Journal of Experimental Psychology
1964. Vol. 67, No. 3, 250-255
CONJUNCTIVE AND DISJUNCTIVE CONCEPT FORMATION
UNDER EQUAL-INFORMATION CONDITIONS *
MICHAEL B. CONANT AND TOM TRABASSO*
Stanford University
College students learned sets of conjunctive and inclusive disjunctive
concepts in which the minimum number of choices necessary for solution
was equated. The disjunctive concepts were more difficult to master.
Selection of instances and redundancy in choices indicated that Ss learn
to use a positive-focusing strategy within a conjunctive problem sooner
than they learn to use a negative-focusing strategy within a disjunctive
problem.
In a discussion of the logical structure of concepts, Bruner, Goodnow,
and Austin (1956) made particular
note of the difference between conjunctive and disjunctive concepts.
A conjunctive concept is defined by
the joint presence of several values.
An example of a conjunctive concept
is "blue-eyed and red-haired men."
A disjunctive concept, on the other
hand, is defined by the presence of one
or another value (e.g., "tall or thin
men"). Hunt (1962) makes a further
distinction between inclusive and exclusive disjunctivity. In an inclusive
disjunction, either value or both may
occur; in an exclusive disjunction,
either value but not both may occur.
In studies of concept formation,
Bruner et al. found that .Ss have
difficulty working with disjunctive
concepts; the authors suggested that
this was due, in part, to the necessity
of using negative instances in order to
solve disjunctive problems most efficiently. Indirect evidence for this
hypothesis came from an earlier study
1 Acknowledgment is made to David L.
Brown for his guidance during the execution
of the research. The second author was supported by a postdoctoral research fellowship,
No. 5 F2 MH-18,070-02, from the National
Institute of Mental Health, United States
Public Health Service.
* Now at the University of California, Los
Angeles.
250
by Hovland and Weiss (1953), who
found that 5s are inefficient in using
negative instances to solve conjunctive
problems. Hunt and Hovland (1960)
studied problems in which conjunctive, relational (e.g., larger than), and
disjunctive solutions were equally correct. They found conjunctive and
relational solutions are more frequently offered than disjunctive ones.
These authors suggest that the frequency of each type of solution may
be an indication of its relative difficulty. Recently, Wells (1963) obtained transfer of training from disjunctive concept formation to choice
problems where either a conjunctive
or a disjunctive solution was possible,
indicating that prior learning of concept types is a factor in the difficulty
of concept learning. Wells' training
procedure apparently modified prior
habits of focusing on positive instances
and thereby facilitated disjunctive
concept formation.
The purpose of the present study
was to obtain a direct comparison
between conjunctive and disjunctive
problems given to the same 5s. This
aim was accomplished by constructing
conjunctive- and disjunctive-concept
problem sets which were structurally, perceptually, and informationally
equivalent.
Stimuli.colored figu
cards. On
the left and
varied inde
(large or s
making 16
Procedut
the experiir
before S at
dimensions
and approp
each conji
problems; i
general nat
by having
negative e:
sample con
problem, L
cards as eit
of an unknc
single cards
his choice w
The 5 grot
to class, an
he verbalize
solutions w<
At the o
instructions
This is
think. A
laid out <
one circl<
triangles
instance,
green. T
For pu
cept will
of these
cards has
find out
possible, i
to you.
Then, be:
was given tc
were read:
Anexar
dealing wi
contain I
circles. V
cards her
having b
circles?
That is
have this
concepts
CONCEPT FORMATION AND INFORMATION
LMATION
S1
ve
on
er.
rn
icr
ve
;i953), who
;nt in using
conjunctive
land (1960)
:h conjuncthan), and
equally cornctive and
more fre:ictive ones,
at the fredution may
lative diffi(1963) obj from disn to choice
conjunctive
as possible,
,ing of conic difficulty
Is' training
dified prior
re instances
disjunctive
rsent study
comparison
disjunctive
e 5s. This
Dnstructing
ive-concept
5 structurmationally
METHOD
Stimuli.—The stimuli consisted of two
colored figures drawn on white 3X5 in. file
cards. On each card there was a triangle on
the left and a circle on the right. The figures
varied independently in two dimensions, size
(large or small) and color (red or green),
making 16 different patterns.
Procedure.—The 16 cards used throughout
the experiments were placed in a 4 X 4 array
before S at the start of each problem. The
dimensions and values were described to S,
and appropriate instructions were read before
each conjunctive and disjunctive set of
problems; it was ascertained that 5 knew the
general nature of the relevant concept type
by having him pick out all positive and
negative example cards associated with a
sample concept of each type. To begin a
problem, E presented 5 with 1 of the 16
cards as either a positive or negative example
of an unknown concept. The 5 then selected
single cards and was told each time whether
his choice was a positive or negative instance.
The 5 grouped the selected cards according
to class, and a problem was completed when
he verbalized the concept. Premature verbal
solutions were discouraged.
At the opening of a session, the following
instructions were read:
This is an experiment to see how you
think. As you can see, there are 16 cards
laid out on the table. Each card contains
one circle and one triangle. Circles and
triangles vary both in size and color. For
instance, a circle is large or small and red or
green. The same holds for triangles.
For purposes of this experiment, a concept will be considered to be a certain set
of these cards. A concept about these
cards has been chosen. Your job will be to
find out the concept as efficiently as
possible, in a manner that will be described
to you.
Then, before a set of conjunctive problems
was given to 5, the following three paragraphs
were read:
An example of the type of concept we are
dealing with might be all those cards which
contain both large triangles and green
circles. Would you please point out all the
cards here which have the property of
having both large triangles and green
circles?
That is correct. Four of the 16 cards
have this property, and with the type of
concepts we will now deal with, 4 of the
251
cards will always be examples of the concept, and 12 will not.
Please keep in mind that with the type
of concepts we are now dealing with, just
one property of the circles is required (red,
large, green, or small), and one property
(red, large, green, or small), not necessarily
the same, is required for triangles. In
other words, small red triangles is not a
concept we are dealing with, because this
has two properties of one figure. Each
concept, again, requires that its examples
have just one property of the circles and one
property of triangles. Do you have any
questions about this?
Before the set of inclusive disjunctive
problems, the following three paragraphs
were read:
An example of the type of concept we are
dealing with might be all those cards which
contain either large triangles or green circles
or both. Would you please point out all
the cards here which have the property of
having either large triangles or green
circles or both?
That is correct. Twelve of the 16 cards
have this property, and with the type of
concepts we will now deal with, 12 of the
cards will always be examples of the concept, and 4 will not.
Please keep in mind that with the type of
concepts we are now dealing with, just one
property of circles is required (red, large,
green, or small) and one property (red,
large, green, or small), not necessarily the
same, is required for triangles. In other
words, small red triangles is not a part of
a concept we are now dealing with, because
this has two properties of one figure. Each
concept requires that its examples have
just one property of circles or one property
of triangles. Do you have any questions
about this?
Following each 'set of conjunctive or
disjunctive instructions, all 5s were told:
I will give you a card that is or is not
(and you will be told which) an example
of the concept. Your job will be to try
other cards, one at a time. I will tell you
after each choice whether or not these are
examples of the concept. You may guess
at the concept at any point, but wrong
guesses will result in a subtraction from your
score. However, there is obviously no
penalty for picking particular cards which
are not examples of the concept. Your
score will be dependent on how few cards
252
MICHAEL B. CON ANT AND TOM TRABASSO
you have to try before you are sure of what
the concept is. When you have arrived at
the concept, tell me what it is. If you are
correct, that problem will be finished and
we will go on to another concept. Time is
not a factor, only the number of cards you
have to try. Work efficiently, but do not
hurry. You may take as long as you wish.
You may rearrange the cards during trials
in any way that may be helpful to you.
We will make two rows at the side, one for
cards that are examples of the concept, and
a row for those that are not. Do you have
any questions?
Design.—A 2X2X2 design was used
with the following variables: (a) two sets of
three problems, a conjunctive set (C) and an
inclusive disjunctive set (D); (&) two presentation orders, C-D and D-C; and (c) sex.
Each S was given both C and D sets and was
randomly assigned to one of the two presentation orders.
Problems.—Table 1 summarizes the two
sets of three C and D problems along with the
example card given before each problem for
Exp. I (for the problems of Exp. II, see
below). Given the instructions and the
stimuli, there are 16 possible triangle-value
and circle-value solutions for each problem.
If an example card is positive for a C problem
or negative for a D problem, 12 of these
possible solutions are eliminated. Similarly, if
an example card is negative for a C problem
and positive for a D problem, 4 of the possible
solutions are eliminated. Thus, positive C
instances and negative D instances yield the
most information and their respective selection would lead to the most efficient C or D
problem solving. When the example card was
positive, a C problem could be solved in two
card choices, and a D problem in three or four
choices, depending upon 5"s first card choice.
When the example card was negative, a D
problem required two choices and a C
problem, three or four. The C and D> sets of
three problems each were equated as to the
minimum number of choices necessary for
solution, 8-10 choices for Exp. I, and 9-12
choices for Exp. II.
Experiments.—Two experiments were performed. In the second experiment, the D
problems in Table 1 were made C problems
and vice versa. Thus, all C example cards
became negative and all D example cards
became positive. The second experiment was
conducted to assess whether or not the
particular value-value pairs contributed to
any possible differences observed in Exp. I.
Subjects.—For Exp. I, the 5s were 24
volunteers, 12 males and 12 females, from
psychology classes at Columbia University.
For Exp. II, there were 12 volunteers, 6 males
and 6 females, from the same student
population.
RESULTS AND DISCUSSION
Efficiency was first evaluated by
comparing the number of choices required to solve all three problems of a
concept set. Table 2 summarizes the
analysis of variance for each experiment.
In both experiments, the D set
required more choices to solution than
the C set. In Exp. I, the mean total
choices were 19.04 for the D set and
13.75 for the C set. For Exp. II,
these values were 19.00 and 15.05,
respectively. No other main effect or
TABLE 1
CONJUNCTIVE AND INCLUSIVE DISJUNCTIVE SETS USED IN EXP. I
Concept
Example
Circle
Triangle
Circle
Instance
Cl
C2
C3
G
L
R
and
and
and
L
R
S
S,G
S,G
L, R
L, R
S,R
L.G
Positive
Negative
Negative
Dl
D2
D3
L
G
S
and/or
and/or
and/or
R
G
L
L,G
L, R
S,R
L.G
S, R
Positive
Negative
Positive
S,G
Note.—The values of the stimuli were G (Green), L (Large), S (Small), and R (Red).
Between 5
Sex (S)
Order (C
S XO
Error (b)
Within 5s
Concept
c xs
c xs >
C XO
Error (v
* p < .05
** p < .01
interactior
transfer b
observed,
tions betvi
required o
sets were
quired all
the maxin
S for a D
problem, !
The me
problem
Relative
cannot be
of confoi
example c
trend.
If 5 se
MEAN
Problem
Triangle
Source
Concept
C
D
CONCEPT FORMATION AND INFORMATION
st card choice,
negative, a D
es and a C
and D sets of
ited as to the
necessary for
. I, and 9-12
3nts were periment, the D
e C problems
example cards
sample cards
cperiment was
or not the
^ntributed to
d in Exp. I.
5s were 24
females, from
a University,
iteers, 6 males
ame student
SSION
aluated by
choices reoblems of a
marizes the
each exthe D set
lution than
mean total
D set and
r Exp. II,
and 15.05,
tin effect or
TABLE 2
ANALYSIS OF VARIANCE ON CHOICES TO SOLUTION
Exp. I
Exp. II
Source
F
2.00
.05
1.57
8
15.04
.37
11.81
7.53
12
1
1
1
1
8
92.04
15.05
.21
5.21
12.81
7.20*
1.18
.02
.41
MS
F
df
6.02
20.02
1.69
28.46
.21
.72
.06
11
1
1
1
Error (b)
23
1
1
1
20
Within 5s
Concepts (C)
C XS
CXO
C XSXO
Error (w)
24
1
1
1
1
20
336.02
3.52
35.02
3.69
14.36
23.40**
.25
2.44
.26
Between 5s
Sex (S)
Order (O)
sxo
* p < .05.
**P <.01.
interactions were significant and no
transfer between concept types was
observed. Product-moment correlations between S's number of choices
required on each of the two problem
sets were not significant. No 5 required all cards to solve a problem;
the maximum number selected by an
S for a D problem was 14 and for a C
problem, 11.
The mean number of choices per
problem is reported in Table 3.
Relative transfer within each set
cannot be properly evaluated because
of confounding with the class of
example cards and inconsistencies in
trend.
If S selects instances at random
Instance
Problems
Positive
Positive
Negative
Positive
MS
df
TABLE 3
MEAN NUMBER OF INSTANCES CHOSEN
PER PROBLEM
Negative
Negative
253
1
C
D
Exp. II
Exp. I
Concept
2
4.17 5.21
7.00 5.75
3
1
2
3
4.37 5.48 5.25 4.32
6.29 6.91 5.76 6.33
throughout a problem, then the proportion of positive instances chosen
should be near .25 for C concepts and
.75 for D concepts. However, if S
uses a "positive-focusing strategy"
(cf. Bruner et al., 1956) which is
efficient for C problems and, conversely, a "negative-focusing strategy" which is efficient for D problems,
these respective proportions should be
higher or lower. To study this, each
6"s problem was divided into halves
and the number of positive C and
negative D instances chosen per half
was counted. These Vincentized percentages were nearly equal for the two
experiments and were pooled for the
summary presented in Table 4.
Summing over all problems, the
proportion of positive C choices and of
negative D choices was above chance
expectations. The first- and secondhalf comparisons in Table 4 indicate
that S learns to select, within a
problem, a positive instance under C
conditions more rapidly than a negative instance under D conditions.
The S would appear to solve C concepts sooner since he learns to choose
positive instances within a C problem
254
MICHAEL B. CONANT AND TOM TRABASSO
TABLE 4
PROBABILITY OF A POSITIVE C AND A NEGATIVE D CHOICE
i'.A. »-. >
Proportion per Half
Problems
1
2
3
Total
Positive C
First
Second
.24
.56
.61
.64
.62
.22
.22
.22
Negative D
Trials
72
84
76
232
more rapidly than negative instances
within a D problem. Since the improvement in selection occurs only
during the second half of the problem
and not over successive problems,
there is no apparent transfer of either
a positive- or negative-focusing strategy. The tendency to choose more
informative instances may result from
the information obtained on initial
choices. That is, a positive C card
may be easier to find once 5 has
selected a few cards, be they positive
or negative.
An information analysis was performed on the card choices with
respect to the number of redundant
and nonredundant card selections to
solution. A card choice was defined
as redundant if it could not eliminate
at least one further incorrect solution
beyond those already eliminated by
the example card, preceding card
choices and, if verbalized by 5, wrong
hypotheses. Admittedly, one does
not know exactly what solutions were
tried by an S but this analysis provides
indirect evidence for informational
use.
According to this analysis, nearly
all incorrect solutions were eliminated
by card choices. Eighty-seven percent of the problems were solved with
all incorrect solutions eliminated; the
remainder were solved with three or
less incorrect solutions still possible.
First
Second
Trials
.19
.17
.21
.20
.32
.36
.37
.35
117
92
107
316
The average nonredundant choices
were nearly equal: for Exp. I, they
were 9.04 for C problems and 8.75 for
the D set, and for Exp. II, these
values were 10.57 for the C set and
11.20 for the D set. However, more
redundancy in choice occurred on D
problems. For Exp. I, mean redundant choices were 4.71 for the C set
and 10.29 for the D set, ^(1,20)
= 32.02, p < .01; and for Exp. II,
these values were 4.48 for the C set
and 7.80 for the D set, F (1, 8) = 7.61,
P < .05. No other main effect or
interaction was significant.
The relative frequencies with which
an 5 chose positive and negative instances on the final card for a problem
are summarized in Table 5. The
percentage of these choices which were
redundant is also reported.
The very high proportion of positive
C instances selected as the last card
and the above-chance selection of
TABLE 5
PROBABILITY OF POSITIVE AND NEGATIVE
FINAL CARDS
Concept
Instance
C
D
Choices Redundant Choices Redundant
Positive
Negative
.85
.15
.66
.75
.64
.36
.84
.65
more freqt
redundant
for the D
these selec
redundant
series.
Since th
C and D <
labels S i:
which S's
may be tl
is unfamil
concept,
proffer a
hence, re<
this possi
the first
correct) v
the raro s
ences wei
mean tria
problem 1
was 4.2*
p > .05;
values w
t (ID =
potheses
56% of t
An alt
redundai
basis of
concepts
a positiv
dant th£
a negatr
CONCEPT FORMATION AND INFORMATION
Trials
117
92
107
316
mt choices
xp. I, they
md 8.75 for
. II, these
C set and
/ever, more
arred on D
lean redunr the C set
t, F(l,20)
>r Exp. II,
r the C set
,8) = 7.61,
n effect or
with which
legative inr a problem
ie 5. The
which were
•
i of positive
ie last card
selection of
D NEGATIVE
'ices Redundant
4
16
.84
.65
negative D instances are consistent
with use of positive- and negativefocus strategies within a problem.
Last-choice redundant negatives are
more frequent for the C set, whereas
redundant positives are more frequent
for the D set. This is expected since
these selections are more likely to be
redundant when chosen later in the
series.
Since there is no difference, between
C and D classifications except for the
labels 5 is forced to apply, the form
which S's verbal hypothesis must take
may be the critical variable. If an 5
is unfamiliar with the language of a D
concept, he may be reluctant to
proffer a statement until doubly,
hence, redundantly certain. To test
this possibility, the trial upon which
the first hypothesis (correct or incorrect) was offered was compared in
the two sets and no significant differences were obtained. In Exp. I, the
mean trial of the first hypothesis per C
problem was 3.82 and per D problem
was 4.28, matched t (23) = 1.28,
p > .05; and in Exp. II, the respective
values were 4.55 and 4.69, matched
t (11) = .32, p > .05. Of the first hypotheses offered, 75% of the Cs and
56% of the Ds were correct.
An alternative interpretation of the
redundancy data can be made on the
basis of 5's choice behavior. With C
concepts, a negative choice following
a positive is more likely to be redundant than a positive choice following
a negative. The reverse is true for D
255
concepts. For the C set, the conditional probability of a negative following a positive was .52; whereas for the
D set, the conditional probability of a
positive following a negative was .66.
Therefore, in C learning, the fewer
redundant choices result from both
the lower number of choices to solution and the higher incidence of
informationally rich positive choices.
In D learning, the redundancy results
from the larger number of choices and
the high incidence of low-information
positive choices. An 5 appears to be
more efficient in using positive C
choices than negative D choices, even
though both instance types yield the
same high information. In particular,
comparing Problems Cl and D2 of
Exp. I, where Cl began with a positive
example and D2 with a negative, the
ratio of redundant to nonredundant
choices in D2 was twice that of Cl.
REFERENCES
BRUNER, J. S., GOODNOW, J., & AUSTIN, G.
A study of thinking. New York: Wiley,
1956.
HOVLAND, C. I., & WEISS, W. Transmission
of information concerning concepts through
positive and negative instances. /. exp.
PsychoL, 1953, 45, 175-182.
HUNT, E. B. Concept learning. New York:
Wiley, 1962.
HUNT, E. B., & HOVLAND, C. I. Order of
consideration of different types of concepts.
/. exp. PsychoL, 1960, 59, 220-225.
WELLS, H. Effects of transfer and problem
structure in disjunctive concept formation.
/. exp. Psychol., 1963, 65, 63-69.
(Received April 29, 1963)