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Developmental and Content Effects in Reasoning with Causal Conditionals
Journal of Experimental Child Psychology 81, 235–248 (2001) doi:10.1006/jecp.2001.2652, available online at http://www.idealibrary.com on
Developmental and Content Effects in Reasoning with Causal Conditionals Pierre Barrouillet Université de Bourgogne, Dijon, France
and Henry Markovits and Stéphane Quinn Université du Québec à Montréal, Montreal, Quebec, Canada Two predictions derived from Markovits and Barrouillet’s (2001) developmental model of conditional reasoning were tested in a study in which 72 twelve-year-olds, 80 fifteenyear-olds, and 104 adults received a paper-and-pencil test of conditional reasoning with causal premises (“if cause P then effect Q”). First, we predicted that conditional premises would induce more correct uncertainty responses to the Affirmation of the consequent and Denial of the antecedent forms when the antecedent term is weakly associated to the consequent than when the two are strongly associated and that this effect would decrease with age. Second, uncertainty responding to the Denial of the antecedent form (“P is not true”) should be easier when the formulation of the minor premise invites retrieval of alternate antecedents (“if something other than P is true”). The results were consistent with the hypotheses and indicate the importance of retrieval processes in understanding developmental patterns in conditional reasoning with familiar premises. © 2002 Elsevier Science (USA) Key Words: deductive reasoning; conditional; causality; cognitive development.
Conditional reasoning in adults as well as in children has been the subject of a great number of studies and theoretical proposals (Braine & O’Brien, 1991; Johnson-Laird, 1999; Johnson-Laird & Byrne, 1991, 2001; see Evans, Newstead, & Byrne, 1993, for a review). Conditional reasoning involves making inferences on the basis of a given “if P then Q” sentence (the conditional premise) and additional information (the minor premise) that can either affirm or negate the antecedent (i.e, the proposition P) or the consequent (i.e., the proposition Q). Among the resulting four classical forms of conditional syllogisms, both Modus ponens (P is true) and Modus tollens (Q is false) lead to certain conclusions that are “Q is true” and “P is false” respectively. For example, from the conditional The authors thank Laurence Arnoud for her collaboration in collecting data. Address correspondence and reprint requests to Pierre Barrouillet, Université de Bourgogne, LEAD-CNRS, Faculté des Sciences Gabriel, 6 Bld Gabriel, 21000 Dijon, France. Fax: 03 80 39 57 67. E-mail: [email protected]. 235 0022-0965/02 $35.00 © 2002 Elsevier Science (USA) All rights reserved
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premise “If a dog has fleas, then it will scratch constantly” and the minor premise “A dog has fleas,” we can conclude with certainty that it will scratch (Modus ponens), and given the minor premise “A dog does not scratch constantly” we can conclude with certainty that it does not have fleas (Modus tollens). Alternatively, neither the Affirmation of the consequent (“Q is true”) nor the Denial of the antecedent (“P is false”) leads to certain conclusions, though children and adults frequently conclude that “P is true” from the former and that “Q is false” from the latter. Indeed, from the information that a dog scratches constantly (Affirmation of the consequent), we cannot be certain that it has fleas because it can have some other parasite or a skin disease, and for the same reasons we cannot conclude that a dog will scratch from the fact that it does not have fleas (Denial of the antecedent). We refer to these as uncertain logical forms. Many recent studies have shown that the inferences that children and adults make from conditional premises vary greatly with the content of the premises, particularly in the cases of the Affirmation of the consequent and Denial of the antecedent. Although current theories of reasoning acknowledge the impact of content, most provide no precise framework for understanding how this factor affects inferential behavior. For example, Overton’s (1990) competence-performance model and Braine’s (1990; Braine & O’Brien, 1991, 1998) theory of natural deduction both allow for possible effects of content on reasoning, but neither provide a functional model that can explain and predict specific content effects. The present article focuses on a theory that attempts to account for both content and developmental effects in conditional reasoning and presents an experiment that tested two predictions concerning content effects in children’s and adults’ reasoning from causal conditional relations. Markovits and Barrouillet (2001) have recently proposed a developmental reformulation of the mental models theory of Johnson-Laird (1999; JohnsonLaird & Byrne, 1991). They assume that reasoning is a process that is related to the construction of mental models of the state of affairs that the premises describe. These models are held and processed in a limited-capacity working memory. They suppose that, initially, a major premise of the form “If P then Q” (e.g., “If a dog gets fleas, then it will scratch constantly”) is represented by an initial model of the following form: P→Q This model refers to a state of affairs in which the antecedent P and the consequent Q are true and the symbol (between them) indicates the form of the relation between P and Q. When an inference is made by adding a minor premise, other possible models are added to the initial model (referred to as “fleshing out”; Johnson-Laird & Byrne, 1991). Markovits (1993, 2000; Markovits & Vachon, 1990) has suggested that fleshing out could be the result of automatic activation and retrieval of knowledge from long-term memory when premises refer to concrete terms. In this case, the fleshing out process would depend on an individual’s knowledge about the terms involved in the premises.
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Specifically, there is now a great deal of evidence in both the developmental and adult literatures that performance on the Affirmation of the consequent and Denial of the antecedent is determined largely by whether the reasoner has access to potential alternative antecedents (Cummins, Lubarts, & Alksnis, 1991; Cummins, 1995; Klaczynski & Narasimham, 1998; Markovits, 1984; Markovits & Vachon, 1990; Markovits et al., 1996, 1998; Quinn & Markovits, 1998; Thompson, 1994). In terms of the above model, this can be explained quite directly. For example, consider Affirmation of the consequent, for which the minor premise is “Q is true.” The model supposes that if the minor premise and the major premise can cue a potential alternative antecedent (A) sufficiently strongly, then the reasoner will construct the following two models: P A
— —
Q Q
Each line refers to a distinct state of affairs. The first refers to a state of affairs in which the antecedent P and the consequent Q are true, and the second line refers to a state of affairs in which the consequent Q results from an alternative antecedent (A). For example, with a conditional premise such as “If a dog gets fleas, then it will scratch constantly,” the first model would represent a dog that has got fleas and that scratches, whereas the alternative antecedent to scratching could refer to a dog having a skin disease. With these two models, the conclusion that can be derived from “Q is true” (i.e., a dog scratches constantly) is that P (it has fleas) may be true or false, thus there is no certain conclusion. However, if an alternative antecedent is not strongly activated, then the reasoner will reason on the basis of the initial model and conclude that “P is true” (i.e., it has fleas), a commonly made biconditional error. A similar analysis can be made for the Denial of the antecedent. Thus, according to this analysis, reasoning on these two forms is affected by the relative accessibility of possible alternative antecedents, a factor that results in striking content effects, and the amount of cognitive resources available to activate stored knowledge and process the models constructed. This theory has predicted some counterintuitive effects about conditional reasoning. As we have seen, making the correct responses of uncertainty to Affirmation of the consequent and Denial of the antecedent depends on the retrieval of at least one alternative antecedent. One factor determining whether an alternative is in fact retrieved is the relative strength of association between the consequent term of the major premise (Q) and the most highly associated alternative antecedent available in memory. This basic analysis has been used to make the following prediction (Markovits, Fleury, Quinn, & Venet, 1998; Quinn & Markovits, 1998). We assume that when reasoners are given an uncertain logical form, e.g., “If P then Q and Q is true,” they will attempt to retrieve a counterexample, “If A then Q,” from long-term memory. If the consequent term is strongly related to a specific potential antecedent, which we refer to as a “strong antecedent,” then there will be a correspondingly strong probability that it will be activated during the search for counterexamples.
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Similarly, there are elements that are more weakly related to Q, one of which we refer to as a “weak antecedent.” Now, suppose that we compare performance on two similar problems. In one, we use the “strong antecedent” in the major premise, and in the other, we use a “weak antecedent.” On the one hand, reasoners who receive the second of these problems would be expected to have a high probability of activating the “strong antecedent” as an alternative antecedent. On the other hand, reasoners who receive the first of these problems would not be able to use the same “strong antecedent” because this is part of the premise and would have to be able to activate other, less strongly associated, elements in order to generate an alternative antecedent. Thus, we predict fewer correct uncertainty responses to both Affirmation of the consequent and Denial of the antecedent with premises containing strongly associated terms when compared with premises with the same consequent term, but with a weakly associated antecedent. This effect has, in fact, been found with both young children using class-based premises (“If something is a dog, then it has legs”; Markovits et al., 1998) and with adults using causal conditionals (“If a dog has fleas, then it will scratch constantly”; Quinn & Markovits, 1998). Our theory also allows a prediction concerning the developmental pattern of responding to these two types of premises. In many models of information retrieval (e.g., Anderson, 1993; Anderson & Lebière, 1998), the amount of cognitive resources required to successfully retrieve an item from memory varies nonlinearly with the strength of association between the item and the retrieval cues. Retrieving an item that is weakly associated to a given retrieval cue is proportionately more costly in terms of cognitive resources than retrieval of a strongly associated item (Anderson, 1993; Rosen & Engle, 1997). Generally, this means that the relative difference in retrieval probabilities between weakly and strongly associated items will decrease when there are greater cognitive resources available. Developmental data supports the idea of an overall age-related increase in both working memory capacities (Barrouillet & Camos, 2001; Case, 1992; Halford, Wilson, & Phillips, 1998; Swanson, 1999) and the efficiency of retrieval from long-term memory (e.g., Kail, 1992), something that is in fact correlated with reasoning performance (Barrouillet & Lecas, 1999; Janveau-Brennan & Markovits, 1999). Thus, we can conclude that the relative impact of using premises with differing associative strength should decrease with age. All things being equal, it should be relatively more difficult for younger than older reasoners to retrieve a weakly associated alternative compared to retrieval of a strongly associated one. On the basis of this rationale, we predicted in an earlier study using class-based premises (“If an object is a car, then it has a motor”) that the difference in responding to the two types of premises should decrease with age. It was, in fact, found that the difference between strongly and weakly associated premises is present with 8-year-old children, but disappears by 11 years of age (Markovits et al., 1998). However, reasoning with causal premises (“If cause P then effect Q”) is more difficult than with class-based premises (Markovits et al., 1998). Thus, we predict that the difference between strongly and weakly asso-
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ciated premises using causal conditionals should be clearly present among adolescents and that this difference should generally decrease with age. This is the first prediction examined in the present study. To examine this prediction, the same weak and strong causal conditional relations used by Quinn and Markovits (1998) were presented to 12-year-olds, 15year-olds, and adults in a conditional syllogism evaluation task. For each of the four logical forms, participants were asked to choose between two proposed conclusions, either of certainty or uncertainty. We expected that the overall rate of correct uncertainty responses to both Affirmation of the consequent and Denial of the antecedent would increase with age. We also expected that more uncertainty responses would be given with weakly than with strongly associated premises. More importantly, we predicted that this difference would decrease with age. Because we manipulate only access to alternative cases, which generally has little effect on responding to either Modus ponens or Modus tollens (e.g., Cummins, 1995), the rate of correct responses to these should remain unaffected by the strength of the causal relation presented in the conditional premise. The second prediction that was examined here concerns the way that the formulation of the minor premise might affect retrieval. In our general analysis, we considered that the retrieval process is cued by both the conditional and the minor premise. For example, Denial of the antecedent elicits frequent biconditional responses (i.e., from not P conclude not Q) in children as well as in adults when its minor premise is phrased as a syntactic negation of the antecedent (“P is not true”). When given premises of the form “If a dog has fleas, then it will scratch constantly” and “A dog does not have fleas,” individuals frequently conclude that “It will not scratch.” Our theory suggests that, by focusing on the negated term, syntactic negation creates an implicit context in which there is only one relevant cause (i.e., fleas), the other possible causes being irrelevant for the problem at hand. Thus, we hypothesized that one factor in the frequent endorsement of Denial of the antecedent results from the fact that its classic minor premise does not directly activate potential alternatives. Indeed, in some ways, this minor premise invites endorsement of the incorrect Denial of the antecedent inference (i.e., If a dog does not have fleas, and all other possible causes are implicitly set aside, it will not scratch). However, it is possible to reformulate the Denial of the antecedent minor premise in such a way as to reduce these effects by explicitly evoking other alternatives. As an example, take the major premise “If a dog has fleas, then it will scratch.” Suppose that, instead of using the formulation “A dog does not have fleas” (which we refer to as the invited inference condition), we use a formulation of the type “A dog has something other than fleas” (which we refer to as the noninvited inference condition). This latter condition was designed to produce a more explicit and thus more efficient activation of potential alternatives. We predicted that participants should produce fewer uncertainty responses to Denial of the antecedent when given the classic invited inference formulation of the minor premise than from the modified noninvited inference formulation.
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METHOD Participants Seventy-two sixth graders (Mean age ⫽ 12;3 years) and 80 ninth graders (Mean age ⫽ 14;7 years) from an urban high school in Dijon (France) participated in this experiment as volunteers, along with 104 undergraduate psychology students from the Université de Bourgogne, who received a partial course credit to participate. In each age group, the participants were randomly assigned to one of the four experimental conditions defined by the two types of causal conditional relations (either strong or weak) and the two types of minor premise for DA. Material The conditional premises used here were of the form “If cause P then effect Q” and were the same as those used by Quinn and Markovits (1998). The consequent term of these premises was chosen to have a single, strongly associated cause, with other potential causes being relatively weakly associated with the given effect. In order to verify this structure, Quinn and Markovits (1998) asked an independent sample of 25 adults to find as many causes as possible for these effects. They then calculated the relative frequency with which a given cause was produced by the participants. As expected, for each of the four effects used here, there was a single cause that was produced at a very high rate (at least by 72%), with the second most frequently produced cause having a rate of production that was at least 40% lower (see Quinn & Markovits, 1998, for details). The following four conditional premises, in the strongly associated condition, used these four effects with the strongly associated causes as the antecedent term: (a) “If a dog gets fleas, then it will scratch constantly”; (b) “If someone cuts his finger, then his finger will bleed.”; (c) “If there is a drop in the intensity of light, then the pupils of the eyes will dilate”; and (d) “If a photographic film is exposed to light, then it will darken during development.” Half of the participants received these conditional premises; the other half were presented with premises that contained a weakly associated cause as follows: (a) “If a dog has a skin disease, then it will scratch constantly”; (b) “If someone pricks his finger, then his finger will bleed”; (c) “If someone takes a psychotropic drug, then the pupils of the eyes will dilate”; and (d) “If a photographic film is expired, then it will darken during development.” For each of these major premises, four questions corresponding to the four logical forms Modus ponens, Affirmation of the consequent, Denial of the antecedent, and Modus tollens were presented. Half of the participants in both the strongly and weakly associated conditions received the minor premise for Denial of the antecedent in the form of a syntactic negation of the antecedent (invited inference condition), while the other half received a minor premise that suggested some different event (noninvited inference condition). For example, for the conditional premise “If a dog gets fleas, then it will scratch constantly,” the minor premise for Denial of the antecedent was either “A dog does not get fleas” or
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“Something else happened to a dog other than getting fleas” (Table 1). This resulted in a 2 (strength of the relation) ⫻ 2 (type of Denial of the antecedent premise) factorial design. For each of these four experimental conditions, participants were given a 17page booklet on the first page of which they read the following instructions: “On each of the following pages, suppose that the sentence written on the top is true, and then answer the multiple-choice questionnaire within the time allowed.” Each of the 16 following pages presented at the top the instruction “suppose that it is always true that:” followed by a conditional premise in bold type. On the bottom were displayed a minor premise (in bold type) and two possible conclusions, either of certainty or uncertainty. The participants were asked to select one of them. For example, Modus ponens took the following form: Suppose that it is always true that: If a dog gets fleas, then it will scratch constantly. A dog gets fleas, then: 1. It is certain that it will scratch constantly 2. One cannot be certain whether it will scratch constantly or not. To control for an eventual order effect, half of the participants received the conditional premises in the order presented above (i.e., dog, finger, pupil, and film) that was repeated four times in 16 pages of the booklet with four different TABLE 1 Example of Conditional Premises with Strong or Weak Causal Relations and Minor Premises for Affirmation of the Consequent and Denial of the Antecedent in the Invited and Noninvited Conditions (Translation from French) Causal relation Strong
Inference
Premise
Affirmation of the consequent
If a dog gets fleas, then it will scratch constantly. A dog scratches constantly. If a dog gets fleas, then it will scratch constantly. A dog does not get fleas. If a dog gets fleas, then it will scratch constantly. Something else happened to a dog other than getting fleas. If a dog has a skin disease, then it will scratch constantly. A dog scratches constantly. If a dog has a skin disease, then it will scratch constantly. A dog does not have a skin disease. If a dog has a skin disease, then it will scratch constantly. Something else happened to a dog other than having a skin disease.
Denial of the antecedent, invited
Denial of the antecedent, noninvited
Weak
Affirmation of the consequent
Denial of the antecedent, invited
Denial of the antecedent, noninvited
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permutations of the four minor premises, Modus ponens, Affirmation of the consequent, Denial of the antecedent, and Modus tollens, and the other half in the reversed order. Procedure The different booklets were randomly given out to participants during regular class time. They were given 1 min to solve each problem. RESULTS Table 2 presents the percentage of correct responses to the four logical forms (Modus ponens, Modus tollens, Affirmation of the consequent, and Denial of the antecedent) for participants who studied either strong or weak causal relations and who received Denial of the antecedent either in the “invited” or “noninvited” condition. For each of the logical forms, a 3 (age: 12-year-olds, 15-year-olds, and adults) ⫻ 2 (type of relation: weak or strong) ⫻ 2 (type of minor premise for Denial of the antecedent: invited or noninvited condition) analysis of variance (ANOVA) was performed on the number of logically correct responses (of four). Because the access to alternative cases we manipulated has generally little effect on responding to either Modus ponens or Modus tollens, we predicted that the rate of correct responses to these should remain unaffected by the strength of the causal relation presented in the conditional premise. As we anticipated, the strength of the causal conditional relation had no effect on the rates of correct responding on both Modus ponens (81 and 77% for strong and weak relations respectively), F(1, 244) ⫽ 1.24, p ⬎ .10, MSE ⫽ 0.97, and Modus tollens (59 and TABLE 2 Percentage of Correct Responses to the Four Logical Forms, Modus Ponens, Modus Tollens, Affirmation of the Consequent, and Denial of the Antecedent, as a Function of Age, Strength of the Causal Conditional Relation (Weak or Strong), and the Type of Minor Premise for Denial of the Antecedent (Invited or Noninvited Condition).
Logical form Modus ponens
Modus tollens
Affirmation of consequent Denial of the antecedent
Condition for Denial of the antecedent Invited Noninvited Mean Invited Noninvited Mean Invited Noninvited Mean Invited Noninvited Mean
12 years (n ⫽ 72) Weak Strong 71 58 65 53 51 52 58 58 58 53 74 63
72 68 70 54 56 55 32 39 35 44 71 58
Age 15 years (n ⫽ 80) Weak Strong 76 84 80 51 55 53 59 69 64 69 89 79
81 88 84 58 65 61 36 39 38 46 84 65
Adults (n ⫽ 104) Weak Strong 90 85 88 55 52 53 63 69 66 68 87 77
87 89 88 62 60 61 63 55 59 57 83 70
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53%), F(1, 244) ⫽ 2.76, p ⫽ .10, MSE ⫽ 1.32. The sole significant effect was an increase with age of the correct responses on Modus ponens (67, 82, and 88% for the 12-year-olds, 15-year-olds, and adults respectively), F(1, 244) ⫽ 14.89, p ⬍ .001, MSE ⫽ 0.97. As far as the Affirmation of the consequent was concerned, as we anticipated, the rate of correct responses of uncertainty increased with age (47, 51, and 63% for the 12-year-olds, 15-year-olds, and adults respectively), F(2, 244) ⫽ 6.19, p ⬍ .01, MSE ⫽ 1.56, and there were more correct responses from weak than from strong relations (63 and 44% respectively), F(1, 244) ⫽ 23.07, p ⬍ .001, MSE ⫽ 1.56. The interaction between these two factors was marginally significant, F(2, 244) ⫽ 2.34, p ⬍ .10, MSE ⫽ 1.56. The effect of the strength of the relation was stronger in children (12- and 15-year-olds taken together) than in adults (36 and 61% correct responses for strong and weak relations respectively compared to 59 and 66% in adults), F(1, 244) ⫽ 4.52, p ⬍ .05, MSE ⫽ 1.56. Actually, strong relations elicited fewer correct responses to Affirmation of the consequent than weak relations in 12-year-olds (35 and 58% respectively), F(1, 68) ⫽ 9.35, p ⬍ .01, MSE ⫽ 1.62, and 15-year-olds (64 and 38%), F(1, 76) ⫽ 13.94, p ⬍ .001, MSE ⫽ 1.58, but this effect did not reach significance in adults (59 and 66%), F(1, 100) ⫽ 1.65, p ⬎ .10, MSE ⫽ 1.50 (Fig. 1). The same effects of age and strength of the causal relation were observed for the Denial of the antecedent. The rate of correct responses of uncertainty increased with age (60, 72, and 74% for the 12-year-olds, 15-year-olds, and adults respectively), F(2, 244) ⫽ 5.44, p ⬍ .01, MSE ⫽ 1.19, and as we predicted there were more correct responses from weak than from strong relations (73 and 64% respectively), F(1, 244) ⫽ 6.80, p ⬍ .01, MSE ⫽ 1.19. However, these two effects did not interact, F ⬍ 1. There was no significant difference in the rate of correct responses from weak and strong relations in the 12-year-old children (63 and 58% respectively), F ⬍ 1, and in adults (77 and 70%), F(1, 100) ⫽ 1.95, p ⬎ .10, MSE ⫽ 1.26. However, there were more correct responses from weak than from strong relations in the 15-year-old children (79 and 65% respectively), F(1, 100) ⫽ 5.36, p ⬍ .05, MSE ⫽ 1.13 (Fig. 1). Responses to the Denial of the antecedent were also of interest for our second prediction that correct responses of uncertainty should depend on the phrasing of the minor premise. In line with our prediction, minor premises with an explicit negation elicited far fewer correct responses than did minor premises that suggested alternative causes (56 and 81% respectively), F(1, 244) ⫽ 51.78, p ⬍ .001, MSE ⫽ 1.19. This effect did not interact with age, F ⬍ 1, or with the strength of the relation, F(1, 244) ⫽ 2.21, p ⫽ .14, MSE ⫽ 1.19. However, it is noteworthy that strong relations elicited fewer correct responses to Denial of the antecedent than weak relations in the invited inference condition (49 and 63% respectively), F(1, 244) ⫽ 8.38, p ⬍ .01, MSE ⫽ 1.19, whereas this effect disappeared when modified minor premises were used (79 and 83%), F ⬍ 1. In fact, there was no age group in which the strength of the relation had a significant effect on Denial of the antecedent when the minor premise suggested alternative causes, Fs ⬍ 1.
244 BARROUILLET, MARKOVITS, AND QUINN FIG. 1. Rates of correct responses of uncertainty to the logical forms Affirmation of the consequent (AC) and Denial of the antecedent (DA) as a function of age and strength of the causal relation between antecedent and consequent (strong or weak) (*p ⬍ .05).
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DISCUSSION In summary, these results were consistent with our hypotheses. As we predicted, correct uncertainty responses for both Affirmation of the consequent and Denial of the antecedent were more frequent from weak than strong relations, and this effect was modulated by the age of the participants. In 12-year-old children, this difference was significant only for Affirmation of the consequent, and it was significant for both Affirmation of the consequent and Denial of the antecedent inferences in 15-year-old children, while performance remained unaffected by the strength of the causal relation in adults. Moreover, when the minor premise of Denial of the antecedent strongly suggests other possible causes (noninvited inference condition) rather than the mere absence of the cause involved in the antecedent (invited inference condition), the retrieval of alternative causes is strongly enhanced and in fact very efficient. In this case, the rate of correct uncertainty responses reached 87% in adults, which is far higher than what is usually observed on this inference (see Evans, Newstead, & Byrnes, 1993, for a review of several studies). The enhanced efficiency of this retrieval process explains the fact that the strength of the relation involved in the conditional premise had only a small impact on Denial of the antecedent reasoning when this inference is not invited by the wording of the minor premise. Thus, this pattern of results is consistent with our theory that assumes that both the wording of the minor premise and the strength of the causal relation affect the same retrieval process of alternative antecedents from long-term memory. Mental models theory assumes that reasoning on the basis of an “If P then Q” sentence and a minor premise involves the construction of an initial mental model of the state of affairs described by the conditional premise and, when needed, a further fleshing out of this initial representation (Johnson-Laird & Byrne, 1991; Johnson-Laird, 1999). According to Markovits and Barrouillet (2001), in the case of conditionals using familiar terms this fleshing out would rely on a process of activation and retrieval of knowledge from long-term memory. Thus, the production of inferences should depend both on individuals’ capacity to activate the appropriate knowledge and on the nature of the cues that trigger this retrieval process, that is, the content of the premises. As predicted, weak relations elicited more correct responses for both Affirmation of the consequent and Denial of the antecedent than strong relations. In addition, the modified minor premise led to an increase in the rate of correct responding to Denial of the antecedent. These two facts are consistent with the idea that correct responses of uncertainty to Affirmation of the consequent and Denial of the antecedent depend on the efficiency of retrieval of alternative antecedents from long-term memory. As anticipated, these content effects were modulated by developmental factors. Indeed, the difference between strong and weak relations was generally greater in children than in adults. It should be noted, however, that the youngest participants manifested this effect only for Affirmation of the consequent, whereas the effect was significant for both inferences in older children. Though this unexpected result could be con-
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sidered as being at odds with our theory, we would claim that it is actually in line with it. The activation and retrieval processes are triggered by the content of the conditional premise, and also by the minor premise, and the minor premises of Affirmation of the consequent and Denial of the antecedent could differ in the efficiency with which they activate alternative antecedents. Indeed, the minor premise of Affirmation of the consequent (i.e., “Q is true”) directly matches alternative cases of the form A ⫺ Q, whereas the minor premise of Denial of the antecedent does not when it is phrased using a syntactic negation. Our theory assumes that young individuals have limited cognitive resources that constrain the efficiency of the retrieval process. As a consequence, in young participants, the minor premise of Affirmation of the consequent should be a better cue to retrieve alternative antecedents than the minor premise of Denial of the antecedent. This explains why the strength effect was more pronounced for Affirmation of the consequent than for Denial of the antecedent in our youngest group, a fact already observed by Janveau-Brennan and Markovits (1999). Older children have greater cognitive resources and they can activate and retrieve a greater amount of information (Janveau-Brennan & Markovits, 1999). Thus, they may have benefited more from the cue provided by the weak relations whatever the type of minor premise they were presented with, and they manifested a significant effect on both Affirmation of the consequent and Denial of the antecedent. Finally, adult participants were able to activate and retrieve alternative cases whatever the strength of the causal relation under study, and the strength effect did not reach significance in this group. This study demonstrated that, among many other factors, the strength of causal conditional relations plays a critical role in determining the developmental pattern of specific content effects. Weak causal relations elicit more conditional responses than strong relations in adolescents, in the same way as weak class-based relations elicited more conditional responses in young children (Markovits et al., 1998). These two phenomena can be accounted for by the constraints inherent to the same process of retrieval from long-term memory. Indeed, the efficiency of this process depends, among other things, on the base-level activation of the items to be retrieved (Anderson, 1993; Anderson & Lebière, 1998), and the relative baselevel activation of different items pertaining to the same class of knowledge depends in turn on the structure and organization of this knowledge in long-term memory. As Quinn and Markovits (1998) pointed out, the strength of the association between antecedent and consequent differs from factors such as familiarity, which is also known to increase the tendency to respond in ways that correspond to normative logic (Markovits, 1986). Indeed, strong relations are probably more familiar than weak relations, but they lead to poorer performances. Thus, content effects would depend primarily on the structure of long-term memory, i.e., on a reasoner’s general cognitive architecture, as Markovits and Barrouillet (2001) suggested. The effect of the number of alternative causes reported by Cummins et al. (1991) and Janveau-Brennan and Markovits (1999) is in line with this theoretical framework. Indeed, the greater the number of alter-
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native causes, the higher the probability to retrieve one of them from memory and to give an uncertainty response to Denial of the antecedent and Affirmation of the consequent. Moreover, it has been argued that this theory accounts also for some content effects in reasoning from artificial conditional relations (Barrouillet & Lecas, 1998, in press). Thus, though we are far from a complete understanding of the psychological processes that underpin conditional reasoning, our theory does permit precise predictions of specific content effects and accounts for their developmental evolution. REFERENCES Anderson, J. R. (1993). Rules of the mind. Hillsdale, NJ: Erlbaum. Anderson, J. R., & Lebiere, C. (1998). The atomic components of thought. Hillsdale, NJ: Erlbaum. Barrouillet, P., & Camos, V. (2001). Developmental increase in working memory span: Resource sharing or temporal decay? Journal of Memory and Language, 45, 1–20. Barrouillet, P., & Lecas, J. F. (1998). How can mental models account for content effects in conditional reasoning? A developmental perspective. Cognition, 67, 209–253. Barrouillet, P., & Lecas, J. F. (1999). Mental models in conditional reasoning and working memory. Thinking and Reasoning, 5, 289–302. Barrouillet, P., & Lecas, J. F. (in press). Content and context effects in children’s and adult’s conditional reasoning. Quarterly Journal of Experimental Psychology. Braine, M. D. S. (1990). The “natural logic” approach to reasoning. In W. F. Overton (Ed.), Reasoning, necessity and logic: Developmental perspectives. Hillsdale, NJ: Erlbaum. Braine, M. D. S., & O’Brien, D. P. (1991). A theory of if: Lexical entry, reasoning program, and pragmatic principles. Psychological Review, 98, 182–203. Braine, M. D. S., & O’Brien, D. P. (1998). Mental logic. Mahwah, NJ: Erlbaum. Case, R. (1985). Intellectual development: Birth to adulthood. New York: Academic Press. Case, R. (1992). The mind’s staircase: Exploring the conceptual underpinnings of children’s thought and knowledge. Hillsdale, NJ: Erlbaum. Cowan, N., Nugent, L. D., Elliot, E. M., Ponomarev, I., & Saults, J. S. (1999). The role of attention in the development of short-term memory: Age differences in the verbal span of apprehension. Child Development, 70(5), 1082–1097. Cummins, D. D. (1995). Naives theories and causal deduction. Memory & Cognition, 23, 646–658. Cummins, D. D., Lubart, T., Alksnis, O., & Rist, R. (1991). Conditional reasoning and causation. Memory & Cognition, 19, 274–282. Evans, J. S. B. T., Newstead, S. E., & Byrne, R. M. J. (1993). Human reasoning: The psychology of deduction. Hillsdale, NJ: Erlbaum. Halford, G. S. (1993). Children’s understanding. Hillsdale, NJ: Erlbaum. Halford, G. S., Wilson, W. H., & Phillips, S. (1998). Processing capacity defined by relational complexity: Implications for comparative, developmental, and cognitive psychology. Behavioral and Brain Sciences, 21, 803–864. Janveau-Brennan, G., & Markovits, H. (1999). The development of reasoning with causal conditional. Developmental Psychology, 35, 904–911. Johnson-Laird, P. N. (1999). Deductive reasoning. Annual Review of Psychology, 50, 109–135. Johnson-Laird, P. N., & Byrne, R. M. J. (1991). Deduction. Hillsdale, NJ: Erlbaum. Johnson-Laird, P. N., & Byrne, R. M. J. (2001). Conditionals: A theory of meaning, pragmatics, and inference. Submitted for publication. Kail, R. (1992). Processing speed, speech rate, and memory. Developmental Psychology, 28, 899–904. Klaczynski, P. A., & Narashimham, G. (1998). Representations as mediators of adolescent deductive reasoning. Developmental Psychology, 5, 865–881.
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Lecas, J. F., & Barrouillet, P. (1999). Understanding of conditional rules in childhood and adolescence: A mental models approach. Current Psychology of Cognition, 18, 363–396. Markovits, H. (1984). Awareness of the ‘possible’ as a mediator of formal thinking in conditional reasoning problems. British Journal of Psychology, 75, 367–376. Markovits, H. (1993). The development of conditional reasoning: A piagetian reformulation of the theory of mental models. Merrill–Palmer Quarterly, 39, 133–160. Markovits, H. (2000). A mental model analysis of young children’s conditional reasoning with meaningful premises. Thinking and Reasoning, 6, 335–347. Markovits, H., & Barrouillet, P. (2001). The development of conditional reasoning: A mental model account. Developmental Review, doi:1006/drev.2000.0533. Markovits, H., Fleury, M. L., Quinn, S., & Venet, M. (1998). The development of conditional reasoning and the structure of semantic memory. Child Development, 69, 742–755. Markovits, H., & Vachon, R. (1990). Conditional reasoning, representation, and level of abstraction. Developmental Psychology, 26, 942–951. Markovits, H., Venet, M., Janveau-Brennan, G., Malfait, N., Pion, N., & Vadeboncoeur, I. (1996). Reasoning in young children: Fantasy and Information retrieval. Child Development, 67, 2857–2872. Overton, W. F. (1990). Competence and procedures: Constraints on the development of logical reasoning. In W. F. Overton (Ed.), Reasoning, necessity, and logic: Developmental perspectives (pp. 1–32). Hillsdale, NJ: Erlbaum. Quinn, S., & Markovits, H. (1998). Conditional reasoning, causality, and the structure of semantic memory: Strength of association as a predictive factor for content effects. Cognition, 68, B93–B101. Rosen, V. M., & Engle, R. W. (1997). The role of working memory capacity in retrieval. Journal of Experimental Psychology: General, 126, 211–227. Thompson, V. A. (1994). Interpretational factors in conditional reasoning. Memory & Cognition, 22(6), 742–758. Received February 26, 2001; revised November 7, 2001
Developmental and Content Effects in Reasoning with Causal Conditionals Pierre Barrouillet Université de Bourgogne, Dijon, France
and Henry Markovits and Stéphane Quinn Université du Québec à Montréal, Montreal, Quebec, Canada Two predictions derived from Markovits and Barrouillet’s (2001) developmental model of conditional reasoning were tested in a study in which 72 twelve-year-olds, 80 fifteenyear-olds, and 104 adults received a paper-and-pencil test of conditional reasoning with causal premises (“if cause P then effect Q”). First, we predicted that conditional premises would induce more correct uncertainty responses to the Affirmation of the consequent and Denial of the antecedent forms when the antecedent term is weakly associated to the consequent than when the two are strongly associated and that this effect would decrease with age. Second, uncertainty responding to the Denial of the antecedent form (“P is not true”) should be easier when the formulation of the minor premise invites retrieval of alternate antecedents (“if something other than P is true”). The results were consistent with the hypotheses and indicate the importance of retrieval processes in understanding developmental patterns in conditional reasoning with familiar premises. © 2002 Elsevier Science (USA) Key Words: deductive reasoning; conditional; causality; cognitive development.
Conditional reasoning in adults as well as in children has been the subject of a great number of studies and theoretical proposals (Braine & O’Brien, 1991; Johnson-Laird, 1999; Johnson-Laird & Byrne, 1991, 2001; see Evans, Newstead, & Byrne, 1993, for a review). Conditional reasoning involves making inferences on the basis of a given “if P then Q” sentence (the conditional premise) and additional information (the minor premise) that can either affirm or negate the antecedent (i.e, the proposition P) or the consequent (i.e., the proposition Q). Among the resulting four classical forms of conditional syllogisms, both Modus ponens (P is true) and Modus tollens (Q is false) lead to certain conclusions that are “Q is true” and “P is false” respectively. For example, from the conditional The authors thank Laurence Arnoud for her collaboration in collecting data. Address correspondence and reprint requests to Pierre Barrouillet, Université de Bourgogne, LEAD-CNRS, Faculté des Sciences Gabriel, 6 Bld Gabriel, 21000 Dijon, France. Fax: 03 80 39 57 67. E-mail: [email protected]. 235 0022-0965/02 $35.00 © 2002 Elsevier Science (USA) All rights reserved
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premise “If a dog has fleas, then it will scratch constantly” and the minor premise “A dog has fleas,” we can conclude with certainty that it will scratch (Modus ponens), and given the minor premise “A dog does not scratch constantly” we can conclude with certainty that it does not have fleas (Modus tollens). Alternatively, neither the Affirmation of the consequent (“Q is true”) nor the Denial of the antecedent (“P is false”) leads to certain conclusions, though children and adults frequently conclude that “P is true” from the former and that “Q is false” from the latter. Indeed, from the information that a dog scratches constantly (Affirmation of the consequent), we cannot be certain that it has fleas because it can have some other parasite or a skin disease, and for the same reasons we cannot conclude that a dog will scratch from the fact that it does not have fleas (Denial of the antecedent). We refer to these as uncertain logical forms. Many recent studies have shown that the inferences that children and adults make from conditional premises vary greatly with the content of the premises, particularly in the cases of the Affirmation of the consequent and Denial of the antecedent. Although current theories of reasoning acknowledge the impact of content, most provide no precise framework for understanding how this factor affects inferential behavior. For example, Overton’s (1990) competence-performance model and Braine’s (1990; Braine & O’Brien, 1991, 1998) theory of natural deduction both allow for possible effects of content on reasoning, but neither provide a functional model that can explain and predict specific content effects. The present article focuses on a theory that attempts to account for both content and developmental effects in conditional reasoning and presents an experiment that tested two predictions concerning content effects in children’s and adults’ reasoning from causal conditional relations. Markovits and Barrouillet (2001) have recently proposed a developmental reformulation of the mental models theory of Johnson-Laird (1999; JohnsonLaird & Byrne, 1991). They assume that reasoning is a process that is related to the construction of mental models of the state of affairs that the premises describe. These models are held and processed in a limited-capacity working memory. They suppose that, initially, a major premise of the form “If P then Q” (e.g., “If a dog gets fleas, then it will scratch constantly”) is represented by an initial model of the following form: P→Q This model refers to a state of affairs in which the antecedent P and the consequent Q are true and the symbol (between them) indicates the form of the relation between P and Q. When an inference is made by adding a minor premise, other possible models are added to the initial model (referred to as “fleshing out”; Johnson-Laird & Byrne, 1991). Markovits (1993, 2000; Markovits & Vachon, 1990) has suggested that fleshing out could be the result of automatic activation and retrieval of knowledge from long-term memory when premises refer to concrete terms. In this case, the fleshing out process would depend on an individual’s knowledge about the terms involved in the premises.
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Specifically, there is now a great deal of evidence in both the developmental and adult literatures that performance on the Affirmation of the consequent and Denial of the antecedent is determined largely by whether the reasoner has access to potential alternative antecedents (Cummins, Lubarts, & Alksnis, 1991; Cummins, 1995; Klaczynski & Narasimham, 1998; Markovits, 1984; Markovits & Vachon, 1990; Markovits et al., 1996, 1998; Quinn & Markovits, 1998; Thompson, 1994). In terms of the above model, this can be explained quite directly. For example, consider Affirmation of the consequent, for which the minor premise is “Q is true.” The model supposes that if the minor premise and the major premise can cue a potential alternative antecedent (A) sufficiently strongly, then the reasoner will construct the following two models: P A
— —
Q Q
Each line refers to a distinct state of affairs. The first refers to a state of affairs in which the antecedent P and the consequent Q are true, and the second line refers to a state of affairs in which the consequent Q results from an alternative antecedent (A). For example, with a conditional premise such as “If a dog gets fleas, then it will scratch constantly,” the first model would represent a dog that has got fleas and that scratches, whereas the alternative antecedent to scratching could refer to a dog having a skin disease. With these two models, the conclusion that can be derived from “Q is true” (i.e., a dog scratches constantly) is that P (it has fleas) may be true or false, thus there is no certain conclusion. However, if an alternative antecedent is not strongly activated, then the reasoner will reason on the basis of the initial model and conclude that “P is true” (i.e., it has fleas), a commonly made biconditional error. A similar analysis can be made for the Denial of the antecedent. Thus, according to this analysis, reasoning on these two forms is affected by the relative accessibility of possible alternative antecedents, a factor that results in striking content effects, and the amount of cognitive resources available to activate stored knowledge and process the models constructed. This theory has predicted some counterintuitive effects about conditional reasoning. As we have seen, making the correct responses of uncertainty to Affirmation of the consequent and Denial of the antecedent depends on the retrieval of at least one alternative antecedent. One factor determining whether an alternative is in fact retrieved is the relative strength of association between the consequent term of the major premise (Q) and the most highly associated alternative antecedent available in memory. This basic analysis has been used to make the following prediction (Markovits, Fleury, Quinn, & Venet, 1998; Quinn & Markovits, 1998). We assume that when reasoners are given an uncertain logical form, e.g., “If P then Q and Q is true,” they will attempt to retrieve a counterexample, “If A then Q,” from long-term memory. If the consequent term is strongly related to a specific potential antecedent, which we refer to as a “strong antecedent,” then there will be a correspondingly strong probability that it will be activated during the search for counterexamples.
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Similarly, there are elements that are more weakly related to Q, one of which we refer to as a “weak antecedent.” Now, suppose that we compare performance on two similar problems. In one, we use the “strong antecedent” in the major premise, and in the other, we use a “weak antecedent.” On the one hand, reasoners who receive the second of these problems would be expected to have a high probability of activating the “strong antecedent” as an alternative antecedent. On the other hand, reasoners who receive the first of these problems would not be able to use the same “strong antecedent” because this is part of the premise and would have to be able to activate other, less strongly associated, elements in order to generate an alternative antecedent. Thus, we predict fewer correct uncertainty responses to both Affirmation of the consequent and Denial of the antecedent with premises containing strongly associated terms when compared with premises with the same consequent term, but with a weakly associated antecedent. This effect has, in fact, been found with both young children using class-based premises (“If something is a dog, then it has legs”; Markovits et al., 1998) and with adults using causal conditionals (“If a dog has fleas, then it will scratch constantly”; Quinn & Markovits, 1998). Our theory also allows a prediction concerning the developmental pattern of responding to these two types of premises. In many models of information retrieval (e.g., Anderson, 1993; Anderson & Lebière, 1998), the amount of cognitive resources required to successfully retrieve an item from memory varies nonlinearly with the strength of association between the item and the retrieval cues. Retrieving an item that is weakly associated to a given retrieval cue is proportionately more costly in terms of cognitive resources than retrieval of a strongly associated item (Anderson, 1993; Rosen & Engle, 1997). Generally, this means that the relative difference in retrieval probabilities between weakly and strongly associated items will decrease when there are greater cognitive resources available. Developmental data supports the idea of an overall age-related increase in both working memory capacities (Barrouillet & Camos, 2001; Case, 1992; Halford, Wilson, & Phillips, 1998; Swanson, 1999) and the efficiency of retrieval from long-term memory (e.g., Kail, 1992), something that is in fact correlated with reasoning performance (Barrouillet & Lecas, 1999; Janveau-Brennan & Markovits, 1999). Thus, we can conclude that the relative impact of using premises with differing associative strength should decrease with age. All things being equal, it should be relatively more difficult for younger than older reasoners to retrieve a weakly associated alternative compared to retrieval of a strongly associated one. On the basis of this rationale, we predicted in an earlier study using class-based premises (“If an object is a car, then it has a motor”) that the difference in responding to the two types of premises should decrease with age. It was, in fact, found that the difference between strongly and weakly associated premises is present with 8-year-old children, but disappears by 11 years of age (Markovits et al., 1998). However, reasoning with causal premises (“If cause P then effect Q”) is more difficult than with class-based premises (Markovits et al., 1998). Thus, we predict that the difference between strongly and weakly asso-
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ciated premises using causal conditionals should be clearly present among adolescents and that this difference should generally decrease with age. This is the first prediction examined in the present study. To examine this prediction, the same weak and strong causal conditional relations used by Quinn and Markovits (1998) were presented to 12-year-olds, 15year-olds, and adults in a conditional syllogism evaluation task. For each of the four logical forms, participants were asked to choose between two proposed conclusions, either of certainty or uncertainty. We expected that the overall rate of correct uncertainty responses to both Affirmation of the consequent and Denial of the antecedent would increase with age. We also expected that more uncertainty responses would be given with weakly than with strongly associated premises. More importantly, we predicted that this difference would decrease with age. Because we manipulate only access to alternative cases, which generally has little effect on responding to either Modus ponens or Modus tollens (e.g., Cummins, 1995), the rate of correct responses to these should remain unaffected by the strength of the causal relation presented in the conditional premise. The second prediction that was examined here concerns the way that the formulation of the minor premise might affect retrieval. In our general analysis, we considered that the retrieval process is cued by both the conditional and the minor premise. For example, Denial of the antecedent elicits frequent biconditional responses (i.e., from not P conclude not Q) in children as well as in adults when its minor premise is phrased as a syntactic negation of the antecedent (“P is not true”). When given premises of the form “If a dog has fleas, then it will scratch constantly” and “A dog does not have fleas,” individuals frequently conclude that “It will not scratch.” Our theory suggests that, by focusing on the negated term, syntactic negation creates an implicit context in which there is only one relevant cause (i.e., fleas), the other possible causes being irrelevant for the problem at hand. Thus, we hypothesized that one factor in the frequent endorsement of Denial of the antecedent results from the fact that its classic minor premise does not directly activate potential alternatives. Indeed, in some ways, this minor premise invites endorsement of the incorrect Denial of the antecedent inference (i.e., If a dog does not have fleas, and all other possible causes are implicitly set aside, it will not scratch). However, it is possible to reformulate the Denial of the antecedent minor premise in such a way as to reduce these effects by explicitly evoking other alternatives. As an example, take the major premise “If a dog has fleas, then it will scratch.” Suppose that, instead of using the formulation “A dog does not have fleas” (which we refer to as the invited inference condition), we use a formulation of the type “A dog has something other than fleas” (which we refer to as the noninvited inference condition). This latter condition was designed to produce a more explicit and thus more efficient activation of potential alternatives. We predicted that participants should produce fewer uncertainty responses to Denial of the antecedent when given the classic invited inference formulation of the minor premise than from the modified noninvited inference formulation.
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METHOD Participants Seventy-two sixth graders (Mean age ⫽ 12;3 years) and 80 ninth graders (Mean age ⫽ 14;7 years) from an urban high school in Dijon (France) participated in this experiment as volunteers, along with 104 undergraduate psychology students from the Université de Bourgogne, who received a partial course credit to participate. In each age group, the participants were randomly assigned to one of the four experimental conditions defined by the two types of causal conditional relations (either strong or weak) and the two types of minor premise for DA. Material The conditional premises used here were of the form “If cause P then effect Q” and were the same as those used by Quinn and Markovits (1998). The consequent term of these premises was chosen to have a single, strongly associated cause, with other potential causes being relatively weakly associated with the given effect. In order to verify this structure, Quinn and Markovits (1998) asked an independent sample of 25 adults to find as many causes as possible for these effects. They then calculated the relative frequency with which a given cause was produced by the participants. As expected, for each of the four effects used here, there was a single cause that was produced at a very high rate (at least by 72%), with the second most frequently produced cause having a rate of production that was at least 40% lower (see Quinn & Markovits, 1998, for details). The following four conditional premises, in the strongly associated condition, used these four effects with the strongly associated causes as the antecedent term: (a) “If a dog gets fleas, then it will scratch constantly”; (b) “If someone cuts his finger, then his finger will bleed.”; (c) “If there is a drop in the intensity of light, then the pupils of the eyes will dilate”; and (d) “If a photographic film is exposed to light, then it will darken during development.” Half of the participants received these conditional premises; the other half were presented with premises that contained a weakly associated cause as follows: (a) “If a dog has a skin disease, then it will scratch constantly”; (b) “If someone pricks his finger, then his finger will bleed”; (c) “If someone takes a psychotropic drug, then the pupils of the eyes will dilate”; and (d) “If a photographic film is expired, then it will darken during development.” For each of these major premises, four questions corresponding to the four logical forms Modus ponens, Affirmation of the consequent, Denial of the antecedent, and Modus tollens were presented. Half of the participants in both the strongly and weakly associated conditions received the minor premise for Denial of the antecedent in the form of a syntactic negation of the antecedent (invited inference condition), while the other half received a minor premise that suggested some different event (noninvited inference condition). For example, for the conditional premise “If a dog gets fleas, then it will scratch constantly,” the minor premise for Denial of the antecedent was either “A dog does not get fleas” or
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“Something else happened to a dog other than getting fleas” (Table 1). This resulted in a 2 (strength of the relation) ⫻ 2 (type of Denial of the antecedent premise) factorial design. For each of these four experimental conditions, participants were given a 17page booklet on the first page of which they read the following instructions: “On each of the following pages, suppose that the sentence written on the top is true, and then answer the multiple-choice questionnaire within the time allowed.” Each of the 16 following pages presented at the top the instruction “suppose that it is always true that:” followed by a conditional premise in bold type. On the bottom were displayed a minor premise (in bold type) and two possible conclusions, either of certainty or uncertainty. The participants were asked to select one of them. For example, Modus ponens took the following form: Suppose that it is always true that: If a dog gets fleas, then it will scratch constantly. A dog gets fleas, then: 1. It is certain that it will scratch constantly 2. One cannot be certain whether it will scratch constantly or not. To control for an eventual order effect, half of the participants received the conditional premises in the order presented above (i.e., dog, finger, pupil, and film) that was repeated four times in 16 pages of the booklet with four different TABLE 1 Example of Conditional Premises with Strong or Weak Causal Relations and Minor Premises for Affirmation of the Consequent and Denial of the Antecedent in the Invited and Noninvited Conditions (Translation from French) Causal relation Strong
Inference
Premise
Affirmation of the consequent
If a dog gets fleas, then it will scratch constantly. A dog scratches constantly. If a dog gets fleas, then it will scratch constantly. A dog does not get fleas. If a dog gets fleas, then it will scratch constantly. Something else happened to a dog other than getting fleas. If a dog has a skin disease, then it will scratch constantly. A dog scratches constantly. If a dog has a skin disease, then it will scratch constantly. A dog does not have a skin disease. If a dog has a skin disease, then it will scratch constantly. Something else happened to a dog other than having a skin disease.
Denial of the antecedent, invited
Denial of the antecedent, noninvited
Weak
Affirmation of the consequent
Denial of the antecedent, invited
Denial of the antecedent, noninvited
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permutations of the four minor premises, Modus ponens, Affirmation of the consequent, Denial of the antecedent, and Modus tollens, and the other half in the reversed order. Procedure The different booklets were randomly given out to participants during regular class time. They were given 1 min to solve each problem. RESULTS Table 2 presents the percentage of correct responses to the four logical forms (Modus ponens, Modus tollens, Affirmation of the consequent, and Denial of the antecedent) for participants who studied either strong or weak causal relations and who received Denial of the antecedent either in the “invited” or “noninvited” condition. For each of the logical forms, a 3 (age: 12-year-olds, 15-year-olds, and adults) ⫻ 2 (type of relation: weak or strong) ⫻ 2 (type of minor premise for Denial of the antecedent: invited or noninvited condition) analysis of variance (ANOVA) was performed on the number of logically correct responses (of four). Because the access to alternative cases we manipulated has generally little effect on responding to either Modus ponens or Modus tollens, we predicted that the rate of correct responses to these should remain unaffected by the strength of the causal relation presented in the conditional premise. As we anticipated, the strength of the causal conditional relation had no effect on the rates of correct responding on both Modus ponens (81 and 77% for strong and weak relations respectively), F(1, 244) ⫽ 1.24, p ⬎ .10, MSE ⫽ 0.97, and Modus tollens (59 and TABLE 2 Percentage of Correct Responses to the Four Logical Forms, Modus Ponens, Modus Tollens, Affirmation of the Consequent, and Denial of the Antecedent, as a Function of Age, Strength of the Causal Conditional Relation (Weak or Strong), and the Type of Minor Premise for Denial of the Antecedent (Invited or Noninvited Condition).
Logical form Modus ponens
Modus tollens
Affirmation of consequent Denial of the antecedent
Condition for Denial of the antecedent Invited Noninvited Mean Invited Noninvited Mean Invited Noninvited Mean Invited Noninvited Mean
12 years (n ⫽ 72) Weak Strong 71 58 65 53 51 52 58 58 58 53 74 63
72 68 70 54 56 55 32 39 35 44 71 58
Age 15 years (n ⫽ 80) Weak Strong 76 84 80 51 55 53 59 69 64 69 89 79
81 88 84 58 65 61 36 39 38 46 84 65
Adults (n ⫽ 104) Weak Strong 90 85 88 55 52 53 63 69 66 68 87 77
87 89 88 62 60 61 63 55 59 57 83 70
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53%), F(1, 244) ⫽ 2.76, p ⫽ .10, MSE ⫽ 1.32. The sole significant effect was an increase with age of the correct responses on Modus ponens (67, 82, and 88% for the 12-year-olds, 15-year-olds, and adults respectively), F(1, 244) ⫽ 14.89, p ⬍ .001, MSE ⫽ 0.97. As far as the Affirmation of the consequent was concerned, as we anticipated, the rate of correct responses of uncertainty increased with age (47, 51, and 63% for the 12-year-olds, 15-year-olds, and adults respectively), F(2, 244) ⫽ 6.19, p ⬍ .01, MSE ⫽ 1.56, and there were more correct responses from weak than from strong relations (63 and 44% respectively), F(1, 244) ⫽ 23.07, p ⬍ .001, MSE ⫽ 1.56. The interaction between these two factors was marginally significant, F(2, 244) ⫽ 2.34, p ⬍ .10, MSE ⫽ 1.56. The effect of the strength of the relation was stronger in children (12- and 15-year-olds taken together) than in adults (36 and 61% correct responses for strong and weak relations respectively compared to 59 and 66% in adults), F(1, 244) ⫽ 4.52, p ⬍ .05, MSE ⫽ 1.56. Actually, strong relations elicited fewer correct responses to Affirmation of the consequent than weak relations in 12-year-olds (35 and 58% respectively), F(1, 68) ⫽ 9.35, p ⬍ .01, MSE ⫽ 1.62, and 15-year-olds (64 and 38%), F(1, 76) ⫽ 13.94, p ⬍ .001, MSE ⫽ 1.58, but this effect did not reach significance in adults (59 and 66%), F(1, 100) ⫽ 1.65, p ⬎ .10, MSE ⫽ 1.50 (Fig. 1). The same effects of age and strength of the causal relation were observed for the Denial of the antecedent. The rate of correct responses of uncertainty increased with age (60, 72, and 74% for the 12-year-olds, 15-year-olds, and adults respectively), F(2, 244) ⫽ 5.44, p ⬍ .01, MSE ⫽ 1.19, and as we predicted there were more correct responses from weak than from strong relations (73 and 64% respectively), F(1, 244) ⫽ 6.80, p ⬍ .01, MSE ⫽ 1.19. However, these two effects did not interact, F ⬍ 1. There was no significant difference in the rate of correct responses from weak and strong relations in the 12-year-old children (63 and 58% respectively), F ⬍ 1, and in adults (77 and 70%), F(1, 100) ⫽ 1.95, p ⬎ .10, MSE ⫽ 1.26. However, there were more correct responses from weak than from strong relations in the 15-year-old children (79 and 65% respectively), F(1, 100) ⫽ 5.36, p ⬍ .05, MSE ⫽ 1.13 (Fig. 1). Responses to the Denial of the antecedent were also of interest for our second prediction that correct responses of uncertainty should depend on the phrasing of the minor premise. In line with our prediction, minor premises with an explicit negation elicited far fewer correct responses than did minor premises that suggested alternative causes (56 and 81% respectively), F(1, 244) ⫽ 51.78, p ⬍ .001, MSE ⫽ 1.19. This effect did not interact with age, F ⬍ 1, or with the strength of the relation, F(1, 244) ⫽ 2.21, p ⫽ .14, MSE ⫽ 1.19. However, it is noteworthy that strong relations elicited fewer correct responses to Denial of the antecedent than weak relations in the invited inference condition (49 and 63% respectively), F(1, 244) ⫽ 8.38, p ⬍ .01, MSE ⫽ 1.19, whereas this effect disappeared when modified minor premises were used (79 and 83%), F ⬍ 1. In fact, there was no age group in which the strength of the relation had a significant effect on Denial of the antecedent when the minor premise suggested alternative causes, Fs ⬍ 1.
244 BARROUILLET, MARKOVITS, AND QUINN FIG. 1. Rates of correct responses of uncertainty to the logical forms Affirmation of the consequent (AC) and Denial of the antecedent (DA) as a function of age and strength of the causal relation between antecedent and consequent (strong or weak) (*p ⬍ .05).
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DISCUSSION In summary, these results were consistent with our hypotheses. As we predicted, correct uncertainty responses for both Affirmation of the consequent and Denial of the antecedent were more frequent from weak than strong relations, and this effect was modulated by the age of the participants. In 12-year-old children, this difference was significant only for Affirmation of the consequent, and it was significant for both Affirmation of the consequent and Denial of the antecedent inferences in 15-year-old children, while performance remained unaffected by the strength of the causal relation in adults. Moreover, when the minor premise of Denial of the antecedent strongly suggests other possible causes (noninvited inference condition) rather than the mere absence of the cause involved in the antecedent (invited inference condition), the retrieval of alternative causes is strongly enhanced and in fact very efficient. In this case, the rate of correct uncertainty responses reached 87% in adults, which is far higher than what is usually observed on this inference (see Evans, Newstead, & Byrnes, 1993, for a review of several studies). The enhanced efficiency of this retrieval process explains the fact that the strength of the relation involved in the conditional premise had only a small impact on Denial of the antecedent reasoning when this inference is not invited by the wording of the minor premise. Thus, this pattern of results is consistent with our theory that assumes that both the wording of the minor premise and the strength of the causal relation affect the same retrieval process of alternative antecedents from long-term memory. Mental models theory assumes that reasoning on the basis of an “If P then Q” sentence and a minor premise involves the construction of an initial mental model of the state of affairs described by the conditional premise and, when needed, a further fleshing out of this initial representation (Johnson-Laird & Byrne, 1991; Johnson-Laird, 1999). According to Markovits and Barrouillet (2001), in the case of conditionals using familiar terms this fleshing out would rely on a process of activation and retrieval of knowledge from long-term memory. Thus, the production of inferences should depend both on individuals’ capacity to activate the appropriate knowledge and on the nature of the cues that trigger this retrieval process, that is, the content of the premises. As predicted, weak relations elicited more correct responses for both Affirmation of the consequent and Denial of the antecedent than strong relations. In addition, the modified minor premise led to an increase in the rate of correct responding to Denial of the antecedent. These two facts are consistent with the idea that correct responses of uncertainty to Affirmation of the consequent and Denial of the antecedent depend on the efficiency of retrieval of alternative antecedents from long-term memory. As anticipated, these content effects were modulated by developmental factors. Indeed, the difference between strong and weak relations was generally greater in children than in adults. It should be noted, however, that the youngest participants manifested this effect only for Affirmation of the consequent, whereas the effect was significant for both inferences in older children. Though this unexpected result could be con-
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sidered as being at odds with our theory, we would claim that it is actually in line with it. The activation and retrieval processes are triggered by the content of the conditional premise, and also by the minor premise, and the minor premises of Affirmation of the consequent and Denial of the antecedent could differ in the efficiency with which they activate alternative antecedents. Indeed, the minor premise of Affirmation of the consequent (i.e., “Q is true”) directly matches alternative cases of the form A ⫺ Q, whereas the minor premise of Denial of the antecedent does not when it is phrased using a syntactic negation. Our theory assumes that young individuals have limited cognitive resources that constrain the efficiency of the retrieval process. As a consequence, in young participants, the minor premise of Affirmation of the consequent should be a better cue to retrieve alternative antecedents than the minor premise of Denial of the antecedent. This explains why the strength effect was more pronounced for Affirmation of the consequent than for Denial of the antecedent in our youngest group, a fact already observed by Janveau-Brennan and Markovits (1999). Older children have greater cognitive resources and they can activate and retrieve a greater amount of information (Janveau-Brennan & Markovits, 1999). Thus, they may have benefited more from the cue provided by the weak relations whatever the type of minor premise they were presented with, and they manifested a significant effect on both Affirmation of the consequent and Denial of the antecedent. Finally, adult participants were able to activate and retrieve alternative cases whatever the strength of the causal relation under study, and the strength effect did not reach significance in this group. This study demonstrated that, among many other factors, the strength of causal conditional relations plays a critical role in determining the developmental pattern of specific content effects. Weak causal relations elicit more conditional responses than strong relations in adolescents, in the same way as weak class-based relations elicited more conditional responses in young children (Markovits et al., 1998). These two phenomena can be accounted for by the constraints inherent to the same process of retrieval from long-term memory. Indeed, the efficiency of this process depends, among other things, on the base-level activation of the items to be retrieved (Anderson, 1993; Anderson & Lebière, 1998), and the relative baselevel activation of different items pertaining to the same class of knowledge depends in turn on the structure and organization of this knowledge in long-term memory. As Quinn and Markovits (1998) pointed out, the strength of the association between antecedent and consequent differs from factors such as familiarity, which is also known to increase the tendency to respond in ways that correspond to normative logic (Markovits, 1986). Indeed, strong relations are probably more familiar than weak relations, but they lead to poorer performances. Thus, content effects would depend primarily on the structure of long-term memory, i.e., on a reasoner’s general cognitive architecture, as Markovits and Barrouillet (2001) suggested. The effect of the number of alternative causes reported by Cummins et al. (1991) and Janveau-Brennan and Markovits (1999) is in line with this theoretical framework. Indeed, the greater the number of alter-
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native causes, the higher the probability to retrieve one of them from memory and to give an uncertainty response to Denial of the antecedent and Affirmation of the consequent. Moreover, it has been argued that this theory accounts also for some content effects in reasoning from artificial conditional relations (Barrouillet & Lecas, 1998, in press). Thus, though we are far from a complete understanding of the psychological processes that underpin conditional reasoning, our theory does permit precise predictions of specific content effects and accounts for their developmental evolution. REFERENCES Anderson, J. R. (1993). Rules of the mind. Hillsdale, NJ: Erlbaum. Anderson, J. R., & Lebiere, C. (1998). The atomic components of thought. Hillsdale, NJ: Erlbaum. Barrouillet, P., & Camos, V. (2001). Developmental increase in working memory span: Resource sharing or temporal decay? Journal of Memory and Language, 45, 1–20. Barrouillet, P., & Lecas, J. F. (1998). How can mental models account for content effects in conditional reasoning? A developmental perspective. Cognition, 67, 209–253. Barrouillet, P., & Lecas, J. F. (1999). Mental models in conditional reasoning and working memory. Thinking and Reasoning, 5, 289–302. Barrouillet, P., & Lecas, J. F. (in press). Content and context effects in children’s and adult’s conditional reasoning. Quarterly Journal of Experimental Psychology. Braine, M. D. S. (1990). The “natural logic” approach to reasoning. In W. F. Overton (Ed.), Reasoning, necessity and logic: Developmental perspectives. Hillsdale, NJ: Erlbaum. Braine, M. D. S., & O’Brien, D. P. (1991). A theory of if: Lexical entry, reasoning program, and pragmatic principles. Psychological Review, 98, 182–203. Braine, M. D. S., & O’Brien, D. P. (1998). Mental logic. Mahwah, NJ: Erlbaum. Case, R. (1985). Intellectual development: Birth to adulthood. New York: Academic Press. Case, R. (1992). The mind’s staircase: Exploring the conceptual underpinnings of children’s thought and knowledge. Hillsdale, NJ: Erlbaum. Cowan, N., Nugent, L. D., Elliot, E. M., Ponomarev, I., & Saults, J. S. (1999). The role of attention in the development of short-term memory: Age differences in the verbal span of apprehension. Child Development, 70(5), 1082–1097. Cummins, D. D. (1995). Naives theories and causal deduction. Memory & Cognition, 23, 646–658. Cummins, D. D., Lubart, T., Alksnis, O., & Rist, R. (1991). Conditional reasoning and causation. Memory & Cognition, 19, 274–282. Evans, J. S. B. T., Newstead, S. E., & Byrne, R. M. J. (1993). Human reasoning: The psychology of deduction. Hillsdale, NJ: Erlbaum. Halford, G. S. (1993). Children’s understanding. Hillsdale, NJ: Erlbaum. Halford, G. S., Wilson, W. H., & Phillips, S. (1998). Processing capacity defined by relational complexity: Implications for comparative, developmental, and cognitive psychology. Behavioral and Brain Sciences, 21, 803–864. Janveau-Brennan, G., & Markovits, H. (1999). The development of reasoning with causal conditional. Developmental Psychology, 35, 904–911. Johnson-Laird, P. N. (1999). Deductive reasoning. Annual Review of Psychology, 50, 109–135. Johnson-Laird, P. N., & Byrne, R. M. J. (1991). Deduction. Hillsdale, NJ: Erlbaum. Johnson-Laird, P. N., & Byrne, R. M. J. (2001). Conditionals: A theory of meaning, pragmatics, and inference. Submitted for publication. Kail, R. (1992). Processing speed, speech rate, and memory. Developmental Psychology, 28, 899–904. Klaczynski, P. A., & Narashimham, G. (1998). Representations as mediators of adolescent deductive reasoning. Developmental Psychology, 5, 865–881.
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