Reasoning with conditional sentences

JOURNAL OF VERBALLEARNINGAND VERBALBEHAVIOR10, 219--225 ( 1971 )

Reasoning with Conditional Sentences JOHN E. TAPLIN Department of Psychology, University of Adelaide, South Australia 5001 The nature of the truth function when reasoning with conditional sentences was investigated through an analysis of the evaluations of the truth of four deductive arguments by 56 naive adult Ss. This analysis suggested that only 44.7 ~ of these Ss consistently assessed the validity of these arguments in a truth-functional manner. Of these Ss, the majority appeared to utilize a truth function which was the same as that for a biconditional in propositional calculus. Only two Ss consistently produced responses which would be logically predicted for the conditional. This finding is similar to previous findings with children in Piaget's concrete operational period.

Most experimenters in human reasoning and study by the present author in which Ss were conceptual behavior (e.g., Inhelder & Piaget, required to discriminate between a conjunc1958; Haygood & Bourne, 1965; Laughlin, tive, a conditional, and a biconditional sentence •968) have assumed that a sentence expressed conventionally expressed with reference to in the form Ifp then q is logically equivalent to a two attribute values of a display of cards conditional sentence given in symbolic form, bearing geometrical designs. None of the 12 p-+q, having a truth-table function where pq Ss was successful in making this discrimination, is true, p~ is false, pq is true, and fig is true. The and it would seem that for these naive Ss validity of this assumption is open to question. no difference exists between the truth functions For example, in an unpublished investigation of the conditional and biconditional sentences. of c9ncept learning involving different logical However, the truth function of both these connectives including the conditional, the connectives did not appear to be like that for present author found that, on attribute- the biconditional in propositional calculus, learning, clarification was required of the but more like conjunction where pq is the type example of a conditional rule when it was only true category. A few Ss did question given in the instructions in the form I f a card whether the conditional and biconditional is green, then it must be square to be a member specifications were "necessary and sufficient," of the concept. On rule-learning, not one S whether perhaps more information was verbalized the conditional rule in the conven- needed on both of these sentences in order to tional form, but instead gave their solutions in know the value of cards which were P. terms of conjunctions, disjunctions, or negaCloser agreement for these results may be tions. found with Wason (1966), who has proposed Matalon (1962) and Peel (1967) have that individuals are not limited by the rules of suggested that the conditional tends to be propositional calculus, but rather assume that interpreted as a biconditional whose English a conditional sentence can have three truth translation is p if and only if q. For the bi- values, namely, true, false, and irrelevant. conditional, the truth table function in Thus, for the sentence l f p then q, pq is true, propositional calculus consists of pq as true, p~ is false, and ff with either q or 4 is irrelevant. p~ false, pq false, and/~4 true. Partial support The notion of a three-valued logic is not new, for this suggestion comes from an unpublished and its feasibility and implications are 9 219 © 1971 by Academm Press, Inc.

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currently being considered by a number of logicians (e.g., Woodruff, 1970). In an experiment in which Ss were set the task of selecting instances of p,/5, q, q necessary to determine whether a given conditional sentence was true or false, Wason (1968) has obtained evidence which supports his suggestion (see above) for pq, p~, and/5~ contingencies, but the majority of his Ss evaluated the pq contingency as false. Few Ss made the contrapositive inference, namely, that /5 follows from q, when trying to show that the conditional sentence was true or false. This latter finding was not so evident when the conditional sentence was expressed in the form either/5 or q (Wason & Johnson-Laird, 1969), but no difference was found for/5/f~, never p without q (Johnson-Laird & Tagart, 1969), and all ps are q (Johnson-Laird & Wason, 1970). Wason (1968) has postulated that the infrequency of the contrapositive inference is due to a set toward truth or correspondence between sentences and states of affairs. The necessity of this postulate is not clear, and a more parsimonious account would be that, if the truth value of the/sq contingency for a conditional sentence is irrelevant, then the contrapositive inference does not convey any information about the truth of the conditional sentence, and thus is also irrelevant. The present experiment was conducted to examine the nature of the truth-table function for conditional sentences in human reasoning

with various types of deductive arguments. Inferences about the truth table were based on the assumption that, if it is known that a given conditional sentence is true, and the truth value of either the antecedent or the consequent is also known, then the truth values for the given conditional sentence may be derived from judgments regarding the validity of a conclusion involving the consequent or antecedent, respectively. Five basic types of deductive arguments were used, and each type was replicated 12 times to allow for variations in the plausibility of conclusions (Wilkins, 1928; Suppes, 1965) and ambiguity of connection between antecedent and consequent (Peel, 1967). A description of these five conditional arguments can be seen in Table 1. Four of the five types of arguments examined were: (1) affirming the antecedent and (4) denying the consequent, which are logically valid arguments, and (3) affirming the consequent and (2) denying the antecedent, which are logically invalid arguments. The fifth type had the same form as affirming the antecedent except for a negated conclusion, thus making it logically invalid. This last type was included because preliminary study using four replicates of each of the first four types of argument with 43 Ss revealed a tendency to affirm the validity of the conclusion in all cases, a tendency which might be accounted for simply in terms of a response bias toward affirmation. A consistent denial of the validity of the conclusion for this last

TABLE 1 FIVECONDITIONALARGUMENTSEXAMINED Argument (type of problem) 1 Affirmingthe antecedent 2 Denying the antecedent 3 Affirmingthe consequent 4 Denying the consequent 5 Affirmingthe antecedent with negated conclusion

Conclusion

Logical validity

First premise

Secondpremise

p--~q p~q p---~q p~q

p /~ q

q ~ p ,5

Vahd Invalid Invalid Valid

p--)-q

p

~

Invahd

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CONDITIONAL REASONING

type, however, would reduce the likelihood of this kind of explanation, and enable inferences to be made about the truth-table function of conditional sentences from the other types of argument. Assuming an affirmative bias explanation is not applicable, the following inferences can be made. For affirming the antecedent, a subjective decision (response) of "vahd" would suggest that the S has a p q truth-table category which is true, and a p q category which is false; an "invalid" response, on the other hand, might indicate that either p q is false or pq is true or both. For denying the antecedent, an "invalid" response by S might mean that both p q a n d / ~ have the same truth value, either true or false, or pq is false and p q is true; a "valid" response, however, would indicate that pq is true and p q is false. For affirming the consequent, an "invalid" response would suggest that p q and fiq have the same truth value, be it true or false, or p q is false andffq is true; a "valid" response, on the other hand, would mean that qp is true and/~q is false. For denying the consequent, a "valid" response would indicate that pq is false and p~ is true; however, an "invalid" response might suggest that either Pq is true or fig is false or both. A summary of these inferences can be found in Table 2. It can be seen that two inferences are drawn about the truth values of each truth-table

contingency. Thus, if it is the case that conditional sentences are considered in a truthfunctional manner, then we can expect significant correlations in performance between at least one and possibly both of affirming the antecedent and affirming the consequent, and affirming the antecedent and denying the consequent. For the same reason, significant correlations may also be expected between one or both of denying the antecedent and affirming the consequent, and denying the antecedent and denying the consequent. METHOD Subjects

The Ss were 56 undergraduates from the subject pool of first-year psychologystudents at the University of Adelaide. Selection of Ss was random with the one exception that none of the Ss, according to their student record cards, was currently or had been previously enrolled in a course in which instruction in logic was gwen. Problems

The problems consisted of five basic types. In the first type, affirming the antecedent, S was given Ifp then q, with p as true, and was asked whether q necessarily followed. In the second type, denying the antecedent, S was given I f p then q, with p as true, and was asked whether q necessarily followed. In the third type, affirming the consequent, S was gwen I f p then q, with q as true, and was asked whether p necessarily followed. In the fourth type, denying the consequent, S was given I f p then q, with ~ as true, and was asked whether p necessarily followed. In the fifth type,

TABLE 2 INFERENCESTO BEDRAWN(T OR F) AnOUTTRUTH-TABLEVALUESFORA CONDITIONALSENTENCEFROMRESPONSESMADETOFOURDEDUCTIVEARGUMENTS Response Argument (type of problem)

Valid

Invalid

1 Affirmingthe antecedent 2 Denying the antecedent

pq (T) andp~ (F)

pq (F) or p# (T) or both

pq (F) and • (T)

3 Affirmingthe consequent

pq (T) and pq (F)

4 Denying the consequent

Pq (F) and/~ (T)

pq (T/F) andp# (T/F) or pq (T) and p# (F) pq (T/F) and/~q (T/F) or pq (F) and pq (T) p~ (T) or P# (F) or both

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affirming the antecedent with negated conclusion, S was given I f p then q, with p as true, and was asked whether ~ necessarily followed. The Ss gave their answer to each problem by checking one of "yes" or "no," and were also required to show their confidence in their answer immediately below according to one of three categories: very confident, moderately confident, not confident. The majority of conditional sentences were taken from various logic textbooks. A wide range of topics was involved and an independent judge was used to eliminate any sentences whose conclusions were obviously plausible or implausible. The sentences contained a mixture of instances where the two propositions, p and q, were causally related and where the connection between the propositions was either arbitrary or ambiguous (Peel, 1967). For example, an instance of a conditional sentence in which the two propositions were causally related was I f food is constantly supphed to them, then the very fiercest creatures live peacably together. An example of a

conditional sentence where the connection between the propositions was arbitrary was I f Mrs. Elton is' a snob, then Mrs. Bates ts a bore. Design

Sixty problems were presented to S, 12 problems for each problem type. The order of problems was randomized with the aid of a Latin square so that each problem type occurred once every five problems. All Ss did all problems in the same order at their own pace. The task took approximately 30 min. for each S. Procedure and Instructions

The procedure was given in the instructions on the front of the test booklet and read in part as follows: Inside this cover are 60 sets of three statements. Each statement within a set is labeled a, b, or c. In this test you are to assume that statements a and b are known to be true. The question you are required to answer is whether statement c necessarily follows from statements a and b, assuming that these two statements (a and b) are known to be true. Indicate your answer by crossing out either "yes" or "no", whichever is mcorrect, at the end of each question. Then indicate the confidence you have in your answer by checking one of the 3 categories underneath very confident ? moderately confident ? not confident ? When S has finished reading the instructions, he was allowed to begin. The Ss were reminded that they should answer all questions. RESULTS T h e t o t a l n u m b e r o f logically correct responses m a d e b y each S over the 12 p r o b l e m s

for each p r o b l e m type was calculated. A t w o - w a y ( p r o b l e m t y p e x Ss) analysis o f variance was p e r f o r m e d on the d a t a a n d a significant effect f o u n d for p r o b l e m type, F(4,220)=48.84, p<.001. A post hoc analysis using N e w m a n - K e u l s multiple-range c o m p a r i s o n s revealed no significant differences between P r o b l e m s T y p e s 2 and 3, a n d between I a n d 5, with all o t h e r c o m p a r i s o n s significant, p < .01. Thus, the o r d e r o f p r o b l e m types in terms o f increasing n u m b e r o f correct responses was 2 a n d 3, 4, a n d 1 a n d 5. The results for each p r o b l e m type were a n a l y z e d to d e t e r m i n e w h e t h e r Ss were b e h a v i n g in a r a n d o m fashion in p r o d u c i n g their responses. A single-sample chi square was used to c o m p a r e the d i s t r i b u t i o n o f the frequency o f correct responses with a b i n o m i a l d i s t r i b u t i o n where the p r o b a b i l i t y o f a correct response is equal to the p r o b a b i l i t y o f an incorrect response. Significant differences were o b t a i n e d between the observed distrib u t i o n a n d the r a n d o m d i s t r i b u t i o n for all p r o b l e m types, p < .001. F r o m a b i n o m i a l test, 91.1 ~ 17.9 ~ , 28.6 ~ , 48.2 ~ , a n d 9 6 . 4 ~ o f the Ss d i s p l a y e d a consistent tendency to r e s p o n d correctly, a n d 0 ~ , 4 8 . 2 ~ , 4 2 . 9 ~ , 14.3 ~ , a n d 0 K o f Ss consistently r e s p o n d e d incorrectly for P r o b l e m Types 1, 2, 3, 4, a n d 5, respectively. T h e frequency d i s t r i b u t i o n o f correct responses for each p r o b l e m type is shown in T a b l e 3. I t has been p r e d i c t e d earlier t h a t perform a n c e m a y show some c o r r e l a t i o n between certain p r o b l e m types a n d this was assessed b y a S p e a r m a n r a n k correlation. This analysis i n d i c a t e d t h a t Ss w h o were g o o d p e r f o r m e r s on P r o b l e m T y p e 1 were also g o o d p e r f o r m e r s on T y p e 3, r h o = . 2 7 0 , p , < . 0 5 , T y p e 4, r h o = .278, p < .05, a n d T y p e 5, r h o = .488, p < .01 ; a n d Ss who were g o o d p e r f o r m e r s on P r o b l e m T y p e 2 were g o o d p e r f o r m e r s o n T y p e 3, r h o = .779, p < .01, b u t p o o r perf o r m e r s on T y p e 4, r h o = - .622, p < .01. A n e x a m i n a t i o n was also m a d e o f the p r o b l e m s used for each p r o b l e m type in o r d e r to d e t e r m i n e w h e t h e r some p r o b l e m s were

CONDITIONALREASONING

223

TABLE 3 OBSERVEDFREQUENCYOF CORRECTRESPONSESFOR EACHPROBLEMTYPEWITHFXPECTEDFREQUENCYAND ASSOCIATEDPROBABILITYOF CORRECTRESPONSESACCORDINGTO A BINOMIALDISTRIBUTIONWHEREp = q = ½ Number of correct responses

Problem type

Binomial distribution Probability

0

1

2

1

0

0

0

0

2 3 4 5

14 10 2 0

8 11 3 0

3 0 0 0

2 3 3 0

4.08 < .05

3

4

5

6

7

8

9

10

11

12

1

1

0

2

0 3 a 0

4 4 6 0

3 1 8 1

9 3 2 0

1

3

2

3 5 2 1

0 3 1 2

1 5 8 4

15 3 4 7 17

31 6 4 10 31

6.78 10.84 12.65 10.84 6.78 .12 .19 22 .19 .12

more difficult than others. A chi-square test yielded no significant departures from a rectangular distribution for each type, indicating that there were no real differences between problems for number of correct responses. The confidence that the S had in his answer to each problem was measured by assigning the scores 1, 2, and 3 to the confidence categories very confident, moderately confident, and not confident, respectively. An analysis of variance on the confidence ratings totaled over the 12 problems in each problem type for each S revealed a significant difference in confidence for different types, F(4, 2 2 0 ) = 38.59, p < .001. Differences between individual types were tested by the N e w m a n - K e u l s multiple-range comparisons method. This post hoc analysis showed that Ss were least confident on Problem Type 4. Higher confidence ratings were obtained on Type 3 than Type 4, p < . 0 5 , Type 2 than Type 3, p <.05, and Types ! and 5 than Type 2, p < .01. An examination was also made of the confidence ratings for different problems within each type, and a chl-square single-sample test revealed no significant difference between problems within each problem type. The degree of correspondence between performance and confidence was investigated by a Spearman rank-correlation test. It was found that the more confident the S was, the

4.08 < .05

fewer correct responses he made on Type 1, rho = - .376, p < .01, and on Type 5, rho = - .465, p < .01, but no significant correlations were obtained for other problem types. DISCUSSION F r o m these results it can be seen that all but 3.6% of the Ss tended to consistently reject the conclusion as invalid (i.e., respond correctly) for affirming the antecedent with a negated conclusion (Type 5). This finding enables us to tentatively disregard an explanation of these results given merely in terms of an affirmative response bias, although it should be admitted that this test is insufficient to completely eliminate this possibility from consideration on problems other than affirming the antecedent. It is also evident that most Ss were not making their judgments at r a n d o m for any of the four types of argument examined. The results indicate that performance on affirming the antecedent is logically better than on denying the consequent, which is better than denying the antecedent and affirming the consequent. They also suggest that the distribution of scores for Ss on affirming the antecedent is unimodal, that Ss are uniform in their judgments of the validity of this argument. This convergence is lacking, however, on denying the antecedent, affirming the consequent, and denying the consequent,

224

TAPLIN

where the distribution of responses seems at least bimodal, and possibly trlmodal. On denying the antecedent, 48.270 of the Ss consistently affirmed the validity of this argument, while 17.970 denied its validity consistently; on affirming the consequent, 42.9 ~ of the Ss affirmed the vahdity of the argument, and 28.6 ~ denied its validity; and on denying the consequent, 48.270 of the Ss regarded this argument as valid, and 14.3 70 judged it as invalid. Related to this is the observation of a trade-off between performance and confidence on affirming the antecedent, while no such relationship has been obtained for the other arguments, probably as a result of the divergence of responding on these arguments. This finding suggests that no truth-table function exists for conditional sentences which is common to all individuals. For example, Wason's (1968) derived truth function of pq true, Pq false, pq false, and /54 irrelevant, would predict valid responses to affirming the antecedent and affirming the consequent, and invalid responses to denying the antecedent and denying the consequent. On only one argument, affirming the antecedent, is this prediction confirmed for the majority of Ss, however. Nearer to describing the present results is Matalon's (1962) and Peel's (1967) suggestion of a truth function in which pq is true, P4 false,/Sq false, andp4 true, since this function predicts valid responses on all four arguments. It, therefore, appears that either a number of truth functions have been employed, or, alternatively, some other criterion was used which was not truth-functional in judging the validity of arguments. The prediction was made earlier that, if Ss were evaluating these arguments in a truth-functional manner, correlations could be expected between one or both of affirming the antecedent and affirming the consequent, and affirming the antecedent and denying the consequent. Since the predicted significant, although in some cases small, correlations have been found on three

out of four of these comparisons, it seems that, perhaps Ss are behaving according to some truth-function when reasoning with conditional sentences. Examination of the results of individual Ss indicates that 37.570 of the Ss consistently produced affirmative judgments of the validity of all four arguments; 3.6 700affirmed the validity of affirming the antecedent and denying the consequent, and denied the validity of denying the antecedent and affirming the consequent; 3.670 said "yes" to affirming the antecedent, and "no" to the other arguments; 3.6 70 gave affirmative responses to affirming the antecedent and denying the antecedent, and negative responses to the remaining arguments; while the rest of the Ss, 51.77o, were not consistent in their responding on one or more of the arguments. From this analysis we may infer that only 44.7 700 of the Ss were behaving consistently in a truthfunctional fashion when reasoning with the conditional. The last two groups of Ss listed above cannot be classified by a truth-function using the kind of inferences given in Table 2. According to the inferences summarized in Table 2, it would appear that 37.5 ~ of the Ss have a truth-function of pq true, Pq false, pq false, and P4 true; 3.6700 possess a truth function where pq is true, Pq is false, pq is true, andp4 is true; and 3.6 ~ of the Ss have a truth function which reads pq true, Pq false,/Sq true, and/54 false. It will be realized that the most common truth function obtained is the same as the truth function given in propositional calculus for the biconditional. It will also be noted that a few Ss are in fact employing the truth function for a conditional sentence as found in the propositional calculus. The finding that the most common truthfunction utilized by these adult Ss, average age 18.7 years, is the same as that found by Matalon (1962) with children between 9.25 and 11.5 years, median 10.17 years, and Peel (1967) with chddren between 5 + and 11 + years, raises important implications for Piaget's theory of intellectual development.

225

CONDITIONAL REASONING

According to Piaget, from the age of approximately 11-12 years onward, Ss develop formal operations. By this it is meant that Ss use the logical forms of propositional calculus and understand the logical relations formalized in the propositional calculus, including the conditional. Doubts that propositional logic provides the essential structure of the final stage of logical development and that Piaget's model completely describes the difference between the formal operational period and its predecessor, the concrete operational stage, have already been expressed (Parsons, 1960), and the present results can only add further weight to these objections with respect to conditional reasoning. It is of interest to note that no difference was found between problems either in terms of logicality of response or confidence in this response for any argument. This finding is worthy of note in view of Peel's (1967) distinction between conditional sentences where the two propositions are causally connected, arbitrarily connected, and not connected. No special precautions were taken in the present study to control for this factor because of difficulties in objectively assessing the nature of the connection between propositions in a number of sentences. However, since no difference was found between problems, it would appear that this factor did not significantly affect evaluation of any argument in the present study. How then are the findings in the present study to be reconciled with those of Wason (1968) and the unpublished findings of the present author reported earlier in this paper ? It would seem possible that the different truth functions derived occur because of the different truth values assigned to the conditional sentence itself in the two types of experiments. In the present experiment the conditional sentence is given to be true; in the previous investigations the nature of the task required the S to determine whether the

conditional was true or false. Further work is needed to show whether the truth status of the conditional sentence may influence the kind of truth function inferred concerning it. REFERENCES

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