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AMBIGUITY AS A PREDICTOR OF
SYLLOGISTIC DIFFICULTY

 

presented by
Thomas M. Steinfatt

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ABSTRACT
AMBIGUITY AS A PREDICTOR OF SYLLOGISTIC DIFFICULTY

BY

Thomas Martin Steinfatt

Bettinghaus, Miller, and Steinfatt (1970) investi-
gated the relationship between the cognitive style variable
of dogmatism and the ability to evaluate syllogistic valid-
ity. Their results suggest that different syllogistic forms
may vary in difficulty and that the meaningfulness of the
syllogistic terms, and the dogmatism of the persons judging
the syllogisms, may affect this difficulty.

The present study sought to extend the Bettinghaus
._E._l. findings by examining the relationship of syllogistic
ambiguity and judgmental difficulty. A set of 21 logically
unique syllogistic premise combinations was selected to
represent all possible syllogistic forms. Each of these
premise combinations was presented to 207 subjects with high
meaningful and with low meaningful trigrams as the terms of
the syllogisms. Two conclusions appeared with each premise
combination, at least one of which could have produced an

atmosphere effect.

 

Thomas Martin Steinfatt

It was hypothesized that the degree of difficulty of
judgment of the validity-invalidity of a syllogistic conclu—
sion would increase directly with the ambiguity of the prem—
ise combination associated with that syllogism. Ambiguity
was defined by considering a source who encodes a message
which is capable of being transformed into a syllogism. The
receiver is assumed to be unaware of the original relation-
ship between the terms of the syllogism which led the source
to encode the message. In judging the validity of the syllo-
gism, the receiver must attempt to reconstruct this source-
observed relationship by analyzing the premises.

Given any two premises involving three terms, there
are between two and eighty possible relationships which
could obtain between the three terms, assuming the truth
of the premises. The ambiguity of a premise combination is
defined as this number: the number of possible relation—
ships between the terms of a syllogism from which the prem-
ise combination of the syllogism could have originated. The
receiver must sort through each of these relationships to
determine if the conclusion is valid. As the number of
relationships increases, so does the probability that the
receiver will fail to consider, or will confuse, one or more
possible relationships, and thus will judge the validity-

invalidity of the syllogism incorrectly.

Thomas Martin Steinfatt

The results support the hypothesis. The main effect
for syllogistic forms is significant at the .001 level,
indicating that there are differences in difficulty among
different syllogistic forms. A correlation of .695 was
obtained between the observed rank order and the order pre-
dicted by the ambiguity hypothesis for the 21 syllogistic
forms.

Term meaningfulness was found to interact with the
syllogistic form variable such that for certain forms, low
meaningful content was easier to judge, while for others,
low meaningful content was more difficult. These differences
were significant for only three of the 21 items at the .01
level.

An atmOSphere-type effect was observed with five of
the 21 items. Dogmatism did not enter into any significant

effect.

AMBIGUITY AS A PREDICTOR OF SYLLOGISTIC DIFFICULTY

BY

Thomas Martin Steinfatt

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Communication

1971

DEDICATION

To Walter Louis Steinfatt, my father.
He was a great bridge player.

And to John Shawanibin, my friend.
He was a great hockey player.

ii

ACKNOWLEDGMENTS

Many people have assisted in producing this disser—
tation. Dr. Gerald Miller and Dr. Erwin Bettinghaus, my
major professors, made many helpful suggestions. I grate-
fully acknowledge their assistance in helping me to think
through the various parts of the study and to bring into
focus the major results.

Dr. Hideya Kumata and Dr. Lawrence Messe, the other
members of my committee, also reviewed the manuscript and
offered suggestions which were incorporated into the disser—
tation. I eXpress my sincere appreciation to them for their
help.

Dr. Estelle Zannes and Mr. Eugene Davis of The
Cleveland State University made available the students of
the basic speech course as subjects for this study. I
express my thanks to them and to the-Department of Speech
of The Cleveland State University for this positive attitude

toward research.

****

iii

TABLE OF CONTENTS

Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . Vi
Chapter
I. INTRODUCTION AND RATIONALE . . . . . . . . . . l
Dogmatism Factor . . . . . . . . . . . . . . 5
Meaningfulness Factor . . . . . . . . . . . 6
Syllogistic Form Factor . . . . . . . . 9
II. METHOD AND PROCEDURES . ._. . . . . . . . . . 15
Subjects . . . . . . . . . . . . . . . . . . 15
Test Administration . . . .H. . . . . . . . 23
Dogmatism Prediction . . . . . . . . . . . . 24
General Design and Analysis . . . . . . . . 25
III. RESULTS . . . . . . . . . . . . . . . . . . . 26
Test of Major Hypothesis . . . . . . . . . . 26
Individual Comparisons . . . . . . . . . . . 29
Dogmatism and Meaningfulness Factors . . . . 32
Diagrams in the Test Booklets . . . . . . . 36
IV. DISCUSSION . . . . . . . . . . . . . . . . . . 37
Ambiguity Hypothesis: Alternative
Explanations for Non-Conforming Items . . 37
Ambiguity and the Existence of a Valid
Conclusion as Predictor Variables . . . . 43
Ambiguity as a Predictor with Only
Invalid Conclusions . . . . . . . . . . . 46
Premise Combination Type, Figure and
Items with Equal Ambiguity . . . . . . . . 48

Meaningfulness Main.Effect and the
Meaningfulness by-Syllogistic Form
Interaction . . . . . . . . . . . . 49
Syllogistic Form Main Effect and the
Meaningfulness by Syllogistic Form
Interaction . . . . . . . . . . . . . 52

iv

Summary . . . . . . . . . . .
Concluding Remarks . . . . . .
REFERENCES . . . . . . . . . . . . . . .
Appendix
I. DEFINITION OF THE SYLLOGISTIC FORM FACTOR

II.

Dogmatism Main Effect .

Diagrams in the Test Booklets and.

Ambiguity . . . .

The Atmosphere Effect .

GENERAL METHOD FOR DETERMINING AMBIGUITY

Page
54
55
56
64
67

69

71

82

Table

LIST OF TABLES

Ambiguity of 21 syllogistic forms . . . .

Three factor anova for scores of 207
subjects . . . . . . . . . . . . .

Percentage correct and rank order for 21
syllogistic forms . . . . . . . . . .

Significant differences between pairs of
21 syllogistic forms . . . . . . . .

Twenty-one forms ordered by invalid items
only . . . . . . . . . . . . . . . . .

Twenty-one-forms ordered after removal of
possible atmosphere effect . . . . . .

Twenty—one forms ordered by invalid items

only after removal of possible atmosphere
effect . . . . . . . . . . . . . . .

vi

Page

20

27

3O

31

47

59

61

CHAPTER I

INTRODUCTION AND RATIONALE

As McGuire points out, ”the 'RATIONAL MAN' CONCEPT,
long out of fashion in the behavioral sciences, is now
undergoing a remarkable revival in the study of cognition"
(1960, p. 65). This study extends that revival by attempt—
ing to establish a baseline on the difficulty of judging
the validity-invalidity of different syllogistic forms over
different levels of meaning for subjects of varying degrees
of dogmatism.

Bettinghaus, Miller, and Steinfatt (1970) investi—
gated the relationship between the cognitive style variable
of dogmatism and the ability to evaluate syllogistic valid-
ity. They found support for the hypothesis that low dog-
matic persons are better able to judge the validity of
logical syllogisms than are high dogmatic persons when such
syllogisms originate from positive and negative sources. A
prediction of a second order interaction between dogmatism,
validity, and source evaluation was also confirmed. These
researchers used a pretest to determine which of nine valid
and nine invalid syllogistic forms would be answered cor-
rectly by about half of all subjects. The terms used in

these pretest syllogisms were abstract; specifically, they

were the capital letters A, B, and C. Two valid and two
invalid forms were observed to have between 45 percent and
54 percent correct responses from all subjects with the
abstract terms.

When more meaningful terms were employed in the main
test (words and phrases concerning the war in Vietnam) the
percentage of correct responses for each of the four forms
changed radically from that obtained with abstract terms
in the syllogisms. This was true deSpite rather elaborate
counterbalancing of the eXperimental conditions intended to
equalize the probability of correct judgments on valid and
invalid syllogisms. Subjects who judged syllogisms contain-
ing meaningful terms gave significantly more correct answers
for valid than for invalid items.

The four forms found to be of near median difficulty
when using abstract terms were: (1) the Third figure
(Ambrose and Lazerowitz, 1948, p. 265), containing §I_
premises and an Q_conclusion; (2) the Third figure, contain-
ing E;.premises and an I_conclusion; (3) the Second figure,
containing E; premises and an g_conclusion; and (4) the
First figure, containing II_premises and anII conclusion.
Syllogisms (l) and (3) are valid and (2) and (4) are invalid.
The percentage of correct responses to syllogisms with high
meaningful terms using each of these forms was: (1) ElggJ
78 percent; (2) EIIQ, 77 percent; (3) glgg, 60 percent; and

(4) I111, 22 percent. Thus, the significant main effect for

valid over invalid items was not general, but was due almost
exclusively to syllogisms of the III; figure type. However,
the Bettinghaus EE.§L- study was not designed in such a way
that the relationship between correct judgments of validity-
invalidity and a syllogistic form factor could be analyzed,
since their design purposely included only four syllogistic
forms.

As mentioned above, Bettinghaus _E;_L- also found
that high dogmatic subjects were significantly less able
to judge syllogistic validity-invalidity than low dogmatics
when the syllogisms originated from positive and negative
sources. By inspecting the percentages of correct responses
given by high and low dOgmatic subjects, they discovered
that on certain syllogisms with high meaningful content, low
dogmatics gave many more correct responses than high dogmat—
ics but on most items the low dogmatics did only slightly
better than the high dogmatic subjects. The syllogisms
showing large differences in favor of the low dogmatics were
of the IIII figure and the EII§_figure. This finding sug-
gests the possibility of an interaction between dogmatism
and a syllogistic form factor, but gives few clues as to the
nature of such an interaction since only four syllogistic
forms were involved and three of these employed the same
premise combination (EI).

To summarize, the form of the syllogism employed in

the Bettinghuas et al. study appears to have affected the

validity-invalidity factor to the extent that a significant
validity main effect occurred when high meaningful terms
were employed in the syllogisms, and high and low dogmatic
subjects appear to have been affected differentially by
differing syllogistic forms. Although the predicted sig—
nificant second order interaction between dogmatism, source,
and validity does not appear to have been enhanced or ham-
pered by the form of the syllogisms employed, the discrep-
ancies in percentage of correct judgments reported between
syllogisms employing abstract terms and those employing more
meaningful terms suggests that checking a given syllogistic
form for ease of judgment of validity with abstract content
is not an adequate control for difficulty of judging syllo-
gistic validity when the terms of the syllogism are highly
meaningful.

These findings suggest that three factors and their
relationships need study before the Bettinghaus §£_§l,
results can be fully understood. These variables are dog—
matism, syllogistic form, and term meaningfulness. An
understanding of the relationships between these three
variables is of interest in itself, and information con—
cerning these relationships and the criterion variable of
judgment of syllogistic validity-invalidity may be necessary
to control for the differential effects, if any, of varying
syllogistic forms and high and low term meaningfulness in

future studies with high and low dogmatic persons.

Dogmatisszactor

As mentioned above, Bettinghaus gt QL. hypothesized
and found a relationship between the degree of dogmatism of
a subject and his ability to judge the validity-invalidity
of non-abstract syllogisms. Since this personality trait
appears to be related to judgmental ability, a study at—
tempting to establish baselines for the ability to judge
syllogisms should take it into account. In discussing the
distinction between high and low dogmatic subjects, Rokeach
states that the more closed the belief system of an individ-
ual, the more

the relation among beliefs should depend on such
irrelevant considerations [as outside authority]
rather than on considerations of logical consis-
tency. Isolation between parts [as in a closed
system] reflects a tendency not to relate beliefs
to the inner requirements of logical consistency,
but to assimilate them wholesale, as fed by one's
authority figure. On the other hand, the more
Open the system, the more should the person
address himself to objective structural require—
ments--that is, logical relationships. . . (1960,
p. 61).

This statement led Bettinghaus et al, to hypothesize
the relationship stated above, but the hypothesis was limited
to syllogisms originating from positive and negative sources.

Since positive and negative sources were not linked
to the items used in this study, no dogmatism effect was
predicted or expected. Still, the inclusion of dogmatism

as a factor in this experiment seemed justified both by the

Bettinghaus eg_al. findings, which suggest a possible

interaction of dogmatism and syllogistic form on particular
syllogistic forms, and by way of establishing the necessity
of positive and negative sources in producing a dogmatism

effect.

Meaningfulness Factor

Bettinghaus gt al, found striking differences in
the judgmental behavior of the same subjects when confronted
with the task of judging syllogisms with meaningful content
‘as Opposed to judging those same syllogisms with the capital
letters A, B, and C used as terms. The use of these capital
letters as terms was an attempt to determine the relative
difficulty of the syllogistic forms themselves without the
intrusion of meaningful content on the subjects' judgmental
processes. While apprOpriate for their study, this test of
the relative difficulty of syllogistic forms has two fea-
tures which reduce its generalizability. First, the letters
A, B, and C may be more meaningful than Bettinghaus g£_al.
assumed them to be, since no norms were available on the
meaningfulness of these letters. Second, the syllogistic
forms chosen for use were only those valid forms which had
a specifically invalid counterform and those counterforms.
This latter problem is discussed below and in Appendix I in
some detail in arriving at a more general syllogistic form

factor. The former problem is one of term meaningfulness.

Many studies have shown that the nature Of the terms
Of a prOposition may have a direct bearing on the probabil—
ities Of a given subject judging it to be valid or invalid
(Bettinghaus g§pal., 1970; Feather, 1964; Janis anthrick,
1943; Lefford, 1946; Morgan and Morton, 1944; Parrott, 1967;
Thistlethwaite, 1950; Thouless, 1959). Much Of this litera-
ture indicates that many subjects attempt to judge the truth
Of the conclusion in terms Of its consequence for their own
beliefs, rather than its logical validity. McGuire (1960)
refers tO this distinction as being between "wishful think-
ing" and "logical thinking."

Henle and Michael take issue with this distinction,
going so far as to refer to the findings Of Morgan and
Morton (1944) as "artifacts," since Morgan and Morton assume
that the influences Of attitudes on the judgment Of syllo-
gistic forms are arbitrary in nature, "that attitudes
Operate blindly, indifferent tO the nature Of the material
on which they act" (1956, p. 125).

There are at least two distinctions involved in
these studies that merit clarification. Most Of the
researchers cited above have employed syllogisms with some
form Of symbolic content as a control condition, along with
the same syllogisms used as Vehicles for expressing state-
ments which have attitudinal implications in an experimental
condition. At least two separate variables are involved in

moving from symbolic content tO attitudinal content:

meaningfulness and attitude. Symbolic content is usually
assumed tO be Of low meaningfulness and to be non-attitudinal
in nature. Words and phrases involved in the terms Of syllo-
gisms in experimental conditions are usually Of the high
meaningful, attitude relevant type. Differences found
between these two types Of conditions are related to changes
in two independent variables, meaningfulness and attitude
arousal, which are confounded in the eXperimental conditions.

Morgan and Morton seek to draw this distinction
implicitly when they differentiate between syllogisms which
contain nothing in their terms to arouse a response based on
personal convictions and syllogisms whose terms relate to
the personal Opinions and fears Of an individual (1944,
pp. 58-59). Wilkins also approaches this distinction when
she comments that "ability to dO formal syllogistic reason—
ing is much affected by a change in the material reasoned
about. The easiest material is the familiar and concrete.
The most difficult is the unfamiliar" (1928, p. 77). Morgan
and Morton's distinction is attitudinal while Wilkins' is
one Of meaningfulness.

As Henle and Michael point out, the effect Of the
attitudinal variable is specific to the relationship between
the attitude held by the subject and the attitude eXpressed
in the syllogism. Thus, the present study will not attempt
to investigate this attitudinal variable; rather, it will

investigate the effect Of the meaningfulness variable. To

what extent does a change in the meaningfulness Of the terms
Of a syllogism, without an attempt to relate the meaningful-
ness tO relevant attitudes Of the subjects, affect the
ability Of those subjects tO judge the validity-invalidity
Of a syllogistic fOrm?

Morgan and Morton (1944) suggest that no differences
based on meaningfulness alone will occur across syllogistic
forms, but little further research bears on this point. NO
general meaningfulness effect is predicted or expected, but
on the basis Of the Bettinghaus §E_al, findings, a meaning-
fulness effect which is specific tO certain syllogistic
forms might be expected. Present data dO not enable the
researcher to predict the particular forms on which this
effect should Occur. If such an effect were strong enough,
an interaction between meaningfulness and syllogistic form
should Occur. However, due tO the incompleteness Of avail-
able data, nO formal hypothesis about this effect seemed

justified at this time.

Syllogistic Form Factor

The derivation Of the syllogistic form factor and
the method Of choosing its levels are contained in Appendix
I. Numerous writers have commented in detail on the nature
Of logical thought and its place in human reasoning. Henle
(1962) summarizes many Of these sentiments in discussing the
relationship between logic and thinking. Unfortunately,

little attention is given in her work, or in the work which

10

she cites, tO a logical form-based mechanism which might
determine the difficulty Of judgment Of a given syllogistic
form. What might such a mechanism be?

The difficulty Of a given task is related to the
length Of the task and the cOmplexity of the task: i.e.,
the number Of errors associated with the performance Of a
given task is Often a function Of the number Of separate
parts Of the task and the complexity involved in the per-
formance Of each part. There are, Of course, many other
factors, such as fatigue and ability, which affect number
Of errors but consider the number Of parts to the task and
the complexity Of each. Logicians Often use Venn or Euler
diagrams to illustrate the relationships involved between
the terms Of a syllogism. The number Of such relationships
can be shown to be different for different syllogisms, or
more specifically, for different premise combinations. One
possible mechanism involved in judging the validity-
invalidity Of a given syllogism would be for the subject to
consider the number Of ways the three terms Of a syllogism
could combine and then tO examine the conclusion for valid-
ity in terms Of these combinations. Given such a mechanism,
the number Of such combinations might be used as an index of
the number Of parts Of, as well as the complexity Of, the
task Of judging syllogistic validity.

This idea may be approached from another direction.

Consider the two premises of a syllogism and ask the question,

11

"How did these two premises originate?" Any possible
premise combination could arise from more than one situation
involving a relationship between the three terms Of the
premises. In this sense, all premise combinations are
ambiguous. It is not possible to learn from an examination
Of the premises the exact situation Of relationship between
the terms which led to the formulation Of the premises.
Judgment Of the validity—invalidity Of a syllogism involves
the listing and sorting Of the possible relationships
between the terms as indicated by the premises, and the
comparison Of the conclusion with each Of the possible
relationships to see if it must hOld true. Thus, it is not
unreasonable tO expect that the difficulty Of judging the
validity-invalidity Of a syllogistic conclusion is directly
related to the number Of possible relationships between the
terms Of a syllogism that the premises allow. These rela-
tionships may be conceptualized as all the possible situa—
tions from which the premises could have arisen. The larger
the number Of such relationships, the more complex is the
task; therefore, difficulty Of judgment should increase
accordingly. The number Of such relationships will be

referred tO as the ambiguity Of the premise combination.

 

Given differences in ambiguity, the major hypothesis Of this

study may be stated as follows:

12

The degree Of difficulty Of judgment Of the

validity-invalidity Of a syllogistic conclu-

sion will increase directly with the ambigu-

ity Of the premise combination associated

with that syllogism.

TO illustrate the grounds for the hypothesis, con-
sider the ambiguity Of the A5; and the ggg conditions. How
many different relationships are there between X, Y, and Z
which allow the AAl_premises tO be true? Ag; is "all Y is
X; all Z is Y." There are four possible situations which a
source with an intent tO communicate could have examined and
truthfully stated about each Of the four that "all Y is X;
all Z is Y." These four are: (l) X, Y, and Z are in iden-
tity with each other; (2) X and Y are in identity with each
other and Z is contained within them; (3) Y and Z are in
identity with each other and are contained in X; and (4)

Z is contained in Y and Y is contained in X. There are no
other possibilities which are different, as defined. Thus,
the ambiguity of the AAI condition is four. Continuing,
the ggg condition is "all X is Y; nO Z is Y." How many
situations could a source have examined and come up with
these two statements, assuming that they are true? This

is part Of the question the receiver Of the syllogism asks
himself in order tO determine its validity or invalidity.
The answer here is two, and the receiver must check each of

these with the conclusion to determine its validity (assum—

ing a naive receiver who is unfamiliar with a logician's

 

13

rules and shortcuts). The two are: (l) X and Y are in
identity and Z is outside Of them, and (2) X is contained
within Y and Z is outside Of Y. The ambiguity Of AEg is)
thus)two. The major hypothesis Of the study states that
more errors should be expected in the AA; condition than in
the ggg condition, since the Aél,is more ambiguous than the
Afig.

In actual practice it may not be the case that dif-
ferences in level Of difficulty show up with small numerical
changes in ambiguity, changes say, from two to four. The
larger the increase in ambiguity, the more such differences
should become apparent. A general method for determining
the ambiguity Of any premiSe combination is given in Appen-
dix II, and a list Of the ambiguity associated with each Of
the premise combinations used in this study is found in
Chapter II, Table 1.

In summary, this study was generally concerned with
variables affecting the ability Of an individual to judge
the validity or invalidity Of a logical syllogism. Twenty-
One logically unique syllogistic premise combination types
were used in the study. A representative Of each Of these
combinations was selected and these 21 premise combinations
and conclusions became the 21 levels Of a syllogistic form
factor. A term meaningfulness factor was utilized to deter-
mine the effect Of low and high meaningfulness Of the terms

Of the syllogism on the ability tO judge syllogistic

14

validity. Finally, subjects were grouped into three levels
Of a dogmatism factor: high, middle, and low. The major
hypothesis Of the study stated that the difficulty Of
judging the validity-invalidity Of a given syllogistic form
will vary directly with the ambiguity associated with the
premise combination Of that form. While an interaction
between meaningfulness and syllogistic form might be
suspected, insufficient information existed tO justify a
formal prediction. Due to the absence Of positive and
negative sources, dogmatism was not expected tO produce

a significant effect.

CHAPTER II

METHOD AND PROCEDURES

Subjects

A total Of 229 gs participated in the study. §s
were students enrolled in the basic speech course at The
Cleveland State University. A total Of 22 Of the 229 §s
were discarded before the data were analyzed. Three Of
these were Observed by proctors to be marking responses
without looking at the test booklet; six were discarded for
marking undefined responses On the answer sheet, and the
remaining 13 were discarded for failure tO respond tO all
items. This last number was minimized by checking each test
bOOklet for completion as it was handed in. Thus, data from
207 Of the original 229 subjects were analyzed.

The test bOOklet consisted Of two parts. The first
part contained 42 premise combinations, each premise combi-
nation having two conclusions stated below it for a total Of
84 items. Part Two was the 20 item short form Of the dogma-
tism scale (Rokeach, 1960; Troldahl and Powell, 1965). Part
One and Part Two were separated by a single sheet Of paper
which instructed the subject to continue on. Each part

contained a machine scored response sheet.

15

16

The 42 premise combinations consisted Of two sets Of
21 premise combinations, One set containing high meaningful
terms and the second set containing low meaningful terms.
These terms were all CVC trigrams Obtained from Archer (1960)
in the following manner. Nine low meaningful and nine high
meaningful terms were chosen so they could be rotated in
three groups Of three, the trigrams in each group changing
in each rotation. The criteria used for choosing the nine
low meaningfulness trigrams were: (a) the first letter Of
each trigram must be different, (b) the middle letter Of
each trigram must not occur more than twice in the middle
position (since there are only six vowels, three Of them
must repeat at least once in nine trigrams), (c) the third
letter Of each trigram must not Occur more than twice in the
third position for the nine terms, and (d) nO letter may
occur more than three times in the nine trigrams. The nine
trigrams with the lowest collective average meaningfulness
on criteria (a) through (d) were selected. These four
criteria were intended tO select nine low meaningful items
which are easily recognizable as different from each other,
so that nO confusion between terms would Occur. If the nine
items lowest in meaning had been chosen, they would all have
begun with X_and would have appeared very similar. The nine
low meaningful items chosen by the above method were XOM,

RYW, TEJ, ZUF, YIV, VUQ, FAJ, PYB, and QIH. On Archer's

 

.9“

17

100 point scale, their meaningfulness ranges from 6 to 10,
and averages 8.67.

The same four criteria were used to choose the high
meaningful trigrams. In this case the nine trigrams with
the highest collective average meaningfulness were selected.
The criteria provided more leeway in the choosing Of the
nine high meaningful trigrams than in the nine low meaning-
ful items, since there are many CVC trigrams with a meaning—
fulness score Of 100. The nine high meaningfulness items
chosen were those which met the above criteria and which
were spaced relatively evenly throughout the alphabet in
terms Of their initial letter. These nine were BOY, FAR,
HIM, JUG, NOD, PAL, TUB, WET, and ZIP, all Of which have a
meaningfulness score Of 100. The 21 items in which the high
meaningful trigrams were used formed the High Meaningful
condition, and the 21 items employing the nine low meaning-
ful trigrams formed the Low Meaningful condition.

Both the high and the low meaningful trigrams were
alphabetized by first letter tO form two lists. A corre-
spondence was established such that for each usage Of the
trigram BOY in a premise combination or conclusion in a high
meaningful item, the trigram FAJ appeared in exactly the
Same position in the corresponding low meaningfulness item,
since BOY and FAJ are the respective first trigrams in terms

Of alphabetization. This correspondence was used for the

18

respective second, third, etc. pairs Of low and high mean-
ingfulness trigrams throughout the test booklet.

Twenty-one unique types Of premise combinations were
used in the study. Within each type, several forms are
logically equivalent; i.e., they are reducible to each other
and have the same set Of term relationships from which they
could have arisen. As indicated in Chapter I, one premise
combination from each unique type was used to represent that
type. The following method Of selecting these particular
premise combinations was employed. For those cases having
only one premise combination Of that type, that premise
combination was used. This accounted for 8 of the 21 types.
Of the remaining 13 types, all premise combinations which
are the logical equivalent Of each other contain the same

two premise types; i.e., a given premise combination, such

 

as an AE, can be logically equivalent only tO another AE

(or EA) premise combination. The distinctions between
premise combinations within a given type are due to either
difference Of figure or difference Of premise order. The
premise orders in which the letter representing the first
premise Occurs first in the alphabet were arbitrarily chosen
on that alphabetical basis. For example, A§_was chosen over
EA, Within the possible figures Of a given premise combina-
tion type, it was desirable to employ one premise combina-
tion involving the same figure across all such premise

combination types in which that figure is possible, in order

19

to control for a possible figure effect. That is, if figure
one were used tO represent one unique set Of AA_type premise
combinations while figure three were used tO represent a
unique set Of A; type premise combinations, where figure one
could have been used with both types, any differences found
between these two unique types would be confounded with
premise figure. With the eight unique premise combination
types for which there is only one case, such "confounding“
is not really confounding, but rather an inherent difference
between the premise combination types. Its introduction in-
tO the comparison between types in which such "confounding"
is not an inherent condition Of a given type does amount to
true confounding Of two distinct variables: figure,and
premise combination type. For each unique premise combina-
tion type, the premise combination with the lowest figure
number was arbitrarily chosen for this experiment.

It should be noted that the term "arbitrary“ is used
in a very restricted sense here, as it is with the choice of
premise order. The possible alternatives in each case were
reduced tO a minimum level by the demands Of the eXperiment
and the constraints imposed by the control Of extraneous
variables in the experimental design. The arbitrariness Of
choice exists only within this minimum level. Thus, the
only alternative tO choosing the premise combination with
the lowest figure number, given the above assumptions, would

be to choose the premise combination with the highest or

 

.9”)I

20

second or third highest figure number in each case. These
would work equally well.

The 21 premise combinations chosen by the above
method tO represent the syllogistic form factor Of this
eXperiment are contained in Table 1. In addition, the table
lists the ambiguity score for each premise combination. The
order Of ambiguity is also the predicted order Of difficulty

for each Of the combinations.

Table 1. Ambiguity Of 21 syllogistic forms

 

 

 

 

Premise Combinations Ambiguity Score
A§_Second 2
AA_First 4
§§_First 5
A§_First 6
AA_Third 8
AA_Second 10
§g_Third ll
AQ_Second 12
EI_First 12
5g Third 14
§Q_First 15
A; First 16
AQ_First 18
AQ_Fourth 18
A;_Second 20
QQ_First 67
IQ_First 70
QQ_Second 73
QQ_Third 73
IQ_Third 76
I;_First 80

 

21

Note that in those cases where the ambiguity Of two
Of the premise combinations is equal, the equality Of ambi-
guity does not imply logical equality. The two premise
combinations are logically unique in each case. The number
Of relationships is equal but they are different sets Of
relationships, as defined in Appendix II.

There are six possible conclusions in a syllogism,
two A, two 9, one §_and one I, With 42 premise combinations,
if each conclusion appeared with each premise combination,
there would be a total Of 252 items. Since it was desirable
that each §_judge each item for control purposes, and since
fatigue effects increase rapidly with difficult judgmental
tasks, it was decided tO employ only two Of the six possible
conclusions for each premise combination. This resulted in
a total Of 84 items.

In a pretest with a group Of 10 undergraduate stu—
dents at the University Of Michigan, it was determined that
the average time required for 80 items Of this type was
between 20 and 40 minutes, but that this time increased to
40 tO 70 minutes with 100 items. For 40 items the time
ranged from 10 tO 20 minutes. On this basis, it was deter—
mined that judging more than two conclusions for each prem—
ise combination might increase the fatigue effects tO an
unacceptably high level.

The use Of two conclusions was considered more

desirable than the use Of one, in order to reduce error

22

variance due to such variables as guessing. For those
premise combinations having two valid conclusions under

the logical system explained in Appendix I, both conclusions
were used. The valid conclusion Of each premise combination
having one valid conclusion was used along with an invalid
conclusion. The six possible invalid conclusions were
rotated through the items in which they appeared so that
each appeared an equal number Of times. One conclusion
appearing with each syllogistic form was designed tO check
for a possible atmosphere effect (Woodworth and Sells,
1935).

In order tO control for possible practice and
fatigue effects associated with serial position, and in
order to control for possible effects of the immediately
preceding premise combination, six different orderings Of
the premise combinations were used. The two conclusions
for each premise combination appeared in the same order for
high and low meaningful items across all six forms Of the
test booklet. The 42 premise combinations were randomly
ordered in the first form by means Of a table Of random
numbers. The remaining five forms were ordered so that no
premise combination was immediately preceded by the same
premise combination more than once and so that the average
rank order Of each premise combination was between 19 and 23

out Of the 42 positions.

 

23

Test Administration

The six forms were distributed such that every sixth
person in the order Of seating chosen by the subjects
received a given form. .The class instructor introduced the
administrator as a person who would give instructions for
filling out the test booklet. When six assistants finished
passing out the booklets, gs were asked by the administrator
to look at the instructions, which were then read aloud by
the administrator. The instructions contained a single
example Of a syllogism, one which did not appear in the test
booklet.

gs were instructed tO judge whether each conclusion
had tO be true given that the premises were true. The
instructions emphasized that the premises were always to be
assumed true and that the judgmental task was to separate
those conclusions which had to be true from those which
might be true or might be false. gs were told tO assume the
existence Of all terms and were also told that the defini-
tion Of "some" as used in the test items was "at least some”
and not "only some." They were instructed not to gO back
and change answers on an item once they had begun the next
item and not to lOOk back to determine how they had re—
sponded to any previous items. The instructions stated that
they should allow about 20 tO 30 seconds per conclusion
judged and that they could write on the test booklet if they

wished.

 

24

gs were told that the results Of the test would have
no bearing on their grades in the class. In order to pro—
vide motivation they were told by the administrator that the
purpose Of the test was to compare the performance of groups
Of undergraduates from different universities. They were
also told that they need not put their name on the test
booklet or on any answer sheet, but that they could put
their name on the answer sheet for the syllogisms if they
wanted to know how they had done.

The dogmatism items followed the syllogisms and had
their own brief instructions. These items were represented
as "statements which peOple have different Opinions about”
so that there were "no correct answers.” NO reason was

given for including these 20 items in the test booklet.

Dogmatism Prediction

The apparent source Of the test items was the test
administrator. The administrator was introduced simply as
a person who would tell the gs what they would be doing
during the class hour. NO mention was made of his occupa-
tion or of any other personal data. According tO interviews
with 12 gs immediately after the administration Of the test,
the administrator was seen as either an instructor or gradu—
ate student, but not as a person holding the rank Of assis-
tant professor or above. Two §s stated that they believed
the administrator was the employee Of an educational testing

service.

 

25

The Bettinghaus gt gl. study associated a positive
or a negative source with each item used, by placing the
name Of the source and a brief description Of his position
above each item. In the present study, no sources were
given for the syllogisms, so nO direct linking of positive
and negative sources with items Occurred. As indicated
earlier, Rokeach's theory would not predict a dogmatism
effect without positive and negative sources being linked
to the items. Thus, a dogmatism effect was not predicted

for this eXperiment.

General Design and Analysis

The analysis Of the data involves the factors Of
dogmatism, meaningfulness, and syllogistic form in a
3 x 2 x 21 design. Scores in the three dogmatism levels
were treated as independent and scores in the two meaning-
fulness levels and 21 syllogistic form levels were treated
as correlated, since each subject participated in each of
the meaningfulness and syllogistic form conditions. In
addition, all test booklets were examined for writing and
figuring so that subjects who used paper-and—pencil-figuring
in making their judgments could be analyzed separately from

those who did not, in a second analysis.

 

CHAPTER III

RESULTS

Test Of Major Hypothesis
The major hypothesis of this study is that the
degree Of difficulty Of judgment of the validity-invalidity

of a syllogistic conclusion will increase directly with the

 

ambiguity Of the premise combination associated with that
syllogism. As an initial test of this hypothesis the data
were subjected to a three factor analysis of variance with
scores on the dogmatism levels assumed independent and
scores On the meaningfulness levels and syllogistic form
levels assumed correlated (Steinfatt, 1970, pp. 31, 33).

The result of Hartley's test was not significant, indicating
that the equality of variance assumption is satisfied.

The results of the analysis of variance are sum—
marized in Table 2. The main effect for syllogistic form
was significant (p < .001). This effect indicates that
subjects gave significantly more correct responses to some
syllogistic forms than to others. While the second—order
interaction was not significant, the first—order interaction
involving syllogistic form and meaningfulness was signifi—

cant at the .001 level. The effect of this interaction on

26

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27

 

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28

the syllogistic form factor is described later in this
chapter and discussed in Chapter IV.

The significant effect for syllogistic form jus-
tified a further test of the hypothesis by computing a
Spearman rank order correlation between the predicted ranks
and the Obtained rankings for the combined meaningfulness
levels. Ambiguity is not distributed normally in the pOpu—
lation Of syllogistic forms. Thus, the bivariate normal
assumption Of a Pearson product moment correlation could not
be met and a rank order correlation was necessary. Either
Spearman's rho or Kendall's tau could have been used and the
Spearman method was chosen due tO its greater familiarity to
most researchers. The power-efficiency Of each of these
when compared with the Pearson £_is 91 percent.

The Spearman rho between the predicted ranks and the
obtained ranks is .695 with an N_Of 21 which is significant
at the .01 level (Siegel, 1956, pp. 202-213, 284). This
result offers further support for the hypothesis.

Spearman rhos were also calculated between the
predicted ranks and the Obtained ranks for high meaningful-
ness and low meaningfulness items separately to determine if
the predictive power Of the ambiguity measurement is enhanced
by the use of either high or low meaningful terms in the
syllogism. Rho for the high meaningfulness and ambiguity
orders is .700 and .687 for the low meaningfulness and

ambiguity orders, both of which are significant at .01.

 

29

This indicates that the hypothesis holds to approximately
the same degree when either high meaningful or low meaning-
ful terms are used in the syllogisms. These rankings are

all listed in Table 3.

Individual Comparisons

The significant F for syllogistic form, along with
the significant rho between predicted and Obtained ranks,
suggests a breakdown Of significance between individual
syllogistic forms. Table 4 contains a 21 x 21 triangular
matrix of the significance Of the difference between all
pairs Of means for each of the 21 syllogistic forms. All
differences below the heavy line are significant at the .01
level by the Newman-Keuls method (Winer, 1962, pp. 77-89,
309-310). The large number Of significant differences is
due to the size Of the significant F for syllogistic form.
(A significant difference between a pair Of means indicates
that that difference is reliable at the stated level Of
significance; i.e., that it is a non—chance difference
occurring in groups of subjects drawn from the pOpulation
of this study. For example, Aglyis reliably easier to judge
than.A§2_for groups drawn from this pOpulation, as indicated

in Table 4.

 

 

 

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31

 

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32

,Doqmatism and Meaningfulness
Factors

The scores Of the 207 subjects on the dogmatism test
ranged from 20 to 94 with a median score Of 60. Subjects
were assigned tO dogmatism levels by using the tertiary
points Of the data as cutoff levels. These points occurred
at scores Of 55 and 65, respectively, on the dogmatism scale.
Ten subjects achieved a score Of 55. These were randomly
assigned tO the low dogmatic and middle dogmatic levels,

five in each level. Three subjects achieved a score of 65

 

on the dogmatism scale. These three were randomly assigned
to the middle and high dogmatism levels, two to the middle
group and one to the high group. Subjects with dogmatism
scores below 55 were assigned to the low dogmatism level,
while subjects with dogmatism scores of 56 tO 64 were
assigned to the middle dogmatism level and subjects with
scores above 65 were assigned to the high dogmatism level.
This manipulation resulted in three dogmatism levels with
69 subjects in each level.

The validity Of these cutoff points, 55 and 65 on
the dogmatism scale, is partially verified by their reli—
ability, as in Bettinghaus §£_gl. The similar cutoff points
indicated by their data on the dogmatism levels of their
subjects were 56 and 65, respectively. Numerically, the
midpoint of the 20 item dogmatism test, with scores ranging
from 20 to 120, is 70. The median on the scale is 60 for

both Bettinghaus §t_§1, data and the data Of the present

33

study. Thus it seemed reasonable to assign subjects to
dogmatism levels using 55 and 65 as the dividing points
between low, middle, and high dogmatics.

As mentioned above, Table 2 indicates that the
second-order interaction involving all three factors of this
study is not significant. While the two first order inter—
actions involving dogmatism are also non-significant, the
remaining first-order interaction between meaningfulness and
syllogistic form is significant at the .001 level. The main

effects for meaningfulness and dogmatism are not significant.

 

Thus, as eXpected, dogmatism does not enter into any of the
significant main or interaction effects. The significant
interaction between meaningfulness and syllogistic form
indicates that the meaningfulness of the terms of the syllo—
gisms was Of differential importance across syllogistic
forms. The mean number Of correct responses to the high and
low meaningfulness items for each syllogistic form is given
in Table 3, above. The interaction of meaningfulness with
syllogistic form is graphed in Figure 1.

TO determine if any differences between high and low
dogmatic subjects might be significant if the effect Of the
middle dogmatic subjects were removed, a second three factor
analysis Of variance was conducted using data from only high
dogmatic and low dogmatic subjects. It produced no changes
in significance level Of any Of the interaction or main

effects, including dogmatism.

34

 

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35

(As a check on the effects of dogmatism and syllogis—
tic form with only high meaningful terms in the syllogisms,
a third analysis of variance was calculated using the scores
of all 207 subjectson the high meaningfulness items, an
analysis similar to a test for simple main effects. While
the dogmatism x syllogistic form interaction was not signif-
icant, the dogmatism factor approached significance (p < .10).
A E_test between high and low dogmatic subjects using high
meaningful items only still yields a probability level of
alpha which is not significant. The syllogistic form main
effect was significant at the .001 level (F = 34.01, df =
20, 4080). This indicates that the simple main effect for
high meaningful items does not differ substantially from the
overall meaningfulness main effect.

Individual comparisons were also conducted between
the high and low meaningfulness means for each of the 21
syllogistic forms. Differences which are significant at the
.05 level by the Newman-Keuls method are indicated by an
asterisk in Figure l. A significant difference indicates
that the use of high meaningful terms in that syllogistic
form resulted in a significant change in the difficulty of
judgment of that form over that difficulty which occurred

when low meaningful terms were used.

36

Diagrams in the Test Booklets

 

The test booklets of all of the 207 subjects were
examined for marks of any kind. Thirty-five of the 207
booklets contained pencil marks which appeared to be either
Venn diagrams or the letters "A," "B," and "C," with "equal
to'l or "not equal to" signs relating them. In some cases
both.Venn type diagrams and the letters occurred near dif—
ferent items in the same test booklet. The means for each
of the 126 cells of the design were calculated for the data
from these 35 subjects and visually compared with the means
calculated for the remaining 172 subjects. The means were
similar in all but two of the 126 cases, and these two
differences were not large. Thus, no analysis of variance

was calculated using marking as a factor.

 

CHAPTER IV

DISCUSSION

Ambiguity Hypothesis: Alternative
_Explanations for Non—Conforming
223%

The significant main effect for the syllogistic form
factor and the significant rhos between the obtained orders
and the order predicted by the ambiguity hypothesis provide
support for that hypothesis. Apparently, the ambiguity of
the premise combination is directly related to the diffi—
culty of judgment of syllogisms containing that premise
combination. This is the major finding of this study. It
may be useful to analyze those items for which the ambiguity
predictions did not hold.

While most of the obtained ranks correspond quite
closely to the predicted ranks, there are two forms which
are distinctly out of line. These are the §§l_form, which
was predicted third but finished fifteenth, and the A;1_form,
which was predicted twelfth but finished first. Elimination
of these two forms from the correlation would increase the
overall rho from .695 to .851. Both forms finish near their
overall position when only high and only low meaningfulness

items are used to determine order. The EEl form finishes

37

'l
J; l’

38

eleventh, fifteenth, and fifteenth, with low, high, and low
plus high meaningful items respectively, while the §;l_form
finishes third, first, and firSt. An analysis of the dis-
tribution of responses to the items involving each of these
forms indicates no particular response patterns such as an
atmosphere effect (Woodworth and Sells, 1935) which might
account for their departures from predicted positions. The
conclusions for the ALL form were A_and ;_type conclusions.
Just which of these two should benefit from any atmosphere
effect is unclear with an A; premise combination. The A
conclusion was marked correctly by 142 subjects in the low
‘-meaningfu1 condition and by 139 subjects in the high mean-
ingful condition. The ;_conclusion was marked correctly by
153 subjects in the low meaningful condition and by 173
subjects in the high meaningful condition. The A conclusion
is invalid and the l_conclusion is valid in §;l,

The conclusions for the §§l_form were §_and g_types,
both invalid. The E conclusion was marked correctly by 101
subjects in the low meaningful condition and by 77 subjects
in the high meaningful condition, while the ;_conc1usion was
marked correctly by 103 and 96 subjects, respectively, under
those conditions. Thus, any atmosphere effect that might
result in incorrectly marking the invalid §_conclusion as
valid occurs only with high meaningful terms, and even then
the effect is not large. Since less than half of the 207

subjects responded correctly to any of the four conclusions

El
.9“

39

involving the.§§1_form, its lower-than-predicted finish
cannot be attributed to any one conclusion.

If the §§1_form were the only form that did not
conform to prediction, a possible explanation might be that
an excess of difficulty of judgment exists in conditions
with multiple universal negatives, over the amount predicted
by the ambiguity hypothesis. This may in fact be the case
with the §§l_form, although it does not explain why such
difficulty should occur with multiple universal negatives.

The A;g_form was included in this eXperiment because
it is not the logical equivalent of the ALL form. Nonethe-
less, both forms contain A;_premise combinations and'it is
interesting to compare the predicted and actual rank order
finish of each. A;l_was predicted twelfth and finished
first. ggg_was predicted fifteenth and finished seventeenth.
Thus, 5;; finished very close to the position predicted by
the ambiguity hypothesis, while All finished far from its
predicted position. Yet only two changes can account for
this difference: the change in the actual terms used (ZUF,
YIV, FAJ, ZIP, WET, and BOY versus HIM, JUG, NOD, QIH, RYW,
and TEJ), and the change of the position of the middle term
in the first premise from first in the first figure to last
in the second figure. Assuming the meaningfulness manipula-
tion was prOperly controlled, attention focuses on the
second change. This positional change for the middle term

results in a logical difference between AIl and A12 and a

u":

40

difference in ambiguity of 16 for A;1 and 20 for ALg, The
logical difference results in a valid ;_conc1usion for 5;;
while Al; has no valid conclusion.

Bettinghaus §E_g1.found a strong main effect for
validity indicating that valid syllogisms were less diffi4
cult to judge correctly than invalid syllogisms (1970,

p. 242). Could it be that ALL moved up in rank because it
has a valid conclusion? If this were true, then in general:
(a) syllogisms with valid conclusions should finish ahead of
their predicted positions, (b) syllOgisms with two or more
valid conclusions should finish even further ahead of their
prediction positions than syllogisms with only one valid
conclusion since if two conclusions were valid for a form
both were used, (c) syllogisms without a valid conclusion
which were predicted to finish ahead of syllogisms with a
valid conclusion should finiSh with a lower than predicted
rank since they would be pushed down from their predicted
positions by the rise of the syllogisms with valid conclu-
sions, and (d) if the above changes occur, they must be
concomitant with a larger number of subjects responding
correctly to the valid conclusion than to the invalid con-
clusion in forms in which a single valid conclusion exists.
Condition (d) simply demands that the changes actually occur
because of valid conclusions, and not just to syllogisms

which happen to have valid conclusions.

 

41

Eight forms used in this eXperiment have at least
one valid conclusion. .Four of these finish ahead of their
predicted position, three finish below their predicted
position, and one finishes in its predicted position. The
sum of the predicted ranks for the eight forms with a valid
conclusion is 51 for a mean predicted rank of 6.375, while
the sum of the obtained ranks for these items is 39 for a
mean obtained rank of 4.875. Thus, by this measure, items
with valid conclusions tend to finish higher than predicted.
Some of this effect is produced by the All_form itself:
however, since if this item were excluded, the mean of the
predicted ranks would be 5.571, and the mean of the obtained
ranks would be 4.429. At best, statement (a) appears to be
partially correct for these data.

Almost the reverse of statement (b) appears to be
true. The two syllogistic forms with more than one valid
conclusion finished below their predicted rank, while four
of the six forms with only one valid conclusion finished
above their predicted rank. Thus, an effect as in statement
(b) did not occur with the data of this experiment.

Four syllogistic forms with no valid conclusions
were predicted to finish above one or more forms with a
valid conclusion. Three of these four did finish lower in
rank than predicted, and the fourth finished as predicted.
Each of the shifts was due at least in part to forms with
valid conclusions finishing higher than predicted. Thus

statement (c) appears to be confirmed by the data.

42

.Four of the six forms with a single valid conclusion
finished above their predicted position. Two of these four,
_A§l and AAQ, had more subjects respond correctly to the
invalid conclusion than to the valid conclusion. The
remaining two, g;l_and AQQJ had a larger prOportion of
subjects respond correctly to the valid than to the invalid
conclusion. Thus, statement (d) is correct for only two of
the four forms.

What conclusions are implied by the preceding dis—
cussion? The effect of having at least one valid conclusion
on the difficulty of judging the validity-invalidity of a
syllogistic form is certainly not a simple one. In four of
six cases the effect appears to be one of decreased diffi—
culty. Yet in two of these four cases this decreased dif—
ficulty is apparently due more to the invalid than to the
valid conclusion. In the other two cases, the number of
correct responses to the invalid conclusion was larger than
expected, though smaller than the number correct for the
valid conclusions. The effect is to decrease the difficulty
of forms with a valid conclusion, but to do so as much
because of a decreased difficulty of the invalid conclusion
as of the valid conclusion. To further complicate the pic-
ture it appears that two valid conclusions are no better
indicator of decreased difficulty than one valid conclusiOn.
In general, these results imply that the quality of having

at least one valid conclusion is directly associated with

 

a".

43

an obtained ranking which is higher than that predicted on
the basis of the ambiguity of the syllogistic form alone.
The reasons why this is true are unclear.

Ambiguity and the Existence of

a Valid Conclusion as Predictor
Variables

 

 

Thus far this discussion has treated ambiguity and
syllogistic forms with a valid conclusion as if they were
independent predictors of the ease of judging syllogistic
validity. The two variables are, of course, related.
Ambiguity is the number of possible different relationships
between the terms of a syllogism which are allowed by the
premise combination of that form. As this number increases,
it becomes increasingly more difficult to make a true state-
ment concerning a way in which the two non-middle terms are
consistently related over the entire set of these relation-
ships. The greater the ambiguity, the less the possibility
of a valid conclusion. In some cases, such as ggl, there is
no valid conclusion even when there are only five possible
relationships between the terms. This occurs when all six
of the possible conclusions are ruled out by the relation-
ship-between-the-relationships. The point is that the
chances of all six being ruled out are generally greater
with a larger number of relationships. A threshOld appar-
ently occurs with an ambiguity level of about 17, since no
syllogism with an ambiguity greater than this has a valid

conclusion. Thus, ambiguity is a non-perfect predictor of

 

44

the existence of a valid conclusion for a given syllogistic
form, and the existence of such a conclusion for a form
increases the probability of predicting the ambiguity of the
form correctly. By combining ambiguity and validity in a
predictive formula, the accuracy of the prediction could
have been increased, at least in this experiment. For
example, if the predicted position were obtained by ranking
those items with a valid conclusion in a first group by
their ambiguity order and those items without a valid con-
clusion by their ambiguity order in a second group below the
first, the rho between predicted and obtained ranks would
increase from .695 to .833. Of course, such a post hoc
prediction has no status in terms of hypothesis testing, but
a future study might profitably test such a hypothesis.

This discussion began as an analysis of why two
items departed so far from their predicted ambiguity posi-
tions. Given the preceding predictive suggestion, attention
should again be focused on these two items. Form gg; is the
highest ranking form which has no valid conclusion when only
ambiguity is used in the ordering. Form ALI is the lowest
ranking form with a valid conclusion in this same ambiguity
ordering. Combining these two facts with the suspicion that
having a valid conclusion has the effect of making the syl-
logism easier to judge gives a partial eXplanation for the

failure of the ambiguity prediction in these two cases.

45

The question now arises as to the usefulness of the
ambiguity prediction. To what extent does it predict order
within the two categories of syllogistic forms, with and
without valid conclusions? For the 13 forms without valid
conclusions, rho between the predicted ambiguity order and
the obtained order is .654. For the eight forms with at
least one valid conclusion the rho value is -.095. This
difference means that ambiguity Operated as a good predictor
for the forms without a valid conclusion but had essentially
no predictive power, either positive or negative, for the
forms with valid conclusions, when these eight forms are
considered alone.

The question of interest becomes one of whether
ambiguity actually predicts only for items with no valid
conclusions or whether there was some particular factor in
this experiment which led to the observed outcome. The most
obvious difference between the forms with a valid conclusion
and those without is that the conclusions of those without
were all invalid while the conclusions of those forms with
at least one valid conclusion were sometimes valid. It
would have been possible to make all of the conclusions to
all of the forms invalid, but that would have led to a major
response set control problem. An alternative would be to
rank the 21 forms using only the responses on invalid con—
clusions to yield the obtained positions. The rho between

these rankings and the predicted ranks would then be free of

46

the influence of the valid conclusions themselves, but
would not suffer from control problems resulting from
using no valid conclusions. If interest is in general
application to all syllogistic forms with both valid and
invalid conclusions, the data of Table 3, Table 4, and
Figure 1 (Chapter III) should be used. The correlation
betWeen the predicted and obtained ranks is .695 on a
straight ambiguity prediction basis, or .833 if the 21
items are ordered first by whether they have a valid con-
clusion and then by ambiguity order. If interest is only
in invalid conclusions to syllogistic forms, then Table 5
and the correlations obtained from it are applicable.

Ambiguity as a Predictor with
Only Invalid Conclusions

 

 

Table 5 contains the 21 forms in the order obtained
by using the data from invalid conclusions only, except for
forms AA; and Ag; for which no data were available. The rho
between the order obtained by the ambiguity prediction alone
and the 21 items ordered by data from invalid conclusions
only, with the two exceptions, is .687. If only the 19
items for which data from invalid conclusions are available
are used, rho is .619. This indicates that ambiguity alone
predicts level of difficulty about as well using data from
invalid conclusions only as it does when data from both valid
and invalid conclusions are used. The principal difference

obtained by using invalid conclusions only in place of valid

47

Table 5. Twenty—one forms ordered by invalid items only

 

 

 

 

 

 

Predicted Actual Syllogistic Table 3 Percentage
Order Order Form Order Correct
4 1 AAA 2 78.50
5 2 AA3 3 75.85
2 3 AA1* 4 70.41
12 4 All 1 67.87
1 5 AE2* 6 65.58
10 6 Ag; 5 64.73
13 7 Ag; 7 '59.18
6 8 AAA 9 53.86
11 9 E9; 11 52.05
7 10 Ag; 12 47 95
8 ll AQA 8 47 10
16 12 99A 13 46 62
18 13 99A 14 46 50
3 14 EA; 15 45 53
20 15 A93. 16 44 20
15 16 AAA 17 42 03
l7 17 Ag; 18 41 67
14 18 A93 19 38 89
9 19 EL; 10 38 l6
19 20 99; 20 37 92
21 21 111 21 34 66

 

*No data from invalid conclusions.

48

plus invalid is in the ordering of certain forms such as All
and Agg_which move down three ranks, and Eli, which moves
down nine ranks. The rho between the two obtained orders,
using invalid conclusions only as opposed to valid plus
invalid, is .922.
Premise Combination Type, Figure
and Items with Equal Ambiguity

The scores on the 21 items were visually inspected
to determine if the variables of premise combination type
and figure, when examined individually, might have had an
effect on the difficulty of judgment of the items, and to
determine if the results obtained with different items of
the same ambiguity level are similar. There appear to be no
particular differences between premise combinations within
the First figure, the only figure in which all premise cOm—
bination types appeared, which are not generally explained
by considering ambiguity and the existence of a valid con—
clusion. Thus, inspecting all items of the same figure
produced no new insights about possible variables affecting
difficulty of judgment. Similarly, inspecting the scores on
all items of the same premise combination type across fig—
ures adds little explanatory information. Neither the
variable of figure nor the variable of premise combination
type considered individually appears to account for any

pattern of responses to the syllogisms.

 

49

Three pairs of premise combinations have equal
ambiguity: A92_and-§;AJ ambiguity of 12, A9A_and A91,
ambiguity of 18, and 999_and 99;, ambiguity of 73. A92_and
§l1_are quite close to each other in eighth and tenth posi-
tions, as expected because of their equality of ambiguity.
A9;_and A9A_finish far apart in seventh and nineteenth
positions, while 992_and 99§_finish fairly far apart in
fourteenth and twentieth positions. The latter two results
would not be eXpected on the basis of an ambiguity predic-
tion. Apparently, the predictive power of ambiguity is not
large for items with the same ambiguity, and it can be seen
by inspection of Table 3 (Chapter III) that this is also
true for items of similar ambiguity. The predictive effi-
cacy of ambiguity is in terms of its overall predictions
among all 21 forms. This is especially true when ambiguity
is combined with knowledge of the exiStence of a valid con-
clusion for the particular forms.

.Meaningfulness Main Effect and
the Meaningfulness by Syllogistic
Form Interaction

The main effect of the meaningfulness manipulation
was not significant. With a possible range of zero to 42,
the mean of the High Meaningfulness cOndition was 22.41
while the Low Meaningfulness mean was 22.58. This finding
was not predicted, but it does not conflict with the results
of previous studies. Morgan and Morton (1944) found no

significant differences in the ability of subjects to select

 

’.

50

the logical conclusion from a group of five conclusions when
the terms of the syllogism were "X," "Y," and "Z," as
Opposed to when the terms were meaningful terms which were
not related to the personal beliefs of the subject. Only
"when an issue is injected which relates to the personal
Opinions, wishes, fears, or convictions of an individual [is
the number of errors related] to the meaning involved in the
terms of the syllogism" (1944, p. 59). Bettinghaus §£.QL-
used abstract terms in their pretest and meaningful terms
involving the subjects' beliefs concerning the war in Viet-
nam in their main test. These researchers found marked
differences in the ability of subjects to judge the validity
of syllogisms under these two conditions. The present study
used low meaningful terms in one condition and high meaning-
ful terms in another, but in neither of these conditions
were the "personal Opinions, wishes, fears, or convictions"
of the subjects brought into play. Thus, the Bettinghaus

t 1. results and the results of this study conform to the

(D

findings of Morgan and Morton (1944).

The differences obtained in the studies by Lefford
(1946), Thistlethwaite (1950), Janis and Frick (1943),
Feather (1964), and Parrott (1967) also follow this pattern.
Across syllogistic forms, differences in the difficulty of
judgment of syllogistic validity—invalidity vary with the
meaningfulness Of the terms of the syllogism only if this

meaningfulness is related to relevant beliefs of the subjects.

 

d”-

51

Within the syllogistic form factor this pattern does
not always emerge. In the present study, the interaction of
meaningfulness and syllogistic form was significant. Thus,
for particular syllogistic forms a change from low meaning-
ful terms to high meaningful terms can have an effect on
judgmental difficulty even when that change does not involve
the emotions or relevant beliefs of the subjects. Some
forms are apparently easier to judge with high meaningful
content than with low, while the reverse is true for others.

Figure 1 (Chapter III)'indicates which differences in term

meaningfulness were significant for the 21 syllogistic forms.

The diagram reveals no general rule, unless many exceptions
to it are allowed. Given many exceptions, there is a slight
tendency for easier items to be still easier with high mean-
ingful content, and for more difficult items to be even

more difficult with high meaningful content. Three of the
21 differences are significant by the Newman-Keuls method
(Winer, 1962, pp. 77-89, 309-312). The low meaningful Ag;
and E9; syllogistic forms were significantly more difficult
than their high meaningful counterparts (p < .05, Newman-
Keuls). The A9; high meaningful syllogism was significantly
more difficult than was the low meaningful l9l_syllogism

(p < .05, Newman-Keuls). The remaining differences between

high and low meaningful pairs are not significant.

 

d":

52

When the scores of the high meaningful items are
analyzed alone the syllogistic form factor remains signif-
icant at the .001 level, although the order of difficulty
of the forms changes as can be seen from Table 3.
Syllogistic Form Main.Effect and
the Meaningfulness by Syllogistic
Form Interaction

The main effect for syllogistic form is significant

and the effect is obvious from Figure 1 (Chapter III). The

statement of this significant effect must be tempered by the

 

nature of the Meaningfulness by Syllogistic Form interaction
described above. Table 4 (Chapter III) indicates which dif-
ferences among all possible pairs of differences between the
21 syllogistic forms are significant by the Newman-Keuls
method. Differences between forms more than five ranks
apart in their order of difficulty are significant in all
cases (p < .01, Newman-Keuls). This means that given a
syllogistic form which finished in rank 5, the chances are
99 out of 100 that a comparable group of subjects will make
more errors on it than on the form that finished in rank k:§
and fewer errors on the form in rank A than on the form
which finished in rank kié, Much stronger statements may be
made if the actual forms involved are specified. Consider
form A9AJ which finished in rank 7. Table 4 (Chapter III)
indicates that it is significantly more difficult than all

forms which ranked above it in number of correct responses

53

and significantly less difficult than all forms that ranked
below it, with the exception of form A9g_which finished in
the next lowest rank (p < .01, Newman-Keuls). -Similar state-
ments concerning each of the 21 forms may be derived from
Table 4. Since any use of these forms would almost surely
involve some type of content, statements of differences
derived from Table 4 should be examined in light of Figure 1
(Chapter III) where content may be described as either low

or high meaningful. It should be noted that in most cases

of reversal of the meaningfulness effect between forms, for

 

example, the case of the major reversal between forms AAA,
EAAJ and 99;, clearly illustrated by Figure 1, Table 4 indi-
cates that the differences between these forms over high and
low meaningful term content are not significant. 'When no
reversals of the meaningfulness effect occur, as with forms

AE2, A01, A02, and AA2 (see Figure 1), no problems of inter-

 

pretation of the syllogistic form effects arise.

Three problem cases must be mentioned in which the
significant difference listed in Table 4 is dependent on the
meaningfulness of the terms of the syllogism. The A§g_form
is listed as significantly different from forms AIl, AEl,
.AAA, and AAA, which rank above it. This is true only for
low meaningful content or fOr combined low and high meaning—
ful content. It is not true for high meaningful content
alone. No problems occur with forms finishing below A99.

The second problem occurs with form EIl, which is listed as

54

significantly different from.E03, 001, and 002. This is
true for low meaningful and combined low and high meaningful
content, but not for high meaningful content alone. The

final problem occurs with form E01, which is listed in

Table 4 as significantly different from 001, 002, EEl, I03,

 

_A;;, and ggl, This is true only for high meaningful items
and items with combined high and low meaningful content. It
is not true for items with low meaningful terms only, as a
glance at Figure 1 will reveal. All other differences
listed as significant in Table 4 are significant for low,
high, and combined low and high meaningfulness levels.

The significant Meaningfulness by Syllogistic Form
interaction is the source of the three problem areas listed
above, and is a good example of why significant main effects
should not be reported unless significant interactions in—
volving them are fully explicated. Once the nature of the
interaction is understoOd, the main effect may be reported

and understood in light of that interaction effect.

Dogmatism Main Effect

The dogmatism main effect was not significant. No
dogmatism effect was predicted, since one would be eXpected
only if the syllogisms were perceived by subjects as orig-
inating from positive or negative sources. Such perceptions
were unlikely, given the experimental conditions of this
study. It is interesting to compare the present result with

the significant dogmatism effect obtained by Bettinghaus 99

 

55

AA. using positive and negative sources, and their finding
of a non—significant dogmatism effect with neutral sources
and neutral syllogisms (1970, pp. 241-242). These combined
results follow the predictions of dogmatism theory; specif-
ically, that only when the source is perceived as an author-
ity, either positive or negative, will dogmatism figure as
a relevant personality variable.

The mean number of correct out of 84 possible judg-
ments for low, middle, and high dogmatics was 45.70, 45.90,
and 43.36, respectively. The difference between only high
and low dogmatics is not significant. To the extent that
judgment of syllogistic validity may be seen as a test of
intelligence, these data correspond to Rokeach's findings

that dogmatism and intelligence are not correlated.

Diagrams in the Test Booklets
and Ambiguity

As reported in Chapter III, the cell means obtained
using only the 35 subjects who marked on their test booklets
were similar to the means of the subjects who made few or no
marks. Since the ambiguity measure is based on a diagram—
matic conception of logical judgment, it seemed worthwhile
to analyze the scores of these 35 subjects further to deter-
mine if ambiguity alone is a better predictor of final rank-
ing of syllogistic form difficulty for subjects who use
diagrams in responding to the validity—invalidity of a

syllogism than it is for subjects in general. The rho

56

between the predicted ambiguity rankings and the rankings
obtained for these 35 subjects is .683 as compared with .698
for the 173 non-marking subjects. Thus, the use of diagrams
by subjects does not appear to result in a general increase
in predictive power for ambiguity used alone. In terms of
responding correctly to the items, the use of diagrams may
have been slightly helpful. The average percentage correct
for all 207 subjects is 53.55 percent. For the 35 marking
subjects this figure increases to 54.72 percent. This dif—
ference is not significant, but the cases available are
limited. An alternative interpretation might be that sub-
jects who are better at logical judgmental tasks have a
tendency to use diagrams in determining their responses to
syllogistic conclusions, but again, this tendency is non-

significant.

The Atmosphere Effect

The atmosphere effect was first identified by
Woodworth and Sells (1935). They hold that the premises of
a syllogism create an atmosphere which subjects perceive and
carry into their judgments of syllogistic validity. This
effect would occur in two ways. First, an A, g, A, org;
premise should influence subjects to mark an A,.§, A, or 9,
conclusion, respectively, as valid. Second, an §.or 9

premise should create a negative atmoSphere leading to in-

creased acceptance of §_or 9 conclusions, and an 9_or A

57

premise should lead to an increased acceptance of 9_or A
conclusions through a someness atmosphere. The present
study attempted to equalize any atmosphere effect across
syllogistic forms. To determine whether an atmosphere
effect did influence the scores on different syllogistic
forms to a different extent, the scores were inspected for
atmosphere influences. With many items as they appear in
the test booklets, it is not clear which conclusions would
be expected to benefit from an atmosphere effect. -For
example, consider the Ag premise combination with A_and 9_
conclusions. By the atmosphere hypothesis, the A premise
should increase the number of subjects responding "valid"
to the A conclusion, and the §_premise should do the same
for the 9_conclusion by creating a negative atmosphere.
Suppose one of these effects occurs but not the other. Was

there an atmosphere effect? Morgan and Morton (1944, p. 40)

use the definition given above of atmosphere effect yet they

do not conform to this definition when they list which
effects are expected to be atmospheric. For an A§_premise

combination they list only E_as an atmosphere-type conclu—

sion (1944, pp. 42-43). Thus, the exact predictions for the

atmosphere effect appear to be equivocal. To avoid this

problem perhaps only obvious cases of a predicted atmosphere

effect, such as an AA premise combination with an.A conclu-

sion, should be considered. All items were examined for any

atmosphere-type effect. This examination revealed that for

58

all but six of the 21 syllogistic form items there was no
way to predict an atmosphere effect that was internally
consistent. The data for these 15 items revealed, at best,

very small effects of the type which on post hoc analysis

 

might be termed atmospheric.
The six forms for which an atmosphere effect could

be predicted were: AAl, AA2, EEl, 111, 001, and 002. A

 

test of the significance of the difference between two pro-
portions was conducted for each of theSe forms, comparing
the prOportion of responses marked valid on the suspect
conclusion with the prOportion of responses marked valid on
the non—suspect conclusion for each form. The non-suspect
conclusion for each form was used as a measure of the pro-
portion of successes to be expected for that particular form.
Five of the six differences were significant, with only the
difference for §§A_failing to reach significance.

To what extent did these differences affect the rank
order position of the five items? With item.AAA the direc-
tion of the atmosphere effect serves to increase the number

of correct responses. With items AA2, 111, 001, and 002,

 

 

the direction serves to increase the number of incorrect

responses. The removal of the atmospheric conclusions has
the effect of lowering AAA in position and increasing AAA,
111, 001, and 99A, Table 6 lists the 21 forms, using only
data from non-atmosphere suspect conclusions. Rho between

the ambiguity predicted order and this order is .603.

a":

59

Table 6. Twenty—one forms ordered after removal of possible
atmosphere effect

 

 

 

Predicted Actual Syllogistic Table 3 Percentage
Order Order Form Order .Correct
12 1 All ‘ 1 73.31
4 2 ,A§A_ 2 72.10
5 3 AAA_ 3 72.10
6 4 AAA 9 71.26
10 5 A9A_ 5 68.72
1 6 AAA 6 65.58
2 7 AAA 4 64.73
16 8 99A_ 13 63.29
13 9 A9A_ 7 59.18
8 10 A9A_ 8 57.85
18 11 99A_ 14 56.28
9 12 ,AAA_ 10 53.50
11 13 99A_ 11 52.05
7 l4 §9A_ 12 47.95
3 15 99A_ 15 45.53
20 16 A9A' 16 44.20
15 17 “AAA 17 42.03
l7 l8 A9A. 18 41.67
21 19 .AAA 21 40.34
14 20 A93_ 19 38.89
19 21 003 20 37.92

 

60

That the differences between this order and the order of
finish including the atmosphere suspect items are small is
indicated by the rho between them of .939.

Note that the six items involved in the reordering
on the basis of the existence of a valid conclusion are
independent of the six items involved in the atmosphere
reordering. Table 7 lists the 21 forms in the order ob-
tained after removal of both the valid conclusion items and
the atmosphere suspect items. Rho between the ambiguity
order and the Table 7 order is .590, Which is still signif—
icant at the .01 level. Between the Table 7 order and the
initial order given in Table 3 (Chapter III) rho is .844.

The use of the Table 3 order as Opposed to the
Table 5 order was discussed above. Similar comments apply
to the use of Tables 6 and 7. Table 6 was obtained by
ordering the items without the influence of a naturally
occurring atmosphere effect bearing upon five of the items
used in the ordering or on the percentage correct statistics.
In future use of these data, if the premise combinations are
to be employed with conclusions that are free of an atmo-
sphere effect, then the order of Table 6 is apprOpriate.

If conclusions are used in which an atmosphere effect is
possible, Table 3 order is the apprOpriate choice. If only
invalid conclusions which do not involve an atmosphere

effect are used, Table 7 should be employed.

0'“?

PI

 

61

Table 7. Twenty-one forms ordered by invalid items only
after removal of possible atmosphere effect

 

 

 

 

Predicted Actual Syllogistic - Table 3 Percentage
Order Order Form Order Correct
4 1 AAA_ 2 78.50
5 2 AAA 3 75.85
6 3 AAA 9 71.26
12 4 AAA_ 1 67.87
1 5 AE2* 6 65.58
2 6 AAA* 4 64.73
10 7 A9A_ 5 64.73
16 8 99A 13 63.29
13 9 A9A_ 7 59.18
18 10 99A_ 14 56.28
11 11 99A_ 11 52.05
7 12 .§9A_ 12 47.95
8 13 A9A_ 8 47.10
3 14 99A_ 15 45.53
20 15 A9A_ 16 44.20
15 16 ,AAA 17 42.03
l7 l7 A9A_ 18 41.67
21 18 AAA. 21 40.34
l4 l9 A9A 19 38.89
9 20 AAA_ 10 38.16
19 21 003 20 37.92

 

*No data from invalid conclusions.

62

Several additional comments on the atmosphere effect
seem apprOpriate. While not appearing in an identifiable
form on most items, the atmosphere effect did influence the
rank order position and percentage cOrrect of five syllo—
gistic forms. One problem with the atmosphere effect is
that while it is a name for an actual response tendency by
certain subjects who are judging the logical validity of
certain syllogistic conclusions, it may be regarded as a
spurious source of variance which should be avoided. Such
is not the case. Since atmOSphere can affect the responses
of subjects to syllogistic items, a ranking and listing of
percentage correct achieved by using atmosphere prone con-
clusions with some premise combinations but not with others,
would not be a prOper ranking for use with either all pos-

sible conclusions or with only non-atmospheric conclusions.

2
Therefore, the effect must be controlled by giving it equal
chance to Occur in each form. For each premise represented
in a premise combination of this study, a conclusion which
could have produced an atmosphere effect appeared at least
once with the premise combination. If the control used were
one of eliminating all possible atmospheric conclusions, the
results would apply, obviously, only to syllogistic judg-
ments of non-atmospheric items, and not to syllogistic

judgment in general. For this reason the rankings of

Table 3 and the additional information supplied by Figure l

a".

I'—

63

and Table 4 (Chapter III) are the preferred rankings and
percentage statistics for most uses involving syllogistic
judgment.

A second problem of the atmosphere effect is that it
confounds two distinct sources of variance in its conception.
The first source is the atmosphere effect itself: for exam—
ple, the effect of using a conclusion beginning with the
word "all" following two premises which begin with the same
word. The second source is the inherent difficulty of
judging the validity—invalidity of an-A conclusion, for
example, in an AAA syllogism. The two must be kept con-
ceptually distinct.

The effects of these two sources of variance could
be investigated using Operations similar to the following.
The second variance could be estimated using the word "all"
(or "none" or "some”) to begin both premises, but substi-
tuting a word or phrase with similar meaning in place of
"all” in the conclusion. For example, "all X is Y, all Y
is Z, every single member of Z is a member of X." The
variance of the first source would be difficult to estimate
directly, since it should not involve the actual relation-
ship Of “X" and "Z," and "all X is Y, all Y is Z, all J is
K" might involve a plausibility control problem, among
others. The first variance could be eStimated by subtract—
ing the effect of the second source of variance from the

overall effect of "all X is Y, all Y is Z, all Z is X."

 

n0":

64

Thus, the removal of the atmosphere.effect as in Tables 6
and 7 removes not just the effect itself but also the effect
of the conclusions which is confounded with it, another

reason for using Table 3 (Chapter III) in most applications.

Summary

The principal points of this discussion are as
follows:

1. The ambiguity hypothesis predicting the order
of difficulty of the forms is supported. An analysis of the
two forms which depart markedly from the predicted order
leads to the conclusion that the ambiguity of a given form
plus knowledge of the existence or non-existence of a valid
conclusion for that form may result in greater predictive
accuracy of the order of difficulty Of the form than ambi—
guity alone, if difficulty is measured using both valid and
invalid syllogisms. This latter hypothesis was not formally
tested. The existence or non-existenCe of a valid conclu—
sion is related to the ambiguity of a given syllogistic form.
The choice of Table 3 (Chapter III) or Table 5 (Chapter IV)
as a rank order list of syllogistic forms depends upon the
way in which they will be used. Table 5 should be used only
when all conclusions are invalid. Ambiguity predicts diffi-
culty about as well with invalid conclusions only, as with

valid and invalid conclusions.

i.) H I

65

2. An analysis across premise types and figures
yields little new information. Analysis of the items of
equal ambiguity reveals that ambiguity when used alone
appears to be useful as a predictor of overall order of
difficulty, rather than as an accurate predictor of order
of finish between pairs of items.

3. A general meaningfulness effect did not occur
in this experiment, while one did occur between the pretest
and the main test of the Bettinghaus gE__A, study. These
two results are in accord with the Morgan and Morton (1944)
prediction that only when relevant beliefs of the subjects
are involved does a change in term meaningfulness result in
a change in apparent difficulty or syllogistic judgment
across all syllogistic forms.

4. For particular syllogistic forms, a change in
meaningfulness from low to high, as defined in Chapter II,
can result in a change in apparent difficulty without invok-
ing the relevant beliefs of subjects. The particular forms
for which the invoking of relevant beliefs appears to be
unnecessary in producing a change in apparent difficulty
from a change in meaningfulness are A§A_and §9A, where low
meaningful terms are more difficult than high meaningful
terms, and A9A, where high meaningful terms are more diffi—
cult to judge than low.

5. There is a main effect for syllogistic form, and

many individual comparisons between the levels of this

66

factor are significant by fairly conservative procedures.
These effects must be interpreted in light of the signif-
icant Meaningfulness by Syllogistic Form interaction which
restricts the between-form differences to particular mean-
ingfulness conditions for certain forms listed in the text.

6. The non—significance of the dogmatism main
effect is consistent with Rokeach's theory of dogmatism
(1960) and with previous results such as those of Betting-
haus §A_AA, Dogmatism as a personality variable appears to
be relevant in judging syllogistic validity only when the
syllogisms originate from a positive or negative source.

To the extent that the ability to judge syllogistic validity
is a measure of intelligence, this study Offers support for
Rokeach's findings that dogmatism and intelligence are
uncorrelated variables.

7. Predictions based on ambiguity alone are no
better for the responses of subjects who used diagrams in
determining their responses than they are for all subjects.
The use of diagrams by a subject was asSOciated with a
slight increase in the number of items marked correctly, but
this difference is not significant.

8. An effect which could be called an atmosphere
effect was found in five forms. Tables 6 and 7 (Chapter IV)
give the order and percentage correct for the 21 forms cor—
rected for atmosphere effect and for atmosphere plus the

existence of a valid conclusion, respectively. Table 3

 

67

(Chapter III) remains the preferred table except when
naturally occurring atmosphere effects and the effects of
the existence of a valid conclusion are to be excluded from
the relative and absolute difficulty ratings. The exact
predictions concerning which items should produce an atmo-
sphere effect are unclear. To the extent that an atmosphere
effect exists it should be viewed as one of several legit-
imate sources of variance which influence subjects' scores
on the various syllogistic forms. Moreover, the atmosphere
effect involves two separate sources of variance: the
effect relating specifically to the use of the same word

in the conclusion as in the premises ("all," "none," "some")
and the effect of the difficulty of the conclusion itself

apart from the effect of the same word.

Concluding Remarks

Appendix I of this dissertation sets forth several
assumptions which limit the generalization of the findings.
First, it is possible to generalize from a given form to its

logical equivalents logically, but it is not known if the

 

behavioral effects will correSpond. Thus, the results for
the AA premise combinations m§y_generalize to §A_premise
combinations, but no data are provided in the present study
to verify this possibility. Similarly, no data are provided
herein to indicate the soundness of regarding AAA, for exam—

ple, according to the results for AEl, even though AE3 is

68

the logical equivalent of AAA, Tests of such behavioral
generalizations must await future research.

This study sought to establish a baseline on the
difficulty of judging the validity-invalidity of different
syllogistic forms. It did so for two reasons. First, in
studies which employ the judgment of syllogistic forms as
a dependent variable, it is usually necessary to know the
relative or absolute difficulty of different syllogistic
forms. Second, the results are interesting in themselves
in terms of a message variable of logical style. Little is
known of the effects of different styles on message percep—
tion. HOpefully, this study has added to the knowledge of

the effects of syllogistic styles in communication messages.

REFERENCES

4.) 1' ‘

 

 

REFERENCES

Ambrose, A., and Lazerowitz, M. Fundamentals 9A Symbolic
Logic. New York: Rinehart, 1948.

Archer, E. A re—evaluation of the meaningfulnéss of all
possible CVC trigrams. Psych. Monog., Whole No.
497, 1960. .

Bettinghaus, E., Miller, G., and Steinfatt, T. Source
evaluation, syllogistic content, and judgments
of logical validity by high and low dogmatic
persons. g, Pers. 999. Psych., 1970,16: 238-244.

Feather, N. Acceptance and rejection of arguments in
relation to attitude strength, critical ability,
and intolerance of inconsistency. A. A9. S_O£.
Psych., 1964, 69: 127—136.

Glaze, J. A. The association value of nonsense syllables.
A, Genet. Psych., 35: 255-269.

 

Henle, M. On the relation between logic and thinking.
Psych. Rev., 1962, 69: 366-378. '

 

Henle, M., and Michael, M. The influence of attitude on
syllogistic reasoning. g, Soc. Psych., 1956, 44:
115-127.

Janis, I., and Frick, F. The relationship between attitudes
toward conclusions and errors in judging logical
validity of syllogisms. .1- Exp. Psych., 1943, 33:
73-77.

Krueger, W. The relative difficulty of nonsense syllables.
9, Exp. Psych., 1934, 17: 145-153.

 

Lefford, A. The influence of emotional subject matter on
logical reasoning. 9, Gen. Psych., 1946, 34: 127-
151.

McGuire, W. A syllogistic analysis of cOgnitive relation-
ships. In M. Rosenberg §£_AA. (eds.), Attitude
Organization and Change. New Haven: Yale Univer.
Press, 1960, pp. 65-111.

69

70

Morgan, J., and Morton, J. The distortion of syllogistic
reasoning produced by personal convictions. g, Soc.
Psych., 1944, 20: 39-59.

Noble, C. An analysis of meaning. Psych. Rev., 1952, 59:
421-430.

Parrott, G. The effects of premise content on accuracy and
solution time in syllogistic reasoning. Unpublished
M.A. Thesis, Michigan State University, 1967.

Rokeach, M. The Qpen and Closed Mind. New York: Basic
Books,‘l960. ‘

Scheffe, H. The Analysis 9A_Variance. New York: Wiley,
1960.

 

 

Siegel, S. Nonparametric Statistics. New York: McGraw-
Hill, 1956.

Steinfatt, T. Computational formulas for the analysis of
variance. Unpublished paper, Department of
Communication, Michigan State University, 1970.

Thistlethwaite, D. Attitudes and structure as factors in
the distortion of reasoning. g, A9, Soc. Psych.,
1950, 45: 442-458.

Thorndike, E., and Lorge, I. The Teacher's Word Book 9£_
30,000 Words. New YOrk: Columbia Univ. Press,
1944.

 

Thouless, R. Effects of prejudice on reasoning. British
g. Psych., 1959, 50: 290-293.

Troldahl, V., and Powell, F. A short-form dogmatism scale
for use in field studies. Social Forces, 1965, 44:
211-214.

Wilkins, M. The effect of changed material on the ability
to do formal syllogistic reasoning. Archives 9:
Psych., 1928, 16: No. 102.'

Winer, B. Statistical Principles AA_EXperimental Design.
New York: McGraw-Hill, 1962.

Woodworth, R., and Sells, S. An atmosphere effect in formal
syllogistic reasoning. 9, Exp. Psych., 1935, 18:
451—460.

 

APPENDIX I

DEFINITION OF THE SYLLOGISTIC FORM FACTOR

APPENDIX I

DEFINITION OF THE SYLLOGISTIC FORM FACTOR

The syllogistic form factor is the central focus of
this study. A syllogism contains three statements: two
premises and a conclusion. There are two terms in each
statement and three distinct terms in each syllogism, each
term being used in two statements. The term employed in
both premises is called the middle term. The relationship
between each of the two remaining terms and the middle term
is stated in the premises, one relationship in each premise.
The conclusion states a relationship between the two non-
middle terms. The validity of the syllogism is determined
by asking the question "must the relationship stated in the
conclusion follow from the relationships stated in the prem-
ises?" If the answer is yes, the syllogism is valid. If
the answer is no or not necessarily, the syllogism is
invalid.

Four distinct types of relationships between the
terms of a syllogism are possible. These are designated as
A, A, A, and 9, 'An A relationship is an AAA relationship.
Letting A, A, and A_represent the terms, "all X is Y“ is an

,A relationship. Given two distinct terms, there are two

71

 

 

 

72

distinct A relationships: "all x is Y" and "all Y is X."
An.§_relationship is a Aggg_relationship; for example, "no
X is Y." There is only one such.§_re1ationship possible
between any two terms X and Y: "no Y is X" is logically
equivalent to "no X is Y.’i An A relationship is a §ng
relationship; for example, "some X is Y." There is only

one A relationship possible between two distinct terms since
"some X is Y" is the logical equivalent of "some Y is X."

An 9 relationship is a some are not relationship, such as

”some X are not Y." Given two distinct terms there are two

 

unique 9_re1ationships possible between them: “some X are
not Y" and "some Y are not X."

The premises of a syllogism may be formed by using
any two relationships of the A, A, A, and 9 type, one a
relationship between the middle term and an X term and the
other a relationship between the middle term and the Z term,
where X is not in identity with Z. The conclusion of a
syllogism is a relationship of the A, A, A, or 9 type
between the X and the Z term. For convenience, this study
will use X to refer to the non-middle term in the first
occurring premise, Y to refer to the middle term, and Z to
refer to the non-middle term in the second occurring premise.
Note that no reference has been made to subject or predicate
terms or, correspondingly, to minor or major premises. If
the major-minor premise distinction is made, the non-middle

term of the major premise becomes the predicate term and

73

occurs.1ast in the conclusion, and the non-middle term of
the minor premise becomes the subject term and occurs first
in the conclusion. The major premise must be the first
premise and the minor premise the second premise. This
study does not consider these distinctions, as they add
little to the discussion while making it more restrictive.
Thus, there are no requirements placed on the premises other
than those stated above, and they will be referred to as
first premise and second premise depending only on their

order of occurrence.

 

The figures of a syllogism are determined by the
order in which the terms of the premises occur. If Y is the
middle term, the order case in which Y occurs first in the
first premise and second in the second premise is designated
the First figure. For example, "all Y is X" and "all Z is
Y" is a First figure premise combination. Adding any con—
clusion (A, A, A, or 9, involving X and Z) would make them
First figure syllogisms. The Second figure occurs when Y
appears second in both premises. The Third figure occurs
when Y appears first in both premises, and the Fourth figure
when Y appears last in the first premise and first in the
second premise. If‘X is always the non-middle term of the
first premise and Z is always the non-middle term of the
second premise, then these four figures are the only pos-
sible orders in which X, Y, and Z can combine. Note that

these figures may be defined without reference to the form

.9“

74

or content of the conclusion, except to require that the
conclusion involve an A, A, A, or 9 relationship between
X and Z.

In summary, there are two premises involving three
distinct terms in each syllogism with four figures or orders
of presentation of these terms. Each first premise may be
'A, A, A, or 9_and each may be associated with an A, A, A, or
9 second premise, yielding 16 possible premise combinations.
Since each of these 16 may occur in any of four figures
there are a total of 64 possible distinct premise combina-
tions. Note that the ordering of the terms in the four
figures accounts for the two A relationships and the two 9_
relationships so that there are in fact four and not six
possible relationships in the premises of the form A, A, A,
or 9 when using the figure system. The figures do not
account for this distinction in the conclusion. Thus, for
each of the 64 possible premise combinations there are six
possible conclusions or 384 possible syllogisms. Since this
is a rather large number, certain simplifying assumptions
will be introduced so that these 384 syllogisms may be
reduced to a more manageable total.

First, instead of referring to syllogisms proper,
this paper will often discuss premise combinations with the
understanding that there are six syllogisms for each of the

64 premise combinations, one syllogism corresponding to each

 

.9!)

75

conclusion. This simply assumes a relationship between each
premise combination and its set of conclusions.

The second assumption is that X, Y, and Z, the terms
of the syllogism, are each non-empty: that there exists at
least one element i which belongs to the set X, at least one
element j which belongs to the set Y, and at least one ele-
ment k which belongs to the set Z. This is an assumption
which is not a part of many logical systems (for example,
Ambrose and Lazerowitz, 1948, and most basic logic texts).

The reasoning behind it is as follows. Judging the
validity of a syllogism assumes the truth of both premises
for the purpose of making that judgment. If this were not
so, no conclusion could be judged with respect to its valid-
ity or invalidity, since "all X is Y" might then mean "not
all X'is Y." This is a postulate of consistency. Within
the frame of reference of a particular syllogism, a premise
cannot be both true and untrue at the same time. If the
assumption is that the premise is "true" in order to allow
a judgment of the validity-invalidity of the conclusion with
respect to the premises, then within the context of the
judgment of the syllogism, that assumption must not be
violated or there will be a logical contradiction.

The set Z cannot be empty in A and 9 premises. For,
the contradiction of "some X is Y” is "no X is Y." Let Z
equal the null set, and consider the statement "some Z is

Y." "Some Z is Y" implies that there exists an element k

 

76

belonging to Z with the property that k belongs to Y. But
"Z is the null set" implies that there exists no element k
such that k belongs to Z. Thus, there exists no k with the
prOperty that k belongs to Z and that k belongs to Y.
Therefore, "no Z is Y," which is a logical contradiction of
"some Z is Y." It can be concluded that Z (and by the iden—
tical reasoning X and Y) may not equal the null set in;A
premises, nor in syllogisms involving A premises, when Z

is the first term of the I premise.

Let Z equal the null set and consider "some Z are

 

not Y." "Some Z are not Y" implies that there exists a k
belonging to Z with the prOperty that k belongs to not-Y.
If Z equals the null set, then there exists no k belonging
to Z. Therefore, there exists no k belonging to Z such that
k belongs to not-Y, and this contradicts a direct implica-
tion of "some Z are not Y." Since the reasoning for X and
Y is identical, it can be concluded that Z, X, and Y cannot
equal the null set when any of them are used as the first
term in an 9 premise.

The above two paragraphs demonstrate the necessity
for the existence of the first term of an A_or 9 premise.
This existence in turn implies the existence of the second
term in the A case, since for some Z to be Y there must be
some Y. This implication does not hold for the second term
of an 9_premise, nor can it be shown that A and A premises

imply the existence of their terms. So in a given syllogism,

77

some terms are forced into existence by the premises and
some are not. This introduces a problem in the definition
of ambiguity of the premise combination. The existence or
non—existence of the terms of a syllogism changes the number
of relationships which are possible between the three terms.
If the terms of the A premise and the first term of an 9
premise are assumed to exist while non-existence is an
allowable state for the terms of A and A premises, the
number of relationships, and thus the ambiguity, of premises
of the A and A form is increased over the ambiguity of 9 and
especially A premises. Thus, for purposes of prediction
from ambiguity of the premises, consistency in existability
is assumed, and since some terms must exist, all terms will
be assumed to exist within the logical system used in this
study.

The third assumption is that the order of occurrence
of the premises is unimportant: that an AA premise combina-
tion, for example, may be treated as an AA premise combina—
tion, and vice versa. By the definitions used in this paper,
such reversal is acceptable on strictly logical grounds.
Since the major—minor premise distinction has not been
adopted, the only difference between AA and AA in the logi—
cal context is the positioning of X and Z. Thus, this
assumption reduces to the assumption that the symbols X and
Z are interchangeable and have been used with the first

premise and second premise only for convenience. Logically,

 

l'“:

78

this is a justifiable assumption. PsycholOgically, this
might not hold true. Logically, by the system used in this
study, "all X is Y, no Y is Z" is equivalent to "no Y is Z,
all X is Y." But a person receiving the former premise
combination in a message might not react to it in the same
way he would to the latter. This assumption means that in
the logical context, results concerning AA_will generalize
to AA, to continue the example. In the psychological con-
text we must caution that while such generalizations seem

logical, they will have been tested only indirectly by this

 

experiment and a direct empirical test may be warranted in
the future to determine if such generalizations seem to hold.

The effect of the third assumption is to reduce the
64 possible premise combinations to 40 logically distinct
combinations.

The fourth assumption is very similar to the third.
It concerns the fact that the four figures involve logical
duplication in certain premise combination conditions due
to: (l) the reversibility of A_and A premises, and (2) the
duplication of premise order. Duplication of premise order
is also discussed in the second assumption above. As an
example of premise duplication, consider the four figures
in the AA premise combination case. These are AAA, "all
Y is X, all Z is Y"; AAA, "all X is Y, all Z is Y"; AAA,
”all Y is X, all Y is Z": and AAA, "all X is Y, all Y is Z."

Inspect the AA2 and AA3 cases. It can be seen by inspection

79

that these two cases are distinct from each other in the
logical sense, since "all Y is X" is not logically equiva-
lent to "all X is Y." The same may be said for their second
premises. Now inspect the AAA case and compare it with the
AAA_and AAA cases. The first premise of the AA_case "all Y
is X" distinguishes it from the AAA case where the first
premise is "all X is Y." Further, if the two premises are
reversed in order for either case (but not both cases) and
the convention of using X as the non-middle term of the
first premise and Z as the non—middle term of the second
premise is continued, it can still be seen that the AAA and
_AAA cases are logically distinct. The same Operations may
be performed on the AAA and AAA cases to demonstrate that
they, too, are logically distinct. Now consider the AAA and
AAA_cases. First, reverse the order of the premises in the
AAA case, and then substitute X for Z and Z for X in accor-
dance with the convention. The two premise combinations now
both read "all Y is X, all Z is Y." Therefore, the AAA case
is logically equivalent to the AAA case.

Continuing with the fourth assumption consider an
example of A_and A_duplication. It will be recalled that
"no X is Y” is the logical equivalent of "no Y is X," and
that "some X is Y" is the logical equivalent of “some Y is
X." This reversibility dOes not hold for A and 9 statements.
Consider the four figures of the AA case. Using the same

procedure as is outlined above under premise duplication,

IF

 

80

it can be seen by inspection that the __A_case is distinct
from the AAA case, since "all Y is X" is not equivalent to
"all X is Y." But in cases AAA_and AAA, the first premise
is "all Y is X" and the second premise is an A_involving Y
and Z. Since I'no Y is Z" can be read (logically) “no Z is
Y," both the AAA and AAA_cases may be read "all Y is X, no
Z is Y" and thus AAA is the logical equivalent of AAA, By
the same reasoning, AAA can be shown to be the logical
equivalent of AAA.

The assumption of duplication and equivalence has
been justified on logical grounds, just as in the third
assumption. The same reservation in generalization dis-
cussed for the third assumption holds for the fourth assump-
tion. Logical equivalence may not result in psychological
equivalence. Logically, what holds for the AAA case must
hold for the AAA case, but the psychological ability to
judge the validity of syllOgisms involving these two premise
combinations may or may not be equal. This is a type of
psychological equivalence which will have to be confirmed
or denied by future research.

The effect of the fourth assumption is to reduce the
number of logically distinct premise combinations from 40 to
21. It will be demonstrated below that a given set of these
21 are not further reducible and, thus, that there are 21
logically unique premise combination types given the logical

system and assumptions outlined above.

 

.1?"

 

81

It should be noticed that the set Of logically
unique premise combinations is not a unique set since, for
example, either AAA or AAA could be included in the set,
but not both. The particular set of 21 logically unique
premise combinations used in this study to represent the
21 unique cases is specified in Chapter II. These 21
premise combinations constitute the 21 levels of the

syllogistic form factor.

 

If

 

.9":

APPENDIX II

GENERAL METHOD FOR DETERMINING AMBIGUITY

F!

q":

 

APPENDIX II

GENERAL METHOD FOR DETERMINING AMBIGUITY

The ambiguity of a premise combination is the number
of different relationships between the three terms of the
combination which would allow the conditions implied by the
premise combination to be true. For any two relationships
of X, Y, and Z, one relationship is different from another
if and only if there exists a specific subset of X-union-Y-
union-Z which may be called W, with the property that W is
empty in one relationship and non—empty in the other rela—
tionship. The condition that W must belong to the union of
X, Y, and Z means that any changes in the relationships
which occur outside of that union, such as a change from
part—of—outside to all—of—outside, are not to be considered
since they will have no bearing on the conclusions which may
be drawn from premises concerning X, Y, and Z.

A general method for determining the number of sit—
uations from which a given premise combination could have
arisen (ambiguity) is as follows. Consider the relationship
of the middle term with either of the other terms, say X.

Y and X have exactly five different ways of combining.

These are (a) disjoint, (b) intersecting but neither

82

 

83

contained within the other, (c) identity, (d) X contained

in but less than Y, and (e) Y contained in but less than X.
No other different combinations of X and Y exist, given that
neither X nor Y can be the null set, and using a definition
of the difference between two sets which is a direct analogy
to the definition of the difference between three sets,
given above.

Now consider the ways in which the third term can
combine with each of the five combinations of X and Y. The
addition of the third term presents a problem that is best
solved by dividing each of the five combinations of two
terms into discrete, non—overlapping spaces. For case (a),
Z may contain all, part, or none of X, all, part, or none of
Y, and will either contain some area outside of X—union-Y or
will not. Note that all-of—outside is not a separate Option
for Z since all—of-outside is not different from part—of—
outside by the definition of different. The number of pos-
sible Euler diagrams or relationships for case (a) is given
by multiplication since for each Way in which Z may combine
with X it may also combine in a given way with Y and also
combine or not combine with the area outside of X-union-Y.
Consequently, the number of different relationships between
X, Y, and Z for case (a) is three (from X) times three
(from Y) times two (from outside) minus one (since Z cannot
be empty) or 17. The reasoning is similar for the remaining

four cases.

ll,

84

For case (b), Z may combine with all, part, or none
of X—minus—the—intersection-ofFX—and-Y, all, part, or none
of the—intersection—of—X-and-Y, all, part, or none of Y—
minus—the—intersection-of—X—and—Y, and part or none of the
area outside of X—union—Y. The number of different rela-
tionships for case (b) is given by multiplying three times
three times three times two, minus one, or 53 possible
different relationships for case (b).

Case (c) is the simplest case. Z may combine with
all, part, or none of XY and with part or none of outside.
Three times two, minus one, gives five possible different
relationships for case (c).

Case (d) is similar to case (b) and case (e) where
the problem must be phrased in terms of spaces that both
cover the field and are disjoint. Z may combine with all,
part, or none of X—minus-Y, all, part, or none of Y, and
with part or none of outside. Multiplying, three times
three times two, minus one, equals 17 different relation—
ships for case (d). Since case (e) is identical to case (d)
with X and Y reversing positions, the number of diagrams for
case (e) will also be 17. Adding the totals for each case
gives 109 total possible relationships between the three
terms of a syllogism, given the assumptions of this study.
This number is the upper limit of ambiguity, although the
ambiguity of no single premise combination is that high in

itself.

If

85

Using the above method with each of the 64 possible
premise combinations discloses that the set of relationships
implied by the premises of each premise combination which
was eliminated by the simplifying assumptions of this paper
(see Appendix I) are in identity with the set of relation-
ships of the particular premise combination which represents
the eliminated premise combinations. This is an independent
verification that a set of 21 logically unique premise com-
binations are in fact the logical equivalents of the premise
combinations which they represent. This method also demon-
strates that a set of 21 logically unique premise combina-
tions is truly logically unique, for, the set of relation-
ships corresponding to each one of a given set of 21
logically unique premise combinations is different from
each of the other 20 sets of relationships. The number of
such relationships is occasionally the same for two of the
21, but the elements of each set are different. The appli-
cation of the above methOd to each premise combination is
slightly different, but the general method is to list those
ways in which X and Y can combine which apply to the first
premise, and then to inspect each of these for the prOper
multipliers with the second premise.

The AAA case will serve as an example. The first
premise, "all x is Y," is covered by the (c) and (e) cases.
Cases (a), (b), and (d) do not apply. When inspected with

the second premise of "no Z is Y," case (c) yields the

[F

 

.g H I

 

86

following results. of the three possibilities with XY (all,

part, or none) only none can apply to "no Z is Y." Of the

 

two possibilities with the outside, either AQAA or AQAA may
apply. By the existence assumption, the none-none case is
ruled out. Thus, the number of relationships possible with
case (c) is one times two, minus one, which equals one.
Case (e) yields the following. Of the three possibilities
with Y-minus—X, only Egg; is allowable. Of the three with

X only none is allowable. Of the two with outside, both are

 

allowable. The none—none—none condition is again out.
Multiplying, one times one times two, minus one, equals one.
Adding across cases gives one plus one which equals two.
Thus, the ambiguity of AAA is two. While this is the
simplest case, the principles involved remain constant
across all premise combinations. The reason for setting up
this counting method instead of simply enumerating the pos-
sible relationships with diagrams will become clear to the
person who attempts to enumerate an AA premise combination
without it. The ambiguity associated with each of the 21

premise combinations is listed with each in Chapter II.

.F

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