Conditional Reasoning

Transcription

1 JOURNAL OF VERBAL LEARNING AND VERBAL BEHAVIOR 18, (t979) Conditional Reasoning SANDRA L. MARCUS AND LANCE J. RIPS University of Chicago Several techniques have been developed to examine how people understand conditional sentences. However, these techniques have produced different conclusions about/f.., then. The l~resent experiments examined two of them, the Conditional Evaluation task and the Conditional Syllogism task, to determine the cause of these differences. In Experiment 1, we collected Evaluation and Syllogism judgments from the same subjects and for the same sentences. The results showed more material conditional responses in the Evaluation than in the Syllogism task. Experiment 2 replicated this finding in a situation where subjects' reaction times were recorded. From these studies, we propose a quantitative model to explain the Evaluation-Syllogism difference in terms of the more complex processing requirements imposed by the Syllogism task. Scientific inquiry, as well as everyday planning and decision making, requires us to formulate and verify contingencies among objects and events. Conditional sentences (e.g., If water is heated to 212 degrees, it boils) provide one way to encode such contingencies, and researchers have therefore looked at a range of tasks to find out how such sentences are understood. Unfortunately, previous studies have turned up only equivocal evidence about the way conditionals are comprehended, different experimental procedures pointing to different representations for these propositions. In particular, two of the main procedures--the Conditional Syllogism task (e.g., Taplin, 1971; Taplin & Staudenmayer, 1973) and the Conditional Evaluation task (e.g., Johnson-Laird & Tagart, 1969; Wason & Shapiro, 1971)--have led to substantially different conclusions about the underlying meaning of conditionals. We would like to thank Joel Angiolillo-Bent, Stephen Schacht, and Robert Sternberg for their comments on a preliminary version of this manuscript. We also acknowledge support from National Science Foundation Grant BNS and NIMH Grant 1-F31 MH Correspondence should be addressed to Lance Rips, Behavioral Sciences Department, University of Chicago, 5848 South University Ave., Chicago, Ill Consequently, our aim in the present paper is to explain how such discrepancies arise. We examine the Conditional Syllogism and Conditional Evaluation tasks in detail below, but before doing so, we need to consider the nature of conditional sentences more generally. In the present context, we will mean by "conditional sentence" one of the form If A, C, where A is called the "antecedent" clause of the conditional and C is called the "consequent." In the following example, Jones gets a raise is the antecedent clause, while he'll buy a Buick is the consequent: (1) If Jones gets a raise, he'll buy a Buick. In elementary logic texts, most sentences of the form If A, C are represented as material conditionals, symbolized A ~ C. The truth of propositions of this kind is completely determined by the truth of its component propositions, A and C. In particular, a material conditional (MC for short) is false when A is true and C is false, and is true otherwise. For example, the sentence about Jones, translated as an MC, will be false if Jones gets the raise but doesn't buy the car, but in all other circumstances will be true. Table 1 summarizes this by showing the truth value of an MC as a function of the truth of its antecedent and consequent /79/ /0 Copyright i979 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

2 200 MARCUS AND RIPS TABLE 1 TRUTH FUNCTIONS FOR If A, ~hen C Truth value of A Truth value of C Truth value of If A, C MC MB Partial True True True True True True False False False False False True True False Irrelevant False False True True Irrelevant A second truth function in propositional logic that figures centrally in the following discussion is the material biconditional (MB), often represented as A -- C. As shown in Table 1, propositions whose main connective is an MB are true when both A and C are true or when both are false. For instance, Conditional (1) Will be true in case Jones gets the raise and buys the car or in case Jones doesn't get the raise and doesn't buy the car. Otherwise, the sentence will be false. The relationship between English conditionals and the MC or MB connective is a controversial one both in the psychological and philosophical literature (for arguments against identifying MC with if see Anderson & Belnap, 1975; Stalnaker, 1968; Strawson, 1952; and Wason & Johnson-Laird, 1972; for opposing arguments see Braine, 1978; Clark, 1971; Grice, 1967). We return to this issue in the General Discussion, but in the meantime, we would like to preserve a neutral stance. Our use of "MC" and "MB" in the following, is therefore meant to refer to the relevant truth functions themselves or to responses by subjects that are related to the truth functions in specified ways. They do not necessarily denote the underlying meaning of /f. The Conditional Syllogism Task Syllogisms have been previously used by Taplin and Staudenmayer (1973; Staudenmayer, 1975; Taplin, 1971) to discriminate MC from MB responses to conditionals. Subjects are asked to judge the validity of simple arguments, for example, whether the final sentence of the argument below (the conclusion of the syllogism) necessarily follows from the truth of the first two sentences (the premises): (2) If there is an A, there is a C. There is an A. There is a C. In Taplin and Staudenmayer's experiment, the first premise was always a conditional sentence, as in the example above. Different problems were formed by varying the second premise and the conclusion. In some problems, the antecedent of the conditional (i.e., There is an A) appeared as the second premise of the argument, and the consequent (There is a C) appeared in the conclusion, as in (2); in other problems, the antecedent appeared in the conclusion, and the consequent in the second premise. Additionally, the second premise and the conclusion could independently have a positive form (e.g., There is an A) or a negative form (e.g., There is no A). These two variations yield eight distinct argument types, which are listed in Table 2. We will have occasion to refer to these arguments individually, and we will do so by placing in angle brackets the second premise of the argument followed by, its conclusion. In this notation, (A,Not-C) will denote the syllogism If A, C; A; Therefore, not C (Syllogism 2 in Table 2),

3 CONDITIONAL REASONING 201 TABLE 2 VALIDITY JUDGMENTS FOR CONDITIONAL SYLLOGISMS Validity judgment Validity judgment Syllogism for MC for MB 1. If A, C Valid (Always true) ~ Valid (Always true) A (Follows) b (Follows) C (True) c (True) 2. If A, C Invalid (Never true) Invalid (Never true) A (Doesn't follow) (Doesn't follow) Not-C (False) (False) 3. If A, C Invalid (Sometimes true) Invalid (Never true) Not-A (Doesn't follow) (Doesn't follow) C (False) (False) 4. If A, C Invalid (Sometimes true) Valid (Always true) Not-A (Doesn't follow) (Follows) Not-C (False) (True) 5. If A, C Invalid (Sometimes true) Valid (Always true) C (Doesn't follow) (Follows) A (False) (True) 6. If A, C Invalid (Sometimes true) Invalid (Never true) C (Doesn't follow) (Doesn't follow) Not-A (False) (False) 7. If A, C Invalid (Never true) Invalid (Never true) Not-C (Doesn't follow) (Doesn't follow) A (False) (False) 8. If A, C Valid (Always true) Valid (Always true) Not-C (Follows) (Follows) Not-A (True) (True) a Predicted response for the alternatives Always, Sometimes, or Never True on the basis of the premises. b Predicted response for the alternatives Follows or Doesn't Follow from the premises. c Predicted response for the alternatives True or False on the basis of the premises. while (Not-C,A) will denote If A, C; Not C; Therefore, A (Syllogism 7). On the MC function, only two of the eight syllogisms are valid: those of Type (A,C) (called "modus ponens" arguments) and those of Type ~Not-C,Not-A) ("modus tollens" arguments), as shown in Table 2. But empirically, only a relatively small proportion of subjects identify just these syllogisms as valid. In the experiments cited above, this proportion varies from 7 to 44~ across different conditions. A somewhat more popular decision is to regard as valid, not only (A,C) and (Not-C,Not-A), but also Types (Not-A,Not- C) and (C,A) (sometimes called "denying the antecedent" and "affirming the consequent," respectively). As Taplin and Staudenmayer have pointed out, these results are consistent

4 202 MARCUS AND RIPS with the MB truth function since all four of the above syllogisms are valid in propositional logic if the first premise is translated as an MB (see Table 2). In the experiments reported by Taplin and Staudenmayer, somewhere between 19 and 75~ of subjects fell into this MB response category. The Conditional Evaluation Task The second technique of interest was developed by Johnson-Laird and Tagart (1969) to determine under what circumstances a conditional will be judged true. Subjects are given a sentence like the following one: (3) If there is an A on the left side of a card, there is a C on the right side. They are then asked to decide for each of a set of test cards, whether the card indicates that the conditional is true or false, or whether it is irrelevant to the truth of the conditional. The test cards themselves are constructed to vary the truth of the antecedent and consequent according to the four possibilities shown in Table 1. Thus, a subject might receive a test card containing an A and a C (antecedent and consequent true), a card with an A and a Y (antecedent true and consequent false), a card with an X and a C (antecedent false and consequent true), and a card with an X and a Y (antecedent and consequent false). If subjects in this experiment use the MC truth function to classify the cards, their responses should match the relevant entries in Table 1. That is, the card containing an A and a Y should indicate the falsity of the conditional, while the remaining cards imply its truth. Similarly, if subjects use the MB truth function, the A-C and X-Y cards entail the truth Of the conditional, while the A-Y and X-C cards prove it false. We can think of the Evaluation task as a kind of Conditional Syllogism in reverse. Where Syllogisms call for a deduction from a conditional to one of its component propositions, the Evaluation task requires an inference from the components to the conditional itself. This relationship can be brought out by comparing the Syllogism in (2) to that of (4): (4) There is an A on the left side of the card. There is a C on the rightside of the card. If there is an A on the left side of the card,, there is a C on the right side. This argument corresponds to the Evaluation problem in that subjects are asked to decide whether the disposition of symbols on the test card [described in the premises of (4)] makes the conditional true. Given this relationship between the Syllogism and Evaluation tasks, it is somewhat paradoxical that the results for the two kinds of experiment are quite different. But while the Syllogism responses are consistent with an MB or MC truth function, as just discussed, the Evaluation experiment supports yet a third hypothesis. Johnson-Laird and Tagart found that most of their subjects decided that a test card made the conditional true when it made the antecedent and consequent true, that a card made the conditional false when it made the antecedent true and the consequent false, and that a card was irrelevant when it made the antecedent false (no matter what the truth of the consequent). For these subjects, the conditional's truth was only a partial function of the truth of its constituents, a pattern that Kneale and Kneale (1962) and Wason and Johnson-Laird (1972) label the "Defective" truth function. We refer to it as the "Partial" truth function in Table 1. A Comparison of the Two Techniques To compare the outcomes of these experiments, we need to ask how subjects who respond according to the Partial function in the Evaluation task would perform on the Conditional Syllogisms. Logical considerations suggest that these subjects shoum pro-

5 CONDITIONAL REASONING 203 duce MC responses to the Syllogisms 1, and we would therefore expect the total proportion of MC and Partial responses in the Evaluation task to equal the proportion of MC responses to the Syllogisms. But a look at the data shows that this is not the case. For example, in comparing the results of Johnson-Laird and Tagart's (1969) Evaluation experiment with the Syllogism data of Taplin and Staudenmayer (1973, Experiment 2), we find that 83~ of the subjects in Johnson-Laird and Tagart's study produced MC or Partial responses while only 42~o of Taplin and Staudenmayer's produced this pattern. The Evaluation and Syllogism problems, like other reasoning tasks, involve at least two processing components: (a) comprehension of the critical statements in the problem (particularly, in this case, the conditional statement) and (b) inferences based on the interpreted statements. Because of the similarity between the conditionals used in the experiments just cited, there seems little reason to suspect that comprehension could account for differences in the results. But it is tempting to speculate that the separate inferential requirements of the tasks may have been responsible. Indeed, the relatively large number of subjects who provided statistically inconsistent responses to the Syllogisms implies that these problems called for fairly difficult logical decisions. (Taplin & Staudenmayer considered a response to a given syllogism "statistically inconsistent" if it occurred no more often than chance across replications of the problem.) In what follows, we report two studies that provide a more precise comparison of the Evaluation and Syllogism techniques. In the first of these, subjects solve Syllogism and Evaluation problems for the same conditional sentences. Experiment 2 extends these results using reaction time, as well as choice probabilities, as a dependent measure. Finally, we attempt to provide a quantitative model for the response distributions and reaction times to the Syllogisms. The model uses subjects' performance in the Evaluation task to predict the Syllogism responses, but it gives an account of factors that contribute to the extra difficulty of the latter problems. To the extent that this model is successful it provides us with an explanation of the difference between the studies cited above, yielding a more unified account of conditional reasoning. 1 It is easy to see that these subjects should respond valid to Syllogisms (A,C) and (Not-C,Not-A) and invalid to Syllogisms {A,Not-C) and {Not-C,A). The real question arises with respect to Syllogism (C,A) and (Not-A,Not-C) since these syllogisms differentiate the MC from the MB patterns. For {C,A) the likely choice is invalid. In order for the subject to decide otherwise, he would have to believe not only that C's appear whenever A's do, but also that A's appear whenever C's do. But such an interpretation would be inconsistent with our subject's performance on the Evaluation task. If A's and C's must always co-occur in order for the conditional to be true, then an instance for which the antecedent is false and the consequent true should disconfirm the conditional rather than being irrelevant to it. For this reason, the subject should respond Invalid to (C,A) and, by the same argument, Invalid to (Not-A,Not-C) as well. It can also be shown that the subject should respond Invalid to (Not-A,C) and (C,Not-A) to keep from contradicting his Evaluation judgments. The resulting pattern of responses is precisely the MC pattern in Table 2. EXPERIMENT 1 Since our motive was to compare the Conditional Evaluation and Conditional Syllogism tasks, we presented both problems to each of our subjects. However, in addition to this basic task variation, the experiment included two other factors to provide some generality for the results. One of these concerned the propositional content of the conditional and the other the response format for the Syllogisms. Our use of varied content was prompted by earlier research suggesting that the subject matter of conditionals alters the way they are comprehended (Legrenzi, 1970; Staudenmayer, 1975). For example, Legrenzi

6 204 MARCUS AND RIPS presented his subjects with a mechanical device in which a ball-bearing could roll through one of two channels, thereby lighting either a red or a green lamp. Subjects were then asked to judge whether test instances (events like the ball rolling left and the red lamp being lit) were compatible with the rule If the ball rolls to the left, the green lamp is lit. By contrast with Johnson-Laird and Tagart's subjects, most of whom used the Partial truth function, most of Legrenzi's used the MB pattern shown in Table 1. (For discussion of this result see Marshall, 1978; Rips & Marcus, 1977; Wason & Johnson-Laird, 1972.) We therefore selected conditionals that had been shown in previous research to affect the proportion of MB versus MC or Partial evaluations. One of these conditionals was similar to the card problem of Johnson-Laird and Tagart (If there's a B on the left side of the card, then there's a I on the right), and a second to Legrenzi's machine (If the ball rolls left, the red light flashes). The third conditional was taken from an earlier study of our own (Rips & Marcus, 1977) and concerned descriptions of tropical fish (If the fish is red, then it is striped). This context had been found to yield a slightly larger percentage of MC responses than either the card or machine stimuli. The second variation in this experiment concerns the response format for the Syllogisms and was included on the basis of evidence that this factor also affects the number of MC and MB responses (Taplin & Staudenmayer, 1973). For two-choice situations, where subjects are asked if the conclusion "followed" or "didn't follow" (Taplin, 1971 ) or if the conclusion was "true" or "false" on the basis of the premises (Taplin & Staudenmayer, 1973, Experiment 1), the MB pattern was dominant. However, with three choices, where subjects judged if the conclusion was "always true (false)," "sometimes, but not always true (false)," or "never true (false)," responses were mostly MC (Taplin & Staudenmayer, 1973, Experiment 2). To explore this effect, we divided our subjects into three groups, with each group assigned to one of the above formats. Method At the beginning of the testing session, subjects were told that the goal of the experiment was to discover how people reasoned about certain problems. One of the conditionals was then read, and the subjects were asked to evaluate a list of test instances with respect to it. When subjects had finished, they were next presented with the eight syllogisms corresponding to the same conditional and were asked for their judgments of the validity of these arguments. This procedure for the Evaluations and Syllogisms was then repeated for the two remaining conditionals. Thus, each subject provided both evaluation and validity judgments for all three conditionals. Although the Evaluation task always preceded the Syllogisms within a given problem pair, all six possible presentation orders of the pairs were employed, with an equal number of subjects assigned to each order. Evaluation task. To provide a context for the problem, a short description was first read to the subject. In one case, subjects were told that the discussion concerned cards with letters (A, B, or C) on one side and numbers (1, 2, or 3) on the other. In a second situation, they were told that the problem was about a machine in which a ball could roll through one of three holes (to the right, left, or straight ahead), and light one of three lamps (yellow, blue, or red). And in the third setting, they were told that the problem centered on a group of tropical fish that could be of one of three colors (red, blue, or green) and could have one of three kinds of markings (striped, spotted, or plain). During this part of the procedure, to make the experiment comparable to those of Johnson-Laird and Tagart (1969) and Legrenzi (1970), materials were presented to illustrate the appropriate setting: either a deck of cards with letters and numbers, a model machine, or a set of pictures of fish.

7 CONDITIONAL REASONING 205 For each conditional, subjects evaluated nine test instances, formed by combining the values of the two attributes described above. The test instances were read to the subjects one at a time in random order, and subjects wrote their responses (Consistent or Inconsistent) in an answer booklet. The task was self-paced, and no indication was given to subjects about the accuracy of their answers. In the Evaluation task, we asked subjects whether test instances were "consistent" or "inconsistent" with the conditional (similar to Legrenzi's "compatible" or "incompatible"), rather than asking whether the instance made the conditional true (as did Johnson-Laird & Tagart). These two types of judgments are not equivalent since one proposition may be logically consistent with another (i.e., noncontradictory) without making it true. For example, the proposition Chicago is in Illinois is consistent with, but does not entail, an unrelated fact like Apples are red. However, in the Evaluation paradigm, test instances that make the conditional true will also be consistent with it, and instances that make the conditional false will be inconsistent. The advantage of consistency judgments is that it avoids the need for a third response category, Irrelevant. For although one proposition may make another true or false or be irrelevant to its truth, two propositions must be either consistent or inconsistent. The MC and Partial functions in Table 1 will appear as a single pattern for the consistency judgments (i.e., Inconsistent when the antecedent is true and the consequent false, and Consistent in all other contingencies), a pattern that we will also refer to as an MC response below. This change simplifies a comparison of the results across tasks. Syllogism task. For each conditional, eight syllogisms were composed, corresponding to the eight forms in Table 2. The following examples illustrate one of the forms (Type (A,C)) for the three conditionals: (5) If there's a B on the left side of the card, then there's a 1 on the right side. There's a B on the left side. There's a 1 on the right side. (6) If the ball rolls left, then the red light flashes. The ball rolled left. The red light flashed. (7) If the fish is red, then it is striped. The fish is red. The fish is striped. A printed version of each syllogism was given to subjects which they followed while the experimenter read the problem aloud. The syllogisms were presented one at a time in random order; and subjects responded to each by recording their validity judgment in the answer booklet. As we have noted, the form of the judgment depended on the group to which the subject had been assigned. While all subjects were told to regard the first two sentences as true, Group 1 was asked to decide if the third statement necessarily followed from the first two, Group 2 was asked if the third statement was true or false on the basis of the first two, and Group 3 was asked if the third statement was always true, sometimes true, or never true on the basis of the first two. The test was again self-paced, with no feedback being given regarding the answers. The order in which the conditionals were presented was balanced across response groups. Eighteen subjects served in each of the three groups, all of them having been recruited through an advertisement in the University of Chicago student newspaper. The subjects were either students (both undergraduates and graduates) or nonstudents of comparable age. Mathematics majors and those who had taken a course in formal logic were excluded from participation. Subjects were paid $2 for a

8 206 MARCUS AND RIPS session lasting 30 to 45 minutes and were tested individually or in groups of two or three. Results and Discussion Comparison of Evaluation and Syllogism performance. Even though the Syllogism and Evaluation problems were used in a withinsubjects design, our study replicated earlier ones in showing large differences in the proportion of MC and MB responses across tasks. A subject in this experiment contributed three sets of nine Evaluation judgments, one set each for the machine, card, and fish problems. In our analysis, we take each set to constitute a single response unit, classifying these responses as MC or MB according to the scheme in Table 1. The resulting distribution of responses is shown in Table 3. Considering all of the Evaluations, we find that 69.8% of the responses were MC and 7.4% MB. A further 4.9% of the subjects adopted a response pattern somewhat similar to MB, which resulted from an attempt to pair values of the attributes mentioned in the description of the problem setting. For instance, given the conditional that described the machine (If the ball rolls left, the red light flashes), these subjects paired the left channel with the red lamp, the right channel with the blue lamp, and the center channel with the yellow lamp (or, alternatively, left with red, right with yellow, and center with blue). These three combinations were judged to be consistent with the conditional and all others inconsistent. By contrast, in a true MB response all five of the combinations left-red, right-blue, right-yellow, center-blue, and center-yellow are consistent. However, we note that the tendency to pair values, which produced the deviant responses, would have resulted in a true MB had we used two, rather than three, values of each attribute (e.g., two channels and two lamps). In addition, subjects using this response pattern should produce MB responses in the Syllogism task (Rips & Marcus, 1977). For these reasons, we grouped such TABLE 3 PERCENTAGE RESPONSES IN EACH TRUTH FUNCTION CATEGORY Evaluation response Syllogism response Propositional Logically content MC MB Other MC MB contradictory Other Experiment 1 Fish (41.7)" (11.1) (44.4) (2.8) Cards (30.6) (13.9) (52.8) (2.7) Machine (25.0) (22.2) (52.8) (0.0) All contexts (32.4) (15.7) (50.0) (1.9) Experiment 2 Fish Cards , Machine All contexts a Responses for Control experiment in parentheses.

9 CONDITIONAL REASONING 207 responses with MBs in Table 3 and in the analyses to be reported. A small number of the remaining responses are listed as "other" in Table 3, composed mainly of conjunctive truth functions (i.e., cases in which a Consistent response was made when both A and C were true and an Inconsistent response otherwise). For the Syllogisms, subjects completed three sets of problems, each set being composed of the eight types of arguments shown in Table 2. We again took each set as a response unit and identified them as MC or MB by means of the Table 2 patterns. Here, 32.1~ of the responses were MC and 12.3~ MB, as shown in Table 3. The majority of responses (52.5~o), however, were logically contradictory in the sense that no single truth function could account for them (see Staudenmayer, 1975, for an exposition of the way such contradictory responses are identified). Again, most of the residual ("other") responses can be classified as Conjunctions. We can get some idea of these contradictions by considering responses to the individual Syllogisms. These data are displayed in Table 4, where we have calculated response proportions for subjects receiving the threechoice format (Always, Sometimes, or Never True) in the first three columns and for subjects receiving the two-choice options (True or False; Follows or Doesn't Follow) in the last two columns. These proportions represent only those trials (82.1~) on which the subject had made an MC or MB response to the immediately preceding Evaluation problem. Thus, we can be fairly sure that thes~ distributions are not the result of deviant interpretations of the conditional itself. (Entries in parentheses are predictions from a model to be discussed below.) According to both the MC and MB patterns, Syllogism Types (A,C) and (Not-C,Not-A) are valid while Syllogisms (A,Not-C) and (Not-C,A) are invalid. In fact, while performance is nearly perfect for (A,C) and (A,Not-C), 13.6~ of the responses to (Not-C,A) and 39.6~o of responses to (Not-C,Not-A) are in error. Part of the difference in error rate between the two valid syllogisms is likely to be due to negation since Syllogism (A,C) contains no negatives while Syllogism (Not-C,Not-A) contains two. Numerous studies have documented the pi'oblems negative propositions create for carrying out commands or answering questions (e.g., Carpenter & Just, 1975; Clark & Chase, 1972; Wason & Johnson-Laird, 1972), and it is therefore plausible to suppose that negatives interfere with solving syllogisms as well. Negation, however, cannot explain the difference between (A,Not-C) and (Not-C,A), (F(1,53) = 21.12, p <.01), since they each contain a single negative. A complete account of these data will therefore have to incorporate other factors, and we will return to the problem of identifying them in the General Discussion. Replication of the Syllogism task. In this experiment, we presented the Syllogisms after the corresponding Evaluations. This was done to promote a better understanding of the problems since all possible values of the problem dimensions are used in the Evaluation task (e.g., all of the numbers and letters in the card problems) while only a single pair of values is mentioned in the Syllogisms. However, it can be argued that subjects' strategy in the Syllogisms was altered by their experience with the Evaluations. For this reason, we replicated the syllogism portion of this experiment, omitting the Evaluation pretest. The procedure in this replication was similar to that described above, except that the Always, Sometimes, and Never format was used with a single group of 36 subjects. Subjects' validity judgments, classified as MC or MB, are shown in parentheses in Table 3, and the results parallel those of Experiment 1. Over all contexts, 32.4~ of responses were MC and 15.7~ MB, very close to the previous values of 32.1 and 12.3~ for MC and MB responses in the first experiment. Similarly,

10 208 MARCUS AND RIPS TABLE 4 SYLLOGISM RESPONSE PROPORTIONS FOR EXPERIMENT 1 Three-choice response format a Two-choice response format b Always Sometimes Syllogism true true Doesn't Never Follows/ follow/ true True False 1. If A, C A (.91) c ' (.09) (.00) (.95) (.05) 2. If A, C A (.00) (.09).(.91) (.05) (.95) Not-C 3. If A, C Not-A (.04) (.73) (.23) (.19) (.81) 4. If A, C Not-A (.19) (.73) (.08) (.28) (.72) Not'C 5. If A, C i C {:22) (.78) (.00) (.31) (.69) 6. If A, C ,18.82 C (.04) (.73) (.23) (.19) (.81) Not-A 7. If A, C Not-C (.00) (.13) (.87) (.05) (.95) A 8. If A, C Not-C (:51) (.41) (.08) (.66) (.34) Not-A anz-44. b n= 901 c Prediotions of model in parentheses. 50.0~o of the responses were contradictory, replicating the 52.5~o found earlier. A further indication of the reliability of these data is a correlation of.98 with the three-choice responses in Table 4. Effects of content and response format. As can be seen in Table 3, the propositional content of the conditional influenced the relative proportion of MC and MB responses in both tasks. With respect to the Evaluations, the machine problems produced approximately equal numbers of MC and MB responses (40.7 and 35.2~o, respectively). But for the remaining conditionals, MC responses

11 CONDITIONAL REASONING 209 predominated, the proportions being 81.5~o MC and 0.0% MB for the card problems and 87.0~ MC and 1.8% MB for the fish problems. The Syllogisms, too, showed an increase in MC responses and a decrease in MB responses from the machine to the fish and card problems. Here MC is equal to MB for the machine Syllogisms (27.8~ in both cases), but MC responses outnumber MB on the cards (29.6 vs 7.4~) and on the fish problems (38.9 vs 1.8%). Since we are primarily interested in the proportion of MC versus MB responses, MCs in both tasks were given a score of + 1, MB responses -1, and all remaining responses 0. In terms of these scores, the effect of propositional content was a reliable one, (F(2,90) = 35.31, p <.01), as was the interaction between content and task (F(2,90)=11.76, p <.01). Table 3 shows that this interaction is principally due to the large number of contradictory responses for the Syllogisms. By contrast with the large effects of content, our results provide little evidence for an effect of three-choice versus two-choice response format. Given the three-choice situation, 35.2~ of Syllogism responses were MC, and while these proportions decreased as expected to 22.2~o for the True/False format, they increased slightly to 38.9~ for Follows/Doesn't Follow. MB responses comprised 11.1~o of judgments for the Always/Sometimes/Never choice, increasing to only 13.0~o for both the True/False and the Follows/Doesn't Follow decisions. These differences were far from significant (F < 1, using the same scoring as in the preceding paragraph), and they contrast with the results of Taplin and Staudenmayer (1973), who found that the three-choice situation increased MC responses at the expense of MB. Possible reasons for this difference are taken up in the following section. Comparisons with earlier studies. Our subjects' evaluations of the card problem are in good agreement with those of Johnson-Laird and Tagart (t969). As we have noted, MC and Partial truth functions in the earlier study should correspond to MC responses in this one because of our change to consistency judgments. In fact, 83.3~o of Johnson-Laird and Tagart's evaluations were of either the MC or Partial type, closely matching the 81.5~o MC responses just reported. There were few, if any, MB responses in the previous study and none in our own card problems. But this congruence between experiments breaks down when we compare the Evaluation data for the machine problem to those of Legrenzi (1970); for although Legrenzi obtained 73.3~ MB responses, we found only 35.2~. This gap is narrowed somewhat if we consider the data from those subjects in our experiment who received the machine problem at the very beginning of the experimental session, that is, before the Evaluations based on the cards or fish. But still only 50.0% of the responses were MB. A procedural difference between Legrenzi's experiment and our own may account for these divergent results. In the previous study, Legrenzi appears to have demonstrated the operation of the machine to his subjects before they performed the task, and this demonstration may have convinced them that each channel was connected to a single lamp (e.g., the left channel to the green lamp and the right channel to the red one). When this kind of explicit information is provided, the number Of MB responses does indeed increase (Rips & Marcus, 1977). However, in the present study no such information was available to subjects. 2 With respect to the Syllogisms, we have already mentioned that, unlike Taplin and Staudenmayer (1973), we obtained no effect of 2 Another salient difference between Legrenzi's experi~ ment and our own is that the machine used in the previous experiment contained two channels and two lamps, while in the present study there were three of each. One possibility, therefore, is that this change also contributed to the difference in the proportion of MB responses. However, in an earlier experiment (Rips & Marcus, 1977), we obtained no support for this hypothesis when the number of channels and lamps were varied explicitly.

12 210 MARCUS AND RIPS response format. We note that the major difference between these results and our own is the proportion of MB responses in the two-choice setting. For while Taplin and Staudenmayer found 75.0~o MB responses, we found but 13.0~o in the comparable condition. This difference is also reflected in a much higher incidence of contradictory responses in our experiment, 50.0~o, compared to Taplin and Staudenmayer's 15.3~o for the same condition. Several aspects of our experiment may have been responsible for this divergence, including: (a) interference produced by the other conditionals (i.e., by the machine and fish problems); (b) the smaller number of Syllogism trials given to subjects; or (c) different methods of aggregating responses. Of these possibilities, the first seems somewhat unlikely. When we examine those sessions in which the card problems were presented before the remaining items, we find approximately the same results as in the combined data. Interference from the machine and fish conditionals can therefore be dismissed. Factors (b) and (c), however, are more plausible. Subjects in Taplin and Staudenmayer's experiment received 12 replications of each of the Syllogism types, and this extra practice may have led to more stable response strategies. With respect to factor (c), subjects whose answers were statistically inconsistent across replications for some of the eight Syllogism types were analyzed solely on the basis of the remaining Syllogisms. In our own study, subjects saw the eight Syllogisms for a given conditional only once, and all eight of these responses were used in deciding whether the sequence followed the MC or MB pattern. We can evaluate this difference in scoring methods in Experiment 2, since in that study we too repeated syllogism trials. EXPERIMENT 2 TO supplement our knowledge of the Syllogism task, we recorded reaction times (RTs), as well as response frequencies in this experiment. The study was tailored along the lines of Experiment 1, including the use of the same three conditional sentences, presented in a sequence of alternating Evaluation and Syllogism problems. But this time, the Syllogisms were presented one at a time in a tachistoscope in order to monitor subjects' RTs. In addition, to obtain stable latencies, we repeated the eight Syllogisms in six successive (independently randomized) blocks of trials. Method For the Syllogism problems, the subjects' task was to read the argument, to decide if the third statement necessarily followed from the first two, and to indicate their response by pressing a button labeled "yes" if it followed or a button labeled "no" if it didn't. To begin a trial, a subject pushed the center button of a three-button response panel with his or her right forefinger, and for a 2-second interval following this button press, a fixation point was displayed in one field of the tachistoscope. Immediately after this interval, a syllogism appeared with its first premise beginning at the place where the fixation point had been. The subject registered his response to the syllogism by pressing the appropriately labeled button with his right forefinger. The subject had been instructed to make his button press as fast as possible without making any mistakes. Half of the subjects indicated a Yes response by pressing a button on the right of the response panel and indicated a No response by pressing a button on the left. For the remaining subjects, the Yes button was on the left and the No button on the right. This second button press terminated the stimulus display and stopped a clock that had been activated when the syllogism appeared. The interval between trials lasted about 10 seconds, during which time the

13 CONDITIONAL REASONING 21 1 experimenter recorded the subject's decision and his latency. The subject was informed of his reaction time after each trial but was given no feedback concerning the accuracy of his response. Subjects were acquainted with the reaction time task in a series of 16 practice trials, presented at the very beginning of the session. These trials involved three-statement arguments, constructed along the lines of the Conditional Syllogisms, but using the connective or rather than if... then. The content of these practice arguments did not overlap that of the experimental trials, but the procedure on practice trials was otherwise identical to test trials. The individual syllogisms were typed in lower-case Orator capitals on 6 x 9 in. white cards. The three sentences composing the problem were typed one below the other and were left-adjusted. However, because of the length of the conditional sentence, it was split into two lines with the break occurring between clauses (e.g., If the ball rolls left,/then the red light flashes). The second line of the conditional appeared 6ram (.37 of visual angle) below the first and was indented 8 mm (.50 ). The second premise was typed 14ram (.88 ) below the bottom line of the conditional, and the conclusion a further 14 mm below the second.premise. Each line measured 3 mm (. 19 ) vertically and varied from 40 mm (2.50 ) to 95ram (5.95 ) horizontally. As in the first experiment, the cards, machine, and fish problems were presented in all six possible orders, with 4 subjects assigned to each order. The 24 subjects who served in this experiment were right-handed, had not participated in Experiment 1, were not mathematics majors, and had never taken a course in logic. They were paid $3 for a single session lasting approximately 1.5 hours. Results and Discussion The Evaluation responses are displayed in Table 3, where they can be compared to the very similar results from Experiment 1. On the average, 75.0~o of them can be classified as MC and 11.1~o as MB, comparable to the previous values of 69.8~ MC and 12.3~ MB. Moreover, the propositional content of the conditional affects the MC MB difference in the way we have come to expect. That is, for the machine problems, this difference is comparatively small (50.0~ MC vs 29.2~ MB), increasing for the card problems (83.3~ MC vs 4.2~ MB) and for the fish problems (91.7~o MC vs 0.0% MB). This effect is clearly a reliable one, F(2, 36)= 12.42, p <.01. The Syllogism data present a more interesting contrast with those of the previous experiment. While MB responses are about equally frequent in the two studies (12.3~o in Experiment 1 and 17.2~o in this experiment), MC responses show a decrease (from 32.l to 16.7~). This unusual behavior of the MCs appears also in a comparison of propositional content. Although MB responses decrease as expected from the machine to the cards and fish problems, MC responses stay relatively constant instead of increasing as they had in Experiment 1. It is natural to assume that the smaller number of MC responses was due to the reaction-time conditions under which they were made. Successful MC responses appear to require more processing time than MB responses in this experiment (2735 vs 2587 milliseconds), although this difference was only marginally significant, F(1, 142)= 3.46,.05 <p<.10. Subjects may prefer to produce a fast but logically contradictory response rather than take the extra time needed for a correct MC. In the above analysis we have considered each block of eight syllogisms as constituting a single response unit. However, it is also possible to analyze these problems in a second way that is similar to the scoring system used by Taplin and Staudenmayer (1973). Recall that these authors eliminated any Syllogism type for which a subject gave a statistically inconsistent response, classifying the subject on only the consistent types. In the present experiment, consistency (at the.05 level by a

14 212 MARCUS AND RIPS binomial test) requires that the subject provide the same response on all six repetitions of a given type. If we discard inconsistent problems and reclassify the remaining ones according to the method of Taplin and Staudenmayer (1973, Table 2), we find little change from the Syllogism data recorded for this experiment in Table 3. For MB responses, these proportions are 41.7, 4.2, and 4.2~ for the machine, card, and fish problems, while MC responses appear in 12.5~o of all three problem types. Thus, the two methods of analyzing the data present nearly the same picture of the underlying response distributions. This result implies that the major difference between Taplin and Staudenmayer's data and our own--the larger proportion of MB responses in the previous study---cannot TABLE 5 RESPONSE PROPORTIONS AND MEAN RTs mr EXPERIMENT 2 a Response proportion Mean RT (msec) Syllogism Doesn't Doesn't Follows follow Follows follow If A, C A (0.98) b (0.02) (1896) (2089) C If A, C A (0.02) (0.98) (2037) (2230) Not-C 3. If A, C Not-A (0.11) (0.89) (2311) (2480) 4. If A, C Not-A (0.23) (0.77) (2452) (2630) Not-C If A, C C (0.39) (0.61) (2170) (2362) A IrA, C C N0trA If A, C Not-C A (0.11) (0.89) (2311) (2480) (0.03) (0.97) (2311) (2374) 8. If A, C Not-C (0.60) (0.40) (2452) (2630) Not-A a n = 248. b Predictions of model in parentheses.

15 CONDITIONAL REASONING 213 be blamed on different scoring methods. Furthermore, repeating the Syllogisms in the present experiment did nothing to increase the frequencies of MBs. So, of the potential explanations of this difference mentioned above, none seems to explain it adequately, and we must leave this problem open. Data for individual Syllogisms are displayed in Table 5 and are in good agreement with the results of Experiment 1. The correlation between response proportions in the two studies is.98. (To ensure stable latencies and response proportions, we have included in Table 5 data from only the last four of the six blocks of trials.) Once again, negatives interfere with Syllogism performance. Notice, in particular, that mean RTs across both True and False responses increase with the number of negatives contained in the problem, from 2071milliseconds for Syllogisms with no negatives to 2467 milliseconds for Syllogisms with one negative and to 2584milliseconds for those with two, F(2, 46)=19.04, p<.01. However, as in Experiment 1, negatives are not the only source of difficulty. For example, although Syllogisms (A,Not-C) and (Not- C,A) each contain a single negative, RTs are faster for the former (2177milliseconds) than for the latter (2483 milliseconds), F(1, 23) = 16.85, p <.01. These general features of the Syllogism data are incorporated in the model to which we now turn. GENERAL DISCUSSION A Model for Conditional Syllogisms In" Experiment 1 we found that 82.l~ of the Evaluation responses could be classified as either MC or MB while, by contrast, only 44.4~oOf the Syllogisms could be interpreted in either of these ways. Similarly, in Experiment 2, 86.1~o of Evaluations were either MC or MB, while only 33.8~o of the Syllogisms fell into these categories. Our review of earlier research on conditionals turned up roughly the same pattern. That is, in general, the Syllogism task produces too few MC or MB responses and too many logically contradictory responses in comparison with the Evaluation task. We have already suggested one possible reason for this difference--the presence of negatives in some of the Syllogism problems. To test this suggestion, as well as other possible factors, we develop a theory of Syllogistic reasoning within which they operate. Such a theory is embodied in the fourstage model outlined in Figure 1, which we call the Consistency model. We can think of it as a composite of two sets of assumptions: structural assumptions about the informationprocessing stages needed for correct reasoning, and error assumptions about how reasoning could go astray. In what follows we discuss both types of assumptions in turn. 3 Processin 9 stages. To understand the four stages of the model, it is useful to consider a sample syllogism for which all of the stages are relevant. We will take the modus tollens argument in (8) as our example (Syllogism Type (Not-C,Not-A)): (8) If there's a B on the left side of the card, then there's a 1 on the right side. There isn't a 1 on the right side. There isn't a B on the left side. Since the syllogism is a valid one on both the MC or MB response patterns, subjects who accept either of these interpretations and who reason correctly should respond that it is Always True (given a choice of Always, Sometimes, or Never True), that it is True on the basis of the premises (given the True-False choice), or that it Follows from the premises (given a choice of Follows or Doesn't Follow). In the following description we focus on the Always, Sometimes, or Never response format 3 The model presented here resembles one that we have suggested earlier (Rips & Marcus, 1977). The major differences concern the first-stage encoding process and the reaction time assumptions.

16 214 MARCUS AND RIPS yes ; conclusion = Q I Encode premises and conclusion Does second premise I equal P? I n Is conclusion consistent I with first and second prem ses? l yes yes; conclusion = not Q no J2e.po.,--'', no Is negation of conclusion( with first I I and second premises? I FI6. 1. A four-stage model for verification of conditional syllogisms (see text for exposition). (see also Figure 1). However, the description applies equally to the binary choices with a simple modification: The Sometimes and Never categories must be merged into a single response corresponding to False or to Doesn't Follow. Always True will correspond to the True and Follows responses. The first stage of the model comprises the preliminary encoding of the syllogism. Thus, in this step subjects must choose an interpretation for the conditional and must provide a representation for the second premise and conclusion. The result of this process (especially, how the conditional is comprehended) critically affects the response to the syllogism. We will assume that the interpretation of the conditional will functionally resemble the MC or MB rules shown in Table 1, and we will refer to these as MC or MB interpretations. However, we again caution against the idea that the mental representations should be identified with the corresponding connectives in propositional logic, an issue to which we return in the final section of this discussion. Following encoding, subjects execute Stage 2, comparing the second premise to the antecedent and the conclusion to the consequent. If both comparisons result in a match, the subject makes an immediate Always True response. If the second premise alone matches, the subject makes a Never True response. A complete mismatch in this stage sends the subject to Stage 3, as in the sample problem. This stage was designed to account for the ease with which subjects can process Syllogisms of Types (A,C) and {A,Not-C) [such as

17 CONDITIONAL REASONING 215 example (5) Given a conditional like If there's a B on th'e left side, there's a 1 on the right side, together with the premise that There's a B on the left side, it seems to follow immediately that There's a 1 on the right side. Syllogisms of Type (A,C) (modus ponens) seem to be among the most common uses of conditionals in informal argtunentation (see Grice, 1967), and it is not unreasonable to think that a separate strategy, like that of Stage 2, would be reserved for it. The third stage of the model provides a decision regarding the consistency of the conclusion and the two premises. In our example, Stage 3 would determine that the second premise and conclusion could be true at the same time as the first premise; that is, that a card containing neither a B nor a 1 is compatible with the conditional rule If B, then 1. A decision that the three propositions are consistent, as in this case, leads to further processing in Stage 4, while inconsistent propositions (as in (Not-C,A) Syllogisms) produce a Never True response. Note that this stage is not sufficient to produce an Always True answer to the problems, since the conclusion of a Syllogism can pass the consistency test without being Always True. However, given our other assumptions, this stage is required for the three-choice situation as a way of distinguishing Syllogisms that are Never True from those that are Sometimes or Always True. Moreover, Wason and Johnson- Laird (1972) have found in other reasoning tasks that subjects often verify a proposition needlessly when falsification is required to test its validity. Stage 3 could be said to provide a type of verification for the Syllogisms, while Stage 4 provides the analogous falsification step. In the final stage, the conclusion of the Syllogism is negated, and this negative conclusion is examined for consistency with the premises, as in Stage 3. In the case of (8), negating the conclusion produces a doubly negated proposition such as It's not true that there isn't a B on the left side, and this proposition, or its affirmative counterpart (There's a B on the left side), can be compared to the premises. Here, the second premise and negated conclusion describe a card containing a B but no 1, which is clearly inconsistent with the conditional. As shown in Figure 1, this decision leads to the Always True response, the correct answer for Syllogisms of this type. If the negated conclusion had instead been consistent with the premises, the appropriate response would have been Sometimes True. Stage 4 provides the crucial step in the validity judgments for most of the Syllogisms. Indeed, validity of any argument can be defined as inconsistency of the negation of the conclusion and the premises (see, e.g., Jeffrey, 1967). We note that the last two stages have been described quite globally, rather than in terms of "elementary information processes (e.g., retrieval and comparison operations). A more detailed account of these stages can be given (Rips & Marcus, 1977), but such an analysis is not necessary for our current purpose of predicting the Syllogism data. Possibilities for errors. It is easy to verify that-the model of Figure 1 classifies syllogisms in just the same way as Table 2. In one sense, then, the model represents an ideal subject, one who makes no inference mistakes in dealing with syllogisms (i.e., no errors in Stages 2-4). However, our comparison of the Evaluation and Syllogism tasks has suggested that inference errors are common, and the model must provide some account of them. There are several potential sources of error that may be useful in this respect. First, subjects may terminate prematurely before they have completed all the stages needed for a correct validity judgment. This could presumably happen after any of the stages except the last, and we will assume (on the basis of preliminary model fitting) that in such a contingency subjects guess at the answer with equal probability for the two-choice formats and respond Sometimes True (as a type of Don't Know response) in the three-choice

18 216 MARCUS AND RIPS situation. To incorporate these assumptions into the model, we need three probability estimates, corresponding to early termination after Stages 1, 2, and 3, and we label these parameters Pt, P.2, and PB, respectively. Second, negative s included in the Syllogisms may cause errors in ithe consistency judgments of Stages 3 and.4 (evaluating the propositions as consistent when they are inconsistent, or the reverse error). We would expect the probability of such errors to increase with the total number of negatives, although it is unclear what the exact form of this function should be. In practice, we have found that a negative exponential function provides reasonable fits, and so we will take. the error probability to be (9) P(incorrect decision in Stages 3 or 4) = 1 - exp (-nq), where n is the total number of negatives and q is a parameter of the model. Note that negating the conclusion in Stage 4 increases the negatives by one, and this means that the probability of an error in Stage 4 will be higher than in Stage 3 for the same syllogism. Finally, the antecedent of the conditional appears on some trials as the second premise and on others as the conclusion of the syllogism. Having the antecedent in the second premise (and the consequent in the conclusion) seems easy to deal with since the order of these propositions matches that of the conditional itself. However, Syllogisms in which the propositions appear in the opposite order (Types (C,A), (C,Not-A), (Not-C,A), and (Not-C,Not-A)) may prove confusing, leading subjects to reverse the logical role of the second premise and conclusion (Marshall, 1978). We will let the parameter r stand for the probability of such a reversal. As an example of the way these probabilities combine, we can again consider the syllogism in (8). Let us suppose that in Stage 1 a given subject adopts the MC interpretation of the conditional premise. He must then execute Stage 2 (with probability 1-pl), where he finds that the second premise does not match the antecedent. He should then proceed to Stage 3 (with probability l-p2) and determine the consistency of the conclusion with the premises. The probability of his doing this correctly, despite the two negatives in the problem, is exp(-2q). In the last step, our subject must execute Stage 4 (which he does with probability 1-p3), correctly negate the conclusion (l-r), and determine that these propositions are inconsistent [exp(-3q)]. Thus, the probability of a correct Always True response for Problem (8), given the MC interpretation is: (10) P(Always True) =(1 - pl)(1 - p2)(1 - p3)(1 -r) x exp ( - 2q) exp ( - 3q) 3 =exp(-5q)(1-r) FI (1-pi). i=1 Of course, Always True responses could also be made through a combination of errors. The formula in (10) reflects only those responses generated by the full four-stage process. Similar predictions can be derived for the MB interpretation of the conditional and for the other Syllogism types. Fits of the Model to Experiment 1 To fit the model formally, we need some idea of the way subjects interpret the conditional in Stage 1. Here, the Evaluation data come in handy, since we can assume that the Evaluation responses reflect the way subjects understand the conditional premise. For example, we will assume that on those trials on which subjects provided an MC response for the Evaluations, they also adopted an MC interpretation of the conditional in the following Syllogisms. This assumption does not commit us to the view that the Evaluation data reflect only encoding of the conditional. The Evaluation task clearly involves reasoning and is subject to logical errors of its own. However, reasoning about an Evaluation problem is simpler than reasoning about a Syllogism because of the absence of factors

19 CONDITIONAL REASONING 217 like negation. Hence, a successful MC or MB response to the Evaluations is likely to provide a good clue to subjects' interpretation of the conditional. Reasonable fits of the model will help substantiate this assumption. On the other hand, mistakes in the Evaluation task tell us little about subjects' understanding of the conditional, and for this reason we limit ourselves to the 82.1~ of Syllogism trials for which a successful MC or MB response had been given in the preceding Evaluation problem. These are the data that we have examined in Table 4. Using these estimates for subjects' decisions in Stage 1, we fit our model to the Table 4 results using Chandler's (1969) STEPIT program to minimize deviations of predicted from observed proportions according to a leastsquares function. The resulting predicted values of our model are shown in parentheses in Table 4 and, in general, the fit of the model is quite good: the root-mean-square deviation (RMSD) is.025 with 19 df, and the percentage of variance accounted for is 99.5~o. Our estimates indicate reasonable values for each of the parameters described above. The probability of early termination after Stages 1, 2, and 3 is estimated by p i =.02, P2 =. 19, P3 =. 13, respectively. The negation parameter, q, equals.039, so the probability of an incorrect consistency judgment in Stages 3 or 4 ranges from.038 for situations involving one negative to. 110 for those involving three (this last case arises only in Stage 4 when the original problem contains two negatives). Finally, the probability of reversing the role of the second premise and conclusion is r=.12. (It is also possible to fit the model separately for subjects giving MC and MB Evaluations. However, the small number of MB responders leads to unstable parameter estimates in this situation.) One way to see how much these parameters buy us is to compare the above fits to those of a model that assumes perfect reasoning performance. That is, we can take the probability of the errors just discussed to be zero and refit the model using the same base rate estimates of MC and MB interpretations for Stage 1. Ignoring reasoning errors in this way produces an RMSD of.172 (df=24) and a percentage of variance accounted for of 88.4~o. To the extent that our model improves on this fit, we have justification for our idea that mistakes in reasoning can account for observed differences between the Evaluation and Syllogism experiments. Fits of the Model to Experiment 2 Qualitative predictions. The model we have developed makes a number of predictions about RTs for validity decisions. These predictions follow from the fact that some of the Syllogisms can be processed completely in two stages, others in three, while still others need all four stages. Before turning to a full statement of the model, we therefore consider the number of stages necessary for a correct MC or MB response. In what follows, we derive predictions separately for valid and invalid Syllogisms since the physical response (a Yes vs a No button press) is different in these two cases. First, suppose that a subject adopts the MC interpretation of the conditional in Stage 1. As can be seen in Table 2, only Syllogisms (A,C) and (Not-C,Not-A) are valid under this interpretation. Of these two, Syllogism (A,C) should take less time to confirm, since it can be judged valid after Stage 2. Syllogism (Not- C,Not-A), on the other hand, requires all four stages (a Syllogism of this type was used in introducing the four-stage model). We can check this prediction with the help of Table 6, which lists mean RTs for correct MC and MB responses. (For the moment, we ignore the possibility that some of the "correct" responses are the result of combinations of errors.) Comparing RTs for MC Syllogisms (A,C) and ~Not-C,Not-A), we find that the prediction is upheld: (A,C) ~ took 2152 milliseconds to confirm, while (Not-C,Not- A) took 3067 milliseconds, F(1,91)=19.03, p<.01.

20 218 MARCUS AND RIPS TABLE 6 MEAN RTs FOR SUCCESSFUL MC AND MB RESPONS~ ~ Syllogism MC MB 1. If A, C A. 2. If A, C A Not-C 3. If A, C Not-A C 4. If A, C Not-A Not-C 5. If A, C C A 6. If A, C C Not-A 7. If A, C Not-C A 8. If A, C Not-C. Not-A RT expressed in milliseconds. p <.01. Note, however, that Syllogism (C,A) presents a difficulty since RTs for this argument are slightly faster than for Syllogism (Not-C,A). This problem with (C,A) crops up again as we inspect the MB responses. Among the valid MB arguments, Syllogism (A,C) is again predicted to be faster than Syllogisms (Not-A,Not-C), (C,A), and (Not-C,Not-A), these latter problems requiring the full four-stage process. Table 6 shows RTs for Syllogism (A,C) to be 2301 milliseconds, and the mean RT for the three other Syllogisms is 2530 milliseconds. The difference between these values is only marginally significant, however, F(1,84)=2.88,.05<p<.10, and this appears to be largely due to fast RTs for (C,A). Thus, for both MC and MB responses, (C,A) is overpredicted by the model. A glance at the different Syllogism forms shows that Syllogisms (A,C) and (C,A) are the only arguments that contain no negative propositions. If negatives take extra time to encode (Clark & Chase, 1972), it would explain why (C,A) is faster than other arguments using the same number of stages. However, this does not yet explain why (C,A) takes no longer than (A,C) among the MB responses. This problem is examined further below.4 Turning to the invalid MB arguments, we find that the model delivers a more accurate account of the results. Since processing of (A,Not-C) is complete in two stages, it should For invalid MC responses, we can distinguish three groups of Syllogisms: Syllogism (A,Not-C), which should be processed in two stages; Syllogism (Not-C,A), which should take three stages; and the remaining invalid Syllogisms, which need all four stages. In accord with the prediction, RT increases from 2332milliseconds for (A,Not-C) to 2649 milliseconds for (Not-C,A) and increases further to a mean of 2919 milliseconds for the other Syllogism types. A contrast shows these differences to be significant, F(1,91)=12.52, 4 A related problem concerns the differences between valid and invalid MB Syllogisms. As we have noted, the valid Syllogisms (Not-A,Not-C), (C,A), and (Not- C,Not-A) each require four-stage processing, while the invalid Syllogisms (Not-A,C), (C,Not-A), and (Not- C,A) require only three stages. Yet the former average 2530 milliseconds and the latter 2810 milliseconds. Part of this difference may be due to difficulty in making a False response; however, it seems unlikely that the response could explain the entire effect. A comparison of Syllogism (A,C) with Syllogism (A,Not-C) yields a True-False difference of only 76milliseconds for the same MB responders. A possible explanation of the relatively fast RTs for the valid Syllogisms, based on a simple matching strategy, is discussed in Footnote 5.

21 CONDITIONAL REASONING 219 consume less time than Syllogisms (Not- A,C), (C,Not-A), or (Not-C,A), which need three stages. As predicted, RT for (A,Not-C) is 2377milliseconds, while the remaining invalid arguments average 2810milliseconds, F(1, 84)= 10.31, p<.01. Thus, except for the troublesome Syllogism (C,A) all four sets of RT predictions appear correct. However, confining ourselves to correct MC and MB responses means that we must ignore most of the data from Experiment 2--nearly twothirds of the Syllogism answers fell outside these two categories (see Table 3). To provide a more complete account, we must therefore return to the explicit statement of the model. Quantitative predictions. To deal with the RTs, our model must incorporate parameters representing processing times for the individual stages. The simplest way to proceed would be to allot a parameter to each of Stages 1-4, but there is a problem with this method when applied to the present data. In order to estimate separate parameters for Stages 1 and 2, we must rely on erroneous responses to Syllogisms (A,C) and (&Not-C) since these are the only explicit cases in which subjects execute the first stage without the second. However, Table 5 shows that the obtained RTs for incorrect responses to these syllogisms (2129 milliseconds) are somewhat slower than for correct responses (2042 milliseconds). This is contrary to what we would expect if errors were based on early termination after Stage 1. It is difficult to know how seriously this result should count against the model since the total number of error responses is very small (only 13 observations). But whatever their cause, we are left without a way to separate processing times for Stages i and 2. As a result, we fit a parameter tl+; for the combined duration of the first two stages. In addition, this component includes the time to perform any background operations, such as execution of the physical response, that are common to all trials. A second parameter, t a.4, was used to represent the duration of Stage 3 or Stage 4, since we.have assumed that.very similar processes are accomplished by these final stages. We also allowed an extra temporal increment, t n, for encoding each of the negatives in the problem statement, and a final parameter, tf, for the additional time needed to execute a False response beyond that required for a True response. As an example of our latency predictions, a correct True response for Syllogism (Not-C,Not-A), based on the entire four-stage process, should take tl t3,4 + 2t~ milliseconds. These four time parameters were combined with the probability parameters described above, and the entire set was fit simultaneously to the reaction times and response proportions in Table 5. For consistency, we merged our former Pl and P2 into a single parameter Pl+2 to parallel t1+2, so that for purposes of model fitting, we have collapsed Stages 1 and 2 into a single combined stage. To accommodate the two different kinds of dependent variables, we normalized both RTs and response proportions, fitting the model by minimizing deviations from these z-scores. As in our modeling of Experiment 1, we used the Evaluation task to estimate the proportion of MC versus MB encodings in Stage 1. The resulting parameter values were t 1 +2 = 1896, t3. 4= 143, t~= 141, tf= 193 milliseconds, P1+2=.04, p3=.00, q=.06, and r=.28. The near-zero values of the termination parameters, PI+Z and P3, reflect the accurate performance of our subjects on many of the more complex syllogisms. This in turn is probably a consequence of the practice our subjects received during Experiment 2. The predictions generated by our model are the parenthesized entries in Table 5. Although the fit is fairly good for the response proportions, difficulties arise from the RTs. The model accounts for 90.0~0 of the variance in the response proportions, but only 26.1~o of the variance in the latencies. Lack of fit is partly attributable to the small number of observations in some of the means (particularly, True RTs for Syllogisms (Not-A,C), (C,Not-A), and (Not-C,A)). The reliability

22 220 MARCUS AND RIPS of the RTs can be estimated at.568 (Winer, 1971, pp ), and correcting for attenuation increases the percentage of variance accounted for to 45.7~o. The most serious problems for the model are the deviations for True responses to Syllogism (C,A) and for both True and False responses to Syllogism (Not-C,Not-A), since in both cases mean RTs are based on substantial amounts of data. The first of them, the model's overprediction of True (C,A) responses, is already familiar from our inspection of correct MBs. As can be seen in Table 5, mean RT in this condition (1813 milliseconds) is faster than in any other, including that of Syllogism (A,C). This strongly suggests that MB responders (for whom True is the correct response for Syllogism (C,A)) have found a way to short-circuit the third stage of the Consistency model. Such a strategy is easily possible since, on the MB interpretation, the validity of a syllogism does not depend on the order of its second premise and conclusion. Because both (A,C) and (C,A) are valid, subjects may extend the second-stage process to both, responding True when the second premise and the conclusion are each of positive polarity, s The poor performance of our model for Syllogism (Not-C,Not-A) presents more of a puzzle. The main difficulty is that True responses for this syllogism take considerably 5 This idea could be extended to provide a general algorithm for MB responses. According to this procedure, a True response can be made whenever the second premise and the conclusion are both negative or both positive propositions, and a False response whenever one is positive and the other negative (Taplin, Note 2). If we assume that positive propositions take less time to encode than negative ones, this model can explain the data for True MB Syllogisms (Table 6). RTs for (A,C) and (C,A) are approximately equal (2301 and 2264milliseconds, respectively), and both are faster than those for (Not- A,Not-C) and (Not-C,Not-A) (2669 and 2657milliseconds). On the other hand, it is difficult to see how this theory can account for RTs to False syllogisms, which vary fi'om 2377 milliseconds for (A,Not-C) to 3005 milliseconds for (Not-C,A). longer than responses to other syllogism types that utilize the same number of stages. For example, the True (Not-C,Not-A) decision takes 242 milliseconds longer than False responses to Syllogism (Not-A,Not-C) (also correct for MC responders), despite the fact that both require four stages and that both contain two negatives. To compound this problem, RTs are relatively short (2245 milliseconds) for False decisions on (Not-C,Not- A). While this could be attributed to early termination, such a possibility is inconsistent with subjects' very accurate performance on other four-stage syllogisms. Again, there appears to be something unique about (Not- C,Not-A) that our model doesn't capture. Although we have no well-motivated explanation for these results, we note an analogy between validity decisions for this (Not- C,Not-A) argument and some results in the sentence verification literature. In such experiments, RTs for "true negative" propositions (e.g., Apples are not blue) are typically longer than for "false negatives" (e.g., Apples are not red). in the present study, Syllogism (Not- C,Not-A) can also be considered a kind of true negative in the sense that it requires a true response despite the presence of negative propositions in its second premise and conclusion. In the same way, Syllogism (Not-A, Not-C) is similar to a false negative for MC responders. In both cases, the negatives may bias subjects toward a false response, a bias which must then be reversed for Syllogism (Not-C,Not-A) at some cost in time. Such a mechanism could presumably be added to the Consistency model, though admittedly such an addition does not follow in any principled way from the assumptions that we have introduced earlier. Comparisons to other models. The only alternative model for RTs in these problems is the Transitive Chain theory proposed recently by Guyote and Sternberg (Note 1). This model is an attempt to explain reasoning in a variety of two-premise problems, and its representational and processing assumptions are corn-

23 CONDITIONAL REASONING 221 plex. However, with regard to Conditional Syllogisms, the model can be described in our own terminology by two assumptions: First, all subjects begin each problem with an MB interpretation of the conditional. With a given probability (a parameter of the model), they attempt a second solution using MC. Second, in both solution steps, they first try to deduce the conclusion using the rule of modus ponens (or, in the case of an MB interpretation, modus ponens and affirming the consequent). If this strategy fails, they then use modus tollens according to a second parameter. RTs are derived by estimating a constant encodingresponse time and additional times for the deduction steps. The duration of the deduction depends on whether modus ponens or modus tollens is employed, whether the deduction involves a negative, and whether the second premise contains the antecedent or the consequent. A direct comparison of the Consistency model and the Transitive Chain model is difficult in the context of Experiment 2. The Transitive Chain model predicts perfect performance on all but three of the Syllogisms, and this means that no RT estimates are available for the Doesn't Follow response to Syllogism (A,C) and the Follows response to (&Not-C), (Not-C,A), (C,Not-A), and (Not-C,A) since these responses are not supposed to occur. Our attempts to fit the model to the remaining RTs yield good fits but unstable parameter values (nine time parameters must be estimated for the 11 means). In short, the present data cannot decide between the two models, and the Transitive Chain theory remains an interesting alternative. The Meaning of If At the outset we mentioned that rival conclusions about /f's meaning have been drawn from the results of different experimental procedures. Having examined two of these procedures, the Evaluation and Syllogism tasks, we are now in a better position to make sense of this problem. Our evidence is consistent with the view that comprehension of the conditional premise is similar in the two tasks and that differences across tasks are attributable to processing limits. The model described in the previous sections is an attempt to make these limits explicit. In this respect, our point of view is similar to that of Johnson-Laird and Wason (1970), who use processing assumptions to explain differences between the Evaluations and a third type of conditional problem, the Selection task. However, this leaves open the question of how conditional sentences are understood in the first place. One obvious possibility is that the interpretation of conditionals is always equivalent to the MC connective, with observed variations being due to processing mistakes or to pragmatic factors that override the underlying MC meaning (Braine, 1978; Gels & Zwicky, 1971; Grice, 1967). But while there is ample evidence for reasoning errors and for the influence of pragmatics (Fillenbaum, 1978), there is remarkably little data to support MC as the basic interpretation. In addition, there are several uses of conditionals for which MC is not appropriate. For example, no truth functional connective can capture the sense of conditionals used to express logical entailments (e.g., the /f of mathematical theorems), since the truth of such statements is not established by the truth of its antecedent and consequent (Anderson & Belnap, 1975). Similarly, counterfactual conditionals have traditionally posed a problem for the truth-functional analysis since the antecedents of such statements are invariably false, giving us no way to distinguish true counterfactuals (e.g., If this water were heated w 212 degrees, it would boil) from false ones (e.g,, If this water were heated to 212 degrees, it would not boil). (See Goodman, 1965.) An equally extreme view of/fis that it serves as a logical wild card with no single meaning of its own, in some contexts equivalent to MC, in others to MB, and in still others to a partial,

24 222 MARCUS AND RIPS strict, or counterfactual conditional. While such a theory can account for the available data (particularly if we also allow for processing errors), it is obviously unparsimonious. Although if may be ambiguous, there are at least strong similarities among its meanings which such a theory cannot explain. In light of these problems, a more reasonable alternative would be to adopt a single representation for /f but to reject a strictly truth-functional approach. On this account we can take the truth of a conditional to depend, not on the current truth of the antecedent and consequent, but on their truth in relevant test situations (cf. Lewis, 1973; Stalnaker, 1968). To illustrate, we can again consider Legrenzi's example If the ball rolls to the left, the green lamp is lit. Translated as an MC, the conditional will be true whenever the ball doesn't roll left or whenever the green lamp is lit. For example, the conditional will be true of a device in which the green lamp is always tit independent of the ball's position. But such an interpretation seems far removed from the customary meaning of the sentence. Instead, the truth of the statement, like the truth of other hypotheses, depends on whether the antecedent and consequent are true or false under conditions that provide a fair test. Such a test would not be carried out in an environment where, for example, the ball's position or the lamp's illumination was fixed. As in any experiment the relevance of the test environment will depend on our current knowledge of the workings of the apparatus, on possible confounding factors, and so on. Our claim is that this everyday knowledge of what constitutes an unbiased test enters our comprehension of conditionals at the ground level and determines the situations in which we are willing to concede that the conditional is true. This theory can easily explain the effects of propositional content that we have observed, and together with explicit models of inference, it provides the basis for a more comprehensive theory of propositional reasoning. REFERENCES ANDERSON, A. R., & BELNAP, N. D., JR. Entailment: The logic cf relevance and necessity. Princeton, N.J.: Princeton Univ. Press, Vol. 1. BRA1NE, M. D. S. On the relation between the natural logic of reasoning and standard logic. Psychological Review, 1978, 85, CARPENTER, P. A., ~ JUST, M. A. Sentence comprehension: A psycholinguistic processing model of verification. Psychological Review, 1975, 82, CHANDLER, J. P. STEPIT; Finds local minima of a smooth function of several parameters. Behavioral Sciences, 1969, 14, CLARK, M. Ifs and hooks. Analysis, 1971, 32, CLARK, H. H., ~ CHASE, W. G. On the process of comparing sentences against pictures. Cognitive Psychology, 1972, 3, FILLENBAUM, S. How to do some things with /f. In J. W. Cotton & R. L. Klatzky (Eds.), Semantic.factors in cognition. Hitlsdale, N.J.: Lawrence Erlbaum Associates, GELS, M. L., & ZWICKY, A. M. On invited inferences. Linguistic Inquiry, 1971, 2, GOODMAN, N. Fact, fiction, and forecast. Indianapolis: Bobbs--Merrill, nd ed. GraCE, H. P. Logic and conversation. William James Lectures, Harvard University, JEFFREY, R. C. Formal logic." Bs scope and limits. New York: McGraw--Hill, JOHNSON-LAIRD, P. N., & TAGART, J. How implication is understood. American Journal of Psychology, 1969, 82, JOHNSON-LAIRD P. N., & WASON P. C. A theoretical account of insight into a reasoning task. Cognitive Psychology, 1970, 1, KNEALE, W., & KNEALE, M. The development of logic. London: Oxford Univ. Press (Clarendon), LEGRENZl, P. Relations between language and reasoning about deductive rules. In G. B. Flores D'Arcais & W. J. M. Lever (Eds,), Advances in psycholinguistics. Amsterdam: North-Holland, LEWIS, D. K. Counterfactuals. Cambridge, Mass.: Harvard Univ. Press, MARSHALL, C. g. Conditional reasoning." The semantics and logic of the natural conditional. Unpublished doctoral dissertation, Stanford Univ RiPs, L. J., & MARCUS, S. L. Suppositions and the analysis of conditional sentences. In M. A. Just & P. A, Carpenter (Eds.), Cognitive processes in comprehension. Hillsdale, N.J.: Lawrence Erlbaum Associates, STALNAKER, R. C. A theory of conditionals, In N. Rescher (Ed.), Studies in logical theory. Oxford: Blackwell, 1968.