Journal of Experimental Child Psychology 128 (2014) 37–51

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On the road toward formal reasoning: Reasoning with factual causal and contrary-to-fact causal premises during early adolescence Henry Markovits Département de Psychologie, Université du Québec à Montréal, Montréal, Québec H3P 3P8, Canada

a r t i c l e

i n f o

Article history: Received 9 January 2014 Revised 28 June 2014

Keywords: Logical reasoning Cognitive development Information processing Deduction Adolescence Content

a b s t r a c t Understanding the development of conditional (if–then) reasoning is critical for theoretical and educational reasons. Here we examined the hypothesis that there is a developmental transition between reasoning with true and contrary-to-fact (CF) causal conditionals. A total of 535 students between 11 and 14 years of age received priming conditions designed to encourage use of either a true or CF alternatives generation strategy and reasoning problems with true causal and CF causal premises (with counterbalanced order). Results show that priming had no effect on reasoning with true causal premises. By contrast, priming with CF alternatives significantly improved logical reasoning with CF premises. Analysis of the effect of order showed that reasoning with CF premises reduced logical responding among younger students but had no effect among older students. Results support the idea that there is a transition in the reasoning processes in this age range associated with the nature of the alternatives generation process required for logical reasoning with true and CF causal conditionals. Ó 2014 Elsevier Inc. All rights reserved.

Introduction Conditional (if–then) reasoning is one of the most important and widely studied forms of logical reasoning. One of the most important uses of conditionals, and conditional inferences, is the creation of a hypothetical link between two propositions (categories, actions, events, etc.). Such a conditional

E-mail address: [email protected] http://dx.doi.org/10.1016/j.jecp.2014.07.001 0022-0965/Ó 2014 Elsevier Inc. All rights reserved.

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allows mathematicians and scientists to create a theoretical world in which the hypothesized relation is true in order to examine the potential consequences and then compare these with empirical data. This is one of the key components of formal thinking (Inhelder & Piaget, 1958). Understanding whether an inference is logically necessary must be done without using knowledge or belief. However, developmental studies of the ability to make logical inferences with conditional relations clearly show a complex pattern of dependence on empirical knowledge. For example, children find it much easier to make logical inferences with very concrete premises (‘‘If an animal is a dog, then it has four legs’’) than with causal premises (‘‘If a rock is thrown at a window, then the window will break’’). This suggests that there is an equally complex developmental progression preceding the ability to reason abstractly. In the current study, we specifically focused on the transition between reasoning logically with factual causal conditions and the ability to reason logically with contrary-to-fact (CF) conditionals during early adolescence. Conditional reasoning involves reasoning on the basis of a given ‘‘if P then Q’’ premise (where P is the antecedent term and Q is the consequent term). There are four basic inferences that can be made from a given if–then premise by affirming or denying the antecedent or consequent term. Two of these lead to logically certain conclusions. The most direct of the four inferences is called modus ponens (MP), from the Latin term meaning ‘‘affirms by affirming,’’ and involves the premises ‘‘If P then Q’’ and ‘‘P is true,’’ leading to the logical conclusion that ‘‘Q is true.’’ The modus tollens (MT) inference, from the Latin term meaning ‘‘denies by denying,’’ involves the premises ‘‘If P then Q’’ and ‘‘Q is false,’’ leading to the logical conclusion that ‘‘P is false.’’ The two remaining inferences do not allow any certain conclusion. The first of these is the affirmation of the consequent (AC), which involves the premises ‘‘If P then Q’’ and ‘‘Q is true.’’ Take the following example: ‘‘If a rock is thrown at a window, then the window will break. Suppose that a window is broken.’’ In this case, the conclusion that ‘‘a rock was thrown at the window’’ is not logically certain because something else might have broken the window. The second of these is the denial of the antecedent (DA), which involves the premises ‘‘If P then Q’’ and ‘‘P is false.’’ Similar to the analysis of the AC inference, the possible conclusion that ‘‘Q is false’’ is not certain. A major problem in understanding the development of conditional reasoning is the very strong variation in the kinds of inferences made by children and adults when premise content is varied. Children as young as 6 or 7 years can reason logically on the AC and DA inferences with some category-based premises; for example, ‘‘If an animal is a dog, then it has legs’’ (Markovits, 2000; Markovits & Thompson, 2008). By contrast, even adults do not consistently give the logical response to these same inferences when reasoning with true causal conditionals; for example, ‘‘If a rock is thrown at a window, then the window will break’’ (Cummins, Lubart, Alksnis, & Rist, 1991). Individual and developmental differences in reasoning with concrete premises up to middle adolescence are related to working memory, retrieval efficiency, and inhibitory capacity (De Neys & Everaerts, 2008; JanveauBrennan & Markovits, 1999; Klaczynski & Narasimham, 1998; Markovits & Barrouillet, 2002; Simoneau & Markovits, 2003). These suggest that a key component of the development of reasoning abilities in this age range involves the way that information about premises is used during reasoning. Premises used in inferential problems explicitly present the relation between antecedent and consequent terms and the potential truth value of one of these (e.g., ‘‘P implies Q’’ and ‘‘Q is true’’). However, there are other implicit forms of information that are relevant to reasoning. The first refers to potential alternatives to the antecedent term. For example, take the premise ‘‘If a rock is thrown at a window, then the window will break.’’ In this case, ‘‘throwing a chair at a window’’ is an example of an alternative antecedent because it is a concrete example of another way to break a window. Having more potential alternative antecedents in long-term memory increases the probability of producing the logically correct response to both the AC and DA inferences in children as well as adults (e.g., Cummins, 1995; Cummins et al., 1991; Daniel & Klaczynski, 2006; Janveau-Brennan & Markovits, 1999; Klaczynski & Narasimham, 1998; Markovits & Vachon, 1990; Thompson, 1994). Using our previous example, being able to easily retrieve information such as ‘‘throwing a chair’’ allows people to readily conclude that ‘‘If a window is broken, it is not necessarily true that a rock was thrown at the window.’’ A second type of information refers to what Cummins (1995; see also Cummins et al., 1991) called disabling conditions. These are conditions that potentially disable the connection between the

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antecedent and consequent terms. For the causal conditional ‘‘If a rock is thrown at a window, then the window will break,’’ conditions such as ‘‘the window is made of Plexiglass’’ are disabling conditions. The numbers of disabling conditions that readily come to mind are associated with the tendency to reject the MP inference with concrete premises (Cummins, Lubart, Alksnis, & Rist, 1991; De Neys, Schaeken, & D’Ydewalle, 2002). There is evidence that retrieval of disablers during reasoning also depends on memory processes (De Neys, Schaeken, & d’Ydewalle, 2003a; Simoneau & Markovits, 2003). The most striking form of disabler is associated with CF premises, that is, conditional relations that are factually false. In this case, the disabling condition corresponds to the ‘‘true’’ conditional relation. This is so clear that very young children find it very difficult to inhibit this form of knowledge. For example, when given a premise such as ‘‘If a feather is thrown at a window, then the window will break,’’ most children will conclude that ‘‘If a feather is thrown at a window, the window will not break’’ (Dias & Harris, 1988, 1990). The ability to correctly accept this kind of inference increases with age up to middle adolescence (Markovits & Vachon, 1989). This difficulty can be overcome by creating an imaginary context into which CF reasoning problems are inserted (Dias & Harris, 1988, 1990; Markovits & Vachon, 1989). It is important to note that using such a context increases not only acceptance of the MP inference but also acceptance of the AC and DA inferences (Markovits, 1995; Markovits et al., 1996). In other words, imaginary contexts act by generally restricting access to memory-based information, which leads to increased logical responding to the MP inference but also to increases in the acceptance of the AC and DA inferences. Contrarily, asking reasoners to produce alternative antecedents to a specific premise, which decreases acceptance of these inferences, can also result in increased rejection of the MP inference (Markovits & Potvin, 2001). Increasing access to information results in better logical performance on the AC and DA inferences but results in decreased logical performance on the MP inference, whereas decreasing information access leads to the opposite effect. Understanding the basic logic of conditionals, thus, requires two opposing forms of information access: (a) the ability to accept the premises as true, leading to the certainty of the MP inference, which requires inhibiting retrieval of disabling conditions (e.g., Simoneau & Markovits, 2003) and (b) the ability to understand the uncertainty of the AC and DA inferences, which requires increased access to alternative antecedents. (Note that these components do not refer to performance on the MT inference. As we will see later, MT inferences are difficult to interpret.) Complicating this analysis is the fact that logical performance is affected not only by information retrieval processes but also by the semantic class of the premises used for reasoning. Children as young as 7 or 8 years can accept the MP inference and understand the uncertainty of the AC inference when reasoning with category-based conditionals (e.g., ‘‘If an animal is a dog, then it has four legs’’) (Markovits, 2000; Markovits & Thompson, 2008). The ability to reason logically with familiar causal conditionals (e.g., ‘‘If a rock is thrown at a window, then the window will break’’) is later developing and is not found consistently until 10 to 12 years of age (Janveau-Brennan & Markovits, 1999; Markovits & Vachon, 1989). Logical reasoning with CF causal premises (e.g., ‘‘If a feather is thrown at a window, then the window will break’’) is not reliably observed until later during adolescence (14–15 years of age) (Markovits & Vachon, 1989). Finally, the ability to reason logically with abstract premises (‘‘If P then Q’’) appears only during early adulthood (Markovits & Lortie-Forgues, 2011; Markovits & Vachon, 1990; Venet & Markovits, 2001). Understanding how information retrieval processes and developmental effects related to different classes of semantic content can be incorporated into a single model, thus, is a key problem in understanding the development of logical reasoning. Critical to any such model is the idea that information related to alternative antecedents and disabling conditions is not presented with a given inference but rather must be generated by the reasoner during the inferential process. The key component of logical reasoning, thus, is the ability to generate alternative antecedents while inhibiting retrieval of disabling conditions, allowing simultaneous acceptance of the MP inference and rejection of the AC (and DA) inference. A recent study (Markovits & Lortie-Forgues, 2011) presented a developmental model that attempts to explain the transition between reasoning with familiar concrete categories and fully abstract reasoning in terms of the degree of abstraction of the process of alternatives generation,

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which specifically refers to the ability to generate potential alternative antecedents based on the given if–then relation while inhibiting retrieval of disabling conditions. This applies Karmiloff-Smith’s (1995) representational redescription model to alternatives generation. Karmiloff-Smith’s model supposes that people initially learn and become expert in using an algorithm in very simple concrete contexts. Additional experience results in development of more abstract representations of this algorithm, which allows its use in more general contexts. The representational redescription model applied to conditional reasoning suggests the following sequence in the development of the process of alternatives generation. The initial form, found in quite young children, involves retrieving alternative category members when reasoning with classes and properties. For example, children as young as 7 or 8 years can maintain in memory an if–then relation of the kind ‘‘If an animal is a dog, then it has four legs,’’ allowing understanding of the certainty of the MP inference. With this kind of premise, children are also able to retrieve examples of animals that have four legs that are not dogs, (e.g., cats), allowing them to understand the uncertainty of the AC inference (Markovits & Thompson, 2008; Markovits et al., 1996). A more complex form of alternatives generation is related to reasoning with causal conditionals; for example, ‘‘If a rock is thrown at a window, then the window will break.’’ This involves generating an ad hoc category such as ‘‘things that can break windows,’’ one that must be produced on-line during reasoning (Cummins et al., 1991; Janveau-Brennan & Markovits, 1999). A more advanced form is provided by alternatives generation with CF premises such as ‘‘If a feather is thrown at a window, then the window will break’’ (Markovits & Lortie-Forgues, 2011; Markovits & Vachon, 1989). This involves maintaining the certainty of the MP inference (‘‘If a feather is thrown at a window, then the window will certainly break’’) and simultaneously constructing the ad hoc category of ‘‘things that can break windows if feathers can do so.’’ The final, and most abstract, level is required by premises with components that have no concrete referents, that is, abstract reasoning. Table 1 gives a synthesis of this progression. As can be seen from the empirical evidence, there is a clear progression in the age at which children and/or adolescents are able to reason logically with different kinds of premises. The representational model explains this progression by the nature of the alternatives generation process required for each type of premise. Critically, the model suggests that these can interact in different ways depending on the developmental level examined. Although there are not, to our knowledge, any other explanatory models that can account for the observed developmental differences in logical reasoning, it is useful to examine one potential alternative framework. It could be argued that these differences are simply related to familiarity. Younger children can reason logically with more familiar content. Transitions between premise categories could then depend on use of analogy to allow transfer of logical reasoning between more familiar and less familiar content. This would then imply that any procedure that promoted more familiar reasoning processes should result in a corresponding improvement in less familiar reasoning. By contrast, the representational model suggests a more complex pattern. In an initial study examining this model, Markovits and Lortie-Forgues (2011) looked at interactions between familiar causal or CF causal reasoning and abstract reasoning among late adolescents and young adults. Results show that participants who reasoned with familiar causal premises (leading to relatively high levels of logical

Table 1 Developmental patterns of conditional reasoning by premise category and alternatives generation Age level

Premise category

Alternative

Example

7–8 years

If category P then property Q (categorical) If cause P then effect Q (familiar causal)

Category A that also has property Q Cause A can also lead to Q

14–16 years

If cause P then effect Q (CF causal)

20+ years

If X then Y (abstract)

Cause A can also lead to Q (where A is not necessarily a cause of Q) A could lead to Y (A could be anything)

If an animal is a dog, then it has four legs. Cats also have four legs. If a rock is thrown at a window, then the window will break. Chairs can also break windows. If a feather is thrown at a window, then the window will break. Dust can also break windows (if feathers can). If Dolx, then Blax. Something else could also lead to Blax.

10–12 years

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responses) showed no improvement in subsequent abstract reasoning. However, reasoning with CF premises (which resulted in lower levels of logical responses) did improve abstract reasoning. In addition, promoting use of the alternatives generation process with familiar content also had no effect on abstract reasoning, whereas doing so with CF content produced improved reasoning. In other words, simple generalizing from reasoning with familiar content cannot explain the observed developmental transitions. Instead, a more complex analysis of interaction between the alternatives generation process and premise content is required to understand these transitions. The specific aim of the current study was to extend our analysis of relations between alternatives generation and reasoning to an earlier developmental period. We specifically examined the period between 11 and 14 years of age. The previous cited studies suggest that children at this age reason very well with familiar causal premises. The representational model supposes that children who are reasoning with such premises will spontaneously attempt to generate appropriate alternatives to such premises. Failure to reason logically would be due to children’s inability to successfully generate alternatives or to inhibit disabling conditions. This in turn is explicable by the cognitive demands of this process, the difficulty of actually retrieving and constructing the relevant post hoc category, and the like. Interventions that attempt to encourage use of the alternatives generation process required for familiar causal premises should have no effect on reasoning with these kinds of premises because this process should normally be deployed at a very high level at any rate. By contrast, children of this age are just developing the ability to generate CF causal alternatives. Thus, although these children are potentially able to generate such alternatives while inhibiting disabling conditions, they will find it very difficult to spontaneously initiate this process when reasoning with CF premises. Interventions that encourage use of the alternatives generation process for CF premises, thus, should improve levels of logical reasoning. Because previous results (Markovits & LortieForgues, 2011) suggest that CF alternatives generation does not become spontaneously deployed at a very high level before late adolescence, this improvement should occur throughout the age range examined here, although we make no more specific predictions. In short, this model predicts that interventions designed to increase use of alternatives generation processes should have no impact on the ability to reason logically with familiar causal premises in this age range. By contrast, interventions designed to increase the generation of CF alternatives should increase levels of logical reasoning with CF causal premises across the age range examined. A second prediction concerns potential interactions between reasoning with true premises and reasoning with CF premises. The key to this analysis is the idea that deploying an alternatives generation strategy requires a form of metacognitive engagement (Moshman & Franks, 1986; Thompson, Prowse Turner, & Pennycook, 2011). Markovits, Brunet, Thompson, and Brisson (2013) recently provided evidence showing that when people are encouraged to use a lower level reasoning strategy (by restricting available time), and they do not metacognitively recognize that this strategy is inappropriate, they will not change to a higher level strategy that is otherwise available to them. Now, we hypothesized that preadolescents will find it easy to activate a generation of alternatives strategy when reasoning with true causal premises but will find activation of this strategy to be quite hard when reasoning with CF premises. Thus, if first asked to reason with CF conditionals, preadolescents will find it very difficult to activate the alternatives generation strategy and less likely to generate the necessary level of metacognitive engagement. This will create a corresponding suppression effect on the generation process that will transfer to reasoning with true causal conditionals. By contrast, reasoning with factual conditionals first will activate the alternatives generation strategy at a very high level and facilitate metacognitive engagement. Thus, we predict that among the younger adolescents, initial reasoning with CF causal conditionals will result in a general decrease in logical reasoning. However, as older adolescents become more efficient at deploying the alternatives generation strategy with CF premises, and as the overall efficiency of the generation process increases, this suppression effect should diminish. It should be noted that previous results show this same pattern of interaction when order effects between reasoning with abstract premises and reasoning with familiar causal premises are examined (Markovits & Vachon, 1990). To examine these predictions, reasoning with factual and CF causal conditionals during the transitional period between primary and secondary schooling levels (11–14 years of age) was examined. Participants were given two blocks of reasoning problems, one presenting true conditionals and the

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other presenting CF conditionals, with the order of the two blocks systematically varied. Half of these participants were given a simple generation task before reasoning involving either CF alternatives (CF prime) or true alternatives (true prime) generation. The other half were given a no-prime condition with the reasoning problems presented first (although the CF and true alternatives generation tasks were also given after the reasoning problems). It should be noted that predictions solely concern performance on the MP, AC, and DA inferences. How performance on the MT inference (‘‘P implies Q’’ and ‘‘Q is false’’; thus, ‘‘P is false’’) can be considered within the current context is not discussed. Of the four conditional inferences, MT is the most complex. The logically correct response is to accept this inference. Paradoxically, younger children accept the MT inference more often than older children (O’Brien and Overton, 1980), suggesting that correct responding can be done in different ways. In fact, Markovits, Doyon, and Simoneau (2002) studied relations between conditional inferences and working memory and found clear evidence that generating the correct response to MT can be done in two different ways. The first involves using a low-cost biconditional strategy that involves accepting all four inferences. The second involves a more complex and cognitively costly process that corresponds to our definition of logical reasoning because it is associated with both accepting the MP inference and rejecting the AC and DA inferences. Thus, rates of acceptance of the MT inference cannot be unambiguously associated with logical reasoning, and this form is not considered. Method Participants A total of 535 students participated in this study. Of these, 125 were in Primary Grade 5 (average age = 11.1 years; 63 girls and 62 boys), 137 were in Primary Grade 6 (average age = 12.1 years; 71 girls and 66 boys), 143 were in Secondary Year 1 (average age = 13.0 years; 66 girls and 77 boys), and 130 were in Secondary Year 2 (average age = 14.1 years; 70 girls and 60 boys). All students were French speaking and came from middle- to lower middle-class neighborhoods. Materials Booklets were constructed using combinations of the following components. Factual causal conditional reasoning problems Participants first were given the following instructions (translated from the original French): Imagine that you have gone to a planet where everything happens exactly the same way as here on Earth. You must consider that everything that is written on the top of each of the next pages is always true. Some situations will be presented, and you must choose the response that you think is the logical one. On the top of the next page appeared the following: Suppose that it is true that: If it rains outside, then the sidewalk will be wet. For each of the following questions, choose the most logical response. Directly below this, participants were given four inferences, each of which had three possible responses. These corresponded to the MP, DA, AC, and MT inferences. The first was presented in the following way: 1. It is raining outside. a. It is certain that the sidewalk will be wet. b. It is certain that the sidewalk will not be wet. c. It is not certain whether the sidewalk will be wet or not.

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The three other inferential problems used the same format. On the subsequent page, participants were given the major premise ‘‘If a person cuts their finger, then the person will hurt.’’ On the final page, participants were given the major premise ‘‘If a person lets an object fall to the ground, then there will be a noise.’’ For each major premise, four inferential problems corresponding to the MP, DA, AC, and MT inferences were presented directly following the premise. CF conditional reasoning problems Participants first were given the following instructions (translated from the original French): Imagine that you have gone to a planet where everything happens very differently from here on Earth. You must consider that everything that is written on the top of each of the next pages is always true. Some situations will be presented, and you must choose the response that you think is the logical one. On the top of the next page appeared the following: Suppose that it is true that: If a feather is thrown against a window, then the window will break. For each of the following questions, choose the most logical response. Directly below this, participants were given four inferences, each of which had three possible responses. These corresponded to the MP, DA, AC, and MT inferences. The first was presented in the following way: 2. A feather is thrown against a window. a. It is certain that the window will break. b. It is certain that the window will not break. c. It is not certain whether the window will break or not. The three other inferential problems used the same format. On the subsequent page, participants were given the major premise ‘‘If a person breaks their arm, then the person will be happy.’’ On the final page, participants were given the major premise ‘‘If a child helps the teacher, then the child will be punished.’’ For each major premise, four inferential problems corresponding to the MP, DA, AC, and MT inferences were presented directly following the premise. CF alternatives generation task On the top of the first page appeared the following instructions: Imagine that you have gone to a planet where everything happens very differently to here on Earth. Answer the following questions. Following this, participants were given three generation problems. These were as follows: On this planet, if you clean a sweater with ketchup, the sweater will become clean. Can you imagine other ways of cleaning a sweater on this planet? Give as many responses as you can. On this planet, if you jump into a lake with freezing water, you will warm up. Can you imagine other ways of warming up on this planet. Give as many responses as you can. On this planet, eating French fries is good for your health. Can you imagine other things that you can do that are good for your health on this planet? Give as many responses as you can. Each problem was followed by six double-spaced lines. Factual alternatives generation task On the top of the first page appeared the following instructions: Imagine that you have gone to a planet where everything happens exactly the same as here on Earth. Answer the following questions.

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Following this, participants were given three generation problems. These were as follows: On this planet, if you put ketchup on a sweater, the sweater will become dirty. Can you imagine other ways of making a sweater dirty on this planet? Give as many responses as you can. On this planet, if you jump into a lake with hot water, you will warm up. Can you imagine other ways of warming up on this planet? Give as many responses as you can. On this planet, eating French fries is bad for your health. Can you imagine other things that you can do that are bad for your health on this planet? Give as many responses as you can. Each problem was followed by six double-spaced lines. Booklets were constructed, in equal numbers, by each of the following combinations of these components. CF priming condition: 1. CF alternatives generation task + CF reasoning problems + factual reasoning problems 2. CF alternatives generation task + factual reasoning problems + CF reasoning problems Factual priming condition: 3. Factual alternatives generation task + CF reasoning problems + factual reasoning problems 4. Factual alternatives generation task + factual reasoning problems + CF reasoning problems No-prime condition: 5. CF reasoning problems + factual reasoning problems + factual alternatives generation task 6. Factual reasoning problems + CF reasoning problems + factual alternatives generation task 7. CF reasoning problems + factual reasoning problems + CF alternatives generation task 8. Factual reasoning problems + CF reasoning problems + CF alternatives generation task Procedure Booklets were distributed randomly within classes. Students were given verbal directions to read the instructions carefully and to take as much time as they needed to respond to the questions. Results We first calculated mean numbers of correct responses to the four logical forms as a function of condition, grade level, and premise type, as summarized in Table 2 (the two younger and two older grades are grouped together in order to reduce table length). Our analysis of logical reasoning suggests two basic components. One of these requires the ability to inhibit potential disabling conditions in order to accept the MP inference. We first examined this component. The clearest index of an inability to inhibit disabling conditions, and one of the more frequent errors in logical reasoning with CF premises (Markovits & Vachon, 1989), is what we refer to as empirical errors. These imply that the major premise is considered to be false despite instructions to accept the truth of the premises and leads to reasoning of the following kind: ‘‘If P then Q’’ and ‘‘P is true’’; thus, ‘‘Q is false.’’ We performed an analysis of variance (ANOVA) with number of empirical errors on the MP inference as a dependent variable with premise type (CF or true) as a repeated measure and grade, condition (CF prime, true prime, or no prime), and order (true premises first or CF first) as independent variables. This indicated significant main effects of grade, F(3, 490) = 8.34, p < .001, partial g2 = .049, condition, F(1, 490) = 4.66, p < .001, partial g2 = .019, and premise type, F(1, 490) = 179.09, p < .001, partial g2 = .268, as well as significant interactions involving Grade  Premise Type, F(3, 490) = 5.20, p < .001, partial g2 = .031, Condition  Premise Type, F(2, 490) = 5.06, p < .03, partial g2 = .020, and Grade  Premise Type  Condition  Order, F(6, 490) = 2.74, p < .02, partial g2 = .032. Post hoc analyses were performed using Student–Newman–Keuls tests with p = .05. Analysis of the Grade  Premise Type interaction showed that the number of empirical errors with the true premises remained close to 0 with no change by grade, whereas the systematic grade-related decline in empirical errors was concentrated on CF premises (see Fig. 1).

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Table 2 Mean numbers of logically correct responses (max = 3) for each of the four logical forms with true and CF premises as a function of grade level and condition Condition

Real premises Premise order

Primary level CF priming CF first Real first Real priming

CF first Real first

No prime

CF first Real first

Secondary level CF priming CF first Real first Real priming

CF first Real first

No prime

CF first Real first

CF premises MP

DA

AC

MT

MP

DA

AC

MT

2.58 (0.79) 2.42 (0.66) 2.50 (0.72) 2.37 (0.77) 2.39 (0.73) 2.30 (0.78)

0.24 (0.71) 0.67 (0.99) 0.41 (0.87) 0.66 (0.94) 0.45 (0.92) 0.59 (0.98)

0.73 (1.10) 1.67 (1.22) 1.13 (1.18) 1.89 (1.16) 1.24 (1.17) 1.60 (1.17)

2.30 (1.02) 2.67 (0.69) 2.50 (0.80) 2.57 (0.65) 2.44 (0.92) 2.51 (0.84)

1.97 (1.29) 1.94 (1.39) 1.69 (1.40) 1.44 (1.42) 1.02 (1.21) 1.52 (1.41)

0.52 (0.76) 0.81 (1.01) 0.31 (0.64) 0.86 (0.87) 0.64 (0.76) 0.75 (0.89)

0.67 (0.92) 0.97 (1.14) 0.63 (1.01) 1.25 (1.16) 0.73 (0.88) 0.88 (1.08)

1.91 (1.10) 1.71 (1.07) 1.78 (1.10) 1.56 (1.08) 1.18 (1.13) 1.48 (1.01)

2.71 (0.59) 1.86 (0.99) 2.60 (0.69) 2.14 (0.96) 2.45 (0.80) 2.11 (0.80)

0.55 (0.85) 1.17 (1.22) 0.80 (1.05) 0.97 (1.24) 1.03 (1.21) 0.83 (1.01)

1.55 (1.36) 2.09 (1.17) 1.57 (1.29) 1.86 (1.19) 1.92 (1.09) 2.22 (0.94)

2.44 (0.67) 2.09 (1.04) 2.27 (0.91) 2.31 (0.93) 2.08 (0.96) 2.28 (0.93)

2.45 (0.94) 2.06 (1.23) 1.94 (1.29) 1.91 (1.26) 1.72 (1.34) 1.54 (1.39)

0.53 (0.86) 1.14 (1.06) 0.69 (0.89) 1.00 (1.12) 0.78 (0.86) 1.14 (1.05)

1.06 (1.22) 1.74 (1.19) 1.47 (1.25) 1.61 (1.30) 1.39 (1.28) 1.63 (1.22)

2.16 (0.92) 1.56 (0.95) 1.86 (1.05) 1.76 (1.03) 1.63 (1.16) 1.70 (1.13)

Fig. 1. Mean numbers of empirical errors on the MP inference with CF and true premises as a function of grade level.

Analysis of the Condition  Premise Type interaction showed that condition had no effect on empirical errors with true premises, which were close to 0 overall (M = 0.07, SD = 0.38). Numbers of empirical errors with CF premises were significantly lower in the CF prime condition (M = 0.60, SD = 1.09) than in the no-prime condition (M = 1.06, SD = 1.27). There was no significant difference between numbers of empirical errors in the true prime condition (M = 0.88, SD = 1.27) and those in

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the other two conditions. Analysis of the Grade  Premise Type  Condition  Order interaction did not reveal any significant deviations from these patterns. Thus, as expected, most empirical errors on the MP inference occurred with CF premises, and these showed a continuous decrease over the age range examined here. Importantly, priming with CF alternatives produces a significant decrease in such errors compared with the no-prime condition. Our basic hypothesis is that encouraging the alternatives generation process should improve overall logical reasoning with CF premises. Certainly, the observed decrease in empirical errors is consistent with this. However, previously mentioned results suggest that a decrease in empirical errors might be due simply to a global decrease in information retrieval, which would result in a corresponding increase in the acceptance of the AC and DA inferences (Markovits, 1995). To examine this possibility, we calculated the number of times that participants accepted the AC and DA inferences. We then performed an ANOVA with number of acceptances as a dependent variable with premise type as a repeated measure and grade, condition (CF alternates first, real alternates first, or reasoning first), and order (true premises first or CF first) as independent variables. This indicated significant main effects of grade, F(3, 488) = 3.60, p < .05, partial g2 = .025, and order, F(1, 488) = 17.11, p < .01, partial g2 = .008, as well as a significant Grade  Premise Type interaction, F(3, 488) = 3.78, p < .05, partial g2 = .027. Analysis of response patterns showed that the level of acceptance of the AC and DA inferences decreased with age, with the greatest decrease being found with true premises. Overall levels of acceptances were higher when the CF problems were given first. Critically, no effect of priming was found. This analysis shows that the increase in logical reasoning on the MP inference with CF premises cannot be accounted for by a global decrease in information retrieval. Our specific hypothesis is that encouraging CF alternative generation will result in an improvement in overall logical reasoning, that is, in the ability to respond with uncertainty to the AC and DA inferences while accepting the major premise as true (i.e., accepting the MP inference). To examine this, we calculated a logical reasoning score in the following way. For each of the true and CF premises, we gave 1 point when participants correctly made the MP inference and, at the same time, gave an uncertainty response to the AC inference. We also gave 1 point when they accepted the MP inference and, at the same time, gave an uncertainty response to the DA inference. Thus, this score measures the frequency with which participants were able to both accept the premise as true (inhibit disabling conditions) and, at the same time, respond with uncertainty to the AC and DA inferences (generate alternative antecedents). This gave a score varying between 0 and 6 for each of the two premise types. We then performed an ANOVA with number of logical responses as a dependent variable with premise type as repeated measure and grade, condition (CF alternates first, real alternates first, or reasoning first), and order (true premises first or CF first) as independent variables. This indicated significant main effects of grade, F(3, 488) = 11.76, p < .001, partial g2 = .067, order, F(1, 488) = 4.51, p < .05, partial g2 = .009, and premise type, F(1, 488) = 69.71, p < .001, partial g2 = .125, as well as significant interactions involving Grade  Order, F(3, 488) = 2.89, p < .05, partial g2 = .017, and Premise Type  Condition, F(2, 488) = 7.58, p < .001, partial g2 = .030. Post hoc analyses were performed using Student–Newman–Keuls tests with p = .05. Analysis of interactions were performed using the Tukey test with p = .05. Analysis of the Condition  Premise Type interaction showed the following pattern. There was no effect of condition on numbers of logical responses to the true premises. More logical responses to CF premises were produced in the CF prime condition (M = 1.25, SD = 1.72) than in the no-prime condition (M = 0.85, SD = 1.53). Number of logical responses in the true prime condition did not differ from either of these (M = 1.09, SD = 1.72) (see Fig. 2). More logical responses were produced to the true premises than to the CF premises in the true prime condition (true: M = 1.66, SD = 1.64; CF: M = 1.09, SD = 1.72) and in the no-prime condition (true: M = 1.72, SD = 1.63; CF: M = 0.85, SD = 1.53). No significant difference was observed in the CF prime condition (true: M = 1.49, SD = 1.63; CF: M = 1.25, SD = 1.72). We then analyzed the Grade  Order interaction (see Fig. 3). We first examined patterns of change in performance as a function of grade. When true premises were presented first, total number of logical responses (combined over both premise types) remained relatively constant, with the only significant difference being between Primary Grade 5 and Secondary Grade 1. In contrast, when CF premises were presented first, no difference was observed for the two youngest grades, but performance signif-

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Fig. 2. Mean numbers of logical responses to true and CF premises as a function of condition (no prime, true prime, or CF prime).

Fig. 3. Total numbers of logical responses as a function of grade level and order of reasoning problems (CF first or true first).

icantly increased between these and Secondary Grade 1 students, with Secondary Grade 2 students showing a significant increase over the latter. Differences between the two orders generally mirror this pattern. Numbers of logical responses were greater in the true first condition than in the CF first condition for both of the two younger grades. No significant difference was found with Secondary Grade 1 and Secondary Grade 2 students.

Discussion The results of this study provide a detailed picture of an important intermediate phase in the development of formal reasoning in adolescents. Previous results show that by 10 or 11 years of age, children can reason quite well with conditional premises that describe familiar true causal relations. Specifically, they are able to accept the MP inference, and simultaneously reject the AC and DA inferences, at a fairly high level. These same children are only starting to have the ability to reason logically with conditional premises using CF premises (Markovits & Vachon, 1989). The representational redescription model (Karmiloff-Smith, 1995) that is proposed suggests that the ability to reason logically with conditional inferences goes through a succession of phases characterized by an increasingly gen-

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eralized ability to (a) inhibit disabling conditions leading to accepting the MP inference and, at the same time, to (b) generate alternative antecedents that allow understanding the uncertainty of the AC and DA inferences (which we refer to as the generation of alternatives strategy). Our basic hypothesis claims that the youngest children at this age level have a well-established ability to spontaneously use the generation of alternatives strategy for true causal conditionals when reasoning. Errors in logical reasoning are due to failure to successfully generate relevant information because reasoning requires constant use of working memory capacity in order to retain premises in memory, among other things. Thus, for example, someone given the inference ‘‘If a rock is thrown at a window, the window will break. A window is broken’’ might wrongly conclude that ‘‘a rock was thrown at the window’’ simply because the individual was able to successfully retrieve the possibility that a chair could also break a window while maintaining premises in memory. At this age, children are beginning to establish the ability to use an alternatives generation strategy with CF premises. The representational model implies that children will not easily deploy this strategy during reasoning with CF premises. Thus, someone given the inference ‘‘If a feather is thrown at a window, the window will break’’ might wrongly conclude that ‘‘a feather was thrown at the window’’ because the individual was unable to even attempt to retrieve other ways of breaking a window (if a feather can do so). Development during this period is characterized by the construction of an increasingly well-established representation of the CF alternatives generation process, allowing children to more easily deploy this strategy. The results of this study provide different forms of support for this model. First, we used two forms of an alternatives generation task (CF and true) in attempting to encourage increased use of alternatives generation and, thus, improve logical reasoning (see Fig. 2). This task simply asked participants to consider a given conditional relation as true and then to produce potential alternatives to the antecedent. The conditionals used were different from those used in the reasoning problems. Our first hypothesis was that at these ages all children would have well-established representations of the alternatives generation process required for true causal conditionals. Thus, encouraging alternatives generation should have no effect on reasoning with such premises, a prediction that was indeed confirmed. Now, one possibility here is that there was a ceiling effect. However, both statistical analyses and inspection of results show that there was a clear developmental increase in logical responding to true causal conditionals, with the overall levels of logical responding remaining relatively low even among the oldest students. As stated previously, this can be attributed to the relative efficiency with which reasoners are able to successfully retrieve potential alternatives when the appropriate strategy is activated. This reflects the difficulty of retrieving information while, at the same time, using working memory capacity to maintain premises in memory and to inhibit disabling conditions (De Neys & Van Gelder, 2009; Janveau-Brennan & Markovits, 1999; Markovits & Quinn, 2002; Markovits et al., 2002). By contrast, encouraging the generation of CF alternatives results in a clear increase in logical reasoning with the CF premises at all ages. This procedure had two effects. First, it decreased the rate of empirical errors on the critical MP inference, for example, reasoning such as ‘‘If a feather is thrown at a window, then the window will break. A feather is thrown at a window. The window will not break.’’ More important, it increased the production of overall logical responses (requiring simultaneously accepting the MP inference and giving uncertainty responses to the AC and DA inferences). Although we made no specific prediction about the effects of encouraging the generation of true alternatives here, it should be noted that true alternatives produce levels of empirical errors and logical responses that are not significantly different from either the no-generation condition or the CF alternatives condition. In other words, this procedure produced an intermediate level both of decreases in empirical errors and of increases in logical responses, but once again only with the CF conditionals. Thus, there is some advantage to encouraging additional use of even true alternatives generation when reasoning with CF premises, although this is not as clear-cut as CF alternatives. Both forms of alternatives generation encourage thinking about different possibilities, and their effects are consistent with the general idea that help with this kind of thinking is useful with CF reasoning. The third set of results that are consistent with this model is the interaction between reasoning with CF premises and reasoning with true premises (see Fig. 3). Our basic hypothesis here, supported by previous results looking at interactions between abstract and concrete reasoning (Markovits &

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Vachon, 1990), is that using a form of reasoning for which the alternatives generation process is not well established will produce a suppression effect on the generation process for reasoning where this would normally be deployed at a high rate. Consistent with this, when true premises are given first, there was little developmental improvement across grades. In contrast, when CF premises were presented first, there was a clear developmental improvement in overall logical reasoning. More specifically, among the Primary Grade 5 and Grade 6 students (11–12 years of age), receiving the CF premises first produces significant global decreases in logical responding compared with receiving the true premises first. This difference disappears among the Secondary Grade 1 and Secondary Grade 2 students (13–14 years of age). Although this result is consistent with our hypothesis, it should be acknowledged that there are alternate explanations. For example, the younger participants might find it more difficult to maintain their concentration over a long series of problems, although previous studies have not shown any such difficulty. One final point here concerns the different developmental patterns related to encouraging alternatives generation on CF reasoning and those related to the interaction between reasoning with CF and true premises. Encouraging use of the CF alternatives generation strategy produced a consistent relative increase in logical reasoning with CF premises across all of the ages examined here. This suggests that even the oldest children in this sample were not able to spontaneously generate CF alternatives at a very high level, which is supported by the relatively low level of logical responses to the CF reasoning problems at all ages. Although there was no specific prediction made as to the developmental effect of encouraging CF alternatives generation, this result suggests that the procedure used to encourage alternatives generation targeted participants who were not too far from spontaneously using CF alternatives at all ages. By contrast, the order effect shows a clear developmental pattern. Reasoning with CF premises results in a decreased overall level of logical reasoning in the younger participants, but this interaction disappears in the older participants. Our basic hypothesis is that the negative effect of CF reasoning is due to a suppression effect, where failure to deploy an alternatives generation strategy on the CF problems carries over to familiar reasoning. However, because both the level of spontaneous use of CF alternatives generation and the efficiency of both CF and true alternatives generation increase with age, this suppression effect should be visible mostly among reasoners who are in the early stages of CF reasoning, which is indeed what is observed. The overall pattern of results, thus, provides support for the idea that between 11 and 14 years of age, a transition occurs that is characterized by the increasing ability to reason logically with CF premises. This latter ability depends on increasing use of a CF alternatives generation strategy that allows reasoners to accept CF premises as true and, at the same time, to generate potential alternatives. Combined with previous results (Markovits & Lortie-Forgues, 2011), this provides a detailed picture of how conditional reasoning develops between late preadolescence and late adolescence that is consistent with a multiphase sequence of increasingly abstract representations of the alternatives generation process. To provide a broader context for these results, it is useful to note that although the alternatives generation task that we use here is derived specifically from studies examining conditional reasoning (Cummins, 1995; Cummins et al., 1991; De Neys, Schaeken, & D’Ydewalle, 2003b; Janveau-Brennan & Markovits, 1999; Markovits, 1986), this task is almost identical to some measures of divergent thinking (Greenberger, O’Connor, & Sorensen, 1971). These generally ask people to generate different ways of accounting for a specific situation; for example, ‘‘It has rained and the street is wet. What else could explain this?’’ In fact, the model that we use suggests that logical reasoning requires both accepting a given conditional relation and, at the same time, having the ability to generate implicit possibilities (alternatives), which resembles the underlying definition of divergent thinking. Previous results have also shown that the ability to reason logically is related to people’s capacity to generate alternatives that are semantically distant from the premises (Markovits & Quinn, 2002). This corresponds to one of the key components of divergent thinking, which is the ability to generate ideas that are conceptually removed from each other. The results of the current study can be seen as support for the idea that logical reasoning requires some form of divergent thinking and, thus, creativity (Nusbaum & Sylvia, 2011). This is in turn potentially related to findings that priming additive or subtractive counterfactual thinking mindsets in adults can improve creative or analytic reasoning (e.g., Markman, Lindberg, Kray,

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& Galinsky, 2007). In this context, the presented model suggests that divergent thinking with CF premises is particularly important to the development of reasoning in this age range. Acknowledgments This study was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada to H.M. We thank the teachers, administrators, and children of the schools involved for their cooperation. References 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. Daniel, D. B., & Klaczynski, P. A. (2006). Developmental and individual differences in conditional reasoning: Effects of logic instructions and alternative antecedents. Child Development, 77, 339–354. De Neys, W., & Everaerts, D. (2008). Developmental trends in everyday conditional reasoning: The retrieval and inhibition interplay. 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