the system actions are observed. Then
the psychological question becomes: is this difference in intuitive
content the determining factor in the development of operations,
or, on the contrary, is the underlying operational structure its de-
terminant? To answer the question, we must compare the results
for the two problems stage by stage.
In both cases the system processes throughout stage I are ex-
plained by an assimilation to the subject's own action, pulling and
pushing, etc. But, since the balance is noticeably symmetrical,
there is a more rapid equalization of distances and weights (as
intuitive regulations without operations) in that experiment.
At substage II-A operational equalization of weights occurs in
both cases, and the subject understands that distance plays a role
in the case of the balance and that inclination is relevant in the
wagon problem. But the subject cannot combine these heterogene-
ous factors with weight (except in certain special cases).
HAULING WEIGHT ON AN INCLINED PLANE 197
At substage II-B the subject discovers the inverse correspond-
ence between the weight and distance for the case of the equi-
librated balance (lighter corresponds to further from the axis and
heavier to closer to it) and discovers as well the fact that the
larger the inclination, the heavier the counterweight needed to
balance the wagon (in the second case). In the latter case, the
coordination between weight and inclination gives rise to the
structuring of the concept of work more work is required to raise
the same weight to a greater than to a lesser height. But in both
cases the coordination remains qualitative. Certain compensations
are understood (heavier = less far, and heavier = less high), but
there is no possibility of solving the more general problem be-
cause of the indeterminacy of logical multiplication.
At substage III-A the subject discovers the metrical proportion
in the balance problem and looks for the same proportion in the
case of the wagon, discovering the qualitative law coordinating
the three factors (weight, counterweight, and inclination). The
time lag in the second case is due to the fact that the height has
to be dissociated from the angle (measured in degrees); half of
the height is not 45 but 33.
At substage III-B the metrical proportion is finally discovered
in the case of the wagon and is explained directly in terms of
work. On the other hand, in the balance problem the metrical law
found at substage III-A appears as a compensation system that
is self-sufficient as long as the two weights are hung up at the
same time; it does not occur to the subject to invoke either height
or work in his explanation. But when the weights are presented
successively with a suspension apparatus that brings out the
alternating differences in height, the subject discovers the inverse
proportion between the weights and the height attained. In the
wagon problem he explains the equilibrium in terms of the equal
amounts of work needed.
Thus, it is clear that between the two lines of development
there are a set of intuitive differences which result from the nature
of the apparatus and from the questions asked of our subjects. So
it is all the more striking that we find the same operational
mechanism underlying the apparent divergences. In both cases,
after the same preoperational representations of stage I and the
same initial operations at substage II-A, the inverse correspond-
198 THE FORMAL OPERATIONAL SCHEMATA
ence is discovered at substage II-B. In both cases the operational
schema of equilibrium is established only when the I N R C group
comes into play at the level of formal or prepositional operations.
And in both cases this leads to the schema for proportions and
compensations in their general form. Thus the differences at the
intuitive level only give rise to slight differences in timing within
stages II and III, while the over-all progression of organization
is the same.
13
The Projection of Shadows 1
IN ADDITION to the usual problem of the formal operations needed
to establish the table of possibilities that allows the discovery and
verification of a law, the present research raises a question about
the formal operational schema relative to proportionality. But
we are dealing with a new type of proportionality. Whereas the
proportions in the problems of the balance and of hauling a
weight on an inclined plane derive from a model of physical
equilibrium, the proportions we shall study in connection with the
projection of shadows are of an essentially geometrical nature.
They denote relationships between distances and diameters in a
physical phenomenon that can be explained in terms of simple
projective geometry.
The problem we have set for ourselves is to discover whether
the proportions involved in the present experiments will be dis-
covered at stage III, as in our previous experiments, and whether
or not this discovery is a function of the I N R C group. If it is,
one must think of the I N R C group in a more general sense than
in the earlier problems.
The law to be discovered in this experiment is extremely
iWith the collaboration of Vinh Bang, research assistant, Institut des
Sciences de 1'fiducation; B. Reymond-Rivier, research assistant, Institut des
Sciences de Tfiducation; and F. Marchand.
199
200
THE FORMAL OPERATIONAL SCHEMATA
simple. Rings of varying diameters are placed between a light
source and a screen. The size of their shadows is directly pro-
portional to the diameters and inversely proportional to the dis-
tance between them and the light source. Specifically, we ask the
subject to find two shadows which cover each other exactly, using
two unequal rings. To do so he need only place the larger one
further from the light, in proportion to its size, and there will be
compensation between distances and diameters.
The stage I reactions need not be presented for this problem.
The preoperational subjects do not understand the formation of
shadows and in another work we have described the representa-
tions of shadow typical of ^-/-year-old children in sufficient
detail to make it unnecessary to take up the question here. 2
FIG. 11. The projection of shadows involves a baseboard, a screen
attached to one end of this, a light source, and four rings of varying
diameters. The light source and the rings can be moved along the base-
board. The subject is asked to produce two shadows of the same size/
using different-sized rings.
2 See Piaget, The Child's Conception of Physical Causality, Chap. VIII,
nos. i and 2, and Play, Dreams, and Imitation in Childhood (Norton, 1951),
Chap. EL
THE PROJECTION OF SHADOWS 201
Stage II. Discovery of the Role of the Size (II-A),
then of Distance (II-B)
The II-A child knows that the size of shadows depends on the
size of the object, but his knowledge goes no further:
PEL (7 ; 10) predicts correctly that a ring 10 cm. in diameter will pro-
duce a larger shadow than a ring of 5 cm., etc. "If I move it to this side,
where will the shadow be?" "There" [accurate]. "Does it stay the
same size or does it get bigger or smaller?" "It's the same."
BAR (8 ; 8) starts with the same reactions. Then, through experiment,
he discovers that the shadow of the same ring varies in size with the
distance. He is then asked to produce a single shadow using 4 un-
equal rings: he places the 20 cm. circle at 70 cm. distance, the 10 cm.
at 41 cm., the 5 cm. at 23 cm., and the i cm. at 11 cm.
It is possible to order serially the sizes of the rings and the sizes
of the shadows and to formulate accurate correspondences at
equal distances. In fact we find an accurate serial ordering of
distances, but the subject does not relate this to the size of the
shadows. He starts out with the assumption that the distance does
not modify size (PEL). When corrected by the experiment, he
expects haphazard transformations and does not find regular
correspondences (BAR).
On the other hand, at substage II-B the subjects no longer think
of light as being "everywhere" without rays having a determinate
direction, and they begin to predict the effect of divergent rays.
At least, they establish an empirical correspondence between the
decreasing sizes of the shadow thrown by the same object and
the increasing distances from the light source. In other words,
they understand that the closer the object is to the screen, the
smaller the shadow.
MAND (9 ; 6): "As it advances [toward the light], it [the shadow] al-
ways becomes bigger, because when it is closer [to the screen] it gets
smdler 9 and when ifs further away [from the screen] it gets bigger"
NOV (10 ; 5): "You have to put the smallest ring in front [toward the
light source], because it keeps getting bigger" [cf. the cone of light
202 THE FORMAL OPERATIONAL SCHEMATA
rays]. Then, in order to obtain equal shadows he puts the 5 cm. ring
at 44 cm,, the 10 cm. at 55, the 15 cm. at 56, the 20 cm. at 57.
OLI (10 ; 2) puts the 5 cm. ring at 10 and the 10 cm. at 19, then 15 at
38 and 20 at 50. "Why did you put them that way?" "Because with
those [5 and 10 diameters] it's bigger [because closer to the light] and
those [15 and 20 diameters] get smaller."
CHRI (10 ; o) puts the 5 cm. ring at 2 cm., 10 at 15, 15 at 29, and 20 at
42 cm., "because the smallest have to be further back to make the same
size as the big ones: you have to put them at almost equal distances
apart: 9
DEL (11 ; 11) puts the 5 cm. ring at 11 cm., 10 at 21 cm., 15 at 32, and
20 at 43. Then 5 at 41, 10 at 52, 15 at 63, and 20 at 74 cm.
MAU (11 ; 10) puts the 5 cm. ring at 55, 10 at 63, 15 at 71, and 20 at
80 cm.
The qualitative correspondence between the shadow sizes and
distance is clearly formulated, but with two peculiarities which
are extremely instructive for analysis of the opposition between
concrete level compensations obtained through logical multiplica-
tion and the true proportions based on multiplicative compensa-
tions found at the formal level.
The first of these peculiarities is that, although the stage II-B
subjects know how to construct inverse correspondences, they
prefer direct ones. Consequently they tend to calculate the dis-
tances by starting from the screen rather than from the light
source (cf. "it advances" for MAND, "further back" for CHRI; these
expressions are relative to the screen). Nevertheless, certain sub-
jects reach an intuitive understanding of the light cone (cf. NOV,
"it keeps getting bigger"), but when they measure the distances
they do so from the screen so that they can make a direct corre-
spondencethe larger the distance (from the screen) the larger the
shadow (the 5 and 10 cm. rings are "bigger" than those of 15 and
20, says ou).
Nevertheless, they clearly understand the compensation be-
tween the distance and the size of the ring. However, the second
and most important peculiarity of these reactions is that at this
stage the attempts at metrical quantification to which this com-
THE PROJECTION OF SHADOWS 203
pensation gives rise cannot be interpreted as derived from a true
proportion i.e., multiplicative relationships. Rather, they derive
from any constant additive differences whatsoever in the serial
orders and correspondences. For example, after a gap of 11 cm.
between the first two rings, NOV places the following rings i cm.
apart (54, 55, 56, 57 cm.). Likewise MAU puts 8 to 9 cm. between
rings at distances of 55 to So cm., and emu calculates constant
differences of 13 to 14 cm. OLI seems to approach an understand-
ing of proportionality, but he makes a simple dichotomy between
the large and small circles. Then he distinguishes additive dif-
ferences of 9 and 12 cm. within each set, but a one-to-two ratio
between the two. As for DEL, whose chronological age would put
him at stage II, his initial proportion is more or less accurate, but
on the second trial he regresses to an arbitrary additive difference
of 11 with 41 cm. as the starting point.
Stage III. Proportionality in the Correspondences
(III-A), then Generalization and Formulation of the
Law (III-B)
At substage III-A an inverse metrical proportionality between
distances and diameters first appears, but it is not yet generalized
to all possible cases. The subject measures the diameters and the
distances and looks for a metrical hypothesis based on the diver-
gent structure of light rays, taking into account the distance be-
tween the light source and the first ring (the smallest or the
largest):
CHE (12 ; 8) measures the rings and finds that tibeir diameters differ by
5 cm. He concludes that one must "find a distance between them
which is a multiple of 5." He places them correctly in proportion to
size.
DUG (12 ; i), after having placed the 20 cm. ring at 83, says: "Now you
have to count from here to there [to the light source] and divide by 4."
Then he puts the 5 cm. ring at 21, the 15 cm. at 61, but the 10 cm.
at 51. *And if I only give you three rings?*' [5, 10, and 15]. "You
have to count from the largest and divide by three''
204 THE FORMAL OPERATIONAL SCHEMATA
WAL (12- ; 4) divides the shadow of the large ring by the distance, ob-
taining 50/40 = 1.25, and looks for multiples of 1.25 as differences.
Then he places the 5 cm. ring at 25 and the 10 cm. at 50: "It's half I"
Next, he places the 15 cm. at 75 and the 2,0 cm. at 100: "It works. It's
always 5 cm. more [for the rings] and the lengths are always 25 cm.
more. It's the same scale" "Can you find me another distance with the
shadow the same for all rings?" He places them with 26-cm. distances
between each ring and moves the light back 4 cm. "I mean without
moving the light/' "I don't think you can. You cant enlarge the scale."
The experimenter places the 15 cm. ring at 46; the subject then puts
the 10 cm. at 23, then the 5 cm. at 7, the 10 cm. at 30, the 15 cm. at
53 and the 20 cm. at 76.
It is clear that at this level the subject assumes proportionality
from the start. But the proportion is only found in one or two
instances and is not yet generalized to all cases. All of the pre-
viously developed relationships of concrete serial ordering and
correspondence are coordinated in an organized view of the
whole; all of the relationships are subordinated to the geometrical
representation of divergent rays (in the experiment the subject's
goal is to control his placement of the rings) and the representa-
tion is correctly given the property of proportionality.
In sum, at stage III-A the subject begins to calculate distances
from the light source rather than from the screen, and in his cal-
culation he takes the distance between the light source and the
first ring into account and not simply the distances between the
rings (two new operations not present at stage II).
But he is satisfied when he has verified his hypothesis on a
single case and does not yet conceive of the relationship as
changeable and as capable of taking a series of equivalent forms.
In other words, he does not yet look for the general law, defined
as a system of necessary relations which are adequate to account
for the obtained result.
However, at substage III-B the law is generalized and made
explicit:
WAH (14 ; i): *Y0t* can take any distance as long as the ratio is the
same."
MIC (14 ; 6): "Since the diameters all have regular differences, the dif-
ferences between the distances have to be the same." Then he places
THE PROJECTION OF SHADOWS 205
the rings of 5, 10, 15, and 20 cm. at distances of 8, 16, 24, and 32 cm.,
respectively; next, lie takes another arbitrary distance and finds the
proportion in the same way: "The distances have to have the same
relation to each other as the rings."
FAU (15 ; 6): "For the shadow to be equal, the same with two rings, the
fraction of the axes [distances] has to be equal to the fraction of the
two rings." Then: "The shadow is never smaller than the actual ring."
HUE (15 ; 6) : "The angle made by the light rays gets wider and wider.
For the light to make twice the size [of the shadow], it takes twice
the distance" etc.
MART (16 ; 2) begins getting the rings to coincide: "You have to put
the largest the furthest away 9 and the ratio between the diameters of
the rings and the distances has to be the same" He is successful in dis-
covering the proportion.
GUY (16 ; 6): "It should send out a ray like this [he shows us a conic
form] from the small ring. I think that the first ring will give a shadow
whose outline will depend on a kind of ray that increases in size. . . "
Thus there is a difference between the set of these children and
the set of the substage III-A subjects. From the start their formu-
lations are dependent upon a hypothesis that is both explanatory
and general, and at this stage the hypothesis no longer deals only
with the divergent light rays but includes a conception of the cone
itself. Thus, proportionality is implied by the explanatory schema
itself and holds, as WAH says, at "any distance at all."
But we cannot forget that proportionality was anticipated be-
fore this final view of the whole was constructed. The proportions
are deduced from the whole figure only after the child under-
stands the divergence of light rays, but at substage III-A the
proportionality was discovered without having first projected this
figure. (Moreover, this may often happen at substage III-B.)
What, tihen, is the nature of this proportionality, which does not
stem from a mechanical schema lilce the proportions in the earlier
chapters but is accompanied from the start by geometrical repre-
sentations (the divergence of rays of light, then the shape of the
cone)? In a sense, the stage III subjects discover proportionality
because they have access to prepositional logic and, therefore, are
206 THE FORMAL OPERATIONAL SCHEMATA
able to understand and transform the equality of two products.
This is possible only when the subject can state that a given in-
crease in the diameters of the rings combined with a given in-
crease in distance can give the same results as other combinations
of increases or decreases. On the other hand, it is clear that this
equality of products is understood only as an instrument which
enables the subject to express a multiplicative compensation be-
tween changes in the diameters and the distances.
Let us designate increases in diameter and distance by p and q
respectively, and decreases in diameter and distance by p and q,
Let r be the conservation of the size of a shadow and f its
modification; let r be an increase in shadow size and f a decrease
(thus f = r v f). Then the subject will state the following propo-
sitions. First, in the case of modification of diameters and dis-
tances, conservation r presupposes either the simultaneous in-
crease or the simultaneous decrease of both:
r D[(p.q)]v(p.q). (i)
Second, combinations p.q and p.q always correspond to modi-
fications of the shadow:
(p.q)v(p.q)Dr . (2)
But in two opposite senses:
(p.<7)Dr,and (3)
(p.q) -Dr. (3a)
Finally, the same results can be obtained either by increasing
the diameter or by diminishing the distance and vice versa:
r D (p v 9H.*., r D [(p.q) v (p.q) v (p.q)] (4)
with exclusion of p.q , and
r D (p v <jrH.e., r D [(p.q) v (p.q) v (p.q)] (5)
with exclusion of p.q .
Actually, if r implies (p-q) or (p.q), the reciprocal is not true
and (p.q) or (p.q) can imply either r or f when the diameters and
the distances are modified.
In this way, reasoning by implication reveals to the subject that
the same products can result from either of two different causes;
THE PROJECTION OF SHADOWS 207
thus, he discovers the qualitative schema of proportionality. From
(i) he concludes:
(p. 9 ) = R(p.$), (6)
from which, by reciprocity of cross-products:
? = R . (7)
p q "'
And, from (2), he concludes:
(p.q)=E(p.q), (8)
from which, by reciprocity of cross-products:
But we remember that this proportion (9) corresponds to the
. , nx x i n
numerical proportion = - .
ny y:n
In a general way, the discovery of proportionality in this par-
ticular case results from an understanding of multiplicative com-
pensations. Even at substage II-B, the child is aware of the fact
that a change in the diameter of the circles can be compensated
by a change in distances, but he is unable to interpret this com-
pensation except by an additive formula (equality of differences).
If he is to assign the true multiplicative form to the compensa-
tion, the child must simultaneously distinguish and coordinate
two kinds of general transformations: transformations by inver-
sion, which cancel the modification in question, and transforma-
tions by reciprocity, which compensate it without canceling it
But this is exactly what prepositional operations enable the sub-
jects to do at stage III. In distinguishing two independent vari-
ables, each of whose modifications can be canceled (inversion)
but which also can compensate each other without cancellation
(reciprocity), subjects get to make effective use of a group of four
transformations (of course, they are not aware of this), and the
discovery of proportionality is a direct consequence.
It is striking to note that expression (p.q) in proposition (3)
increasing the diameter and decreasing the distance is the correl-
ative of expression (p v q) in proposition (4), just as expression
(p.q) m proposition foa) is the correlative of expression (p v q) in
208 THE FORMAL OPERATIONAL SCHEMATA
proposition (5). Each of these pairs of propositions (3) and (4) or
(3a) and (5) is linked to an inverse result of the other (r or f).
Thus, we have the group:
l(p v q)
R(pvq)
C(p.g).
Hence, the possible proportion:
pvg pva , Ix Ex
*- 2 -=*- = - 1 - thus =r-r- (11)
p.q p.q Cx NX ^ J
where x is (p v q).
But it is unlikely that propositions (10) and (11) actually play
a part in the subjects' reasoning, although they involve nothing
more than propositions (3), fea), (4), and (5), which are them-
selves direct expressions of the stage III statements. Thus we are
dealing with an example of a structure which is merely potential
but which is implied by the actual reasoning we have observed.
On the other hand, one can say that the I N R C group does play
a part in the subjects' thought processes in a simpler "unary"
form, and that this accounts for proportions (7) and (9). If the
increase in diameter is set forth as the identical transformation
(I = p), then its decrease is the inverse transformation (N = p).
But the increase in distance from the light source compensates
for the increase in diameter without canceling it Consequently, it
plays the part of the reciprocal transformation (R = q). Finally,
the negation of the reciprocal produces the same effect as the in-
crease in diameter and thus plays the role of the correlative
(C = q). The following proportions result:
=| or R | (proposition [9]).
The foregoing analysis demonstrates that the I NRG group
has a more general function than that of explaining mechanical
equilibrium. It comes into play when two distinct reference sys-
tems (as in relative motions: see Chap. 17) have to be coordinated,
as we shall see in Part III of the present work. As we see in this
chapter, it even operates in the coordination of changes in two
independent variables when multiplicative compensation of their
THE PROJECTION OF SHADOWS 209
effects is possible (as opposed to the additive compensations first
formulated at the concrete operational stage).
NOTE. It should be noted that logical proportionality is not tied up
only with the I N R C group but can also be derived from the general
structure of proportionality found in the lattice (in a way which seems
to us to involve the I N R C group as well; see J. Piaget, Essai sur les
transformations des operations logiques, pp. 166-68). A lattice is a
partially ordered set of inclusions (and in logic, of propositions and
implications as well), such that for any two elements of the set, x and y,
there is always a least upper bound UB ( = the smallest element
which includes both x and t/) and a greatest lower bound LB ( = their
intersection). Now, there is a proportional relationship in any lattice
such that = y^r . For example, in logic, the proportional rela-
tionship is *--*- z= - . Passing from the lattice in propositional
p <pv q * * -
logic to the lattice in whole numbers, we know that LB = the greatest
common divisor (GCD) and UB = the least common multiple (LCM).
We then have:
GCD y i % 6 13
- = y ;;, , . for example = or = TT- .
x LCM ' * 4 is 20 60
But, in the shadow problem, the subject could find the logical
proportion directly:
p pvq
where p = (p.qv p.q) and q = (p.q v p.q); and (p, q, and theii
negations carry the same meaning that they do in propositions
[1] to [11]).
However, we have no evidence that the stage III subjects actu-
ally resort to the lattice structure in the solution of the shadow
problem, for this structure is psychologically manifested in quali-
tative reasoning by an explicit utilization of combinatorial opera-
tions, which is not the case here. Moreover, from the formal
standpoint, proportion (12) can be reduced to proportions which
derive from the I NRG group, whereas the converse of this
statement is not true. 3
8 See Piaget, Les Transformations des operations logiques, p. 225. (Not
transl.)
14
Centrifugal Force
and Compensations 1
THE SCHEMA of proportionality has been examined in several
forms, both in connection with the equilibrium schema (Chaps.
11 and 12) and independently of it (Chap. 13). We now have to
examine one more case in order to define the relationship between
the proportionality schema and the schema of multiplicative com-
pensation. Here we are not speaking of compensation in the most
general sense of the term, in which it is synonymous with reversi-
bility. Rather, we are referring to compensation between hetero-
geneous factors x and t/, such that an increase in the value of one
gives the same result as an increase or decrease in the value of
the other. We have already come across compensations of this
type: in the flexibility problem (Chap. 3); in the balance problem,
where distances and weights compensate each other; in the prob-
lem of traction on an inclined plane, where inclination and
weights are involved; and finally, in the case of shadow projection,
where diameters and distances compensate each other. Still, we
thought it worth while to analyze a new example, one in which
two possibilities are open to the subject. He can construct metrical
proportions (which he could not in the flexibility problem), and
he can isolate the factors that determine equilibrium in terms of
the collaboration of M. Meyer-Gantenbein and L. Vergopoulo,
Institut des Sciences de Tfiducation.
210
CENTRIFUGAL FORCE AND COMPENSATIONS 211
the "all other things being equal" method (which he could not do
in the traction problem). Our aim is to discover whether, psy-
chologically, proportions carry with them the idea of compensa-
tion or whether it is the other way around.
Three metal balls of different weights are placed on a disc at
three different distances from its center. The disc is rotated faster
and faster until the balls roll off the disc because of centrifugal
force. The problem is to predict in what order they will leave
their initial positions and why. Obviously, the law of centrifugal
force is a complex one i.e., f = mv 2 /r where m = mass, r = the
radius (distance from the center), and v 2 = r 2 ^ 2 (where w = the
angular speed). When o 2 is replaced by r 2 ^ 2 , / = mw 2 r is obtained.
But, since the speed of the disc is constant with the initial accel-
eration, the subject need isolate only factors m and r i.e.> need
understand only the following two relationships: a ball is dis-
placed sooner in direct proportion to its weight and later in in-
verse proportion to the distance from the center. Consequently,
a problem of compensation arises. A heavy ball placed at a point
nearer the center may move at the same time as a lighter one
closer to the periphery. (The three weights are calculated in such
a way as to compensate exactly for the three distances.)
Stage 1. Preoperational Interpretations
Subjects under 7 years refer to all possible factors to explain the
order of succession of movements, including among others the
size and the distance:
COQ (6 ; 11): "One takes off on one side, the other on the other side,
because they don't want to go on the same side."
PAV (6 ; 2): "The ball on the third circle wiU take of -first because it's
nearer the edge. [Large and medium on the same circle. 1 Together. It
doesn't make any difference if they're bigger [experiment]. The big one
took off first because it's heavier. The heavy ones always go first."
CUM (6 years): "They rolled because it turns. I want to put on the little
ones because the big ones roll [experiment]. The big one takes off
first."
212 THE FORMAL OPERATIONAL SCHEMATA
Thus, even at this level the subject is able to account for the
two factors, for he accepts anything he sees and lacks any opera-
tional caution (cf. PAV who goes from "never" to "always" when
he has observed a single case!). More accurately, at this stage the
child does observe the facts but does not have a sufficiently devel-
oped set of inclusions or relationships between "all" and "some"
to establish laws. Moreover, he tends to believe that everything
that occurs has to be as it is. This assumption is both the principle
from which he generalizes and a failure to distinguish between
the moral and the physical.
Substage II-A. Partial Correspondences
When concrete operations appear, the child can correctly order
serially the sizes (or weights) of the balls and formulate the cor-
respondence with the take-off order, but only in those cases in
which the distances are equal (independently of the subject's
manipulations). He also discovers the correspondence between
the take-off order and the distances when the weights are equal.
But the multiplication of the two relationships appears only in
exceptional cases. Moreover, it never occurs when compensation
is involved i.e., when the two factors do not vary in the same
direction.
GUI (7 ; 4): "They aU moved . . . the big ones moved away and
pushed the little ones" Prediction: "They won't move because I put a
big one on [he turns the disc]. They took off anyway [new trial]. That
one stayed because it's too light" "Why?" "The light balls stay; some-
times they take off because it turns fast."The experimenter places
large balls of equal weights at distances D 3 and D 2. 2 "[D 3] was
first, [D 2] second."- *Why?"- fe Because it's always first when it is at"
[i].~*Why?"-7* doesn't have far to go/'-The experimenter puts
W2 at D 3 and W i at D 2.-"W2 will be first [He turns the disk],
Both at the same time; the small one [W 2] a little before." 'Does that
surprise ycm?"-"No. The big ones go faster [another trial]. The little
one [W 2] was first and the big one [W i] next, but they got there at
the same time. The big one went first and pushed the little one [actu-
2 Distances D i, Da, and D 3 are numbered from the center in that order;
balls P i, P 2, and PS are numbered in order of increasing weight.
CENTRIFUGAL FORCE AND COMPENSATIONS 213
ally both took off at about the same time on opposite sides]. The big
one was first because it is bigger [another trial: more or less simul-
taneous]. The little one goes first because it only has a little way to go"
MON (8 ; o) prediction: "They will go all over the place if I turn it: it's
the same for all of them [experiment]. The nearer ones go first . . .
those there [near the edge: D 3] go first." He is given two different
weights: "The middle ones go off first because they are lighter [ex-
periment]. No, it's the other way around. 9 ' "Con you do it again?"
-*7/ it's like that once, it will be alwaysf-We put Wi at D 3 and Wz
at D i."No, I'm pretty dumb. It [W i] took off before because [W 2]
is nearer the pivot."
MOR (8 ; o) after observations of this sort arrives at the following
logical multiplication: "How are you going to place the balls [W i,
W 2, and W 3] if you want them to take off one after the other?" [He
puts W 3 at D 3, W 2, at D 2, and W i at D i.] "Because the big one is
heavier and ifs further forward, the second is smaller and further
back, and the third is smaller and still further back"
These facts are relevant to the problem of compensation. When
the experimenter dissociates the factors in order to make the
child's task easier, the child discovers what part each plays. The
balls take off according to weight, and those closest to the
periphery are displaced before the most distant. But the subject
could not have found the two laws by himself, since the "all other
things being equal" method does not come into play before stage
III. Also, when, after his independent discovery of the two fac-
tors, the subject is asked to compare balls of different weights
and at different distances, he runs into a number of difficulties.
First, the subject is unable to gauge the simultaneity of take-
off when balls are displaced in opposite directions. (Also, the
experimental apparatus causes some minor discrepancies along
this line.) In other words, the subject is not yet able to eliminate
deviations due to uncontrolled factors.
Secondly, logical multiplication is inadequate for a solution
to the problem. At this level subjects are able to use logical
multiplication effectively when two factors reinforce each other
(cf. MOR). But when the two factors work against each other
simultaneously, the child brings one and then the other into his
explanation, but he fails to understand that they can compensate
214 THE FORMAL OPERATIONAL SCHEMATA
each other. Nor does he look further for a multiplicative product.
This fact raises a problem must we assume that multiplication
of relations remains incomplete and operates only in the intui-
tively favorable cases, or do we have to consider the possibility
that the child regresses when faced with the indeterminate result
of the multiplication "heavier X closer" and "lighter X further
away*? One could argue against the second explanation on the
grounds that the product is just in front of his eyes, since, in fact,
there is compensation. On the other hand, the child fails to per-
ceive it because he does not understand it, or he sees it without
understanding that compensation is involved. Thus, in this par-
ticular case, it is Likely that the operational mechanisms govern-
ing the multiplication of relations are already complete for those
situations in which the product is determinate, but his operational
mechanisms do not yet appear in a form that can be generalized
to products that admit of three possibilities (heavier X closer =
the heavier ball takes off after the lighter ball or before it or at the
same time because of compensation).
Substage II-B. The Beginning of
Concrete Compensation
Unless the problem is simplified by the experimenter, at this level
the subject cannot always explain why two balls of different
weights can take off at the same time if they are placed at differ-
ent distances from the center. He cannot isolate the variables
without help; thus, he cannot conceive of the compensation of
two opposing factors based on possible combinations. But when
the experimenter simplifies the task by varying the factors one at
a time, the child can discover both the role of distance and the
part played by weight. Then, he begins to understand compensa-
tion and in some cases even predicts it:
(9 ; 3). W 3 at D 2 and W 3 at D 3: "They didnt take off at the
same time because one is far away and the other up close" "And
when they are like that?" [W 3 at D i and W 2 at D 3].-*T/iflf one
[W 3! will go first because it's bigger and the other is smaller [experi-
ment: contrary result]. Ifs because it [W 2] is closer to the edge. 9 '
'"What makes it change?' '-"The weight and the size "-'At that?" [W i
CENTRIFUGAL FORCE AND COMPENSATIONS 215
at D 3 and W 2 at D 2].- "Together, because if you had put this one
[W i] with a big one [W 3] the big one would go -first, but with [W 2]
it goes at the same time because it is little and it has a little distance
and the other is big and it has a longer distance." "And?" [W 3 at D 3,
W 2, at D 2, and W i at D i].-"That one [W 3] will go first because
it's closer to the edge and it is bigger; [W 2] is further from the edge
and smaller and [W i] is less close [than W 2] and less big!' "What
do you have to do to make them both go at the same time?" He puts
W 3 at D i; W 2 at D 2 and W i at D 3 "because the smallest one has
a small distance" [to cover].
CRO (10 ; 2). Experiment: W 3 at D 2 and W 3 at D 2. "They witt go
together because they are both the same size." "And?" [ W 3 at D i and
W 3 at D 2]. "The furthest from the center went off first, because it's
nearer the edge."-[W$ at D i and Wz at D i.] "Which one will take
off first?" [W 3] "first because it's heavier"-[W 3 at D 2 and W 2 at
D 3?] 'Both together because that one is smallest but closer to the edge,
the other one is bigger but further from the edge, and the biggest is
furthest from the edge"
Thus the main advance over substage II-A is that, once the sub-
ject knows the respective roles of the two factors, he completes the
coordination (or logical multiplication) of weights and distances.
Of course, the multiplication is performed correctly when the
results are cumulative (Wi X E>i, W 2 X E>2, and W 3 X DS). But
some subjects explain or even predict (CRO) for case WjDs,
W 2 D 2 , and W 3 Di, in which the three balls take off at the same
time because the three factors compensate each other. In these
cases the operation is:
( + w) X ( + D) = before ( = W) X ( + D) = before
( _ w) X ( - D) = after ( = W) X ( - D) = after
( + W) X ( = D)= before W
( = W) X ( = D) = equality
where : W = =t heavy, it D = more or less distance from the
center.
So once again we encounter the same double-entry table we
found for work relationships (Chap. 12), density, etc., and again
we encounter it at substage II-B.
216 THE FORMAL OPERATIONAL SCHEMATA
One may be tempted to believe that the concept of multiplica-
tive compensation is acquired at this substage. But two circum-
stances contradict this assumption. First, as we have stated above,
the subject does not yet succeed in isolating the factors by him-
self. He does perform the operation correctly when the data are
prepared in advance, but this is not the same as a spontaneous
organization of deductive proof based on his previous structuring
of the relevant elements (the latter is the general procedure of
experimental method and requires the formal combinatorial sys-
tem). Second, the operation of logical multiplication the only one
available to the concrete level subject remains indeterminate in
the cases ( + W) X ( D) and ( W) X ( + >) There may
be compensations in these cases, but they do not follow from
these products. The result cannot be determined completely with-
out making use of proportions, and proportions necessitate formal
thought.
Stage III. Spontaneous Isolation of Variables and
Compensation by Proportionality
At substage III-A the subject can organize the experiment with-
out outside help, and he can anticipate compensations by using
a system of prepositional operations. Still, his deductions are
incomplete:
CHAM (10 ; 7, advanced) begins haphazardly by placing W 3 at D 3,
W 2 at D 2 and W i at D i: "They left one after the other!' "Does
that surprise you?" "No. The "biggest should move after [the others!
because it turns more slowly. . . . Oh! No. It's the other way around.
The closer it is to the edge, the faster it turns." To test his hypothesis,
lie puts W 3 at D 3, W 3 at D 2, and W 3 at D i; then, on a new trial,
W i at D 3 and W i at D 2. He varies the distances at equal weights
and uses the extreme weights for the counterproof. "Those [D 3] took
off first. The middle ones go next' [D 2,]. Then, after several new trials
for distance, he discovers that when he puts W 3 at D i and W i at
D 3, they talce off at the same time, but that when he puts W 3 and
W 2, at D 3, "the big one goes first." Next, he compares the problem
to one of equilibrium: "It's a little like a balance scale fin the sense
of movements required for balancingl; it's a question of equilibrium.
CENTKIFUGAL FORCE AND COMPENSATIONS 217
If [W 3] and [W i] are at [D 3] they have no equilibrium; if they
are at \D 2] there is a little more; if it's at [D i] it's still better in
equilibrium; and if it's in the center it's completely in equilibrium. The
closer it [the ball] is to the center, the slower it goes. . . . But if [W$]
is at [D i] and [W i] at the center, the big one goes first" "What
determines the result?" "The size and the holes a little; no, only the
size because the holes are all the same; and the force with which the
ball is thrown off." "What determines the equilibrium?* "The size
of the balls and their places. If you put the same size [W i] at [D 3]
it moves before [W i] at the center. If you want them to go at the
same time, you have to put a big one here" [at D i when W i is at
DEF (11 ; 2) also discovers the role of the weight and distance. He is
asked to predict the result when W"3 is at D i and W 2, is at D 2:
"They will go at the same time. The big one has a larger distance to
cover, but it is heavier and heavier things go faster. 9 ' "And?" [W 3
at D i, W 2, at D 2, and W i at D 3]. "They witt go at the same time.
The big one is heavier but has a larger distance to cover, so it comes
out the same."
vis (12 ; 9), after the same train of reasoning: "That's a compensation."
DUB (13 ; 4), after he has discovered the two factors by himself: "Can
you make the balls go at the same time? 9 * "You have to form a pro-
portion: the weight and distance [he puts W 3 at D i and W 2 at D 3].
They didn't go together because of the difference in weight and the
difference in distance. They have to counterbalance each other exactly."
It is worth noting that the subjects adopt a new attitude toward
compensation at exactly the point when they become able to
isolate the variables (without hel 
