Interpretation of abstract conditional sentences in deductive reasoning

JOURNAL OF VERBAL LEARNING AND VERBAL BEHAVIOR, 12,

530-542 (1973)

Interpretation of Abstract Conditional Sentences in Deductive Reasoning 1 JOHN E. TAPLIN 2 AND HERMAN STAUDENMAYER

University of Colorado Sentences of the form ifp then q have been defined m logic to have a fixed and singular meaning expressed by a condmonal truth function. For natve subjects, however, this is only one of its possible meanings. Sometimes an abstract sentence of this form is interpreted in this way, but on other occasions xt has a bicondltional truth-functional meaning, while on still other occasions it has a meaning not found in the propositional calculus at all because part of the truth table is irrelevant. In the present study it is suggested that these various meanings are due in part to different inferences by the subject about how the sentence is supposed to be used in the experimental task, brought about by relatively small differences in the procedure employed It is argued that such inferences are important for any theory of the interpretation of sentences and provide further evidence to indicate that the meaning of a sentence is not solely a function of the intrinsic properties of the sentence itself but is also dependent upon context as well. Until recently, studies o f the psychology o f thinking have maintained that h u m a n deductive reasoning cannot be represented adequately by a logical model. One principal source o f evidence cited in support of such an assertion comes f r o m experiments which have shown that reasoning is influenced in general by the degree o f abstractness or concreteness o f the material (Wilkins, 1928) and, in particular, by the attitude held by the subject toward the conclusion of the process o f reasoning (e.g., M o r g a n & Morton, 1944; K a u f m a n n & Goldstein, 1967). Thus, Morgan and M o r t o n (1944) suggested that "the only circumstance under which we can be relatively sure that the inferences o f a person will be logical is when they lead to a conclusion which he has already accepted" (p. 39). Henle (1962) has argued convincingly that the presence of errors in reasoning due to su.ch

factors is not sufficient to refute logic as a model. To support her argument she attempts to conceive of a process which obeys the rules o f logic which can account for the errors obtained in these studies. First, she suggests the errors in dealing with the presented material m a y be due to a failure to accept the logacal nature of the task, that is, to distinguish between a conclusion which is logically true and one which is factually correct or one with which the subject agrees. Second, errors m a y be due to the restatement of a premise or conclusion so that the intended meaning is changed. In such cases, Henle contended that the validity o f the reasoning should be judged in relation to the arguments actually employed, not the ones intended. Third, errors may be due to the omission of a premise, and fourth, to the intrusion o f additional premises. Henle suggested that when the 1 This research was supported by Research Grants intended premises are replaced by the actual MH 14314 from the National Institute of Mental propositions utilized, the judgments o f the Health and GB-340-77X from the National Science subject m a y be found to be logically correct. Foundation, awarded to Lyle E Bourne, Jr, to whom This justification of logic as a model o f the authors express their sincere appreciation reasoning was aimed at accounting for the z Requests for reprints should be sent to John E. effects of beliefs and attitudes in producing Taphn who is now at the Department of Psychology, errors. It may also help to explain why errors Claremont Graduate School, Claremont, Ca 91711. Copyright © 1973 by Academic Press, Inc. 530 All rights of reproduction m any form reserved.

Printed m Great Britain

ABSTRACT CONDITIONAL SENTENCE REASONING

should occur when the reasoning material is abstract (such that the subject is unlikely to hold any preconceived notions about the truth or falsity of the conclusion). Ceraso and Provitera (1971) have demonstrated that many errors in categorical syllogistic reasoning with abstract material are due to misinterpretation of the premises to refer to simpler class relations, although their results showed that some errors were also a consequence of an incomplete analysis of the logical structure of the syllogism. The present paper is concerned with a type of deductive reasoning known as sentence reasoning, in particular, reasoning with sentences having the form, i f p then q, wherep and q are any two propositions linked together by the connective, i f . . . then . . . . Using the terminology sometimes employed in logic textbooks, these sentences will be referred to as conditional sentences, and the reasoning which proceeds from them, conditional reasoning. The validity of conditional reasoning is dependent upon the meaning which is assumed for the conditional sentence, which in logic is dependent upon the meaning of the if... then . . . connective. This expression has been traditionally taken by logicians to be eqmvalent to the logical operation symbolized by "-~" (or sometimes "D"). The meaning of this symbol is given in the propositional calculus by a truth function. Consider the sentence p - + q. This sentence has a truthfunctional meaning which simply says when the sentence is true and when it is false as a function of the truth and falsity of the propositions p and q; the truth function is pq true, p4 false,/sq true, and/sq true. This will be called a conditional truth function. Since the sentence, i f p then q, is supposed to be equivalent to p - + q, this has the same truth function. In fairness, it should be pointed out that many logicians (e.g., Reichenbach, 1947; Strawson, 1952) have realized that i f p then q and p - + q may not always be synonymous with one another, that the symbol "-+" may

531

not represent the only meaning of i f . . . then . . . . Nevertheless, it has been usual to read p -+ q and i f p then q as meaning the same thing and to ignore other possible meanings. This has been done for the sake of expediency to remove ambiguity, but at the cost of admitring that the translation may be only a special case (Copi, 1961). If the meaning of the sentence, t f p then q, for a naive human subject is not equivalent top --~ q and thus not expressible by the above truth function, then this may have important consequences for any judgments of the logicality of his reasoning with such a sentence. If the subject assumes a different truth-functional meaning for the sentence, even though his reasoning may be perfectly consistent with his modified interpretation, the end-product of his reasoning is likely not to conform in all instances with the predictions of the propositional calculus. Thus, before passing judgment on whether an individual's thinking is logical or not, it is imperative that we know how he interpreted those linguistic expressions which are supposed to be operative, in the present case, if...

then . . . .

The first suggestion in the psychological literature that the truth-functional meaning assumed in propositional logic for the conditional connective may not be the same as that for naive adult subjects came from Wason (1966). He proposed that the sentence, i f p then q, has instead a truth function where pq is true, pq false, and Pq and p4 irrelevant. Wason (1968) tested his suggestion in an experiment in which subjects were required to choose instances of p,/5, q, q which would enable them to determine whether a given conditional sentence, i f p then q, was true or false, if in this way they could learn the truth value of the other proposition. The results of his experiment supported the predicted truth values for the pq, p(t, and/sq contingencies, but only 26.5~ of subjects appeared to regard the /sq contingency as irrelevant. Additional support for Wason's prediction comes from Taplin (1971b) who found sub-

532

TAPLIN AND STAUDENMAYER

jects unable to discriminate a conditional sentence (i.e., i f p t h e n q) from a conjunctive sentence (i.e., p a n d q) and a bicondltional sentence (i.e., p i f a n d o n l y i f q), when these sentences were given as the necessary and sufficient specifications for three different classes within a stimulus population incorporating ps and qs. For each sentence the class specified was interpreted to be formed by the conjunctionpq. In a different kind of task Taplin (1971a) confirmed Wason's suggestion that the psychological meaning of the conditional connective is not the meaning which has been typically used to represent it in logic texts. In this experiment subjects were presented with several conditional arguments and required to indicate whether the conclusion of each argument necessarily followed or not. Analysis of the evaluations made revealed that only 4 4 . 7 ~ of subjects reasoned in a truth-functional manner. Several different truth functions were inferred for the conditional, the most common (37.5 ~ ) being one wherepq was true, pc] false, pq false, and b q true, which is the bicondltlonal truth function in the propositional calculus; only 3.6 ~ subjects responded in a manner consistent with the predictions from propositional calculus for the conditional. The remaining subjects were either inconsistent in their evaluations of one or more arguments, sometimes responding affirmatively, sometimes negatively, or they were consistent but it was not possible to infer any truth function to describe their evaluations. Comparing the findings obtained by Wason and by Taphn, we may wonder about the reason for the different truth functions inferred for the conditional. One possible reason for the discrepancy may be found by noting the similarity between i f . . . then... and connectives like s i n c e . . , t h e n . . . and because . . . then .... In each case the connective can express some temporal or causal relation between the antecedent and the consequent. However, for the last-men-

tioned connectives the truth value of the antecedent is evidently not in doubt. This means that not all truth-table contingencies are applicable, and it seems plausible to regard pq and p~ as havlng an irrelevant value. The truth function for these connectives, therefore, reads p q true, p~] false,/~q irrelevant, and p q irrelevant, which is also the truth function originally proposed by Wason for the conditional. Now, if a subject were to treat the conditional sentence, i f p t h e n q, in such a way that the antecedent, p, was assumed to be true, then the prediction made by Wason seems inevitable. In a task like that employed by Wason (1968), this assumption is possible. In the syllogistic reasoning task by Taplin (1971a), however, this assumption is unlikely. Indeed, on some arguments, subjects are actually required to assume that the antecedent is false. Thus, the different results obtained in these tasks may be attributable to the different set induced toward the truth value of the antecedent. A second obvious difference between these experiments lies in the methods employed for inferring the truth function of the conditional sentence. In Wason's task the nature of the truth function was derwed from the evaluations of the conditional sentence made by subjects when the values of both the antecedent and the consequent were known. If, when the other side of a card was revealed, the subject said, "that makes it true," then the particular truth-table contingency defined by the terms on either side of the card were classified as true; if he said, "that makes it false," then the contingency was classified as false; if he said, "that has nothing to do with it," or words to that effect, then it was classified as irrelevant. The method adopted in Taplin (1971a) for inferring truth values does not permit any assertion that the value of a particular contingency is irrelevant. If, in fact, subjects were to treat some contingencies as irrelevant, then perhaps this should be evidenced by inconsistency in the responses made to those arguments which depend on these levels of the

ABSTRACT CONDITIONAL SENTENCE REASONING

truth function. This implication is of considerable interest when it is noted the majority of subjects (51.7 %) were found to be statistically inconsistent in their evaluation of one or more of the five conditional arguments presented. This inconsistency, however, has an alternative explanation, related to the use of meaningful sentences for presenting the arguments in this experiment. Several factors may be involved in reasoning with this type of material. First, the subject may take into account the plausibility of the conclusion in making his evaluation, something which has been well established for deductive reasoning with categorical, if not conditional syllogisms. An independent judge attempted to eliminate any conclusions which were obviously plausible or implausible, but it is likely that there still remained some variability between items on this dimension. If so, this would be partly respons%le for the large number of inconsistent subjects found. Second, meaningful conditional sentences may involve different forms of connections between the antecedent and the consequent. This possibility has been suggested by Peel (1967) who distinguished between conditional sentences consisting of two "naturally and causally related" propositions, two "unconnected" propositions, and two "arbitrarily connected" propositions. Unfortunately, Peel d~d not provide any objective criteria for identifying sentences of each type, although in this study he reported to examine only sentences involving two "arbitrarily connected" propositions. Evidence, therefore, on the effect of the type of connection between the antecedent and the consequent of the conditional sentence on the interpretation of the sentence is lacking. Nevertheless, variations of this nature m the sentences presented by Taphn (1971a) may have contributed to the inconsistency in the evaluations observed. Third, there were other aspects of the sentences not controlled, like their length and syntactic structure, which have been shown previously to affect sentence comprehension and which may have affected

533

performance on this task. Evidence for an effect of syntax on syllogistic reasoning has recently been presented by Lippman (1972). In view of the fact that these variables were not controlled in Taplin's experiment, at seems important that this study be replicated, initially using sentences in which their effect is minimized or at least held constant. Subsequent investigations may then examine the role of these variables through successive approximations to sentences as complex as those originally employed. That is, in order to understand fully the processing of conditional sentences in a reasoning task, the approach to be followed seeks to reduce the sentence to its simplest grammatical form and then to increase its complexity systematically. The present report ~s concerned with the first stage of this investigation, namely, reasoning with simple, abstract conditional sentences. The experiments to be described are based on the method and analysis employed by Taphn (1971a). The major modification lies in the use of abstract sentences like I f there is a P, then there is a Q. It seems unlikely that any subject would have some preexisting attitude about the truth or falsity of this sentence or its component propositions. The nature of the semantic relation between the propositions seems arbitrary, yet constant, and the sentence has a simple grammatical structure. Thus, the probable role of the linguistic variables referred to above for the sentences presented by Taphn (1971a) is minimal in this investigation. A minor modification was that the number of conditional arguments examined was extended to all eight possible types listed in Table 1. EXPERIMENT !

In this experiment subjects were presented with the eight forms of the conditional argument shown in Table 1 and asked to evaluate the conclusion of each argument as either "true" or "false." The responses of each subject were analyzed individually. First, it

534

TAPLIN AND STAUDENMAYER TABLE 1 EIGHT FORMS OF CONDITIONALARGUMENTS

Name of argument

Conclusion

Affirming the antecedent

Affirmative Negative Affirmative Negative Affirmative Negative Affirmative Negative

Denying the antecedent Affirming the consequent Denying the consequent

First premise If p, If p, If p, If p, If p, If p, If p, If p,

was determined w h e t h e r the subject was statistically consistent on each o f the eight a r g u m e n t forms. A b m o m i a l test was used for this purpose and the criterion point at which the null hypothesis that the likelihood o f a response o f " t r u e " was equal to the likelihood o f a response o f " f a l s e " was rejected was = .05. I f the subject was f o u n d n o t to be consistent at this criterion level on all eight arguments, then he was classified as statistically inconsistent. Otherwise, the analysis pr o ceed ed to the next step, which was to refer a truth f u n c u o n which could a c c o u n t for the subjects evaluations o f the arguments. T h e inferences m a d e a b o u t the truth values for each truth-table contingency for the conditional sentence represented by ifp then q are shown in Tab l e 2. It can be seen that four estimates were m a d e o f the value o f each truth-table contingency. I f all estimates on each o f the four contingencies agreed, then these constituted the t r u t h function for the c o n d i t i o n al sentence for that subject. I f there was disagreement on the estimated t r u t h value o f any one o f the four contingencies, the subject was classified as statistically consistent but logically contradictory.

Method The subjects were 73 introductory psychology students at the University of Colorado, who took part in this experiment to fulfill a course requirement. Subjects were run m groups of about 20. None of these had received any instruction m logic at the college level.

then q then q then q then q then q then q then q then q

Second premise

Conclusmn

Logical vahdity

p p /5 /5 q q c) O

q 0 q 0 p fi p /5

Valid Invalid Invalid Invalid Invalid Invalid Invalid Valid

Subjects were presented with 96 test items Each stem was contained on a shde and projected onto a screen. Each slide portrayed a set of three statements labeled A, B, and C, where A and B constituted the premises and C the conclusion For example, one typical slide might have been (A) If there is a Z, then there is an H. (B) There is a Z. (C) There is an H. The subjects were instructed to assume that statements A and B are true, and on the basis of this knowledge, to indicate (on a response sheet provided) whether statement C was true or false The test items presented differed in two ways First, the sequence of 96 items included 12 presentations of each of the eight arguments shown in Table 1, randomlzed so that each argument occurred once in each block of eight items. Second, items differed in the pair of letters of the alphabet within the conditional sentence. Letter pairs were randomly assigned with two restrictions: (1) that the antecedent and consequent clauses of a sentence did not contain the same letter; and (2) that no letter appeared more than once within a block of eight Items. Presentation of items was paced, each item being exposed foe 20 sec.

Results and Dtscussion F o r each argument, the frequency distribution o f " c o r r e c t " responses, that is, the response predicted when the condxtional connective has the t r u t h function assumed in the p r o p o si t i o n al calculus, is tabulated in Table 3. A c o m p a r i s o n between these distributions and a b i n o m i a l distribution, in which the p r o b a b i h t y o f a " c o r r e c t " response is equal to the probability o f an " i n c o r r e c t " response is equal to 1/2, revealed sigmficant

535

ABSTRACT CONDITIONAL SENTENCE REASONING TABLE 2 INFERENCES DRAWN IN EXPERIMENT I ABOUTTRUTH-TABLEVALUESFOR A CONDITIONAL SENTENCE FROM RESPONSESMADE TO EIGHT CONDITIONAL ARGUMENTS SubJect's response to conclusion Name of argument

Conclusion

True

False

Affirming the antecedent

Affirmative

pq(T) and p0(F)

either pq(T) and p0(T) or pq(F) and p0(T) or pq(F) and p0(F)

Negative

pq(E) and p0(T)

either pq(T) and pO(T) or pq(Y) and p0(F) or pq(F) and p0(F)

Affirmative

riq(T) and ri0(F)

either riq(T) and ri0(T) or riq(F) and b0(T) or riq(F) and ri0(F)

Negative

riq(F) and P0(T)

either riq(Y) and ri0(T) or riq(T) and ri0(F) or riq(F) and ri0(F)

Affirmative

pq(T) and rig(F)

eltherpq(T) and riq(T) or pq(F)and riq(T) or pq(F) and riq(F)

Negative

pq(F) and riq(T)

either pq(T) and riq(T) or pq(T) and riq(F) or pq(F) and riq(F)

Affirmative

p0(T) and ri0(F)

eitherpO(T) and ri0(T)

Denying the antecedent

Affirming the consequent

Denying the consequent

or or Negative

p0(F) and ri0(T)

p0(F) and ri0(T) p0(F) and ri0(F)

either p0(T) and ri0(T) or p0(T) and ri0(F) or p0(F) and ri0(F)

TABLE 3 OBSERVED FREQUENCYOF CORRECT RESPONSESON EACH OF THE EIGHT ARGUMENTSEXAMINED IN EXPERIMENTI (n = 73) Number of correct responses Name of argument

Conclusion

0

1

2

3

4

5

6

7

8

9

10

11

12

Affirming the antecedent

Affirmative Negative

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

2 1

4 5

67 67

Denying the antecedent

Affirmative Negative

2 47

4 5

0 6

0 1

0 0

0 1

1 1

1 0

1 3

0 1

5 1

9 0

50 7

Affirming the consequent

Affirmative Negative

51 0

5 3

3 1

1 0

0 0

0 0

2 1

1 0

1 1

0 3

3 3

1 9

5 52

Denying the consequent

Affirmative Negative

1 3

1 1

1 1

0 1

0 0

1 1

1 0

0 0

2 1

2 4

5 7

15 13

44 41

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TAPLIN AND STAUDENMAYER

differences in every case, p < .001, suggesting 4 . 2 ~ subjects tended toward a conditional that subjects were not behaving in a random truth function, 11.1~ subjects towards a fashion in producing their responses. An bicondltional truth function, 2 . 8 ~ were examination of the number of "correct" logically contradictory in their evaluations responses on each of the 12 presentations of of the various arguments, while the final 2.8 each argument faded to show any significant were not consistent on sufficient arguments difference between the items for any argu- for a truth value to be estimated for all truthment. The only significant effect was for type table contingencies. A summary of these findof argument, F(7,504) = 171.48,p < .001. The ings may be seen in Table 5. arguments on which most errors were proThese results confirm the findings obtained duced were denying the antecedent with a by Taplin (1971a) using sentences composed negative conclusion, and affirming the con- of meaningful material. Thus, in both experisequent with an affirmatlve conclusion; ments it is evident that subjects do not affirming the antecedent in both its forms evaluate conditional arguments randomly. Rather, many subjects respond m a manner yielded almost no errors. The analysis then focused on the data for consistent with a truth function like those to each individual subject. The first step was be found in the propositional calculus. Very to determine whether or not the subject was few, however, utlhze the truth function statistically consistent in his or her evaluations assumed for a conditional sentence--only of each argument. It was arbitrarily decided 2 . 8 ~ of all naive subjects in the present that any subject who was inconsistent on five experiment and 3 . 6 ~ in Taplin (1971a). In or more of the eight arguments would be each case the most common truth function to eliminated from any further analysis. One be inferred was one identical to the blcondisubject was rejected for this reason. Of the tional in the proposational calculus, that is, remaining 72 subjects, 20.8%0 were found pq true, Pq false, pq false, and/Sq true. to respond inconsistently to one or more Having noted these similarities between arguments. The next step in the analysis was the findings in the present study and m Taplin to take those subjects found to be consistent (1971a), something should also be said about on all eight arguments and seek to infer a truth the differences. The most obvious difference function using Table 2 to describe their hes m the proportion of subjects classified as evaluations of the arguments. In this way, statistically inconsistent. Whereas Taplin 2 . 8 ~ subjects were found to have a truth (1971 a) found the majority of subjects (51.7 ~ ) function identical to the conditional connec- to be inconsistent on one or more of the five tive in the propositional calculus, that is, arguments presented, in the present study where pq is true, Pq false, pq true and/~4 true; using eight arguments the percentage of 63.9 ~ subjects were inferred to have a truth inconsistent subjects was only 20.8 ~ . Correfunction identical to the biconditional in the sponding to the reduction in the number of propositaonal calculus, that is, pq true, Pq inconsistent subjects found is an increase in false,/Sq false, and/54 true; no single truth the number of subjects who apparently function could be found for the remaining utilized a biconditional truth-functional inter12.5 ~ subjects who were classified as being pretation of the connective. The most hkely logically contradictory. source of this effect lies in the type of lexical Those subjects initially found to be statisti- material presented: In Taplin (1971a) the cally inconsistent on one or more arguments sentences were composed of meaningful were analyzed further on only those argu- material, while in the present experiment the ments on which they were consistent, using sentences were abstract. In the meaningful Table 2. This analysis revealed that a further sentences factors like the plausibility of the

ABSTRACT CONDITIONAL SENTENCE REASONING

conclusion, and the syntactic and semantic structure of the sentences may have affected the interpretation of the conditional sentence, which in turn affected the reasoning process. The available evidence does not enable us to specify which of these factors is most responsible for producing the effect. Their role is something to which future research needs to be addressed. From the present experiment it can be concluded that the major source of inconsistency evident in Taplin (1971a) is due to the nature of the sentences employed. There is still a significant proportion of inconsistent subjects even with abstract sentences, but these subjects show similar tendencies to the subjects found to be consistent. Apart from affirming the antecedent, no one argument produced any more or less inconsistency than another, suggesting that the reason for their inconsistency may be related to some uncertainty about the truth value of the iOq and 10c] contingencies, as proposed by Wason (1966). Further examination was also made of the protocols of subjects who displayed a consistent pattern of responding to the different arguments but for whom no single truth functions could be inferred, that is, the subjects classified as consistent but contradictory. Explaining the behawor of these subjects is difficult. One possibility is that they misinterpreted the meaning of the response categories "true" and "false". In logic "true" refers only to a conclusion which is the only conclusion to follow from the premises; any other conclusion is "false." Some subjects, however, seemed as if they may have also regarded a conclusion which sometimes follows as "true." Other subjects appeared to make a different assumption: If the conclusion to an argument form is false, then negating the conclusion makes it true; thus each of the four argument forms examined has a true conclusion and a false conclusion. In logic, however, this may not always be so. It is possible that both the affirmative and the negative forms of the conclusion may be judged as false because

537

neither one always follows from the premises. The psychological literature contains very few references to the investigations of the concept of truth and falsity for naive human subjects. Wason and Johnson-Laird (1969), for example, reported some differences in evaluating sentences as either "true" or "false," w~th more erroneous responses occurring when subjects attempted to prove the sentence was "true." Interpretation of these differences in terms of a concept of truth and falsity is obscure. In the present experiment, it must be said that if any of the present subjects interpreted "true" and "false" differently from the assumed logical meaning of these categories, then the inferences drawn from their responses to the arguments are invalid. EXPERIMENT II

In an attempt to overcome the problem of the meaning of the response categories, a second experiment was performed, identical to the first except for the labelling of the categories. Two groups of subjects were run: The first group was asked to evaluate the conclusions of the arguments as either "always true," "sometimes but not always true," or "never true"; the second group evaluated the conclusions of the arguments as either "always false," "sometimes but not always false," or "never false." It was not clear whether any difference in performance should be expected between the two groups, but in view of the differences observed by Wason and Johnson-Laird (1969) when subjects were asked to evaluate the truth or the falsity of sentences, a further comparison seemed warranted. Apart from this, it was not expected that these categories would produce any significant change in the results for the majority of subjects, only that it might help to interpret the data for subjects who would otherwise be classified as statistically consistent but logically contradictory. Not only does the use of three response

538

TAPLIN AND STAUDENMAYER TABLE 4 INEERENCES DRAWN IN EXPERIMENT I I ABOUT THE TRUTH-TABLE VALUES FOR A CONDITIONALSENTENCE FROM RESPONSES MADE TO EIGHT CONDITIONALARGUMENTS

Subject's response to conclusion Name of argument Affirmmg the antecedent

Denying the antecedent

Affirmmg the consequent

Denying the consequent

Conclusion

Always true Never false

Sometimes true Sometimes false

Never true Always false

Affirmative

pq(T) andp0(F)

pq(T) andpO(T)

Negative

pq(F) andp0(Y)

pq(Y)andp0(T)

eltherpq(F) andp0(T) or pq(F) and p0(F) eitherpq(T) andp0(F) or pq(F) and p0(F)

Affirmative

riq(Y)and ri0(F)

riq(T) and ri0(T)

Negative

riq(F) and ri0(T)

riq(T) and ri0(T)

Affirmative

pq(T) and riq(F)

pq(T) and riq(T)

Negative

pq(F) and riq(T)

pq(Y)and riq(T)

Affirmative

p0(T) and ri0(F)

p0(Y) and ri0(T)

Negative

p0(F) and ri0(T)

pO(T)and ri0(T)

categories have the advantage of clarifying the nature of a subject's evaluation of an argument, but also it has the effect of mcreasing the power of the analysis of each subject's data. The first step in the analysis, as before, was to determine the statistical consistency of each subject's evaluations of each a r g u m e n t using a b i n o m i a l test, where e = .05 a n d the probability that a response will fall in a particular category equals 1/3. The next stage was to infer a t r u t h function for the connective which would describe the consistencies in a subject's evaluations of the conditional arguments. The inferences d r a w n are shown m Table 4. It can be seen in this table that additional i n f o r m a t i o n a b o u t the truth function is now obtained because of the sphtting o f the "false" category employed m the first experiment into two.

Method The stimulus material, instructions, and procedure were the same as that employed m Experiment I,

either pq(F) and ri0(T) or riq(F) and ri0(F) either riq(T) and ri0(F) or riq(F) and riO(F) either pq(F) and riq(T) or pq(F) and riq(F) either pq(T) and riq(F) or pq(F) and riq(F) eltherp0(F) and ri0(T) or p0(F) and ri0(F) either pO(T) and P0(F) or pO(F) and ri0(F)

except for the labeling of the response categories Two groups of subjects were used. The first group was instructed to evaluate statement C as either always true, sometimes but not always true, or never true. There were 76 introductory psychology students from the University of Colorado in this group, none of whom had received any instruction in logic; one subject was rejected for falhng to respond consistently on at least four of the eight arguments. The second group of subjects was instructed to evaluate statement C as either always false, sometimes but not always false, or never false There were 79 introductory psychology students from the University of Colorado m this group, none of whom had received any instruction m logic.

Results and Discussion The total n u m b e r of "correct" responses made by subjects an the two groups for each a r g u m e n t was calculated; the correctness of a response was identified with the prediction from propositional logic. A n analysis of variance on these data revealed no effect of the l a b e h n g of the response categories, n o r any

ABSTRACT CONDITIONAL SENTENCE REASONING

significant interaction between labeling and the type of argument. Once again, however, there was a significant difference between arguments, F(7, 1071) = 65.55, p <.001. Fewest errors were produced on affirming the antecedent; denying the antecedent In both Its forms, and also affirming the consequent, yielded the most errors. After the experiment, all subjects in the group asked to evaluate the falsity of the conclusmn to the arguments were questioned about their trdatment of the response categories. Only 12.7 ~ reported using the given categories. However, 58.2~ indicated they transformed the "never false" to read "always true," they maintained the "always false" category, and renamed the middle category "sometimes (true and sometimes false)." Another 2 4 . 1 ~ said they converted the categories of always, sometimes, and never false into never, sometimes, and always true. The answers given by the remaining 5 . 0 ~ subjects could not be classified. These responses help to explain why no difference in performance was found between those subjects required to evaluate the truth and those subjects required to evaluate the falsity of the conclusions to the different arguments presented. The data for each individual subject were analyzed according to the method described in the introduction to the experiment. The pattern of the results obtained was significantly different from that obtained in Experiment I. For the group asked to evaluate the arguments as either always, sometimes, or never true, out of 75 subjects 36.0~ were inferred to have a conditional truth function (i.e., pq true, pc] false,/Sq true, and pq true), 14.7 ~ a bicondltlonal truth (that is, pq true, pc] false,/Sq false, and/5c] true), 14.7 ~/owere statistically consistent on all eight arguments yet no single truth function could be found to describe their responding, and finally, 34.7 ~ were found to be statistically inconsistent on one or more of the arguments. When this last group of subjects was analyzed only on those

539

arguments on which they were consistent, it was found that they could be divided further into 8 0 ~ who were then inferred to have a conditional truth function, 4 . 0 ~ a biconditional truth function, 2 0 . 0 ~ who were contradictory in the sense that there was no one truth function which could describe their responding, while for 2.7 ~ there was insufficient data with which to infer a value for every truth-table category. For the group asked to evaluate the arguments as either always, sometimes, or never false, the results did not differ significantly from the first group. Out of 79 subjects, 31.6 ~ were inferred apparently to utilize a conditional truth function, 12.7~ the bicondltlonal, 15.2~ were consistent but contradictory, and 4 0 . 5 ~ were rated statistically inconsistent. Further breakdown of the statistically inconsistent group by analysis of only those arguments on which they were consistent revealed 7 . 6 ~ with a tendency toward the conditional, 1 0 . I ~ toward the blconditional, 19.0~ contradictory, with 3.8 ~ unable to be classified. The percentage of subjects in each classification is shown in Table 5. The important difference between these findings and those of Experiment I is seen in the decrease in the proportion of subjects inferred to interpret the connective biconditlonally and in the increase in the proportxon of subjects inferred to interpret the connective according to the conditional in the propositional calculus. This difference is not simply attributable to a possible misinterpretation of the response categories "true" and "false" in the first experiment. The explanation which seems to fit best takes into account the fact that a biconditional interpretation of the connective requires the use of only two of the three response categories, "always" and "never," while a conditional interpretation requires that the subject respond to all three. These observations, combined with the assumption that a naive subject would expect all three given categories to be relevant at some time, suggest a way of explaining the results of

540

TAPLIN AND STAUDENMAYER TABLE 5 PERCENTAGE OF SUBJECTS INFERRED IN EACH CLASSIFICATION

Statistically consistent on all arguments Biconditional

Contradtctory

Total

Condotional

28

63 9

12.5

79.2

4.2

11 1

2.8

(a) always, sometimes, or never true

36.0

14.7

14.7

65.3

8.0

4.0

20.0

27

34.7

(b) always, sometimes, or never false

31.6

12.7

15.2

59 5

76

101

190

38

40.5

Condition Experiment I

Condittonal

Statistically inconsistent on at least one argument Bicondi- Contrational dictory

Other Total 2.8 20.8

Experiment II

Experiment lI. This explanation imphes that the subject is capable of interpreting the connective in more than one way, in the present situation either blconditionally or conditionally. The majority of subjects tend to favor a conditional interpretation because in this way they are able to satisfy their expectation that all available response categories will be relevant. If there is some uncertainty about how to interpret the connective, then this may also explain why so many subjects were found to be inconsistent on one or more arguments, and why the percentage of subjects, for whom no single truth function could be inferred, increased. The protocols for this latter group of contradlctory subjects are confusing. The only identifiable subgroup among them are a few individuals who evaluated every argument except affirming the antecedent as "sometimes true (false)"; affirming the antecedent with an affirmative conclusion was judged as "always true (never false)," while a negative conclusion was judged to be "never true (always false)." Denying the antecedent, affirming the consequent, and denying the consequent were found by Taplin (1971a) to be the arguments which subjects are least confident in evaluating, so perhaps the small group of subjects above adopted a policy of "if in doubt, choose the middle category."

The fact that subjects are less confident on these arguments is consistent with Wason's (1966) proposal that the pq and/Sq contingencies of the truth table have an uncertain value, since all arguments except affirming the antecedent involve the use of these contingencies. GENERAL DISCUSSION

Three different truth-functional meanings have been found for abstract conditional sentences by seemingly small changes in the reasoning task presented. In Wason's sentence verification and evaluation tasks the truth funcnon tended to be pq true and pc] false with /Sq and pc] irrelevant. However, in a syllogistic reasoning task like that employed in the first experiment of this study in which the conclusion was evaluated as being either true or false, the truth function most commonly obtained was the bicondltional, that is one where pq is true, pc] false,/Sq false, and/5C] true. But when the conclusion to an argument was either always, sometimes, or never true as in the second experiment, the most frequent meaning inferred was the conditional truth function, namely, pq true, pc] false,/Sq true, and tic] true. Two factors seem to be involved in producing these various meanings. The first has to do with the complexity of the deductive infer-

ABSTRACT CONDITIONAL SENTENCE REASONING

ences required by each meaning. Other things being equal, it is suggested that subjects will tend to favor that interpretation which simplifies the inferential process. In the syllogistic task, reasoning appears simplest when based on the biconditional truth function. This truth function says that "p imphes q" and that "q implies p"; it also says that "p implies q" and that "q implies p." The conditional, by comparison, is not reversible: it allows only two of these postulates, "p implies q" and "q implies 10;" one cannot say that "q implies p" because q may be followed by eitherp or/5; similarly one cannot say that "/5 implies q" because p may be followed by either q or q. As a consequence, when reasoning with the conditional, one must not only determine whether the given conclusion may follow from the premises, but also whether it is the only conclusion to follow. With the biconditional, on the other hand, only one conclusion can follow any particular set of premises so that it is not required that one determine whether the given conclusion necessarily follows, only that it follows. A truth function like that posited by Wason in which some of the contingencies of the truth table are irrelevant, ~s difficult to apply in this task; it most probably resembles the conditional insofar as it seems necessary to consider alternative possibilities besides the conclusions provided on arguments other than affirming the antecedent. If we examine the task employed by Wason, however, the bicond~tional interpretation appears to require the most complicated set of inferences to establish the truth of falsity of the sentence. In fact, given this interpretation of if p then q, the subject needs to select instances which will test each of the four implications listed above, namely, "p implies q", "q implies p", and so on; that is, he must select each ofp,/5, q, and 4. For the conditional, on the other hand, there are only two implications to be tested, namely, "p implies q" and "q implies/5," so that it is necessary only to selectp and q. For the truth-functional mean-

541

ing suggested by Wason, there appears to be only one logical implication, namely, "p is followed by q," so that in this case the only choice necessary is p. These observations indicate why the biconditional interpretation is frequently obtained in syllogistic reasoning and why Wason's three-valued interpretation tends to be favored in sentence verification. Something more is reqmred to explain why the syllogistic reasoning task in the second experiment reported m this paper yielded more conditional truth functions. The second factor suggested to be involved m the interpretation of these sentences proposes that the different meanings are due to small differences between the reasoning tasks affecting the subjects' perceptions of the way in which the sentences were supposed to be used, that is, the way in which the experimenter intended that they should be used. In Wason's study it is evident that the interpretation of the sentence found is a consequence of the subject assuming the truth of the antecedent at all times. Precisely why such an assumption hke this should be made is hard to determine. But when a procedure is used which counteracts this set toward the truth of the antecedent, as in syllogistic reasoning, a more complete truth function results. To account for the different meanings obtained in the syllogistic task requires two assumptions about the demand characteristics of this situation. It seems plausible to suppose: (1) that the subject expects each of the response categories provided for evaluating the conclusions of the various arguments to be relevant for at least one argument; and (2) that each of the categories should be relevant approximately equally frequently. Thus, in the second experiment in which the response system provides for three kinds of evaluations, the conditional interpretation of the major premise tends to be preferred because this is the only interpretation which permits each of these evaluations to be drawn. In the first experiment, the biconditional interpretation

542

TAPLIN AND STAUDENMEYER

m a y be favored because, unlike the conditional, it produces equal numbers of " t r u e " and "false" responses. It should not be supposed that the importance of inferences about what is an appropriate interpretation o f a sentence are restricted to abstract sentences or to contrived experimental situations. Recently the hnguists, Gels and Zwicky (1971), have proposed that when the sentence ifp then q is used m such a way as to convey either a pre&ction, a promise, a threat, and so on, ~t tends to be interpreted not as a conditional but as a biconditional. In their terms, a sentence like I f you mow the lawn, I'll give you five dollars invites the inference that f f y o u do not m o w the lawn, I will not give you five dollars. Clearly such references m a y have a marked effect on subsequent behavior. In summary, the present study indicates the importance of factors sometimes ignored in discussions o f the interpretation o f sentences, namely, the manner in which the sentence is to be used. Other workers (e.g., Peel, 1967) have suggested that the meaning o f a sentence like ifp then q depends on the type of connection between the propositions p and q. Some evidence consistent with this position has been obtained recently by the authors in an investigation o f the effect o f different nominal class relations between the antecedent and the consequent on the use o f such sentences in a reasoning t a s k ? Nevertheless, to be sure, the meaning o f the sentence is not solely a function of the semantic relations within the sentence. Otherwise, i f . . . t h e n . . . , could be used variously as a conditional, a biconditional, a disjunctive, or some other connective merely by manipulating the structure and content o f the propositions connected, which is clearly not the case. Certainly it would be difficult to account for the present findings with abstract conditional sentences in these terms. It is worth restating the view, therefore, that the interpretation o f a sentence is not solely a function o f the intrinsic properties o f 3 H. Staudenmayer and J. E. Taplin. Semantic effects on conditional reasoning. In preparation.

the sentence itself but is also dependent upon the context in which the sentence is produced. REFERENCES

CERASO,J., ~; PROVITERA,A Sources of error in syllogistic reasoning Cognitwe Psychology, 1971, 2, 400-410 CoPI, I M. Introduction to logic, 2nd edition New York: Macmillan, 1961 GELS,M L, & ZWICKY,A. M. On invited inferences. Lmgmsttc Inqmry, 1971, 2, 561-566. HENLE,M. On the relation between logic and thinking. Psychological Revww, 1962, 69, 366-378. KAUFMANN, H., & GOLDSTEIN, S. The effects of emotional value of conclusions upon distortions in syllogistic reasoning. Psychonomtc Science, 1967, 7, 367-368. LIPPMAN, M. Z. The influence of grammatical transform in a syllogistic reasoning task. Journal of Verbal Learmng and Verbal Behavior, 1972, 11, 424-430. MORGAN,J. J. B., tg..MORTON,J. T. The distortion of syllogistic reasoning produced by personal convictions. Journal of Social Psychology, 1944, 20, 39-59. PEEL, E. A. A method for investigating children's understanding of certain logical connectives used in binary propositional thinking British Journal of Mathematical and Statlstlcal Psychology, 1967, 20, 81-92. REICHENBACH,H. Elements of symbolic logic. New York: Dover, 1947. STRAWSON, P. F. Introductzon to logical theory. London: Methuen, 1952. TAPLIN, J. E. Reasoning with conditional sentences. Journal of VerbalLearning and Verbal Behavior, 1971, 10, 218-225 (a) TAPLIN,J E. An experimental study of human reasoning and conceptual behaviour Unpublished PhD. thesis, University of Adelaide, South Australia, 1971. (b) WASON, P. C. Reasoning. In B. Foss (Ed.), New horizons mpsychology Harmondsworth: Penguin, 1966. WASON, P. C. Reasoning about a rule Quarterly Journal of Experhnental Psychology, 1968, 20, 273-281. WASON, P. C, & JOHNSON-LAIRD,P. N. Proving a disjunctive rule. QuarterlyJournalof Experimental Psychology, 1969, 21, 14-20. WILKINS, M C The effect of changed material on ability to do formal syllogistlcreasoning. Archwes of Psychology, 1928, no. 102. (Received March 6, 1973)