to a Piagetian approach, reasoning with verbal propositions is a formal ability and should not appear until around 12 years (or later). In this perspective, the ...
Copyright 1999 by the American Psychological Association, Inc. 0012-1649/99/S3.00
Developmental Psychology 1999, Vol. 35, No. 4, 904-911
The Development of Reasoning With Causal Conditionals Genevieve Janveau-Brennan and Henry Markovits Universite du Quebec a Montreal A total of 512 children in Grades 1 through 6 received a conditional inference task using causal conditionals (If cause P, then effect Q) and a generation of alternatives task. The inference task used premises for which there were few or many possible alternative causes. Results show a steady age-related increase in uncertainty responses to the two uncertain logical forms, affirmation of consequent (AC) and denial of antecedent (DA), and an increase in production of disabling conditions for modus ponens. More uncertainty responses were produced to AC and DA with premises with many possible alternatives. Individual differences in inference production were related to numbers of alternatives produced in the generation task. Results support the idea that both developmental and individual differences in reasoning can be at least partially explained by differential access to knowledge stored in long-term memory.
clear developmental patterns in reasoning abilities (Byrnes & Overton, 1986; Markovits & Vachon, 1989, 1990; O'Brien & Overton, 1980, 1982; Overton, Ward, Black, Noveck, & O'Brien, 1987; Ward & Overton, 1990). Typically, even very young children (in some cases as young as 4 and 5 years of age; see Hawkins, Pea, Glick, & Scribner, 1984) are able to give correct responses to the two valid forms, MP and MT. However, many studies show that children do not give spontaneously correct answers to AC or DA before 12 years of age (Ennis, 1976; Knifong, 1974; Kodroff & Roberge, 1975; O'Brien & Overton, 1982), and even adults often do not respond correctly to these logical forms (Byrne, 1989; Cummins, 1995; Evans, 1989; Thompson, 1994). Younger children typically give a false biconditional response to these two forms (e.g., "P implies Q, Q is true, then P is true"), although some studies (Byrnes & Overton, 1986; Rumain, Connell, & Braine, 1983) have shown that children as young as 10 years of age can correctly resolve the uncertain forms if they are given explicit "countermanding" of a biconditional interpretation of the premises, involving the addition of a statement that the antecedent is not necessary but only sufficient (e.g., "If it rains, then the grass will be wet. But if the sprinkler is on then the grass will also be wet."). These results have been interpreted in various ways. According to a Piagetian approach, reasoning with verbal propositions is a formal ability and should not appear until around 12 years (or later). In this perspective, the generally poor results obtained with younger children reflect their inability to understand the principles of formal reasoning (Byrnes & Overton, 1986). In contrast to this, some researchers have claimed that young children are capable of reasoning logically but that factors such as context (Dias & Harris, 1988; Hawkins et al., 1984), pragmatic inferences (Rumain et al., 1983), and interpretational factors (Ennis, 1976) may mask their abilities. Recent studies with preadolescent children have shown that performance on classical conditional reasoning problems is strongly affected by information-processing constraints linked among other things to the way that information is structured and retrieved in long-term memory (Markovits, Fleury, Quinn, & Venet, 1998; Markovits et al., 1996). These effects mirror those observed in adolescents (Markovits & Vachon, 1990) and adults (Cummins, 1995; Cummins, Lubart, Alksnis, & Rist, 1991; Thompson, 1994) and suggest that a key element in understanding
Conditional reasoning is one of the cornerstones of advanced thinking. As such, it is probably one of the most extensively studied forms of logical reasoning, both in the adult and the developmental literatures. Conditional reasoning involves making inferences on the basis of some given "if-then" relation. Most studies of conditional reasoning have examined children's abilities to make inferences on the four basic logical forms. Modus ponens (MP) is the logical principle that involves reasoning with the premises "P implies Q, P is true" and leads to the logically correct conclusion "Q is true." Modus tollens (MT) involves reasoning with the premises "P implies Q, Q is false" and leads to the logically correct conclusion "P is false." These two are valid logical forms, because they both lead to a single, logically correct conclusion. Affirmation of the consequent (AC) involves reasoning with the premises "P implies Q, Q is true." Denial of the antecedent (DA) involves reasoning with the premises "P implies Q, P is false." Neither of these forms leads to a single, logically correct conclusion, and we subsequently refer to these as uncertain logical forms. Conditional reasoning is one of the abilities that characterizes Piaget's stage of formal reasoning (Inhelder & Piaget, 1955; Overton, 1990). According to this view, preadolescent children do not have the cognitive abilities required to reason "logically" with purely verbal propositions. Much developmental research on conditional reasoning abilities has in fact focused on the question of whether young children can make logically correct deductions on the basis of conditional premises. In fact, many studies have shown
Genevieve Janveau-Brennan and Henry Markovits, Departement de psychologic Universite du Quebec a Montreal, Quebec, Canada. Preparation of this article was supported by doctoral scholarships from the Fonds pour la Formation de Chercheurs et l'Aide a la Recherche (FCAR) and the Fondation de l'UQAM and by grants from the National Sciences and Engineering Research Council of Canada and from FCAR. We thank Denise Cummins for very helpful comments on an earlier version of this article. Correspondence concerning this article should be addressed to Genevieve Janveau-Brennan, c/o Henry Markovits, Departement de psychologie, Universite du Quebec a Montreal, C.P. 8888, Succ. "A," Montreal, Quebec, Canada H3C 3P8. 904
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the development of reasoning may involve a closer examination of the developmental patterns affecting variation in reasoning performance. Recent empirical work on the kinds of inferences made to conditional premises by adolescents and adults has indicated the existence of a clear pattern of variation tied to the specific content of premises (Markovits & Vachon, 1990; Thompson, 1994). The most precise of these concerns the way that adults make inferences on the basis of causal conditionals ("If cause P, then effect Q"). Cummins (Cummins, 1995; Cummins et al, 1991) has shown that an important factor in explaining how adults reason with causal premises is the number of alternative causes (i.e., Causes A that also lead to Effect Q) that are readily available to reasoners. Specifically, these studies show that adults will tend to produce fewer uncertainty responses to AC and DA for premises for which there are relatively fewer alternative causes available in long-term memory. In addition, these studies show that adults will tend to respond with uncertainty to MP and MT to the extent that they can produce what Cummins referred to as disabling conditions, that is, a condition that allows P to be true while Q remains false. Our recent work examining developmental patterns in conditional reasoning has established that children as young as 7 or 8 years of age are able to generate uncertainty responses to AC and DA when presented with class-based premises that allow for easy access of alternative cases (Markovits et al., 1996). In addition, it has been shown that young children's conditional reasoning performance is affected by the relative associative strength of possible antecedents for class-based premises and that reasoning with causal premises is more difficult than with equivalent class-based premises (Markovits et al., 1998). Although these studies provide evidence for the existence of content-based variation in young children's reasoning, they do not provide a detailed look at how children reason with causal conditionals. The developmental course of such reasoning is interesting in its own right. In addition, a key question that can be asked about causal reasoning concerns the extent to which children's inferential processes are subject to the same kinds of variation that has been observed with adults. To examine this basic question, we gave children aged 6 to 11 years causal premises that varied in the number of available alternative causes (established by pretesting). This factor was introduced to determine whether young children's reasoning is sensitive to relative numbers of alternative causes in the same way as adults. A second aspect that was examined here concerned the possible tendency of young children to spontaneously produce disabling conditions to causal premises. Results of a previous study (Markovits et al., 1998) indicated that children may start to produce disabling conditions unaided toward the end of the age level studied here, despite specific instructions to suppose that the premises were true. To facilitate possible production of disabling conditions, we presented causal conditionals without such instructions, similar to the mode of presentation used by Cummins et al. (1991). A recent study by Vadeboncoeur and Markovits (in press) has in fact shown that varying instructions concerning the extent to which the premise is to be taken as being true does affect production of disabling conditions in adults while not affecting the nature of inferences made to the invalid forms, AC and DA. A second aim of this study concerns an analysis of some of the underlying mechanisms used in reasoning. Markovits (Markovits, 1993; Markovits et al., 1996; Markovits et al., 1998) has proposed
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that an important component of reasoning with "if-then" premises may be the ability to retrieve specific alternative antecedents from semantic memory. According to this view, developmental differences in reasoning, particularly with the two uncertain forms, may be partly accounted for by age-related differences in the efficiency of this retrieval process. Specifically, one of the developmental parameters that may differentiate younger from older children is the relative difficulty they have in activating and retrieving information that is in memory. To provide a possible rough measure of such a process, we used a generation-of-alternatives task, in which we simply asked children to produce as many possible alternative causes as possible for a given "If P, then Q" in a limited time period. The premises used in this task were different from those used in the reasoning task. If efficiency of retrieval is in fact a factor in reasoning performance, then we would expect there to be a relationship between the relative numbers of alternatives produced and performance on the two uncertain logical forms in the reasoning task. Pretest We needed to construct premises that were understandable to young children and that had varying numbers of possible antecedents in the age range examined. To do this, we presented a set of six conditional premises to Grade 2 children. We asked them to generate as many possible alternative antecedents within a time period of 1 min (which had been found to be a period long enough for most children to stop responding at its end). To ensure the obtained difference continued at a later age, we repeated the procedure with the four items with the most and least alternatives with children in Grade 5.
Method Participants. A total of 19 children (11 girls and 8 boys; mean age = 7 years 6 months) in Grade 2 and 20 children (9 girls and 11 boys; mean age = 10 years 4 months) in Grade 5 of a French elementary school in Ottawa, Ontario, Canada, were examined. None of these children took part in any of the other studies. All of the children were native French speakers. Procedure. Children were met individually. The experimenter told them that they were to be given a series of riddles in which they were to be given an "If P, then Q" relation and to name as many ways of causing Q without P as possible in the space of 1 min. The children were told that for some riddles they would be able to find many causes and for some they would only be able to find a few. They were given two examples to ensure that they understood what was asked of them: (a) "If it is a bee, then it has wings. Name as many things as you can that also have wings." and (b) "If it is a tricycle, then it has wheels. Name as many things as you can that also have wheels." At each of the six trials, children chose one of the six premises at random by picking a card from a closed box and were given 1 min to generate possible alternative antecedents. Their responses were recorded.
Results The number of alternative responses was determined by adding up the total number of responses given and then adjusting this in two ways. First, responses that were not correct (i.e., were not possible causes of the consequent) were eliminated. Second, all responses that repeated the same basic idea (falling in water, falling in a puddle of water) were counted as a single alternative.
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The mean number of alternatives produced by children in Grades 2 and 5 is given in Table 1. The two premises with the most number of alternatives and the two with the least in Grade 2 were presented to Grade 5 children. As can be seen, the difference between the two sets remains at the later grade. These four premises were thus used in the following study. Method
Participants A total of 103 children in Grade 1 (mean age = 6 years 4 months; 52 girls and 51 boys), 102 in Grade 2 (mean age = 7 years 6 months; 47 girls and 55 boys), 105 in Grade 3 (mean age = 8 years 5 months; 46 girls and 59 boys), 101 in Grade 5 (mean age = 10 years 4 months; 54 girls and 47 boys), and 101 in Grade 6 (mean age = 11 years 4 months; 43 girls and 58 boys) participated in this study. The children were native French speakers and attended one of five elementary schools in Montreal and Sherbrooke, Quebec, Canada.
Materials Videotapes showing a cartoon character with a black background were constructed (all videotapes were in French; English translations are given in the following) in one of four versions. The character greeted the child and then said that he would have some questions to ask the child. In the first version, which used premises with many possible alternatives, he then proceeded to ask the following questions: "If someone breaks his arm, then he will hurt. Suppose that someone breaks an arm. Will he hurt?" (MP). "If someone breaks his arm, then he will hurt. Suppose that someone hurts. Did he break his arm"?" (AC). "If someone breaks his arm, then he will hurt. Suppose that someone does not break his arm. Does he hurt?" (DA). If someone breaks his arm, then he will hurt. Suppose that someone does not hurt. Did he break his arm"?" (MT). The same cartoon character then reappeared and told the child that he had another set of questions to ask. These were the following. "If someone drops a pot, then there will be a noise. Suppose that someone drops a pot. Will there be a noise?" (MP). "If someone drops a pot, then there will be a noise. Suppose that there is a noise. Did someone drop a pot?" (AC). "If someone drops a pot, then there will be a noise. Suppose that someone does not drop a pot. Will there be a noise?" (DA). "If someone drops a pot, then there will be a noise. Suppose that there is not a noise. Did someone drop a pot?" (AC). The second version was the same as the first except that the major premises were the two with few possible alternatives, that is, the first set of questions used the premise "If someone goes to sleep late, then they will be tired" and the second set of questions used the premise "If the electricity
Table 1 Mean Number of Alternative Antecedents Produced by Children in Grades 2 and 5 to Trial Premises Premise If If If If If If
a person breaks his arm, he will hurt. a person drops a pot, there will be noise. a person plays the flute, there will be music. a person takes a bath, he will be wet. a person goes to sleep late, he will be tired. the electricity goes off, the school will be closed.
Grade 2 Grade 5 5.47 5.47 4.79 4.00 3.37 3.11
5.35 5.80 3.35 3.05
goes off, then the school will be closed." Versions 3 and 4 were identical to Versions 1 and 2, respectively, except that the base content of the problems within each set was inverted. For example, in Version 3, participants were asked to respond to problems using "If someone drops a pot, then there will be a noise" as the major premise for the first four problems, whereas the final four problems used "If someone breaks his arm, then he will hurt" as the major premise. The voices on the tapes were all recorded by someone who was unaware of the aim of the experiment. In addition, the four logical questions were recorded once and were dubbed into all the tapes at the appropriate moment. This was done to eliminate any possible effects of presentation on children's responses.
Procedure Children were examined individually in sessions lasting about 15 min. Half of the children received the generation of alternatives task followed by the conditional reasoning task, the other half received the two tasks in the inverted order. Order of presentation was systematically altered for each successive participant. Generation of alternatives task. The experimenter explained to the child that he or she would be given some statements of the form "If P, then Q." Children were told that they would have to name as many different ways of causing Q as possible in the time allotted (which was 30 s in all trials, a time that was sufficient for children to respond adequately but short enough to generate some time constraints and thus simulate an online process). After this, children were given two practice sessions, using two premises having many alternatives (using the same two premises as those used for practice in the pretest). The children were then given a closed box containing two cards each with one of the premises: (a) If someone takes a bath, then he or she will be wet, and (b) If someone plays the flute, then there will be music. Then the children chose one of the two premises by picking a card from the box. They then took out the second card. In each case, they were asked to name (a) ways that someone can get wet without taking a bath or (b) ways of making music without playing the flute. They were given 30 s for their responses, which were recorded on audiotape. Conditional reasoning task. The experimenter told the children that they would watch a single videotape and that during the tape, they would be asked some questions and that they were to answer these questions. They were also told that they were to give the reasons for their answers. The children were also told that if they did not understand a question, the experimenter would repeat it for them. After that, the experimenter started the videotape, which contained one of the four possible versions described previously. The tape was paused after each question. If necessary, the experimenter would repeat the question (something that happened very infrequently). Children's responses and justifications were recorded by the experimenter. The four videotaped versions were varied systematically from one participant to the next.
Results Children's responses to the four logical forms were coded in the following way. For the two uncertain forms, AC and DA, responses that indicated explicit uncertainty or those that denied the proposed conclusion and were accompanied by an explicit justification that referred to possible alternative antecedents were considered correct. For MT, responses that denied the conclusion were considered correct. For MP, responses that accepted the conclusion were considered correct. See Markovits et al. (1996) for a detailed discussion of this coding scheme. All other responses were considered incorrect. Scores for the two problems that corresponded to the same logical form were combined. Thus, for each of the four logical forms, combined scores varied from 0 to 2. It is useful to
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note that children either responded with uncertainty or gave the conclusion that was consistent with the premises (e.g., P implies Q, Q is true, thus P is true) for all the logical forms. Thus, for MP and MT, responses that were not correct were responses of uncertainty. For AC and DA, responses that were not correct corresponded to a biconditional response.
Development of Causal Reasoning An initial analysis looked at the developmental pattern of responses to the four logical forms. Table 2 indicates the percentage of correct responses for each of the four logical forms as a function of grade level and type of premise. For each of the four logical forms, we then performed an analysis of variance (ANOVA) with number of correct responses to the two problems as the dependent variable and grade level and premise type (few vs. many alternatives) as the independent variables. For AC, this showed significant main effects for grade level, F(4, 492) = 51.98, p < .0001, and premise type, F(l, 492) = 31.25, p < .0001. Post hoc analyses were performed using the Neumann-Keuls procedure. This showed that children in Grades 5 and 6 performed better on AC than did younger children. In addition, children in Grade 3 performed better than children in Grade 1. More correct responses were given for premises having many alternatives than for premises having few alternatives. For DA, there was a significant main effect for grade level, F(4, 492) = 29.38, p < .0001, and a significant Grade Level X Premise Type interaction, F(4, 492) = 2.37, p < .05. Post hoc analyses showed that children at Grades 5 and 6 did better on DA for problems with many alternatives compared with younger children. No other difference was significant. For MP, there were significant main effects for grade level, F(4, 492) = 11.40, p < .0001, and for problem type, F(l, 492) = 298.74, p < .0001. There was also a significant Grade Level X Problem Type interaction, F(4, 492) = 6.91, p < .0001. Post hoc analyses indicate that performance on problems with many alternatives was generally superior to problems with few alternatives. For problems with premises having few alternatives, Grade 6 children responded with fewer correct responses to MP than did children in Grades 1 and 2.
Table 2 Percentage of Correct Responses to the Four Logical Forms (MP, MT, AC, and DA) by Grade Level for Premises With Few Possible Alternatives and With Many Possible Alternatives Logical form MT
MP
AC
DA
Grade
Few
Many
Few
Many
Few
Many
Few
Many
1 2 3 5 6
78.1 76.0 68.8 57.3 51.0
97.3 99.0 100.0 98.1 94.9
82.3 88.0 88.5 75.0 72.1
88.2 93.3 93.9 84.9 87.8
10.4 19.0 34.4 55.2 64.4
17.3 40.4 49.1 82.1 82.7
7.3 10.0 14.6 29.2 42.3
4.6 16.4 13.2 50.9 44.9
Note. MP = modus ponens; MT = modus tollens; AC = affirmation of the consequent; DA = denial of the antecedent.
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For MT, there were significant main effects for grade level, F(4, 492) = 4.37, p < .002, and for premise type, F(l, 492) = 12.34, p < .0005. Post hoc analyses showed that fewer correct responses were given to problems with premises having few alternatives than to problems with many alternatives. These results indicate some clear parameters of developmental change in conditional reasoning with causal premises, one that varies according to the logical form. There was a consistent increase in correct responding on the two uncertain logical forms, AC and DA, over the age range examined, with AC leading to higher levels of performance than DA. For the these forms, more responses of uncertainty were produced with premises having many alternatives than for those with few alternatives. Generally, children in Grades 5 and 6 produced consistently higher levels of uncertainty responses to the uncertain forms than did younger children. The pattern observed for MP and MT was different. The major effect here was a decrease in correct performance on both MP and MT with age. This was particularly marked for responding to MP with premises having few alternatives, in which close to half of the older children (51%) refused to endorse MP. These results are consistent with results obtained with adults by Cummins (1995; Cummins et al., 1991), who found that adults will tend to refuse MP if they are able to generate "disabling conditions," that is, conditions that can empirically disable the causal link between the cause and the effect described in the major premise. For example, take the premise, "If there is a loss of electricity, then the school will close." This relation can be modulated by considerations such as "If there is an emergency lighting system" that would allow for the antecedent to be true and for the consequent to be false. Thus, one possible explanation for the relative decrease in performance on MP with age would be to suppose that older children are more successful in generating disabling conditions. We return to this point when we examine the justifications that the children used.
Children's Justifications In addition to encoding responses to the logical questions, we also asked the children to justify their responses. To get a clearer picture of the overall developmental pattern of justifications, we first divided the participants into two groups: The older children included those in Grades 5 and 6, and the younger children included those in Grades 1, 2, and 3. Justifications were coded using a taxonomy that included 14 categories. Two independent judges each coded about half the justifications from the written transcripts. The two judges also both coded the same subsample of justifications (representing about 20% of the total sample). The overall kappa coefficient for this sample was .83. For each of the four logical forms, only those categories that represented more than 10% of children's responses or that were particularly interesting because of their specific nature were retained for subsequent analysis. For the two uncertain forms, AC and DA, the following categories were retained: 1. Alternatives. Explicit referral to the existence of other possible causes leading to the same consequent (e.g., "There are other ways of making noise" and "Breaking a window also makes noise").
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Percentage of Production of Various Justifications for Responses to the Two Uncertain Logical Forms as a Function of Age and Premise Type (Few and Many) Age level Older
Younger Logical form and justification Affirmation of the consequent Alternatives Repetition Knowledge Biconditional Other Denial of the antecedent Alternatives Repetition Knowledge Biconditional Negation of alternative causes Other
Many
Few
Many
Few
37.5 16.2 7.9 3.7 34.8
22.6 16.1 13.0 7.9 40.4
82.4 6.4 2.9 2.0 6.4
62.0 9.5 10.5 2.5 15.5
12.80 28.05 7.32 24.09 14.9 12.8
11.30 19.18 26.03 19.5 3.4 20.6
49.02 13.73 3.92 9.8 16.7 6.9
36.5 10.00 26.5 8.5 6.0 12.5
2. Repetition. Any repetition of either the major or the minor premise. 3. Knowledge. Use of knowledge about the world or specific personal experience, with the exception of knowledge about alternative causes. 4. Biconditional. A justification that involves assuming that not-P implies not-Q (e.g., "If someone does not drop a pot, then there won't be any noise."). 5. Negation of alternative causes. This involves explicitly denying the possibility that any other cause could have led to the same effect (e.g., "Nothing else was dropped that could make noise."). 6. Other. The relative percentage of the justifications used by the children as a function of age and premise type for the two uncertain forms is given in Table 3. Generally, the kinds of justifications used reflect the same pattern of results obtained for the responses. The key category here is the "alternatives" category, which requires explicit reference to a possible alternative cause for Q. ANOVAs with percentage of production of "alternatives" justifications to AC and to DA as dependent variables and age (younger vs. older) and premise type (few alternatives vs. many alternatives) as independent variables were performed. For AC, this indicated significant main effects of age, F(l, 508) = 158.17, p < .0001, and premise type, F(l, 508) = 27.51, p < .0001. For DA, there were significant main effects of age, F{\, 508) = 107.40,/? < .0001, and premise type, F(l, 508) = 4.08, p < .05. For both AC and DA, older children gave relatively more "alternatives" justifications than did younger children. In addition, relatively more such justifications were produced for premises with many alternatives than for premises with few alternatives. Thus, the pattern of use of the "alternatives" justification is consistent with our hypotheses. For the two certain forms, MP and MT, the following categories were retained: 1. Disabling. Explicit mention of a condition that could allow P to be true without necessarily entailing Q.
2. Alternatives. Explicit referral to the existence of other possible causes leading to the same consequent. 3. Repetition. Any repetition of either the major or the minor premise. 4. Knowledge. Use of knowledge about the world or specific personal experience, with the exception of knowledge about alternative causes or disabling conditions. 5. Biconditional. A justification that involves assuming that not-P implies not-Q (e.g., "If someone does not drop a pot, then there won't be any noise."). 6. Reductio ad absurdum. An explicit argument of the form "Not-Q is true. If P is true, then Q is true, thus not-P must be true." (used only for MT). 7. Negation of alternative causes. This involves explicitly denying the possibility that any other cause could have led to the same effect (e.g., "Nothing else was dropped that could make noise."). The relative percentage of the justifications used by the children as a function of age and premise type for the two certain forms is given in Table 4. For MP, the key category concerns children's explicit use of disabling conditions. An ANOVA with the percentage of "disabling" justifications to MP as the dependent variable and age and premise type as independent variables indicated significant main effects of age, F(l, 508) = 62.12, p < .0001, and premise type, F(l, 508) = 166.81, p < .0001, and a significant Age X Premise Type interaction, F(l, 508) = 39.91, p < .0001. Post hoc analyses showed that more "disabling" justifications were used for premises with few alternatives and that older children produced more such justifications than did younger children for premises with few alternatives. No developmental difference was found for premises with many alternatives. This pattern shows that the developmental decrease in accepting MP conclusions follows that of children's increasing use of disabling conditions. This implies that, consistent with results obtained by Cummins (1995; Cummins et al., 1991), there should be a clear link between these
Table 4 Percentage of Production of Various Justifications for Responses to the Two Certain Logical Forms as a Function of Age and Premise Type (Few and Many) Age level Younger Logical form and justification Modus ponens Disabling Repetition Knowledge Other Modus tollens Disabling Alternatives Repetition Knowledge Biconditional Reductio ad absurdum Negation of alternative causes Other
Older
Many
Few
Many
Few
0 22.87 50.91 25.61
13.01 18.49 43.49 17.81
2.45 28.43 56.37 12.75
37.00 10.50 41.00 5.50
0.91 4.27 10.98 3.96 17.99 8.84 11.28 39.63
0.34 4.79 17.47 9.93 13.01 8.21 2.74 40.07
4.90 8.82 24.51 6.86 10.29 15.69 11.76 15.20
7.50 14.50 15.00 11.50 7.00 6.50 5.50 25.50
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two. To look at this more closely, we examined the relation between responses to MP and production of disabling conditions for both premise types. As expected, the correlation between production of disabling conditions and refusal of the MP inference was high for both types of premise: .70 (p < .001) for premises with few alternatives and .53 (p < .001) for premises with many alternatives. Thus, the observed pattern of responding to MP can be accounted for by differential production of disabling conditions. However, results obtained by Cummins and, in another context, by Thompson (1994) indicate that the factors influencing responses to MP appear to be independent of those influencing responding to AC and DA. Now the fact that children produce "disabling" justifications (and refuse the MP inference) mostly to premises having few alternatives is not consistent with these analyses. There are several possible explanations of this. The simplest would claim that because the premises used in this study were not examined for possible disabling conditions, it might be the case that the premises with few alternatives could also have more disabling conditions. To examine this possibility, we examined 21 children in Grade 2 (mean age = 7 years 3 months; 10 girls and 11 boys) and 22 children in Grade 5 (mean age = 10 years 5 months; 10 girls and 12 boys) attending elementary school in Ottawa, Ontario, Canada. None of the children had participated in the previous studies. We asked the children to give possible disabling conditions ("ways that P could be true without Q being true") for the four premises, with order of presentation of premises counterbalanced between participants. The results of this showed that the mean numbers of disabling conditions for the two premises with few alternatives were 2.24 and 3.32 in Grades 2 and 5, respectively. The corresponding means for premises with many alternatives were 2.62 and 3.14. We performed an ANOVA with numbers of disabling conditions as the dependent variable and grade level as the independent variable with repeated measures across premise type. This indicated only a main effect of grade level, F(l, 41) = 4.01, p < .05. No other effect was significant. These results confirm the developmental pattern that older children are better at producing disabling conditions than are younger ones (even though it is important to note that even the younger children can produce such conditions when prompted). However, the results do not allow an explanation of the difference observed between the two premise types. Another explanation (which was suggested by a reviewer) concerns the possible effects of causal strength. Cummins (1995) chose items that had been rated as having a strong causal relationship between antecedent and consequent terms. It is thus possible that the premises might have varied according to this dimension in addition to relative numbers of possible alternatives. We thus asked 15 university students to rate the causal strength of the four premises used in this study on a 5-point scale. (Given the steady increase with age found in refusing the MP premise, we considered that any such difference should reasonably be observed in older participants.) In fact, the two premises with few alternatives were rated as having a less strong causal relation (3.5) than the premises with many alternatives (4.2). Such a factor might well account for at least part of the observed difference in performance on MP. However, it should also be noted that analysis of justifications implies that the difference in performance on MP is accountable for by differences in production of disabling conditions, something that is not readily explicable by the difference in causal
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strength, unless an interaction between access to disabling conditions and causal strength is assumed. For MT, the pattern of justifications is much more complex than for the other logical forms. Although there is a pattern to the production of "disabling" justifications that is similar to that found for MP, the level of these is much smaller, possibly reflecting the fact that the relative decrease in performance in MT is much smaller than that found for MP. Also to be noted are the relative increases in "alternatives" with age, and the very small number of reductio ad absurdum arguments. The latter is particularly interesting, because this corresponds to the "logically appropriate" justification for MT. However, the large number of justifications used here appears to indicate that children may be using a variety of strategies to respond to MT, something that is not reflected in a simple analysis of performance.
Relation Between Reasoning and Generation of Alternatives The generation-of-alternatives task requires participants to simply generate as many alternatives as they can in a limited amount of time. A first analysis looked at the overall developmental pattern of the number of alternatives. To do this, we calculated the total number of alternatives generated on the two tasks combined, and the average number as a function of age is shown in Table 5. These results clearly show an increase in total numbers with age. We initially examined how individual differences in generating alternatives might be related to reasoning performance by including number of alternatives generated in the generation task as a covariate in ANOVAs with age and premise type as independent variables for each of the four logical forms. These analyses indicated that the number of alternatives was significantly related to performance for AC, F(l, 508) = 20.42, p < .001, for DA, F(l, 508) = 16.99, p < .001, and for MP, F(l, 508) = 6.01, p < .02. There was no relation between performance on MT and numbers of alternatives generated. To provide a more synthetic picture of these effects, we classed the numbers of alternatives that were generated into three categories. Children giving fewer than 8 alternatives were classed as low, children giving 9 or 10 alternatives were classed as intermediate, and children giving more than 11 alternatives were classed as high. This gave a roughly even split of all children (162 low, 164 intermediate, and 186 high). We also classed children into older (Grades 5 and 6) and younger (Grades 1 to 3) groups. Table 6 indicates the percentage of correct responses for MP, AC, and DA as a function of age, premise type, and relative numbers of alternatives generated. For each of the three logical forms for which a
Table 5 Mean Number of Alternatives Generated on the Two Generation Tasks (Combined) as a Function of Age Grade
No. of alternatives 6.43 8.55 9.54 10.97 11.72
JANVEAU-BRENNAN AND MARKOVITS
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correlated with their tendency to accept the MP inference. No relation between number of alternatives and performance on MT was observed.
Table 6 Percentage of Correct Responses to MP, AC, and DA by Premise Type and Age for Children Generating High, Intermediate, and Low Numbers of Alternatives in the Generation Task Age and production of alternatives
Discussion
Logical form N
MP
AC
DA
Many alternatives Older High Intermediate Low Younger High Intermediate Low
63 23 16
94 100 100
84 80 78
48 56 38
29 61 74
100 100 97
57 43 21
26 11 05
Few alternatives Older High Intermediate Low Younger High Intermediate Low
57 29 14
48 62 61
62 64 39
38 40 14
37 51 58
68 77 76
34 21 14
18 08 09
Note. MP = modus ponens; AC = affirmation of the consequent; DA = denial of the antecedent.
significant effect of number of alternatives was found, an ANOVA with age, premise type, and relative numbers of alternatives as variables was performed. The pattern of results for the variables age and premise type repeated what was previously found, and so we report only those results relating to relative numbers of alternatives. For MP, there was a significant main effect of relative numbers of alternatives, F(2, 508) = 5.52, p < .01. Contrast analyses indicated that children producing few alternatives accepted the MP inference significantly more frequently than children producing many alternatives, F(l, 508) = 5.17, p < .05. For AC, there was a significant main effect of relative numbers of alternatives, F(2, 508) = 5.52, p < .01. Contrast analyses indicated that children producing few alternatives produced uncertainty responses significantly less frequently than children producing many alternatives, F(l, 508) = 22.03, p < .001. For DA, there was a significant main effect of relative numbers of alternatives, F(2, 508) = 7.78, p < .001. Contrast analyses indicated that children producing few alternatives produced uncertainty responses significantly less frequently than children producing many alternatives, F(l, 508) = 22.03, p < .001. There was also a significant Age X Relative Numbers of Alternatives interaction, F(l, 508) = 3.78, p < .05. Contrast analyses indicated that this affected only the relative performance of children producing intermediate numbers of alternatives, and no effect was found on the difference between children producing few and many alternatives. Generally, these analyses indicate that the number of alternatives produced by children in the generation task is positively correlated with children's tendency to produce uncertainty responses to the two uncertain forms, AC and DA, and is negatively
This study has examined two basic aspects of reasoning with causal conditionals. First, the results obtained permit a clear picture of the development of such reasoning between the ages of 6 and 11 years. It should first be noted that the reasoning task used here is not logical in the classic sense, because children were not explicitly required to assume the truth of the major premise. Instead we examined the developmental pattern of inferences that are made on the basis of a given causal premise, while allowing for the possibility of children spontaneously producing conditions that could disable the given causal relation. The youngest children (first graders who are 6-years-old) studied here generally produce a pattern of reasoning that corresponds to the classic biconditional pattern often found in young children; that is, they accept the MP and MT conclusions but also conclude with certainty for AC and DA. However, by the second and third grades, children are starting to show the ability to respond with uncertainty to AC, which is the less difficult of the two uncertain forms. It is interesting to note that these children are also starting to produce disabling conditions and denying the MP conclusion in certain cases. Finally, the older children are able to respond with relatively high levels of uncertainty to DA, although this remains more difficult than AC. This increase is accompanied by the production of even greater numbers of disabling conditions and a corresponding decrease in acceptance of the MP conclusion, although only for a subset of the premises used. The pattern of results also indicates that children are indeed sensitive to the relative numbers of alternative antecedents that characterize causal conditionals. As soon as children are able to spontaneously generate a relatively high proportion of uncertainty responses to AC and DA, their responses to these two logical forms vary according to numbers of alternatives. This is particularly striking with AC, because the effect of this factor is present even among the youngest participants. This thus shows that young children appear to be sensitive to numbers of alternatives in the same way as adults (Cummins, 1995; Cummins et al., 1991). In addition, they are able to spontaneously generate disabling conditions to deny the MP conclusion in the same way as adults (Cummins, 1995; Cummins et al., 1991), although it is important to note that the children in this study produced disabling conditions only for premises having relatively few possible alternatives, something that has not been found in adults. Part of the difference between the children and adults' responding here might reflect differences in the causal strength of the premises used in the present study. Generally, the fairly regular changes in responding to causal conditionals and the observed developmental pattern, in addition to the effects of numbers of alternatives, are certainly consistent with the idea that the basic processes of reasoning with concrete content may be similar in both children and adults. This in turn supports the notion that development in this period may be mediated at least partly by increases in the efficiency and speed of these processes (Markovits et al., 1998). Finally, it should be noted that the generation of alternatives (or disabling conditions or both) assumes the presence of some kind of causal theory (Cummins,
REASONING WITH CAUSAL CONDITIONALS 1995). This is the case because the conversion of a causal conditional of the form "If cause P, then effect Q" into something of the form "Other ways of obtaining effect Q" requires the elaboration of a theory-based criterion (e.g., "Hard things can be used to break windows"). The second aim of this study was to explicitly examine the relationship between the kinds of inferences children make to causal conditionals and a fairly rough measure of informationretrieval capacities. Specifically, we looked at the numbers of alternative causes that children were able to generate in a fairly short time period as a measure of the efficiency of the retrieval process. It is important to add that the premises used in this task and in the reasoning problems were different, so that any effect could not be attributable to training with the specific content of the tasks. Despite this, children producing relatively greater numbers of alternatives in the generation task did produce relatively greater numbers of uncertainty responses to AC and DA. It is interesting to note that children who produced relatively greater numbers of alternatives in the generation task also tended to deny the MP inference more frequently. This is consistent with a more general retrieval model, because analysis of justifications indicates that denial of the MP inference is associated with retrieval of disabling conditions from memory. These results combined with those of a previous study (Markovits et al., 1998) suggest that speed and efficiency of retrieval of information from long-term memory can explain at least part of the developmental pattern observed in conditional reasoning performance. Finally, it should be noted that although these results indicate a fair degree of consistency in how children and adults reason with causal conditionals, there are some specific points at which their performance appears to diverge. This is particularly the case with the large difference in performance between DA and AC and with the lack of any effect of disabling conditions on the MT inference. Thus, any general model of reasoning will have to be able to account for both the similarities and the differences in how children and adults reason with these kinds of premises. References Byrne, R. M. J. (1989). Suppressing valid inferences with conditionals. Cognition, 31, 61-83. Bymes, J. P., & Overton, W. F. (1986). Reasoning about certainty and uncertainty in concrete, causal, and propositional contexts. Developmental Psychology, 22, 793-799. Cummins, D. D. (1995). Naive 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. Dias, M. G., & Harris, P. L. (1988). The effect of make-believe play on deductive reasoning. British Journal of Developmental Psychology, 6, 207-221. Ennis, R. H. (1976). An alternative to Piaget's conceptualization of logical competence. Child Development, 47, 903-919.
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Received May 18, 1998 Revision received November 4, 1998 Accepted November 9, 1998
