Drawing inferences from quantified statements: a study of the square of opposition

JOURNAL OF VERBALLEARNINGAND VERBALBEHAVIOR22, 535-546 (1983)

Drawing Inferences from Quantified Statements: A Study of the Square of Opposition STEPHEN E . NEWSTEAD

Plymouth Polytechnic AND

RICHARD A. GRIGGS University of Florida Two experiments are reported which investigated subjects' ability to draw inferences from statements using the quantifiers all, no, some, and somenot. In both experiments, subjects had to evaluate the truth of test statements with respect to given quantified statements. Performance on some inferences accorded well with logic, but on other inferences, especially when the test statement used the converse of the given statement, performance was poorer. In addition, there were large and consistent individual differences in the inferences subjects were willing to draw. Theories of syllogistic reasoning which suggest that quantifiers are interpreted logically and/or consistently are called into question by the present results. The results also suggest a simple explanation for memory confusions between quantifiers.

Ever since Aristotle's work on syllogisms, there has been considerable interest in how people use quantified statements. Current research interest centers on two main themes: how people reason with quantifiers and how people r e m e m b e r quantifiers. It is argued in this paper that this research is difficult to interpret because we have insufficient information on how quantifiers are understood, a deficiency which is in part rectified in the experimental work reported here. Consider, for example, the research that has been carried out on syllogistic reasoning. One suggestion that has been put forward to explain the errors made is the conversion hypothesis (Chapman & Chapman, 1959). This claims that a universally quantified statement such as All A s are Bs The first author was on sabbatical leave at the University of Florida when part of this research was conducted. Address all c o r r e s p o n d e n c e to Stephen E. Newstead, Dept. of Psychology, Plymouth Polytechnic, Drake Circus, Plymouth PL4 8AA, England, or to Richard A. Griggs, Dept. of Psychology, University of Florida, Gainesville, FL 32611.

is incorrectly interpreted as implying its converse, All Bs are A s . The conversion theory is given its most extreme statement in the recent work of Revlin (Revlis, 1975a & b; Revlin & Leirer, 1980); he claims not only that subjects have a tendency to convert statements, but that the converted interpretation is actually the preferred one. Thus, when carrying out a syllogistic reasoning task, subjects actually reason with all statements in their converted form, and only reason with the unconverted form if no conclusion follows. The problem with the conversion hypothesis is in the "circular relation between understanding and logic" (Smedslund, 1970). The hypothesis assumes that, having converted the premises, the subject reasons logically; however, there is no way of knowing that this is the case, and hence no way of knowing whether the errors occur through faulty interpretation of the premises or faulty reasoning. Clearly, it is essential to have independent evidence of conversion occurring during interpretation. Similar problems arise with extended syl-

535 0022-5371/83 $3.00 Copyright© 1983by AcademicPress, Inc. All fightsof reproductionin any formreserved.

536

NEWSTEAD AND GRIGGS

logisms, or set inclusion relationships. A four-term set inclusion paragraph contains the information that All A s are Bs, All Bs are Cs, and All Cs are Ds. Many subjects perform poorly on the converted premises, incorrectly responding as if they are true (Griggs, 1976). The conversion hypothesis can readily explain this finding, but once again there is no way of knowing whether the error arises at the interpretation stage or during reasoning. The research on memory for quantifiers has focused on the question of representation. Some researchers have claimed that the evidence supports an analog model (Holyoak & Glass, 1978) while others favor propositional approaches (Anderson, 1981). In both of these studies, recognition confusions b e t w e e n quantifiers were examined, on the assumption that such confusions can provide information as to how quantifiers are represented. This assumption is w e a k e n e d by the fact that some quantifiers can be logically inferred from each other. Thus s o m e can be inferred from all, and s o m e n o t can be inferred from n o - though, as we shall see, it is not necessarily the case that people will make the permitted inferences. In addition to logical inferences, quantifiers frequently lead to invited inferences. Thus the quantifier s o m e might invite the listener to assume that not all the instances are positive, in other words that s o m e n o t is also the case. While such an inference is against the dictates of logic, it is a plausible one for listeners to draw (Grice, 1975). The existence of such inferences makes research on memory for quantifiers difficult to interpret. For example, if subjects confuse all and some, is this because these are given similar representations in memory, or is it because subjects are making a logical inference and finding it difficult to distinguish between the originally presented sentence and the inference? That subjects frequently do confuse inferences with presented material is extensively illustrated in the work of Bransford and Franks and their colleagues (e.g., Bransford, Barclay &

Franks, 1972). In fact, confusions between a// and s o m e were quite frequent in Holyoak and Glass's (1978) study and in that of Anderson (1981), though the interpretation of this finding is different in the two studies. It seems essential to have more detailed evidence on the types of inference that subjects do draw before firm conclusions about representation in memory can be arrived at. The "correct" interpretation of quantitiers is prescribed in the classical square of opposition (see Fig. 1), which depicts the relationship between all (A), no (E), s o m e (I), and s o m e n o t (0). All and s o m e n o t are contradictories, since the truth of one implies the falsity of the other and vice versa. Thus if the all statement All republics are ungrateful is true, then the s o m e n o t statement S o m e republics are not ungrateful is necessarily false. If the all statement had been false, the s o m e n o t statement would necessarily be true. N o statements (e.g., N o republics are ungrateful) and s o m e statements ( S o m e republics are ungrateful) are also contradictories. All and no statements are contraries; from the truth of one, the falsity of the other can be inferred, but nothing can be inferred about the other from CONTRARIES

A

SUBCONTRARIES A = ALL I = SOME

E = NO O = SOME NOT

F[o. l. The classical square of opposition.

537

DRAWING INFERENCES FROM QUANTIFIERS

the falsity of one of these. S o m e and s o m e n o t are s u b c o n t r a r i e s in that the truth of one can be inferred from the falsity of the other, but nothing can be inferred from the truth of one of them. Finally, s o m e is the s u b a l t e r n of all, and s o m e n o t is the subaltern of no. From the truth of all one can infer the truth of s o m e , but if all is false, nothing can be inferred about s o m e . From the truth of s o m e , nothing can be inferred about all; but from the falsity of s o m e , one can infer the falsity of all. For further discussion of the square of opposition the reader is referred to Cohen and Nagel (1934). Logic also dictates whether quantified statements can be converted or not. We have already seen that it is not valid to convert all statements and the same is true of s o m e n o t statements. On the other hand, n o and s o m e a r e convertible. Table 1 gives the permitted inferences from all, no, s o m e , and s o m e n o t statements. It is, of course, an empirical question as to whether people do draw the inferences presented in this table. There already exists some evidence concerning the interpretation of such quantified statements. One popular technique has been to ask subjects which Venn or Euler diagrams are appropriate for quantified statements of different kinds (JohnsonLaird, 1970; Neimark & Chapman, 1975; Griggs & Warner, 1982; Caplan, Note 1).

Typically, about 75% of adult subjects choose the two correct diagrams for statements such as A l l A s are B s (i.e., A is a subset of B; A is identical to B). This is a little surprising from the point of view of the conversion hypothesis, since one might have expected a tendency to choose just the identity relationship, for which conversion is permissible. In fact, the only study which gives a breakdown of errors reveals that choice of this diagram alone was made by only 1 subject out of 32 (Griggs & Warner 1982). Subjects show great accuracy with n o statements, the vast majority correctly choosing the single correct diagram of two nonoverlapping circles. On the other hand, s o m e and s o m e n o t statements lead to many incorrect responses; in Neimark and Chapman's (1975) study, college students showed only about 30% accuracy on each of these quantifiers. The principal error can be characterized as indicating an unwillingness to generalize from the particular to the universal. Thus subjects given s o m e statements failed to choose the two diagrams for which all statements were applicable; subjects given s o m e n o t statements often failed to choose the diagram for which n o statements were valid. These studies using Venn or Euler diagrams provide evidence that subjects interpret universal statements with considerable accuracy but not particular statements.

TABLE 1 PERMISSIBLE INFERENCES FROM QUANTIFIED STATEMENTS Quantifier given

Inference

All As are Bs

No As are Bs

Some As are Bs

S ome As are not Bs

All As are Bs No As are Bs S o m e As are Bs So me As are not Bs All Bs are As No Bs are As So me Bs are As So me Bs are not As

T F T F U F T U

F T F T F T F T

U F T U U F T U

F U U T U U U U

Note. T = True; F = False; U = U n d e t e r m i n e d .

538

NEWSTEAD AND GRIGGS

However, there are problems with this experimental technique, not least of which is the fact that different quantifiers have different numbers of appropriate diagrams. In general, it is found that the quantifiers with the fewest appropriate diagrams are easiest, a potential confounding factor which it is impossible to avoid. An alternative technique has been used recently by Sternberg and Turner (1981) and Fisher (1981), in which subjects are required to judge the truth value of various quantified statements with respect to a given quantified statement. Sternberg and Turner reported that subjects' responses were 90% correct, but unfortunately they do not provide a breakdown of the errors made, and it is possible that some consistent misinterpretations (e.g., conversions) occurred. Fisher's first experiment does provide a breakdown of errors, but is complicated by the fact that only two responses were permitted, "possibly t r u e " and "necessarily false." This does not permit a test of conversion of all statements, since this would be demonstrated by subjects calling the converted form necessarily true. The present Experiment 1 investigated further the interpretation of quantifiers using a similar technique to these two studies. Subjects were presented with a quantified statement and asked to judge the truth or falsity of a series of other quantified statements with respect to it. In this way it is possible to investigate the extent to which subjects draw the inferences permitted by the square of opposition. It also enables a direct examination of conversion errors, not only with all statements but with other quantifiers as well. EXPERIMENT 1 Method Subjects. The subjects were 96 introductory psychology students at the University of Florida who participated in partial fulf'dlment of a course requirement. Material. Two kinds of questionnaires were used in this experiment. For the first

of these, the True-False (TF) questionnaire, these were the instructions: In this booklet you will fmd a statement written at the top of each page. This statement describes the relationship between two classes of things. Each of these classes is represented by a different letter of the alphabet. You should assume, for the purposes of this experiment, that the statement at the top of each page is true. Written beneath the statement, you will find eight sentences. Your task is to decide whether each of these sentences logically follows from the statement at the top of the page; if it does, circle the letter T (for true) to the right of the sentence. However, if you think the sentence is not true, then circle the letter F (for false). Once you have finished a page, please go on to the next page and do not return to that page.

There followed four pages at the top of which a p p e a r e d an all, no, s o m e , or somenot statement, the order of these being randomly varied in each booklet. Beneath the quantified statement appeared eight further statements corresponding to the all, no, some, and somenot versions of the statement at the top, along with their converses. For example, one of the statements at the top was the A statement All o f the Ms are Zs. Beneath this were the statements All o f the Ms are Zs, All o f the Zs are Ms, None of the Ms are Zs, None o f the Zs are Ms, and so on. To the right of each of these statements were a large T and E As a precaution, half of the questionnaires had the letters reversed in the statement at the top. The second questionnaire, the TrueFalse-Maybe (TFM) questionnaire was different only in that it permitted a third response. The instructions were identical to those for the TF questionnaire, with the addition of this sentence at the end of the second paragraph: " I f the sentence could be either true or false in the light of the statement at the top of the page, circle the letter M (for maybe)." This response was introduced because it was felt that subjects might feel themselves unduly restricted by the binary true-false choice, especially since, as can be seen from Table 1, some responses are in fact indeterminate. Procedure. Subjects were run in 2 groups

539

D R A W I N G I N F E R E N C E S FROM Q U A N T I F I E R S

of 48, each group receiving one version of the questionnaire. Subjects were allowed as much time as they required to complete the questionnaire. Results and Discussion

The results from the TF questionnaire are presented in Table 2, those from the TFM questionnaire in Table 3. Despite the differences b e t w e e n the questionnaires the results were similar and were combined for most of the analyses. In the initial analysis, performance was compared to the dictates of the square of opposition (see Fig. 1). With c o n t r a d i c t o r i e s ( a l l - s o m e n o t and no--some) the truth of one quantifier implies the falsity of the other. Subjects had little difficulty with this inference, overall accuracy being at 96%. Performance was even higher-100%--on c o n t r a r y inferences, where the truth of all or no implies the falsity of the other. In s u b c o n t r a r i e s , the truth of s o m e or s o m e n o t leaves the truth status of the other undetermined. However, accuracy was only 22% on this inference, on the assumption that subjects given the TF questionnaire should respond false to such items. For 75% of the responses, subjects believed that the truth of s o m e or s o m e n o t implied the truth of the other. This is strong evidence for the claim that s o m e implies the e x i s t e n c e of negative i n s t a n c e s , while

s o m e n o t implies the existence of positive

instances. Subjects were surprisingly good at drawing s u b a l t e r n inferences. There was 70% accuracy in inferring s o m e from all and 58% accuracy in inferring s o m e n o t from no. This high level of accuracy is surprising with respect to previous research in the area, where it has been widely assumed that subjects would be unwilling to draw such inferences. It is also surprising with respect to the findings just reported on subcontrary inferences. The paradox is that the s a m e subjects who believe all implies s o m e also believe that s o m e implies the existence of negative instances! It would appear that s o m e is interpreted in its logical sense of at l e a st s o m e in one situation, but in its everyday sense of s o m e b u t n o t all in another. Any explanations of this will obviously be tenuous, but it may be that the invited inference from s o m e to s o m e n o t is a particularly powerful one in everyday language; in contrast, the invited inference from all to n o t s o m e may be much less compelling, and subjects may find it easier to adopt the strict, logical interpretation. Performance on converted statements is of particular interest since this has been the focus of much recent work. Of the subjects, 34% believed that all statements implied the truth of their converses. It is possible to

TABLE 2 PROPORTION OF TRUE RESPONSES GIVEN ON TF QUESTIONNAIRE Quantifier tested Normal A E I O Converted A E I O

Quantifier given A

E

I

O

1.00T(.98) .00F(.00) .73T(.56) .02F(.03)

.00F(.00) .98T(.94) .02F(.08) .69x(.59)

.00F(. 17) .00F(.02) 1.00T(.98) .94F(.97)

.00F(.02) .02F(. 13) .83F(.92) .98"r(.98)

.33F(.37) .08F(. 10) .65T(.79) .4ff(.57)

.02F(.05) .54x(.59) .21F(.33) .54T(.71)

.10F(.14) •10F(.08) .69W(.90) .63F(.09)

.13F(.11) .15F(. 17) .65F(.88) .65F(.81)

Note. The superscripts indicate the logically correct response. Figures are from Experiment 1; figures in parentheses are from Experiment 2. A = all; E = No; I = some; O = somenot.

540

NEWSTEAD AND GRIGGS TABLE 3 RESPONSES GIVEN ON TFM QUESTIONNAIRE

Quantifier Tested Normal A T M F E T M F I T M F O T M F Convened A T M F E T M F I T M F O T M F

A

E

I

O

*1.00 .00 .00

.00 .00 *1.00

.00 *.40 .60

.00 .08 *.92

.00 .00 *1.00

*.96 .02 .02

.02 .02 *.96

.06 *.17 .77

*.67 .10 .23

.04 .06 *.90

*1.00 .00 .00

.65 *.31 .04

.02 .06 *.92

*.48 .19 .33

.58 *.35 .06

*.98 .02 .00

.35 *.52 .13

.02 .46 *.52

.06 *.60 .33

.06 *.54 .40

.06 .40 *.54

*.46 .48 .06

.08 .33 *.58

.04 *.44 .52

*.40 .46 .15

.48 .19 *.42

*.50 .44 .06

.45 *.49 .06

.13 *.52 .35

*.23 .58 .19

.31 *.63 .06

.50 *.44 .06

Note. Asterisks indicate the logically correct response. A = all; E = no; I = some; O = some not.

check for consistency in these subjects by examining performance on converted s o m e n o t statements. A subject who converts A l l A s are B s should believe S o m e B s are n o t A s to be false. Of the converters, 73% made this r e s p o n s e , as o p p o s e d to 42% o f the n o n c o n v e r t e r s ( t h o s e w h o c h e c k e d converted all statements as false). This difference is significant, X 2 = 5.55, p < .05. Thus there is evidence for the existence of a group o f subjects who genuinely believe that all s t a t e m e n t s i m p l y the t r u t h o f t h e i r con-

verses, and r e s p o n d c o n s i s t e n t l y on that basis. However, this group is small smaller than would p e r h a p s have b e e n p r e d i c t e d from Revlin and L e i r e r ' s (1980) model. According to their model, subjects consider the converted form o f the statement before the n o n c o n v e r t e d one; if this w a s true of the subjects in the present experiment, then all of t h e m should h a v e b e e n c o n s i s t e n t converters ! Conversion is valid with no statements, but many subjects seemed not to realize this, since a c c u r a c y was only 50%. The subjects who failed to convert no statements seemed to genuinely believe that S o m e B s are A s is consistent with the statement N o A s are B s . Of the 48 nonconverters, 30 thought that S o m e B s are A s was either true or maybe true in the light of the statement N o A s are B s . By contrast, only 8 of the 48 converters made such a response, a difference which is statistically significant, X 2 = 21.17, p < .001. H e n c e this is evidence that there exists a group of subjects who consistently fail to convert no statements. This is a little surprising since, for instance in r e s e a r c h using Venn diagrams, it has been widely assumed that no is almost always correctly interpreted. C o n v e r s i o n is valid with s o m e statements, but only 58% of subjects did make such c o n v e r s i o n s . Only 40% o f subjects were accurate on c o n v e r t e d s o m e n o t statem e n t s , w h e r e c o n v e r s i o n is not logically valid. Again, errors are more prevalent than previous research would have led us to believe. T h e o c c u r r e n c e o f such logical errors seems at first sight inconsistent with much p r e v i o u s r e s e a r c h , b u t this d i s c r e p a n c y stems largely from the way in which the results are analyzed. For example, Sternberg and Turner (1981) report 90% accuracy on a task similar to the present one. If we consider performance only on nonconv e t t e d r e s p o n s e s t a t e m e n t s , which is directly comparable to one of Sternberg and Turner's analyses, there is 80% accuracy in the present experiment. Since Sternberg and

DRAWING INFERENCES

FROM QUANTIFIERS

541

Turner instructed their subjects in the log- tions as in Experiment 1 (the General inical meaning of quantifiers, for instance that structions); the other half had L o g i c insome means some and possibly all, and since structions. These latter were the same as errors in which s o m e n o t was inferred from the General instructions with the following s o m e and vice versa account for a major paragraph added: proportion of the present errors, the results N o t e that the task is to decide w h e t h e r s e n t e n c e s seem quite consistent. follow logically from other s e n t e n c e s ; thus the In Experiment 2 we try to isolate the t e r m s m u s t be interpreted in their logical s e n s e s . source of some of these logical errors. The A c c o r d i n g to logic, " s o m e " m e a n s " a t l e a s t o n e . " " S o m e A s are B s " m e a n s that at least one instructions used in Experiment 1 did not A is a B, a n d does not rule out the possibility stress that this was a logical task, and subthat e v e r y single A is a B. " S o m e A s are not jects may have responded according to B s " m e a n s that at least one A is not a B, a n d e v e r y d a y linguistic usage rather than does not rule out the possibility that not a single according to logic. Thus Experiment 2 invesA is a B. Please b e a r in m i n d t h e s e logical interpretations w h e n filling in y o u r a n s w e r s . tigated the effects of giving subjects instructions about the logical interpretation After completing the questionnaire, subof quantifiers. In addition, Experiment 1 jects were given a set inclusion passage. used abstract material (letters of the al- Subjects were instructed to assume that the phabet); in Experiment 2 the effects of using premises in the passage were true, and to more realistic material were investigated. indicate whether a series of 20 statements Finally, subjects were given a set inclusion which appeared below were true or false task after completing the questionnaire, to with respect to the passage. The subjects check if subjects showing conversion errors who had received abstract material in the on the questionnaire were those who made questionnaire were given an abstract set insimilar errors on this task. clusion passage: All the N s are Zs; All the Z s are Ls; All the L s are Qs; All the Qs are EXPERIMENT 2 Ks. The 20 test items consisted of all the Method possible permutations of the letters. There S u b j e c t s . Sixty-four first year psychol- were four true adjacents (the original premogy students at Plymouth Polytechnic re- ises) such as All the N s are Zs; four false ceived £0.75 for their participation. None (converted) adjacents, such as All the Z s had taken a course in logic. are N s ; six valid (true) remote inferences, Material. Questionnaires based on the TF three at an inferential distance of one, e.g., condition of Experiment 1 were con- All the Z s are Qs, two at an inferential disstructed. Half of the questionnaires used tance of two, e.g., All the Z s are Ks, and letters of the alphabet, as in Experiment 1, one at an inferential distance of three, All while the other half used concrete items the N s are Ks; and six invalid (false) remote similar to those used by Johnson-Laird and inferences which were the converted forms Steedman (1978), for example, All the bee- of the valid inferences. Next to each test keepers are artists, S o m e o f the musicians statement appeared a large T and E Subare not inventors. Subjects were told that jects who had received the concrete version these statements referred to a group of of the questionnaire were given a set inclupeople gathered in a room. Although this sion passage about a mythical tribe called material is concrete it is fairly neutral and the Fundalas, followed by 20 test items. would not be expected to lead to reProcedure. Subjects were run in a single sponding on the basis of prior experience, group and were allowed as much time as for example, on the basis of the believ- they required to complete the tasks. Of the ability of the relationship described. Half of subjects, 16 were given the Abstract mathe questionnaires had the same instruc- terial with General instructions, 16 the Ab-

542

NEWSTEAD AND GRIGGS

stract material with Logic instructions, 16 the Concrete material with General instructions, and 16 the Concrete material with Logic instructions. Allocation of subjects to conditions was random. Results and Discussion

The results were surprisingly similar across the four versions of the questionnaire, suggesting that the pattern of responding found in Experiment 1 is fairly stable over different instructional sets and types of material. The results, collapsed over the different versions of the questionnaire, are presented in parentheses in Table 2. There were just three significant differences between the questionnaires, all of which concerned the type of instructions given. Subjects given the Logic instructions were more likely to deduce s o m e from all, X2 = 8.59, p < .05, suggesting that subjects were following the instructions given. Subjects given Logic instructions were also more likely to infer all when given s o m e and n o given s o m e n o t , X 2 = 9.02 and X 2 = 9.44, respectively, p < .05. Such inferences are not valid, but were clearly cued in by the instructions saying that s o m e did not preclude all and s o m e n o t did not preclude n o . It is paradoxical that in this case the Logic instructions actually led to l e s s logical performance; since these instructions were similar to those used in previous studies (e.g., Anderson, 1981; Sternberg & Turner 1981) it is questionable what effect the instructions may have had in these other experiments. Overall, the pattern of inferences was very similar to that found in Experiment 1. What is more, there were similar individual differences in conversion errors. Of the subjects, 37% believed that all implied its converse, and 83% of these converters believed that converted s o m e n o t statements were false given an all statement; by contrast only 21% of the nonconverters believed this to be false, a difference which is highly significant, X 2 = 22.67, p < .001. Of the subjects, 42% failed to convert n o statements

and 58% of these believed that converted s o m e was true given a n o statement; this response was rarer (16%) with subjects who correctly converted n o statements, and again this difference was significant, X 2 = 11.80, p < .001. In Experiment 2 there were rather more subjects (90%) who correctly converted s o m e statements, but there were also rather more (81%) who incorrectly converted s o m e n o t statements. The results from the set inclusion task are presented in Table 4. There was no difference between the two kinds of material used, or between subjects given General or Logical instructions prior to the questionnaire, hence the results are collapsed over these conditions in this table. The data were analyzed using an analysis of variance in which the factors were Truth (whether an item was logically true or not) and Distance (number of inferential steps). There was a highly significant interaction between Truth and Distance, F (3,189) = 20.48, p < .001. This interaction is typical of previous research into this task (e.g., Griggs, 1976; Griggs & Warner, 1982); performance on true items tends to decline over inferential distance, while performance on false items tends to improve over inferential distance. On the basis of performance on the questionnaire task, subjects can be divided into those who tended to convert all statements

TABLE 4 PROPORTION CORRECT ON SET INCLUSION TASK IN EXPERIMENT 2 Inferential steps 0

1

2

3

All subjects True False

.97 .55

.75 .64

.73 .73

.69 .72

Converters True False

.98 .39

.70 .58

.70 .72

.61 .74

Nonconverters True False

.97 .63

.77 .67

.74 .73

.73 .71

DRAWING

INFERENCES

FROM

QUANTIFIERS

543

and those who did not. The performance of view involves those subjects who consisthese two subgroups on the set inclusion tently converted all items, that is, believed task is given in Table 4. Although both that false adjacents were true. These subgroups produced significant Truth by Dis- jects were significantly more likely than tance interactions, the nature of the inter- nonconverters to say that converted action is different for the two groups. In s o m e n o t items were false, U = 40, nl = nz particular, converters show poor logical = 13, p < .01, Mann-Whitney test. Thus performance on false adjacent items, tending we have evidence here of just the kind of to call these true rather than false; hence consistency that was obtained in the questhese subjects commit errors of conversion tionnaire used in Experiments 1 and 2. It is on the set inclusion task as well as on the also of interest that 13 out of the 40 subjects questionnaire, significantly more so than could be classified as converters, very simnonconverters, U = 590.5, nl = 23, n2 ilar to the proportion found in Experiments 39, p < .05, Mann-Whitney test. This is 1 and 2. strong evidence that errors on set inclusion Since conversion errors were common tasks are due in part to subjects misinter- with all the quantifiers used in the present preting the premises, and incorrectly be- studies, it might be thought that it is the lieving them to be convertible. It is inter- same subjects making such errors throughesting, however, that several subjects out. This was checked by examining perclassed as nonconverters on the question- formance on the TF questionnaires in Exnaire task committed conversion errors on periments 1 and 2, giving a total of 110 the set inclusion task. It is probable that the subjects. Table 5 lists the probabilities of very complexity of the set inclusion task converting other test items for subjects who induces conversion errors. converted or failed to convert a particular The results from Experiment 2 thus pro- test item. It should be remembered when vide further evidence for the existence of a examining this table that conversion is valid group of subjects who consistently convert with n o and s o m e , but invalid with all and all statements. It might be objected that the s o m e n o t . It is clear from this table that it is consistency across the questionnaire and set not the same subjects making conversion inclusion task is artifically high because of e r r o r s throughout this task; rather there the similarity and proximity of the two tasks. seems to be a tendency for the same subHowever, we have evidence of similar con- jects to convert all items irrespective of logsistency in a rather different experiment ical validity. This tendency to convert, since carried out by Tom Gruenenfelder at In- it seems independent of logic, is probably diana University, which will be reported a response bias, and clearly is subject to only briefly here. Subjects were given concrete set inclusion passages (one of which TABLE 5 was the Fundalas passage) in which there PROBABILITY OF CONVERTING TEST STATEMENTS BY was some additional, irrelevant material. CONVERTERS AND NONCONVERTERS OF The subjects, 40 Indiana University underEACH QUANTIFIER graduates, were required to remember each All No Some Somenot passage after studying it for as long as they All c o n v e r t e r s -.69 .92 .72 pleased, and were then given an intervening All n o n c o n v e r t e r s -,49 .75 .76 task involving mental arithmetic. The test No c o n v e r t e r s .44 -.84 .79 items were presented individually on a N o n o n c o n v e r t e r s .25 -.77 .67 computer screen, and involved not only the Some c o n v e r t e r s .40 .59 -.93 .14 .48 -.38 20 possible items using all, but also another Some n o n c o n v e r t e r s .34 .61 .90 -60 items using n o , s o m e , and s o m e n o t . The Somenot c o n v e r t e r s .45 .55 -result of interest from the present point of Somenot n o n c o n v e r t e r s ,39 =

544

NEWSTEAD AND GRIGGS

individual differences. This raises the important question of whether other response biases, such as the well-known belief bias effect in syllogistic reasoning (e.g., Janis & Frick, 1943) are subject to similar individual differences. GENERAL DISCUSSION

Perhaps the most important theme to emerge from these experiments concerns individual differences in the interpretation of quantified statements. Thus in both experiments there was evidence for a proportion of s u b j e c t s - - a b o u t one-third--who consistently converted all statements. Even more s u b j e c t s - - o v e r 6 0 % - - i n c o r r e c t l y converted s o m e n o t statements. Illicit conversion of these quantifiers has been suggested by Chapman and Chapman (1959) and others to be a source of error in syllogistic reasoning. The present experiments provide strong evidence that such errors do occur--perhaps the first time such direct evidence has been produced. However, theories such as Chapman and Chapman's suggest that conversion errors are general and widespread; the present experiments indicate that only a certain proportion of subjects make such errors. This is particularly damaging to Revlin and Leirer's (1980) claim that subjects reason with the converted form of the premises first; at best this theory can explain the performance of that proportion of subjects who are converters, and Dickstein (1981) has provided evidence that it is inadequate on other grounds. Nearly half the subjects in Experiments 1 and 2 failed to realize that no statements logically imply their converses; and about one-fourth failed to convert s o m e statements. Failure to convert no and s o m e has not previously been considered as a source of error in syllogistic reasoning. However, it could well explain some errors that are made. For example, Johnson-Laird and Steedman (1978) found that 25% of subjects

thought that no valid conclusion followed from the premises N o B s are A s ; A l l B s are Cs. I f subjects failed to realize the convertibility of no statements, .then arguably no valid conclusion w o u l d follow. The fact that conversion errors are so clearly indicated as a source of error in syllogistic reasoning poses problems for other theories, since many of them assume that premises are interpreted logically. This is so in the case of Sternberg and Turner's (1981) transitive chain theory, and Johnson-Laird and Steedman's (1978) analogical theory. While the present findings do not prove such theories wrong, since these are primarily theories of processing rather than comprehension, they do render the theories incomplete. Indeed, interesting questions arise as to the possibility of individual differences in the adequacy of these models. For example, Johnson-Laird and Steedman (1978) found a strong tendency for subjects to draw A-C conclusions given a first premise expressing the relationship between A and B and a second premise expressing the relationship between B and C. It is probable that a subject who converts the In'st premise may demonstrate this conclusion bias to a smaller extent than a subject who does not convert it. It is interesting that the frequency of occurrence of conversion errors on all statements should be so consistent across the experimental work reported here, since very different populations were used. Approximately one-third of subjects made such errors in Experiments 1 and 2, and in the set inclusion experiment reported briefly in the discussion of Experiment 2. Previous work on set inclusions is consistent with such individual differences. For example, Mynatt and Smith (1979) found that 9 out of 26 subjects consistently used strategies involving symmetry, that is, involving conversion errors. Other research has found rather more errors of symmetry, frequently over 50% (e.g., Griggs, 1976). It may well be that, as suggested in Experiment 2, set inclusion

DRAWING INFERENCES FROM QUANTIFIERS

tasks induce conversion errors in subjects who would not make them on simpler tasks. The present results suggest a simple explanation for the confusion errors that occur between quantifiers, to which so much importance has been attached by Holyoak and Glass (1978) and Anderson (1981). It seems that subjects tend to confuse together quantifiers that can be inferred from each other. Subjects readily inferred s o m e from all, which could explain why these quantifiers were so frequently confused in both the Holyoak and Glass and the Anderson study. Similarly, subjects readily inferred s o m e n o t from n o , and believed that s o m e n o t and s o m e implied each other. Thus confusions b e t w e e n these quantifiers would be expected and were, in general, found. However, according to the present hypothesis, such confusions should not be symmetrical. For example, although s o m e can be inferred from all, the reverse does not hold true; Tables 2 and 3 confirm that subjects seldom draw the reverse inference (though they were more likely to when given the Logic instructions in Experiment 2). The predicted asymmetries have been found in previous research. Holyoak and Glass (1978) found that subjects gave fairly high confidence ratings (2.90) for the quantifier s o m e when all was the quantifier presented; this is considerably higher than the rating of 2.38 given to all when s o m e was presented. Similarly, Anderson (1981) in his first experiment found a false alarm rate of 0.399 for s o m e when all had been studied, but only 0.267 for all when s o m e had been studied. Hence the present results indicate the need for caution in interpreting the results from studies of memory for quantifiers. Many of the findings seem to result from the inferences that subjects draw rather than from the underlying representation used. Further, there seem to be substantial and quite consistent individual differences in the way in which quantifiers are interpreted. Individual differences in representation have recently come to light in other areas, for

545

instance in sentence-picture verification tasks (MacLeod, Hunt & Mathews, 1978). Clearly such findings render general modelbuilding in this area a rather dangerous exercise.

Note added in proof. Since the submission of this paper, I. Begg and G. Harris (Journal o f Verbal Learning and Verbal Behavior, 1982, 21, 595-620) have published an experiment similar to our Experiment 1. The results of the two experiments are quite consistent.

REFERENCES ANDERSON, J. R. Memory for logical quantifiers.

Journal of Verbal Learning and Verbal Behavior, 1981, 20, 306-321. BRANSFORD, J. D., BARCLAY, J. R., & FRANKS, J. J. Sentence memory: A constructive vs. interpretive approach. Cognitive Psychology, 1972, 3, 193-209. CHAPMAN, L. J., ~¢ CHAPMAN,J. P. Atmosphere effect re-examined. Journal o f Experimental Psychology, 1959, 58, 220-226. COHEN, M. R., ~¢ NAGEL, E. An introduction to logic and scientific method. New York: Harcourt, 1934. DICKSTEIN, L. S'. The meaning of conversion in syllogistic reasoning. Bulletin of the Psychonomic Society, 1981, 18, 135-138. FISHER, D. L. A three-factor model of syllogistic reasoning: The study of isolable stages. Memory and Cognition, 1981, 9, 496-514. GRICE, H. P. Logic and conversation. In P. Cole & J. L. Morgan (Eds.), Syntax and semantics, Volume 3: Speech acts. New York: Seminar Press, 1975. GRIGGS, R. A. Logical processing of set inclusion relations in meaningful text. Memory & Cognition, 1976, 4, 730-740. GRIGGS, R. A., & WARNER, S. A. Processing artificial set inclusion relations: Educing the appropriate schema. Journal of Experimental Psychology: Learning, Memory and Cognition, 1982, 7, 51-65. HOLYOAK, K. J., & GLASS, A. L. Recognition confusions among quantifiers. Journal o f Verbal Learning and Verbal Behavior, 1978, 17, 249-264. JANIS, I., & FRICK, E The relationship between attitudes towards conclusions and errors in judging logical validity of syllogisms. Journal of Experimental Psychology, 1943, 33, 73-77. JOHNSON-LAIRD,P. N. The interpretation of quantified sentences. In Flores D°Arcais & W. J. M. Levelt (Eds.), Advances in psycholinguistics. New York: American Elsevier, 1970.

546

NEWSTEAD AND GRIGGS

JOHNSON-LAIRD, P. N., & STEEDMAN,M. The psychology of syllogisms. Cognitive Psychology, 1978, 10, 64-99. MACLEOD, C. M., HUNT, E. B., & MATHEWS,N. N. Individual differences in the verification of sentence-picture relationships. Journal of Verbal Learning and Verbal Behavior, 1978, 17, 493-507. MYNATT, B. T., ~¢ SMITH, K. H. Processing of text containing artificial inclusion relations. Memory and Cognition, 1979, 7, 390-400. NEIMARK,E. D., & CHAPMAN,R. H. Development of the comprehension of logical quantifiers. In R. J. Falmagne (Ed.), Reasoning: Representation and process. Hillsdale, N.J.: Eflbaum, 1975. REVLIN, R., & LEIRER, V. O. Understanding quantified categorical expressions. Memory and Cognition, 1980, 8, 447-458. REVLIS, R. Formal syllogistic reasoning: Logical decisions from a complex data base. In R. J. Falmagne (Ed.), Reasoning: Representation and process. Hillsdale, N.J.: Eflbaum, 1975. (a)

REVLIS, R. Two models of syllogistic reasoning: Feature selection and conversion. Journal of Verbal Learning and Verbal Behavior, 1975, 14, 180195. (b) SMEDSLUND, J. Circular relation between understanding and logic. Scandinavian Journal of Psychology, 1970, 11, 217-219. STERNBERG, R. J., • TURNER, M. E. Components of syllogistic reasoning. Acta Psychologica, 1981, 47, 245-265.

REFERENCE NOTES 1. CAPLANJ. S. Play-it-safe: A new motto and model for behavior in syllogistic reasoning tasks. Paper presented at the Eastern Psychological Association, New York, April, 1981.

(Received December 21, 1982)