Elsevier

Cognition

Volume 75, Issue 3, 15 June 2000, Pages 237-266
Cognition

Conditional reasoning by mental models: chronometric and developmental evidence

https://doi.org/10.1016/S0010-0277(00)00066-4Get rights and content

Abstract

The aim of this article is to verify two predictions resulting from the mental models theory of conditional reasoning. First, the denial of antecedent (DA) and modus tollens (MT) inferences should take longer to verify than modus ponens (MP) and affirmation of consequent (AC) because the former require subjects to flesh out the initial model whereas the latter do not. This prediction was confirmed in two reaction time experiments in adults. In line with Evans' proposal (Evans, J. St. B. T. (1993). The mental model theory of conditional reasoning: critical appraisal and revision. Cognition, 48, 1–20), there was a strong directionality effect: inferences from antecedent to consequent (MP and DA) took less time to verify than the inferences in the opposite direction (AC and MT). Second, the development of conditional reasoning should result from the increasing capacity to construct and coordinate more and more models. As a consequence, the pattern of conditional inference production should evolve with age from a one-model conjunctive pattern (production of MP and AC more frequent than DA and MT) to a three-model conditional production pattern (higher production rate for MP and MT than for DA and AC). This prediction was confirmed using an inference production task in children, adolescents, and adults.

Introduction

It has recently been suggested that it is not possible to come to an empirical choice between the two main theories of human reasoning (i.e. Johnson-Laird and Byrne's mental models and mental logic theories) because both of them lack sufficient precision (Evans & Over, 1997). This article presents three experiments which aimed at verifying predictions resulting simply from the mental models theory (Johnson-Laird and Byrne, 1991, Johnson-Laird et al., 1992, Johnson-Laird et al., 1994) for the processing of if then conditional statements. The first two experiments compared reaction times in order to endorse the conclusions drawn from different conditional syllogisms in adults. The third experiment adopted a developmental approach and studied age-related development in the frequency of conditional inference production from childhood to adulthood.

The mental models theory postulates that deductive reasoning is a three-step process. The first step is thought to consist of a comprehension process which leads to the construction of a representation (a mental model) of the state of affairs the premises describe. According to the theory, this mental model would have the same logical structure as the described state of affairs. When there are several premises to be considered, and thus several mental models to be coordinated, the theory postulates coordination heuristics which preserve the information provided by the premises. The second step consists of the production of a putative conclusion from the resulting model. A given item of information would be considered to be a conclusion by the subject if it is both true in all the models constructed and not explicit in the premises. These first two steps are simply a matter of comprehension and do not embody any reasoning process per se. In contrast, in the third step, the subject would try to falsify the reached conclusion by constructing alternative models of the premises. If there is no alternative model in which this conclusion does not hold, then it is necessarily true, but if there is such a model, the conclusion must be judged to be uncertain.

The main prediction of the theory is that the greater the number of models a given reasoning requires, the harder it should be (i.e. prone to error, experienced as difficult by the subject, and taking a long time), because both the construction and manipulation of mental models would be carried out in a capacity-limited working memory. Thus, subjects would tend to reduce the cognitive load resulting from constructing and coordinating models by making some part of the information to be represented implicit. They would then construct simplified models, called initial models. If required by the current situation, the initial model could be fleshed out to provide a complete representation of the situation described.

These proposals can be illustrated in the field of the conditional reasoning and the deductive inferences permitted by if then statements. The mental models theory proposes that an if p then q premise would be represented by the following initial modelpqwhere each line represents a distinct model. The first line in the diagram denotes an explicit model of the situation in which the antecedent and the consequent are true. The three dots denote an implicit model standing for other possibilities that make the conditional true. This implicit model would contain a mental footnote indicating that it refers to possibilities in which the antecedent of the conditional is false. Mental footnotes can be ephemeral and soon forgotten (Johnson-Laird, Legrenzi, Girotto, Legrenzi & Caverni, 1999). If the footnote is retained, then reasoners can, in principle, flesh out the initial model, leading either to a biconditionalpq¬p¬qor a conditional interpretationpq¬p¬q¬pqwhere ¬ is a propositional-like tag which denotes negation. Thus, the set of mental models for if p then q represents all the possible situations in which the statement is true.

In a conditional syllogism task, the statement if p then q constitutes the major premise, with which four minor premises resulting either from the affirmation or the negation of the antecedent or the consequent can be coordinated. In formal logic, only the affirmation of antecedent (p), known as modus ponens (MP), and the negation of consequent (not q), known as modus tollens (MT), lead to certain conclusions which are q and not p, respectively. On the other hand, neither the affirmation of consequent (q, AC) nor the denial of antecedent (not p, DA) lead to a firm conclusion, although the subjects frequently endorse or produce the conclusions p from the minor premise q (for AC) and not q from not p (for DA, see Evans, Newstead & Byrne, 1993, for a review).

Thus, the endorsement or the production of both MP (deduce q from p) and AC (deduce p from q) would be possible from the initial model of if p then q, because it contains explicit representations of both p and q. The subject would then consider the information this model links with the information given by the minor premise to be the conclusion (i.e. q for MP, and p for AC). Note that q is a correct conclusion for MP because even within a complete (i.e. fleshed out) representation, there is only one model representing p, and this model links q with p, whereas the conclusion p for AC should be rejected because it does not hold in a complete three-model representation (i.e. there is a model linking q with not p).

According to Johnson-Laird (1993), the subjects endorse conclusions that hold in only some models because they fail to construct all the possible models. They are deductive satisficers, and when they come up with one conclusion that fits their available beliefs, they would not tend to search for others. The denial inferences DA (deduce not q from not p) and MT (deduce not p from not q) require subjects to flesh out the initial model because it does not contain explicit representations of not p and not q. Whether the resulting interpretation is biconditional (two models) or conditional (three models), the conclusion not p necessarily follows from not q (MT). If only two models are constructed (biconditional interpretation), the minor premise not p brings about the conclusion not q (DA), but this conclusion does not hold in a complete conditional representation.

Thus, the endorsement or the production of MP and AC would require only one model whereas MT and DA would require two models. The mental models theory therefore predicts that among the logical inferences MP and MT, MP should be easier to produce than MT because the latter requires a fleshing out process. Of the fallacious inferences AC and DA, AC should be produced more often than DA because the latter requires two models whereas the former is supported by the initial model of the conditional. However, empirical data contradict the prediction of a more frequent endorsement and production of AC than DA. When a wide range of experiments are considered, it appears that AC and DA are endorsed and produced with equal frequencies (Evans, 1993, Evans et al., 1993). O'Brien, Braine and Yang (1994) have used this fact to argue against the mental models theory and have claimed that the mental logic has no problem in accounting for it.

The theories relating to the mental logic (Braine, 1990, Braine and O'Brien, 1991, Braine and Rumain, 1983, Rips, 1994) propose that deductions are permitted by a set of inference schemas (formal rules of inference) stored in long term memory. These rules would apply to propositional representations resulting from a comprehension process of the premises. As far as if is concerned, these theories (e.g. Braine & O'Brien, 1991) presuppose that two rules are available. The first would correspond to MP: when two propositions of the form if p then q and p, respectively, are available, the rule derives a conclusion of the form q. The second is aimed to introduce if by making it possible to state that if p then q for any proposition q which can be derived from a supposition p. Such a model predicts, as does the mental models theory, that MT should be more difficult than MP. The latter would be directly achieved by applying an available schema while the former requires a reasoning strategy by reductio ad absurdum. As far as DA and AC are concerned, these inferences would result from pragmatic principles or invited inferences. There is no reason to suppose one of them to be more difficult than the other. That is probably the reason why O'Brien et al. (1994) have claimed that mental logic has no difficulty in accounting for their equal frequency of production.

Evans (1993) has suggested modifying the mental models theory in order to make it compatible with this phenomenon and has suggested two explanatory factors. The first would be a negative conclusion bias leading to a more frequent acceptance of negative than affirmative conclusions. Now, this bias favors DA acceptance (negative conclusion not q) and impedes AC (affirmative conclusion p) when both the antecedent and the consequent of the major premise if p then q are affirmative propositions. The second factor would be that the models of the conditional are directional in nature and are oriented from the antecedent to the consequent. According to this proposal, MP and DA (minor premises p and not p, respectively) would be forward inferences and hence easier to endorse than the corresponding backward inferences AC (for the first model, minor premise q) and MT (for the second model, minor premise not q, see Ormerod, Manktelow & Jones, 1993, for an account of the directionality of models resulting from if p then q conditionals). Thus, both the negative conclusion bias and the directionality of models would favor DA (which is a forward inference leading to a negative conclusion), and impede AC (which is backward inference with a positive conclusion), thus leading to equal frequencies of endorsement although the former requires one more model than the latter.

Two comments could be added to the Evans (1993) proposals. First of all, the mental models theory permits predictions not only of the frequency of acceptance of the different inferences as a function of the number of models they require, but also of the time needed for these acceptances. Thus, according to the theory, the inferences which require one model only (MP and AC) should be faster than those which require two models (DA and MT), because it should take longer to construct two models by means of the fleshing out process compared to one model provided by the initial model of the conditional. Marcus and Rips (1979) have already studied conditional syllogisms using RT measures. However, they used only three different conditional premises which elicited major differences in the rates of endorsement of inferences. Second, it is possible that the number of models required to produce a given inference has a strong effect on its frequency of production only if the construction and maintenance of the models lead to a cognitive overload. Now, it is possible that the number of models required to endorse DA (i.e. two models) is not high enough to exhaust the cognitive resources of adult subjects. Thus, the effect of the higher cognitive load induced by DA compared to AC (two models rather than one) might be superseded by the opposite effects of both the negative conclusion bias and the directionality of models.

However, the cognitive load effect should be all the stronger the lower the subject's cognitive resources. For example, many authors have suggested that children have lower working memory capacity than adults (Case, 1985, Case, 1992, Halford, 1993, Pascual Leone, 1988). If this is the case, young children should produce MP and AC more often than DA and MT because of their limited working memory capacity.

The general hypothesis that the process of inference production is constrained by the number of models required has been tested in three experiments. The first two experiments used a reaction times measure paradigm in which adult subjects were asked to evaluate conditional syllogisms. We hypothesized that the endorsement of a given inference should take all the longer the higher the number of models it requires. The third experiment used a developmental approach and the fact the number of models the children can construct and use for reasoning increases with age (Barrouillet, 1997, Barrouillet and Lecas, 1998, Barrouillet and Lecas, 2000, Lecas and Barrouillet, 1999). We hypothesized that the production rates of the four conditional syllogisms should evolve with age according to the number of models they require.

In Experiment 1, we asked adult subjects to judge conditional syllogisms presented on screen. For each syllogism, a first screen presented a conditional premise of the form if p then q preceded by a short text. According to the mental models theory, the interpretation of this conditional premise would lead to the construction of an initial model as in Eq. (1). A second screen presented two propositions (a minor premise and a conclusion) of the form p and q, q and p, not p and not q, and not q and not p for MP, AC, DA, and MT, respectively. The participants had to judge the validity of the conclusion in light of both the conditional and the minor premises. Reaction times (RTs) for the acceptance of each type of inference were recorded. According to the mental models theory, RTs for MP and AC endorsement, which require one model, should be shorter than those for DA and MT, which make it necessary to flesh out the initial model. The Evans (1993) proposed directionality of models predicts that, among the initial-model inferences, MP should be faster than AC, and that among the second-model inferences, DA should be faster than MT, because MP and DA are forward inferences from antecedent to consequent whereas AC and MT are backward inferences from consequent to antecedent.

The Braine and O'Brien (1991) model does not predict this pattern of results. Although this model predicts longer RTs for MT than for MP, it does not predict longer RTs for DA than for AC because both inferences result from pragmatic principles and there is no reason to expect any difference in RT or frequency between them. As O'Brien et al. (1994) have claimed that mental logic has no difficulty in accounting for the equal frequencies of AC and DA production, we can suppose that mental logic predicts equal RTs in the same way.

The aim of Experiment 2 was to discount the alternative hypothesis that MP and AC took shorter than DA and MT because the latter require the processing of negative sentences. The same design as in Experiment 1 was used but both the minor premise and the conclusion were expressed using either synonyms of the antecedent and the consequent of the conditional premise for MP and AC, or antonyms for DA and MT. As for Experiment 1, we predicted the same effect of the number of models to be constructed and the same directionality effect of models.

The aim of Experiment 3 was to verify the effect of the number of models to be constructed on the frequency of inference production in childhood and adolescence. Several experiments conducted in our laboratory suggest an age-related evolution of the interpretation of if p then q statements, from a conjunctive-like interpretation resulting from the construction of a single model of the form p·q, to a biconditional interpretation based on a two-model representation (p·q and ¬p·¬q), followed by an interpretation based on the complete three-model representation described by Johnson-Laird and Byrne (1991) (Barrouillet, 1997, Barrouillet and Lecas, 1998, Barrouillet and Lecas, 2000, Lecas and Barrouillet, 1999). In line with the proposals of Johnson-Laird and Byrne (1991), the growth in the number of models that children can construct and maintain depends on a related increase in working memory capacity (Barrouillet & Lecas, 1999). Accordingly, the rate of production of each type of inference should evolve as a function of the number of models the children can construct.

Modus ponens, which is based on the initial model, should be frequently and constantly produced across the three levels of interpretation previously described. Modus tollens, which is based on the second model, should be rare in children who construct one model only, and then more frequent when children are able to construct two or three models. In contrast, the frequency of production of AC should be high in children who construct one or two models and then decline when three models can be constructed, because the third model would represent instances of the form ¬p·q which prevent subjects from concluding p from q. Finally, the frequency of production of DA which requires subjects to flesh out the initial model should be low in children who construct one model only, and should then increase when children are able to construct two models, and finally decline when the third model can be constructed (Table 1).

Thus, the mental models theory predicts a distinct developmental pattern for each of the four inferences: MP frequency should remain high and stable across development, MT frequency should increase with age, AC frequency should decrease, and DA frequency should first increase and then decrease. Therefore, there should be a developmental level where MP and AC will be produced more frequently than DA and MT. This gap between the one-model and the two-model inferences should disappear when children become able to construct two models. Finally, a third developmental level should correspond to the capacity to construct a complete three-model representation, where MP and MT should be more frequent than AC and DA.

Once again, this set of predictions does not follow from mental logic theories. Of course, these theories predict a high and stable rate of production for MP across development, because this inference corresponds to a schema stored in long term memory. They also predict an age-related increase in MT production because this inference would require a reasoning strategy based on hypothesis production, and one can suppose that the older the children, the smarter the implementation of this strategy would be. As we have already seen, AC and DA would result from pragmatic principles. According to Braine and O'Brien (1991), the development of the understanding of conditional sentence would be based on an increasing age-related capacity to set aside the pragmatic principles in situations which require logical treatment (e.g. laboratory tasks). If this is the case, we should therefore expect a frequent production of these inferences in young children and then a progressive decrease in the production rate of both DA and AC. The mental models theory predicts such a developmental trend for AC but it differs with regard to DA, for which it predicts an inverse U-shaped production rate curve (see above).

Section snippets

Experiment 1

The aim of his experiment was to verify that the endorsement of inferences based on the initial model of the conditional (i.e. MP and AC) is faster than that of inferences which require subjects to flesh out this initial model (i.e. DA and MT). For all the inferences, a major premise of the form if p then q was presented in a first screen. In order to control for a possible negative-conclusion-bias effect, all the conditional premises we used had an affirmative antecedent and consequent (e.g.

Experiment 2

In this experiment, both the minor premise and the conclusion included either synonyms of the terms involved in the conditional premise for AC and MP, or antonyms for DA and MT. For example, for a conditional premise like ‘if it is a high tree, then the mushrooms are toxic’, we proposed ‘it is a tall tree’ and ‘the mushrooms are harmful’ as the minor premise and conclusion, respectively for MP, the reverse for AC, and ‘it is a small tree’ and ‘the mushrooms are good’ for DA and MT. Thus,

Experiment 3

In this experiment, third, sixth, and ninth graders and adults had to produce conclusions from the four types of conditional syllogism.

General discussion

In discussing the mental logic–mental models controversy, Bonatti (1998, p. 435) stressed that “one way to make progress is to concentrate on a small domain that both theories address, and thoroughly examine which theory seems to be more likely to capture human reasoning”. Conditional reasoning is one of these small domains, and the results of the three experiments presented here were more compatible with the mental models theory than with its rival. Of course, it could be claimed that many of

Acknowledgements

The authors would like to thank Arnaud Desarménien for collecting the data of Experiment 1.

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